The Knowable and the Unknowable: Modern Science, Nonclassical Thought, and the ''Two Cultures''

The Knowable and the Unknowable



THE Knowable AND

THE Unknowable

Modern Science,

Nonclassical Thought, and the

"Two Cultures"

Arkady Plotnitsky

Ann Arbor

THE UNIVERSITY OF MICHIGAN PRESS

Copyright © by the University of Michigan 2002

All rights reserved

Published in the United States of America by

The University of Michigan Press

Manufactured in the United States of America

O Printed on acid-free paper

2005 2004 2003 2002

4 32 1

No part of this publication may be reproduced,

stored in a retrieval system, or transmitted in any form

or by any means, electronic, mechanical, or otherwise,

without the written permission of the publisher.

A CIP catalog record for this book is available from the British Library.

Library of Congress Cataloging-in-Publication Data

Plotnitsky, Arkady.

The knowable and the unknowable : modern science, nonclassical

thought, and the "two cultures" / Arkady Plotnitsky.

p.

cm. - (Studies in literature and science)

Includes bibliographical references and index.

ISBN 0-472-09797-0 (cloth : alk. paper) - ISBN 0-472-06797-4

(paper : alk. paper)

1. Quantum theory. 2. Science-Philosophy. 3. Literature and

science. I. Title. II. Series.

QC174.12

501-dc21

Copyright © by the University of Michigan 2002 All rights reserved Published in the United States of America by The University of Michigan Press Manufactured in the United States of America i§ Printed on acid-free paper

.P6 2001

2001005506

2005

2004

2003

2002

4 3 2 1

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, or otherwise, without the written permission of the publisher. A CIP catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Plotnitsky, Arkady. The knowable and the unknowable : modern science, nonclassical thought, and the "two cultures" I Arkady Plotnitsky. em.- (Studies in literature and science) p. Includes bibliographical references and index. ISBN 0-472-09797-0 (cloth: alk. paper)- ISBN 0-472-06797-4 (paper: alk. paper) 1. Quantum theory. 2. Science-Philosophy. 3. Literature and science. I. Title. II. Series. QC174.12 .P6 2001 501-dc21 2001005506

Und darum: Hoch die Physik! Und hoher noch

das, was uns zu ihr zwingt,-unsre Redlichkeit!

-NIETZSCHE

The significance of physical science for philosophy

does not merely lie in the steady increase of our

experience of inanimate matter, but above all in

the opportunity of testing the foundation and

scope of some of our most elementary concepts.

-BOHR



Contents

Acknowledgments

ix

Preface

xiii

Outline of the Chapters

xix

Chapter 1. An Introduction to Nonclassical Thought

1

Chapter 2. Quantum Mechanics, Complementarity, and

Nonclassical Thought

29

Chapter 3. Versions of the Irrational: The Epistemology of

Complex Numbers and Jacques Lacan's

Quasi-Mathematics

109

Chapter 4. "But It Is Above All Not True": Derrida, Relativity,

and the "Science Wars"

157

Chapter 5. Deconstructions

201

Conclusion

235

Notes

243

Bibliography

291

Index

301



Acknowledgments

First, I would like to thank many mathematicians and scientists for their

contribution to this project through their own work and thought and

through numerous discussions of and direct comments on specific subjects,

scientific and philosophical, considered in this study. What is right about

mathematics and science in this study would not be possible without them,

and I hope that they (and other mathematicians and scientists) will forgive

me for what must be improved and refined, especially if they find something

that is erroneous here. I have tried hard to avoid such errors, but there are

no absolute safeguards, not even for mathematicians and scientists them-

selves, especially in dealing with quantum theory, as the nearly century-long

debate concerning it would demonstrate. I am especially grateful to David

Mermin for many exchanges, which were indispensable in helping me to

shape both my understanding of quantum mechanics and its presentation

here. I am further grateful to him for his specific commentaries on several

key arguments of this study. My special thanks also to John Archibald

Wheeler for a wonderful conversation on things quantum in Philadelphia in

April 1998. From a distance of decades, I would like to thank Ludwig D.

Fadeev, Misha Gromov, and Vladimir A. Rokhlin, with whom I studied in

the Mathematics and Mechanics Department at Leningrad State University

(now St. Petersburg State University). They are among the greatest scientists

in the world in their respective fields of quantum theory and topology, and

I was extraordinarily fortunate to have studied with them. I also thank Gus-

taaf Cornelis, Kurt Gottfried, Barry Mazur, David Reed, Philip Siemens,

Joshua Sokolar, and Stephen Weininger. The participation in two recent

conferences-the NATO Advanced Research Workshop "Decoherence and

Its Implications in Quantum Computation" in Mykonos, Greece, in 2000;

and "Quantum Theory: A Reconsideration of Foundations" at Vaixjo Uni-

versity, Vaixjo, Sweden, in 2001-was exceptionally helpful in refining the

presentation of quantum mechanics and its epistemology in this study. I am

grateful to Tony Gonis and Patrice E. A. Turchi for inviting me to the first

and to Andrei Khrennikov and Christopher A. Fuchs for inviting me to the

x Acknowledgments

second. I am also grateful to Tony, Andrei, and Chris for productive

exchanges, and to Chris, in addition, for discussions of, in his phrase,

"quantum foundations in the light of quantum information theory."

I owe a very special debt of gratitude to Jacques Derrida for his work and

thought, invaluable conversations, and his support of my work. I am also

grateful to him for his support of this study itself, which, while it most

specifically offers a commentary on his work in the context of the current

intellectual and cultural debates, extends a broader exploration of the rela-

tionships between this work and the philosophical problematics of quantum

physics and other areas of modern mathematics and science.

My position as a fellow at the Center for Interdisciplinary Studies in Sci-

ence and Cultural Theory of Duke University had a special role in my work

on this project, which in part developed from several public lectures I was

invited to give as part of the 1996-97 series "Reconfiguring the Two Cul-

tures." The series, which took place amid the debates known as the "Science

Wars" and involved some of the leading practitioners in philosophy, his-

tory, and sociology of science, further helped to shape the conception of this

project. I am grateful to Barbara Herrnstein Smith, the director of the Cen-

ter, for the opportunity to be a part of this series and to work at the Center.

I am also grateful to her for many helpful discussions and support of this

project. I would like to thank in addition Roy Weintraub in the Economics

Department at Duke, and to thank Fredric Jameson for his help and support

as a colleague and as Chair of the Literature Department at Duke and also

for offering me an invaluable opportunity to work on and to teach the sub-

jects addressed in this study.

There are many others to whom this study is indebted. It would not be

possible to mention all of them. I would like to mention especially Uzi

Awret, Mario Biagioli, Claudia Brodsky Lacour, Jean-Michel Rabate,

Samuel Weber, and Silke-Maria Weineck.

I would also like to thank my students, graduate and undergraduate,

including science majors, at Duke and Purdue. Teaching courses and semi-

nars on the "two cultures" was instrumental to my work on this study.

This and related projects were generously supported by several fellow-

ships and grants. I regret that essays on de Man and quantum epistemology;

on biology and information; on Emmy Noether's work in algebra and the

nature of abstract thinking in mathematics; and on modern mathematics

and science and gender theory, which have also been supported by these fel-

lowships, could not be included in this study. They have appeared or will

appear elsewhere.

Acknowledgments * xi

I was a fellow at the Center for the Humanities at Oregon State Univer-

sity in the spring of 1994. I am grateful to the Center and to Oregon State

University for this support and to the members of the Center and other fel-

lows, as well as to faculty members at Oregon State University, for a most

stimulating time there. I would like to express my thanks to Wendy Madar.

Peter Copek, who founded the Center and was its director until his death in

June 2001, when this book was in press, offered his generous support and

made my time at the Center especially productive. I am saddened that he did

not live to see the book.

In 1994-95 I was a William S. Vaughn Visiting Fellow at the Robert

Penn Warren Center for the Humanities at Vanderbilt University. I am

grateful to the Center and to Vanderbilt University for this special honor, as

I was the first William S. Vaughn Fellow appointed. I also thank the partic-

ipants, humanists and mathematicians and scientists, of the seminar "Sci-

ence and Society" held there during that year.

In 1995-96, I was a fellow at the Center of the Humanities at the Uni-

versity of Utah. The fellowship enabled me to do a significant portion of the

research on the subjects of this study, for which I am grateful to the Center

and the university. I would like to offer my special thanks to Esther

Rashkin, whose support was invaluable.

The research support from Purdue University in the summer of 2000, the

fall of 2000, and the summer of 2001, and the appointment as a University

Faculty Scholar provided major help at the final stages of this work. I am

grateful to Tom Adler, the Chair of English, and Margaret R. Rowe, the

Dean of the School of Liberal Arts, for their help and support. I am also

grateful to my colleagues in the English Department at Purdue, in particular

to Victor Raskin for many productive discussions and to Pat Sullivan for

helpful suggestions concerning the book.

This book appears as part of the series Studies in Literature and Science,

which is published in association with the Society for Literature and Science

(SLS). I thank the Society and its members for their role in shaping this

study. SLS conferences provided both a valuable intellectual and scholarly

ambiance and a forum for some of the ideas of this study. I am grateful to

the editorial board of the series, co-chaired by N. Katherine Hayles and

Stephanie Smith, for including my book. I would especially like to thank

Katherine Hayles for her support of the project, which was instrumental for

its progress and completion.

I am grateful to the editorial board of the University of Michigan Press

and to LeAnn Fields, the executive editor at the Press, who supported the

xii * The Knowable and the Unknowable

project with exemplary attention and care at every stage. I thank Marcia

LaBrenz for editorial supervision of the book, Anne Taylor for copyediting,

and others at the Press who helped to bring the book to its published form.

A copy of the photograph of Bohr's drawing was kindly provided by the

Emilio Segre Visual Archives of the American Institute of Physics and is

reproduced with their permission, which is greatly appreciated and grate-

fully acknowledged.

Finally, my more personal thanks to Nina, Marsha, Inge-Vera, and Paula.

An earlier version of chapter 4, "But It Is Above All Not True: Derrida,

Relativity, and the Science Wars," and the accompanying exchange with

Richard Crew, from the Mathematics Department of the University of

Florida, appeared in Postmodern Culture 7.2 (January 1997) and 8.2 (Jan-

uary 1998) (published electronically). A portion of chapter 3 was published,

in an earlier version, as "Lacan and Mathematics," in Lacan in America,

edited by Jean-Michel Rabate (New York: Other Press, 2001), and a por-

tion of chapter 1 is included in "Disciplinarity and Radicality: Quantum

Theory and Nonclassical Thought at the Fin de Siecle and the Philosophy of

the Future," in Disciplinarity at the Fin-de-Siecle, edited by Amanda Ander-

son and Joseph Valente (Princeton, NJ: Princeton University Press, 2001).

Preface

This study offers an exploration of the relationships between modern math-

ematics and science-in particular quantum mechanics, arguably the most

controversial scientific theory of the twentieth century-and what I here call

nonclassical thinking and the theories, nonclassical theories, to which this

thinking gives rise. This thinking and these theories radically redefine the

nature of knowledge by making the unknowable an irreducible part of

knowledge, insofar as the ultimate objects under investigation by nonclassi-

cal theories are seen as being beyond any knowledge or even conception,

while, at the same time, affecting what is knowable. Thus, according to Niels

Bohr's nonclassical understanding of quantum mechanics, as expressed in

the statement to which I continue to return throughout this study, "we are

not dealing with an arbitrary renunciation of a more detailed analysis of

atomic phenomena, but with a recognition that such an analysis is in princi-

ple excluded" (Bohr's emphasis). It is this impossibility, in principle, of any

analysis of the phenomena considered by nonclassical theories beyond cer-

tain limits (which nonclassical theories establish as well) that defines these

theories. By the same token, this impossibility also defines "the unknowable"

of my title as that which is placed by such theories beyond the limit of any

analysis, knowledge, or conception, while, again, having shaping effects

upon what can be known. Indeed, as will be seen, in these circumstances, the

very concept of "phenomenon," as relating to these objects, or the concept

of "object," requires a special reconsideration and redefinition, which Bohr

was compelled to undertake in the case of quantum mechanics.

By contrast, classical theories, as understood here, consider their primary

objects of investigation as, at least in principle (it may not be possible in

practice), available to conceptualization and, often, to direct or, at least

sufficiently approximate, representation by means of such theories-in

short, as knowable. This is "the knowable" of my title. Classical thinking

does not deny that there are things that are, in practice or even in principle,

beyond theory or any knowledge. In contrast to nonclassical theories, how-

ever, classical theories are not concerned with the irreducibly unknowable

xiv * Preface

or its effects upon the knowable. The irreducibly unknowable, if allowed, is

placed strictly outside their limits, rather than is seen, as it would be in non-

classical theories, as a constitutive part of knowledge. Thus, most of classi-

cal physics, such as classical, Newtonian, mechanics, can be and customar-

ily is seen as classical theory in this sense, in contrast to quantum mechanics

in Bohr's or other nonclassical interpretations. It is a separate question

whether quantum mechanics could be interpreted classically, or, conversely,

classical mechanics nonclassically. As will be seen, the cases of classical and

quantum physics are not symmetrical as concerns their respective resistance

to classical interpretation. This resistance is much greater and is perhaps

even impossible to overcome in the case of quantum mechanics. In any

event, on the view adopted by the present study, the knowable and the clas-

sical are one and the same. By the same token, classical theories become a

pathway to establishing the existence of and the link to the unknowable,

and they have also contributed and often led to the emergence of nonclassi-

cal thinking historically. Indeed classical theories provide not only a path-

way to the unknowable but, by definition, the only such pathway. For how

could we otherwise know about the unknowable, or, more crucially, how

could we rigorously establish or conjecture the existence of the unknowable

in this radical sense, rather than only imagine it, did the unknowable not

have manifest effects upon what we can know? These manifestations, how-

ever, or these effects of the unknowable, cannot be properly understood by

classical means and instead require nonclassical theories. Accordingly, non-

classical theories theorize both the knowable and the unknowable, found in

nonclassical situations, and their (nonclassical) relationships. This different

(from that of classical theories) relationship between the knowable and the

unknowable is just as crucial to understanding nonclassical theories and

their place in intellectual history or culture as the radical nature of the non-

classical unknowable itself. Indeed both, this relationship and the nonclassi-

cal unknowable, define each other.

The nonclassical theories and ways of thinking specifically discussed in

this study are those exemplified, in various ways and to various degrees, in

the works of Niels Bohr, Werner Heisenberg, Jacques Lacan, and Jacques

Derrida. This study devotes a chapter to each of them (a little less in the case

of Heisenberg, whose work, however, is prominent throughout the book).

Bohr's interpretation of quantum mechanics, known as complementarity,

serves as the primary paradigm of nonclassical theory for this study. The

ideas of a number of other figures-such as Karl Friedrich Gauss and Bern-

hard Riemann, on the side of mathematics and science, and Friedrich

Nietzsche, Georges Bataille, Maurice Blanchot, Emmanuel Levinas, and

Preface * xv

Gilles Deleuze, on the philosophical side-will be addressed as well. While

extraordinary in their own right, these ideas indicate the broad historical

and conceptual range of nonclassical thinking and of the interactions

between nonclassical thinking and mathematics and science. The argument

of this study is that these interactions proceed in both directions. Modern

mathematics and science, from at least the early nineteenth century to quan-

tum physics and beyond, contain elements of nonclassical thinking and

sometimes borrow these elements from other areas of human inquiry. Reci-

procally, nonclassical thought elsewhere often depends on modern mathe-

matics and science and their philosophically nonclassical aspects.

Although some among the mathematical and scientific subjects to be con-

sidered here are complex, no disciplinary knowledge of mathematics and

physics is required for understanding my argument. I have tried to introduce

these subjects for nonscientific readers and to be as clear and accurate as

possible in my exposition of them; and I have tried to do the same for the

nonscientific subjects discussed here. While the book is not a primer on the

nonscientific subjects anymore than it is on the mathematical and scientific

subjects in question, my aim is to make the nonscientific material sufficiently

available to the reader, including possible scientific readers, just as it is to

make the mathematical and scientific parts of the book available to nonspe-

cialists. However, the character of our "two cultures," as C. P. Snow

famously called them, the humanities and the sciences (mathematics

included), makes the situation to which this project belongs (and that it

indeed addresses), and, accordingly, this task itself asymmetrical. This

asymmetry persists, even though there may be more symmetry than is often

thought and even though there are, and have always been, arguably, begin-

ning at least with Plato, more than two cultures involved, or perhaps both

more than two and less than one. Partly real and partly imaginary, the

"Snow divide" persistently and perhaps unavoidably reenters this multiplic-

ity and this less than unity. The Greeks might have introduced this split

when they invented mathematics, arguably the first science in the full sense

of Snow's argument, since mathematics appears to have managed to place

itself apart from philosophy, poetry and the arts, politics, and to some

degree even language, although it could not be born or exist without them.

But then, neither this type of invention nor this type of divide could have a

single origin, a point or even a culture of unique emergence, or have

occurred one single time, even leaving aside large-scale cultural entities (that

is, cultural multiplicities), for example, Babylonian, or, later, Arab mathe-

matics and astronomy, or the always partly evolutionary nature of such

events. All of these-emergences of new sciences, the many (more than two

xvi * Preface

and less than one) cultures that give them birth, the two cultures and divides

to which they give rise, and so forth-occur all the time, sometimes without

involving mathematics and science. The complexity, the irreducible nonsim-

plicity, of these dynamics makes it difficult and ultimately impossible to

establish once and for all (in many cases, even provisionally) what defines

each culture and what divides them. In the case of Snow's two cultures,

however, the divide persists. It is equally difficult to say whether the Snow

divide will ever allow a "dream of great interconnections," of which Bohr

speaks and which requires greater cultural multiplicity, to become much

more than a dream. One of the persistent effects of the Snow divide is the

asymmetry, just invoked, of the ways in which we discuss the two cultures.

The nature of this asymmetry, or of the Snow divide itself, is outside the

scope of this study, although, thematically and in practice, it could not be

avoided either. In any event, in view of this asymmetry, while nonclassical

theories in other fields of inquiry in turn require as rigorous and careful

treatment as possible, the presentation of mathematical and scientific ideas

places greater demands on a project like the one undertaken here and,

accordingly, at certain points on the book's readers.

I stand by my argument and claims concerning mathematics and physics.

As far as quantum mechanics qua physics is concerned, most of my claims

will be supported by arguments offered in Bohr's works (with some of

Heisenberg's ideas added on), obviously, in turn given a particular interpre-

tation, and, hence, also entailing a particular interpretation of Bohr's inter-

pretation of quantum mechanics. This role of interpretation or reading is

unavoidable, even by classical, let alone nonclassical, standards of interpre-

tation, however careful and rigorous one tries to be. Most of my arguments,

moreover, would apply whether or not one agrees with Bohr's interpreta-

tion of quantum physics, although I argue this interpretation to be at the

very least viable and effective, even if not inevitable, however troubling or

even epistemologically unacceptable it may be for some. The latter was

actually the view of Albert Einstein, who ultimately found quantum

mechanics itself consistent and effective but epistemologically unpalatable

in view of its nonclassical implications (his view of Bohr's complementarity

is more complex and ambivalent).

While, in accordance with the outline just given, conceived more broadly,

the argument of this book is also a response to both long-standing and more

recent debates concerning the two cultures. The most recent stage of these

debates also involves what has become known as the "Science Wars," fol-

lowing the appearance of Paul R. Gross and Norman Levitt's book, Higher

Superstition: The Academic Left and Its Quarrels with Science (1994) and

Preface * xvii

Alan Sokal's hoax article published in the journal Social Text (1996). A

more recent book, Impostures intellectueles (1997), coauthored by Sokal

and another theoretical physicist, Jean Bricmont, first published in France

and later in England and the United States under the title Fashionable Non-

sense: Postmodern Intellectuals' Abuse of Science (1998), and hosts of

related publications have expanded these debates still further, both intellec-

tually and politically, and indeed geographically, in particular to the French

intellectual scene. One of the aims of this study is to contribute to more pro-

ductive approaches to understanding the relationships among the various

disciplines involved in these debates and to a better understanding of the

debates themselves. A more sustained understanding of the nature and

significance of nonclassical thought in mathematics and science, on the one

hand, and in the humanities and social sciences, on the other, is, I would

argue, crucial to this task.



Outline of the Chapters

The following outline is designed to help the reader navigate through this

study and perhaps choose an alternative trajectory or sequence for follow-

ing its argument(s). For example, one might, after reading chapter 1 (which

serves as a general introduction to the book) and portions of chapter 2

(which contains a comprehensive introduction to its epistemological argu-

ment in the first section, while the details of quantum mechanics and com-

plementarity are given in subsequent sections), follow a two cultures and

Science Wars trajectory extending to chapters 3, 4, and 5 (especially the first

section of the latter). Conversely, one may proceed to chapters 2, 3, and 5

(especially the second section of the latter) for the conceptual discussion of

the relationships between nonclassical thinking and quantum mechanics,

complex numbers, and other areas of modern mathematics and science. On

the other hand, the actual sequence of the chapters is designed to make the

overall argument as comprehensive as possible.

Chapter 1: An Introduction to Nonclassical Thought

This chapter explains the key terms of the book and sketches the broad lin-

eaments of nonclassical thinking in mathematics and science and elsewhere.

Chapter 2: Quantum Mechanics, Complementarity,

and Nonclassical Thought

This chapter offers an extended discussion of quantum mechanics and

Bohr's complementarity as a nonclassical theory. It considers both Bohr's

own key concepts and some of the key aspects of the century-long debate

concerning the epistemology of quantum theory, especially those involved

in Einstein's criticism of it, and eventually leading to and, as Bell's theorem,

extending the famous argument of Einstein, Podolsky, and Rosen (hereafter

referred to as EPR, following a long-standing convention) concerning the

possible incompleteness and nonlocality of quantum theory. (Nonlocality

xx * Outline of the Chapters

has to do with a possible instantaneous physical action at a distance, incom-

patible with Einstein's relativity theory.) The EPR argument played a crucial

role in Bohr's thinking and indeed forced, but also enabled, him to refine his

earlier argument concerning complementarity. The chapter also comments

on some of Heisenberg's work (a subject that takes center stage in chapter

5). Finally, using Bohr's epistemology of quantum physics as a paradigmatic

example, the chapter argues that, in certain circumstances, the nonclassical

epistemology becomes a necessary condition of disciplinarity, scientific or

philosophical, rather than inhibiting it, as many opponents of nonclassical

thought believe.

My discussion of Bohr is somewhat more technical than the rest of the

book. Or rather (since following it does not require a technical knowledge

of physics), this discussion is especially detailed as regards specific features

of quantum physics. Beyond the more general asymmetry indicated in the

Preface, there are several reasons for pursuing this approach.

First, my aim is to bring out the philosophical content of Bohr's interpre-

tation and its radical nature to the maximal degree possible. The most rad-

ical aspects of his interpretation have rarely been given their due even in sci-

entific and philosophical literature. One of the main reasons for this neglect

is that the more subtle and sometimes minute nuances of Bohr's argument

are often missed or misunderstood. His argument, however, crucially

depends on and explores these nuances, as do other key philosophical argu-

ments in question in this study. Accordingly, some of the features of quan-

tum mechanics considered here may be unfamiliar to the general reader,

while the presentation of others departs from their renditions in literature

on the subject.

Second, specific details and nuances of Bohr's interpretation are funda-

mental to the general argument of this study concerning the nature and via-

bility of nonclassical theory wherever it emerges.

Third, a careful consideration of these details and nuances is necessary in

order to argue (this is my disciplinarity argument) that nonclassical theory

can be fully rigorous, scientifically and philosophically, and fully consistent

with the disciplinary practice of science as currently constituted.

Chapter 3: Versions of the Irrational: The

Epistemology of Complex Numbers and Jacques

Lacan's Quasi-Mathematics

This chapter considers Lacan's work in the context of modern mathematics,

which extends from imaginary and complex numbers, such as the square

Outline of the Chapters * xxi

root of -1, usually designated as "i," and related developments in late-eigh-

teenth- and early-nineteenth-century mathematics to our own time, includ-

ing that used in quantum mechanics (which, in fact, crucially depends on the

role of complex numbers). This mathematics will be considered under the

broad heading of non-Euclidean mathematics, of which (the more familiar)

non-Euclidean geometry is a part as well. This chapter will delineate this

concept and explain its relationships to nonclassical thought well beyond,

and sometimes against the grain of, Lacan's work.

The context itself just invoked is, as is well known, explicit in Lacan's

work, which deliberately and rather liberally (sometimes too liberally)

deploys ideas borrowed from these and related areas of mathematics. While

conceived much more broadly (as is this study as a whole), chapter 3 is a

response to some of the questions and debates concerning Lacan's use of

mathematics posed during the Science Wars. Lacan's work has occupied a

special place in academic and intellectual debates for quite a while now,

whether one speaks of psychoanalysis, philosophy, literary studies, or, most

recently, the Science Wars. In Fashionable Nonsense, Lacan's is arguably

presented as the most notorious case of the postmodern abuse of science.

The argument of the chapter is applicable to Lacan's deployment of

mathematical ideas borrowed from a variety of fields, such as mathematical

logic and topology (the latter will be considered in some detail here in the

general context of non-Euclidean mathematics rather than in the context of

Lacan's work). This chapter, however, deals directly only with imaginary

and complex numbers and Lacan's argument linked to them. My analysis

primarily concerns, first, the way mathematics is used in Lacan and why it

is so used, not the mathematical accuracy of his mathematical references.

This accuracy may, admittedly, be sometimes improved upon, although, all

things considered, Lacan is not as bad as some of his recent scientific critics

think. Indeed, sometimes he even displays a better sense, if not knowledge,

of mathematics or, at least, of something in mathematics than do these crit-

ics. Second, it concerns the philosophical (rather than more specifically psy-

choanalytical) dimensions of Lacan's work. I argue that the structure of

philosophical concepts (in Deleuze and Guattari's sense) is where Lacan's

usage of mathematics most fundamentally belongs and is the best perspec-

tive from which this usage can be meaningfully considered. I would like to

emphasize that my argument is not a defense or endorsement of Lacan's use

of mathematics or of his psychoanalytical, philosophical, or other ideas

(personally, I am inclined to be critical of Lacan), but instead a kind of epis-

temological case study. Ultimately the discussion of complex numbers and

non-Euclidean mathematics is the conceptual center of this chapter.

xxii * Outline of the Chapters

Chapter 4: "But It Is Above All Not True": Derrida,

Relativity, and the "Science Wars"

This chapter examines the Science Wars and related debates concerning the

two cultures, in part by specifically considering the role and treatment of

Derrida's work in these discussions. Similarly to the preceding chapters, it

also offers a discussion of the substantive connections between Derrida's

ideas and modern mathematics and science, including relativity, which was

the primary subject of the Science Wars exchanges on Derrida. More

broadly it addresses the question of reading nonscientific texts, such as Der-

rida's, when these texts engage or relate to mathematics and science and

philosophically reflect, and reflect on, fundamental conceptual conjunctions

of scientific and nonscientific fields. This argument is extended in the next

chapter in conjunction with some of Heisenberg's conceptual arguments

concerning quantum mechanics, which may, with due qualification and cau-

tion, be seen as deconstructive. The chapter, thus, brings together the ques-

tions of the ethics of intellectual discussion and of the philosophical content

of modern mathematics and science and their relationships with nonclassi-

cal philosophy.

Chapter 5: Deconstructions

The first section of this chapter discusses the representation of mathematics

and science, most especially quantum physics, in some of the Science Wars

criticism, specifically in Gross and Levitt's and Sokal and Bricmont's books.

In part as a contrast, the second section considers the positive significance of

nonclassical thought, in particular Derrida's work, for mathematics and sci-

ence, here specifically in conjunction with Heisenberg's discussion of quan-

tum mechanics in his 1929 lectures at the University of Chicago.

The scholarly and intellectually unacceptable treatment of the humanities

and the social sciences in the Science Wars books just mentioned is by now

recognized by commentators. It is a less realized and still less, if at all, com-

mented upon fact that these books contain significant problems in their rep-

resentation of mathematics and science themselves. The first section of this

chapter is concerned primarily with these latter problems, some of which

are discussed in earlier chapters as well, especially in chapter 3 in the con-

text of Lacan and complex numbers. As I argue there, Sokal and Bricmont

are almost worse on complex numbers than they are on Lacan, and, at

points, worse than Lacan is. I would further contend that an adequate treat-

Outline of the Chapters * xxiii

ment, positive or critical, of the radical philosophical work considered in

this study requires a rigorous and nuanced treatment both of this work itself

and of mathematics and science. The Science Wars books in question not

only, by and large, uniformly fail in the first task but often also, and, given

that the authors are scientists, less forgivably, fail in the second as well.

By way of a contrast and extending the argument of the preceding chap-

ter, the second section, closing this study, brings together Heisenberg's 1929

lectures and Derrida's work as both instances of deconstruction in Derrida's

sense. Heisenberg's argument is seen as establishing a kind of Kant-Derrida

axis in the epistemology and, to some degree, even physics of quantum the-

ory and as posing significant questions concerning the relationships between

deconstruction and nonclassical epistemology in quantum mechanics and,

by implication, in Derrida's own work.

Conclusion

Taking as its point of departure Bohr's final thoughts (expressed literally on

the last day of his life), the conclusion offers a brief, codalike commentary

on the two cultures and on the ethics of intellectual discussion, an ethics

defined by the necessity of communicating those ideas that bring the cul-

tures involved-say, mathematics and science, on one side, and the human-

ities, on the other-to the limits of both what is known and unknown, or

unknowable, to them. In the case of nonclassical theory, found, I argue here,

in both cultures, the unknowable reaches arguably the farthest known lim-

its. This, however, may enable us to open more effective channels of com-

munication and even ethical relationships between our two cultures, or,

again, always more than two and less than one.

Abbreviations

BCW

Niels Bohr. Niels Bohr: Collected Works. 10 vols.

Amsterdam: Elsevier, 1972-96.

PWNB

Niels Bohr. The Philosophical Writings of Niels Bohr. 3 vols.

Woodbridge, CT: Ox Bow Press, 1987.

QTM

John Archibald Wheeler and Wojciech Hubert Zurek, eds.

Quantum Theory and Measurement. Princeton, NJ: Princeton

University Press, 1983.



Chapter 1

An Introduction to Nonclassical Thought

The Classical, the Nonclassical, and the Quantum

Throughout this study, classical theories will be understood as considering

their principal objects available to conceptualization and, often, to direct

representation in terms of particular properties of these objects, their behav-

ior, and the relationships between them. Indeed, these features define such

objects as objects of classical theories, since these objects may be idealized

from some other objects, some of whose other properties, moreover, are dis-

regarded by the theory, in the way, for example, classical physics abstracts

certain key physical properties from other properties of material bodies it

studies. Thus, classical mechanics, the part of classical physics that deals

with the motion of individual physical objects or systems composed of such

objects, is or (this qualification is crucial) may be interpreted as such a the-

ory. It fully accounts, at least in principle, for its objects and their behavior

on the basis of physical concepts and abstracted or idealized measurable

quantities of material objects corresponding to them, such as the "position"

and "momentum" of material bodies. Other possible properties of actual

physical objects involved, say, planets moving around the sun, are disre-

garded by classical mechanics, which thus deals with idealized objects. The

equations of classical mechanics allow us to know the past state (or to oper-

ate under the assumption of such knowledge) and to predict the future state

of the system under investigation at any point once we know it at a given

point. Other areas of classical physics, such as thermodynamics and statisti-

cal physics, chaos theory, and electromagnetism (a wave, rather than parti-

cle, theory), can be shown to be, or to be interpretable as, classical in the

same sense. While, thus, in general an idealization, within its proper limits

(short of relativity and quantum physics), classical physics offers an excel-

lent approximation of the behavior of material bodies in nature and enables

most of the technology currently in use, including that used in quantum

measurement.

Classical physics is, thus, by definition, realist and usually causal. Or,

2 * The Knowable and the Unknowable

again, it may be and usually is interpreted as such for most purposes of its

analysis and use; that is, one can combine theory (specifically mathematical

formalism) and experimental data so as to construct models, classical mod-

els, each comprising a set of idealized objects, whose causal behavior the

theory describes. I shall consider the concepts of "causality," which relates

to the nature of the processes themselves in question, and "determinism,"

which relates to our ability to predict the outcome of such causal processes,

in detail in the next chapter. In general, classical theories, as here defined,

need not entail causality. Classical physics, however, is virtually uniformly

causal, although not always deterministic, while nonclassical theories are

neither causal, nor deterministic, nor realist as concerns their ultimate

objects. Within its proper scope, classical physics offers both excellent

descriptions of the natural objects it considers (or ultimately constructs as

such) and from which it idealizes the objects of physical theories, and excel-

lent predictions of the outcome of experiments it performs upon natural

objects. In Bohr's nonclassical interpretation, complementarity, quantum

mechanics allows only for the latter, not for the former, and indeed rigor-

ously disallows even the possibility of constructing a model of the classical

type with respect to the ultimate objects it considers, since it only describes

the effects of the interaction between these objects and measuring instru-

ments. This stronger prohibition is crucial, since, in principle, classical mod-

els need not be seen as describing, even approximately, the behavior, let

alone the ultimate nature, of actual physical objects (even though they can

be and often are seen as so doing), but only as serving to predict outcomes

of experiments. In other words, in question here is a rigorous impossibility

of applying classical-like models, rather than merely abandoning such mod-

els. The key aspects of the classical situation in physics just outlined can be

generalized to classical theories elsewhere. Reciprocally, classical physics

may itself be seen as derived from classical theories elsewhere and also, in

part correlatively, as a refinement of everyday experience and language (no

longer applicable to physical objects at the quantum level), a point often

made by both Bohr and Heisenberg.1 Thus, the classical and the knowable

of my title are one and the same, denoting that which is available to knowl-

edge, representation, conceptualization, theorization, and so forth. Indeed,

according to this view, what is knowable is classical, and only what is clas-

sical is, in all rigor, knowable.

By contrast, the ultimate "objects" of nonclassical theories are irre-

ducibly, in practice and (this defines the difference between classical and

nonclassical thinking) in principle, inaccessible, unknowable, unrepre-

sentable, inconceivable, untheorizable, undefinable, and so forth by any

An Introduction to Nonclassical Thought * 3

means that are or ever will be available to us, including, ultimately, as

"objects" in any conceivable sense of the term. Hence, they cannot be

assigned any conceivable attributes, such as those conceived by analogy

with objects of classical theories. For example, it may not be, and in Bohr's

interpretation is not, possible to assign the standard attributes of the objects

and motions of classical physics to the ultimate objects of quantum physics.

It may no longer even be possible to speak of objects or motions (such as

particles or waves, for example), which, however, does not imply that noth-

ing exists or everything stands still. The latter, naturally, is itself a classical

physical attribute, a refinement of everyday experience. But then, how else

can we even conceive of such attributes, given the present understanding of

classical theories? For, in this understanding, only classical theories or, more

generally, thinking could allow us such an attribution. Thus, the ultimate

objects of nonclassical theories are not their objects insofar as one means by

the latter anything that can actually be described by such a theory. The

impact of such objects on what the theory can account for is crucial, how-

ever, and this impact cannot be described classically, which is what makes a

nonclassical description necessary in such cases.

This statement requires the following further qualification in view of the

fact that the situation is subtler than just presented as concerns the ultimate

"efficacity" of the impact of the unknowable objects in question upon what

we can know-of the effects of the nonclassical upon the classical. Here and

throughout this study, I use the term "efficacity" in its dictionary sense of

power or agency producing effects but, in this case, without the possibility

of ascribing this agency causality, which point is especially significant in

quantum theory and in Bohr's work. The nonclassical inaccessibility, as

here understood, must be seen as referring to the objects, or the efficacious

processes leading to the effects in question, as they are defined by a given

nonclassical theory, even though ultimately we may not be able to access

such objects by any conceivable means, rather than only by means of this

theory itself. This inaccessibility does not refer to what can be further linked

to such objects in nature (the existence of such a link would be assumed in

physics, or at various levels in other natural sciences) or in mind (a link cus-

tomarily assumed in philosophy), or in other domains, such as society, pol-

itics, or culture, which a given theory would consider. Upon what happens

at these levels as such, or how it can possibly be seen by other theories, non-

classical theory does not make any claim, including the claim that it is inac-

cessible, unknowable, unrepresentable, inconceivable, untheorizable,

undefinable, and so forth. Accordingly, from the nonclassical perspective,

the ultimate or, as it were, the ultimate "ultimate objects" of nonclassical

4 * The Knowable and the Unknowable

theories are inaccessible even as inaccessible, unknowable even as unknow-

able, unrepresentable even as unrepresentable, inconceivable even as incon-

ceivable, untheorizable even as untheorizable, and so forth. In other words,

such un-objects may not correspond in any way to the objects of a given

nonclassical theory, even though and because this theory conceives of its

objects as unavailable to description or conception in terms of the theory or

indeed in any terms (such as "objects"). This latter view, too, must be seen

as a (possibly ultimately inadequate) form of idealization, in the way it

would be, for example, in Bohr's interpretation of quantum mechanics (this

is in part why it is an interpretation). This idealization is, moreover, not an

approximation (of the kind classical theories often pursue), but instead a

kind of irreducible rupture from whatever may physically exist. In all rigor,

following and radicalizing Immanuel Kant, we may not even be able to

speak in terms of the existence in space and time, or in terms of any specific

form of materiality we can conceive of, since they may not be applicable,

even in the sense of the remotest analogy. From this perspective, nonclassi-

cal theories entail two irreducible ruptures or discontinuities (hence there is

no causality, which is, conceptually, a form of continuity). The first rupture

is that between the knowable effects and their unknowable efficacity, for

which the ultimate objects of a given theory are responsible. The second is

that between these objects, qua objects of the theory, and that-those un-

objects-to which these objects can possibly be linked in nature, as in

physics, or elsewhere. The second rupture, obviously, adds new complexity

to the concept of the nonclassical. This complexity may appear excessive,

and, indeed, one need not engage with it for most practical workings and

applications of nonclassical theories, and even for many epistemological

arguments concerning them vis-a-vis classical theories. As will be seen, how-

ever, at certain points this part of the nonclassical conceptual and epistemo-

logical architecture becomes crucial and, in fact, irreducible, for example, in

considering the question of interpretation in quantum (or, for that matter,

classical) physics. It is of course also necessary for a fully rigorous

specification and understanding of the nature of nonclassical theories as

they are defined by this study.

There may, again, be a link between such un-objects and the objects of a

given nonclassical theory, and indeed what happens at these more remote

levels may be and usually is ultimately responsible for, or is the ultimate

"efficacity" of, everything at stake in the theory in question. For example,

this type of link and this type of ultimate efficacity enter quantum theory

through the experimental data of quantum physics, for which certain un-

objects, as just defined, are responsible and through which an interpretation

An Introduction to Nonclassical Thought * 5

of quantum mechanics may conceive of quantum objects nonclassically, as

irreducibly inconceivable, in the way Bohr's complementarity does. In terms

to be developed in the next chapter, this view is part of Bohr's "model" and

(this distinction will be properly established in the next chapter as well)

"interpretation" of quantum mechanics, rather than necessarily a view of

what ultimately happens in nature at the level of its ultimate constituents.

Strictly speaking one should put quotation marks around "nature," "ulti-

mate," "constituents," or, when used, "quantum." Complementarity does

not claim the ultimate validity of this idealization for anything in nature,

beyond its role in the argument for the completeness (an exhaustive account

for the data) and consistency of quantum mechanics (thus interpreted)

within its proper scope. It would be tempting to say in these circumstances

(and one finds such claims sometimes) that, at the ultimate level of the quan-

tum constitution of nature, there are no particles, no atoms, and so forth,

rather than only that we cannot see the objects of quantum mechanics in

these terms from the perspective of a particular interpretation, such as com-

plementarity. It appears, however, more prudent to adopt the present, more

cautious, view, which, in addition, suffices for the purposes of my argument.

As will be seen, Bohr argues that "there is no quantum world" in the pres-

ent sense. Or, at least, this statement and complementarity in general may

be interpreted or adjusted in accordance with this view, since Bohr, on occa-

sion, appears to make or to be inclined to make somewhat stronger claims.

Accordingly, the unknowable of my title are the unknowable objects of

nonclassical theories, coupled to the unknowable (or the unknowable

unknowable) unobjects, as just explained. While, however, "knowledge"

or, conversely, "the unknowable" appear to be almost maximally capacious

terms to be deployed in this context and often indeed subsume other denom-

inations deployed by this study, such as (in)accessible, (un)representable,

(un)conceptualizable, and so forth, it does not appear possible to fully con-

tain this situation, even negatively, by any single term. Indeed, as will be

seen, this impossibility is itself a rigorous consequence of the nonclassical

view and is correlative to the nonclassical character of the unknowable in

question.

The view just outlined, thus, accommodates the possibility that other the-

ories may define the un-objects of nonclassical theories differently and may

differently relate to or idealize them as their objects, for example, along

more classical and specifically realist lines. The ultimate viability of such

alternative approaches is a different question and may become a subject of

criticism or assessment, for example, vis-a-vis the nonclassical view. On the

other hand, given that our physical theories are manifestly incomplete, mov-

6 * The Knowable and the Unknowable

ing beyond the scope of the standard (nonrelativistic) quantum mechanics,

to which Bohr's complementarity rigorously applies, may change the way

we need to idealize quantum objects. For the moment, however, nonclassi-

cal epistemology appears to hold, at least in terms of an interpretation, for

most available quantum theories and even for relativity, or their extensions,

such as higher-level field theories, quantum gravity, or string theory.2 The

same type of argument would apply to nonclassical theories elsewhere, and

outside mathematics and science the space of possibilities is perhaps even

more ambiguous and uncertain. Perhaps! There is plenty of epistemological

ambiguity in mathematics and science as well.

In quantum theory, in Bohr's complementarity and related nonclassical

interpretations, we cannot ascribe conventional properties (such as "posi-

tion" and "momentum" of classical mechanics) or any physical properties

describing their spatial-temporal behavior to quantum objects qua quantum

objects, such as elementary particles, which we now see as the ultimate con-

stituents of matter. Or, again, in accordance with the qualifications just

given, such interpretations see such objects as an "idealization" of the ulti-

mate constitution of matter as unknowable. Being a particle or, conversely,

a wave would of course itself be defined by a set of such properties, and,

accordingly, as Bohr stressed already in his introduction of complementar-

ity, the terms "particle" and "wave" cannot be applied to quantum objects

otherwise than provisionally (PWNB 1:56-57). Nor, at the nonclassical

limit, can the terms "quantum" or "objects" be applied, or, ultimately, any

conceivable term or concept. Some of these complexities transpire already in

Einstein's so-called special relativity of 1905, which deals with the propaga-

tion of light in a vacuum. According to this theory, this speed (in a vacuum)

is always constant, the famous c, regardless of the state of motion of the

source, which cannot be accelerated so as to change the emitted light's speed

relative to it. It follows from the latter fact that in the case of light itself such

classical properties as time dilation cannot apply. Were it possible (it is not)

to install a clock on a photon, according to relativity the time shown by this

clock would stand still. Relativity may, however, be seen as a classical the-

ory in other respects, specifically causality. Remarkably, quantum theory

can predict, primarily in statistical terms, the outcome of the experiments

involved as well as classical statistical physics, which enables its extraordi-

narily successful functioning as a physical theory. Classical or classical-like

theories failed to do this, or at least they failed to do so when quantum

mechanics was introduced. (As will be seen, the question whether such a

classical-like account is in principle possible is under debate.) Classical

physics may be statistical, too, but not quite in so radical a way as quantum

An Introduction to Nonclassical Thought * 7

physics appears to be. From this perspective Einstein's famous pronounce-

ment "God does not play dice" may well be true, but only in the sense that,

as Bohr observes on several occasions, it is not clear in what sense one can

even speak of dice (ultimately a classical statistical game) when confronting

the statistical "game" of quantum mechanics. (The concept of "game," too,

may no more be applicable here than any other.) Eventually (in 1953, one

year before he died) Einstein came to accept this point, even though he

would still, apparently for this very reason, clearly prefer God's playing dice

rather than the games of quantum physics and its predictions, effective as

they may be in practice, which Einstein had always acknowledged.3

As will be seen, in Bohr's interpretation, this possibility of quantum-

mechanical predictions is seen as enabled by the interactions between quan-

tum objects and measuring instruments. The instruments themselves, or

more accurately, those parts of them through which we register outcomes of

measurements, are described (idealized) in terms of classical physics and

classical epistemology. As a result, in this interpretation, the role of tech-

nology becomes constitutive and irreducible in quantum mechanics (and, by

implication, giving the term "technology" its broader meaning, in nonclas-

sical theories elsewhere), while it may be seen as merely auxiliary and ulti-

mately dispensable in classical physics. Indeed, one could also define non-

classical theories through the irreducible role of technology in them and,

conversely, classical theories by the auxiliary and ultimately dispensable

functioning of technology there.

It follows that in this interpretation quantum mechanics does not

describe, either through its mathematical formalism (as classical mechanics

often does) or otherwise, even in principle and as an idealization, the actual

properties and behavior of its objects in the way classical mechanics

describes the behavior of its objects. Indeed, as I said, it is the latter view

that becomes the idealization of nature at the quantum level according to

complementarity. By the same token, it is neither causal nor, more crucially,

realist in any sense hitherto available. Nor, in the view here adopted, are

other nonclassical theories.

Obviously, classical theories, too, involve things that are, at least at cer-

tain moments, unknowable and inconceivable to them, while nonclassical

theories enable new knowledge, indeed often knowledge that would be

impossible without the intervention of nonclassical theories. Nonclassical

theories change the nature of the unknowable and of the relationships

between the knowable and the unknowable, as against classical theories,

and, consequently it is what we can know and conceive of that are different

in nonclassical theories. The ultimate knowledge concerning the objects of

8 * The Knowable and the Unknowable

nonclassical theories becomes no longer possible, while the existence of

these unknowable objects or what they idealize and their impact upon what

we can know is indispensable. "Ultimate," however, is, in turn, a crucial

qualifier here and throughout this study, where it features prominently (and

it is of course subject to the same ultimate inapplicability at the ultimate

level of description). For, in the first place, classical theories or ways of

thinking in general are often extraordinarily effective and sometimes indis-

pensable across a broad spectrum of theoretical thinking and other human

endeavors, or indeed in everyday life. Second, indeed as a corollary of the

definitions given here, they are equally indispensable in nonclassical theo-

ries, since they serve as a pathway, indeed the only pathway, to establishing

the existence of and the connections to the unknowable. Or, more accu-

rately, classical theories allow us to handle the classically manifest effects of

the unknowable in question in nonclassical theories, which unknowable

cannot be inferred or treated otherwise. By the same token, some among

such knowable effects, specifically certain configurations of such effects, or,

it follows, the emergence (efficacity) of each of these effects cannot be prop-

erly explained by means of classical theories and require nonclassical theo-

ries. The latter are able to use them, while leaving the ultimate nature of the

efficacity of these effects unknowable and inconceivable even at the level of

idealization. The models at stake in such theories and the way they con-

struct their objects require this irreducible unknowability in order to

account for the effects in question. Indeed, "even" is not altogether appro-

priate here, since, as I explained above, this (irreducibly unknowable) char-

acter of any such efficacity is itself now seen as an idealization, a nonclassi-

cal idealization, and, in the case of Bohr's complementarity, part of the

model it considers. Nonclassically, one does not make even (now "even" is

necessary) this claim concerning the ultimate inaccessibility of anything, any

more than any other claims, as regards more remote levels, such as that of

the ultimate constitution of nature qua nature (again, to the degree this term

applies) in quantum physics. The unknowable ultimate constituents of

nature as "quantum objects" and "quantum processes" (or any forms of

"efficacity" they entail) are idealizations of quantum mechanics as comple-

mentarity-are part of its "model." These idealizations are, however,

argued to be consistent with available data and prediction that the theory

makes by using its mathematical formalism. These remote levels may be

seen as part of, or indeed as, the ultimate "efficacity" of everything to which

a given nonclassical theory relates. For example, as I said, the data in ques-

tion in quantum mechanics may be seen or, again, may be idealized as

linked to such "efficacities," while the efficacities themselves are seen or ide-

An Introduction to Nonclassical Thought * 9

alized as irreducibly unknowable according to a nonclassical view without

making any ultimate claim of unknowability upon them. It follows that the

term "efficacity" or "effects" is as provisional as any term here deployed

(for example, "elementary particles" of quantum physics) at the level of the

nonclassical unknowable.

While the ideas of Bohr and several other thinkers, such as Nietzsche and

Derrida are significant here, the conception just outlined is especially

indebted to the work of Georges Bataille, who had a powerful influence on

most other nonclassical authors considered here, and to his concept of

"unknowledge" (nonsavoir). According to Bataille, "it would be impossible

to speak of unknowledge [ultimately even as 'unknowledge'] as such, while

we can speak of its effects."4 Reciprocally, "it would not be possible to seri-

ously speak of unknowledge independently of its effects."5

The conjunction of both propositions equally defines Bataille's and

Bohr's complementary epistemology, as well as a number of other nonclas-

sical or near nonclassical conceptions, such as Derrida's "differance" or

Foucault's "power" (both in turn indebted to Bataille).6 To cite Bohr's state-

ment, with which I began this study, "in quantum mechanics [as comple-

mentarity], we are not dealing with an arbitrary renunciation of a more

detailed analysis of atomic phenomena, but with a recognition that such an

analysis is in principle excluded" (PWNB 2:62; Bohr's emphasis). (The term

"phenomena" in this statement should be given Bohr's special sense, neces-

sitated by his nonclassical view, the sense to be explained in chapter 2.) This

impossibility, however, does not preclude, but instead enables, a rigorous

analysis of the effects of this unknowable efficacity. Bohr, especially in his

later writings, often, and at crucial junctures, spoke in terms of effects-

"typical quantum effects," "the peculiar individuality of quantum effects,"

and so forth-using them very much in the present sense. In particular, non-

classically we cannot speak of "the quantum world" itself (for example, as

"the quantum world") but only of the effects of the interaction between

"quantum objects" and measuring instruments, which interaction initiates

the efficacity of these effects. This interaction is itself quantum (and thus

depends on the quantum aspects of the constitution of the measuring instru-

ments) and hence is unavailable to classical or, again, any treatment, even

though the effects of this interaction are available to classical physical and

epistemological treatment. The character of the (knowable) effects in ques-

tion irreducibly precludes the knowledge or conception, specifically on any

model of what is knowable, of their ultimate efficacity or, at least, the ulti-

mate nature of their efficacity. In other words, the (irreducibly) unknowable

itself in question is no more available to nonclassical theories than to classi-

10 * The Knowable and the Unknowable

cal theories. Nonclassical theories, however, allow us rigorously to infer the

existence of this unknowable (rather than merely imagine its existence) on

the basis of the phenomena that they consider, and explain its significance

for what we can know, and utilize the (manifest) effects of this interaction,

while this cannot be done by means of classical theories. Accordingly, non-

classical theories are defined by the interaction between what is knowable

(the previous list of related terms is presupposed) by, it follows, classical

means and what is unknowable (the same parentheses apply) by any means,

classical or nonclassical.

It is worth noting (since misunderstanding can occur on this point) that

nonclassical epistemology, as here understood, does not refer to a kind of

temporary state of affairs whereby something (such as the efficacity of cer-

tain effects in question in quantum mechanics) that is not known, accessi-

ble, or conceivable now could eventually become conceivable, accessible, or

known. This possibility would make this "unknowable" into something

that could in principle become known and, hence, classical in the present

sense. By contrast, as I have stated at the outset, the nonclassical unknow-

able refers to something that cannot be known or conceived not only by any

means that is now available but also by any means that could ever be avail-

able. Such a conception, to return to Bohr's locution, "is in principle

excluded." The nonclassical view of knowledge is not defeatist or nihilistic,

as Nietzsche, arguably the first thinker of the ultimate limits of both the

nonclassical and nihilism (but, importantly, not a nihilist himself), was first

to understand. Bohr makes this point clear as well in relation to quantum

theory, when he points out that his "argumentation does of course not

imply that, in atomic [quantum] physics, we have no more to learn as

regards experimental evidence and the mathematical [or other theoretical]

tools appropriate for its comprehension." Indeed, Bohr adds "it seems

likely that the introduction of still further abstractions into the formalism

will be required to account for the novel features revealed by the explo-

ration of atomic processes of very high energy" (PWNB 3:6). The history of

quantum physics has demonstrated and continues to demonstrate just that,

as all of its experimental and theoretical findings so far appear to be con-

sistent with Bohr's epistemology. In other words, new knowledge at the

level of the effects of the unknowable is generated all the time. The

unknowable itself, however, remains irreducible under nonclassical condi-

tions. This irreducibility of the unknowable is one of the (permanent)

effects of the nonclassical efficacities.

On this view, the question could only be whether or not the phenomena

in question in quantum physics (either as coupled to a particular mathemat-

An Introduction to Nonclassical Thought * 11

ical formalism or in themselves) or in analogous theories elsewhere in fact

require epistemologically nonclassical interpretation and treatment, rather

than whether such phenomena will eventually become better known so as to

enable any further access to the ultimate objects of a given theory and the

ultimate efficacity of these phenomena. (As will be explained later, these

phenomena are constituted by knowable effects.) Nonclassically, the

unknowable is to remain as irreducible in the future as it was at the initial

stages of nonclassical theories-unless, of course, we change our very con-

ception of what it means to know or to conceive so as to leave space for the

irreducibly unknowable. This argumentation would be consistent with the

nonclassical view, such as Bohr's, in the case of quantum mechanics. Indeed,

to some degree this change may be seen as in practice taking place in quan-

tum physics, whether the practitioners themselves like it (or even recognize

it) or not. It transpires, for example, when the results of experiments are

being discussed directly in terms of those elements of the quantum-mechan-

ical formalism that do not refer to space-time processes at the level of quan-

tum objects (which reference is nonclassically impossible) but only to out-

comes of experiments, already performed or possible.7

One may hope, as did Einstein or Erwin Schrodinger and, following

them, many others who also thought the epistemological cost exorbitant or

even unacceptable, that such phenomena might at some point no longer

require an appeal to nonclassical thought and theories. In other words, the

hope here is that the unknowable aspects of the phenomena in question in

quantum physics could be configured classically. The better thinking of that

type, such as that of Einstein and Schrodinger, was far too complex and

sophisticated to expect that the unknowable could be altogether eliminated

in physics, only that its nonclassical character could be. Hence, Einstein and

Schrodinger did not think that they could abandon the search for what they

would (from a more classical perspective) see as a more complete and

(which is in fact what greater completeness ultimately meant for them) epis-

temologically more palatable conception. Many still continue to resist or do

not perceive, to begin with, this potential irreducibility of the nonclassical

unknowable in quantum mechanics or elsewhere, and some question the

necessity or superiority of the standard quantum mechanics itself as an

inevitable or even valid account of the data it considers. Thus, most propo-

nents of David Bohm's so-called hidden-variables theories (there are several

versions) belong to the latter group and, accordingly, see Bohm's theory as

an alternative to quantum mechanics. These theories lend themselves more

naturally to a classical-like epistemology, albeit at the cost of their own

problems, in particular their nonlocality.8

12 * The Knowable and the Unknowable

Is then quantum mechanics itself, that is, the experimental data in ques-

tion in it and the mathematical formalism that accounts for it, uncircum-

ventably nonclassical, something that ultimately disallows classical-like

interpretation? It may well be, and, while exercising caution, Bohr and oth-

ers were and many are inclined to think so. Here, however, I only argue for

the nonclassical nature of Bohr's complementarity, or, even more cau-

tiously, for the nonclassical nature of the present interpretation of comple-

mentarity as an interpretation of quantum mechanics, since this interpreta-

tion may be challenged as well, just as, again, may be Bohr's claims (in

whatever interpretation). I do not find currently available more classical-like

readings of Bohr sufficiently compelling to be seen as having already offered

such a challenge or alternative.9 In any event, the potential effectiveness (or

ineffectiveness) of the present argumentation does not depend on the fact

that it concerns and extends only a particular interpretation of quantum

mechanics rather than the (ultimate) character of nature itself. As I have

explained earlier, upon the latter, this interpretation, by definition, makes

no claim in any event by virtue of its nonclassical character, ultimately, not

even a claim that such claims are ultimately impossible. We do not ulti-

mately know whether what is behind the nonclassical unknowable is ulti-

mately knowable or unknowable. My contention is only that Bohr's inter-

pretation is at least as consistent and comprehensive as any, indeed more so

than most, even if not all (although the latter possibility is not inconceivable

to the present author). My argument here, however, need not go that far. It

is of course crucial that the nonclassical interpretations of quantum

mechanics, or (with due qualifications) nonclassical theories and interpreta-

tions elsewhere, remain consistent (logically and with respect to the data

specifically in question in a given theory or otherwise relevant to it) and

complete, even by classical criteria. This requirement is, I argue, amply

satisfied by Bohr's interpretation, arguably more so than by most other

interpretations, but at least as much as by most opposing interpretations on

the current scene. This view also assumes that quantum mechanics itself

remains logically consistent and complete within its proper scope, that is, as

Bohr puts it, that it employs "a logically consistent mathematical formal-

ism" and that one cannot demonstrate that the consequences of this for-

malism exhibit "the departure from experience" or that the predictions

based on this formalism "do not exhaust the possibilities of observation"

(PWNB 2:57).

This status of quantum mechanics is by and large accepted now. Most

arguments against quantum mechanics and even Bohr's interpretation

(although the latter continues to generate much discontent) now center on

An Introduction to Nonclassical Thought * 13

possible and, to those who are more classically minded, epistemologically

more classical alternatives, such as along the lines of Bohm's theories. While

it has attracted some public attention, the latter is a small minority view in

the physics community (for the reasons to be explained later). As will be

seen, however, there many other (more) classically oriented approaches to

quantum mechanics and its interpretation. How viable such interpretations

are is, again, a separate question, which I shall not, and for my purposes

need not, fully address here. I shall comment on some of these attempts later

in this study, especially since meeting some of Einstein's challenges in fact

requires an interpretation, and, as I shall argue, with Bohr, complementar-

ity does meet them.10 The classical-like attempts at interpreting quantum

mechanics or offering classical-like alternatives to it, such as Bohm's, bear

significantly on the discussions and controversies that have surrounded non-

classical epistemology throughout the twentieth century and have by now

extended into the twenty-first.11

Geneologies, Disciplinarities, and Interconnections

"Nonclassical" itself is a relatively new term, which, as here defined, may be

related to and deployed alongside such terms as "poststructuralist,"

"deconstructive," and "postmodernist" used more prominently in similar

contexts, mostly outside mathematics and science, in current discussions.

On the other hand, "classical" is a capacious and widely used denomina-

tion, for example, in Michel Foucault's work, and relates to a wide and

diverse set of practices, much wider than does "nonclassical." It may be

argued, however, that the view of the "classical" here adopted is sufficiently

fundamental to relate to, if not to subsume, a large spectrum of such denom-

inations, including Foucault's.12 It goes without saying that it is not a ques-

tion of introducing yet another uniquely fundamental conceptual opposi-

tion that would fully master the field(s) in question and to which all other

theoretical descriptions would be subordinate. At this point, largely thanks

to nonclassical theories from Nietzsche to Derrida and beyond, it is difficult

to sustain the assumption that this is possible or, again, ultimately possible.

But, as Derrida has stressed from the outset of his deconstructive project

(associated with the deconstruction of binary oppositions and with the

impossibility of the kind of mastery just described) and as Heraclitus per-

haps already knew, such oppositions may be inescapable.13 They become

even more effective once we understand the more complex dynamics under-

lying their emergence and functioning, which we may need to do nonclassi-

14 * The Knowable and the Unknowable

cally. They are certainly crucial for Bohr's interpretation of quantum

mechanics. There certain mutually exclusive (and hence never applicable

simultaneously) or, in Bohr's terms, complementary features of physical

description are both necessary for a comprehensive theoretical account and

strictly correlative to the nonclassical nature of Bohr's interpretation, called

complementarity in view of this correlation. In general, such oppositions

may be more or less traditional, such as mind/nature, language/ thought,

logic/intuition, and so forth, or more or less new, such as the classical and

the nonclassical, as here defined, speaking for the moment of fundamental

aspects of theorizing knowledge. They may be more descriptive, as, for

example, are those used in quantum physics, say, between the wave and the

particle descriptions. Besides, classical theories are themselves irreducible in

nonclassical theorizing. Hence, the relationships between classical and non-

classical thought cannot be strictly oppositional or mutually exclusive,

although the two types of epistemology thus designated are irreducibly dif-

ferent. They are not complementary, however, since they are not applicable

to the same types of objects. For in Bohr's interpretation, nonclassical epis-

temology applies exclusively to "quantum objects" (to the degree the latter

denomination itself applies), while classical epistemology applies to certain

parts of measuring instruments. The role itself of measuring instruments in

the constitution of the data of quantum mechanics and in the physical

description it provides can never be neglected or idealized away so as to

deal, even ideally, with objects themselves.

I call the thinking in question nonclassical rather than, say, "postclassi-

cal" (the term employed in similar contexts previously, including by this

author) for the following reason. It is true that its most radical forms may

be argued to be relatively recent. In science, we find them in quantum

physics or modern biology and genetics and in certain areas of modern

mathematics and mathematical logic. In the humanities, we encounter them

beginning more or less with Nietzsche and then extending to the authors

previously mentioned, many of whom, such as the authors discussed here,

follow and develop Nietzsche's ideas (the list of nonclassical thinkers is

actually rather limited). Indeed, it may be argued that nothing like quantum

mechanics appears to have been even remotely imaginable before it came

onto the scene. It appears difficult to trace the quantum-mechanical episte-

mology, in its full measure, prior to the emergence of quantum theory, and,

then, as will be seen, some of its nonclassical effects are strangers still in

their quantum-mechanical specificity. In this sense the term "postclassical"

is not out of place. Certain key elements of nonclassical thinking could,

however, be traced throughout the earlier history of theoretical thinking in

An Introduction to Nonclassical Thought * 15

mathematics, science, and philosophy, which compels one to speak of non-

classical rather than postclassical thought.

In philosophical, if not physical, terms, the history of nonclassical think-

ing concerning causality extends at least to David Hume and Immanuel

Kant or indeed to earlier critics of Sir Isaac Newton and (which is not quite

the same) Newtonianism. Nietzsche (in general no friend of Kant) speaks of

"Kant's tremendous question mark that he placed after the concept of

'causality'-without, like Hume, doubting its legitimacy altogether. Rather,

Kant began cautiously to delimit the realm within which this concept makes

sense (and to this day we are not done with this fixing of limits)" (emphasis

added).14 This is not that far from Bohr's agenda, at least from the causality

part of it ("the realm within which this concept makes sense" was delimited

by him as that of classical physics); and Bohr might well have been aware of

this aspect of Kant's project, and, possibly, of Nietzsche's assessment of it.s5

On several occasions he speaks of complementarity as "a rational general-

ization of the idea of causality" (PWNB 2:41), an important point to which

I shall return in chapter 2. One might also cite Ludwig Wittgenstein's state-

ment (intriguingly, in turn, immediately following an elaboration on Kant)

in his Tractatus Logico-Philosophicus, published first in 1922 amid the tur-

moil of quantum theory prior to the invention of quantum mechanics, a

statement that might have been known to Bohr at some point in his lifelong

work on complementarity. Wittgenstein says: "What can be described can

happen too, and what the law of causality is supposed to exclude will not let

itself to be described either" (translation modified).16 Obviously, both

causality and reality are at stake here and are mutually implicated. Bohr's,

or Nietzsche's, epistemological agenda may well be more radical and more

(radically) nonclassical than Kant's, especially in view of the questioning of

the limits of the concept of reality it involves. Kant's project, however,

remains significant in this context as well, as Bohr's appeal to the concept of

phenomenon would indicate, and may indeed require a rereading from this

quantum-mechanical perspective. Ultimately, the history of nonclassical

thought, at least of some of its key ingredients, can be traced as early as the

pre-Socratic thought, as, to qualify with Blanchot, "we reconstitute it now,"

for example, in the Heraclitean becoming or the Pythagoreans' discovery of

irrational magnitudes, which I shall discuss in chapter 3. Most modern non-

classical thinkers mentioned here, Bohr among them, credited the pre-

Socratics with at least as much.

This tracing sometimes appears to allow for nonclassical interpretations

of some among such earlier theories as a whole. Such interpretations (what-

ever the degree of their viability) pose, first, the question of a more rigorous

16 * The Knowable and the Unknowable

genealogy of nonclassical thought and, second, the question of how classi-

cal, for example, how causal or realist, certain classical theories (wherever

they are found) really are or even can be. Or, again, more accurately, the

question is to what degree one can challenge in nonclassical directions clas-

sical interpretations of classical theories in physics or elsewhere or of the

thought of certain philosophical figures, beginning with some pre-Socratics

and Plato. At the same time, even as the tracing just indicated takes place, at

its radical limits, nonclassical thinking and, perhaps especially, some of its

consequences and implications are hardly more accepted by, or acceptable

to, a large majority of the contemporary intellectual (including scientific)

community than they have ever been and are often resisted with increased

vehemence and passion.17 This resistance can be easily exemplified by recent

debates, particularly those involving responses to the thought of the figures

just mentioned, including

by some mathematicians and

scientists,

specifically during and in the wake of the Science Wars. It is this resistance

that is largely responsible for the continuing application of the characteriza-

tion "radical" to nonclassical theories by their critics.

On the other hand, one can, and I here shall, use the term "radical" in the

sense of fundamental and far-reaching, going to the ultimate roots of the

question of knowledge, which ("going to the root") is the etymological root

of the word "radical." Indeed, as I shall argue, the question may well be

whether contemporary philosophical thought, however rigorous and radical

(it is, as I shall argue here against recent critics just mentioned, both), is yet

rigorous and radical enough for what is at stake at the philosophical limits

of modern physics, in particular in quantum theory. The philosophical con-

sequences of new physics are sometimes more radical than any "postmod-

ernism" can imagine, and they compelled Bohr to invoke repeatedly "the

epistemological lesson of quantum mechanics." These consequences are, it

is true, often in turn resisted by (classically minded) scientists and philoso-

phers in spite of their recognition of those aspects of new physics that (from

the nonclassical perspective) would lead to such consequences, and some-

times as a result of their own work. The work of both Einstein and

Schrodinger, and more recently the work of John S. Bell (of Bell's theorem,

to be explained in chapter 2), in quantum physics offers major examples of

the latter case, and their views or even names are often appealed to as the

source of authority in support of arguments against nonclassical aspects of

quantum mechanics.18 The resistance itself to the consequences and impli-

cations of new physics is much more general and more widespread.

Throughout its history, however, physics has been an extraordinarily fer-

tile ground for questioning our philosophical assumptions. As Bohr argues,

An Introduction to Nonclassical Thought * 17

"The significance of physical science for philosophy does not merely lie in the

steady increase of our experience of inanimate matter, but above all in the

opportunity of testing the foundation and scope of some of our most elemen-

tary concepts" (PWNB 3:1). Bohr's writing on quantum mechanics or

Galileo's earlier work demonstrates this point especially powerfully, but the

work of most major figures in the history of physics would be nearly as

indicative here. This role of physics is in part why Nietzsche said: "Und

darum: Hoch die Physik! Und hoher noch das, was uns zu ihr zwingt,-unsre

Redlichkeit!" (And this is why: long live physics! And even more so that

which forces us to turn to it-our integrity) (translation modified).19

Nietzsche's remark may be seen as part of his more general argument for

putting everything we claim-in science, philosophy, or ethics-to "the

inexorable test of experiment," to use Heisenberg's description of Bohr's

understanding of any philosophy of nature "in our day and age."20

Nietzsche most likely had in mind such contemporary developments as elec-

tromagnetism and, especially, thermodynamics, which, at the time, put con-

temporary physical and philosophical conceptions to a severe test indeed.

Twentieth-century physics, and much of its mathematics and science in gen-

eral, continues to test our most fundamental philosophical concepts, indeed

all our concepts, and forces us to develop new ones. Reciprocally, nonclas-

sical philosophy may help physics and, sometimes, even force physics' own

integrity to turn to this philosophy. In other words, both, reciprocally, have

tested and advanced, and indeed created each other throughout their his-

tory, and, in the process, have productively shaped the ambient culture

around them, and continue to do so.

Nor, accordingly, does nonclassical epistemology necessarily negatively

affect our cultural, including political, practices in general, as has been

equally often and equally uncritically claimed as well, again, especially in

the Science Wars debates. It can of course do so sometimes, but that is

equally true about classical theories. Some of the greatest misconceptions

and some of the most uncritical arguments against nonclassical theories

emerge in view of misunderstanding and sometimes ignorance of the com-

plexity of the connections between knowledge and practice. This complex-

ity equally defines classical and nonclassical knowledge, naturally, without

eliminating certain key differences in either the nature of knowledge or

practice, or in the relationships between them. As Nietzsche pointed out on

many occasions, it is quite naive to think that "right knowledge" necessar-

ily leads to "right action," even though and because he also knew that even

the greatest thinkers, from Socrates and Plato on, and indeed on occasion

himself, are tempted to assume this. Or at least, they, and we, wish that such

18 * The Knowable and the Unknowable

were the case, for example, in the Science Wars, on both or, again, all sides

of it; and, naturally, it does not follow that we could not or should not, or

have no ground to, assess and criticize such action or knowledge. We can

and we should. We may have no absolute or ultimate grounds for so doing,

but these are not the only grounds. Nor is it ever guaranteed that the conse-

quences of what appears, with whatever degree of certainty, as right action

would always or necessarily be beneficial either, even in the short run,

although often they may be. Even if classical theories or classical epistemol-

ogy of ethical or political action were in fact always true, adopting them

would not guarantee the success of these actions themselves. Could we

indeed assume that we can even define either "right knowledge" or "right

action" with any certainty at any given moment, let alone once and for all?

The latter assumption, too, is something that Nietzsche and other nonclas-

sical thinkers mentioned here radically question and indeed deny. It, again,

goes without saying that, as Nietzsche specifically stressed, it is not a matter

of denying, "unless [one is] a fool," that many actions considered good or,

conversely, bad or evil by most accepted (classically based) standards can in

fact be good, or bad or evil, at any given moment. They may be so for the

most part but never always or once and for all, let alone for everybody. It is

rather a matter of rethinking why they are or may be so, or when and for

whom, and they may in fact be such for "other reasons than hitherto," and

understanding these reasons may well require nonclassical thought.21

A number of the major figures, including those considered here, targeted

by the Science Wars critics are often seen by the latter, as well as by many

others, as responsible for the unproductive undermining of disciplinary sta-

bility and theoretical, scholarly, and intellectual norms and rigor. I would

argue this view to be mistaken or at least lacking in discrimination. One

could not deny differences between the work and attitudes of these thinkers

themselves and the fact that the reasons and effectiveness of nonclassical

questioning do vary in their work. According to the view and practice of

many among them, however, specifically those discussed here, radicality is,

or at a certain point becomes, the condition of the continuity of disciplinar-

ity and discipline (in either sense) in their fields. In these cases one finds what

may even be called, strange as it may sound to their recent (and some ear-

lier) critics in relation to these thinkers, an extreme disciplinary conser-

vatism, to be discussed in chapter 2. I refer by this phrase to an often

extreme reluctance to abandon classical argumentation and doing so only

under the extreme pressure of intellectual, scholarly, and disciplinary rigor.

A departure from a given preceding (classical) configuration of thought is

enacted, first, after exhausting the possibilities it offers for a new configura-

An Introduction to Nonclassical Thought * 19

tion, which may in fact arise in part from within the old one. Second, it is

enacted under the extreme pressure of maintaining and even conserving

significant and even defining disciplinary aspects of the old configuration. In

the case of new physics (relativity and quantum mechanics) Heisenberg

argues (rightly) as follows: "Modern theories did not arise from revolution-

ary ideas which have been, so to speak, introduced in the exact sciences

from without. On the contrary they have forced their way into research

which was attempting consistently to carry out the programme of classical

physics-they arise out of its very nature. It is for this reason that the begin-

ning of modern [twentieth-century] physics cannot be compared with the

great upheavals of previous periods like the achievements of Copernicus."22

Even the point concerning the time of Copernicus might require further

qualification, since there was plenty of continuity in that revolution as well.

However, this point does suggest (again, rightly) that there are other

configurations, other views, and other effects of theoretical practice in what-

ever field one considers. Thus, one finds more manifestly or (it may be

difficult to be certain) perhaps more manifest radical "moves"-more pro-

nounced and "speedier" departures from particular forms of disciplinarity.

One can think, for example, of the cases of Nietzsche, Bataille, Deleuze, and

Lacan as different from those of Bohr, Heisenberg, and Derrida. These dif-

ferences notwithstanding, however, one might still argue for analogues, if

not equivalents, of disciplinary conservatism among the thinkers on the first

list, and indeed one finds arguments to that effect in their work. Thus, for

Nietzsche and (differently) Deleuze, one's sense of the "discipline" (in either

sense) and theoretical rigor in fact requires an enactment of a much broader

and deeper transformation, and indeed redefinition, of a given disciplinary

configuration or field. In the process a given disciplinary history, such as

that of philosophy or, especially in Lacan's case, psychoanalysis, becomes

refigured as well. Bataille's is a still different and somewhat more complex

case. His strong sense of philosophical or even, in a certain sense, almost sci-

entific rigor is pronounced in spite and sometimes because of the strange

nature of his texts. In these cases one also confronts more complex discipli-

nary and interdisciplinary configurations than in the case of mathematics or

science. The latter are far from free from these complexities either. The spec-

trum of disciplinary and interdisciplinary configurations in the cases in

question is much broader, however, and, accordingly, one cannot avoid

specificity and limitations in making the kind of argument I am making

here. The case of mathematics and science, or, again, specific cases, such as

that of Bohr's work, may be disciplinarily and interdisciplinarily less com-

plex than those involving the work of figures, such as those just mentioned,

20 * The Knowable and the Unknowable

elsewhere. Or at least this type of complexity may be kept at bay more eas-

ily in the disciplinary practice of mathematics and science rather than, say,

in fully understanding Bohr's work. This specificity, however, also allows

one to make a stronger argument (applicable in other cases just mentioned):

in certain circumstances, extreme epistemological radicality is the condition

of the continuation of disciplinarity and even arises as the outcome of

extreme disciplinary conservatism. It is primarily in order to make the

strongest possible case that the present analysis bypasses certain extrasci-

entific complexities of Bohr's work and focuses on the relationships between

the radical epistemology of quantum mechanics and the disciplinary

specificity of physics there.23

These aspects of the work of nonclassical theorists are entirely, and per-

haps inevitably, missed by most Science Wars critics. On the other hand, a

meaningful criticism of some nonclassical thinkers in question may be that

they are not always sufficiently familiar with mathematics and science not

so much in order to offer a better commentary on mathematics and science

(sometimes this is the case as well) but to test the epistemological limits of

both or of their own theories. A deeper knowledge of mathematics and sci-

ence, or at least of their more fundamental conceptual and epistemological

aspects and limits, may in fact entail more radical philosophical concep-

tions. This knowledge, along with and reciprocal with his extreme episte-

mological conservatism and extraordinary persistence and patience, is per-

haps what enables Bohr to go further than some of these thinkers do. This

is of course of no help to the Science Wars critics, who often have just as lit-

tle patience for Bohr and Heisenberg as they do for Lacan or Derrida. One

of their greatest problems is also this lack of patience, which is rarely helped

by anxiety, although this impatience has other sources, too. This, it is true,

may also be said about many of those on the other side, on other sides, of

the debates in question in the Science Wars and beyond. One must of course

be as critical as possible, but, again, rigorously and patiently so, for other-

wise criticism is rarely effective or even meaningful, even when we deal with

lesser or outright bad work, if we want to criticize it, especially in public.

For one thing, we may, on the second, or third, look, discover that we were

ourselves wrong in our assessment, or at least we may want to make sure

that we were right. And then, the scientific critics in question themselves

sometimes have less knowledge, especially philosophical knowledge (but

sometimes even disciplinary knowledge) of mathematics and science, than

would be necessary to offer a meaningful criticism of nonclassical theories

or, as I shall argue in chapter 5, to tell us something profound about math-

ematics and science themselves.

An Introduction to Nonclassical Thought * 21

There are of course also alternative trajectories leading to nonclassical

limits. As Nietzsche observed in The Birth of Tragedy, "the periphery of the

circle of science [in the broader sense of German Wissenschaft] has an

infinite number of points, ... from which one gazes into what defies illumi-

nation [and sees] how logic coils up at these boundaries and finally bites it

own tail" (emphasis added).24 It may be, however, that it is in our encoun-

ters with nature that we test these limits most severely, as Bohr says; and this

is why Nietzsche invokes physics in this way, even though, while indebted

to proto-nonclassical physics mentioned earlier, his own nonclassical think-

ing has its greatest sources elsewhere. In particular, as I said, it would be

difficult, perhaps impossible, to dream up something of the kind quantum

mechanics tells us out of our imagination alone. As John Archibald Wheeler

once said of what it shows or un-shows to us: "What could one have

dreamed up out of pure imagination more magic-and more fitting-than

this? "25 Accordingly, against the Science Wars critics and their anxieties and

fears, one might urge a maximal engagement with modern mathematics and

science on the part of contemporary humanities, philosophy in particular,

even at the risk of getting certain technical details wrong here and there.

One must of course try to be as careful as possible as concerns the claims

and arguments of mathematics and science, and I shall try to exercise

utmost caution in this respect throughout this study. This maximal engage-

ment with mathematics and science may well be necessary if one is to reach

the limits of knowledge in these fields, and the threshold of the unknown,

and even the unknowable, which, it may be argued, defines all significant

knowledge. It appears to be the nature of our knowledge, perhaps especially

in mathematics and science, that new knowledge, for example, that leading

to solving outstanding problems (of meaningful difficulty), is only possible

by opening new horizons of the unknown, by opening the space of problems

of ever greater complexity. This process makes the life of knowledge more

difficult, but it also makes this life possible.

The reasons and circumstances for the Science Wars and related criticism

of nonclassical and related radical thought (that of scientists, such as Bohr

and Heisenberg, ultimately became a target as well) are many, given the

diversity of the participants involved, on both or all sides, and the immense

complexity of the contemporary cultural scene, where these confrontations

take place. The targets, both figures and ideas, are many and diverse, and so

are the reasons for their being targeted. Some of them are obvious enough,

sometimes all too obvious, while others are quite complex. It may be, how-

ever, that, to quasi-psychoanalyze this situation a bit, the greatest anxiety, if

not outright fear (although sometimes this fear nearly cries out, too), of

22 * The Knowable and the Unknowable

many such critics, mathematicians and scientists or others, is this: "the inex-

orable test of experiment," both in the narrow sense of scientific experiment

of Heisenberg's statement and in the broad sense Nietzsche gives to the sit-

uation, will prove nonclassical thought uncircumventable, from mathemat-

ics and science to political practice. It may even become the source of better

technical practice rather than only a philosophy or epistemology of mathe-

matics and science. This practice, it is true, at the moment need not, by and

large, depend on the possibly irreducible nonclassical nature of certain the-

ories, such as and in particular quantum physics, in which one can be (and

most physicists are) quite successful without paying any tribute or even

attention to anything nonclassical. If there is one thing that the major target

figures of the Science Wars-such as Lacan, Derrida, Deleuze, Luce Irigaray,

and Bruno Latour-share, it is their fundamental questioning of the limits

of classical theories and ways of thinking (the degree of success of their chal-

lenges varies, of course). I leave aside here many other authors targeted by

the Science Wars critics, some of whom may well merit criticism (as some-

times do the authors just listed) but, by and large, may still be seen as missed

targets of the Science Wars critics. These critics are rarely, if ever, in a posi-

tion to properly discriminate between what is and what is not a proper tar-

get of criticism, even on science, or sometimes even in science.

Nonclassical thought defines one of the most significant and controver-

sial strata of modern philosophical thinking, and its role in the humanities

is not unlike that of quantum theory in the discipline of physics. The works

of the figures to be specifically considered here are among the most contro-

versial, including, again, in recent discussions concerning the relationships

between modern mathematics and science and contemporary humanities.

Indeed, the very application of the denomination "philosophy" to their

work is controversial. But then so is, one might argue, the denomination

itself. Here I would like to adopt Deleuze and Guattari's understanding of

philosophy in What Is Philosophy? They see philosophy as a creation of

new concepts, indeed concepts that are forever new, thus defining it as, in

Nietzsche's famous phrase, always "the philosophy of the future." The term

"concept" itself must be, and here will be, used in the particular sense

Deleuze and Guattari give to it, rather than in any common sense of it,

specifically that of an entity established by a generalization from particulars

or "any general or abstract idea," as they argue, via Georg Wilhelm

Friedrich Hegel.26 A philosophical concept in this sense is an irreducibly

complex, multilayered structure or architecture-a multicomponent con-

glomerate of concepts (in their conventional sense), figures, metaphors, par-

ticular elements, and so forth. So understood and thus constructed, philo-

An Introduction to Nonclassical Thought * 23

sophical concepts often entail an engagement of diverse disciplines, fields of

inquiry, and human endeavors-in particular, in addition to philosophy

itself, mathematics and science, on the one hand, and literature and art, on

the other.

Accordingly, some interdisciplinarity may be unavoidable here, insofar

as the introduction of nonclassical theories affects the relationships between

different disciplines and the debates concerning these relationships over the

course of this century. An invocation of the currently fashionable and (in

part given the nature of the fashion) risky term "interdisciplinarity" may be

misleading. The present study pursues certain specific-nonclassical-epis-

temological configurations rigorously shared by different fields, where they

may have different roles to play, rather than certain interactions between

such fields themselves for their own sake, be they more or less rigorous or

more or less loose. Such interactions and our views of them have sometimes

been too loose and superficial or, one might say, lacking in discipline (in

either sense), often in the name of interdisciplinarity. It may be appropriate

and opportune to cite Bohr in introducing his philosophical essays: "The

following articles present the essential aspects of the situation in quantum

physics and, at the same time, stress the points of similarity it exhibits to our

position in other fields of knowledge beyond the scope of the mechanical

conception of nature. We are not dealing here with more or less vague

analogies, but with an investigation of the conditions for the proper use of

our conceptual means of expression. Such considerations not only aim at

making us familiar with the novel situation in physical science, but might on

account of the comparatively simple character of atomic problems be help-

ful in clarifying the conditions for objective description in wider fields"

(PWNB 2:1-2; also 3:1).

First of all, as will be seen, in Bohr's framework all proper references to

the data in question in quantum physics are "objective" in the sense of being

unambiguously defined and unambiguously reportable, and hence not sub-

jective. As he says, "in complementary description all subjectivity is avoided

by proper attention to the circumstances [of measurement] required for the

well-defined use of elementary concepts" (PWNB 3:7). At stake is, thus,

also the clarification of the rigorous conditions of the disciplinary function-

ing of physics as a mathematical science of nature, as Bohr understood it,

and (again, keeping due differences in mind) of other theoretical fields. This

(rather than the appeal to objective reality at the quantum level) appears to

be the primary reason for Bohr's usage of this, in turn risky and often

abused, term "objective," especially in his later works. There he also

responds to charges, not well founded, against the subjectivist, irrational,

24 * The Knowable and the Unknowable

and mystical character of quantum mechanics and complementarity itself.

(As will be seen, the term "irrational" has a more complex status in Bohr's

works.) One can of course hardly avoid hearing overtones of Kant's critical

philosophy in Bohr's invocation of "the conditions for the proper use of our

conceptual means of expression," even though the nonclassical character is

far more decisive here (however one reads or rereads Kant from this non-

classical perspective, as one perhaps ought to, as will be seen in chapter 3).

An excellent example of Bohr's point on "our conceptual means of expres-

sion" is his extraordinary statement on the (nonclassical) character of

chance and probability in quantum mechanics as complementarity, which is

of crucial significance in the nonclassical context and which I shall discuss

as such in chapter 2. Bohr says: "It is most important to realize that the

recourse to probability laws under such [quantum-mechanical] circum-

stances is essentially different in aim from the familiar application of statis-

tical considerations as practical means of accounting for the properties of

mechanical systems of great structural complexity [as in classical statistical

physics]. In fact, in quantum physics we are presented not with intricacies of

this kind, but with the inability of the classical frame of concepts to com-

prise the peculiar feature of indivisibility, or 'individuality,' characterizing

the elementary processes" (PWNB 2:34; emphasis added). The underlined

proposition is remarkable, and it extends well beyond the question of

causality (or even that of reality) and may be read as defining the essence of

nonclassical thought in quantum mechanics and elsewhere. One may adjust

it a bit to make my point more apparent: "In these theories we are presented

not with usual, if complex, intricacies of the classical kind, but with the

inability of the classical frame of concepts to comprise the peculiar features

characterizing the processes ultimately in question in nonclassical theories."

A very different frame of concepts may thus be required for such processes,

some of them classical and some new, nonclassical, as was indicated earlier

and, accordingly, or again reciprocally, very different (objective) conditions

of theoretical practice.

It is this type of epistemological convergence, rather than broader com-

parative-interdisciplinary networks (more commonly engaged with in most

recent works on the relationships between mathematics and science and the

humanities), that is my main concern in this study. A broader cultural per-

spective may be helpful, however, and, as a background and within the

agenda just indicated, such a perspective will sometimes be adopted (with

caution) in this study. Indeed, Bohr, too, sometimes spoke more ambitiously,

even if not in print, of his "dream of great interconnections." I shall now

An Introduction to Nonclassical Thought * 25

briefly sketch an outline of this perspective. Many lines and points of inter-

sections-names, ideas, projects, and so forth-will have to be omitted. The

broad and, hopefully, persuasive lineaments should be apparent, however.

Physics, specifically a comprehensive theory of quantum data, problem-

atic ever since Max Planck's introduction of quantum theory in 1900, may

have been Bohr's or Heisenberg's main concern in the development of their

interpretations of quantum mechanics, and its specificity must always be

kept in mind. Their work and their specific interpretations of quantum

mechanics can, however, be positioned in relation (I am not saying that they

are fully defined thereby, as some have argued) to a more general intellectual

and cultural configuration that emerged around 1900 in the wake of, on the

one hand, certain developments in mathematics and science and, on the

other, certain trends in philosophy, literature, and the arts. Many of these

trends may and, I would contend, sometimes must be seen as nonclassical in

the present sense.

On the side of literature and the arts, they include (to give a limited list)

such literary icons as Franz Kafka, James Joyce, Ezra Pound, and Samuel

Beckett; Arnold Schonberg and serial music; and Pablo Picasso and Cubism

and Marcel Duchamp, in the visual arts.

On the side of philosophy, Nietzsche's work was especially significant,

both in its own terms and in shaping the ideas of Heidegger and, then, the

thinkers in question in this study or the related work of such figures as

Georges Bataille, Maurice Blanchot, and Emmanuel Levinas. There are,

however, other key links. Thus, while the work of Theodor Adorno, Walter

Benjamin, and Ludwig Wittgenstein is outside the scope of this book, the

historico-theoretical configuration in question encompasses their work. The

work of Imre Lakatos, Thomas Kuhn, and Paul Feyerabend and their fol-

lowers in what has become known as science studies and related develop-

ments, offers a new understanding (sometimes bordering on nonclassical

thought) of scientific practice, which, it may be added, has been a subject of

some especially heated exchanges during the Science Wars.

On the side of mathematics and science, key areas include relativity and

quantum physics and their extensions, and more recently chaos and com-

plexity theories, and modern biology and genetics. Most prominent in

mathematics have been investigations into the foundational questions and

mathematical logic leading to Kurt Godel's incompleteness theorems. Other

areas-topology, modern analysis and algebra, number theory, and alge-

braic geometry-extend and join ideas developed in the works of earlier

figures such as Karl Friedrich Gauss, Evariste Galois, Bernhard Riemann,

26 * The Knowable and the Unknowable

and Henri Poincare.27 These developments have important links to other

areas of mathematics and science that shape the intellectual landscape con-

sidered in this study, as will be especially seen in chapter 3.

There are many well-known connections between the authors and fields

just mentioned. Duchamp had a major interest in non-Euclidian geometries

and multidimensional spaces, in part via Poincare's mathematical and philo-

sophical ideas. Bohr's thinking appears to have been influenced by or may be

connected to Cubism, while the latter was in turn affected by previous devel-

opments in physics and mathematics. The emergence of Duchamp's key

ideas, too, coincides (but is not coincidental) with the emergence of quantum

physics and the modern technology of scientific experiments, such as that

used in quantum physics. The significance of quantum physics for Kafka,

Joyce, Rainer Maria Rilke, Virginia Woolf, and many others has been well

documented and is, by now, the subject of considerable scholarship. There

exists a significant interface between Heisenberg's physics and philosophical

views and Heidegger's philosophy. The significance of quantum physics and

its epistemology in the work of Bataille, Blanchot, Levinas, Derrida, and Paul

de Man, and their many contemporaries and key predecessors, is undeniable

and irreducible, even if not always immediately manifest, which does not

diminish its impact. The influence of Riemann's ideas on Deleuze's philoso-

phy is immense. More generally, the impact of mathematics and science on

modern French thought is one of the most extraordinary phenomena and

one of the most controversial subjects in recent history.

At the same time, however, disciplinary specificity must be rigorously

respected and taken into account in exploring these connections or the work

of these figures in their respective fields. Accordingly, one also needs a care-

ful discrimination as concerns what can and cannot be analytically trans-

ferred from one field to another, whether one proceeds from the humanities

to mathematics and science, or from mathematics and science to the human-

ities (or the social sciences). The present study adopts this view and tries to

implement it throughout.

Indeed, one rarely deals here even with unambivalent local determina-

tions, let alone with a single organizational synthesis, but instead with net-

works of connections and, sometimes, ruptures, and hence something that

can only be seen as partially rhizomatic in Deleuze and Guattari's sense. On

the other hand, the parallel-interactive horizontal networkings of different

fields and areas of inquiry that are involved, rather than a vertical subordi-

nation to or encompassing by any given field, bring the present view close to

Deleuze and Guattari's horizontal rhizomatics, juxtaposed by them to the

arborial (vertical) subordination. Now, while heterogeneous, the fields and

An Introduction to Nonclassical Thought * 27

even conceptions involved are also interactive-they are heterogeneously

interactive and interactively heterogeneous-which is what enables and

maximizes productive interdisciplinary practices, specifically productive

crossings between science and the humanities. Heterogeneity is irreducible,

thus making it unavoidable that at least certain areas of different disciplines

cannot be brought together, now or, conceivably, ever. But interactiveness

is unavoidable, too, and sometimes indeed gives rise to great interconnec-

tions, of which we can at least dream, as Bohr did. Accordingly, one can

never be certain what will or will not be brought together or when. Nor can

one be certain whether the bringing together, or conversely separation (we

must use both), will work. This study's bet is that we might benefit by bring-

ing together a few more things than are together now, even though it

appears to be far less certain which ones.

Nonclassical epistemology, which also applies to these interconnections,

appears likely to be here to stay, at least for a while, even if the situation or

the configuration of opinion will eventually change in physics and return us

to a more classical view of the ultimate constitution of nature. (Hence, I

gave my "for now" qualification earlier). This is possible, given the manifest

incompleteness and diversity of physics as it is currently constituted. The

validity of Bohr's interpretation depends on the present stage of quantum

physics and other experimental facts of physics, in particular, as will be

seen, those pertaining to relativity theory. One cannot of course be certain

concerning even the more or less immediate survival of nonclassical episte-

mology either, even within the still relatively limited domain of its accep-

tance, as discussed earlier here. For, even leaving aside the persistence of

classical views and resistance to nonclassical ones, the configurations of

opinion are always in flux on all fronts and in all areas. One may detect a

certain resurgence of classical views in general and specifically in the debates

concerning quantum physics, even while the debate about quantum-

mechanical epistemology continues and in part of course because it contin-

ues. On the other hand, yet greater epistemological surprises may also be in

store in physics or elsewhere in mathematics and science (modern biology,

for example), or elsewhere, hard as it appears to imagine, given the strange

things quantum mechanics and nonclassical epistemology in general tell us,

or at least make some of us hear.



Chapter 2

Quantum Mechanics, Complementarity, and

Nonclassical Thought

... que ne se montrent pas mais sont agissantes.

-MARCEL PROUST

... that do not reveal themselves but are nonetheless

efficacious

This chapter offers a comprehensive introduction to quantum mechanics

and complementarity as a nonclassical theory. Its nonclassical character is

reflected in Bohr's statement, which occurs in his reply to the argument of

Einstein, Podolsky, and Rosen and which guides my argument in this chap-

ter. (Both articles were published under the same title, "Can Quantum-

Mechanical Description of Physical Reality be Considered Complete?")

Bohr invokes: "a final renunciation of the classical ideal of causality and a

radical revision of our attitude towards the problem of physical reality,"

and, it appears, a final renunciation of the classical ideal of reality is also at

stake. The main reason is that complementarity places the ultimate objects

of quantum mechanics, quantum objects, beyond the reach of quantum the-

ory and beyond all knowledge and conception. It is, as we have seen, this sit-

uation that defines nonclassical theory. The chapter proceeds as follows.

The first section serves as an introduction. It offers a brief outline of the key

junctures in the history of quantum theory and then introduces the key epis-

temological aspects of both quantum mechanics and complementarity, as an

interpretation of quantum mechanics. The second section offers a compara-

tive epistemological discussion of classical and quantum physics. The third

section considers in detail complementarity and, through the nonclassical

optics of complementarity, the essential features of quantum physics, begin-

ning with the double-slit experiment, an archetypal quantum-mechanical

experiment. It also discusses Bohr's new nonclassical concept of atomicity,

which no longer applies at the level of nature (quantum objects) but instead

at the level of technology (measuring instruments), one of Bohr's most rad-

ical and most extraordinary concepts. The fourth section considers uncer-

tainty relations and the nonclassical nature of quantum probability. The

30 * The Knowable and the Unknowable

fifth section addresses the Einstein, Podolsky, and Rosen experiment and the

question of locality of quantum physics (its compatibility with relativity).

Locality has become the central issue in more recent debates concerning

quantum theory and, perhaps inevitably, has entered the current public dis-

cussions and debates surrounding nonclassical epistemology, and

specifically the Science Wars, to be discussed in chapter 5. The sixth section

is concerned with the "disciplinarity" argument, thus bringing together

epistemological radicality and scientific disciplinarity in quantum mechanics

and, by implication, in nonclassical theories elsewhere. As will be seen,

according to Bohr, by virtue of its radical epistemology, complementarity in

fact "provides room for new physical laws [those of quantum mechanics],

the coexistence of which might at first sight appear irreconcilable with the

basic principles of science," that is, unless we find a proper interpretation,

such as complementarity, of the new situation we encounter in quantum

mechanics. In other words, given the data in question in quantum mechan-

ics, complementarity is a (perhaps necessary) condition of scientific discipli-

narity and indeed of the main task of science, as we understand it now-the

discovery of new physical laws-rather than being in conflict with the basic

principles of science, as some of its critics contend.

Introduction: The Nature(s) of the Quantum

Quantum physics was inaugurated in 1900 by Max Planck's discovery,

widely seen as the single greatest discovery in twentieth-century physics,

that radiation, such as light (very high frequency electromagnetic waves),

previously believed to be a continuous (wavelike) phenomenon in all cir-

cumstances, can, under certain conditions, have a quantum or discontinu-

ous character. Planck made his discovery in the course of his attempt to for-

mulate and then to interpret the radiation law for the so-called black body,

whose model is a heated piece of metal with a cavity, then an outstanding

problem of classical physics. The limit at which this discontinuity appears is

defined by the specific frequency of the radiation of the body and a univer-

sal constant of a very small magnitude, h, now known as Planck's constant,

which Planck himself termed "the quantum of action" and which turned

out to be one of the most fundamental constants of all physics. The indivis-

ible (energy) quantum (quantity) of radiation in each case is the product of

h and the frequency v, E = hvy.1 Planck's work and related developments rad-

ically transformed physics and our sense of the limits of our knowledge, sci-

entific and philosophical, and its claims upon nature and mind. The trans-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 31

formation took a while, as did a more adequate understanding of quantum

phenomena themselves. We are hardly finished with either now, as we have

entered the twenty-first century. The debate concerning quantum mechanics

itself continues, even if mostly for epistemological reasons, which have to do

primarily with resistance to the nonclassical aspects of it. No end appears to

be in sight.2

Quantum physics began to expand its scope and accumulate complexity,

ultimately insurmountable by means of classical physics, from the outset,

beginning with the scope and complexity of quantum phenomena them-

selves.3 While crucial, the subatomic discontinuity or, at least in Bohr's

interpretation, even the particlelike nature of quantum objects is only an

approximation of the ultimate complexity of quantum phenomena. The

particlelike understanding of light, a concept more radical than Planck's dis-

continuity of radiation (assuming that Planck himself went even that far,

which is far from clear), emerged more or less immediately following

Planck's discovery, primarily due to Einstein. The idea met with consider-

able resistance on the part of the theoretical physics community, including

Bohr at the time. Eventually, sometime around 1920, it won the physics

community over in the wake of Arthur Compton's experiments. In these

experiments photons would interact in a particlelike manner with other par-

ticles. It also was around that time that the particles of light were named

"photons." The light acquired a more explicitly dual, wave and particle,

character each manifest in different experimental circumstances, which clas-

sical physics did not appear able to explain. More or less around the same

time, the more complex and pervasive nature of the wave-particle duality at

the quantum level became apparent as well. Both radiation, such as light,

wavelike according to the classical view, and particles (or what were classi-

cally seen as such), such as electrons, may manifest their existence, if not

themselves (this qualification has proved decisive for Bohr), in both wave-

like and particlelike phenomena under different circumstances. The wave

aspect of particles was originally proposed by Louis de Broglie in 1923 and

confirmed soon thereafter. All ultimate constituents of nature acquired a

dual wave-particle character, which continues to be the case, although our

understanding of each of these physical aspects itself as applied to quantum

objects has changed in the wake of quantum mechanics.4 At the same time,

it did not appear possible to ever observe both types of phenomena together.

On the one hand, this circumstance appeared to make the situation para-

doxical from a classical viewpoint, according to which it would be consid-

ered in terms of properties of quantum objects themselves: What are, ulti-

mately, quantum entities, particles or waves? Or how does one combine

32 * The Knowable and the Unknowable

such incompatible, mutually exclusive, features as properties of the same

objects? On the other hand, the situation eventually suggested, especially to

Bohr, a way out of the paradox, almost a blessing: Since such incompatible

observational effects are always mutually exclusive and can never be simul-

taneously observed, the paradox in fact disappears! This mutual exclusiv-

ity was then used by Bohr to define the complementarity of certain features

of physical description, the concept introduced in the so-called Como lec-

ture of 1927, "The Quantum Postulate and the Recent Development of

Atomic Theory," published (in a revised version) a year later (PWNB

1:52-91). Eventually complementarity came to designate Bohr's overall

interpretation of quantum mechanics and, then, a general philosophical

conceptuality.

It took more than two decades to sort out the initial complexities by

means of quantum mechanics, also called at the time "the new quantum the-

ory," as opposed to "the old quantum theory," developed primarily by, in

addition to, Planck, Einstein, Bohr, and Arnold Sommerfeld. At least it

became possible to formulate a theory, which was, as mechanics, analogous

in functioning to Newtonian mechanics, although, as became quickly

apparent, not in character. Quantum mechanics was introduced in 1925-26

by Heisenberg and Schrodinger in two different versions, which were

proved to be mathematically equivalent, although, one might argue, poten-

tially different epistemologically, as Heisenberg's matrix mechanics lends

itself more naturally to the nonclassical view and was indeed initially pre-

sented by Heisenberg in virtually nonclassical terms. It was developed in the

work of Max Born, Pascual Jordan, Paul Dirac, Wolfgang Pauli, and (pri-

marily in terms of interpretation) by Bohr.s This theory was nonrelativistic

and dealt with the motion of electrons at speeds significantly slower than

those of light, although the initial work on relativistic quantum theory,

quantum electrodynamics (known as QED), was virtually contemporary.

Dirac introduced his famous relativistic equation for the electron in 1928.

Schrbdinger's equation described the motion of the electron in the standard

quantum mechanics (insofar as one can speak of motion here).

Quantum mechanics, however, brought with it new epistemological

complexities, even as it has offered a degree of resolution or at least closure

of the problems posed by Planck's discovery (which the old quantum theory

failed to solve) and even as it became in mathematical-theoretical terms the

standard theory. The majority of even the most resilient critics, Einstein and

Schrodinger among them, acknowledged that quantum mechanics brought

with it considerable improvements as concerns the predictive capacity of

quantum theory. What bothered these critics and even some proponents

Quantum Mechanics, Complementarity, and Nonclassical Thought * 33

was a manifest deficiency of the explanatory-descriptive capacity of quan-

tum mechanics with respect to quantum objects themselves. The situation

was roughly as follows.

The old quantum theory dealt reasonably well (albeit not perfectly) with

statistical matters and was or, rather, seemed to have been analogous to

classical statistical physics. What was lacking was the mechanics describing

individual objects that would underlie the manifest statistical behavior of

multiplicities in a manner similar to Newtonian mechanics, which explains

causally the motion of individual molecules in the classical statistical theory,

according to the modern view, which emerged during the first decades of

this century, courtesy primarily of Einstein. Both Bohr's 1913 theory of the

hydrogen atom (and much of his work that followed it prior to quantum

mechanics) and Einstein's 1916 work on the so-called induced and sponta-

neous emission did deal with individual quantum processes.6 While, how-

ever, describing well the atomic spectra in terms of discontinuous (quan-

tum) jumps of electrons in the atom from one energy level to another, Bohr's

theory did not account for the mechanism of transition as classical physics

would. Einstein's argument introduced the potentially irreducible probabil-

ity consideration into the individual quantum processes (rather than those

involving large multiplicities of objects and interactions), a defining feature

of quantum theory ever since and yet one more of Einstein's revolutionary

insights. These works, thus, further showed that the true mechanics was

lacking, before quantum mechanics revealed that such a "true" (classical-

like) mechanics was perhaps no longer possible and might even in principle

be precluded (it is in Bohr's interpretation). It became clear almost immedi-

ately in the wake of Planck's discovery of his law, however, that, contrary

to Planck's original argument (accordingly incorrect in this respect, as Ein-

stein was among the first to show), this law was incompatible with a classi-

cal-like underlying picture. Indeed, for that reason, the way of statistical

counting is different in classical and quantum statistics. Planck's counting

was correct, even though part of his physics was wrong, an error that, as

Einstein observed, was most fortunate for physics.7

The new quantum mechanics, which was expected to resolve these prob-

lems, was, however, nothing like classical physics. Certain aspects of the old

quantum theory, in particular, again, Bohr's 1913 theory of the atom and

most of Einstein's work, were harbingers of the new, and to many unap-

pealing, features that quantum mechanics was to retain and enhance rather

than eliminate, as some hoped it would. Skipping for the moment even

(ever?) greater epistemological complexities, the new quantum theory could

only predict, mostly statistically, the outcome of certain individual events,

34 * The Knowable and the Unknowable

such as collisions between particles and a silver-bromide photographic

screen, but it appeared unable to describe the motion of quantum objects in

a manner analogous to classical physics. (As will be seen, it does make some

exact predictions, which, however, does not fundamentally change the epis-

temology in question, and may even be seen as reinforcing it.) In short, it

would predict the outcome of the experiments in question (classical-like the-

ories would fail to do so), but it would not describe the behavior of physical

objects in the way classical physics would. Nor would it predict in the same

way either. Far from having eliminated chance from the theory, quantum

mechanics gave chance an even more radical character by making it irre-

ducible both in practice and in principle even in dealing with individual,

rather than only collective, behavior. Indeed, the collective behavior could

in certain circumstances be subject to theoretical formalization and (statisti-

cal) predictive law; the individual behavior is fundamentally, irreducibly

lawless. Quantum mechanics does not proceed in the way classical statisti-

cal physics does, from causal and deterministic individual behavior to sta-

tistical collective behavior. Instead, it combines a relatively ordered collec-

tive behavior and irreducibly lawless individual behavior, a combination

that makes possible excellent statistical predictions but leaves little, if any

(none in Bohr's interpretation), space to the description of the actual physi-

cal processes in space and time responsible for such predictions. We can

describe the impact, or effects, of quantum processes upon our measuring

instruments in a reasonably (indeed, in Bohr's interpretation, strictly) classi-

cal manner, but, in accordance with nonclassical epistemology, not the ulti-

mately efficacity of such effects. In other words, while in classical physics

observable phenomena (in the usual sense) can be properly related to the

observable properties of actual objects, in quantum mechanics, in Bohr's

interpretation, observable phenomena can only be correlated with the

behavior of quantum objects. It is not possible to establish the correlata of

such correlations at the quantum level-that is, any properties of quantum

objects themselves under investigation or of their behavior or those of quan-

tum aspects of measuring instruments interacting with quantum objects.

(This interaction makes measurement possible in quantum physics).8 This

impossibility eventually led Bohr to his redefinition of the term "phenome-

non" in terms of the effects of the interactions between quantum objects and

measuring instruments upon the latter. Similar considerations compelled

Heisenberg to speak, anticipating Bohr's subsequent arguments, of a "new

kinematics" necessary for quantum mechanics in his first paper on quantum

mechanics and other earlier works. Traditionally, the term "kinematics"

refers to the attributes of motion itself, such as positions (coordinates) or

Quantum Mechanics, Complementarity, and Nonclassical Thought * 35

time, or velocities of the body. Dynamic properties are irreducibly depen-

dent on kinematics but are defined by the presence of mass of the bodies, as

are momentum or energy. Heisenberg's "new kinematics," by contrast,

referred to what is observable in measuring instruments under the impact of

quantum objects rather than to the attributes of these objects themselves.

Accordingly, insofar as kinematical and dynamic properties (of whatever

kind) are involved in quantum mechanics, they are only those of certain

parts of measuring instruments.9 Bohr's comment on Heisenberg's

approach in the Como lecture is worth citing: "The new development [of

quantum theory] commenced in a fundamental paper by Heisenberg, where

he succeeded in emancipating himself completely from the classical concept

of motion by replacing from the very start the ordinary kinematical and

mechanical quantities by symbols which refer directly to the individual

processes demanded by [Planck's] quantum postulate" (PWNB 1:70-71).

(The phrase "individual processes" must be seen as referring to what is actu-

ally observed in measuring instruments rather than to quantum processes

themselves, on which I shall comment later.) The revolutionary, emancipa-

tory rhetoric aside, when Bohr says "fundamental" (or essential), he means

fundamental-foundationally irreducible. Bohr immediately adopts the

concept of new kinematics itself in his Como lecture, whose first section-

which also offers Bohr's own (much more elementary than Heisenberg's

but, in some respects, epistemologically less radical and less sophisticated)

derivation of uncertainty relations-is entitled "Quantum of Action and

Kinematics" (PWNB 1:57-62; also 1:48-51).

Thus, the complexity of quantum phenomena opened the way to, even if

(the question is, again, under debate) not definitively entailed, nonclassical

interpretations of these phenomena and of quantum mechanics as a theory

that accounts for them, such as Bohr's complementarity and related inter-

pretations, often assembled under the rubric of "the Copenhagen interpre-

tation." Not all of these reach the nonclassical limits in question in this

study. The use of the denomination "Copenhagen interpretation" requires

caution, given the differences between them and the thought of the different

figures involved. This includes those who are considered, and consider

themselves, close to Bohr. It is not always easy to ascertain the degree of this

proximity or difference in some of the latter cases, such as Heisenberg and

Pauli, or (still differently) Dirac and John Von Neumann, since their episte-

mological views are not explicated to the same degree as Bohr's. I would

argue, however, that, once considered in all of its aspects, Bohr's interpreta-

tion (in the present reading or "interpretation") is unique and is the only

one that is fully nonclassical epistemologically, with the possible exception

36 * The Knowable and the Unknowable

of the epistemology that transpires (but is not philosophically worked out)

in Heisenberg's early work, as just indicated. One must also exercise caution

as concerns complementarity itself. It was introduced by Bohr in the Como

lecture in 1927 and developed by him through his life. In a way, it was his

life, as he in fact said: "It was, in a way, my life."10 In the process, however,

complementarity had undergone considerable evolution and refinement

before it reached its ultimate and ultimately nonclassical version (by the late

1940s), to which I primarily refer here.

Sometime between 1935 and 1938, in part under the impact of Einstein's

criticism of quantum mechanics and specifically of the EPR 1935 argument

concerning the possible incompleteness of quantum mechanics as a physical

theory, Bohr rethought the very concept of phenomenon (as applicable in

quantum physics) in epistemologically nonclassical terms.11 This

redefinition was prepared by Bohr's earlier rethinking of the irreducible role

of measurement in quantum mechanics and led him to his arguably most

refined and epistemologically radical (nonclassical) version of complemen-

tarity. Phenomena would no longer refer, in the way they do or can be

defined to do in classical physics, to physical properties of quantum objects

or their behavior, or what is inferred as such on the basis of the experimen-

tal effects in question (PWNB 2:64). Instead, they would refer to experi-

mentally observed effects (the term persistently used in his writings hence-

forth) upon, in terms of their physics, classically described measuring

instruments under the impact of their quantum interactions with quantum

objects. It would perhaps be more accurate, given the history of the term, to

see "phenomena" as referring to the representations of such effects, and

Bohr's usage can indeed be adjusted accordingly. I shall, however, continue

to follow Bohr's usage, although some qualification will be necessary at cer-

tain junctures.12 The rigorous specification of each experimental arrange-

ment is essential, since it is now itself considered part of each phenomenon

and is one of the reasons for the complementary (mutually exclusive) char-

acter of some of them.

Complementary features of description are now defined by the mutual

exclusivity of some among such arrangements, considered, nevertheless, all

necessary for a comprehensive description of quantum phenomena, in

accordance with Bohr's original definition of such features. In such experi-

mental arrangements quantum-mechanical effects, such as, most famously,

those associated with "waves" or "particles," manifest themselves-as

macroscopic effects, pertaining to or manifest in certain (classical) proper-

ties of certain parts of measuring instruments, rather than as properties of

quantum objects themselves.13 Each such effect-say, a spot left by a parti-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 37

cle on a photographic plate (differently patterned multiplicities of such spots

define wave and particle effects) or a displacement of a certain part of a

measuring apparatus under the impact of a collision with a quantum object

(a particle effect)-is a classical physical object. As such it becomes a phe-

nomenon in Bohr's sense to be described in terms of classical physics. That

is, it is classical insofar as we ignore the actual "efficacity" of its emergence,

defined by the interaction (itself quantum in nature) between quantum

objects and measuring instruments, or, in view of qualifications given ear-

lier, what is so idealized. Once we consider this efficacity, the situation

becomes as follows. We can observe a change in the state of certain parts of

measuring instruments and measure the pertinent classical variables, say, a

change in momentum, in a perfectly classical manner. Or we can use such

parts as a classical frame of reference, where we can, say, register the posi-

tion of a spot left by a "particle" (using this denomination and indeed this

conception by analogy with classical processes, an analogy ultimately inap-

plicable here) colliding with a screen attached to it. We cannot, however,

classically account for the actual physical process that led to such changes in

the conditions of the measuring instruments, nor use such occurrences for

theoretical predictions based on the formalism of classical physics. Any

attempts to use classical physics in handling such quantum effects, from

Planck's own interpretation of his discovery to the introduction of quantum

mechanics in 1925, had failed, as did those attempts (during the same

period and, I would argue, subsequently) to interpret quantum-mechanical

mathematical formalism in a classical-like manner. Classical physics could

neither explain the sum total of these effects nor offer theoretical predictions

concerning the appearance of each. Nor, it follows, could it describe the

efficacity of each such effect. The latter, however, cannot, by definition, be

done nonclassically either, and hence by means of quantum mechanics in

Bohr's nonclassical interpretation.

From this perspective quantum objects themselves must be seen (or,

again, idealized) as "entities" different from either particles or waves, while

giving rise to one or the other type of phenomena (but never both types

together) by virtue of their interaction with measuring instruments. Each

type of phenomena (but never both together) appears (in either sense) as the

effect of these interactions, in specific and always mutually exclusive cir-

cumstances, which can be rigorously defined and, whenever necessary, set

up experimentally. In other words, the appearance of the phenomena of

each type uniquely depends on a particular type of experimental setup; and

we can always arrange for such a setup and expect the appearance of the

corresponding type of phenomena (although not the same outcome for each

38 * The Knowable and the Unknowable

experiment). We can, however, never combine both types of phenomena so

as to ascertain, even in principle, all characteristic phenomenal properties in

question or construct any experimental setup that would enable us to do so

(in the way it can always be done, at least in principle, in classical physics).

Thus, we can observe either the wavelike effects or the particlelike effects of

the interaction between quantum objects and measuring instruments, but

never both simultaneously. We can neither avoid phenomena of either

type-either with wavelike effects or with particlelike effects-nor combine

the phenomena of both types at any point. In some respects the latter is true

in classical physics as well. There, however, these two descriptions apply

(directly) to the distinct and rigorously separable types of objects: particles

are always particles, waves are always waves. A difference in the experi-

mental setup would not change the nature of either type of object. By con-

trast, in quantum mechanics, in this interpretation, it is the (particular) dif-

ference in the experimental setup that defines the different-the wavelike or

the particlelike-character of the observable phenomena for the same type

of "objects," while the latter are seen as, ultimately, unobservable as such,

outside their interactions with measuring instruments. I emphasize "type"

because one cannot say that the complementary phenomenal effects in ques-

tion in any given set of experiments could pertain to the same object, even

though, technically, we can sometimes either repeat the same type of exper-

iment or a (complementary) alternative experiment on the "same" object,

were we to see the situation classically. In quantum physics a strict (rather

than statistically treatable) repetition of the same experiment is not possible.

Nor, in Bohr's interpretation, could we think in terms of the same object in

this situation. Each phenomenal effect/efficacity configuration must be seen

as different, unique each time, as concerns both the effect and the efficacity

under consideration.

Of course, we need instruments in classical physics as well, even if it is

only a human eye observing, say, the moon moving in the sky, famously

invoked by Einstein in this context (I shall comment on "Einstein's moon"

presently). However, in classical physics, such interactions can always be

neglected or compensated for without affecting the soundness of classical

description, which is not possible in quantum physics, at least in Bohr's

interpretation. One can get a preliminary idea of the situation by imagining

that each time in order to observe the moon (or what would be left of it) we

would need to shut something on the scale of Jupiter into it and study the

debris left. One must, however, also keep in mind the limited nature of this

analogy. While helpful, it can also be misleading, since, as will be seen later

in this chapter, the key epistemological features here discussed need not

Quantum Mechanics, Complementarity, and Nonclassical Thought * 39

depend on this (or any) type of physical "disturbance" of quantum objects

themselves, itself a misleading way of expression, as Bohr persistently points

out (PWNB 2:63-64, 73; 3:5). For the moment, as I said, we can neither

observe a proper fusion of phenomena of different types nor conceive of a

single underlying quantum configuration that (even while itself unobservable

or even inconceivable) would itself possess both attributes as its coexisting

aspects or as its coexisting effects. Indeed, in Bohr's interpretation, the con-

cept of an "underlying quantum configuration" is strictly inapplicable, how-

ever one conceives of such a configuration, whether in terms of particles,

waves, or still otherwise. While each of the efficacities in question is nonclas-

sically unknowable, each of them is also as individual as each of their effects,

and both, efficacities and effects, reciprocally define each other. In this sense

such efficacities are in turn subject to complementary relationships.

Thus, while always mutually exclusive in quantum physics, the two types

of phenomena in question, "wave effects" and "particle effects," are both

necessary for a comprehensive overall quantum-theoretical description; in

other words, they are complementary in Bohr's definition. As will be seen,

however, the application of the term "the wave-particle complementarity"

requires further qualification, and indeed it ultimately loses its significance

(which has always been limited) for Bohr. The wavelike effects observed in

quantum-mechanical experiments must instead be seen in terms of collec-

tivities of individual phenomena, defined differently from the particlelike

collective effects and phenomena by virtue of the mutually exclusive exper-

imental circumstances in which such different collectivities would arise. In

other words, the wave-particle complementarity has a more complex archi-

tecture, which is also fundamentally related to the statistical nature of quan-

tum mechanics. These considerations to some degree justified Heisenberg's

initial suspicions and eventually diminished Bohr's enthusiasm concerning

Schrbdinger's wave mechanics. Schrbdinger's equation itself can be seen as

relating to this more complex situation rather than to any space-time wave

propagation or even strictly wavelike phenomenal effects.

By the same token, however, in this interpretation (in contrast to certain

other interpretations), the mathematical formalism of quantum theory, such

as Schrbdinger's equation, refers, in a particular statistical way (to be con-

sidered later), to these effects and only to them, a view, again, anticipated by

Heisenberg in his original paper introducing quantum mechanics and related

early work. Accordingly, this formalism is no longer seen as referring to the

attributes, such as, say, "positions" or "velocities," and behavior, such as

motion, of quantum objects themselves in the way classical physics would

refer to and indeed (this is sometimes forgotten) define its objects, such as

40 * The Knowable and the Unknowable

those, suitably idealized, moving in the macroworld around us. While it

defines classical physics, this type of idealization or model is no longer possi-

ble in quantum mechanics in this interpretation. It became apparent imme-

diately in the wake of quantum mechanics that one could never apply all of

the classical particle properties, or, conversely, all of the classical wave prop-

erties, in quantum physics but only some of them sometimes, as stressed by

Heisenberg.14 Instead, a different idealized model, correlated with the math-

ematical formalism of quantum mechanics, may be defined. In this model the

only physical attributes actually referred to are those pertaining to idealized

measuring instruments rather than, as in classical physics, those of quantum

objects themselves. Indeed, one would not be able to apply all the necessary

classical properties (such as position and momentum) within a given experi-

ment even in this case, although considered in themselves, outside the con-

text of quantum-mechanical experiments, the relevant parts of measuring

instruments may be seen as possessing all such properties. In other words,

such properties would come into play in the classical manner in the classical

interactions between these measuring instruments and classical objects. It

also follows that the situation just described pertains to the quantum inter-

action between quantum objects and the quantum aspects of the measuring

instruments, the interaction initiating the efficacity of the effects in question

in Bohr's interpretation.15s

One can, thus, define the key variables involved in a perfectly classical

manner as referring to the physical properties of measuring instruments.

Unlike in classical physics, however, such variables can never be defined

simultaneously, which is, in Bohr's interpretation, the ultimate meaning of

Heisenberg's uncertainty relations, more customarily seen as establishing

the unsurpassable limits, defined by Planck's h, upon simultaneous exact

measurement, rather than, as in Bohr, definition, of such variables. This is

also why we can never simultaneously apply all the classical particle prop-

erties even to the parts of measuring instruments impacted by quantum

objects. The concept of the particle in classical physics in fact or in effect

depends on the presence of both such attributes, a fact sometimes forgotten

by those who use the concept. The quantum-mechanical situation just out-

lined enables Bohr to give a rigorous physical meaning to uncertainty rela-

tions in his interpretation (which, thus, may also be seen as an interpretation

of uncertainty relations) in terms of this mutual exclusivity or, in his terms,

complementarity. Such descriptions are both necessary for a comprehensive

account but can never apply simultaneously, since they entail mutually

exclusive experimental conditions under which the corresponding (comple-

mentary) phenomena or effects appear. There appears to be no experimen-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 41

tal arrangement that would allow us to circumvent this difficulty. Bohr

made a virtue out of necessity and used this fact as a way to resolve these

problems, which, he argued, arise only if one adopts the epistemologically

classical view of the situation.

This, it goes without saying, is not to say that quantum objects or, again,

more accurately, that which compels us to speak in such terms do not exist,

for example, when we are not there to interact with them by means of our

technologies and build our theories, classical or nonclassical, about them.

Quite the contrary; in Bohr's interpretation the existence of "quantum

objects" of some sort and of their actions, such as those producing certain

effects, is an inevitable inference given the data (i.e., "effects") in question.

To put it another way, the particular ("strange") architecture of the data or,

one might say in the spirit of quantum information theory, information that

one encounters in quantum physics compels one to infer the existence of cer-

tain (quantum) objects of nonclassical nature; or, again, these data, includ-

ing as coupled to the mathematical formalism of quantum mechanics, can

be interpreted in this way. In other words, the point here is that such attrib-

utes as "quantum" (in any sense, such as that of discontinuity, individual-

ity, atomicity, or whatever, including complementarity) or indeed any

attributes at all do not apply to the ultimate constituents of nature, as we

understand them now. When Bohr speaks of the "behavior of quantum

objects," this phrase should be read in this general sense of the existence of

something that is beyond the classical level of description and that (in the

interaction with measuring instruments) produces observable effects. It

should not be read as implying an ascription of conventional physical or

perhaps any other attributes to this behavior anymore than to quantum

objects, ultimately the attribute of "behavior" itself included. We do not

and cannot know or conceive of what sort of objects quantum objects may

be, nor can we ultimately, in all rigor, think of them as "objects" (a rela-

tively recent philosophical concept), let alone attribute physical properties

to them or their behavior, including motion, by analogy with classical

physics. This, however, need not mean that nothing exists or that everything

stands still, as Parmenides was perhaps first to surmise, thus helping Plato to

ground his view of the world.16

Abraham Pais reports the following conversation with Einstein on

"objective reality" (or the lack thereof) in quantum theory: "We often dis-

cussed his notions of objective reality. I recall that during one walk, Einstein

suddenly stopped, turned to me and asked whether I really believed that the

moon exists only when I look at it."17 Einstein's target is Bohr's argument

that quantum mechanics disallows one to speak unambiguously of any

42 * The Knowable and the Unknowable

attributes of quantum objects but instead relates to the effects of these

objects upon the measuring instruments involved, as discussed here. Quan-

tum mechanics, however, or complementarity, does not tell us that there is

nothing "out there," for example, where we see the moon, when there is

nobody to look at it. (There are very few who would believe this, even

among the most committed antirealists.) Instead it tells us that it is far from

certain that, once there is nobody to look at it, it would be the moon or any-

thing we can possible think of, now or ever, say, a collection of elementary

particles. And then, which moon? We might do well to follow, to its perhaps

ultimate limits, Juliet's admonition to Romeo: "By yonder blessed moon I

vow.... O, swear not by the moon, th' inconstant moon, / That monthly

changes in her circled orb, / Lest that thy love prove likewise variable"

(Romeo and Juliet, II.ii.109-11). The moon is never quite there, at least as

anything quite the same, even when we look at it, but it is also not quite up

to us to create it by merely looking at it, although we can change it at least

a little by so doing. To cite Proust's comment: "the trees, the sun and the sky

would not be the same as what we see if they were apprehended by creatures

having eyes differently constituted from ours, or else endowed for that pur-

pose with organs other than eyes which would furnish equivalents of trees

and sky and sun, though not visual ones." Perhaps such "organs" (even if

this denomination could apply) would not furnish even that much, in any

event nothing equivalent. It is, now not surprisingly, at this juncture that

Proust invokes those "ideas that do not reveal [show] themselves but are

none the less efficacious [agissantes]."18 It is also clear that while he says

"ideas" he starts his meditation with the physical world and sees two paral-

lels, and indeed, interactive dynamics, material and phenomenal (in the

usual sense of appearance to the mind), at work. Both are always at work in

nonclassical epistemology as, in this case, it moves us from Einstein's moon

to Bohr's atom.

While logically consistent and fully compatible with the experimental

data in question, Bohr's nonclassical solution of the dilemma and, hence, his

interpretation of quantum mechanics carry to some, Einstein in particular,

an exorbitant and even unacceptable epistemological cost. In his reply to the

EPR article, Bohr speaks of "the necessity of a final renunciation of the clas-

sical ideal of causality and a radical revision of our attitude towards the

problem of physical reality" (QTM, 146). As a result of this radical revision,

we also confront a final renunciation of the classical ideal of reality, rather

than only that of causality, at the ultimate level of description, which

appears to have ultimately troubled Einstein most. Nonclassically, more-

over, the suspension of realism entails not only the impossibility of describ-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 43

ing physical properties of quantum objects and processes but also the

impossibility of ascribing such properties to them and the impossibility of

applying any hitherto available concept of reality to quantum objects them-

selves. Indeed, in the absence of classical reality (and perhaps there is no

other reality) one can hardly speak of causality. The point was well

expressed by Schrodinger, who, as Einstein, never accepted quantum

mechanics, to the creation of which he contributed so much, as earlier did

Einstein. Or at least both did not accept the epistemology or "doctrine," as

Schrodinger referred to it (claiming further that it was "born of distress"

resulting from the impossibility of finding a more classical-like theory

accounting for quantum data), here in question, and both appear to have

doubted whether quantum-mechanical data and mathematical formalism

could avoid it. "If a classical [physical] state does not exist, it can hardly

change causally," Schrodinger said in his famous "Schrodinger cat para-

dox" paper (1935).19 In other words, causality, however conceived, is

merely one of such attributes, as ultimately is "change," "process," and so

forth, or, crucially, their opposites, such as "permanence" or "system."

Ultimately, no concept of reality that is, or ever will be, available to us

might be applicable to our description of the "quantum world" or to the

ultimate "objects" in question in nonclassical theories elsewhere.20 The

quotation marks become obligatory under these conditions, since the terms,

however conceived, of this sentence ("quantum," "world," "object,"

"things," even if seen as "things-in-themselves") may be no more applicable

that those, however conceived, of the preceding sentence. "Ultimately,"

however, is again a crucial qualifier here since both concepts, that of reality

and that of causality, and, as I said, classical ways of thinking in general,

apply and are extraordinarily effective across a broad spectrum of theoreti-

cal work and other human activities, or in everyday life. They are also nec-

essary for the functioning of nonclassical theories, since the latter must

work with the classical (knowable) effects and, hence, rely on classical ways

of thinking in handling these effects, even though their efficacity remains

ultimately unknowable.

Equally, by definition, however-this is a crucial twist of nonclassical

epistemology!-the ultimate nature and character of this efficacity are not

available to nonclassical theories either. The latter, however, rigorously

account for the overall configuration of the effects in question and, in the

case of quantum mechanics, predict measurable quantities associated with

such effects. This capacity ensures the status of quantum mechanics as, in

Galileo's words (defining modern physics), "a mathematical science of

nature" and, thus, also guarantees a disciplinary continuity between quan-

44 * The Knowable and the Unknowable

turn and classical physics. We need nonclassical theories for these purposes

rather than only in order to introduce the irreducibly unknowable that ulti-

mately defines them. Indeed, the latter conception itself emerges by virtue of

the rigor of nonclassical thought, such as Bohr's, and, reciprocally, ensures

the rigor of theoretical knowledge, and even "provides room for new phys-

ical laws [here the laws of quantum mechanics], the coexistence of which

might at first sight appear irreconcilable with the basic principles of science"

(QTM, 146). This argument enables us to read the two statements, which

respectively, open and close Bohr's argument in his reply to EPR, together:

"the necessity of a final renunciation of the classical ideal of causality and a

radical revision of our attitude towards the problem of physical reality ...

provides room for new physical laws, the coexistence of which might at first

sight appear irreconcilable with the basic principles of science" (QTM, 146,

150; emphasis added). In other words, the nonclassical epistemology of

quantum mechanics provides room for what can be properly known in

accordance with the disciplinary rules of physics, above all, for the discov-

ery of new laws of nature, the ultimate goal of all physics.

From Classical to Quantum Physics, from Classical

to Nonclassical Epistemology

In physics the difference between classical and nonclassical (to distinguish it

for the moment from "nonclassical" in the epistemological sense of this

study) theories emerged without a priori ontological and epistemological

considerations concerning the objects of investigation or the character of the

theories involved. It was defined in terms of physics itself (that is, in terms

of the constitution of the data and the theories accounting for these data) as

the difference between classical, often also called Newtonian, physics and

the new physics, which emerged around 1900 and departed from classical

physics in significant ways. In addition to Newtonian mechanics, the key

areas of classical physics included such nineteenth-century developments as

molecular theories, including thermodynamics, and Maxwell's electrody-

namics, at the intersections of which quantum physics emerged in the work

of Planck. All these theories were subject to complex development,

refinement, interpretation and reinterpretation, as well as continuous

debate, extending to our time and sometimes joining the debate concerning

quantum physics. A critical investigation of Maxwell's theory, most espe-

cially by Einstein, led to relativity theory. The new physics specifically

Quantum Mechanics, Complementarity, and Nonclassical Thought * 45

referred at the time, first, to this theory or, as it became known later, special

relativity theory, developed in the work of, in addition to Einstein, Henrik

Lorentz, Henri Poincare, and others, in contrast to general relativity theory,

Einstein's (non-Newtonian) theory of gravitation, developed by him around

1915. The latter extended relativity theory to spaces whose geometry was

affected, curved, by gravity, and became a major part of new physics as

well. Second, the new physics referred to quantum theory, eventually lead-

ing to quantum mechanics.

The classical physical theories just mentioned, however, are also classical

in the epistemological sense of this study and hence are realist, as well as

causal (which, as I said, in general need not be the case classically). Or, more

accurately, they may be and (in contrast to quantum theory) commonly are

interpreted as such. At least this can be, and commonly is, done at the level

of idealization, whereby the properties of material bodies and their behav-

ior, such as motion, are treated as physically measurable mathematical

quantities abstracted from other properties that material bodies and their

behavior possess. One can see such idealized configurations of bodies and

motions as models that classical theories provide, such as those mathemati-

cally described by the equations of Newtonian mechanics.21 Following

Schrodinger, we may call these models "classical models" and the ideal itself

of the possibility of such a description "the classical ideal." Both have

proved to be extraordinarily effective in classical physics and have been

adopted in other fields of science and elsewhere as classical theories in the

broader sense of this study. To cite Schrodinger in his "cut paradox" article:

In the second half of the previous [nineteenth] century there arose, from the

great progress in kinetic theory of gases and in the mechanical theory of heat,

an ideal of the exact description of nature that stands out as the reward of

centuries-long search and the fulfillment of millennia-long hope, and that is

called classical. These are its features. (QTM, 152)

It is worth interrupting the quotation for the moment to make the fol-

lowing observation. Schrodinger is well aware of the ideal nature of the clas-

sical ideal even in classical physics, as well as its possible inapplicability at

the ultimate level of description in view of quantum theory (ten years old

and well accepted by the time), although Einstein's criticism renewed his

classical hopes (QTM, 153-54). However, as these sentences and his argu-

ment as a whole make clear, his appreciation of the ideal itself was as pro-

found as were his concerns for the possible loss of this ideal in view of quan-

tum mechanics. To resume Schrbdinger's description of the "features" of

the classical ideal:

46 * The Knowable and the Unknowable

Of natural objects, whose observed behavior one might treat, one sets up a

representation-based on the experimental data in one's possession but with-

out handcuffing the intuitive imagination-that is worked out in all details

exactly, much more exactly than any experience, considering its limited

extent, can ever authenticate. The representation in its absolute determinacy

resembles a mathematical concept or a geometric figures which can be com-

pletely calculated from a number of determining parts: as, e.g., a triangle's

one side and two adjoining angles, as determining parts, also determine the

third angle, the other two sides, the three altitudes, the radius of the inscribed

circle, etc. Yet, the representation differs intrinsically from a geometric figure

in this important respect, that also in time as fourth dimension [this repre-

sentation] is just as sharply determined as the figure is in the three space

dimensions. Thus, it is a question (as is self-evident) always of a concept that

changes with time, that can assume different states; and if a state becomes

known in the necessary number of determining parts, then not only are all

other parts also given for the moment (as illustrated for the triangle above),

but likewise all parts, the complete state, for any given later time: just as the

character of a triangle on its base determines its character at the apex. It is

part of the inner law of the concept that it should change in a given manner,

that it should continuously run through a given sequence of states, each one

of which it reaches at a fully determined time. That is its nature, that is the

hypothesis, which, as I said above, one builds on the foundation of intuitive

imagination. (QTM, 152)

The classical representation in question, or "image" or "model," as

Schrodinger also calls it, is both causal and realist (in view of the determi-

nation of all parts from a sufficient given set of parts). That is, the systems

that are conceptualized according to this scheme (hence, Schrbdinger's

appeal to a "concept that changes with time") are used as a representation

or an idealization of "natural objects, whose observed behavior one might

treat . . . based on the experimental data in one's possession." The classical

ideal may or may not be fulfilled to various degrees by particular physical

theories and models. Some among such theories and models may be suc-

cessful (or successfully refined), while others may fail and may be aban-

doned. Newtonian mechanics is the primary and arguably still most suc-

cessful manifestation of the classical model and ideal at work (other

theories, primarily of a statistical nature, that Schrodinger mentions here,

often depend on it as well). Consider it, for example, as applied to the falling

bodies or (physically the same thing) the motion of planets around the sun.

We can idealize such objects and behavior, as massive material points (cor-

responding to the approximate centers of mass of the actual bodies

involved) moving along continuous trajectories described by well-specified

equations, so as to apply a causal and realist description to map this ideal-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 47

ization. In this case the "determining parts" would be such variables and

measurable quantities as "position" and "momentum," whose determina-

tion at any given point and the equations of classical mechanics (which

mathematically represent the idealized system in question) would ensure a

causal evolution of the system. This argumentation is not undermined by

the fact that once more than two bodies are involved (say, the sun, the earth,

and the moon), the behavior of the system becomes, in general, chaotic,

which prevents deterministic predictions. The behavior itself is considered

causal, and, as I will explain presently, chaos theory is classical in the pres-

ent definition. Nor is it significantly undermined by the fact that Newton's

theory ultimately fails even on this scale and needs to be replaced by Ein-

stein's general relativity, since the latter is classical within these limits. The

situation becomes more complicated once very massive bodies, such as neu-

tron stars or black holes, enter the picture (even when one leaves aside the

quantum aspects of their constitution, ultimately necessary for theorizing

such objects) or when cosmological considerations are involved. Thus clas-

sical physics fails both on very large scales and on very small ones when

quantum theory takes over.

To a large degree, as a disciplinary practice, classical physics is an

engagement with such models and a relating of them to the experimental

data. Classical or nonclassical theories in physics are more complex forma-

tions, which include both models and interpretations, since the status of

classical models as a description of nature may be interpretation dependent,

even while the classical ideal itself is maintained. In particular, an interpre-

tation of a given classical model could establish, or at least consider, the

relationships between this model and the experimental data or the behavior

of the natural objects in relation to the model. Thus, in the case of planetary

motion the question is relatively straightforward, at least in a good first

approximation, insofar as the realist and causal models describe and predict

the behavior of material bodies involved, as just explained. In other cases

the situation may be more complex since the relationships of such models to

the behavior of the natural objects involved are far less direct.

It is also worth keeping in mind (this is often forgotten or remains unper-

ceived) that setting up a classical, or of course nonclassical, model itself also

requires interpretation, even as it refers to the behavior of idealized systems.

In particular, one may need an interpretation of a given mathematical for-

malism, which would relate, say, a given set of equations to the motion of

idealized physical objects. (This was to become a problem in the case of the

equations of quantum mechanics, such as Schrodinger's equation.) This

type of interpretation may precede and give rise to an idealized model rather

48 * The Knowable and the Unknowable

than serve to mathematize a given ideal situation, or both processes may be

(and mostly are) in complex reciprocal relationships. It follows that classi-

cal models are not strictly mathematical models, since nonmathematical

(physical and philosophical) considerations and idealizations are involved,

such as those concerning the "motion," specifically causal motion, of

"material bodies." These are not inherently mathematical concepts, even

though they have helped mathematics to develop and refine some of its con-

cepts, in particular those of differential calculus. We may speak of math-

ematizable models, insofar as the behavior of the objects covered by a given

classical model can be mapped by mathematical objects, such as those used

by Galileo or (these are not quite the same) Newton and his successors,

especially Joseph Louis Lagrange and Sir Rowan Hamilton, who gave clas-

sical mechanics its modern form. Their work laid the mathematical and, in

some respects (insofar as nature itself is seen as describable in terms of dif-

ferential equations), epistemological foundations for much of the subse-

quent physics, such as Maxwell's electrodynamics, and then relativity and

quantum mechanics and other quantum theories, whose equations are of

the Hamiltonian type. As will be seen, however, the variables to which these

equations apply are mathematically very different from those used in classi-

cal physics.

In contrast to classical physics, quantum mechanics proved resistant to

classical-like interpretation from the outset. At the very least, it appears to

be conducive to, even if it does not entail, nonclassical interpretations, such

as Bohr's. Einstein's relativity may be seen as occupying an intermediate

position between classical and quantum physics, especially insofar as it

remains a causal theory. As I said, however, just as quantum theory, general

relativity may be, and may ultimately need to be, seen as epistemologically

nonclassical.22 Initially the problems of quantum theory concerned primar-

ily determinism (the predictive capacity of the theory) and causality (the

causal nature of the processes in question). These, as I said, appeared lack-

ing even in the case of the individual quantum processes, such as, say, an

electron's motion around the nucleus of an atom, and even at the level of

idealization and models. In other words, as Schrodinger was to lament in

the article cited earlier, a classical-like model and a fulfillment of the classi-

cal ideal appeared all but impossible, which, in his view, indicated a neces-

sary, or near, abandonment of quantum mechanics. The models to which

the standard quantum theory lends itself and that are mathematically

described by the formalism of quantum mechanics (such as Schrodinger's

equation) did not, and still do not, appear available to a classical and,

specifically, causal treatment. Indeed, most classical-like and, specifically,

Quantum Mechanics, Complementarity, and Nonclassical Thought * 49

realist interpretations of quantum mechanics (as opposed to Bohmian

mechanics) do suspend causality and determinism as well. In other words,

we do not seem to be able to configure what the mathematical formalism of

the theory describes in terms of a classical-like system, and ultimately any

physical system, at the level of quantum objects themselves. Early stages of

quantum theory, from Planck on, prior to and in the immediate wake of

quantum mechanics were characterized by such attempts. They still con-

tinue but, to this author, without much success at the very least at the level

of causality (realist noncausal interpretations are a more complex matter).23

Heisenberg and then Bohr made a virtue out of necessity by taking advan-

tage of this potential impossibility, which, for Heisenberg, necessitated a

new "kinematics" of quantum theory, as explained earlier, the "kinemat-

ics" dealing with the effects of quantum processes upon measuring instru-

ments, rather than with motions of quantum objects.

In relation to nature at the level of its ultimate constituents, quantum

objects, or, again, what we infer as such from the data in question in quan-

tum mechanics, Bohr's complementarity is both a model and an interpreta-

tion-an interpretation of the relationships between quantum mechanics in

this model and nature. As concerns the quantum-mechanical formalism, as

accounting for these data, it is a model, an idealization, insofar as it deals

with classically idealized measuring instruments and certain types of effects

upon them. Quantum objects and processes themselves cannot be seen as

idealized in this model, since there is nothing the model says about them

(indeed, it rigorously prohibits saying anything about them)-for many,

beginning with Einstein, a distressing, even if not logically inconceivable,

feature of the model and the resulting theory, almost a betrayal of the task

of physics as science. Or, as qualified earlier, quantum objects are idealized

as unidealizable by this model, while what they relate to in nature is not ide-

alized even as unidealizable, inconceivable, and so forth. For, as was indi-

cated in chapter 1, complementarity only deals with this type of idealization

(as the nonclassical unknowable) as part of its model rather than attributing

this conception to whatever actually happens in nature at the level of its ulti-

mate constituents. (Accordingly other models or interpretations of quantum

mechanics can idealize the situation differently.) Complementarity, how-

ever, is also a consistent and comprehensive-complete within its proper

limits-interpretation of quantum theory or, again, of a certain model of

quantum theory, as represented by the standard quantum-mechanical for-

malism. Bohr's argument is that such an interpretation is at least possible,

even if not inevitable, but only-this is the nonclassical epistemological

essence of complementarity as a model-if one suspends any claim whatso-

50 * The Knowable and the Unknowable

ever concerning the physical attributes of quantum objects and behavior, in

other words, if one establishes the model just outlined. That is, comple-

mentarity is entailed by or (these relationships are, again, reciprocal)

entails a physical theory, a model, of the interactions between quantum

objects and measuring instruments in which measuring instruments are

(idealized as) described in terms of classical physics. More accurately, one

should, again, speak of describing in terms of classical physics those parts

of measuring instruments through which we register outcomes of their

interaction with quantum objects. This interaction is, accordingly, itself

quantum but is, again, capable of producing manifest classical effects. By

contrast quantum objects themselves (or, it follows, the quantum aspects of

the interaction in question) are seen as inaccessible and, hence, not subject

to any model. The effects of these interactions are rigorously correlated

with the formalism of quantum theory, which predicts them, just as classi-

cal physics predicts the outcomes of classical experiments, while, however,

dealing with the properties and behavior of the objects themselves that it

considers. (The role of measuring instruments can, at least in principle, be

disregarded or compensated for in classical physics.) Thus, as was dis-

cussed earlier, the only natural (material) reality described, in a suitably

idealized way, by Bohr's complementarity model would be that of the clas-

sical macroscopic effects of the interactions between these objects and mea-

suring instruments upon the latter, rather than that of the ultimate objects,

quantum objects, of quantum mechanics and their behavior or quantum

aspects of the interaction between these objects and measuring instru-

ments. Bohr's model prohibits a description of the quantum to the point of

the inapplicability of any concept of reality, or, it follows, causality, or

indeed any conceivable concept to quantum objects or behavior, ulti-

mately, (given that even this is an idealization) even the concept of incon-

ceivability itself.

Bohr's argument itself could of course be challenged, for example, by

considering alternative models and interpretations of quantum theory; dif-

ferent forms of mathematical formalism and theories potentially accounting

for this data (such as Bohmian mechanics); and so forth. While I do not

think, for the reasons to be explained later, that such challenges have so far

been successful in undermining the consistency and effectiveness of Bohr's

argument, I am not arguing that such challenges are impossible or not

potentially viable. My point at the moment is that, in Bohr's case, an inter-

pretation of the relationships between the model and nature is built into the

nature, the character, of the model itself-a unique and elegant way of

doing physics and philosophy alike.

Quantum Mechanics, Complementarity, and Nonclassical Thought * 51

One may also put this a little differently and perhaps more strongly.

Bohr's interpretation of quantum mechanics prohibits doing in principle

what quantum mechanics itself, as a physical theory, cannot, it appears, do

in practice.24 This formulation differentiates Bohr's complementarity from

other interpretations that hold the same view as to what quantum mechan-

ics actually delivers, or does not deliver, as a theory or a model. As an epis-

temologically nonclassical theory, complementarity establishes uncircum-

ventable limits upon what we can, in principle, analyze or know or, again,

even conceive of, now or, again, ever. This is why Bohr said that, "in quan-

tum mechanics [as complementarity] we are not dealing with an arbitrary

renunciation of a more detailed analysis of atomic phenomena [again, in

Bohr's sense], but with a recognition that such an analysis is, in principle,

excluded" (PWNB 2:62; Bohr's emphasis). On the basis of the present read-

ing of Bohr and the present interpretation of complementarity, I am com-

pelled to see this statement as referring to quantum mechanics as comple-

mentarity, viewed as a particular model and interpretation of quantum

mechanics, as just explained. Quantum mechanics may and has been inter-

preted otherwise. It is, again, another question of how effective such alter-

native interpretations are. Bohr, too, could be interpreted differently, in par-

ticular as making a stronger claim here. How adequately or effectively is

another question, too. I shall return to these questions later in this chapter.

On the point itself made by Bohr here complementarity, thus, goes fur-

ther than most other interpretations even in interpreting what quantum

mechanics actually can or cannot do in practice rather than only in further

prohibiting in principle what it cannot in practice accomplish. Some among

such alternative interpretations also see the quantum-mechanical formalism

as referring to, even if not describing, the behavior of quantum objects, as

opposed to, as in Bohr, strictly the effects of their interaction with measur-

ing instruments upon the latter. In complementarity, quantum-mechanical

mathematization leaves a much greater nonmathematical residue that one

encounters in classical physics, even though quantum mechanics employs as

rigorous a mathematical scheme as classical physics does. This residue per-

tains, first, to the inaccessible and nonmathematizable efficacity of the

effects in question, and hence the ultimate objects of theory, and, second,

correlatively, to the rules, such as Max Born's rule for using the wave func-

tion for calculating the probability of quantum-mechanical events. These

rules link the mathematical formalism to the experimental data in question

in quantum mechanics and, thus, enable its workings as a physical theory.

But, while extraordinarily effective, they remain ad hoc and exterior to the

theory; that is, they have neither mathematical nor physical justification

52 * The Knowable and the Unknowable

from within the theory itself, and (this is my point here) in Bohr's interpre-

tation they cannot have.

Certain views of classical theories, such as Galileo's (in some contrast to

Newton's), may not be concerned with such justifications either. This, how-

ever, is not quite the same as not being able to offer such a justification at all

from the first principles (which can be done reasonably well in classical

physics) or, especially, to claim it as rigorously impossible or prohibited, as

Bohr was compelled to do in his interpretation of quantum mechanics. In

his 1913 theory of the atom, Bohr, too, only saw such a justification as

impossible at the stage of the theory then, while quantum mechanics, in

Bohr's interpretation, made it impossible rigorously. Classical physics was

eventually able to offer such a justification (or at least what was accepted as

such) in the case of Galileo's physics, while in quantum mechanics this does

not appear to have been possible so far, even in interpretations short of non-

classical limits of complementarity.25

I shall now explain the main physical reasons, first, for the epistemologi-

cally classical nature, or, again, the possibility of the epistemologically clas-

sical interpretation of classical physics, and then the physical reasons for

why the situation is different in quantum mechanics.

Classical physics, such as Newtonian mechanics, is, or can be interpreted

as, ontologically, realist because it can be seen as fully describing all the

(independent) physical properties of its objects necessary to explain their

behavior. At least, again, such is the case for idealized systems or at the level

of classical models, when, more immediately, the properties in question are

abstracted from other properties of the objects comprising a given system.

The resulting model can then also serve as a model for more complex and

less tractable systems. In these latter cases, similarly to quantum mechanics,

in most interpretations, classical physics would account for the observable

effects or consequences of the workings of such systems, but, unlike quan-

tum mechanics (in strictly nonclassical interpretations) the ultimate efficac-

ity of these effects is amenable to the same classical treatment. Classical

physics also is or may be interpreted as, ontologically, causal because the

state of the systems it considers (these systems may, again, be idealized and

function as models for real systems) at any given point is assumed to be

determined (in the past) by and to determine (in the future) its states at all

other points. It is also (usually), epistemologically, deterministic insofar as

our knowledge of the state of a classical system at any point allows us to

know, again, at least in principle and in ideal cases (but also as a good

approximation in most actual cases) its state at any other point. Description

of this kind was, as Bohr observes, found to have a very wide scope, extend-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 53

ing (with due nuances) to (statistical) kinetic theories of gases and electro-

magnetic (classically, wavelike) phenomena, and specifically Maxwell's

wave theory of the latter, and, then, albeit, again, ultimately more ambiva-

lently, to Einstein's relativity (PWNB 3:1-2).26

Causal physical theories or models need not be deterministic in the above

sense. Thus, both classical statistical physics and, differently, chaos theory

(which is, in most of its forms, classical and is sometimes a direct extension

of Newtonian mechanics) are causal, or, again, at least they are assumed to

be and allow for such an interpretation. Rigorously, we must see them in

terms of idealizations and models as well, even if only because the quantum

aspects of the objects considered by such theories indicate the very different

nature of the ultimate reality, or the lack thereof, to which such models

apply.27 These theories, however, are not deterministic even in ideal cases in

view of the great structural complexity of the systems they consider. This

complexity blocks our ability to predict the behavior of such systems, either

exactly or at all, even though we can write equations that describe them and

assume their behavior to be causal. Indeed, the latter assumption is often

necessary in these cases. For similar reasons, it would be difficult to make

Newtonian mechanics truly deterministic, or even realist, in most actual

cases, which need to be suitably idealized or modeled (which is not always

possible in practice) for the equations of Newtonian mechanics to do their

job. In principle, however, as an idealization, or a model, it is a causal and

deterministic theory, or can be interpreted as such, while classical statistical

theory, or chaos theory, is (while causal) not deterministic even as an ideal-

ization. In general, it does not follow that deterministic theories are realist,

since the actual behavior of a system may not be mapped by our description

of it, even though we can make reasonably good predictions concerning that

behavior. Classical mechanics (or chaos theory) is, however, also realist

insofar as such a mapping is assumed to take place, at least as an idealiza-

tion or a good approximation that classical models provide. By contrast,

classical statistical physics is not realist insofar as its equations do not

describe the behavior of its ultimate objects, such as molecules of a gas. It is,

however, usually based on the realist assumption of an underlying nonsta-

tistical multiplicity, whose individual members in principle conform to the

strictly causal laws of Newtonian mechanics.

As I have indicated, I here expand the denomination "realist" to theories

or models that are approximate in this sense or further to theories that pre-

suppose an independent reality that cannot be mapped or even approxi-

mated but that is assumed to possess attributes and structure by analogy

with, or, one might say, on the model of, classical models. Indeed, the latter

54 * The Knowable and the Unknowable

is our only source of such conceptions, for example, those responsible for

the assumption that quantum objects possess independently existing prop-

erties ("elements of reality," Einstein liked to call them) and behave in the

manner of those of classical physics, or may be idealized or modeled accord-

ingly, or for the assumption that somebody plays or does not play quantum

dice. Most realist theories and models in physics hitherto available may be

described by the presupposition that their objects in principle possess inde-

pendently existing physical attributes, whether we can or cannot, in practice

or even in principle, ever describe or approximate them. The (structured)

physical reality thus is assumed to be ultimately independent of observation

and measurement, but subject to conceptual, theoretical, and, sometimes,

quantitative approximation, which physics can undertake as a mathemati-

cal theory of this reality. Some see this presupposition as epistemologically

necessary for the practice of physics or science in general. On this definition,

realist and classical theories are obviously one and the same and are treated

as such here. Accordingly, realist theories, simultaneously, both make the

systems that they consider subject to representation (with the qualifications

here offered) and yet treat the behavior of these systems as, at least in prin-

ciple, independent of our interactions with them. We can guess, at least

approximately, how reality is organized, for example, with our intuition

and imagination, which may itself be seen to be part of the epistemology of

the classical ideal, as it was seen by both Einstein and, as was discussed,

Schrodinger and many of their predecessors and followers.

From this viewpoint, insofar as it was not describing the physical space-

time behavior of quantum objects themselves even, and indeed in particular,

at the level of idealized models, quantum mechanics was almost not physics

for Einstein, at least not the way he practiced it with such extraordinary

skills and results. Epistemologically, he was not altogether mistaken, for

none of the features of classical theories just sketched is, by definition, pos-

sible in quantum mechanics, at least in nonclassical interpretations, at the

level of the ultimate "reality" in question. It is this irreducible and irre-

ducibly unbridgeable rupture, a properly quantum discontinuity (perhaps

the only true discontinuity in quantum physics), that defines the difference

between nonclassical thinking, found in quantum physics, and any realism.

As I have indicated, there is in fact a double discontinuity here: first, there is

a discontinuity within the model itself, since it configures or idealizes quan-

tum objects as something that is beyond any physical space-time (or indeed

any other) description; and second, between this idealization and whatever

may actually happen in nature. It appears, at least in the present context,

Quantum Mechanics, Complementarity, and Nonclassical Thought * 55

that it was the first discontinuity that was especially crucial and vexing to

Einstein and Schrbdinger, and many of their followers, who would hardly

find the word idealization suitable here. Einstein spoke of "Jacob's pillow,"

which, he thought, provided comfort to those in Copenhagen and Gottin-

gen, two main centers of new quantum theory, perhaps especially in view of

the almost mysterious (in either sense) predictive power of the theory. As I

shall explain, however, while this power may indeed be seen as mysterious,

insofar as we do not know and cannot even think in terms of any underly-

ing space-time physical mechanism through which such predictions occur, it

cannot be seen as mystical, in the sense of postulating some unknown

(divine or otherwise metaphysical) single agency behind such predictions.

However sophisticated realism may be (clearly the conceptions just out-

lined may and, in such cases as Einstein's and Schrbdinger's, did entail con-

siderable complexities in terms of idealizations, models, correspondence

with reality, and so forth) and however useful or indeed indispensable it is

in science, including, within certain limits, in quantum mechanics, realism

will always remain naive in this sense. Perhaps no (mere) refinement of com-

mon experience, of everyday life, which gave rise to the conception of clas-

sical physics (extending already from Aristotle's Physics), can be anything

but naive or at least at, in turn, an unbridgeable distance from nonclassical

thinking. As I have indicated earlier, both Bohr and Heisenberg (rightly) see

classical physics as refinement of common experience. To be sure, from

Galileo on, this refinement reaches far beyond Aristotle, but not even nearly

far enough (which may in fact not be possible classically) for quantum

physics, if not relativity.

One might, however, also question why one should expect nature at

these levels to follow the classical ideal, even as an ideal or model, which is

something that emerges at a very different scale and in very different cir-

cumstances of experience, or indeed something that is merely a refinement,

however exquisite, of common experience. For example, why should we

expect it to play or not play dice, Einstein's famous, if, as we have seen,

finally removed, objection to quantum mechanics (PWNB 2:47)? Of course,

as Heisenberg observes in commenting on Bohr's complementarity, "the

concept of 'observation' belongs, strictly speaking, to the class of ideas bor-

rowed from the experience of everyday life. It can only be carried over to

atomic phenomena when due regard is paid to the limitations placed on all

space-time description by the uncertainty principle."28 Heisenberg here

refers to Bohr's initial version of complementarity, which, as will be seen, is

not the best way to handle the situation. In Bohr's post-EPR version, any

56 * The Knowable and the Unknowable

observation is always strictly classical, and uncertainty relations merely

make necessary two mutually exclusive (complementary) classical observa-

tions or phenomena.

Quantum mechanics in Bohr's interpretation is neither causal nor deter-

ministic nor realist in any of the senses described earlier, most specifically in

the sense of the possibility of assigning any specific form (particlelike or

wavelike, or other) of independent physical reality to quantum objects or

processes. The latter may, at most, be said to be real or, again, may relate to

something that exists (even when we are not there to observe it), but this

existence or this "reality" appears to prevent us from conceiving of the way

in which it exists. This is how we are compelled to idealize these objects and

processes in the nonclassical model of quantum mechanics as complemen-

tarity. Ultimately, such concepts as "objects" and "processes" may not be

applicable either, however we define them, even at the level of idealization

or model, and indeed, in the case of complementarity, specifically at this

level. In other words, complementarity idealizes quantum "objects" and

"processes" as something to which no possible physical description or con-

ception is applicable. The reasons for this are as follows.

It is not only that the state of the system at a given point gives us no help

in predicting its behavior or in allowing us to assume it to be causally deter-

mined, if unpredictable, at later points (lack of determinism and causality)

(PWNB 1:65-66).29 More radically, even this state itself cannot, at any

point, be unambiguously defined on the model (in either sense) of classical

physics (radical nonrealism). The lack of causality is merely an immediate

consequence, as Schrodinger noted in the comments cited earlier. This

impossibility of an unambiguous definition of the state of the system is cor-

relative to, and indeed is Bohr's interpretation of, Heisenberg's uncertainty

relations, arguably the single most defining quantitative law of quantum

mechanics, reconceived by Bohr, nonclassically, in terms of the interaction

between quantum objects and measuring instruments, as indicated earlier.30

Most immediately, uncertainty relations express the strict quantitative lim-

its, absent in classical physics, on the simultaneous joint measurement of the

so-called, by analogy with classical physics, conjugate variables, which

define the motion of classical objects, such as "position" or "coordinate"

(q) and "momentum" (p). These limits are expressed by the famous formula

AqAp

h, where h is Planck's constant and A designates the precision of

measurement (the same type of formula holds for time and energy). The

increase in precision in measuring one such variable inevitably implies

equally diminished precision in measuring the other. Bohr's post-EPR ver-

sion of complementarity gives a more radical (strictly nonclassical) interpre-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 57

tation to uncertainty relations, which prohibits even assignment or unam-

biguous definition of physical properties, such as a position or a momen-

tum, ultimately even if each is taken by itself, to quantum objects and

behavior, rather than inhibits their joint measurement.

Both situations, classical and quantum-mechanical (including in a non-

classical interpretation), can be mathematized in accordance with the

requirements of, in Galileo's terms, modern mathematical sciences of

nature. This point is crucial to Bohr and part of his argument, cited earlier,

that although and, as will be explained, indeed because it radically suspends

both causality and reality, "quantum mechanics provides room for new

physical laws, the coexistence of which might at first sight appear irrecon-

cilable with the basic principles of science." As could be expected, the two

respective mathematical schemes, that of classical and that of quantum

mechanics, are irreducibly different. In particular, the quantum-mechanical

mathematical formalism involves a form of algebra of noncommuting sym-

bols whereby the product of the magnitudes involved depends on the order

of multiplication (i.e., PQ does not equal QP), even though this formalism

can be converted into that of classical physics by assuming Planck's constant

h to be equal to zero at the classical limit. This noncommutativity disap-

pears in the process as well, for the very simple reason that in the quantum-

mechanical case PQ - QP = ih/2n, where i is the square root of-1, an imag-

inary number, which will be discussed in chapter 3. The role of complex

numbers, which may be shown to be irreducible in the standard quantum-

mechanical formalism, is another crucial difference between quantum-

mechanical and classical formalism. It can also be shown that in order for

this scheme to lead strictly to uncertainty relations such mathematical

objects must in fact be infinite-dimensional, as were Heisenberg's original

matrices, infinite tables related to quantum data. The equations themselves

describing these quantities as variables were formally the same as those of

classical mechanics (Hamilton's equations). Physically, once we can assume

that h equals zero, the symbolic quantities P and Q (usually still called

"momentum" and "position" by analogy with classical physics) become the

regular momentum and position variables of classical physics and subject to

its equations rather than to the quantum-mechanical mathematical scheme

or schemes (as we have seen, there are several mathematically, although not

always epistemologically, equivalent). This is a manifestation of the so-

called correspondence principle, introduced by Bohr in the context of "the

old quantum theory" but successfully used by him and others, especially,

again, by Heisenberg in his pioneering work on quantum mechanics,

throughout the history of quantum theory.31

58 * The Knowable and the Unknowable

Bohr's Atoms: Phenomenology and Epistemology

of Complementarity

My argument here is based primarily on the post-EPR version of comple-

mentarity and still more specifically on the version of it presented in Bohr's

"Discussion with Einstein on Epistemological Problems in Atomic Physics,"

originally published in 1949 in the so-called Schilpp volume honoring Ein-

stein.32 Bohr's 1935 reply to EPR was arguably the most decisive work in

developing Bohr's ideas in the form considered here, although there is still

earlier evidence of this shift, around 1930, under the impact of his earlier

exchanges with Einstein. The 1938 Warsaw lecture is crucial as well, as it

introduces Bohr's concept of phenomenon, although the latter is implicit in

Bohr's reply to EPR (BCW 7:301-22). According to Bohr himself, however,

"Discussion with Einstein" elucidates Bohr's response to Einstein's argu-

ments and "give[s] a clearer impression of the necessity of a radical revision

of basic principles for physical explanation" (PWNB 2:61). "Discussion

with Einstein" becomes Bohr's arguably most comprehensive and consistent

exposition of his interpretation of quantum mechanics, although, as I men-

tioned earlier, Bohr's final exposition of complementarity (in its post-EPR

version) in "Quantum Physics and Philosophy: Causality and Complemen-

tarity" in 1958 may be the most definitive. To the present reader, however,

it lacks the comprehensiveness and nuance of "Discussion with Einstein,"

which in addition explicates and refines or even corrects, and sometimes

reinterprets, his earlier works, such as the Como lecture of 1927. The dif-

ferences between different post-EPR arguments are not negligible but are

not even nearly as significant as those between them and those prior to the

EPR arguments or, in any way, prior to Einstein's critique from 1927 on.

These nuances are sometimes essential for understanding Bohr's epistemol-

ogy and its nonclassical nature, although they are not always addressed or

carefully sorted out by Bohr's readers, who are not always helped by Bohr

either, since he tends to minimize these differences and to reread his earlier

works in terms of later ones.

Thus, in referring to his earlier publications (now assembled in PWNB 1)

in his 1935 reply to EPR, Bohr says: "I shall therefore be glad to use this

opportunity to explain in somewhat greater detail a general viewpoint, con-

veniently termed 'complementarity,' which I have indicated on various pre-

vious occasions" (QTM, 145). This is not altogether wrong, but he is not

quite right either. While certain key features, specifically the mutual exclu-

sivity of complementary descriptions, are indeed preserved, it is just not

quite the same version of complementarity.33 The Como lecture introduces

Quantum Mechanics, Complementarity, and Nonclassical Thought * 59

complementarity as follows: "The very nature of the quantum theory thus

forces us to regard the space-time coordination and the claim of causality,

the union of which characterizes the classical theories, as complementary

but exclusive features of the description, symbolizing the idealization of

observation and definition respectively" (PWNB 1:53-54). While the con-

cept, or one might say, the descriptive concept, of complementarity itself is

in place, the complementarity of coordination and causality was to disap-

pear rather quickly from Bohr's writings, and for good reasons. This partic-

ular claim is incorrect, at least vis-a-vis Bohr's own ultimate argumentation.

There cannot be such a complementarity because there is never causality in

quantum mechanics, even at the level of effects and phenomena, manifest in

measuring instruments, of the latter version, where this absence of causality

becomes quite clear. And there is none in Bohr's later writings. The comple-

mentarity of observation and definition is gone as well, although it is not

problematic in itself and is used effectively in different contexts in the Como

lecture. The disappearance of this complementarity has more to do with

Bohr's different, more experimentally rather than philosophically oriented,

presentation in his subsequent writing and, possibly, with the comparative

lack of enthusiasm in response to Bohr's Como formulation of complemen-

tarity on the part of the physics community, although the latter problem

persisted, as Bohr indeed candidly admitted (PWNB 2:63). Instead, Bohr

will speak of "a final renunciation of the classical ideal of causality" (QTM,

145) and of complementarity itself as "a rational generalization of the...

ideal of causality" (PWNB 2:41; emphasis added). The Como lecture also

speaks of "a consistent theory of atomic phenomena, which may be consid-

ered as a rational generalization of the causal space-time description of clas-

sical physics" (PWNB 1:87), but, besides a significant nuance in the formu-

lation itself, it still refers to the complementarity of coordination and

causality, which is manifestly absent in Bohr's later writing. The space-time

coordination becomes complementary to the applications of the laws of

conservation of momentum and energy, thus entailing (this point becomes

central) the complementarity of different experimental arrangements or

phenomena, where the corresponding measurements and applications of

physical laws become mutually exclusive, as considered earlier. The role of

measuring instruments, while seen as irreducible in the Como lecture, is not

quite yet fully worked out so as to bring it to a nonclassical level.34 One can

see this transition to his new version of complementarity in Bohr's thought

even prior to the EPR argument, but clearly under the impact of Einstein's

earlier arguments, in the extraordinary, and unfortunately never published,

lecture of 1931, "Space-Time Continuity and Atomic Physics" (BCW

60 * The Knowable and the Unknowable

6:361-70). This lecture may be seen as "a previous occasion" where "a gen-

eral viewpoint, conveniently termed 'complementarity,' " as it appears in his

reply to EPR, was indeed "indicated" by Bohr there, but not in any of his

published works to which he referred in his reply.

Bohr's Como approach contained great benefits and ideas as well, begin-

ning with complementarity but far from ending there, and the Como lecture

remains germane to Bohr's thought and our understanding of it, including as

it is expressed in "Discussion with Einstein." The epistemological harvest is

tremendous. Perhaps most significant, even though not fully worked out, are

the irreducible role of measuring instruments and its radical (ultimately non-

classical) consequences, in particular, the understanding that "radiation in

free space as well as isolated material particles are abstractions, their proper-

ties on the quantum theory being definable and observable only through

their interactions with other systems" (i.e., measuring instruments) (PWNB

1:57). This is just a few inches away from his ultimately nonclassical view,

whereby no properties of any kind can be attributed to quantum objects

themselves, which leads, in the lecture itself and especially later on, to the

whole array of extraordinary conceptual innovations.

In particular, Bohr's argument, in "Discussion with Einstein" and other

late essays, concerning "the peculiar individuality of quantum effects," cru-

cial to his epistemology and making it fully nonclassical, is a refinement of

his argument in the Como lecture. The latter presents its argument in terms

of Planck's h-Planck's "quantum of action"-and "the quantum postu-

late" (PWNB 1:53) rather than in terms of "effects" and "phenomena," as

in the later works. By the time of "Discussion with Einstein," Bohr argues

that "the peculiar individuality of quantum effects presents us, as regards

the comprehension of well-defined evidence, with a novel situation unfore-

seen in classical physics and irreconcilable with conventional ideas suited

for our orientation and adjustment to ordinary experience [on which classi-

cal physics is based]. It is in this respect that quantum theory has called for

a renewed revision of the foundations for the unambiguous use of elemen-

tary concepts" (PWNB 2:62).

What is this "individuality," what makes it so peculiar, and why

"effects" (a term that itself became essential for Bohr), rather than quantum

objects themselves? To some degree, the preceding analysis explains the sec-

ond concept. The analysis to follow will explain the first and will link both

together. It is, again, the Como lecture that introduces the subject of and the

very term "individuality," by way of a hesitant parenthesis. Bohr says:

"Notwithstanding the difficulties . . . involved in the formulation of the

quantum theory, it seems that its essence . . . may be expressed in the so-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 61

called quantum postulate, which attributes to any atomic process an essen-

tial discontinuity, or rather individuality, completely foreign to classical

physics and symbolized by Planck's quantum of action" (PWNB 1:53;

emphasis added). This hesitant shift from "discontinuity" to "individuality"

(eventually further supplemented by "indivisibility" and "wholeness") is the

opening move in the development of one of Bohr's most radical and innov-

ative concepts. It took him twenty years and much further development of

his interpretation of quantum mechanics before this concept crystallized at

the time of "Discussion with Einstein." By then it came to designate certain

individual phenomena, in Bohr's sense of the registered effects of the inter-

action between quantum objects and measuring instruments upon the latter

rather than anything applicable to quantum objects. Indeed, Bohr's concept

of phenomenon may be as correlative or even equivalent to his view of

"individual quantum effects." I shall now explain why.

The strange features of quantum effects and the spectacular experiments

that manifest them are described in most standard accounts of quantum

physics, including in Bohr's writings cited here. It may, however, be useful

to consider the double-slit experiment-the "archetypal" quantum-mechan-

ical experiment-which may be argued to contain all of the key physical and

epistemological features of quantum mechanics and the questions it poses. I

shall discuss the double-slit experiment following Bohr or, in any event,

consistently with Bohr's understanding (there are alternative accounts).

The arrangement (see figure 1) consists of a source; a diaphragm with a

slit (A); at a sufficient distance from it a second diaphragm with two slits (B

and C), widely separated; and, finally, at a sufficient distance from the second

diaphragm a screen, say, a silver bromide photographic plate. (It is conve-

nient to consider, as Bohr does, an intermediate diaphragm with one slit

between the source and the two-slit diaphragm, which does not change the

essence of the situation.) A sufficient number (for a full effect it must be very

large, say, one million) of elementary particles, such as electrons or photons,

emitted from a source, is allowed to pass through both diaphragms and leave

their traces on the screen. Provisionally, I speak for the moment in terms of

quantum objects themselves; strictly speaking, we can only observe certain

effects on the screen or physically equivalent macrophenomena (also in

Bohr's sense). Indeed, which is the main point here, the attempts to appeal to

the behavior of quantum objects themselves in explaining the effects in ques-

tion ultimately lead to problems. Two setups are considered: in the first we

cannot know through which slit each particle passes; in the second we can,

at least in principle (a qualification of considerable importance).

If both slits are open and, crucially, no arrangements, such as particle

62 * The Knowable and the Unknowable

Both slits are open, and no counters are installed (interference pattern)

B

Source

A

C

Counters are installed (no pattern)

Source

B

C

Counter

Figure 1.

counters, are made that would allow us to establish through which slit each

particle that hits the screen passes, a wavelike interference pattern will

emerge on the screen. In principle (there may be practical limitations as to

what kind of setup can be actually realized), this pattern will emerge regard-

less of the distance between slits or the time interval between the emissions

of the particles, say, one hour, an eternity at the quantum scale of events.

Particles will "arrange" themselves in a pattern even though the next emis-

sion occurs after the preceding particle is destroyed after colliding with the

screen. This fact is, of course, remarkable, although routine by the stan-

dards of quantum physics, which shows and "tells" us stranger things.

More accurately, one should speak here of a pattern analogous to the traces

Quantum Mechanics, Complementarity, and Nonclassical Thought * 63

that would be left by classical waves in a corresponding medium, say, water

waves on the sand. In other words, if quantum objects were classical waves,

which, however, they are not in this interpretation, they would leave this

type of pattern. This pattern is the actual manifestation and, according to

Bohr's and most standard interpretations (there are alternative views, espe-

cially in Bohmian mechanics), the only possible manifestation of the "wave"

character of the quantum world. This pattern will appear whether we deal

with what would be prior to the advent of quantum physics classically seen

as wavelike phenomena, such as light, or particlelike phenomena, such as

electrons. In this type of interpretation at least, one can speak of "wave

propagation" or of any attributes of the classical-like phenomenon of wave-

propagation (either associated with individual particles or with their behav-

ior as a multiplicity) prior to the appearance of these registered marks only

by convention or symbolically, or, in the sense to be explained later, alle-

gorically. According to this interpretation, the same is also true as concerns

the attributes of classical particle motion, in particular trajectories. As I

said, on this view, quantum objects must be seen as entities other than either

particles or waves, or indeed anything, since the term and any notion of

"entity," or, again, "objects" or "quantum" (in any conceivable sense)

could no longer apply to them. It is also worth keeping in mind that we see

on the screen only classically manifest traces of quantum objects. The latter

themselves are destroyed in the process of this, in Bohr's terms, "irreversible

amplification" of all our encounters with quantum objects to the classical

level, a process, it may be added, of extraordinary complexity, which com-

plexity is ultimately at stake here (PWNB 2:51; 3:3).

The outcome of this first setup may appear to imply that each particle

would spread, in a wavelike fashion, into a volume larger than the slit sep-

aration or would somehow divide into two and then relocalize or reunite so

as to produce a single effect, a pointlike trace on the plate. (The distance

between slits, too, can be very large relative to the "size" of the particles,

thousands of times as large.) This view or equivalent or analogous views are

sometimes found in literature on the subject, especially, again, in dealing

with Bohmian mechanics. However, whether one subscribes or not to the

particular interpretation under discussion, the standard view of the situa-

tion is more or less as follows. Although having both routes open always

leads to the interference effect, once a sufficient number of particles accu-

mulates, any given particle passing through the slits should be seen as an

indivisible whole (i.e., the corresponding effects upon the measuring devices

are individualized accordingly). There is no evidence that would compel us

to conclude otherwise. Placing a detector in the experiment would always

64 * The Knowable and the Unknowable

confirm this-at the cost of losing the interference pattern, which circum-

stance can, again, be shown to be equivalent to uncertainty relations. In the

so-called delayed choice experiment we can make alternative arrangements,

revealing either the particlelike or the interference pattern, long, in principle

arbitrarily long (light years), after particles passed the slits, while we can

never observe any "spreading" or "division" of single particles (or, again,

the corresponding effects upon the measuring instruments).

If, however, there are counters or other devices that would allow us to

check through which slit particles pass (indeed even merely setting up the

apparatus in a way that such a knowledge would in principle be possible

would suffice), the interference pattern inevitably disappears. In other words,

an appearance of this pattern irreducibly entails the lack of knowledge as to

through which slit particles pass. Thus, ironically (such ironies are charac-

teristic of or even define quantum mechanics), the irreducible lack of knowl-

edge, the impossibility of knowing, is in fact associated with the appearance

of a pattern and, hence, with a higher rather than a lower degree of order, as

would be the case in, say, classical statistical physics. (Chaos theory is yet

something else.) The situation-that is, the fact that the knowledge or even

the possibility in principle of knowing through which slit the particles pass

would inevitably destroy the interference pattern-may be shown to be

equivalent to uncertainty relations (PWNB 2:43-47; QTM, 146-47).

Beyond, and indeed by virtue of, being correlative to uncertainty rela-

tions, the situation can also be given a statistical interpretation, equally

manifesting this apparently inescapable strangeness of the quantum world.

Furthermore, by virtue of this correlation, the statistical nature of quantum

mechanics appears to be irreducible. I shall return to this point later. For the

moment, I shall focus on the statistical considerations involved. I shall fol-

low Anthony J. Leggett's elegant exposition, describing a different but

equivalent experiment, in which instead of slits we consider the initial state

A, two intermediate states B and C, and then a final state E. (The latter is

analogous to the state of a "particle" at the point of its interaction with the

screen in the double-slit experiment.) First, we arrange to block the path via

state C but leave the path via state B open. (In this case we do not attempt

to install any additional devices to check directly whether the object has in

fact passed through state B.) In a large number (say, again, one million) of

trials we record the number of particles reaching state E. Then we repeat the

same number of runs of the experiment, this time blocking the path via B

and leaving the path via C open. Finally we repeat the experiment again

with the same number of runs, now with both paths open. In Leggett's

words, "the striking feature of the experimentally observed results is, of

Quantum Mechanics, Complementarity, and Nonclassical Thought * 65

course, summarized in the statement that . . . the number reaching E via

'either B or C' appears to be unequal to the sum of the numbers reaching E

'via B' or 'via C.' " The probabilities of the outcomes of individual experi-

ments will be affected accordingly. The situation is equivalent to the emer-

gence of the interference pattern when both slits are open in the double-slit

experiment. In particular, in the absence of counters, or in any situation in

which the interference pattern is found, one cannot assign probabilities to

the two alternative "histories" of a "particle" passing through either B or C

on its way to the screen. If we do, the above probability sum law (based on

adding the so-called amplitudes, related to the wave-function, to which one

applies Schrodinger's equation, rather than, classically, adding probabilities

themselves), would not be obeyed and the conflict with the interference pat-

tern will inevitably emerge, as Bohr stressed on many occasions (PWNB

2:46-47; QTM, 146-47). This is also why the ways of counting probabili-

ties are so different in classical and quantum physics, as Planck discovered.

One may also put it as follows. We must take into account the possibility of

a particle passing through both states B and C (and through both slits in the

double-slit experiments), when both are open to it, in calculating the prob-

abilities of the outcomes of such experiments. We cannot, however, at least

in Bohr's interpretation, assume either that such an event in space and time

physically occurs for any single particle, anymore that we can assume that

one can walk into a building simultaneously through two doors, when these

doors are sufficiently far apart. Leggett concludes:

In the light of this result, it is difficult to avoid the conclusion that each

microsystem [i.e., particle] in some sense samples both intermediate states B

and C. (The only obvious alternative would be to postulate that the ensem-

ble as a whole possesses properties in this respect that are not possessed by its

individual members-a postulate which would seem to require a radical revi-

sion of assumptions we are accustomed to regard as basic.)

On the other hand, it is perfectly possible to set up a "measurement appa-

ratus" to detect which of the intermediate states (B or C) any particular

microsystem [particle] passed through. If we do so, then as we know we will

always find a definite result, i.e., each particular microsystem is found to have

passed either B or C; we never find both possibilities simultaneously repre-

sented. (Needless to say, under these (different) physical conditions we no

longer see any interference between the two processes.) . . . (Clearly, we can

read off the result of the measurement only when it has been amplified to a

macroscopic [classical] level, e.g. in the form of a pointer position [of mea-

suring instruments]. )35

The first possibility corresponds to more familiar questions, such as, how

do particles know that both slits are open, or conversely that counters are

66 * The Knowable and the Unknowable

installed, and modify their behavior accordingly? The alternative proposed

by Leggett would be as remarkable (or intriguing) as any "explanation" of

the mysterious behavior of quantum objects. It does, however, indicate a per-

haps more general question, to be discussed later, of how order (such as that

of the interference pattern of the double-slit experiment or of the correlations

found in the EPR experiment) arises from the absolute randomness of indi-

vidual quantum events. Indeed, in this case it may even happen without any

connections between such events, or in any event without any possibility of

establishing such connections, since, as I said, the next particle can be emit-

ted long after the previous one has already been destroyed.

This situation, sometimes also known as the quantum measurement

paradox, is indeed remarkable. Other standard locutions include strange,

puzzling, mysterious (and sometimes mystical), and incomprehensible. The

reason for this reaction is that, if one speaks in terms of particles themselves

(this always appears to be the source of trouble), in the interference picture

the behavior of each particle appears to be "influenced" by the location of

the slits. Or, even more radically, individually, or (which would be even

more troubling) collectively (along the lines of Leggett's suggestion), parti-

cles appear somehow to "know" whether both slits are or are not open, or

whether there are or are not counting devices installed. In short, any attempt

to picture or conceive of this behavior in terms of physical attributes or, ulti-

mately, any conception of quantum objects themselves appears to lead to a

logical contradiction; or to be incompatible with one aspect of experimental

evidence or the other; or to entail strange or mysterious behavior; or to

require more or less difficult assumptions, such as that proposed by Leggett

or, more generally, that of attributing volition or personification to nature

in allowing particles individual or collective "choices," as Bohr points out

(PWNB 2:51, 73); or to imply nonlocality, as instantaneous connections

between spatially separated events, which would be incompatible with rela-

tivity. The latter alternative was first proposed by Einstein and is legitimate

and, arguably, the most rational under this attribution. Or-this is the EPR

argument-if local, quantum theory would be incomplete in this view. Yet

another alternative would be a retroaction in time (also in conflict with rel-

ativity), which is hardly more palatable, at least to this reader and, it

appears, to most physicists, but which is entertained by some, especially

along Bohmian or Bohmianlike lines.36

Bohr, by contrast, sees the situation as indicating what he terms the

"essential ambiguity" (a locution crucial to his writings following the EPR

argument) of ascribing conventional physical attributes, such as wavelike or

particlelike behavior or even identity, or such classical variables as "posi-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 67

tion" and "momentum" (these variables of course define the very concept of

particle in classical physics), to quantum objects themselves or their inde-

pendent behavior. He writes: "To my mind, there is no other alternative

than to admit that, in this field of experience, we are [rather than with prop-

erties of quantum objects] dealing with individual phenomena and that our

possibilities of handling the measuring instruments allow us only to make a

choice between the different complementary types of phenomena we want

to study," the different types of effects of the interaction between quantum

objects and measuring instruments upon the latter (PWNB 2:51). The term

"phenomena" must here be understood in Bohr's sense.37

This inevitability appears to hold insofar as there is no local interpreta-

tion that is essentially different from Bohr's epistemologically. At the very

least, Bohr's interpretation is both consistent with the quantum-mechanical

predictions and local, in contrast, say, to Bohm's hidden-variables theo-

ries.38 In any event, the undesirable features mentioned earlier become

removed without affecting the integrity of the data or formalism of quan-

tum theory, once a Bohr-type interpretation is in place and reference to the

properties of quantum objects themselves is suspended. This is a reasonably

trivial application of the standard logical deduction under the nontrivial and

to some unacceptable, but far from irrational, epistemological assumptions

entailed by Bohr's interpretation. (We keep in mind that this application

takes place and works only if Bohr's interpretation itself applies.) As will be

seen in the next section, a similar argument allows one to ascertain the local-

ity of quantum mechanics in Bohr's interpretation. In general, Bohr's inter-

pretation is defined by its deliberate logical consistency; and Bohr

specifically denies that any departure from ordinary logic (for example, mul-

tivalued logic) is necessary, at least within his interpretation.39

In Bohr's interpretation, a reference, or at least an unambiguous refer-

ence to quantum objects or processes themselves, would remain impossible

even when one speaks of single such attributes rather than, as would be

more common, of a simultaneous attribution of joint properties, involved in

various uncertainty relations, and even at the time when the measurement

takes place. This stronger claim, or, again, the corresponding interpretation,

appears to have emerged in Bohr's arguments in the wake of the EPR arti-

cle, although, as I said, the Como lecture says nearly as much as well. It is

clear, however, that this is what Bohr has in mind, since in speaking of this

ambiguity he never qualifies it by a reference to either joint properties or the

uncertainty relations. As he says in his reply to EPR, "In fact to measure the

position of one of the particles can mean nothing else than to establish a cor-

relation between its behavior and some instrument . . . which defines the

68 * The Knowable and the Unknowable

space frame of reference" (QTM, 148-49); emphasis added). "Its behav-

ior," not its position! This is why the term "phenomena" in the passage

cited earlier by Bohr concerning the double-slit experiments must be under-

stood in Bohr's sense. Otherwise, the radical (nonclassical) nature of Bohr's

argument would be missed, as it often is by his critics; indeed Bohr's state-

ment makes no sense otherwise. The sentence also compels one to argue that

in Bohr's scheme this "behavior" itself, too, cannot be described but can

only be correlated with the behavior of certain parts of measuring instru-

ments, behavior describable by means of classical physics. (The behavior of

certain other parts, where the actual interaction with quantum objects takes

place, is, again, quantum and hence is ultimately indescribable.) It is true

that Bohr speaks here in terms of particles. It is clear, however, that he does

so only provisionally and primarily in arguing with EPR, who use these

terms. It would indeed be difficult to think of particles in considering objects

to which no conceivable attribute of particle-like identity or behavior could

apply. What is referred to by particles is, in all rigor, only a class of properly

correlated measurements or, even more accurately, interactions between

quantum objects and measuring instruments, which we may indeed better

call interacting instruments. As Bohr says a bit earlier in his discussion of the

double-slit experiment:

This point is of great logical consequence, since it is only the circumstance

that we are presented with a choice of either tracing the path of a particle or

observing interference effects [once we have a sufficient number of the parti-

cles hitting the screen], which allows us to escape from a paradoxical neces-

sity of concluding that the behavior of an electron or a photon should depend

on the presence of a slit in a diaphragm through which it could be proven not

to pass. We have here to do with a typical example of how the complemen-

tary phenomena appear under mutually exclusive experimental arrange-

ments . . . and are just faced with the impossibility, in the analysis of quan-

tum effects, of drawing any sharp separation between an independent

behavior of atomic objects and their interaction with the measuring instru-

ments which serve to define the conditions under which the phenomena

occur. (PWNB 2:46-47)

We now see why Bohr needs his concept of phenomena as defined by the

appearance of the particular, specifically complementary, individual effects

recorded in certain parts of measuring instruments under rigorously

specifiable experimental conditions and why this specification must itself be

seen as part of the phenomena (PWNB 2:64). In Bohr's interpretation, the

influence of these conditions can be never eliminated in considering the out-

comes of quantum-mechanical experiments in the way it can, at least in

Quantum Mechanics, Complementarity, and Nonclassical Thought * 69

principle, be done in classical physics. If seen independently of the quantum-

mechanical context of its appearance, each mark on the screen in the dou-

ble-slit experiment would be perceived in the same way or as the same phe-

nomena in the sense of the philosophical (say, Edmund Husserl's)

phenomenology of consciousness. Such a mark would appear as the same

regardless of the difference in the conditions and, hence, outcome ("inter-

ference" or "no interference") of the double-slit experiment. According to

Bohr's understanding, however, each mark is seen as a different individual

phenomenon depending on these conditions, which are always mutually

exclusive in the case, such as this, of complementary phenomena. In classi-

cal mechanics such conditions would of course be the same, assuming that

we deal with particles (in the case of waves these conditions would be the

same as well but there would be no pointlike traces, and different equations,

wave equations, would apply). Quantum-mechanical predictions, including

numerical statistical predictions, crucially depend on this distinction as well.

It also follows that, in the double-slit experiment, rather than dealing

with two phenomena, each defined by a different multiplicity of spots on the

screen, we deal with two distinct multiplicities of individual phenomena,

defined by each spot. Each of the latter is indivisible-two sets of phenome-

nal atoms or atomic phenomena in Bohr's nonclassical sense-depending on

two different sets of conditions of the experiment. One of these sets of con-

ditions will lead to the emergence of the interference pattern, "built up by

the accumulation of a large number of individual processes, each giving rise

to a small spot on the photographic plate, and the distribution of these spots

[following] a simple law derivable from the wave analysis" (PWNB

2:45-46; emphasis added). Each single spot, however, must be, again, seen

as a different individual phenomenon, which depends on the conditions in

which the event occurs. Two different overall patterns, "interference" and

"no interference," pertain, thus, to two (very large) sets of different individ-

ual phenomena. Far from being a matter of convenience, this distinction

between two multiple-spot phenomena and two multiplicities of spot-phe-

nomena is essential for Bohr's meaning and the consistency of his argumen-

tation. First, no paradoxical properties, such as simultaneous possession of

contradictory wavelike and particlelike attributes on the part of quantum

objects themselves, are involved. Second, and perhaps most crucially, in our

analysis we can never mix considerations that belong to complementary

experimental setups in analyzing a given experimental outcome, even when

dealing with a single spot on the screen, as we could, in principle, do in the

case of classical physics. This is not an uncommon error (at least as Bohr's

interpretation and arguments are concerned), including in some of Ein-

70 * The Knowable and the Unknowable

stein's arguments, which could indeed lead to the appearance of paradoxes,

on which point I shall comment in the next section in the context of the EPR

argument and counterfactual logic. The latter, however, disappear once this

rule of complementarity as mutual exclusivity of such considerations is fol-

lowed. Throughout his arguments with Einstein, Bohr stresses in such situ-

ations, which are invoked in most of Einstein's arguments, including of the

EPR type, "we must realize that . . . we are not dealing with a single

specified experimental arrangement, but are referring to two different,

mutually exclusive arrangements" (PWNB 2:57).

Indeed, ultimately, at least in this version of complementarity, every phe-

nomenon is unique, singular (which becomes the meaning of the individual-

ity of a quantum phenomenon or effect in Bohr), and, in this sense, each phe-

nomenon is mutually exclusive vis-a-vis any other individual phenomenon.

Of course, not all such phenomena are "complementary" in the same sense

in which, say, a given position and a given momentum measurement subject

to the uncertainty relations would be complementary. The latter phenomena,

too, however, must be seen as separate individual phenomena in Bohr's

sense, rather than two aspects, actual or even possible, of the same entity

(object or phenomenon). Indeed, in "Discussion with Einstein," Bohr defines

complementary phenomena closer to the idea of the uniqueness of each indi-

vidual phenomenon, without invoking mutual exclusivity, even though the

more standard use of mutual exclusivity involving uncertainty relations

remains Bohr's primary concern. He writes: "evidence obtained under differ-

ent experimental conditions cannot be comprehended within a single picture,

but must be regarded as complementary in the sense that only the totality of

the [observable] phenomena [produces the data that] exhausts the possible

information about the [quantum] objects [themselves]" (PWNB 2:40).

Thus, on the one hand, quantum objects are (or, again, are idealized as)

irreducibly inaccessible to us, are beyond any reach (including, again, as

objects); and in this sense there is irreducible rupture, discontinuity,

arguably the only quantum-physical discontinuity in Bohr's epistemology.

On the other hand, they are irreducibly indissociable, inseparable, indivisi-

ble from their interaction with measuring instruments and the effects this

interaction produces. This situation may seem in turn paradoxical. It is not,

however, once one accepts Bohr's nonclassical epistemology, according to

which the ultimate nature of the efficacity of quantum effects, including

their "peculiar individuality," is both reciprocal (that is, indissociable from

its effects) and is outside any knowledge or conception, continuity and dis-

continuity among them (PWNB 2:39, 62). Indeed, as I have indicated at the

outset, one cannot speak here of a single efficacity of any two effects but

Quantum Mechanics, Complementarity, and Nonclassical Thought * 71

only of the same type of efficacity, as defined by the quantum interactions

between quantum objects and measuring instruments. In this sense, there is,

as I said, also a complementarity to efficacities, along with that of their

effects and, it follows, manifesting itself only through these effects. These

efficacities themselves remain irreducibly, nonclassically inaccessible. Thus,

Bohr's concept of the indivisibility or (the term is used interchangeably)

wholeness of phenomena allows him both to avoid the contradiction

between indivisibility and discontinuity (along with other paradoxes of

quantum physics) and to reestablish atomicity at the level of phenomena

(PWNB 2:40, 72-73; 3:4).40

Indeed, these individual (interactive) phenomena, rather than indivisible

quantum objects as the ultimate atoms of nature, become Bohr's interpreta-

tion of the quantum "atomicity" (in the original Greek sense of being indi-

visible any further) of matter, as against the view prompted by Planck's dis-

covery (thus the view still persists). By contrast, quantum objects themselves

are not assigned and, it is argued, cannot be assigned atomicity anymore

than any other features, properties, images, and so forth, "wavelike" for

example. Since it is impossible to consider the quantum objects indepen-

dently of this interaction, this "indivisibility" makes it impossible to rigor-

ously isolate quantum objects from their phenomenal enclosure, to open, as

it were, the envelop of a phenomenon. In his later works Bohr sometimes

speaks of "closed phenomena" in this sense (PWNB 2:71, 98).

Bohr's phenomena are also indivisible in the sense of being unsubdi-

vidable. That is, in contrast to the situation that obtains in classical

physics, phenomena can never be subdivided so as to make it possible to

disregard or compensate for the interactions between measuring instru-

ments and quantum objects and to allow one to isolate the latter or their

behavior as subject to a classical-like or any physical description in space

and time. As Bohr writes, "the individuality of the typical quantum effects

finds its proper expression in the circumstance that any attempt of subdi-

viding the phenomena will demand a change in the experimental arrange-

ment introducing new possibilities of interaction between [quantum]

objects and measuring instruments," which "reveal the ambiguity in

ascribing customary physical attributes to atomic objects" (PWNB 2:40,

51). In other words, any attempt at a subdivision of a phenomenon can

only produce another indivisible phenomenon or a set of phenomena of

the same nature, and, hence, it will always retain or reinstate complemen-

tarity (rather than allowing a reconstitution of it into a classical-like

wholeness).

By the same token, phenomena become individual, indeed each of them-

72 * The Knowable and the Unknowable

every (knowable) effect and, or rather conjoined with, every (unknowable)

efficacity-is unique and unrepeatable, singular. That, while each time

unknowable, the efficacity of each individual effect remains equally individ-

ual-unique and singular remains crucial to Bohr's epistemology. Accord-

ingly, every event in question in quantum physics is also individual and sin-

gular in the same sense and is in itself not predictable or, more generally, not

comprehended by law. Law, in quantum mechanics, applies only to collec-

tive regularities such as the interference pattern in the double-slit experiment.

Indeed, as we have seen, these collective regularities have a peculiar sequen-

tial character as they pertain to sequences of separate events occurring over

certain time intervals, which, in a certain sense, makes temporality irre-

ducible in quantum laws. Quantum discontinuity becomes quantum individ-

uality in the ultimate sense of the uniqueness of each individual quantum

event. This explains why Bohr uses the term in this context, in preference to

discontinuity, now seen as merely a classically manifest effect of this individ-

uality. Thus, along with the quantum atomicity as indivisibility, the quantum

atomicity as individuality (originally in turn applied to quantum objects) is

now also understood as the individuality of phenomena.

As I have indicated, however, quantum discontinuity, another general

characteristic of "atomicity," must, too, now be seen as something different

from discontinuity at the level of quantum objects, where Planck "found" it,

or where it was seen immediately in the wake of Planck's discovery (and

where it is still often seen). It becomes refigured as the irreducible inaccessi-

bility of quantum objects themselves, to the point of the impossibility of

applying either of these concepts, continuity or discontinuity, or indeed any

conceivable concept, to their "relation" (yet another inapplicable concept),

to the manifest effects of their (quantum) interaction with measuring instru-

ments, which is the efficacity of these effects.

Thus, all of the features of quantum (Planckean) atomicity-individual-

ity, wholeness, and discontinuity-are transferred from the level of atomic

objects (where their application appears inevitably to lead to a conflict with

the experimental data of quantum mechanics or other paradoxes and prob-

lems, as considered earlier) to the level of instrumental, technological phe-

nomena in Bohr's sense. Bohr directly argues that quantum physics reflects

a new feature of atomicity, foreign to the mechanical (atomic) conception of

nature, extending from Democritus's atomism. This argument is invoked as

early as 1932 (PWNB 2:6) and is made throughout his later writings, twice

at the outset of "Discussion with Einstein": "Planck's discovery. . . revealed

a feature of atomicity in the laws of nature going far beyond the old doctrine

of the limited divisibility of matter. . . . Einstein explored with a most dar-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 73

ing spirit the novel features of atomicity which pointed beyond the frame-

work of classical physics" (PWNB 2:33, 71; 3:2). Classical physics is based

on this traditional doctrine, as the latter is in turn reciprocally based on clas-

sical physics or protophysics, such as that of Democritus or Epicurus and

Lucretius. This physics would describe the behavior of its objects as such

(in, at bottom, causal, even if practically statistical or chaos-theoretical,

terms). By contrast, in Bohr's interpretation, the only ultimate "atomic"

entities that can be rigorously described by quantum theory are certain indi-

visible configurations of experimental technology represented by phenom-

ena in Bohr's sense. Planck's, or now Bohr's, quantum postulate itself

becomes a technological concept, a concept defined through the role of mea-

suring instruments. From this perspective, the only "atoms" that can be rig-

orously described by quantum theory are "techno-atoms"-certain indivis-

ible/individual configurations of experimental technology. These "atoms"

are inseparable from quantum objects and make the latter irreducibly inac-

cessible to, irreducibly discontinuous from our knowledge.41

This is one of Bohr's greatest and most extraordinary conceptions to

which one might best apply the term "Bohr's atom." This phrase is usually

applied to Bohr's 1913 semiclassical conception of the hydrogen atom, in

which electrons around the nucleus discontinuously (and hence in conflict

with classical electrodynamics) jumped between energy levels in accor-

dance with Planck's and Einstein's quantum conception of radiation. This

was Bohr's first step on the long road to his nonclassical "atoms" just con-

sidered. Obviously, these "atoms" are not physically atomic, since they

are composed of more elementary (ultimately quantum) physical con-

stituents. They are atomic in the sense that no different form of access to

quantum objects is possible, regardless of how far we can divide and sub-

divide our experimental technology or matter itself. These circumstances

prevent any possibility for quantum objects to appear or to be considered

or even conceived of in any specific, such as "atomic," form indepen-

dently, outside of the instrumental enclosures of specific experiments, of

"(en)closed phenomena."

From this perspective, quantum objects themselves may be even more

radically inaccessible than the interior of black holes. While it must be

applied with caution, the analogy itself is not out of place, insofar as the

"naked" singularity, inevitably (i.e., by virtue of the equations that describe

a black hole) found in the interior of a black hole, can never be seen. Black

holes must, of course, be considered as quantum objects in view of Stephen

Hawking's famous theorems. On the other hand, in certain recent versions

of string theory the ultimate constituents of matter are in fact configured as

74 * The Knowable and the Unknowable

microscopic black holes. This idea is in turn partly due to Hawking, who

suggests that, in general, once considered as quantum objects, black holes

may introduce more radical levels of, at least, indeterminacy (but, it

appears, also of noncausality and arealism) into physics.42 This limit upon

how far our knowledge can penetrate into black holes in no way appears to

limit our advancement of knowledge concerning them. Nor, accordingly, as,

following Bohr, I have stressed from the outset, should we be alarmed by the

presence of similar limits in quantum theory.

Complementarity, Uncertainty Relations,

and Probability

The preceding analysis makes clear why, far from being restricted to the

wave and particle effects (which in a way virtually disappear into other

complementary effects), complementary phenomena are common in and

peculiar to quantum physics. As Bohr writes:

Within the scope of classical physics, all characteristic properties of a given

object can in principle be ascertained by a single experimental arrangement,

although in practice various arrangements are often convenient for the study

of different aspects of the phenomena. In fact, data obtained in such a way

simply supplement each other and can be combined into a consistent picture

of the behavior of the object under investigation. In quantum physics, how-

ever, evidence about atomic objects obtained by different experimental

arrangements exhibits a novel kind of complementary relationship. Indeed, it

must be recognized that such evidence, which appears contradictory when

combination into a single picture is attempted, exhausts all conceivable

knowledge about the object. Far from restricting our efforts to put questions

to nature in the form of experiments, the notion of complementarity simply

characterizes the answers we can receive by such inquiry, whenever the inter-

action between the measuring instruments and the objects forms an integral

part of the phenomena. (PWNB 3:4; emphasis added)

This view makes Bohr's very use of the term "complementarity" peculiar,

since the word "complementary" usually indicates parts of or adding up to a

whole, which is precisely what is by definition impossible under the condi-

tion of complementarity in Bohr's sense. This difference is here captured by

Bohr's reference to "different aspects of the (same) phenomena" in classical

physics, as opposed to two different and indeed mutually exclusive individ-

ual phenomena in quantum physics. Such phenomena are never reducible to

or derivable from a single entity of any kind, which gives a very peculiar

Quantum Mechanics, Complementarity, and Nonclassical Thought * 75

character to the efficacity behind the quantum-mechanical effects. If certain

aspects of the situation seem inconceivable, this is because they in fact are, or

at least are conceived of as inconceivable in Bohr's interpretation.

Ultimately, the most significant complementary phenomena may well be

those related to the measurement of physical variables, analogous or, in

Bohr's view, symbolically analogous, to those of classical physics, such as

position and momentum, or time and energy, which become in quantum

mechanics subject to uncertainty relations. Bohr's complementarity gives

the physical meaning to uncertainty relations in terms of the mutual exclu-

sivity of the corresponding phenomena as considered here. It may be help-

ful to explain uncertainty relations and Bohr's interpretation of them a bit

further here, primarily because there is so much confusion about uncer-

tainty relations, especially in the humanities but sometimes even in special-

ized literature (in part because their meaning is significantly interpretation

dependent).

Most immediately uncertainty relations express strict quantitative limits,

absent in classical physics, on the simultaneous joint measurement of the

conjugate variables, such as "position" or "coordinate" (q) and "momen-

tum" (p): AqAp h (h is, again, Planck's constant, and A is the precision of

measurement). According to Bohr's interpretation, however, uncertainty

relations manifest the impossibility of not only simultaneous measurement

but also the simultaneous determination or unambiguous definition of both

such variables at any point. It is of course how we conceive of the determi-

nation of such variables, even each by itself, that becomes the next question.

Bohr's post-EPR version of complementarity arguably gives the most rigor-

ous interpretation here, although, as I said, in fact or in effect, Heisenberg's

precomplementarity views anticipate Bohr's later (phenomenal) interpreta-

tion. In this interpretation, the physical meaning of the uncertainty relations

is represented by the mutual exclusivity (complementarity) of the phenom-

ena corresponding to observations of the conjugate variables involved in

quantum measurements, but physically pertaining to the parts of measuring

instruments described in terms of classical physics. That is, the variables

themselves involved in actual measurements can refer only to the physical

behavior of certain parts of the measuring instruments rather than to the

behavior of the quantum objects themselves. It is an important and often

missed or unappreciated point that, as indicated earlier, in Bohr's interpre-

tation only certain parts of measuring instruments are subject to classical

physics, while other parts are subject to quantum behavior, which enables

their interaction with quantum objects resulting in the appearance of classi-

cally observable and measurable effects.

76 * The Knowable and the Unknowable

Indeed, as we have seen, the situation is more complicated even in the case

of a single variable in view of this "impossibility of any sharp separation

between the behavior of atomic [i.e., quantum] objects and the interaction

with measuring instruments which serve to define the conditions under

which the phenomena appear" (PWNB 2:39; Bohr's emphasis). For, "under

this circumstances an essential element of ambiguity is [always] involved in

ascribing conventional physical attributes to [quantum] objects" (PWNB

2:40), ultimately even if one talks of a single such attribute under all condi-

tions. Accordingly, uncertainty relations "cannot. . . be interpreted in terms

of [any] attributes of [quantum] objects referring to classical pictures"

(PWNB 2:73). Hence, no "position" or "momentum" impact of the interac-

tions between quantum objects and measuring instruments upon the latter,

in all rigor, measures a classical-like variable of "position" or "momentum"

of the quantum object involved. In this sense, it would not be quite accurate

to speak, as Bohr does in his reply to EPR, of the momentum exchange

between a particle and the apparatus, unless in provisional or symbolic

terms, since no momentum as such, or indeed particleness (in part defined by

the application of such variables), can be attributed to a particle (QTM,

146). The latter might have been the case for Bohr, who subsequently refined

most such formulations in any event. In this view, uncertainty relations have

as much, if not more, to do with reality (or the lack thereof) as with deter-

minism and causality (or the lack thereof) in quantum physics, and, as will

be seen presently, they have almost nothing to do with the accuracy of mea-

surement. The name "unknowability relations" was proposed for them.43

From this perspective, a more accurate explanation of the meaning of

uncertainty relations in Bohr's (post-EPR) interpretation is as follows. The

data recorded in certain parts of our measuring instruments as a result of

their interactions with quantum objects is of the same type as the data

resulting from the measurement of classical objects in their interaction with

measuring instruments. In this sense, this macrolevel data is "objective" or

"realist." In classical physics, however, we can, at least in principle, always

measure both variables in question simultaneously and disregard or com-

pensate for (so as to avoid its significance for our theoretical description) the

interaction between the objects in question and measuring instruments. By

contrast, it is never possible to do so in quantum mechanics as complemen-

tarity. It follows that we can only measure and, ultimately, unambiguously

define either one or the other variable of that type, say a change in the

momentum of a certain part of the measuring instruments involved, but

never both simultaneously. Hence, classical-like determinism is not possible

even at the classical macrolevel of phenomena or effects, each effect itself a

Quantum Mechanics, Complementarity, and Nonclassical Thought * 77

classically describable physical object. The effects themselves of the interac-

tions between quantum objects and measuring instruments upon the latter

can, however, be described in the realist manner, since any single variable of

either type by itself can always be predicted with a probability equal to

unity. (The latter fact led Einstein to think that something is amiss in quan-

tum theory, that it is perhaps incomplete. Not so, Bohr countered!) Such

variables can only now be seen as defining either the positional coordinates

of the point registered in some part of the measuring instruments or, con-

versely, a change in the momentum of another such part, under the impact

of its interaction with the object under investigation. They, even each by

itself, cannot be seen as referring to the attributes of quantum objects them-

selves. In terms of the corresponding variables of the measuring instruments

involved both variables can never be simultaneously defined. As referring to

quantum objects, not even a single variable can ever be defined.

As Bohr writes, "these circumstances find quantitative expression in

Heisenberg's indeterminacy relations which specify the reciprocal latitude

for the fixation, in quantum mechanics, of kinematical [position] and

dynamical [momentum] variables required for the definition of the state of

a system in classical mechanics." He adds a rather striking sentence: "in this

context, we are of course not concerned with a restriction as to the accuracy

of measurement, but with a limitation of the well-defined application of

space-time concepts and dynamical conservation laws, entailed by the nec-

essary distinction between [classical] measuring instruments and atomic

[quantum] objects" (PWNB 3:5).44 The application itself, referred to by

Bohr, of "space-time concepts" and, in all rigor, even "dynamical conserva-

tion laws" (this case is a bit more subtle) occurs at the level of measuring

instruments rather than quantum objects.

This view gives a new and more radical meaning to Bohr's argument con-

cerning the irreducible discrimination "between the objects under investiga-

tion and the measuring instruments which serve to define, in classical terms,

the conditions under which the phenomena appear" (PWNB 2:50). This

discrimination is, we recall, consistent with the indivisibility of phenomena

themselves. Indeed, Bohr sees "this necessity of discriminating in each

experimental arrangement between [them]" as "a principle distinction

between classical and quantum-mechanical description of physical phenom-

ena" (QTM, 150). For, while "in classical physics the distinction between

object and measuring agencies does not entail any difference in the charac-

ter of the description of the phenomena concerned," in quantum physics it

does (QTM, 150; PWNB 2:50; 3:3). These and related statements by Bohr

do not mean (as it might appear) that, while parts of measuring instruments

78 * The Knowable and the Unknowable

are described by means of classical physics, the behavior (in space and

time) of quantum objects is described by means of quantum-mechanical

formalism. Bohr obviously says the former, but he clearly does not say,

here or elsewhere, and does not mean the latter. Instead, the difference

between the two descriptions is this: classical theory describes the classical

world, specifically certain parts of the measuring instruments; quantum

theory describes or, more accurately, relates, in terms of statistical predic-

tions of the outcomes of experiments, to the interaction between the mea-

suring instruments, described partly classically but interacting quantumly

with quantum objects, indescribable by means of either classical or quan-

tum theory. The totality of possible individual quantum phenomena (or

effects) and specifically the complementary character of some of them can

be comprehensively and consistently accounted for, in terms of statistical

predictions, by the formalism of quantum theory, but not, on this view, by

means of the classical theories. The two types of formalism themselves

remain ultimately incompatible, even though they must coexist in Bohr's

interpretation.

As Bohr points out, "it is true that the place within each measuring pro-

cedure where this discrimination is made is in both cases largely a matter of

convenience" (QTM, 150). The arbitrariness of this place, however, or the

arbitrariness of the "cut," as it is also called, becomes a natural feature of

this interpretation rather than a problem, as it would be, and often has been,

when we assume that the quantum-mechanical formalism describes the

space and time behavior of quantum objects themselves. At this point of his

reply to EPR, Bohr brings into consideration the so-called transformation

theorems of quantum mechanics. The latter mathematically ground the EPR

argument and, according to Bohr, "perhaps more than any other feature of

formalism contribute to secure its mathematical completeness and its corre-

spondence with classical mechanics" (QTM, 145, Note). For "by securing

its proper correspondence with the classical theory the [transformation] the-

orems exclude in particular any imaginable inconsistency in the quantum-

mechanical description, connected with a change of the place where the dis-

crimination is made between object and measuring agencies. In fact it is an

obvious consequence of [Bohr's] argumentation that in each experimental

arrangement and measuring procedure we have only a free choice of this

place within a region where the quantum-mechanical description of the

process concerned is effectively equivalent with the classical description"

(QTM, 150).

What makes this last point (which, thus, also conveys the deeper episte-

mological aspects of Bohr's correspondence principle) so crucial is that

Quantum Mechanics, Complementarity, and Nonclassical Thought * 79

quantum objects are always on the other side of the "cut" and indeed may

be rigorously defined accordingly. This point is, again, intimated by Heisen-

berg's original presentation of his matrix version of quantum mechanics. In

Bohr's interpretation, the predictions of quantum mechanics refer strictly to

classically manifest effects upon the measuring instruments, which can be

rigorously correlated with specific configurations involving quantum

objects. It is only this correlation that enables any possible identification of

such a measurement with the "position" or "momentum" of a quantum

object. The correlation is, according to Bohr, real, or rigorous, while the

identification can only be, in his words, "symbolic," insofar as, in view of

the nonclassical character of Bohr's interpretations, we cannot unambigu-

ously attribute such (or, again, any) properties to quantum objects or their

behavior. At one end, by virtue of their classical nature, the individual

effects in question can be isolated materially and phenomenologically-we

can perceive and analyze them as such-once an experiment is performed,

although they cannot be isolated from their efficacity by even conceiving, let

alone analyzing, their physical emergence. By contrast, at the other end,

quantum objects and processes can never be isolated, either materially (from

the measurement process and measuring instruments) or perceptually or

even conceptually (we cannot in principle conceive of what actually happens

at that level or how it happens).

By the same token, the behavior of quantum objects cannot be seen in

terms of any formalism, (more obviously) classical or (less obviously but

more crucially) quantum. In Bohr's interpretation there is no presupposition

that the quantum-mechanical formalism in any way describes the ("undis-

turbed") quantum objects and processes, or their properties, even single

properties (say, either position or momentum). It does not do so either

before the measurement interference takes place, or between instances of

such interference, or even during measurement, a more common (but ulti-

mately epistemologically less radical) understanding of the situation and of

Bohr's understanding of it. Indeed, in the present view, "measurement" may

not be an altogether fitting term here, since we do not measure anything at

the quantum level but only at the level of "measuring," or, we may say,

interacting or correlating, instruments under the impact of their interaction

with quantum objects. About the latter we obtain only certain irreducibly

indirect or inascribably, effect-like, information. Throughout his writing,

from the Como lecture on, Bohr denies that the quantum-mechanical for-

malism can unambiguously refer to quantum objects and processes in terms

of space-time concepts in the way it can be done in classical physics or even

in relativity, which, as I said, already introduces significant complications in

80 * The Knowable and the Unknowable

this respect (PWNB 2:40-41). As he said, "even in the indeterminacy rela-

tion[s], we are dealing with the formalism which defies unambiguous expres-

sion in words suited to describe classical physical [space-time] pictures"

(PWNB 2:40). Ultimately, this formalism defies all "unambiguous use of

space-time concepts," which are instead "confined to recording of observa-

tions which refer to marks on a photographic plate or to similar practically

irreversible amplification effects" (PWNB 2:51). If one is concerned, as Bohr

(or Einstein) was, with the physical meaning of a theory as referring to the

processes occurring in space and time, in Bohr's interpretation what takes

place in space and time is a particular configuration of phenomenal effects.

Bohr saw the mathematical formalism of quantum mechanics as correlative

to this situation, including as applied to the EPR experiment (QTM, 145). In

other words, the formalism of quantum mechanics predicts only certain

effects manifest in the measuring instruments.

We recall that this formalism is highly symbolic and nonvisualizable in its

very nature, especially by virtue of using complex numbers and infinite-

dimensional mathematical objects. The latter could be related to observa-

tions, always recorded in real numbers, by means of artificial schemes, such

as Born's "square moduli" rule for deriving probabilities from quantum

amplitudes, Von Neumann's projection postulate, and so forth, described in

most standard treatments of quantum mechanics.45 What would it mean to

speak of the physical reality in terms of such objects, of the physical mean-

ing (in space and time) of such mathematical objects and their properties?

Dirac, whose work may be seen as especially responsible for the view that

the quantum-mechanical formalism pertains to quantum object themselves,

clearly denies that the key mathematical operations involved can have any

physical meaning. Accordingly, as I noted, his epistemology may be much

closer (which is not to say identical) to Bohr's than it may appear.46 In other

words, such mathematical objects cannot realistically represent or directly

refer to space-time processes (even noncausal ones), such as wave propaga-

tion or particle motion. As will be seen in the next chapter, they may not

even constitute a mathematical, at least geometrical, phenomena in the

sense of allowing us to conceive of them spatially.47 Indeed, as indicated ear-

lier, the practice (more or less standard since Dirac's and then Von Neu-

mann's versions of quantum-mechanical formalism) of using the Hilbert-

space formalism and notation in explaining "what is actually going on"

may be seen as a manifestation of this impossibility. We may also change

our understanding of what it means to offer a description of a physical

process. Such a change would, however, be consistent with Bohr's view

insofar as one maintains that in speaking in terms of space-time behavior

Quantum Mechanics, Complementarity, and Nonclassical Thought * 81

and classical-like physical variables or concepts we can only refer to the

effects of the (quantum) interactions between (indescribable) quantum

objects and (classically described) parts of measuring instruments. As I have

indicated, Heisenberg's original (matrix) formulation of quantum theory

appears to conform to this view more readily and may be seen as shaping

Bohr's view all along. His "matrices" and the formal (Hamiltonian) equa-

tions, retained from classical physics but now applied to these matrices, relate

to the observable radiation effects rather than to the motion of electrons.

This is what he called the "new kinematics" of quantum mechanics, although

the term "kinematics" may, again, be misleading here, since it suggests a rep-

resentation of motion in terms of mathematical variables. Heisenberg's

matrices do not represent motion or, for that matter, anything physical at the

quantum level. They only enable statistical predictions concerning the out-

comes of experiments related to physical space-time features of certain parts

of the instruments used under the impact of their interactions with quantum

objects. These matrices are also immediately interpretable in terms of

Hilbert-space formalism. Bohr adopted these views early on (PWNB 1:48).

Figure 2 offers an illustration of quantum mechanics in Bohr's interpre-

tation, as compared with both classical physics and with a possible classical

or classical-like interpretation of the quantum-mechanical data, here

specifically uncertainty relations. (A) In the case of classical physics, in an

epistemologically classical interpretation, a simultaneous measurement of

both the position q and the momentum p (corresponding to certain, to use

Einstein's language, "elements of reality") is always possible, at least in

principle. This fact allows us both to explain and indeed to picture (visual-

ize) the process and predict the state of the system (it may consist of many

particles) at any point, or know it at any point in the past, once we know it

at a given point. This knowledge is enabled by the measurement in question.

Classical statistical physics merely complicates this situation in terms of

actual prediction but does not change the epistemology. (B) In the classical-

like interpretation of quantum mechanics or a classical-like version of quan-

tum theory, such as Bohm's mechanics, which retains the uncertainty rela-

tions, we can, given the uncertainty relations, only measure either the

position or the momentum of a given quantum object. However, this

deficiency is seen as merely due to our inability to do so now, or perhaps

even in principle. An underlying existence of some elements of reality corre-

sponding or giving rise to the measured quantities is presupposed and

defines the causal evolution of the system, however unavailable. Epistemo-

logically, this picture is close to that of chaos theory; in terms of prediction

it is closer to classical statistical physics. The latter would of course also be

(A) Classical physics (no uncertainty relations)

No uncertainty relations: both p (momentum) and q (position) can be mea-

sured simultaneously with any given precision (within the capacity of our

measuring instruments), thus allowing us to establish the state of the sys-

tem under consideration and to track and predict the system's behavior at

any point. Both p and q can be seen pertaining to the system itself and as

measurable without affecting the system appreciably.

Measurement:

pandq

pandq

System S

(B) Classical-like view of quantum physics (or Bohmian view)

Uncertainty relations AqAp~ h preclude the simultaneous knowledge of

the state of the system and tracking or predicting its evolution, but both

the state of the system at any given point and its evolution are presupposed

on the model of (A). The causal evolution of the system is defined either

by p and q alone or possible additional ("hidden") variables.

Measurement:

System S

(C) Nonclassical view of quantum physics

Uncertainty relations AqAp

h are seen as entailing the impossibility of

considering even the possibility of a single classical-like state of the system

at any point and, hence, considering of the situation in terms of either (A)

or (B). Every measurement represents a singular, unique situation, which

cannot be linked as in (A) or (B). No underlying properties (including that

of "system") are attributable. The latter circumstance is indicated by quo-

tation marks around systems "S." The punctured lines indicate the inac-

cessible efficacity of each effect of measurement. Both p and q relate to the

properties of measuring instruments only.

Measurement:

"Black holes"

p .

p.

"S1" "S2" "S3" "S4"

Figure 2.

Quantum Mechanics, Complementarity, and Nonclassical Thought * 83

the case in a nonclassical interpretation of quantum mechanics, such as

Bohr's. (C) In the case of nonclassical interpretations of quantum mechan-

ics, however, while we can measure either the position or the momentum of

certain parts of measuring instruments there is no such underlying

configuration comprised by the elements of reality from which such com-

plementary measurements arise. Each situation of measurement, each phe-

nomenon, is isolated, singular-individual and unique-and the efficacity of

the effect defining each such phenomenon is both itself each time unique and

irreducibly inaccessible. In this respect the character of each phenomena is

defined by a kind of "black hole," with an unbridgeable epistemological,

although not physical (although the physics itself cannot ultimately be

ascertained), gap, as explained earlier. Accordingly, no evolution of the sys-

tem can be presupposed in this interpretation, anymore than properties of

quantum objects and processes.

According to Bohr's and most standard interpretations, uncertainty rela-

tions are correlative to the statistical character of quantum mechanics. We

recall that, in Bohr's interpretation, quantum mechanics only predicts the

probabilities of certain occurrences and only predicts such probabilities (as

probable outcomes of certain measurements on the basis of previously per-

formed measurements) rather than describes or accounts for the motion of

quantum objects themselves in the way classical mechanics does for classi-

cal objects. According to Bohr, from the outset of quantum physics in the

situation defined by Planck's and Einstein's discoveries, "there could be no

question of attempting a causal analysis of [quantum] radiative phenomena

[or any phenomena in question in quantum physics], but only, by a com-

bined use of the contrasting [complementary] pictures, to estimate proba-

bilities for the occurrence of the individual radiation processes" (PWNB

2:34). The reasons for this argumentation, from the perspective of Bohr's

ultimate (post-EPR) version of complementarity, are roughly as follows (a

full analysis would require technical considerations beyond the scope of this

study).

As we have seen, the exact (within the capacity of our instruments) pre-

dictions of classical mechanics are possible by virtue of the possibility of a

simultaneous joint determination of both the position and the momentum

of a given object at any point. In quantum physics such a joint determina-

tion is never possible (regardless of the capacity of our instruments) in view

of uncertainty relations and complementarity (the variables themselves,

moreover, now pertain to the measuring instruments rather than to quan-

tum objects). Any measurement of both such variables would require at

least two identically set up experimental situations performed on two dif-

84 * The Knowable and the Unknowable

ferent quantum objects (again, to the degree that we can speak of them as

"objects"), say, an emission of a particle, its passing (or not passing)

through a slit, and hitting (or not hitting the screen) in the double-slit exper-

iment. (The measurement of position or momentum can be performed in

terms of the double-slit experiment.) In quantum physics, however, the out-

comes of such experiments can never be quite sufficiently controlled so as to

be the same even if we repeat the measurement of the same variable. For, to

begin with, a repetition of any given setup can never be strictly identical,

even in principle, as they are in classical physics, since the kind of control of,

say, an emission of a particle that we can have in classical physics is itself

subject to uncertainty relations. According to Bohr: "The very fact that rep-

etition of the same experiment . . . in general yields different recordings per-

taining to the object immediately implies that a comprehensive account of

experience in this field must be expressed by statistical laws" (PWNB 3:4,

11-12). Thus, in contrast to classical physics, at least two experiments on at

least two different objects are always necessary for anything to be predicted

along the lines of classical physics (a fact that becomes crucial in the EPR

argument and Bohr's reply). In fact, and in part correlatively, a large num-

ber of repeated experiments are necessary, which makes the statistical char-

acter of the quantum-mechanical predictions unavoidable. This argument

could be adjusted so as to refer only to the probabilities of individual exper-

iments. A verification of how good such predictions are would of course

require multiple experiments of the same type.

Quantum theory and especially complementarity require, and depend on,

the concept of the individual physical event. The individuality of such events

is essential, in the strict sense of being irreducible. It is in part this concept

that defines quantum mechanics as quantum, even though, according to

Bohr, it has to be and, in his interpretation, is given a complex architecture,

defined by the concept of "phenomenon" in Bohr's sense. At the same time

and by the same token, in contrast to classical physics, quantum mechanics

offers us no laws that would enable us to predict with certainty the outcome

of such individual events, or when some of them might occur. Similarly to

classical statistical physics, the laws of quantum mechanics rigorously pre-

dict the statistical behavior of quantum collectivities, which, as I have noted,

have, moreover, a peculiar sequential character. But, in contrast to classical

statistical physics, the laws of quantum mechanics also rigorously allow for

the irreducible individuality, the irreducible "un-lawfulness" or "lawless-

ness" of individual quantum-mechanical events. It follows of course that the

character of the behavior and predictions themselves cannot be identical,

since, as I have already noted, one can no longer see the situation in terms

Quantum Mechanics, Complementarity, and Nonclassical Thought * 85

of proceeding from the causal and deterministic individual behavior or

events to the statistical collective behavior. Instead the irreducibly lawless

individual occurrences somehow "conspire" to give rise to relatively

ordered (statistically correlated) collective behavior and irreducibly lawless

individual behavior. The character of this "conspiracy" is just as ultimately

unavailable (and hence is irreducibly "conspiratorial") as that of any

efficacious dynamics of quantum-mechanical events; indeed, it is the same

efficacious dynamics. By the same token, the statistical-predictive character

of quantum mechanics, too, becomes an aspect of the inaccessible efficacity

of the typical quantum effects. We do not know and ultimately cannot

know why certain probability rules, say, those of Planck's law or Max

Born's "square moduli" rule for calculating quantum-mechanical probabil-

ities from the wave function, in fact apply, but they do. The fact that we can

make such excellent statistical predictions is as mysterious as all other, often

famously strange, quantum-mechanical effects-mysterious, but in this

view, not mystical, insofar as, by very virtue of nonclassical epistemology,

one cannot assign a single agency, however in itself unknowable, behind

these effects.

From this viewpoint, one can indeed hardly speak of anyone playing dice,

a game, after all, subject to the classical-like statistical rules, which, as we

have seen, Einstein indeed was willing to accept after all, perhaps for this rea-

son. But that "[the Lord] should gamble according to definite rules, that

[was] beyond him," he said in 1953, one year before his death (QTM, 8), one

presumes, according to the extraordinary and extraordinarily precise rules of

quantum mechanics that combine (correlate) an order of events with the

absolute lawlessness of each of them. That is, these rules manage to handle

this mysterious combination, which nature's hand deals us, if indeed we can

apply this card-playing locution anymore than we can speak of anybody

playing dice, although it seems to suit the situation rather better. In Bohr's

words: "It is most important to realize that the recourse to probability laws

under such circumstances is essentially different in aim from the familiar

application of statistical considerations as practical means of accounting for

the properties of mechanical systems of great structural complexity [as in

classical statistical physics]. In fact, in quantum physics we are presented not

with intricacies of this kind, but with the inability of the classical frame of

concepts to comprise the peculiar feature of indivisibility, or 'individuality,'

characterizing the elementary processes" (PWNB 2:34; also 3:4).

Thus, contrary to Einstein's view, according to Bohr, quantum mechan-

ics allows for an interpretation that does "offer an exhaustive description of

the individual phenomena" rather than "merely. . . the means of account-

86 * The Knowable and the Unknowable

ing for the average behavior of a large number of atomic systems" (PWNB

2:61; emphasis added). But it redefines what "individual phenomena" are

or, in Bohr's interpretation, can possibly be, given the data in question in

quantum theory. For that matter, as we have seen here, it redefines every

single term and conceptual grammar of this sentence-from "exhaustive" to

"description" to "individual" to "phenomena" to "accounting" to "aver-

age" to "behavior" to "atomic" to "system"-within an extraordinary non-

classical architecture of complementarity. In view of this redefinition, our

analysis of, in the language of Bohr's reply to EPR, "this entirely new situa-

tion as regards the description of physical phenomena" cannot in principle

appeal to the physical properties of quantum objects or the nature of quan-

tum processes themselves, as Einstein wanted (QTM, 148). Nor, it follows,

can such an analysis appeal to the ultimate efficacity of the effects in ques-

tion in, and accounted for (in terms of statistical prediction) by, quantum

mechanics, since this efficacity is defined by the quantum interaction

between quantum objects and the quantum aspects of measuring instru-

ments. This double (nonclassical) prohibition may be the ultimate meaning

of Bohr's statement that "in quantum mechanics, we are not dealing with an

arbitrary renunciation of a more detailed analysis of atomic phenomena,

but with a recognition that such an analysis is in principle excluded"

(PWNB 2:62; Bohr's emphasis).

Einstein said (in 1936) in the Schilpp volume, Albert Einstein: Philoso-

pher-Scientist, that "to believe this [i.e., that quantum theory offers an

exhaustive description of individual quantum processes] is logically possible

without contradiction," and ultimately he thought that Bohr did most jus-

tice to the problem, even though, as he said, certain aspects of complemen-

tarity eluded him. His reply also contains an exposition of his own views on

the subject. However, he saw the way quantum mechanics works "'as so

very contrary to [his] scientific instinct that [he could not] forgo the search

for a more complete conception'" (PWNB 2:61; emphasis added). By this

point, Einstein saw the situation in terms of the alternative for quantum the-

ory as that between being "complete and nonlocal" versus "local but

incomplete" and appears to (mis)read Bohr along these lines. He presents

Bohr's argument, as he says, "translating [it] into his own way of putting

it," a translation that may in fact be precisely impossible, without losing the

key aspects of Bohr's argumentation, since Einstein still clearly speaks in

terms of elements of reality.48 Thus, he was searching for a local and com-

plete conception of quantum physics. The latter, I argue here, becomes pos-

sible once one accepts Bohr's complementarity and its nonclassical episte-

mology as an interpretation of quantum mechanics, an interpretation that,

Quantum Mechanics, Complementarity, and Nonclassical Thought * 87

accordingly, indeed aims "at characterizing.., the entirely new situation as

regards the description of physical phenomena" rather differently from the

way Einstein would envision (QTM, 150). From this perspective a counter-

argument to Einstein's is that, once the epistemology in question is accepted,

it becomes possible to interpret quantum mechanics as both complete and

local. Of course, as Bohr noted, Einstein's "attitude, even if . . . [it] might

seem balanced in itself" still "implies a rejection of [Bohr's] argumentation

aiming to show that in quantum mechanics, we are not dealing with an arbi-

trary renunciation of a more detailed analysis of atomic phenomena, but

with a recognition that such an analysis is in principle excluded." It is a

rejection on epistemological grounds, even if Einstein saw Bohr's argumen-

tation as compatible with locality, which is, again, not altogether clear

(PWNB 2:62; emphasis added). For Einstein completeness of physical the-

ory implied a possibility of a classical-like model applied to the objects of the

theory themselves, along Schrodinger's lines as considered earlier. Neither

such model nor such application may be possible and indeed did not appear

possible to Einstein on the basis of the standard quantum mechanics.49

For Bohr quantum mechanics is as complete as a theory could be under

the nonclassical conditions, which he saw as necessary for his interpreta-

tion, even if not ultimately unavoidable, and his interpretation reflects the

possibility of this claim. Indeed, from this viewpoint, it is not even clear in

comparison to what such an account would be incomplete, physically

speaking, assuming that no (epistemologically) essentially different local

account is available. Would merely a lack of conformity with certain classi-

cal-like models, whose ultimate correspondence, however indirect (Ein-

stein's position on this was complex), to "physical reality" has never been

established, be enough to see Bohr's conception as lacking in completeness?

Bohr did not think so, which made his interpretation at least sufficient, if

not inevitable.50 No local alternatives were on the horizon at the time, nor

perhaps even now.

The EPR Argument, Quantum Entanglement, and

Locality: A Nonclassical View

The current state of the debate concerning quantum mechanics is dominated

by the arguments concerning the so-called Einstein-Podolsky-Rosen-type

experiments and (this term is due to Schrodinger) "quantum entanglement,"

the existence of a particular type of correlation between certain spatially

separated quantum-mechanical events. Such correlations are not found in

88 * The Knowable and the Unknowable

classical physics, are inherently quantum-mechanical, and are indeed extra-

ordinary in their character, implications, and, by now, possible applica-

tions, such as quantum cryptography and computing. These correlations, a

form of order, pose perhaps most sharply one of the greatest enigmas of

nature at the level of its ultimate constituents and their effects upon what

appear to us as phenomena (in either sense). How does order arise from the

absolute randomness of individual events with which we are concerned in

quantum mechanics? But then, as we have seen, every quantum measure-

ment carries this question with it and, as will be seen presently, is indeed an

event of entanglement or correlations (between quantum objects and quan-

tum aspects of measuring instruments).

Some see the situation as indicating the nonlocality either of quantum

mechanics or of the quantum data itself, for one need not use quantum

mechanics to ascertain the correlations in question, although EPR used it.

This view, I argue, can be avoided, at least given Bohr's interpretation,

which, I also argue, is local. The reasons for this and for my argument in this

section are roughly as follows. Quantum mechanics itself (i.e., a direct

application of the formalism) correctly predicts the data in question, corre-

lations included, without expressly making claims concerning actual prop-

erties of quantum objects themselves. Such claims are, obviously, compli-

cated by uncertainty relations (which are a rigorous consequence of the

theory) but are not simply, short of an interpretation, such as Bohr's, made

impossible by it. Bohm's mechanics has uncertainty relations interpreted in

a classical-like manner. On the other hand, in view of Bell's theorem and

related results, any classical-like theory (such as Bohm's) based on the attri-

bution of properties to quantum objects themselves, that would equally pre-

dict the data of quantum mechanics, could be shown to be nonlocal. At least

it strongly appears to be (there are some residual questions as to how tight

such arguments ultimately are). The situation would be similar to Bohm's

classical-like mechanics, where, however, nonlocality is an explicit conse-

quence of the mathematical formalism. Bell's theorem tells us that any clas-

sical-like (hidden variables) theory, regardless of its mathematical structure,

would inevitably be nonlocal. The argument is further amplified and

nuanced by related theorems, in particular the Kochen-Specker theorem,

mentioned earlier. These are momentous findings, which, on the present

view, support Bohr's argumentation. While not going as far as they do in

rigorously proving the impossibility of local classical-like theories compati-

ble with the data of quantum mechanics, Bohr at the very least (which is

hardly less momentous) found that it is possible to interpret quantum

mechanics as a local theory, if one follows nonclassical epistemology. In

Quantum Mechanics, Complementarity, and Nonclassical Thought * 89

other words, realism and locality indeed appear to be incompatible, given

the data in question, as Einstein's arguments intimated. Accordingly, the

question rests on whether one can or cannot as an alternative interpret

quantum mechanics in a nonclassical and specifically nonrealist manner.

Bohr's complementarity answers this question in the affirmative, since it

prohibits any attribution of anything to quantum objects themselves, even a

single classical-like conjugate variable (which appears to be necessary for

locality), and even when the measurement is performed, as any information

involved pertains strictly to measuring instruments. In other words, he

offers a local and nonclassical interpretation of quantum mechanics as a

complete theory. This argument, accordingly, avoids the alternative, argued

for by Einstein, between quantum mechanics as a local but incomplete the-

ory and a nonlocal but complete theory under the epistemologically classi-

cal conditions of completeness.

This particular way of thinking about the famously strange character of

quantum physics emerged in the wake of Einstein, Podolsky, and Rosen's

article, "Can Quantum-Mechanical Description of Physical Reality Be Con-

sidered Complete?," and Einstein's subsequent arguments on the subject.

The latter refined the EPR argument by more sharply focusing it on the pos-

sible nonlocality of quantum mechanics (rather than on its incompleteness

as a physical theory) and proposed an alternative between seeing quantum

mechanics either as a complete but nonlocal or a local but incomplete the-

ory. This alternative is suggested in the EPR article as well (albeit in a some-

what obscured form) and in Einstein's earlier arguments. By contrast, Bohr

sees the completeness of quantum mechanics as fully compatible with its

locality. Or, at least, he offers an interpretation of quantum mechanics, as

complementarity, that, while preserving quantum entanglement in the sense

just explained, is local. It follows that only entanglement is inevitable in

quantum mechanics and may, accordingly, be seen as part of the quantum-

mechanical data, while nonlocality is not. Schrodinger's "cat paradox"

paper (1935), cited earlier; David Bohm's reformulation of the EPR type of

experiments in terms of the so-called spin of particles (this does not change

the epistemology of the situation) and then his (nonlocal) hidden-variables

theories; John S. Bell's theorem and his discontent with quantum mechan-

ics; Alain Aspect's experiments; and further refinements and extensions of

Bell's theorem, as well as more recent investigations of decoherence and

related questions, are all well-known developments and parts of the debate

in question.51

EPR begin their article by proposing the following, apparently natural

and minimal, criterion of physical reality: "If, without in any way disturb-

90 * The Knowable and the Unknowable

ing a system, we can predict with certainty (i.e., with probability equal to

unity) the value of a physical quantity, then there exists an element of phys-

ical reality corresponding to this physical quantity" (QTM, 138). It may

appear that, as is argued by EPR, this criterion applies, to quantum mechan-

ics as well. Recall that, in view of uncertainty relations, it is only a joint

simultaneous determination or prediction of two variables involved in the

quantum-mechanical physical description, such as "position" and "momen-

tum," that is impossible. Accordingly, a determination or prediction of the

value of a single variable is always possible, with any degree of precision,

including in the EPR experiment, which deals with the measurements of

physical variables pertaining to two spatially separated quantum objects

that have previously been in interaction. Hence, predictions concerning such

variables now take place without "disturbing" a quantum system under

investigation by measurement, that is, without first performing a measure-

ment upon it, which is how quantum-mechanical predictions are made in

more standard cases, as considered earlier. (Bohr speaks more cautiously of

not "interfering" with this system, since in his interpretation there is no clas-

sical-like or otherwise specifiable undisturbed configuration or properties

that are then disturbed in the process.) This can indeed be done in quantum

mechanics for (this remains crucial) a single variable in certain cases, such as

that considered by EPR. It is achieved by means of performing measure-

ments on other systems (in the EPR argument, another particle), which have

previously been in an interaction with the system (such as a particle) under

investigation. For the sake of economy, I, again, speak for the moment of

"particles" rather than, as would be more appropriate nonclassically and as

Bohr does, of "variables involved in the quantum-mechanical physical

description" (QTM, 144, 145). Indeed, as Bohr observes, in a certain sense,

the EPR conditions apply to all quantum-mechanical predictions. In any

quantum measurement, we can predict either the position or the momentum

of a particle after a preceding measurement took place and on the basis of

this measurement, and hence without interfering with the object and with-

out assuming that we can define either quantity independently of measure-

ment (PWNB 2:57). In fact, the particle and those parts of the measuring

instruments involved that interacted with it (and, hence, may be considered

as a quantum system) become entangled in the EPR way (PWNB 2:60). In

the EPR situation, which involves two particles, rather than a particle and a

measuring apparatus, we have a slightly more complicated, but not funda-

mentally different, case (QTM, 149-48; in this order). Accordingly, the

standard situation of quantum measurement may serve as a general model

of the case, as it sometimes does for Bohr. In any event, predictions (limited

Quantum Mechanics, Complementarity, and Nonclassical Thought * 91

by uncertainty relations) concerning a given particle are possible on the

basis of measurements performed on another particle that has previously

been in an interaction with the first particle but, at the time of measurement,

is in a region spatially separated from it. Hence, at the time of determination

in question, there is no physical interaction either between the two particles

or between any measuring apparatus and the second particle. This circum-

stance led some, beginning with Einstein, to conclude that there are some

nonlocal connections involved. Einstein famously called them "spooky

action at a distance." Bohr did not think that such connections are implied

by the circumstances of measurement just described, in part because he saw

them as correlative to the EPR criterion of reality, which he argued possibly

inapplicable in quantum mechanics. Nor did he see quantum mechanics as

nonlocal, at least insofar as it allows for a local interpretation, which he

(correctly) saw complementarity to be.

I shall not consider here EPR's subtle argument and Bohr's equally subtle

reply in detail. To do justice to both would require a much longer analysis,

and it perhaps cannot be done within the limits of a single treatment. The key

point is this. If one accepts that the EPR criterion of reality applies in quan-

tum physics, quantum mechanics can indeed be shown to be incomplete, or

more accurately and more crucially (this is in fact or in effect what EPR

argue) either incomplete or nonlocal, that is, entailing an instantaneous

action-at-a-distance, as just explained. Accordingly, perhaps the only effec-

tive counterargument would be to show that the EPR criterion is ultimately

inapplicable or, at least, need not be applicable in the situation in question in

quantum mechanics. Epistemologically this would mean that quantum

physics rigorously disallows even the minimal form of realism entailed by the

EPR criterion. This is what Bohr argues. Or, again, at least he argues that one

can interpret quantum mechanics accordingly, which suffices for my pur-

poses here, since it makes the nonclassical view of quantum theory at least

viable, even if not inevitable. It is a separate question whether other inter-

pretations of quantum mechanics also allow one to handle these difficulties.

It may be argued that Bohr's was at least the first such interpretation.

As must be clear from the preceding analysis, in Bohr's interpretation one

cannot unambiguously ascribe, as EPR do in accordance with their criterion

of reality, even a single physical attribute (or ultimately even identity) to a

quantum object as such-that is, as considered independently of measure-

ment and hence of our interaction with it by means of experimental tech-

nologies, as considered here. Indeed, EPR argue that one can ascertain a

simultaneous independent reality of both conjugate variables, or more accu-

rately of two "elements of reality," to which such variables can correspond,

92 * The Knowable and the Unknowable

in contrast to Einstein's subtler nonlocality argument. This point is, as I

said, intimated, or at most implied, by EPR, literally in the last sentences of

their paper, but is not developed in their article. This shift from the impos-

sibility of simultaneously ascribing both conjugate variables to the impossi-

bility of ascribing even a single variable at any point to the second (or indeed

the first) particle of the EPR experiment is what changes the argument from

that for the outright incompleteness of quantum mechanics to Einstein's

alternative between "local and incomplete" versus "complete but nonlo-

cal." While Bohr's argument in his reply is directed primarily against the

argument for the incompleteness of quantum mechanics (which is indeed

easier to counterargue), it also explicitly sustains the locality of quantum

mechanics and hence de facto replies to Einstein's second argument. As we

have seen, in Bohr's interpretation any independent consideration of quan-

tum objects apart from measuring instruments, opening or breaking the

wholeness/indivisibility of Bohr's phenomena, is impossible. He makes this

point clear in his reply, and he states it (at least) three times in "Discussion

with Einstein" (PWNB 2:39-40, 52, 61). This separation is impossible even

though we can, in quantum mechanics, predict the outcome of such mea-

surements on the basis of earlier measurements performed on a given object

or contemporaneous measurements performed on other objects, which have

previously been in interaction with the object in question, as indicated ear-

lier. Hence, such measurements would not involve this object itself at the

time of determination of the variables concerned, which is crucial to EPR or

Einstein's other arguments considered here.

These are the circumstances that Bohr has in mind when he says that

"under these circumstances an essential element of ambiguity is involved in

ascribing [any] conventional physical attributes [single or joint] to quantum

objects [themselves]" (PWNB 2:40). Once this interaction is irreducible, he

is able to argue that "a criterion like that proposed by [EPR] contains ... an

essential ambiguity when it is applied to the actual problems with which we

are here concerned" (QTM, 146). This is the same ambiguity. For, in view

of this interaction, we cannot unambiguously ascribe, as EPR, again, do in

accordance with their criterion, independent properties to quantum objects.

Ultimately we cannot do so even for a single such property, let alone both

complementary ones (as EPR, again, do in their argument), or, at the limit,

an independent specifiable identity of any kind (not even that of "object-

ness" or "objectivity") to quantum objects.

Bohr agrees that quantum mechanics allows for and enables such ("at-a-

distance") predictions and by so doing rigorously accounts for quantum

correlations and entanglement. At the same time, however, it does not, in

Quantum Mechanics, Complementarity, and Nonclassical Thought * 93

Bohr's interpretation, imply nonlocality, as some suggest. The entangle-

ment/correlation part of the EPR-type argument can be adjusted so as to

refer only to the outcomes of measurements rather than to quantum objects,

without, however, reinstating nonlocality.52 In other words, entangled

quantum objects (using this term with due nonclassical qualifications) or

(Bohr shuns this language) "states" exist, but this only means (a) that par-

ticular forms of experimental detection (phenomena or effects) are possible

and (b) that the quantum-mechanical formalism allows us to predict such

effects. Accordingly, nonlocality, which is entailed neither by (a) nor by (b),

need not follow, unless of course it is independently derived from the for-

malism itself, which, as I said, does not appear to be the case thus far.53 It

would follow that the efficacities of correlations are not available to quan-

tum-mechanical or any explanation. But then neither are most typical quan-

tum effects, many of which are the effects of correlations. In other words,

Bohr argues that a local but nonclassical (as opposed to nonlocal and clas-

sical) interpretation, or model, to begin with, of quantum mechanics is pos-

sible. This possibility was missed by EPR's argument, although perhaps not

by Einstein, who, however, thought the nonclassical epistemology of com-

plementarity unacceptable.

It has been argued that the question of locality is irrelevant for Bohr and,

more surprising (but also more uncommon), that Bohr's interpretation is in

fact nonlocal.54 It would be difficult to agree with either view, especially the

second one. Bohr's concern with the relationships between quantum

mechanics and relativity is manifest in his reply to EPR and most of his sub-

sequent (and preceding) writings on the subject, including virtually all arti-

cles here cited. He states in a key (but usually ignored) note at the end of his

reply, "the relativistic invariance of the uncertainty relations ensures the

compatibility between the argument outlined in the present article [i.e., his

reply to EPR] and all exigencies of relativity theory" (QTM, 150n). Indeed,

as the argument (mentioned as forthcoming in the same note) of "Discus-

sion with Einstein" makes clear, the proper application of quantum-

mechanical formalism and specifically of the uncertainty relations in fact

depends on general relativity. This discussion between Einstein and Bohr

took place in 1930, well before the EPR article, although, as Bohr observed,

it contained all the key epistemological elements of the EPR argument

(PWNB 2:57).

For Bohr, relativity, and hence locality, was a given, an "axiom" or "a

law of nature," confirmed by experiment, just as the uncertainty relations

were a given, "a law of nature," and a consequence of quantum theory,

confirmed by experiment, or as (with qualifications given earlier) the partly

94 * The Knowable and the Unknowable

classical and partly quantum description of measuring instruments was a

given, corresponding to the fact that on the macrolevel classical physics still

holds. The first challenge, answered by complementarity, was an interpreta-

tion accounting for the peculiar character of the quantum-mechanical data

and formalism, especially as manifest by the wave-particle complementarity

and the uncertainty relations. EPR's or, more properly, Einstein's challenge

was to prove that complementarity is, or may be refined to be, both com-

plete, within its scope, and consistent with relativity. Bohr's ultimate inter-

pretation allows him to see quantum theory and data as fully consistent

with all three-quantum

physics, relativity, and classical physics-and

brings all three together (involving classical physics at the level of measuring

instruments), even if at the cost (again, to some exorbitant) of nonclassical

epistemology. The latter, as I said, also deprives us of the possibility of

knowing how quantum entanglement and correlations are ultimately possi-

ble in physical (space-time) terms, since they must now be seen as classically

observable and measurable effects whose ultimate efficacity is no more

accessible than in the case of other quantum effects, many of which in fact

involve entanglement. In a certain sense, all do, insofar as any measurement

creates an entanglement between the object and the apparatus, as indicated

earlier.

These are the circumstances that enable Bohr to argue that "the apparent

contradiction [found by EPR] in fact discloses only an essential inadequacy

of the customary viewpoint of natural philosophy for a rational account of

physical phenomena with which we are concerned in quantum mechanics."

Instead, in accordance with the preceding discussion, the irreducibility of

this interaction "entails the necessity of the final renunciation of the classi-

cal ideal of causality and a radical revision of our attitude towards the prob-

lem of physical reality" (QTM, 145-46).55 Or, again, at least the corre-

sponding interpretation of quantum mechanics, complementarity, allows

one to effectively reply to the EPR argument. For this view makes it possi-

ble to develop a local interpretation of quantum mechanics, and, as a kind

of philosophical bonus, a new form of epistemology. Thus, quantum entan-

glement and the question of locality indeed make "interpretation" even

more significant in quantum mechanics. For short of a particular, very

specific interpretation, it may not be possible to counter Einstein's argu-

ments. Even if at an epistemological cost (acceptable to Bohr, exorbitant to

Einstein), Bohr's interpretation ensures the compatibility of quantum

mechanics with relativity, the desideratum equally defining his and Ein-

stein's view. Is Bohr's interpretation the only such interpretation currently

available? It may or may not be; the question would requre a separate

Quantum Mechanics, Complementarity, and Nonclassical Thought * 95

analysis, although, as I have indicated, some of its aspects may, in effect,

invade some among the proposed alternatives. Are more radically different

interpretations of quantum theory possible, say, those that Einstein would

accept as "more complete conceptions"? That is, is the epistemology in

question inevitable given quantum theory, or even already quantum data, or

only consistent with it? It is difficult to say, and it is the latter, quite enough

in itself, that is my argument here. Bohr, too, especially in print, appears to

argue primarily for the latter, although he did not appear to think (rightly)

that there were other fully developed such interpretations at the time of his

writing (QTM, 145). This view of Bohr's argument is not inconsistent with

his other claims, such as that "in quantum mechanics, we are not dealing

with an arbitrary renunciation of a more detailed analysis of atomic phe-

nomena, but with a recognition that such an analysis is in principle

excluded," as considered earlier (PWNB 2:62; Bohr's emphasis). As is clear

from Bohr's accompanying elaboration, this claim may be read as applica-

ble within Bohr's interpretation. This interpretation itself is seen by Bohr as

enabling one to view quantum mechanics as a fully consistent and, within its

proper scope, complete physical theory.

Thus, given Bohr's interpretation, it need not follow that quantum

mechanics is incomplete, since, in this interpretation, the EPR criteria are

inapplicable without ambiguity, an argument that Einstein accepted to some

degree. Nor does it follow that it is nonlocal, since, on the one hand, the the-

ory itself is consistent with relativity. On the other hand, all available argu-

ments concerning nonlocality of (the standard) quantum mechanics, begin-

ning with Einstein's, appear to make presuppositions that Bohr's interpretation

does not make (which still appears to be the case), beginning with and per-

haps always amounting to making assumptions concerning the independent

behavior of quantum objects.

I would like to comment in this context on counterfactual (or as it is

sometimes called counterfactual definite) logic in quantum mechanics,

which has been a major part of most arguments concerning quantum non-

locality. Bohr's interpretation would at least inhibit and perhaps prohibit

the applicability to quantum theory of EPR's and most standard (perhaps

all) counterfactual arguments, that is, arguments based on what could, but

actually did not, happen, if certain (unperformed) experiments were "in

fact" (a word perhaps inapplicable under these conditions) performed.

Thus, as I said, in examining an outcome of a given double-slit experiment

we can only argue from within one among the possible complementary

(mutually exclusive) situations, defined by the particular experimental

arrangement at hand, say, when both slits are open and no counters are

96 * The Knowable and the Unknowable

installed, even when we only have a single spot on the screen. Our exami-

nation, however, cannot rely on (counterfactual) considerations of the other

complementary situation, defined by the alternative arrangement, here the

one that would allow us to determine through which slit particles pass. We

cannot draw conclusions concerning the situation at hand by considering

what could, but actually did not, happen to the quantum objects in the alter-

native complementary situation. In the EPR situation the circumstances of

complementarity manifest themselves in the fact that, in practice, two runs

of any EPR type of experiment, and hence two actual physical situations,

are always required in order to confirm the EPR alternative predictions. Just

as in the case of, and epistemologically equivalently to, the double-slit

experiment, there is no single physical situation of which these two situa-

tions would be parts or aspects, or from which they could be derived, of

which they would be effects. Hence, EPR's and Einstein's other nonlocality

arguments cannot avoid counterfactual logic, which, however, need not be

seen as applicable to quantum mechanics. Following Bell's theorem, the

assumption that counterfactual logic applies in quantum mechanics appears

to be correlative to deriving nonlocality from quantum theory or quantum-

mechanical data. (As I have indicated, Bell's theorem itself does not prove

the nonlocality of quantum mechanics but, as we have seen, only of certain,

epistemologically realist hidden-variables theories, which allow for counter-

factual reasoning.) These assumptions, however, may be seen as in turn sup-

plementary rather than structural in quantum physics, as Bohr perhaps

knew, and as both Arthur Fine and David Mermin have emphasized.56

Indeed, most of such arguments rely, directly or implicitly, on attributing

physical variables or quantum-mechanical descriptions to quantum objects

rather than to the outcome of quantum measurements registered in measur-

ing instruments, even though the counterfactual statements themselves may

refer to the latter. Or, again, they rely on the presupposition that both com-

plementary situations may, at least in principle, be derived from a single

common situation, know or unknown (or even unknowable). As I have just

indicated, this presupposition is untenable in Bohr's interpretation and, it

appears to follow (courtesy of Einstein), in any local interpretations of

quantum mechanics. As Mermin observes, considered in terms of strictly

statistical correlations, given a sufficient number of trials, quantum mechan-

ics is local.57 Einstein to some degree realized this point, although he con-

sidered this way out of the dilemma "too cheap": "The interpretation of the

fr-function as relating to an ensemble also eliminates the paradox that a

measurement carried out in one part of space determines the kind of expec-

tation for a measurement carried out later in another part of space (coupling

Quantum Mechanics, Complementarity, and Nonclassical Thought * 97

parts of systems far apart in space)." On the other hand, as Mermin ele-

gantly shows, if one uses counterfactual reasoning, one cannot establish that

quantum predictions, corresponding to these correlations, are strictly local

at the level of individual events, which is of course what Einstein thought,

again, on the basis of arguments using counterfactual reasoning. In short,

Mermin points out, such arguments always appear to use counterfactual

reasoning.58

Here, again, given the possibility of Bohr's interpretation, we face a sim-

ple "either/or" logical choice-of giving up either locality/relativity, a well-

confirmed experimental feature of the physical world (never an option for

Bohr and most other physicists), or certain arbitrary, even if commonsen-

sical and in terms of classical physics natural, assumptions, such as the

possibility of applying counterfactual reasoning. It follows, for example,

that Einstein's alternative for quantum mechanics between "nonlocality

and completeness" and "locality and incompleteness" depends on his use

of counterfactual logic. This offers us the possibility of ascribing to quan-

tum mechanics "locality and completeness," while suspending the applica-

bility of counterfactual reasoning. Bohr's interpretation of course does not

depend on the latter. An intriguing possibility would be to consider giving

up the assumption that certain parts of measuring instruments and, by

implication, the macroworld are described even ideally by classical physics.

The question would require a separate discussion, although it may be

pointed out that the applicability of classical physics at this level is difficult

to contest, and quantum physics has relied on it in other respects as well

throughout its history, beginning with the correspondence principle.59 My

main point at the moment is that Bohr's interpretation allows him to han-

dle Einstein's argument, even if we adjust the EPR argument to nonlocal-

ity, as Einstein did later, rather than only incompleteness. Once relativ-

ity/locality is taken as a given, a (in present terms) nonclassical

interpretation of quantum mechanics offers to Bohr a much better alterna-

tive. His argument is that this interpretation is possible, which also entails

a new version of complementarity.60 It is worth keeping in mind, however,

that Bohr's reply to EPR addresses the EPR argument as such, that is, as an

argument concerning the (in)completeness of quantum mechanics; and it

would be inappropriate to disregard or diminish this fact, or the character

of the EPR initial argument in considering and assessing Bohr's reply, as is

sometimes done.

Obviously, there remains the fact that we make predictions concerning

the future behavior of the second particle as a result of interfering, substan-

tially (by physically, mechanically engaging with measuring devices) and

98 * The Knowable and the Unknowable

unsubstantially (by making meaningful statements, predictions, and so

forth), with the local measuring system associated with the first particle.

That, however, is quite different from and indeed incompatible with any

reading of Bohr's argument as implying nonlocality. The system associated

with the first particle enables us to unambiguously define either one or

another complementary variable (never both simultaneously) assigned, via a

corresponding measurement or phenomenon, to the second particle. A mea-

suring system of the same type may be introduced for the second particle

and, then (once a sufficient number of "identically" prepared experiments is

performed), correlated with the one associated with the first particle, once

this second system is being in turn interfered with, for example, in order to

measure the value of a given variable for the second particle. Such correla-

tions are involved in Bell's theorem and its refinements and the discussions

surrounding these findings and their implications, more recently around

Greenberger-Horne-Zeilinger and Lucien Hardy types of experiments

(which are particularly striking versions of the EPR type of experiments and

subjects of many recent discussions).61 This, however, is not what Bohr

refers to on this occasion, even if what he says is consistent with these "cor-

relations," a point that, as Mermin suggests, Bohr perhaps missed or missed

making in this form.62 Nor, of course, could Bohr have had in mind more

subtle features in subsequent developments around the EPR experiment(s),

but his argument can, I think, be adjusted to these cases as well. My point is

that quantum entanglement and locality are both maintained and can be

experimentally and theoretically understood within the epistemology of

"effects" as discussed here.

How, then, is this possible, especially given the apparent impossibility of

applying Hans Reichenbach's common cause principle, according to which

such correlations should be explained in terms of the preceding common

history of the events involved?63 The answer of Bohr's interpretation is that

we do not know how this ultimately is or can be (physically) possible,

indeed we cannot know, or even conceive of it, since any further analysis

that would, in principle, allow us to do so is, in principle, excluded. The

epistemology of knowable effects whose efficacity is ultimately inconceiv-

able and, hence, the inaccessibility of "the quantum world" itself (to any

knowledge or conceptualization that is or will ever be available to us, for

example, as "quantum" or the "world," in any conceivable sense of either

word) allow one to rigorously maintain both the consistency and locality of

quantum theory. But, then, entanglement is only part of what we cannot

conceive of in quantum theory, beginning with the double-slit experiment.

How do electrons "know," individually or collectively, that both slits are

Quantum Mechanics, Complementarity, and Nonclassical Thought * 99

open to arrange themselves in the interference-like pattern? As I said, quan-

tum theory appears to, and in Bohr's interpretation does, prohibit in prin-

ciple what it does not deliver in practice. But it delivers plenty of what can

be known, too, and different ways in which knowledge, including informa-

tion in its technical sense, can be obtained and processed, for example,

through quantum entanglement, which, among other things, enables the

whole field of quantum cryptography and computing. That it delivers the

impossibility of knowledge or conception has a positive role to play, since

this impossibility often enables knowledge and conceptions that would not

be possible otherwise. This interpretation conforms to all standard require-

ments and desiderata of scientific research, of the disciplinarity of physics

as, to return to Galileo's definition, "a mathematical science of nature,"

and indeed enables us to advance where epistemologically classical

accounts fall short.

"The Basic Principles of Science": Nonclassical

Epistemology and the Disciplinarity of Physics

I close this chapter by considering the implications of Bohr's argument with

EPR and his nonclassical epistemology of quantum mechanics for his views

of the basic principles of science and, hence, for the disciplinary practice of

quantum physics. He writes:

the argument of [EPR] does not justify their conclusion that quantum

mechanical description [sic!] is essentially incomplete. On the contrary this

description, as appears from the preceding discussion [i.e., in Bohr's inter-

pretation], may be characterized as a rational utilization of all possibilities of

unambiguous interpretation of measurements, compatible with the finite and

uncontrollable interaction between the [quantum] objects and the measuring

instruments in the field of quantum theory. In fact, it is only the mutual

exclusion [given this interaction] of any two experimental procedures, per-

mitting the unambiguous definition of complementary physical quantities,

which provides room for new physical laws [i.e., the laws of quantum

mechanics], the coexistence of which might at first sight appear irreconcilable

with the basic principles of science [but is ultimately not]. It is just this

entirely new situation as regards the description of physical phenomena, that

the notion of complementarity aims at characterizing. (QTM, 148)

Bohr, intriguingly, omits "reality" here. This omission may have been

deliberate and is certainly telling. The possible completeness of quantum-

mechanical physical description may no longer allow for reality in EPR's

100 * The Knowable and the Unknowable

sense, that is, an independent physical reality, defined by postulating the exis-

tence, on the classical model, of physical properties of objects (or, again, con-

ceivably, such classical-like objects themselves) as independent of their inter-

action with measuring instruments. Instead, this completeness may be

enabled by Bohr's interpretation of the quantum-mechanical data in terms of

phenomena, as here considered, defined as the overall experimental arrange-

ments within which quantum effects, marks left in our measuring devices,

manifest themselves. That, again, does not mean that a certain level of the

constitution of matter does not exist in some form, but only that the attribu-

tion of physical properties, including that of the individual identity of a par-

ticle, may not be possible at that level. Nor, as we have seen, would it follow

(as some contend) that this suspension of the independently attributable par-

ticle identities, such as those of two "particles" in the EPR situation, in fact

entails nonlocality. Two quantum entities (for the lack of a better word)

involved would still be spatially separate, and, according to Bohr, there is in

the EPR case certainly "no question of a mechanical [i.e., physical] distur-

bance of [one] system under investigation" by our interference with the other

quantum system involved in the EPR thought experiment (QTM, 148). It is

just that, as I have argued here, we cannot attribute independent physical

properties, ultimately, even that of a "particle," to that system. Once we

assume that we can, as Einstein did, nonlocality appears to follow. Einstein's

argument is, accordingly, not logically wrong. His assumption may well be

wrong or at least is not necessary.

Thus, according to Bohr, the irreducible interaction between quantum

objects and measuring instruments, while indeed incompatible with the clas-

sical ideals of causality and reality, is by no means incompatible with "the

basic principles of science." This compatibility, however, is only possible if

one adequately interprets what is in fact available to an unambiguous

account in the entirely "new situation" we encounter in the field of quantum

theory and what this theory actually unambiguously accounts for and how

it accounts for what it accounts for. The essential ambiguity of the EPR cri-

teria arises precisely from their failure to perceive that a complete and local,

albeit nonclassical, interpretation of the situation is possible, however sub-

tle and revealing of new aspects and "mysteries" of the quantum it may be.

In particular, it is, again, the failure to see that "under these circumstances

an essential element of ambiguity [may be] involved in ascribing conven-

tional physical attributes to atomic [quantum] objects [themselves]"

(PWNB 2:41), ultimately single (rather than only complementary) such

attributes, or even seeing such objects as particles. Or, at the very least, they

fail to see that such an assumption is unwarranted, and an alternative view,

Quantum Mechanics, Complementarity, and Nonclassical Thought * 101

such as Bohr's, of the situation is possible, while retaining locality. (EPR,

again, clearly saw a nonlocal alternative.) Bohr's interpretation, which, by

contrast, rigorously follows this requirement, "may be characterized as a

rational utilization of all possibilities of unambiguous interpretation of mea-

surements, compatible with the finite and uncontrollable interaction

between the [quantum] objects and the measuring instruments in the field of

quantum theory." As such it also "provide[s] room for new physical laws

[of quantum mechanics], the coexistence of which might at first sight appear

irreconcilable with the basic principles of science."

Bohr's argument is, thus, as follows: were it not for the irreducibility of

"the finite and uncontrollable interaction between the [quantum] objects

and the measuring instruments in the field of quantum theory," which

grounds Bohr's concepts of phenomena,

(a) EPR would be right: quantum theory would be incomplete, or else

nonlocal (or short of an interpretation that ensures both complete-

ness and locality, at least at the time); and

(b) there would be no room for the laws of quantum mechanics as phys-

ical laws (the same type of parenthesis is required).

The laws of quantum mechanics may appear, in particular to EPR, to be

"irreconcilable with the basic principles of science," at least, ultimately, if

one wants to keep the locality requirement and, hence, compatibility with

relativity. Quantum mechanics, however, accounts for its data as well as

any classical theory does for its data, which is what EPR tried, ultimately

unsuccessfully, to question. It may, thus, depend on which principles one

sees as basic to science, both in general and insofar as such principles can be

applied in the case of quantum physics. Bohr argues as follows. Rather than

a conformity with a particular criterion of physical reality, such principles

are the logical consistency of a given theory, its correspondence with the

available experimental data, and its capacity to "exhaust the possibilities of

observation." According to Bohr, quantum mechanics, within its proper

limits and when given a proper interpretation, conforms to them. Hence

Einstein's argumentation could not be directed toward demonstrating the

inadequacy of quantum mechanics (PWNB 2:56-57).64

Accordingly, along with the reexamination of the classical ideals of

causality and reality necessitated by quantum mechanics, a similar, and

indeed parallel and interactive, reexamination of what constitutes the basic

principles of science appears to be rigorously necessary. For, on the one

hand (at least, again, under the conditions of locality), the EPR criterion of

102 * The Knowable and the Unknowable

reality cannot unambiguously apply to the "entirely new situation as

regards the description of physical phenomena" in question in quantum

mechanics. On the other hand, quantum mechanics itself comprehensively

accounts for these phenomena. Hence, it is as rigorously scientific as any

(classical) mathematical science in every respect (other than causality and

reality). Clearly, one needs to (re)consider what the basic principles of sci-

ence are. This is a crucial point: physics itself, not philosophy, requires this

reconsideration, as Heisenberg observes in the passage cited earlier. Indeed,

according to Heisenberg's view, expressed on the same occasion, "Bohr

was primarily a philosopher, not a physicist, but he understood that nat-

ural philosophy in our day and age carries weight only if its every detail can

be subjected to the inexorable test of experiment."65 This test may entail

the irreducibly nonclassical character of natural philosophy, once one con-

siders nature at the quantum level, the level of its ultimate constituents. The

basic principles of science must be weighed and, if necessary, adjusted

accordingly.

Ironically, however, the basic principles of science, as seen by Bohr, are

in accord with the defining aspects of the project and practice of classical

physics, beginning with Galileo, to whom, as we have seen, Bohr specifically

refers in this context (PWNB 3:1). It is true that Einstein and many others

would see certain other (philosophical) principles as equally basic. Accord-

ingly, Bohr's argument concerning quantum mechanics also suggests that

the basic principles of science qua science may, at a certain point, come into

conflict with those metaphysical principles, however consistent the latter

may be with classical physics. What are the basic principles of science,

according to Bohr's view? What would define the science and the discipline

(in either sense) of physics as, to use Galileo's locution, a (modern) "mathe-

matical science of nature"? There are, as I can see it, more or less four such

principles, which can be described as follows. Further qualifications and

nuances are necessary, in part in view of the massive recent reconsideration

of the nature and the character of scientific knowledge, which, it may be

recalled, itself played a major role in the recent cultural debates, such as, but

again far from exclusively, the Science Wars.66 It may be shown that, in

essence, these formulations are consistent with this reconsideration-at its

best. The works involved are not without their own problems, sometimes as

severe as those of the classical views in question in this reconsideration, with

which, at their best, the principles in question can in turn be correlated. At

the moment, however, I am more interested in arguably a stronger point,

that of the compatibility of nonclassical epistemology even with a reason-

Quantum Mechanics, Complementarity, and Nonclassical Thought * 103

ably traditional view of the basic principles of science, or at least some

among these principles, but perhaps also the most crucial ones.

(1) The mathematical character of modern physics. By this I mean the fol-

lowing, and I think, with both Galileo and Bohr, that this is what modern

physics most fundamentally is. It is the usage of mathematics as a particular

way of offering convincing arguments about certain aspects of and certain

facts pertaining to the physical world rather than necessarily mathemati-

cally representing the ultimate nature or structure of this world.67 The lat-

ter, as we have seen, is rigorously impossible to do in quantum physics, at

least in Bohr's interpretation. This point is crucially implicated in the Bohr-

Einstein debate, and, as I have indicated, there appear to be significant dif-

ferences in this respect between Galileo's and Newton's project and philos-

ophy of science, or of nature itself, as well. For Galileo a science of motion

is a construction of convincing mathematical arguments about certain facts

and aspects of nature, which also brings the resulting epistemology of

Galileo's own project closer to the nonclassical epistemology of comple-

mentarity but does not eliminate significant differences between them, for

example, and in particular, in view of the causality of Galileo's models. One

is almost tempted to say that, while his models are classical, his view of how

they relate to nature is almost, even if not quite, nonclassical. In any event,

for Newton, it is a representation of nature, grounded in the classical realist

claim that nature possesses a structure that can ultimately (at least by God)

be represented mathematically.68 The latter view in fact defines the attitude

of most physicists, including many of those who do quantum physics.

Exceptions appear to be few.

(2) The principle of consistency. These theories (i.e., arguments and inter-

pretations involved), while they may and perhaps, especially in quantum

physics, must exceed their mathematical aspects, must offer logically con-

sistent arguments. Indeed, as we have seen, in his The Interpretation of

Quantum Mechanics-the interpretation, no less!-Roland Omnes bases

his whole interpretation of quantum mechanics around logical consistency.

There are important further nuances to this principle as well, which I must

bypass here. In any event, these theories must be as logically consistent as

anything can be.

(3) The principle of unambiguous communication. These theories, in

their mathematical and nonmathematical aspects alike, must allow, within

the practical limits of the functioning of science, for the (sufficiently) unam-

biguous communication of both the experimental results and theoretical

findings involved. This is also what Bohr's interpretation of quantum

104 * The Knowable and the Unknowable

mechanics "provides room for," in part by virtue of exploring the possibil-

ities of the unambiguous definition of all physical variables and aspects of

physical description involved. (Accordingly, the concepts of "unambiguous

definition" and "unambiguous communication" become especially crucial

for Bohr in the wake of the EPR argument.)

(4) The principle of experimental rigor (based, at least from Galileo on,

on the concept of measurable quantities). These theories must correspond to

and, within their limits, exhaust the experimental data they aim to account

for, although this data is of course itself subject to interpretation. Certainly,

in quantum physics the question of how one interprets its data is as crucial

as, and reciprocal with, that of how one interprets quantum theory. Much

more is to be said on this point as well (even leaving aside the question of

the social construction of theories and related arguments, which would

affect the principles of consistency and unambiguous communication as

well). The principle itself, however, remains crucial.

Physical laws would then be seen and defined as physical laws in accor-

dance with these principles, which-this is my point-define both classical

and quantum physics. In order, however, to rigorously maintain them in the

case of quantum physics, one, according to Bohr, may need to accept the

nonclassical epistemology of quantum physics. If one does so, however, one

must also abandon, at the level of the quantum world, certain other (episte-

mological and ontological) principles, applicable (alongside the basic prin-

ciples of science as just outlined) in classical physics. The (epistemologically)

radical character of quantum physics, as defined by a nonclassical interpre-

tation, becomes rigorously correlated with, if not the condition of, its disci-

plinarity as physics, and thus maintains the continuity with classical physics,

which could otherwise be broken. It is true that one can technically practice

quantum physics while subscribing to the classical philosophy of nature or

of physics, including quantum theory. In question here, however, is a rigor-

ous epistemological justification of such a practice, which, Bohr argues, at

the very least may entail nonclassical epistemology.

According to this view, the departure (still not quite absolute, given that

classical physics may be interpreted "less classically" than classically) from

classical physics occurs at the level of epistemology, not at that of the char-

acter and the practice of physics as a mathematical science or, since, as Hei-

degger argues, both define each other in science, as a mathematical-experi-

mental science.69 The application of mathematics to the ultimate objects of

investigation changes. Mathematical description is not applicable to them at

all, not anymore than any other description. I have stressed throughout that

classical epistemology is not simply abandoned either. Along with classical

Quantum Mechanics, Complementarity, and Nonclassical Thought * 105

science it continues to function within its proper limits and is often part of

nonclassical theories as well, which, again, irreducibly depend on it. Bohr's

nonclassical epistemology itself is of course fundamentally, irreducibly,

reflected in both the specific character of the phenomena observed (unex-

plainable by means of classical physics) and the mathematical formalism

that predicts these phenomena. This is why Bohr often speaks of "the epis-

temological lesson of quantum mechanics." He rigorously connects his epis-

temology to the mathematical-experimental structure of quantum theory,

even if he does not strictly derive it from the latter.

The laws of quantum physics are now seen as the laws of nature only in

the sense of corresponding to the "regularities" arising in our interaction

with nature, specifically by means of experimental technology. The term is

used by Bohr in speaking of "the new types of regularities," which we

encounter as effects of the interaction between the ultimate quantum con-

stituents of nature and our measuring technology and which cannot be

accounted for by classical physics (QTM, 150). In this interpretation, how-

ever, quantum mechanics does not describe the nature or structure of these

ultimate constituents themselves. There is, in this interpretation, nothing

that quantum theory, or, in view of its laws, conceivably any theory, can say

about these constituents as such. Ultimately, as we have seen, not even this

impossibility can be absolutely ascertained about them. Otherwise, quan-

tum mechanics and Bohr's interpretation of it conform to all traditional

principles defining the project and practice of the "mathematical sciences of

nature" from Galileo on. Quantum mechanics at the very least allows for,

even if not necessitates (it may ultimately do this too), compatibility

between "the basic principles of science" and nonclassical epistemology and

was the first theory to do so. One might argue (Bohr does) that Einstein's

relativity would pose some of these questions already, perhaps, once all the

chips or (we may never have all) more chips are in, even all of these ques-

tions.

Indeed, as I have indicated in the preceding chapter, the overall situation

just described could be extrapolated to several nonclassical figures outside

science, discussed (in various degrees) in this study, such as Nietzsche,

Bataille, Levinas, Blanchot, Lacan, Deleuze, de Man, and Derrida. This, it

may be added, also applies to the relationships between the thought of these

thinkers and modern science in its nonclassical aspects. The list of these

figures is not random, and this type of argument may be more difficult in

some of the cases just mentioned, and it will not be applicable at all in still

other cases. As in Bohr's case, however, some of the most radical epistemo-

logical thinking involves the deepest disciplinary concerns insofar as the

106 * The Knowable and the Unknowable

basic principles of their disciplines are at stake. Obviously in some of the

cases just mentioned the disciplinary determination itself-Philosophy? Crit-

icism? Psychoanalysis?-or the stratifications within and the interactions

between such fields are extremely difficult. Accordingly, significant adjust-

ment of the argument would be necessary. I would, however, maintain that,

(a) even in the case of science, nonclassical epistemology is not in conflict

with its basic principles; it is or at certain points becomes the condi-

tions of applicability of such principles; and that

(b) one encounters what I called earlier the extreme disciplinary conser-

vatism of the thinkers in question, that is, contrary to common

claims and some appearances, their extreme reluctance to bring a

radical change in or shift to nonclassical accounts, which they finally

do only at points and in regions where there is really no choice, in the

sense that their discipline (in either sense) in fact requires it. Indeed,

it appears that in such cases one needs to be both an extreme radical

and an extreme conservative, along different lines, and sometimes

even the same, or at least interactive and mutually depending, lines.70

That both of these facts are commonly overlooked largely accounts for

the persistent misunderstanding of the thought of the figures in question and

for the misshapen nature of some of the recent debates in which they figure

prominently.

Einstein deeply understood this second aspect of Bohr's thought, even

though and perhaps because he never accepted his views or quantum

mechanics itself as a way to describe nature. (He did recognize its practical

effectiveness.) In 1949, after a quarter of a century of their debate, he spoke

of some of Bohr's radical (in the present sense) physics as "the highest musi-

cality in the sphere of thought."71 A very good violin player and an admirer

of Haydn, in particular, Einstein might have preferred and had tried to give

this music a more classical shape, or he at least urged others to do so. Nei-

ther he nor others ever succeeded. Bohr, although, by contrast, a bad piano

player, was in every sense a contemporary of Arnold Schonberg. Haydn,

however, let alone Bach, Mozart, and Beethoven, may be much closer to

Schonberg than Einstein thought, as his appeals to Milton's description or,

as the case may be, undescription of chaos in Paradise Lost in his Creation

might indicate, when we find Milton or Satan on the threshold of chaos:

Before thir [Satan's, Sin's, and Death's] eyes in sudden view appear

The secrets of the hoary deep, a dark

Quantum Mechanics, Complementarity, and Nonclassical Thought * 107

Illimitable Ocean without bound,

Without dimension, where lengths, breadth, and highth,

And time and place are lost; where eldest Night

And Chaos, Ancestors of Nature, hold

Eternal Anarchy, amidst the noise

Of endless worth, and by confusion stand.

For hot, cold, moist, and dry, for Champions fierce

Strive here for Maistry, and to Battle bring

The embryon Atoms; they around the flag

Of each his Faction, in thir several Clans,

Light-arm'd or heavy, sharp, smooth, swift or slow,

Swarm populous, unnumber'd as the Sands

Of Barca and Cyrene's torrid soil,

Levied to side with warring Winds, and poise

Thir lighter wings. To whom these most adhere,

Hee rules the moment; Chaos Umpire sits,

And by decision more imbroils the fray

By which he Reigns: next his high Arbiter

Chance governs all. Into this wild Abyss,

The Womb of Nature, and perhaps her Grave,

Of neither Sea, nor Shore, not Air, nor Fire,

But all of these in thir pregnant causes mixed

Confus'dly, and which thus must ever fight,

Unless th' Almighty Maker them ordain

His dark materials to create more Worlds.

(Book II, 890-916)

This extraordinary vision is, I would argue, much closer to that of the

quantum-mechanical (Bohr-like) rather than the chaos-theoretical vision of

the ultimate nature of things, albeit ultimately not quite as radical.72

Accordingly, my point here is not merely, or even primarily, about how to

perform Haydn's music differently (although it is this too). Instead, this is a

statement about Haydn as a classical composer, in either, or neither, sense

of this strange, "confus'dly" mixed, almost quantum-mechanical and non-

classical, word "classical."



Chapter 3

Versions of the Irrational: The Epistemology of

Complex Numbers and Jacques Lacan's

Quasi-Mathematics

One is quite unable to visualize imaginary quantities.

-RENE DESCARTES

Imaginary roots are a subtle and wonderful resort of the

divine spirit, a kind of Hermaphrodite between existence and

non-existence (inter Ens and non Ens Amphibio).

-GOTTFRIED WILHELM LEIBNIZ

These symbols themselves, as is indicated already by the use

of imaginary numbers, are not susceptible to pictorial

interpretation.

-NIELS BOHR

The square root of -1 does not correspond to anything that is

subject to our intuition, anything real-in the mathematical

sense of the term-and yet it must be conserved, along with

all its functioning.

-LACAN

Introduction

Although I had thought from time to time on the subject of this essay, the

beginning of my work on it coincided with the events of the Science Wars,

following Paul Gross and Norman Levitt's book Higher Superstition: The

Academic Left and Its Quarrel with Science and physicist Alan Sokal's hoax

article published in the journal Social Text. Sokal and his coauthor, Belgian

physicist Jean Bricmont, had then just published their book, Impostures

intellectuelles, devoted to the misuse or, as the subtitle of the American ver-

sion (Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science)

indicates, even abuse (alleged by the authors) of mathematics and science by

some leading French intellectuals.1 Jacques Lacan's work appears to be seen

by the authors as arguably the most notorious case of this alleged abuse.

Some of Lacan's statements that they cite were bound to attract special

attention, in particular his identification of the penis with the mathematical

110 * The Knowable and the Unknowable

square root of -1, as it was sometimes (mis)reported in the popular press or

(mis)understood by its readers. The statement, understandably, caused

some puzzlement on the part of those readers, including some in the sci-

entific community, as to whether Lacan was even speaking seriously at all

and, if he was serious, whether there was any way to view this or certain

other statements by Lacan (and other authors attacked in Sokal and Bric-

mont's book) as anything but complete nonsense.

I shall, by way of replying to these questions, also sketch an argument

applicable to Lacan's use of mathematical ideas other than imaginary num-

bers (such as the square root of -1), for example, those he borrows from

topology and mathematical logic. I shall deal directly, however, only with

imaginary and complex numbers and Lacan's argument, leading to the

statement just cited, in "The Subversion of the Subject and the Dialectic of

Desire in the Freudian Unconscious."2 It is worth noting at the outset that,

as a psychoanalytically informed reader would be aware, the "erectile

organ" of Lacan's statement is not the same as the penis. The phrase may

not even quite be seen as referring to the phallus, defined by Lacan in the

same essay as "the image of the penis," but instead as the image of the phal-

lus-the image of the image of the penis. Indeed, whatever the word "penis"

may ultimately refer to, this reference is always mediated by a specific

image, idea, metaphor, and so forth, or indeed by a network of such ele-

ments. This is a significant aspect, owing much to Hegel, of Lacan's psy-

choanalytical epistemology. This part of Lacan's discussion would immedi-

ately indicate to an attentive reader, even one unfamiliar with Lacan's work,

that Lacan's subject is in part a certain psychoanalytical epistemology,

which would, accordingly, have to be addressed if one is to make sense of

Lacan's "equation." One need not, of course, choose to engage with this

epistemology fully or even at all, but one cannot avoid it in commenting on

Lacan's "equation."

On the other hand, by a kind of ironic twist, especially in the context of

the Science Wars debates, this epistemology is related to the key epistemo-

logical questions involved in the concept of complex numbers. These rela-

tionships (rather than more strictly mathematical considerations) may well

have been primarily responsible for Lacan's unorthodox appeal to complex

numbers. As a result, my argument here concerns at least as much the epis-

temology of complex numbers and, by implication, of modern mathematics

in general as Lacan's work. I shall argue this epistemology to be nonclassi-

cal and as such fundamentally analogous to the epistemology of quantum

mechanics as complementarity. As I have indicated earlier, both epistemo-

logical situations may be related, since the nonclassical character of cornm-

Versions of the Irrational * 111

plementarity may be seen as correlated with, even if not predicated upon,

the irreducible role of complex numbers in the mathematical formalism of

quantum mechanics, as opposed to that of classical physics. Complex

numbers are sometimes used there as well, specifically in classical wave

theory, but they do not play the same irreducible role in this case, and the

mathematical solutions of the equations involved do not contain complex

numbers and can be directly related to the results of measurements. Not so

in quantum theory, insofar, as we have seen, the latter needs additional ad

hoc procedures in order to relate such solutions (usually in terms of prob-

abilities) to the results of measurements, since these solutions must be rep-

resented in terms of real, or indeed rational, numbers. We cannot "mea-

sure" (assign length to) complex numbers as such, although we can

associate a "length," a positive real number, to any given complex num-

ber, considered as a vector, a fact that helps us in quantum-mechanical

measurement. (This association is not unique since an infinite number of

complex numbers can have the same "length" of this type.) Nor can we

measure physical quantities with complex numbers.3 I shall comment on

these connections as I proceed.

One has to exercise considerable caution here. My strict argument is that

both quantum mechanics (in its standard version) and complex numbers

may be, and may need to be (it may never be certain that they must be),

interpreted in epistemologically nonclassical terms. I also argue that the

nonclassical epistemology of complementarity can be correlated with the

usage of complex numbers in quantum-mechanical formalism. This, how-

ever, is not the same as saying that the nonclassical epistemology of com-

plementarity is correlated with the nonclassical epistemology itself of com-

plex numbers. It is difficult not to think (and both Bohr and Heisenberg,

among others, make some suggestions in this regard) that the nonclassical

character of both may be more rigorously linked in view of the irreducible

role of complex numbers in quantum mechanics. However, this is a separate

and more complex argument, which can only be indicated rather than prop-

erly pursued here. None of my arguments and claims in this study depends

on this argument. I am unaware of any rigorous argument of that type. Nor

am I aware of any argument concerning the nonclassical epistemology of

complex numbers, to begin with.

If, however, one must exercise considerable caution in the case of the

relationships between the epistemology of complex numbers and the episte-

mology of quantum mechanics one must exercise extraordinary caution

indeed when one moves to the relationships between mathematical and sci-

entific theories, or their interpretations, and nonscientific theories, such as

112 * The Knowable and the Unknowable

those of Lacan or Derrida, and the nature of their respective arguments and

claims. I have stressed this point from the outset, and I shall try to exercise

maximal caution throughout the remainder of this study, which largely

deals with such relationships. Boldness, of course, has its appeal and even

necessity, too, as do "dreams of great interconnections," of which Bohr

spoke; and it is a right balance of boldness and caution, of radical and con-

servative moves, and of wakefulness of rigorous disciplinary specificity and

dream of great interconnections that is most necessary as well as most diffi-

cult. On the other hand, one does not need exceptional boldness or great

dreams to notice significant (nonclassical) epistemological parallels between

the (nonclassical) theories in question or to notice that, to return to Bohr's

formulation, "we are not dealing here with more or less vague analogies,

but with an investigation of the conditions for the proper use of our con-

ceptual means of expression" in different nonclassical situations (PWNB

2:1-2). My aim here is to remain with this premise, or promise, even if in

order to do so one sometimes has to surrender somewhat Bohr's other great

hope, the "dream of great interconnections," that is, to limit one's argu-

ments concerning the scope of these interconnections, which may indeed be

greater than what I shall argue for here.

I shall for the most part bypass the Science Wars debates in this chapter

and consider them in the following chapters. It will, however, become

apparent from my argument in this chapter as well that regardless of any

potential problems in the work of Lacan and other authors under criticism,

the arguments against them by Gross and Levitt, Sokal and Bricmont, and a

few other recent critics in the scientific community cannot be seen as ethi-

cally, scholarly, or intellectually appropriate. Nor can they be seen as in

accord with the spirit of scientific inquiry itself. Obviously, I refer

specifically to the authors just mentioned and not to the views or opinions

concerning these subjects on the part of the scientific community in general.

Indeed, it is my view that such critics as Gross and Levitt or Sokal and Bric-

mont do not represent, and should not be seen as representing, science and

scientists, as a large number of commentaries on these authors by scientists

would testify as well. The criticism by these particular authors is disabled by

(a) their lack of the minimal necessary familiarity with the specific subject

matter, arguments, idiom, and context of many of the works they criticize;

(b) their inattentiveness to the historical circumstances of using mathemati-

cal and scientific ideas in these works; (c) their lack of general philosophical

acumen, which is necessary for understanding most of the works in ques-

tion, whether one sees them positively or critically; and d) their insufficient

expertise in the history and philosophy of mathematics and science or even

Versions of the Irrational * 113

certain areas of mathematics and science themselves. These factors, manifest

in Sokal and Bricmont's "treatment" of Lacan, make any constructive criti-

cism by such critics virtually impossible. Indeed, some of their claims con-

cerning mathematical objects in question and specifically complex numbers

are incorrect, which is far less excusable, and in a way more of a disservice

to mathematics and science in their case than in Lacan's, even assuming that

Lacan is as bad as Sokal and Bricmont make him out to be (which is far

from clear). Lacan is not a mathematician or physicist, as Sokal and Bric-

mont are, nor does he criticize mathematics and science, let alone belliger-

ently attack them, the way Sokal and Bricmont, or Gross and Levitt, attack

him and their other targets.

Lacan's statement under discussion is a part of a complex psychoanalyti-

cal and philosophical conceptual assemblage and of an equally complex tex-

tual network. It makes little, if any, sense without taking both and their con-

text (disciplinary, historical, or other) into account, or without translating

Lacan's ideas into a more accessible idiom, and certainly without taking into

consideration Lacan's immediate subject at the moment. Even such transla-

tions are bound to retain considerable complexity for the general audience.4

The psychoanalytical or philosophical substance of Lacan's argument

requires no mathematics as such, which one can decouple from this argu-

ment by translating Lacan's statements containing mathematical references

into statements free from them. Indeed, sometimes one must do so, includ-

ing in the present case, insofar as "the square root of -1" of Lacan's state-

ment is, I shall argue here, in fact not the mathematical ' -1. (For this rea-

son from here on, I shall use the symbol J for the mathematical square root

and use (L)> for Lacan's "square roots" or other symbols of Lacan's "alge-

bra," such as (L)1, (L)-1, and so forth.) One might argue that Lacan should

have performed this subtraction himself and avoided the use of his quasi-

mathematics in view of the apparent incongruity of bringing these two

images together. Should he have said instead (with, as will be seen, much the

same meaning) something like the following, "the image of the erectile

organ is the constituting element, and the center, of the (psychoanalytic)

subject's phenomenological configuration of desire; and the relationship

between this image and the psychoanalytic reality behind it are profoundly

problematic (to the point of the impossibility of applying any concept of

reality hitherto available here)"? Perhaps. Or, perhaps, were he to keep his

mathematical analogy, he should have done so more cautiously. He could,

for example, have added the following to the just formulated statement:

"The situation could be compared (with caution) to that of the mathemati-

cal complex numbers, where the square root of -1 generates the system of

114 * The Knowable and the Unknowable

complex numbers in an analogous way and with similar epistemological

implications, or to certain aspects of the epistemology of quantum mechan-

ics, which is not to say that we can simply (or at all) identify such objects."

For whatever reasons, he did nothing of that nature. Accordingly, one can-

not especially object to, and could even expect, a degree of unease with

Lacan's statement. One would also do well, however, to recall that the

incongruity, if any, is far less pronounced in the psychoanalytic context in

which the statement was made. Indeed, from the psychoanalytic perspective

one might ask why this conjunction appears incongruous or silly to some or

even most of us. One can also ask, in the psychoanalytic context, a number

of other questions connecting, at least in certain psychoanalytic situations

(for example, in the psychoanalysis of dreams), even the strictly mathemat-

ical ~ -1, let alone Lacan's "square root of -1," (L)E -1, to the erectile

organ. Conversely, one can question (and some have, of course) the value of

the psychoanalytic context or of psychoanalysis itself, whether in general or

as specifically concerns its Lacanian version. Such considerations, however,

would not undermine my argument here; on the contrary, they would sup-

port it. My point is that, while one can (I would think rather easily) decou-

ple mathematics from Lacan's argument, the reverse cannot be done: one

cannot decouple "Lacan" from the "mathematics" he uses. At the very

least, one must be extremely careful in negotiating the psychoanalytical or

philosophical content and context of Lacan's "mathematical" concepts and

statements, such as the one in question here. We may call such concepts and

statements "quasi-mathematical": they are mathematical-like and may

share a common philosophical content or epistemology with mathematics,

but they are not mathematical in the rigorous disciplinary sense.

Admittedly, the task of reading Lacan's texts is not easy even for an

informed reader because of their idiosyncratic character, multiple ironies,

calculated shock effects, unexpected connections, undeveloped links, con-

volutions, fragmented or even spasmodic textual economy, and still other

complications. (These features in part result from the fact that one usually

confronts not written texts but transcripts of oral presentations, which pose

major editorial problems in their own right.) I need not deal with these

problems in any significant measure here. For, while I cannot fully bypass

Lacan's psychoanalytical argument, I need not fully spell out this argument

for my purposes of examining how mathematics is used in Lacan and how

this usage contributes to the more philosophical, rather than psychoanalyt-

ical dimensions of his work.

In addition, since (in contrast to his scientific critics) it is not my aim to

attack or even criticize Lacan, the ethics of the discussion gives me a greater

Versions of the Irrational * 1 15

degree of freedom to bypass a full argument. I do believe that there is an eth-

ical asymmetry here. Criticism obligates us to a much greater rigor of analy-

sis and degree of support of our claims. On the other hand, it is not my aim

to defend Lacan either and it would be misleading to read this chapter as

such a defense; indeed the present argument may even establish proper

grounds for a criticism of Lacan, which may even be more severe than that

found in his Science Wars critics but which may also be meaningful.5 I am

not a follower of Lacan and, while I do find his imagination and some of his

ideas appealing, I do not find all of his texts and arguments either especially

compelling or unproblematic. However, "as becomes a positive spirit," as

Nietzsche would say, I am far more interested in tracing the possibilities of

philosophical interconnections between philosophical or (again, a more

complex question) psychoanalytical concepts and mathematical and sci-

entific concepts-a much worthier subject in my view than the problems in

Lacan's arguments.6

I am also interested in the structure or architecture of philosophical con-

cepts as such; and Lacan's concepts will be considered here as philosophical

concepts. Such concepts often entail an engagement of diverse disciplines,

fields of inquiry, or human endeavors-such as mathematics and science, on

the one hand, and literature and art, on the other. The term "concept" is,

again, used here in Deleuze and Guattari's sense in What Is Philosophy?, as

explained in chapter 1, as a (organized) multicomponent conglomerate of

concepts (in their more conventional sense of generalization from particu-

lars), particular elements, figures, metaphors, and so forth. The psychoana-

lytical dimensions of Lacan's concepts are, again, a separate matter. I will

not be able to consider them in detail here, although they are of course cru-

cial to Lacan's work and are indissociable from the overall architecture of

his concepts. Indeed, the psychoanalytical dimensions of his work are pri-

marily responsible for the particular role of the concept of the erectile organ

in the essay in question (or elsewhere in Lacan).7 Accordingly, I need some

of Lacan's psychoanalysis. My argument, however, concerns primarily the

philosophical component of Lacan's discourse, and the role of mathematics

there will be considered accordingly. I shall return to the question of the

relationships among mathematics, philosophy, and psychoanalysis at the

end of this chapter.

It may be recalled here, by way of justifying this approach, that Lacan's

essay in question was a contribution to a philosophical conference entitled

"La Dialectique." Its first reference is Hegel and his Phenomenology of

Spirit. Hegel is one of the key figures for and (if implicitly) subjects of the

essay, indelibly inscribed in the phrase "the dialectic of desire" of its title. It

116 * The Knowable and the Unknowable

can be shown that Lacan's diagram of signifiers, representing the dialectic of

desire, from which Lacan's "formula" in question is derived, is in fact, or in

effect, a redrawing of Hegel's famous sketch of "Geist as Organism." The

sketch was made as Hegel was thinking through the structure of his dialec-

tic-Geist's architecture in motion, as it were-in Phenomenology. Lacan

(in part via Alexandre Kojeve's reading) rereads this dialectic by using his

"imaginary numbers"-that is, his signifiers-in particular the erectile

organ ("the square root of -1," (L)> -1 of Lacan's system), as the dialectic

of desire. This reading, he would be likely to argue, is in fact more faithful

to the spirit of Hegel or to the letter of Hegel's spirit, just as he argued that

his reading of Sigmund Freud is more faithful to the letter or letter/spirit of

Freud. (He uses his famous pun on "letter" as, among other things, an item

of "postal correspondence" and a "minimal" alphabetical signifier.) It

would hardly be possible to dissociate the dialectic of unconscious desire

from Hegel's dialectic of Geist as self-consciousness at any point. Lacan may

be seen and saw himself as interested in the "topology" of their intercon-

nections. He may also be seen and would have seen himself as trying to

"cut" a more conventional picture of these two dialectical movements as

two sides of Hegel's vision and then reglue them into a kind of Moebius

strip or, given the twisted structure of each movement, into Klein's bottle

(two Moebius strips glued together along their edges). I hope I will be for-

given for this (I would think mild) abuse of topology, especially since I am

merely trying to follow Lacan, and perhaps Hegel. Indeed, these topological

pictures or metaphors, while not without their uses, are rather impoverished

diagrams of Hegel's dialectic. Perhaps all mathematical objects ultimately

are, although one could, if one wants to go quite a bit further in choosing

more appropriate topological objects, say, more complex sheaves or fiber

bundles (objects, I simplify, that are locally straight, Euclidean, but globally

twisted) of which the Moebius strip is one of the simplest examples.8 Simi-

larly, however (and with the same implications), Lacan's topological dia-

grams are also (although not only) his redrawing and regluing of Kojeve's

well-known diagrams of Hegel's dialectic in terms of more complex and

entangled, knotlike, topological structures. These circumstances clearly

indicate the multilayered nature of Lacan's concepts as well as the non-

mathematical or quasi-mathematical character of his mathematics. Accord-

ingly-this is my main point at the moment the structure of philosophical

concepts in the above sense is, I would argue, where Lacan's usage of math-

ematics most fundamentally belongs and perhaps the best perspective from

which this usage can be meaningfully considered.9

Versions of the Irrational * 1 17

Complex Numbers and Nonclassical Epistemology

From this perspective, there is a way, at least one way, to argue that Lacan's

statement in question and the connections (rather than an identification or

even a metaphor) between the erectile organ and the square root of -1,

(L)/ -1, make sense. Ironically, in order to pursue this argument one has

indeed to know something not only about Lacan but also about imaginary

and complex numbers and their history. On that score, at least judging by

their comments on the subject in their book, Sokal and Bricmont appear to

be rather less informed than they should have been and, even more ironi-

cally, in some respects perhaps less informed than was Lacan. They appear

to be taking complex numbers for granted as self-evident mathematical

objects, in the way some physicists indeed treat them in their work, which is

fine for all practical purposes. The situation, however, is more complicated

mathematically and, especially, philosophically and epistemologically,

which is particularly relevant to my argument.

Accordingly, it may be useful to review some basic facts concerning

imaginary and complex numbers as well as numbers in general.10 Given

their crucial role in defining first (real) irrational numbers and then complex

numbers, square roots will be my primary focus here. Let us recall, first, that

the square root (/) is the mathematical operation reversing the square of a

number. The square of 2 is 4, the square root of 4, /4, is 2, or of course -2,

a fact that will be significant here. I hope I will be forgiven for being so ele-

mentary, but I want even those who know nothing, or forgot everything,

about mathematics-unlike Plato we do admit them into the academy these

days-to understand my argument. Besides, things quickly get rather more

complicated. Thus, X/2 is already a far more complex matter than /4, both

mathematically and philosophically, although it is of a rather straightfor-

ward mathematical genealogy. One needs it if one wants to know the length

of the diagonal of a square. This is how the Greeks discovered it, or rather

its geometrical equivalent. If the length of the side is 1, the length of the

diagonal is X/2. I would not be able to say-nobody would-what its exact

numerical value is. It does not have an exact numerical value in the way

rational numbers do: that is, it cannot be exactly represented (only approx-

imated) by a finite, or an infinite periodical, decimal fraction and, accord-

ingly, by a regular fraction-by a ratio of two whole numbers. It is what is

called an "irrational number," and it was the first, or one of the first, of such

numbers-or (they would not see it as a number) mathematical objects-

discovered by the Greeks, specifically by the Pythagoreans. The discovery is

118 * The Knowable and the Unknowable

sometimes attributed to Plato's friend and pupil Theaetetus, although ear-

lier figures are also mentioned. It was an extraordinary and, at the time,

shocking discovery-both a great glory and a great problem, almost a scan-

dal, of Greek mathematics. The diagonal and the side of a square were

mathematically proven to be mathematically incommensurable, their

"ratio" irrational. The very term "irrational"-both alogon (outside logos)

and arreton (incomprehensible) were used-was at the time of its discovery

also used in its direct sense. The discovery, made by the Pythagoreans

against themselves, may be seen as the "Godel's theorem" of antiquity.11 It

undermined the Pythagorean belief that, as everything rational, the har-

mony of the cosmos was expressible in terms of (whole) numbers and their

commensurable ratios (proportions), or fractions, as we came to call them.

This discovery was also in part responsible for a crucial shift from arith-

metical to geometrical thinking in mathematics and philosophy. Eventually

(it took a while) "algebra," a marvelous invention of Arab medieval mathe-

matics (although some of the key ideas are found in earlier Greek figures,

such as Diophantus), entered the scene and in large measure defined mathe-

matical (and perhaps more than merely mathematical) modernity. For,

while the diagonal of the square was self-evidently within the limits of geo-

metrical representation, it was outside those of arithmetical representation,

as the Greeks conceived of it. On the other hand, following Deleuze and

Guattari, one might also see such problems as paths to solutions, that is, as

leading to new, more effective theories. Some Greek mathematicians did so

in this case. So did Einstein in confronting the "problem" of the speed of

light (as independent of the motion of the source); or Heisenberg and Bohr

the "problems" posed by quantum physics; or Godel the "problem" of

undecidability and incompleteness of mathematics. The results were

momentous. As will be seen, the situation is reversed in the case of imagi-

nary and complex numbers: the problem is their geometrical representation,

seemingly natural (especially by now) and mathematically most effective but

epistemologically complicated. For the moment, to cite Maurice Blanchot:

The Greek experience, as we reconstitute it, accords special value to the

"limit" and reemphasizes the long-recognized scandalousness of the irra-

tional: the indecency of that which, in measurement, is immeasurable. (He

who first discovered the incommensurability of the diagonal of the square

perished; he drowned in a shipwreck, for he had met with a strange and

utterly foreign death, in the nonplace bounded by absent frontiers.)12

Blanchot's philosophical point can be linked to Lacan's investigations of

Greek tragedy, which may allow one to consider the question of Lacan and

Versions of the Irrational * 1 19

mathematics, or of Greek thought to begin with, from yet another perspec-

tive-that of the relationships between tragedy and mathematics, and

between poetical order and mathematical or physical (natural) order. The

subject would require a separate treatment. The source of the legend men-

tioned by Blanchot appears to be the first scholium of Book X of Euclid's

Elements, which discusses the Pythagoreans and the discovery of the incom-

mensurables and irrationals and further deepens the meaning and the alle-

gory of the legend itself. According to Thomas Heath's commentary on

Euclid's Elements.

The scholium quotes . . . the legend according to which "the first of the

Pythagoreans [sometimes identified as Hipassus] who made public the inves-

tigation of these matters [of the irrationals and the incommensurables] per-

ished in a shipwreck," conjecturing that the authors of this story "perhaps

spoke allegorically, hinting that everything irrational and formless is prop-

erly concealed, and, if any soul should rashly invade this region of life and lay

it open, it would be carried away into the sea of becoming and be over-

whelmed by its unresting currents." There would be a reason also for keep-

ing the discovery of irrationals secret for the time in the fact that it rendered

unstable so much of the groundwork of geometry as the Pythagoreans had

based it upon the imperfect theory of proportions which applied only to

[rational] numbers. . . . [T]he discovery of incommensurability must have

necessitated a great recasting of the whole fabric of elementary geometry,

pending the discovery of the general theory of proportion applicable to

incommensurable as well as to commensurable magnitudes. (The Thirteen

Book of Euclid's Elements 3:1)

Assuming that the story is indeed allegorical, the allegory in question

may be seen as or as intimating an allegory of allegory itself, insofar as the

latter is seen, along the lines of de Man's understanding of allegory as

defined by the relation to the irreducibly unrepresentable. De Man takes this

understanding to the nonclassical limit considered in this study.13 The ques-

tions addressed in the passage are far from irrelevant to the current (and of

course long-standing) debates concerning rationality, objectivity, truth, and

so forth.

To begin with, irrationality-the inaccessibility to rational representa-

tion (in whatever sense)-can itself be discovered rationally, for example

and in particular, by means of mathematical proof, a paradigmatic rational

argument. This emergence of the irrational (the inaccessible, the unknow-

able, the unrepresentable, the incomprehensible, the inconceivable, and so

forth) at the limit of the rational (the accessible, the knowable, the repre-

sentable, the comprehensible, the conceivable, and so forth) defines the

120 * The Knowable and the Unknowable

project of philosophy throughout its history, from Anaximander to Heideg-

ger and beyond, or in mathematics from the Pythagoreans and the diagonal

to Godel and undecidability.14 In the wake of Heidegger, or indeed Niet-

zsche, who understood this epistemology more profoundly than anyone

before him (and at least as profoundly as anyone since him), the extraordi-

nary critical potential of this situation has been powerfully utilized by such

nonclassical thinkers as Bataille, Blanchot, Levinas, Lacan, Derrida, and de

Man, or of course Heisenberg and Bohr in the case of quantum mechanics.

Indeed, the nonclassical epistemology of quantum measurement, as consid-

ered here, gave especially remarkable shape to these relationships. The (non-

classical) nature of the unknowable we now have to confront may be more

complex and radical than that confronted by the Greeks. The nature of the

debates and, to some degree, the philosophical, or even mathematical and

scientific, concepts involved is not unsimilar. To "be carried away into the

sea of becoming [and of the formless] and be overwhelmed by its unresting

current" still appears to some as (and perhaps indeed ultimately is) a great

threat to "rationality." It is difficult, however, to keep our "irrationalities"

secret, be they those of philosophy or of mathematics and science. Concepts

such as (in)commensurability, (ir)rationality, and (im)measurability, and

their history, ancient or modern (say post-Kantian or post-Cartesian), are

crucial to the idea and history of complex numbers or other areas of math-

ematics invoked by Lacan, in particular foundations of mathematics and

topology. They shaped both the philosophical thought and mathematical

work of most major mathematicians themselves involved in this history.

Lacan and other radical thinkers discussed in recent debates appear to be

more aware of and attentive both to these concepts and their history them-

selves and to the philosophical thought (often in turn quite radical) of the

key mathematical and scientific figures involved than are their recent critics

in the scientific community.

We now call fractions and whole numbers rational numbers. Rational

numbers together with (real) irrational numbers (such as roots of all powers

and still other irrational numbers, such as t, which cannot be represented as

roots or even as solutions of polynomial equations) are called real numbers.

I say "real irrational" numbers because all imaginary and complex numbers

are, by definition, irrational, since, not being real numbers, they cannot be

represented as a ratio of two whole numbers, which is always a real num-

ber. (This fact is sometimes forgotten, including by Sokal and Bricmont,

but, it appears, not by Lacan, whom they criticize for confusing irrational

and complex numbers.) Real numbers can be either positive or negative or

can be zero (the latter, yet another invention of Arab mathematicians, was

Versions of the Irrational * 121

unknown to the Greeks, as were negative numbers).15s The main reason for

using this term is that real numbers are suitable for measurements, in par-

ticular of the length of line segments, straight or curved, in the material

world around us-the world of things that are, or appear to be, real. It is

true that all actual measurements are approximations in terms of rational

numbers. Indeed, irrational real numbers can be rigorously defined in terms

of infinite sequences of rational numbers. We can also represent and visual-

ize them as points on the continuum of a straight line.16 We can do all stan-

dard arithmetic with real numbers and generate new real numbers in the

process-add them, subtract them, multiply them, divide them (except by

0), and so forth. The same is true for rational numbers but (because of divi-

sion) not for whole numbers.

Now "there's the rub"-the square root. If a number is positive, there is

no problem. We can always mathematically define its square root and cal-

culate it to any degree of approximation. However (this is the rub), in the

domain of real numbers the square root can be defined, can be given unam-

biguous mathematical meaning, only for positive numbers. This is so for a

very simple reason (recall that the square root is the reversal of the square):

whether you square a positive or a negative number-that is, multiply any

number by itself-the result is always positive. Thus, 2 times 2 is 4, and -2

times -2 is also 4, and the same is true for 1 and -1-the square of both is

1. In a sense, square roots of negative numbers, such as -4 or -1, do not

exist, at least in the way real numbers exist or appear to exist. This is why,

when introduced, they were called imaginary, and sometimes even impossi-

ble, numbers.

Why worry, then? First, from early on it appeared (correctly) that one

could operate with square roots of negative numbers as with any other num-

bers-add them, subtract them, multiply them, divide them, and so forth.

Moreover, the "impossible" square roots of -1 and of other negative num-

bers appear most naturally as solutions of rather simple algebraic equations,

such as x2 + 1 = 0. This is how J - 1 and other "imaginary quantities" (as

they were called at the time) made their first appearance during the Renais-

sance. Roots of negative numbers naturally emerge throughout mathemat-

ics. In short, on the one hand, mathematics at a certain point appeared to

need to be able to deal with square roots of negative numbers, beginning

with -1. On the other hand, it was clear that such "numbers" could not be

any numbers already available.

It took the mathematical community a while (nearly two centuries) to

accept the mathematical legitimacy, let alone reality (the mathematical real-

ity), of these new entities and rigorously to define them as numbers. Their

122 * The Knowable and the Unknowable

status as mathematical objects has remained in question for much longer,

especially in philosophical terms of their mathematical reality, or as con-

cerns their possible role in describing material reality (for example, in

physics, where they eventually came to play a major role). The resolution

required a great and protracted effort and the best mathematical minds

available. It was achieved by a seemingly simple, especially from our van-

tage point, but in truth, at least at the time, highly nontrivial stratagem-by

formally adjoining J -1 to real numbers. This "simple" resolution amounted

to the introduction of new numbers and of a new kind of number, which

could be manipulated in the manner of all other numbers. These numbers,

eventually called complex numbers, include square roots of negative num-

bers (for which the term "imaginary numbers" was retained) but are in gen-

eral slightly more complicated mathematical entities, although they have

been known for just as long as strictly imaginary numbers. Complex num-

bers in general are written in the form a + bi, where a and b are real num-

bers. In the case of imaginary numbers, a = 0. The number J - 1, called i,

remains the simplest complex number, and it may be seen as the fundamen-

tal element, which, once it is formally adjoined to the old domain of real

numbers, enables one to introduce the new domain of complex numbers.

Just as real (or rational) numbers do, complex numbers form what is in

mathematics called a "field"-a multiplicity with whose elements one can

perform standard arithmetical operations with the outcome being again an

element of the same multiplicity (except dividing by zero). The immediate

mathematical gains are immense, insofar, for example, as one no longer

needs to worry about how to properly define any roots or, more generally,

any solution of polynomial equations. For, unlike the case of real numbers,

they now all belong to the same field. (In mathematical terms, the field of

complex numbers is algebraically closed.) To formally adjoin J - 1 to real

numbers is all one needs in order to have not only x2 + 1 = 0 but every poly-

nomial equation axn + bxn-1 + . . . = 0 be in principle solvable and have

exactly as many solutions as the (highest) degree of the polynomial. The lat-

ter statement constitutes the so-called main theorem of algebra, one of the

most beautiful and important theorems in mathematics.

With the introduction of complex, rather than only imaginary, numbers

it also became possible to represent the whole system on the regular real (in

the mathematical sense) two-dimensional plane. In this representation, the

line representing real numbers serves, symbolically, as the horizontal axis,

while the line representing imaginary numbers (square roots of negative

numbers) serves as the vertical axis in the Cartesian-like mapping of the

plane. The number ~/ -1, or i, would be plotted at the length equal to 1

Versions of the Irrational * 123

above zero on the vertical axis. In this representation the domain of com-

plex numbers is topologically two-dimensional, in contrast to the one-

dimensional domain of real numbers as represented by the line. This repre-

sentation is sometimes called the Argand plane, or the Gauss-Argand plane,

for it was the great (one of the greatest ever) German mathematician Karl

Friedrich Gauss who used it in his efforts to give legitimacy and perhaps

reality to complex numbers. "You made possible the impossible" was a

phrase (which also refers to complex numbers) used in a congratulatory

address on the fifty-year jubilee of his doctorate. In 1977 the German Post

Office issued a stamp illustrating the Gauss-Argand plane to celebrate the

bicentennial of his birth in 1777 (obviously a very lucky sequence of num-

bers). Gauss was also one of the discoverers of non-Euclidean geometry, a

discovery he, however, suppressed for twenty years for fear of being laughed

at by philistines or perhaps for the same reason that made the Pythagoreans

to think it wise to conceal the existence of the irrationals. Gauss did not

think, as it happened rightly, that the human world was quite ready to con-

front the non-Euclidean world. As will be seen, this and related geometrical

and topological aspects of Gauss's work are extraordinary and are highly

relevant to the present study as a whole.

The picture is, however, not without complications, of which, as will be

seen, Gauss might have been more aware than it might appear and which

are crucial for the present argument. Accordingly, I shall discuss these com-

plications at some length, even though I can still offer only a sketch rather

than a full argument. Lacan appears to have seen the epistemology of com-

plex numbers along these lines (even if in a more general and vague way)

when he introduced a certain, a kind of, "square root of -1," or in my nota-

tion here (L) J -1, into his argument concerning the dialectic of desire. I

would like to suggest a reasonable argument explaining that such a view is

at least possible and even reasonable, given the epistemological aspects of,

and questions concerning, complex numbers.

In particular, complex numbers, such as - 1, and their operations, may

be said to be geometrically "represented" and "visualized" via the two-

dimensional (real) plane only as a kind of diagram (the Gauss-Argand

plane)-a schematic illustration, comprehensive (point by point) as it is-

but not in themselves. That is, the Gauss-Argand plane diagram may not be

seen as geometrically fully representing complex numbers and, hence, as

sufficient to visualize them as mathematical objects with their actual indi-

vidual and collective (these are, of course, co-defining) mathematical prop-

erties. Far more diverse and complex geometrical (and topological) means

appear to be necessary to approach complex numbers geometrically; and it

124 * The Knowable and the Unknowable

is conceivable that, in themselves, they may ultimately not be geometrically

or topologically, or (if we refer by "topology" strictly to the mathematical

discipline under this name) more accurately spatially, or otherwise, visualiz-

able, if indeed it is possible to visualize otherwise than spatially. It may be

observed that, while all visualization may need to be seen as in one way or

another spatial, not all mathematically topological configurations, for

example, not even all three- or even two-dimensional configurations, are

visualizable due to the complexity of their spatial structure. In the case of

complex numbers and certain other "geometrical" objects, both spatial and

geometrical visualizations (the conjunction of both defines Euclid's geome-

try) may be in question. "Ultimately" is, again, an important qualifier here,

indicating that at least certain forms of geometrical representation and visu-

alization of some of their aspects are possible and are even necessary. A

much more complex interplay of algebra and geometry, along with analysis

and topology-of the representable and the unrepresentable, of the visualiz-

able and the nonvisualizable, and so forth-than the Gauss-Argand plane is

capable of giving us appears to be necessary. A bit more mathematics and

mathematical epistemology is required to explain this point.

First, the real two-dimensional plane is-this is a mathematical fact-

mathematically not the same object as complex numbers. In the language of

mathematics, these two objects are not isomorphic, insofar as one can

assign, and one can, any algebraic structure to the real plane. The main rea-

son for this is that a real (in the mathematical sense) point on the two-

dimensional plane with Cartesian (now truly Cartesian rather than Carte-

sian-like) coordinates, such as that with coordinates (0,1), diagrammatically

corresponding to i ( - 1) on the Argand plane, is not a number. Alge-

braically, a point on the real plane can only be seen as a vector, usually "pic-

tured" as an arrow, in this case of a definite length (which is significant),

extending from the zero point to a given coordinate point. This structure,

however, may be fully formalized in terms of algebra, which does not

depend on this representation, and generalized accordingly. The two-dimen-

sional real plane itself is seen as having or is given the structure of the so-

called vector space over real numbers, rather than being a field, as are com-

plex numbers or real (or rational) numbers. In the case of vector spaces, for

their elements-vectors-addition (or subtraction) is defined (and it can be

represented in terms of "arrows"), as well as multiplication (or division) by

numbers over which a given vector space is defined, here that of real num-

bers. (Vector spaces can be defined over the field of complex numbers as

well, as are, for example, the infinite-dimensional Hilbert spaces used in

quantum physics.) New vectors are produced by means of such operations,

Versions of the Irrational * 125

and new points on the two-dimensional plane can be located accordingly.

There is, however, no multiplication or division defined for the elements

(vectors) themselves and, accordingly, for the points on the two-dimen-

sional real plane.17 This absence defines the difference between the algebra

of vector spaces, such as that of the real plane naturally endowed with a vec-

tor-space algebraic structure, and the algebra of fields, such as that of com-

plex or real (or rational) numbers. In contrast to the field of complex num-

bers, the field of real numbers is represented by the (real) line as a field and

that of rational numbers, again as a field, by a subset of rational points on

the real line. (As will be seen, epistemologically the situation may be more

complicated even in these cases.) In any event, in the case of fields both mul-

tiplication and division between their elements are defined as well (except,

again, the division by zero), and in the former case, again, they are not. We

can multiply or divide "points" (considered as vectors) on the two-dimen-

sional plane by real numbers, but not among themselves. Thus the point

(1,0) can be multiplied by any given number, say, a, giving us the point

(a,0), but it cannot be multiplied or divided by another point of the plane,

say, the point (0,1). This can only be done for the elements, such as 1 and i,

of the field of complex numbers, which are diagrammatically represented by

such coordinate points on the Argand plane but which are fundamentally

different mathematical objects. These objects are the elements of an alge-

braic structure that is not isomorphic to the two-dimensional real plane,

considered as a vector space, and thus is really not captured by the Argand

plane. Also, importantly, unlike real numbers or real vectors, complex num-

bers as such cannot be assigned lengths and, hence, cannot be used in mea-

surements. In this sense, they are indeed deeply "irrational," insofar as they

cannot be represented as fractions or as anything that can be measured or,

again, approximated by measurement. The length of the vector correspond-

ing to a given complex number (z = x + yi) is equal to the so-called modulus

/z/ of this number defined (by virtue of the Pythagorean theorem) as Iz/ =

x + y2. Geometrically, the modulus is the distance from the point 0 to the

point corresponding to z on the Argand plane, and hence it can be mea-

sured. The modulus is also equal to the product (x + yi) x (x - yi) of a given

complex number by its so-called complex conjugate, a product that is,

accordingly, always a real positive number as well, which is, as we have

seen, significant for quantum mechanics.18

It follows that the complex plane in fact carries two different, noniso-

morphic, algebraic structures: (a) that (an algebraically richer one) of a com-

plex algebraic field of complex numbers; and (b) that of a two-dimensional

real vector space (not a field) over real numbers. The latter is, by definition,

126 * The Knowable and the Unknowable

isomorphic to the real two-dimensional plane, which is given an algebraic

structure as a real object. This vector-space structure is, I argue, available to

our spatial intuition-it is visualizable, indeed visualizable as the two-

dimensional plane. It is the two-dimensional (real) plane. The only way,

however, to give the real plane the structure of an algebraic field is to endow

it with the structure of the field of complex numbers, which would no longer

allow us to see it (in either sense) as a real plane. Indeed, the language of this

last sentence would have to be amended accordingly so as not to speak of

giving the real plane the structure of the algebraic field. This simply cannot

be done on the real plane but only on the complex "line," as it were, since

the complex dimension of this object would, properly mathematically, be

equal to one rather than to two. This structure, I argue, may ultimately not

be visualizable. Visualizing or giving geometrical representation or meaning

even to some of its aspects appears to require much more complex means

than the Gauss-Argand plane. Indeed, the heterogeneity of these means may

exceed the capacity of any single geometrical object, let alone a diagram,

such as the Gauss-Argand plane, which, I argue, may be only, spatially and

geometrically, seen as a real plane.

I here, again, use the term "visualizable" in a sense the way it was used

by Bohr (or Heisenberg) in quantum mechanics, similar to the German

anschaulich, as that which is more or less available to our immediate geo-

metrical/topological intuition or as picturable, as "thinkable" in representa-

tions, images, pictures, and so forth. As will be seen presently, the same term

(Anschaulichkeit) was used by Gauss in the context of complex numbers

and what we would now call topology along proximate epistemological

lines. This double nature or the double structure of the topological space of

complex numbers is, I would argue, the source of the great-I would call it

non-Euclidean in the most general sense of the term-complexity of mathe-

matics of, or involving, complex numbers. One may define the Euclidean

(geometrical/topological) objects in terms of the possibility of visualization

(in the present sense) of such objects themselves and of their topological and

geometrical properties. In terms of the discussion in chapter 2, such objects

and their mathematical treatment provide a model, which may be called

Euclidean mathematics and which is, in this sense, analogous to classical

physics and its model, and indeed from Galileo on served as a source of such

models (with qualifications considered earlier).

From this perspective, the (Euclidean) geometry of the two-dimensional

(flat) Euclidean plane may be the only rigorous realization of this model

(even the three-dimensional Euclidean geometry possesses significant fur-

ther complications, as the relevant parts of Euclid's Elements indicate). Cer-

Versions of the Irrational * 127

tainly, the geometrical and topological properties of non-Euclidean (now in

its conventional sense) geometries pose major difficulties in this respect.

Thus, it does not appear to be possible to construct a global Euclidean rep-

resentation of the Gauss-B61yai-Lobachevsky non-Euclidean (two-dimen-

sional) geometry of negative curvature as an actual real surface in three-

dimensional Euclidean space, although it does have realizations in terms of

non-Euclidean objects and specifically as complex (complex-number-related)

objects.19 The double nature of the complex plane appears to force one to

combine different properties of complex numbers and of their arithmetic,

algebra, analysis, topology, and geometry, since these differently involve,

and are differently involved in, constituting the real and complex properties

of complex numbers. At the same time, it does not appear to be possible, in

practice and, conceivably, in principle, to unify these fields in doing this

mathematics, nor perhaps even to bring together various properties of com-

plex numbers within a single rigorously established conceptual field. We

appear to be dealing with a mathematics that is both heterogeneous and

interactive-both interactively heterogeneous and heterogeneously interac-

tive-but ultimately not unifiable either in practice (as many would admit) or

(an idea that few are willing to entertain) in principle. We can see this het-

erogeneous-interactive mathematics at work, often brilliantly used (by

manipulating various aspects of complex numbers and functions and various

means of analyzing them), beginning with Gauss, Augustin-Louis Cauchy,

F. G. M. Eisenstein, Leopold Kronecker, and, especially, Riemann and

extending, first, to their immediate followers and, then, to twentieth- and by

now virtually twenty-first-century mathematics. In the process diverse and

complex forms of geometrical and topological representations, in particular

the usage of so-called Riemannian surfaces, appear to be necessary. Such rep-

resentations may be more direct but partial or more complete but diagram-

matic (as the Gauss-Argand plane). No single such representation, however,

appears to suffice, thus, at the very least, posing radical questions as to

whether complex numbers can find any comprehensive geometrical repre-

sentation or, hence, are ultimately visualizable at all. Here, accordingly, we

may be dealing with the ultimate suspension of all geometrization/spatializa-

tion ("topologization"), rather than as in the case of certain real two- or

three-dimensional geometrical configurations that are unavailable to visual-

ization merely due to their intricacy, as indicated earlier.

These considerations also pose more general questions concerning the

usage of geometrical or, again, more generally spatial concepts, if not terms,

except metaphorically or by analogy, or allegorically in de Man's sense,

while in fact referring to the objects that can only be algebraically defined or

128 * The Knowable and the Unknowable

indeed rigorously conceived.20 These questions may even apply to the very

idea of geometry, as conventionally understood, or indeed space, beyond

three-dimensional, if not two-dimensional, real spaces, which are properly

the domain of Euclidean mathematics. Infinite dimensional spaces of modern

mathematics, such as Hilbert spaces, used in quantum mechanics, would be

an obvious case here. There are also far more esoteric mathematical objects

available as examples. The point, however, appears to apply even in the case

of far less esoteric structures, such as complex numbers or Hamilton's

quaternions (the first four-dimensional objects of mathematics). Indeed, it

may be argued that Rene Descartes's analytic geometry already poses certain

questions of that type and, by so doing, brings mathematics to the threshold

of the non-Euclidean domain, which is hardly surprising given other (some-

times radical) departures-mathematical, methodological, philosophical,

and conceptual-from Euclid in Descartes's work. Descartes's comments on

complex numbers in his Geometry, cited as an epigraph to this chapter, may

be seen as a further (and perhaps not coincidental) intimation of this non-

Euclidean future of mathematics and specifically of the relationships between

algebra and geometry there.21

It is, of course, true that the spatial-geometrical metaphors and the very

metaphors of space and geometry are pervasive in mathematics or physics,

from quaternions to Hilbert spaces to still more abstract topological spaces.

There are even spaces without points altogether, as in the so-called topos

theory introduced by Alexandre Grothendieck, arguably the furthest exten-

sion of the concept of topology (in the sense of "structure" rather than "dis-

cipline"), which also has important connections to post-Godelian mathe-

matical logic, and, as I have indicated in the preceeding chapter, more

recently (via mathematical logic) to both quantum mechanics and quantum

gravity. (Grothendieck's topology is sometimes slyly referred to by mathe-

maticians as "pointless topology.") Indeed, they are so pervasive that they

no longer appear to function as metaphors but rather almost as a form of

topological, even if not properly spatial, intuition. I would argue, however,

that in all these cases, and even in the case of complex numbers, perhaps at

the limit all numbers, even natural numbers, we are in fact dealing with

"algebra," even if in the name of geometry. That is, in fact we ultimately use

the equivalents or analogues of the algebra associated with, and not of the

geometry of, the two- or three-dimensional spaces of Euclidean mathemat-

ics. This algebra does of course have a certain specificity that arises by virtue

of this association with space and geometry in the latter case and that,

accordingly, affects (rigorously) the "spatial" structures in question, such as

quaternions or vector spaces (of whatever dimensions), and (conceptually-

Versions of the Irrational * 129

metaphorically) our way of talking about them. The space-times of Ein-

steinian relativity would be an especially interesting case here as well. These

spaces are non-Euclidean, even in the case of special relativity, and, hence,

globally pose certain problems of visualization even when we consider their

low-dimensional models. Even the four-dimensional Euclidean space may,

however, be considered from this perspective.22 When we spatially visualize

such objects, we visualize only three- (and perhaps mostly two-) dimen-

sional configurations and then (or in the process) supplement them by alge-

braic considerations or algebraic intuitions.23 The term "algebraic geome-

try"-which names the field that is, arguably, the ultimate extension of the

"geometrization" of algebra and which began with Descartes's analytic

geometry and developed from the late eighteenth century to the present, in

particular in the works of Andre Weil and Grothendieck and their follow-

ers-may be seen as a vast ambient metaphor in this sense.24

These points may be more immediately apparent and more often sug-

gested in the case of the arcane objects just mentioned, perhaps especially

infinite-dimensional spaces. This type of argument was made in this context

by Robert Langlands, who refers to quantum physics as a parallel and as a

case where, as we have seen throughout this study, one has confronted this

situation all along. The mathematics of the infinite-dimensional vector

spaces, or that of complex numbers (over which the vector spaces of quan-

tum mechanics are usually defined), is a routine part of the quantum-

mechanical mathematical formalism and (with qualifications given earlier)

is correlative to the nonclassical nature of at least certain interpretations of

quantum mechanics.25 At this limit we may indeed be more immediately

dealing with things that are inconceivable in any terms and by any means,

beyond irreducibly tentative and ultimately inadequate metaphors or alle-

gories (in de Man's sense), rather than only with that which is spatially

unintuitable or unvisualizable in the above sense. I would argue, however,

that the argument would apply everywhere beyond three dimensions and in

many cases at lower dimensions. In other words, as I have indicated, our

visual/spatial representation or intuition, Anschaulichkeit, does not extend

further than three dimensions, and in this sense modern mathematics is

indeed non-Euclidean even in the most immediate sense. If geometry qua

geometry is indeed the science of space, as David Hilbert said, insofar as we

can conceive of space as its object, this science as such can perhaps only deal

with real spaces (of dimensions below or equal to three) and even, in all

rigor, only with Euclidean spaces. These spaces sometimes allow us to real-

ize partial or local models, more or less rigorous or more or less loose, of

various non-Euclidean objects (either in the narrow sense of non-Euclidean

130 * The Knowable and the Unknowable

geometry or in the broader sense of the present discussion).26 But of course,

as I argue here, the history of mathematics and science has taught us that we

need not be able even to conceive of, let alone spatially or pictorially visual-

ize or otherwise intuit, the objects of mathematics and science in order to do

both. Quantum mechanics and infinite-dimensional spaces, and even more

so their conjunctions, may, again, be especially radical and remarkable

examples, but, with respect to the rigorous conceivability of its objects, the

nonclassical epistemological component may be irreducible in all mathe-

matics. The properties of these objects, such as complex numbers, or of the

relationships between them, may need to be seen in terms of certain effects

of such objects, as defined in the preceding chapter, rather than in terms of

the properties of these objects themselves.

In a certain sense, we may (may!) here face a situation that is epistemo-

logically even more radical than in quantum physics. In the latter case we

can at least ascertain the material character of the efficacity of these effects

in nature, locating this efficacity beyond the furthest classically drawn "cut"

possible with the help of measuring instruments in any given case. By so

doing we associate it with the ultimate quantum constitution of nature, or,

again, what is thus idealized (as quantum or, again, even as "constitution,"

"nature," "matter," "ultimate," or whatever). In other words, we can place

the (unknowable) efficacity of relevant (knowable) effects within the same

idealized configuration of measurement and thus link the whole idealization

to the material nature, even while maintaining the nonclassical character of

this idealization. (The latter qualification is here used both in the sense that

it can be idealized nonclassically and in the sense that, even though non-

classical, it is still an idealization.) In the case of mathematical objects in

question, it does not appear possible to establish any "material" or even

"virtual" (Platonist) location or perhaps even existence for them as objects

in the same way. Such objects may quite simply not exist anywhere, even

though we can effectively provisionally (one is tempted to say rigorously

provisionally or provisionally rigorously) use their definitions and (strictly)

rigorously use the effectlike properties in question in practice.27 Hence, the

efficacity of these effects, as mathematical effects, may be seen as epistemo-

logically more radical or at least more complex than the case of quantum

mechanics. The emphasized qualification is important, since we can think in

terms of, say, material biology of the brain as a sufficiently ultimate efficac-

ity here, say, for the purposes of scientific investigation of the human mind,

and even conjecture quantum aspects to this efficacity, as Roger Penrose

does (evidence is, again, so far slim at best, even at the conjectural level).28

Quantum or not, however, this level of investigation is not of much help

Versions of the Irrational * 131

here. While the subject of lively investigations and debate, it is at the very

early stages even at the most rudimentary levels.29 The possibility of locat-

ing "mathematical effects" of human thought (or whatever their efficacity)

is almost infinitely far away, if possible at all, even by the most optimistic

views. Even this, however, may not get us to mathematical "objects" them-

selves, say, complex numbers, or for that matter any numbers, let alone

infinite-dimensional Hilbert spaces, behind these "effects," if such objects

exist at all in any conceivable sense, in other words if they might be thought

of classically rather than nonclassically. But then, as I said, it is also possi-

ble that the ultimate constitution of nature may be accommodated classi-

cally, once, as I said, more chips are in, say, when we have some form of

quantum theory of gravity.30

It follows from the preceding discussion, however, that a similar "non-

material" efficacity (it could still depend on the materiality of our body, the

brain in particular, in the same way) is ineluctably involved in quantum-

mechanical phenomenology as well. The impossibility of applying any con-

cepts of object, or of quantum, to "quantum" "objects," and, besides, con-

sidering even this view as an idealization of "nature" (we may now

appreciate the epistemological significance of this fact even further), brings

both situations even closer, but it does not quite erase the differences

between both types of "objects," or "un-objects" beyond these objects. It

also follows, however, that in quantum mechanics, at least as complemen-

tarity, one can approach its "inconceivable" objects only by means of math-

ematical "objects" of the same, if not more radically, "inconceivable"

nature, perhaps, again, altogether nonexistent as (mathematical) objects

anywhere, while allowing us to use their mathematical properties. The two

irreducible inconceivable entities in, respectively, nature and mind can be

linked and work, with the help of experimental technology. This is one of

the most radical implications, or indeed applications, of nonclassical episte-

mology, which, however, also links differently nature, mind, and technol-

ogy. (For the sake of graphic comfort, I skip quotations marks, which can

be put just about everywhere in these sentences, and in most epistemologi-

cal elaborations in this chapter, or this book.)

One, however, finds parallels to this situation in mathematics itself, when

one needs to rely on properties of one mathematical object, or one type of

object, in order to understand the properties of another object or another

type of object. I now mean this understanding in the sense of rigorous con-

nections (similar to those between mathematical formalism and quantum

objects in quantum mechanics) rather than in terms of analogies or

metaphors, as discussed earlier. This may become possible if both objects,

132 * The Knowable and the Unknowable

or both types of objects, can be linked in a particular way, as happens, for

example, in "representation theory." I refer by this term to a particular area

of modern mathematics, in relation to which Langlands makes his com-

ments, cited earlier, even though he may not see the situation in epistemo-

logically quite so radical terms, either at the analogical-metaphorical or the

rigorously mathematical level. Representation theory plays a crucial role in

quantum theory and elsewhere in modern mathematics and physics, includ-

ing (although this became apparent somewhat in retrospect) in nineteenth-

century mathematics and its extensions, which is to say most of contempo-

rary mathematics. The analysis of the workings of representation theory

(specifically the so-called group theory and group representation, closely

tied to symmetry considerations) in quantum mechanics, or indeed else-

where, from the epistemological viewpoint is a barely, if at all, touched sub-

ject and is well beyond my scope here. The epistemological situation of rep-

resenting the inconceivable by means of something

that is itself

inconceivable, just outlined, also finds its persistent parallels in nonclassical

epistemology outside the sciences in the work of most of the key figures con-

sidered here, including, as will be seen, Lacan.

In any event, what I call here "non-Euclidean mathematics" operates in

this nonclassical epistemological field (whether or not the practitioners of

this mathematics subscribe to this view), or, again, at least, just as in the

case of quantum mechanics, this type of interpretation of this mathematics

is possible, even if not ultimately inevitable. In some mysterious (inconceiv-

able) way this mathematics depends on and benefits from so doing in view

of the richness of the effects in question and their interconnections, even

though the efficacity of these effects is beyond any conceivable conception.

But then, such is the nature of nonclassical epistemology. It works in this

mysterious way. But, luckily for us, it works, and often where classical the-

ories cannot help us.

In his discussion of Riemann's work in the context of "turning points in

the conception of mathematics" and commenting on the rise and develop-

ment of non-Euclidean geometry, sometimes seen as one of the greatest rev-

olutionary steps in mathematics since the Greeks, Detlef Laugwitz in his

Bernhard Riemann: Turning Points in the Conception of Mathematics par-

allels it with the rise and emergence of the complex numbers. The latter his-

tory is usually seen as more or less evolutionary, or, in Kuhn's terms, as

"normal science." Laugwitz argues that (in contrast to Riemann's concept

of space, a real "turning point in the conception of mathematics") both

were in fact examples of "normal science": "Where is the revolution in this

[the development of non-Euclidean geometry]? Consider an analogous state

Versions of the Irrational * 133

of affairs, the story of the complex numbers. For centuries people assumed

that they could add to the real numbers a 'number' [i] such that i2= -1, and

they thus obtained many results without running into contradictions [a cru-

cial issue in legitimating the non-Euclidean geometries]. In the first half of

the 19th century people exhibit models of the complex numbers [just as

those of non-Euclidean geometries]. This story is qualitatively just the same

as the story of non-Euclidean geometry, yet no one speaks of revolution!"31

While there are some reasons for this conclusion (especially as far as the last

century, or perhaps century and a half, is concerned), this is not altogether

true, certainly not as far as the history of the debate about complex numbers

is concerned. But, in any event, perhaps, at least epistemologically, one

should speak of a revolution in both cases, even though both involved a fair

amount of "normal science," which, however, is usually the case in all rev-

olutions. Indeed, as I argue, both may be seen as part of the same non-

Euclidean revolution, as the concept is defined here. It is worth citing a com-

ment by Misha Gromov, a leading contemporary mathematician, who made

seminal contributions to several key related fields, most especially differen-

tial topology and differential geometry:

Imaginary numbers appear in algebra when we try to take square roots of

negative numbers. . . . Geometric interpretation consists in observing that

two consecutive rotations of the plane by 90 degrees around a fixed point

reverse the directions of the vectors. If we think of the 180-degree rotation

reversing vectors as the geometric counterpart of multiplication of numbers

by -1 reversing the sign, then we are inclined to accept the 90-degree rotation

(of the plane containing the line of real numbers) as the square root of -1. All

this looks childishly simple, why do mathematicians make such a fuss around

it? How can one dare to compare this plain idea to profound philosophical

pronouncements, such as "Cogito ergo sum" of Descartes? But look (as my

colleague David Ruelle once suggested) from another perspective. "Cogito

ergo sum" stayed unperturbed for more than three centuries, like a monu-

ment, a Greek statue, a magnificent piece of art, impervious to the flow of

time, whilst the little speck of dust, the square root of -1, has been growing

and developing for hundreds of years in the minds of mathematicians,

geniuses like Cauchy, Gauss and Riemann, and turned into an evergreen

intensely alive vibrant tree supporting in its branches our sacred knowl-

edge-quantum mechanics-ruling everything we see (and do not see) in this

world.32

Beautiful as this remark is and as pertinent as it is on complex numbers (as

concerns them, it is in accord with the present argument), it is quite mis-

leading as concerns Descartes and "Cogito ergo sum" (assuming that this

statement as such occurs in Descartes, which is at best uncertain). Indeed,

134 * The Knowable and the Unknowable

one may make virtually the same observation concerning it as that Gromov

makes of complex numbers. One can see it (I paraphrase Gromov slightly)

as growing and developing for hundreds of years in the minds of philoso-

phers, such as Kant, Hegel, Husserl, Heidegger, Levinas, and Derrida, and

sometimes physicists and mathematicians, geniuses like Cauchy, Gauss, and

Riemann, among them, and turning into an evergreen intensely alive vibrant

tree supporting in its branches our sacred knowledge, philosophical and

sometimes mathematical and physical, such as quantum mechanics, ruling

everything we see (and do not see) in this world. This is not to deny either

the disciplinary specificity of both conceptions or other differences between

them, for example, insofar as concerns the critique of Descartes's Cogito,

from Kant on (specifically in the figures listed above), which, however,

would only reinforce the present point. (It is not impossible to speak of a cri-

tique of the concept of complex numbers either, including as concerns the

possibility of their geometrical representation.) I want instead to stress par-

allels and sometimes reciprocity between both concepts in the history in

question here and in the thought of scientific figures just mentioned, begin-

ning with the work of Descartes himself (who, as will be seen presently, was

the first to use the term imaginary numbers) and extending to, among oth-

ers, Gauss, Riemann, and Bohr. It would be, I think, quite naive to think

that both concepts could be unequivocally separated in Descartes's thought.

Gromov is right to give complex numbers the place they deserve in intellec-

tual history, but it may not hurt, at this point, to give Descartes's Cogito the

place it deserves, too, including next to complex numbers.

Laugwitz may well be right in arguing that Riemann's broader conceptual

contribution was more profound and fundamental, beginning with conceiv-

ing of mathematics in terms of concepts and his more radical approach to the

concept of space. Riemann here follows both his teacher Gauss (more math-

ematically) and Gottfried Wilhelm Leibniz (more philosophically). As Laug-

witz says: "[The non-Euclidean geometry prior to Riemann] remained

entirely within the framework of Euclidean construction methods, whereas

Riemann left this area and constructed his spaces in a completely different

way."33 I think this is, by and large, correct. I would replace entirely with

often, especially given Gauss's work. I would further argue that Riemann's

and some earlier work, especially again that of Gauss, was so radically revo-

lutionary in this field in part by virtue of their engagement with, or indeed

creation of, non-Euclidean mathematics in the broader sense suggested here.

Their work on complex numbers was among their greatest achievements,

technical and especially conceptual. It was crucial, even uniquely crucial, to

their far-reaching contributions, including in bringing mathematics and, to

Versions of the Irrational * 135

some degree, even physics (to which they made major contributions as well,

and not only of a mathematical nature) to, at least, the threshold of nonclas-

sical thought.

It is of some interest that, in considering the significance of Riemann's

contribution here, Laugwitz refers to a crucial and extraordinary passage by

Gauss on prototopological considerations involved in Gauss's view of pre-

cisely the complex "plane." Laugwitz recognizes this, without, however,

addressing the question of the complex plane itself from this perspective,

which appears to be on Gauss's mind. Riemann, however, appears to have

taken this passage as his point of departure in his topological considerations

of, in Laugwitz's words, "a continuum without a geometric structure, one

of which it was not initially possible to speak of, say, straight lines, distances

between points, and angles." In other words, in our terms, this is a purely

topological structure into which geometrical structure could, or perhaps

could not, be introduced. In principle, in accordance with the considerations

just given, the nature or even existence of such a continuum as a mathemat-

ical object would still remain a question. Even leaving these deeper consid-

erations aside, the geometrical representation as such of complex numbers

was in question, even as late as 1849, fifty years after Gauss's dissertation,

where the considerations of the complex plane were involved in Gauss's

proof of the theorem for which he was awarded his degree. Gauss says:

"The wording of the proof [of his theorem] is taken from the geometry of

position, for in this way it gains the greatest intuitive representability

[Anschaulichkeit] and simplicity. Strictly speaking [in Grunde], the proper

[eigentliche] content of the whole argumentation belongs to a higher, space-

independent [von Riumlichem unabhbindingen] domain of the general

abstract study of magnitudes [Grossenlehre] that investigates combinations

of magnitudes held together by continuity. At present, this domain is poorly

developed, and one cannot move in it without the use of language borrowed

from spatial pictures [Bilder]."34 I modify Laugwitz's or rather Abe Shen-

itzer's (the translator of Laugwitz's book) translation, as the passage obvi-

ously poses considerable philosophical and interpretive difficulties, which

complicates the translation as well. Gauss here introduces, in part following

Leibniz, but more deliberately and decisively, and, one might say, more

properly mathematically, topology into mathematics as the field dealing

with continuous transformations of (mathematical) spaces. Leibniz's

"analysis situs" is still Gauss's term, as it is also that of Riemann and even

Poincare, who may be seen as a founder of topology as a mathematical dis-

cipline similar to algebra and analysis or, of course, arithmetic and geome-

try, earlier. The term "topology" was introduced around Riemann's time,

136 * The Knowable and the Unknowable

however. It may be argued that already the Greeks had a kind of topology

as an area of philosophical, rather than mathematical, investigation, which

was continued in modern philosophy as well and would "invade" mathe-

matics here and there. Indeed, philosophically and epistemologically the

Greeks went rather far in their topology, for example, in Plato's discussion

of what he calls chora, a kind of prespace or protospace, in Timaeus, on

which I shall further comment in the next chapter. Topology (or what one

might so call) was their philosophy of spatiality, sometimes reaching beyond

the spatial. However, for the Greeks, as for the moderns, geometry was the

only mathematics of space. Leibniz was arguably the first to envision a pos-

sibility of topology as mathematics and to see (correctly) the essential

significance of topological (continuity) considerations, however implicit, for

calculus. With Gauss (one can invoke several others among his contempo-

raries) the more systematical mathematical approach to topological consid-

erations enters the history. Riemann takes it further, and for both topologi-

cal thinking is linked to both non-Euclidean geometry and complex

numbers, including, I would argue, specifically along the lines of non-

Euclidean mathematics. The main point at the moment is that spatial-topo-

logical and epistemological considerations concerning complex numbers,

possible difficulties of their geometrical representation on or by a two-

dimensional plane, the Gauss-Argand plane, and so forth are clearly sug-

gested by Gauss and developed by Riemann.

From the perspective of the statement just cited, Gauss's view of geomet-

rical representation or, possibly, unrepresentability of complex numbers by

the Gauss-Argand plane may need to be reconsidered. For this statement

appears to indicate that for Gauss, too, the complex plane may have been

ultimately a diagram used for the sake of possible simplicity and intuitabil-

ity rather than a rigorous geometrical representation. That is, such would be

the case if we can call such a diagrammatic picture the Gauss-Argand plane,

which, at least for Gauss, may have meant a rather different topological/

algebraic/geometrical object, using the term "geometry" here with the

above analysis in mind. The same type of argument may be made concern-

ing Riemann's view of both complex numbers and spatiality of space. There

still remains the question of how far both Gauss's and Riemann's views

reach epistemologically along the lines indicated earlier, that is, as concerns

the possibility of the existence of a kind of topological object indicated here

by Gauss. I cannot address this question here. As the preceding considera-

tions would indicate, however, there may be some surprises in store as con-

cerns their view or that of their contemporaries. Some similarities with

Bohr's views are suggested by Gauss's passage, although one must, again, be

Versions of the Irrational * 137

extremely cautious in sorting out proximities and differences here. Laugwitz

comments on Gauss's passage and Riemann's usage of it as follows: "This

can certainly be regarded as an allusion to manifolds [in the mathematical

sense of the term] coordinized by continua of n-tuple of numbers, as well as

the use of geometric language in a nongeometrical context. Gauss consid-

ered the points of the plane given by real coordinates t, u and introduced an

'algebraic structure,' that of [the field of] complex numbers. Riemann was

to introduce real n-tuples and to a 'metric structure.' "35 This assessment

would, I think, need to be further nuanced and radicalized in accordance

with the preceding analysis. It is clear, however, that Gauss, who anticipates

and prepares Riemann's more strictly geometrical/topological ideas as well,

in fact or in effect introduces a greater than commonly acknowledged com-

plexity of geometrical/topological representation, or unrepresentability, of

complex numbers along with their so-called geometrical representation as

the Gauss-Argand plane. In other words, the emergence of modern geome-

try and topology and the emergence of complex numbers are parallel, as

well as conjoined, mathematically and epistemologically.

Now, in the case of the complex numbers it is indeed the particular alge-

braic structure of the field that poses the main problem, in spite of the pos-

sibility of "representing" it as the Gauss-Argand plane. It is, then, in the

sense and for the reasons just delineated that I see the latter as a "diagram"

rather than as a geometrical object, representing complex numbers in the

way the (real) line represents the real numbers, or perhaps appears to repre-

sent them. Ultimately, these considerations may pose more general and still

further-reaching epistemological questions concerning all numbers. Thus, as

I suggest here, there is no natural (if any) way to geometrically conceive of

necessary arithmetical operations, especially multiplication or division

(multiplication and addition pose somewhat less of a problem here), for

complex numbers as such. There are of course perfectly well-defined

("real," Euclidean) procedures for locating geometrically the outcome of

algebraic operations on complex numbers. Thus, the diagonal of a parallel-

ogram gives the result of the addition (say, 1 + i will be represented by a

point (1,1) on the plane); or using a pair of similar triangles enables one to

locate the multiple of two complex numbers. This is what the Argand plane

enables us to do. These, however-this is my point here-remain diagra-

matic Euclidean "pictures," representing only Euclidean properties of the

plane. The Argand plane diagram as such gives us only this representation.

It does not give us access to algebraic (or analytic) properties of complex

numbers as such, even though it diagramatically represents the one-to-one

(topological) correspondence between the points of the real plane and com-

138 * The Knowable and the Unknowable

plex numbers, and the arithmetical operations performed upon complex

numbers.

It would appear that a strict (rather than diagrammatic) representation

would be naturally possible in the case of the real line and real numbers.

First, one is no longer dealing with two qualitatively different structures

described earlier (i.e., with the complex "plane" as both the field of complex

numbers and the real vector space). Second, all arithmetic of real numbers

(i.e., a field rather than a vector space it possesses as a substructure) appears

to be not only naturally geometrically represented but also, with respect to

all operations, visualizable as the real one-dimensional line. While, how-

ever, the first point, my main point here, will be retained in all circum-

stances, the situation may be more complex even in the case of real num-

bers. In particular, it appears that multiplication (in contrast to addition), as

an operation, may pose certain questions concerning "geometrizability"

even in the case of the real numbers, even though multiplication appears to

be naturally defined there. It was defined by Euclid as the "abbreviation of

addition," ultimately too cryptically and not altogether rigorously in

Definitions 15 and 16 of Book VII of The Elements. The same may be said

as concerns his Proposition 16 there (proving the commutativity of multi-

plication) and the way Euclid handles multiplication geometrically.36 Obvi-

ously, it is not a question of "blaming" Euclid here but only of indicating

that much greater epistemological complexities than we are accustomed to

think of are involved even in the case of the most familiar mathematical

objects. It is also worth recalling the complexity of the Euclidean geometri-

cal construction of arithmetical operations with the rationals and the irra-

tionals on the plane and, of course, that in this case we are dealing with

intervals and their length, not with "real numbers." In other words, there

may be questions, ultimately along the more general lines indicated earlier,

as to whether the (real) line really allows us to geometrically "represent"

real numbers in all their properties and in all of their aspects, and in partic-

ular the operations performed upon them. Adding two intervals (whatever

their lengths) may be conceivable. But what does it mean to multiply an

interval (say of length 2) by an interval (say, again, of length 3)? We know

of course that the result is an interval of the length 6, and we know that it is

the same as to add the first interval to itself three times, or that this product

can be represented as the volume of a plane figure. We can also construct

analogous procedures for other numbers (using rigorous approximations in

the case of the fractions and, then, the irrationals, for example). But would

these really be a geometrization of multiplication on a real line, anymore

(one is almost tempted to add) than in the case of complex numbers and the

Versions of the Irrational * 139

Gauss-Argand plane? Multiplying by negative numbers would introduce yet

further complexities into the geometrization of real numbers and further

parallels with the case of complex numbers. In other words, the line may be

geometrically capturing only the additive but not the multiplicative proper-

ties of the real numbers. Hence, it captures only their vector space but not

their field structure, similarly to the way the real plane captures the vector

space but not the field structure of the complex numbers. In the latter case,

it does so, however, by virtue of the fact that as a vector space the "com-

plex" plane is isomorphic to the real plane, but as a field it is not, which is,

of course, my main point here.

Thus, ultimately, numbers-their algebra or (interactively) their arith-

metic-may be even less geometrically conceivable than we are accustomed

to think even in the simplest cases. This is in part why algebra is such a great

invention. It allows us to rigorously handle and indeed to create the nonvi-

sualizable, the geometrically unthinkable. It may, accordingly, have been

Gauss's algebra (his achievements in this field were phenomenal) and topol-

ogy or prototopology, not the Gauss-Argand plane, that enabled him to

make possible the impossible (at least before Godel came along).

There are, thus, differences in degree (in particular as concerns visualiza-

tion) within the general epistemological problematics of geometrical repre-

sentation or/as visualization. First of all, these differences are responsible for

the difference in attitudes toward the imaginary and complex (as opposed to

the real) numbers, even on the part of mathematicians, especially those

involved in the earlier history of complex numbers. Certainly, the more rad-

ical epistemological complexity of complex numbers, or at least of their visu-

alization, as considered here, helps to explain the ambivalent attitude toward

them on the part of the key figures involved in their discovery or creation.37

Thus, Augustin-Louis Cauchy (1789-1857), a contemporary of Gauss and a

great mathematician in his own right, had reservations concerning the geo-

metrical representation of complex numbers throughout his life. He consid-

ered them as purely symbolic (algebraic) entities and at one point even

attempted a general mathematical definition of "symbolic expression" in

explaining his attitude-to some discontent among his colleagues.38 But, as

we have seen, even, or indeed in particular, Gauss's view of the situation has

a great complexity, as do those of other key nineteenth- and twentieth-cen-

tury (and quite a few earlier) figures involved in this history.

Indeed, as discussed earlier, sometimes quite radical, epistemological

complexities are involved in the case of real numbers, or indeed of all num-

bers. Lacan aside, the question may well be, Do we know what is J -1, or

indeed -1 (which, as I said, gave mathematicians some pause even as late as

140 * The Knowable and the Unknowable

the eighteenth century), or for that matter 1, which may take us all the way

to Parmenides and his undifferentiable "One," perhaps also a reply to

Pythagoreans, as the opening of this question? Gottlob Frege once said that

it is scandalous that we do not know what numbers really are.39 The spatial

representation/visualization of the real numbers as the (real) line may indeed

ultimately be impossible as well, beyond its diagrammatic potential. Either

way, it would not change the point here defined by the double algebraic

structure of the complex numbers, one (a vector-space) isomorphic to the

real plane and the other (a field) not. As a result the representation/visual-

ization of complex numbers as the Gauss-Argand plane may indeed have to

be seen only as the "real" diagram in the above sense, and in this sense as a

visual "fiction," however convenient and perhaps indispensable. It may not

be seen as the geometrical or even topological, spatial representation or

visualization of the complex numbers. The latter, this is my main point here,

appears to be rigorously impossible. Complex numbers as such cannot in all

rigor be seen as represented as points on the two-dimensional real plane and

indeed are epistemologically unavailable as visualizable or, more generally,

geometrical objects. Their properties can be spelled out and rigorously com-

prehended algebraically. In the sense of algebraic representation there is no

epistemological difference between real and complex numbers, although

there are fundamental differences in the algebraic properties of the two

domains. Thus, ultimately, complex numbers may remain not only imagi-

nary but, at least geometrically, strictly unimaginable. They (in the ultimate

structure of their properties and attributes) are certainly not visualizable as

such. Epistemologically, at least in terms of its geometrical representability,

the square root of-1 or, more accurately, the signifier "-1" signals-"rep-

resents"-the ultimate lack of geometrical representation. It is something

that in itself is geometrically unvisualizable or unrepresentable, or, one

might say, geometrically unepistemologizable. It is fundamentally, irre-

ducibly irrational, even though it could be considered rationally.

At the very least (whatever one's view of the epistemology of complex

numbers or their geometrical representation), if Lacan saw complex num-

bers in, epistemologically, this type of way (as he appears to have), he had

reasons to do so. He might not have been aware of or contemplated some of

the nuances and qualifications considered here. As concerns the question of

complex numbers (rather than Lacan) these nuances and qualifications are

significant and must be at least indicated so as to present the actual situation

in a more rigorous way, which is what I have tried to do here. They would,

however, leave plenty of space for making Lacan's looser view of complex

numbers at the very least understandable.

Versions of the Irrational * 141

Lacan's "Numbers" and Nonclassical Epistemology

On the other hand, whatever Lacan's view of complex numbers or mathe-

matics in general, it could appear "wondrous strange" indeed, and to some

outright bizarre, that the theory of complex numbers has anything to do

with the erectile organ. However, given the argument just sketched and

some knowledge of Lacan, it is not so difficult to see that Lacan's "formula"

is in fact not so strange. One must keep in mind the much looser (than in

mathematics) structure of Lacan's system, which, accordingly, cannot be the

subject of the type of argument concerning mathematical complex numbers

given earlier. Reciprocally, however, there is a suspension of certain nuances

of Lacan's system in the discussion to follow, and, sometimes, the arcane

nature of some of its specific features, used here, may, as I have stressed

from the outset, pose difficulties in turn. In this sense, at certain junctures at

least, some knowledge of Lacan might be helpful if one is to appreciate his

overall argument as such. As I also stressed from the outset, however, my

argument here is of a different nature. It concerns the character of Lacan's

analogy, the structure of his concept, and his view of complex numbers

(shaping this analogy), not the (more) arcane specifics of Lacan's concepts

and argument; and, hence, it will remain sufficiently tolerant of the difficul-

ties just indicated.

The epistemological point just made concerning complex numbers-their

ultimate unavailability to visualization and perhaps any geometrical con-

ceptualization, while they seem to be represented as points on the two-

dimensional real plane-gives one a hint. "The erectile organ" may be seen

to be theorized by Lacan as a symbolic object (also in Lacan's sense of the

Symbolic), specifically a signifier (in Lacan's sense), which is epistemologi-

cally analogous to the signifiers one encounters in the case of complex num-

bers, and specifically J -1. Within the Lacanian psychoanalytic configura-

tion, any image, in particular visual image, of the erectile organ, including

that of an "erectile organ," can only be an image of the signifier-the

signifier, not the signified. (I shall further comment on this point presently.)

This signifier itself is fundamentally and irreducibly nonvisualizable. At the

limit, this signifier-that is, the ultimate structure of the signifier designated

as the erectile organ-may be inconceivable by any means. This epistemol-

ogy or de-epistemization, and specifically devisualization-accompanied by

a certain algebraization-is crucial to most of Lacan's key concepts. Indeed,

this signifier is in fact or in effect unnamable, for example, again, as the erec-

tile organ, or the phallus, which, as I said, may not be the same as the erec-

tile organ within the Lacanian economy of subjectivity and desire. That is,

142 * The Knowable and the Unknowable

we can formally, "algebraically" (and allegorically in de Man's sense),

manipulate its image or images, or names, or further formal symbols asso-

ciated with it, just as we can formally manipulate complex numbers within

their mathematical system, which Lacan's "algebra" in part "mimics" (or

emulates) but to which it is not identical. At the same time, however, we do

not really know and perhaps cannot in principle conceive of what the erec-

tile organ really is as a signifier and what its properties are, if one can speak

in terms of properties here. (At least, we cannot do so from within the

Lacanian psychoanalytical situation, defined by this functioning of inacces-

sible signifiers within it.) The image of this signifier, and in particular its

visual image, would, then, be analogous to the geometrical, hence visualiz-

able, representation of complex numbers, and in particular of J -1. The

erectile organ is an analogon of the latter within the Lacanian psychoana-

lytic "system," rather than J -1 as a mathematical imaginary number as

such.40 The situation may even be subtler, insofar as one may need to deal

with further levels of formalization at which the analogy in question actu-

ally emerges. Let me stress that (whether one is within Ferdinand de Saus-

sure's or Lacan's scheme of signification) we are dealing here with the irre-

ducible inconceivability of the erectile organ as the signifier, not the signified

(in Saussure, roughly, the conceptualization "mediating" the relationships

between the signifier and the referent within signification). Its signified and

its referent, whatever they may be, may be in a certain sense even more

"remote" and "inaccessible" or inconceivable. One would still need, how-

ever, to think here in terms of the ultimate inaccessible nature (which is not

to say identity in terms of their functioning) of all three-the signifier, the

signified, and the referent, the latter of which should be considered in the

register of the Lacanian Real. That is, along the lines considered earlier, the

irreducibly inaccessible or inconceivable is represented in terms of some-

thing that is itself irreducibly inaccessible or inconceivable, if one still wants

to think of the latter as an object in any way. It would, again, be more rig-

orous to think in terms of representative (phenomenal) effects or properties

that are associated more directly (although still ultimately as effects) with

such an object rather than to attribute them to this object as such. It follows

that one can further split this situation into a potentially infinite chain, sim-

ilarly to and radicalizing both C. S. Peirce's and Louis Hjelmslef's (glosse-

matic) views of signification, both of which are, along with Saussure's semi-

otics, among Lacan's sources.41

It may be further argued that the (nonclassical) register of the Lacanian

Real is epistemologically analogous to quantum mechanics as complemen-

tarity, specifically with respect to the materiality of the efficacity of the

Versions of the Irrational * 143

effects of the Real and the irreducible and irreducibly nonclassical random-

ness of these effects as considered earlier. This, of course, does not mean

that the Real is physically quantum but only that it is epistemologically

quantumlike in its effects and in its nonclassical material efficacity. It is ulti-

mately the Real that is responsible for the nonclassical features of Lacan's

epistemology. I shall not pursue this argument here, but one can virtually

directly transfer most epistemological aspects of the quantum-mechanical

situation (even the role of formalism) into the Lacanian matrix.42 It may,

however, be tempting to illustrate the quantum-mechanical situation in

quasi-psychoanalytical terms, even if in a more tentative and metaphorical,

rather than more rigorously conceptually and epistemologically, analogous

way. We may compare the strange behavior of quantum objects, or indeed

(since we cannot speak of the latter as such) of the strange effects of this

behavior on our classical world, to the inaccessible workings of the uncon-

scious, as conceived (perhaps more classically) by Freud or (more nonclassi-

cally) by Lacan and Derrida, or de Man, or earlier Nietzsche and Bataille.

We have no access to the unconscious itself. If we had, we might confront

something more similar to "madness" than "reason," using both terms with

utmost caution. Unlike Freud, Lacan does apply the psychoanalytic machin-

ery to psychosis as well, a related but separate issue, which cannot be con-

sidered here. De Man, too, tropes the effects of irony as a kind of mad ver-

tigo.43 But of course we can only understand or even relate, at least

analytically (in either sense), to madness, that is, again, to the effects of

madness, only in terms of and through reason. Or, we might say, that we

can do so only through the "observational instruments" of reason, just as

we can only relate to the (to us, seemingly, "mad") "quantum world"

through its effects upon our classical, reasonable instruments, even though

the ultimate constitution of the latter may be quantum as well. But then,

speaking metaphorically, the ultimate constitution of the mind may be

unconscious, which brings us back to the question, nonclassical and hence

ultimately unanswerable, of how order emerges out of what may at bottom

be irreducibly random or, again, how the ordered effects are constituted of

random ones. In a certain sense, from Socrates and before on, this question

defines the history of our inquiries into human nature, and by the twentieth

century, with quantum mechanics, into material nature as well. Lacan's

Real operates in this regime and is seen as the ultimate and ultimately mate-

rial (in the sense of the preceding chapter) efficacity of all the effects in ques-

tion in Lacanian epistemology and his psychoanalysis.

My main argument concerning Lacan's "numbers"-his "arithmetic"

and "algebra"-may, thus, be summarized as follows. Both the signifier of

144 * The Knowable and the Unknowable

the erectile organ, (L)> -1, in the Lacanian psychoanalytic field, and J -1, i,

in mathematics may be seen as fundamentally formal, symbolic, and ulti-

mately epistemologically nonclassical, de Man would say, allegorical, enti-

ties that enable an introduction of, and may be seen as structurally generat-

ing, two new symbolic systems-that of Lacanian psychoanalysis (his

(re)interpretation of Freud's Oedipal economy) and the field of complex

numbers in mathematics. In each case, the introduction of these new sym-

bols allows one to deal with problems that arise within previously estab-

lished situations but that cannot be solved by their means. The first situation

is a preanalytic situation, or a more naively (for example, by way of mis-

reading Freud, conceivably, to a degree, even by Freud himself) constructed

analytic situation in psychoanalysis, where one needed to, and in the previ-

ous regime could not, approach certain particular forms of anxiety. The sec-

ond is defined by the system of real numbers in mathematics, where one

needed to but could not rigorously define complex numbers in order, for

example, to solve certain polynomial equations. In both cases, the philo-

sophical-epistemological status of these new symbolic systems is complex.

In particular, in question are:

(a) the extent to which such systems represent or otherwise relate to,

respectively, psychological/psychoanalytic and mathematical reality

(with the question of material reality in the background in both

cases-the question of the Real in Lacan's case); and

(b) the extent to which the properties of such symbolic systems and of

their elements, such as what is designated as /- 1 in mathematics, or

the erectile organ in Lacanian analysis, can themselves be accessed or

conceived and specifically visualized by means of images, such as the

geometrical representation of complex numbers or the image we

form perceptibly or configure theoretically (and these are subtly

linked in turn) of the erectile organ in the Lacanian psychoanalytic

situation.44

In other words, while Lacan's statement or even usage of mathematical-

like language may or may not be seen as out of place (and while one might

want them tempered, if not avoided), his argument appears to indicate

significant and ultimately nonclassical epistemological complexities of psy-

choanalytic theory, for which the epistemology of complex numbers may

serve as a "model." At least, there is some space for and value in parallel, if

not for crossing of the terms of discussion. As I said, one may still wonder

why one might prefer to avoid this crossing in this context (the present

Versions of the Irrational * 145

author would) or find it incongruous or bizarre, or even troubling. The lat-

ter reaction, though, seems to me to be taken too far. It is not that big a deal

after all. As Lacan obviously did not have such misgivings, he offered us this

crossing as well, and if one is to address it, critically or not, one must try to

understand the nature of this crossing in Lacan.

As explained earlier, in mathematics these complexities, historically

reflected in the term "imaginary numbers," are not altogether resolved even

now, although since and following Gauss most mathematicians stopped

worrying about complex numbers philosophically. Indeed, as we have seen,

Gauss might have worried more than them and more than they suspected.

Leibniz may well have given the problem its most glamorous expression:

"Imaginary roots are a subtle and wonderful resort of the divine spirit, a

kind of hermaphrodite between existence and non-existence (inter Ens and

non Ens Amphibio)."45 Perhaps Descartes should be given the last word

here. For, among the great many other things he was and the many other

"firsts" to his credit, he was one of the first to give serious consideration to

imaginary roots and their nature and, indeed, was the first to use the very

term "imaginary."46 As the inventor of analytic geometry, which funda-

mentally relates geometrical and algebraic mathematical objects, he, as we

have seen, ventured into more complex relationships between algebra and

geometry than merely a representation of one through the other, and, as I

said, quite possibly both to his concept (also in Deleuze and Guattari's

sense) of cogito. In view of this complexity, Gauss's and Riemann's work

(on both geometry and complex numbers) may be seen as a continuation of,

rather than a departure from, or as much a continuation of as a departure

from, Descartes. His view of complex numbers reflects these complexities.

"One is quite unable," he said, "to visualize imaginary quantities"47

unless the last word is Lacan's, who said in "Desire and the Interpretation

of Desire in Hamlet": "the square root of -1 does not correspond to any-

thing that is subject to our intuition, anything real-in the mathematical

sense of the term-and yet it must be conserved, along with its full func-

tioning" (29). This may need to be more precisely stated, but it is in essence

right, and it is this statement that grounds and guides my analysis here. It

may be seen as an updated rendition of Leibniz's early assessment: "From

the irrationals are born the impossible or imaginary quantities whose nature

is very strange but whose usefulness is not to be despised," although numer-

ous subsequent statements by leading mathematicians might also be cited.48

In the same passage Lacan also speaks of imaginary numbers as "irra-

tional." The passage is cited both in Sokal's hoax article and in Impostures

intellectuelles as an example of Lacan's confusion of irrational and imagi-

146 * The Knowable and the Unknowable

nary numbers. Lacan's usage, however, does not appear to me, due to his

lack of understanding of the difference between real irrational numbers and

imaginary numbers, imputed to him by Sokal and Bricmont. Instead it may

be seen as a reflection of his sense of imaginary numbers as an extension of

the idea of irrational numbers-both in the general conceptual sense,

extending to its ancient mathematical and philosophical origins, as consid-

ered earlier, and in the sense of modern algebra. This view is correct, and it

displays, conceptually, a better sense of the situation on Lacan's part than

on that of Sokal and Bricmont. Their description of irrational and imagi-

nary numbers in their book is hardly edifying as concerns the substance and

the significance of the subject. It is also imprecise and misleading insofar as

it suggests that there is no connection between irrational and imaginary

numbers. Indeed, they claim even more strongly that they have nothing to

do with each other.49 This is simply wrong. The profound connections

between them define modern algebra. Certainly, complex numbers, begin-

ning with i, are irrational numbers as the latter are defined by Sokal and

Bricmont (as unrepresentable by a ratio of two whole numbers): no real

fraction can be found to represent them, since no real number of any kind

can represent them. Conceptually, J -1 is also as "irrational" as J -2 in the

sense that both require one to adjoin to the system square roots that do not

exist within it and for which the "rationality" (now in the sense of compre-

hensibility) of the original system provides. But, for the moment, this is sec-

ondary to the strictly mathematical irrationality of complex numbers. The

latter may seem a minor point, and, while a strange oversight, one can

hardly think that Sokal and Bricmont could be unaware of it in principle.

One might expect two theoretical physicists to be a bit more cautious, given

how these aspects of complex numbers define the way in which they func-

tion, even in strictly technical terms, in quantum mechanics, even if they are

not interested in the epistemological questions arising there. In general, I am

not holding Sokal and Bricmont responsible for their treatment of numbers

as such, inadequate and imprecise as it is, although their explications of

other mathematical and scientific ideas are often sloppy and not very help-

ful to their nonscientific readers and do much disservice to mathematics and

science. (I shall give further examples later.) They are physicists, not mathe-

maticians, historians, or philosophers of mathematics, and it is not their

responsibility to know (or to be precise about) mathematics and the philos-

ophy and history of mathematics. It is, however-and this is hardly a minor

point-their responsibility to know those aspects of all three that they con-

sider in Lacan. Or, assuming that they do, it is their responsibility to care-

Versions of the Irrational * 147

fully consider and appropriately explain these issues as part of their argu-

ment, especially if they want to criticize Lacan.

The erectile organ of the Lacanian, or, Lacan argues (in part "against"

Freud himself), already Freudian, system is, then, analogous to the mathe-

matical square root of -1-analogous, but not identical. As I said, the

Lacanian system obviously cannot be seen as having the same (mathemati-

cal) kind of rigor that the mathematical system of complex numbers does; it

is much "looser," although it is not without its own (form of) rigor. In this

sense, Lacan's "analogy" borrows to a much greater extent, if not alto-

gether, from what may be seen as a certain philosophical (conceptual and

epistemological) enclosure of mathematics rather than from mathematics in

its technical-disciplinary sense. (I am not saying of course that it reproduces

or absorbs this enclosure.) While this enclosure plays its role in and is, in a

certain sense, indissociable from mathematics as a discipline, one must be

careful analytically to differentiate them, especially if one wants to read

mathematics via any general philosophical epistemology (Lacanian or

other). Certainly, insofar as one can see Lacan as borrowing from the math-

ematics of complex numbers, much of the mathematical content of the orig-

inal system is lost in the process. These circumstances, I argue here, indeed

make it possible (and, some would argue, advisable, if not necessary) to

decouple some or even most (although perhaps not all) mathematics in its

disciplinary sense from Lacan's "mathematics." As I also argue here, how-

ever, the very same circumstances also make it impossible (certainly, inad-

visable) to decouple Lacan from his "mathematics," which is what his

recent scientific critics do.

Accordingly, the proper way of conceiving of the situation is as follows.

The erectile organ, or, again, a certain formalization of it, must be seen as

(as defined by and as defining) "the square root of-1," (L)E-1 of the Lacan-

ian system itself. It is an analogon of the mathematical concept of the math-

ematical J -1 within this system, rather than anything identical, directly

linked, or even metaphorized via the mathematical square root of -1.s50 In a

word, the erectile organ is the square root of -1, which I here designate as

the (L)> -1, of Lacan's system; the mathematical ' -1 is not the erectile

organ. There is no mathematics in the disciplinary sense in Lacan's analysis,

but rather only certain structural and epistemological analogies or homolo-

gies with the mathematics of complex numbers, most particularly the fol-

lowing. First is the structural analogy: the erectile organ, as a signifier, or

indeed the signifier (in Lacan's sense), belongs and gives rise to a psychoan-

alytical system different from the standard one or ones (based on misread-

148 * The Knowable and the Unknowable

ings of Freud, conceivably to a degree by Freud himself) and to a different

formalization-(an allegorical) "algebra"-of psychoanalysis, a formaliza-

tion that is more effective both conceptually and in terms of the ensuing prac-

tice. Second is the epistemological analogy: the erectile organ, as a signifier

or, again, the (Lacanian) signifier of this system, while and in a sense because

it governs the economy of the system, can only be approached by means of

tentative, oblique, and ultimately inadequate images, concepts, metaphors,

allegories, and so forth. It is ultimately inaccessible, along with its signified

and its referent. At the limit it is inaccessible even as that which is absolutely

inaccessible but definable in terms of independent properties and attributes.

In order to explain the reasons for seeing Lacan's statement in this way,

I shall indicate some steps and elements of Lacan's logic and "algebra"

(mimicking the algebra of complex numbers). I shall not spell out the struc-

ture of Lacan's key concepts-such as the subject, the signifier, desire, or

indeed the erectile organ-which would require a perusal of a much larger

textual field. This, as I said, may introduce certain difficulties into my argu-

ment, which I shall try to mitigate as much as possible; it may not be possi-

ble to avoid them within my limits here. In this sense, the "summary" to fol-

low may be best seen as a sketch of a possible argument of a Lacanian type

along the philosophical-epistemological lines considered here, and as cen-

tered accordingly, rather than as a rigorous reconstruction or even a recon-

structive summary of Lacan's argument. Most of Lacan's psychoanalytic

argument will be often circumvented here, although it is crucial for Lacan

and, arguably, the very reason for the epistemology or analogy in question.

Accordingly, Lacan's psychoanalytical concepts and formulations will be

used only in order to show that Lacan relates them to the epistemology in

question and will not be explicated or read as such. This, I realize, may be

equally (albeit for different reasons) frustrating both for those who know

Lacan and for those who do not. However, as I said, my argument essen-

tially concerns philosophical and epistemological aspects or even primarily

implications of, and reasons for, Lacan's argument, not its psychoanalytical

substance, rigor, or sustainability. To some degree, I shall read Lacan here

in the way Sokal and Bricmont could, and perhaps should, have tried before

deciding (preferably negatively) whether to engage publicly with Lacan's

arguments. My point in pursuing this analysis is to show that we are deal-

ing here with Lacan's own "algebra," analogized to mathematical complex

numbers, not with the mathematical complex numbers themselves. This

point can be maintained regardless of how one assesses Lacan's uses of

mathematics or his motivations, or his psychoanalytical argument, neither

of which it is, as I said, my aim to defend here.

Versions of the Irrational * 149

In Ecrits, Lacan defines a signifier in general as "that which represents the

subject for another signifier" (316). This "definition" places signification

and its epistemology (or of course phenomenology) in the psychoanalytic,

rather than (disciplinarily) linguistic, context, in particular that of intersub-

jective relations, which will be treated at best in a highly schematic fashion

here.51 It would indeed take a rather long elaboration, almost an exegesis,

to unravel this definition (or virtually any proposition to be cited later) and

to give it a proper sense, which, however, is not important for the present

purposes. What is important is that such a signifier may be conceived as

(represented by) the "1," (L)1, of the system, which Lacan is about to map

by (or relate to) his "algebra." This formalization would reflect a certain

configuration of psychoanalytic signification but not the one in which Lacan

is interested-that is, not without extending it first to the "negative" and

then the "imaginary numbers" of his algebra, in other words, by construct-

ing respectively the "-1," (L) -1, and "the square root of-1," (L)/-1, of his

system. Indeed, he does not appear to see any configuration short of this

extension as meaningfully psychoanalytic, as a possible subject of psycho-

analytic theory and practice. What is, again, important for me here is the

fact of Lacan's formalization itself. Accordingly, first, the signifier "S" is

introduced as the "signifier of a lack in the Other [Autre]." It is, in Lacan's

system, "S" that is "the signifier for which all the other signifiers represent

the subject." This signifier is argued to be "symbolized by the inherence of

(-1) in the whole set of signifiers" (316). This signifier is, thus, (symbolized

as) "the -1," (L) -1, of the psychoanalytical system in question in its "alge-

braic" representation (the "algebra" is, again, that of Lacan). This negation

is also Lacan's "translation" of the negation of Hegelian dialectic, which is,

thus, part of a Lacanian conceptual conglomerate and is placed alongside,

and in a reciprocal conceptual interaction with, his "algebra." I am men-

tioning this point in order to indicate that at stake here is indeed Lacan's

system and its "numbers" rather than mathematical complex numbers; and

Lacan's map or graph of the "economy" of desire offered in the essay clearly

shows this as well. "As such, S is inexpressible, but its operation is not inex-

pressible" (316). In other words, "S" is operationally formalizable (within

Lacan's system), and this formalization itself is expressible by means of a

certain "algebraic" system analogous to mathematical numbers and may

even have a certain visualizable or geometrical model (in the way real num-

bers do). We can, accordingly, "algebraically," or quasi-algebraically,

manipulate such symbols, even if their (signified) content (or, of course, ref-

erents) or, in some cases, even their ultimate structure and nature as

signifiers are inaccessible.52 That the scheme is designed to reflect certain

150 * The Knowable and the Unknowable

epistemological complexities begins to emerge already at this point, insofar

as "S" is inexpressible as such. However, the formal object corresponding to

"S" in Lacan's "algebra"-(L) -1-is analogous to the mathematical -1 and

to its more benign epistemology (it is not without some complications

either, especially given the historical perspective outlined earlier). It is not an

analogon of J -1, which is radically inaccessible (at least to visualization

and geometrical representation) even as a formal object.53 Then, how-

ever, another signifier, "s"-the symbolic square root of -1, (L)> -1-is

"derived." This derivation takes the form of a symbolic analogy with the

algebraic equation for J -1 in mathematics, x2 = -1, whose symbols and

relations are given the psychoanalytic content via the nature of S and s

(320). Ultimately "s" is defined as that which is radically inaccessible,

unthinkable, for the subject-both as such, similarly to "S," and, I would

argue, more radically, at the level of the corresponding element of the for-

mal "algebra" built by Lacan.

There are, thus, two interactive but distinct levels of the functioning and

epistemology of the signifier in Lacan. The first is the more general concep-

tual level (subject, the phallus, lack, and so forth), which is not quasi-math-

ematical. The second is the level of a certain "algebra," which is quasi-

mathematical and at which the analogy between Lacan's "algebra" and

mathematics in fact emerges. The signifier "s" is not immediately equated

with the erectile organ. First, Lacan only argues that a certain radically inac-

cessible signifier is inherent in the dialectic of the subject. Indeed, this

signifier is argued to be the signifier ultimately generating this system, or,

again, yet another formalization of it, as "the square root of -1," (L) J -1,

of this formalization, rather than the "1," (L)1, or "-1," (L) -1, (which is

"S") of the system. It is of course also responsible for the specific character

of this system, just as ' -1 in mathematics is responsible for the fundamen-

tal difference between real and complex numbers. According to Lacan,

"This [i.e., that which is designated or formalized as Lacan's square root of

-1] is what the subject lacks in order to think himself exhausted by his cog-

ito, namely that which is unthinkable for him [although, we might add, it

appears as representable to him]." Here I again suspend the psychoanalytic,

and much of the philosophical content, and cite the statement only to indi-

cate Lacan's epistemological concerns. "But," Lacan asks next, "where does

this being, who appears in some way defective in the sea of proper names,

originate?" (317).

In order to answer this question, Lacan maps the passage from, in his

terms, the Imaginary to the Symbolic order, especially as regards the phallic

imagery. First, thanks to Freud's "audacious step," the phallus is argued to

Versions of the Irrational * 151

acquire the privileged role in the overall economy of signification in question,

via the castration complex. Here one must keep in mind the difference

between Freud and Lacan insofar as the Lacanian economy of the signifier

(replacing Freud's "signified") is concerned, in particular as it relates to the

phallus and the difference between the phallus (as a Freudian signified) and

what Lacan designates as the erectile organ as a signifier. The latter, more-

over, may need to be seen as formalized yet further as "the square root of

-1," (L) ' -1, thus adding yet another "more distant" level of signification.54

Then, moving beyond, if not against, Freud, Lacan argues as follows:

The jouissance [associated with the infinitude involved in the castration com-

plex in Freud] ... brings with it the mark of prohibition, and, in order to con-

stitute that mark, involves a sacrifice: that which is made in one and the same

act with the choice of its symbol, the phallus.

This choice is allowed because the phallus, that is, the image of the penis,

is negativity in its place in the specular image. This is what predestines the

phallus to embody the jouissance in the dialectic of desire.

We must distinguish therefore between the principle of sacrifice, which is

symbolic, and the imaginary function that is devoted to that principle of

sacrifice, but which, at the same time, masks the fact that it gives it its instru-

ment.55ss

The psychoanalytic content of the statement, again, requires long expli-

cation, which may be unavailable to summary. It is clear, however, that it

establishes the centrality of a particular configuration involving the concept

and image of the phallus in the dialectic of desire. It follows, according to

Lacan, that it is the erectile organ-the image or, better, the signifier as an

un-image of the phallus, and thus un-image of the image of the penis (in the

Lacanian Symbolic order)-that is subject to the equation of signification at

issue.56 It is, then, as such and only as such that the erectile organ is the

square root of-1, that is, (L) J/-1-that is, as "the square root" within, and

of, the Lacanian system itself in the Symbolic order of its operation or,

again, more accurately, of a certain formalization of that system. "Thus the

erectile organ comes to symbolize [again, also in Lacan's sense of the sym-

bolic] the place of jouissance, not in itself, or even in the form of an image,

but as a part lacking in a desired image: that is why it is equivalent to the

square root of -1 of the signification produced earlier, of the jouissance that

it restores by the coefficient of its statement to the function of the lack of the

signifier -1" (320).7 As I mentioned, the latter formulation is a psychoana-

lytical-semiotic analogon of the algebraic equation for J -1 in mathematics

x2 = -1. The main point here is that "the signification of the phallus" so

conceived conforms to the economy of the inaccessible signifier. It can be

152 * The Knowable and the Unknowable

shown that neither the signified nor the referent is simply suspended here,

and is in fact conceived of as ultimately inaccessible as well, via the Lacan-

ian Real. For as Lacan says, "if [the erectile organ's role], therefore, is to

bind the prohibition of jouissance, it is nevertheless not [only] for these for-

mal reasons" (320). Instead it is due primarily to a complex materiality, ulti-

mately related to the Real and its epistemology. As I have indicated, the Real

in Lacan's sense may be seen as this (nonclassical) materiality rather than

(classical) reality, or indeed causality, given the irreducibly random nature

of its effects. In particular it may not be conceived of as anything that can be

seen as possessing any attributes (perhaps even the attribute of existence in

any way that is or will ever be available to us), which only emerge at the

level of certain psychoanalytic effects, handled by Lacan's Symbolic or mis-

handled by Lacan's Real. In other words, the register of the Real follows the

nonclassical phenomenology and epistemology of effects of the unknowable

material efficacity, which is the Real. In short, the Real is unknowable in the

nonclassical sense of this study.

Within the Lacanian psychoanalytical situation, the image or the signifier

of the erectile organ is a scandal-in either sense, but most crucially in terms

of its psychoanalytic management, or the difficulty or even impossibility

thereof. In this latter sense it is not unlike what ' -1 in mathematics was

epistemologically at some point, or perhaps as any number still is epistemo-

logically, as Frege said. Lacan's approach is to refigure it as a symbolic (and,

again, epistemologically allegorical) object-specifically in Lacan's sense of

the juxtaposition between the Symbolic and the Imaginary. In the register of

the Imaginary "the signification of the phallus," while conceivably involving

inaccessible signifieds and referents, may be seen as defined by accessible

signifiers, but (in part as a consequence) it is psychoanalytically useless. In

the new system (in the Symbolic register) the Lacanian signifiers themselves,

in particular the erectile organ, are ultimately inaccessible. By the same

token, a symbolic system (in Lacan's sense of the Symbolic) is introduced as

the dialectic of desire and castration, which enables the subject defined by

this system and/as the Lacanian analytical situation to function. The sym-

bolic object itself in question is given a specific formal structure, just as ' -1

is in mathematics, if, again, in Lacan a looser one. From this perspective, the

erectile organ is not a real unity or oneness, positive or negative, neither "1"

nor "-1," the (post-Parmenidean) One, positive or negative, nor even any-

thing merely fragmented, analogous to either the mathematical real rational

or real irrational. Instead it is a "solution" of the psychoanalytic equation

that contains oneness, "1," (L)1, and the negative of oneness, "-1," (L) -1,

as terms but that makes the "solution" itself, while in a certain sense for-

Versions of the Irrational * 153

malizable, inaccessible even at the level of the signifier. To the degree they

offer us any image of it all, all our imaginaries and visualizations ineluctably

"miss" this signifier, along with the signified and the referent-the Real,

keeping in mind the qualifications made earlier.

I am not certain to what degree Lacan's epistemological ideas were

derived from the epistemology of mathematical complex numbers, or other

areas of mathematics and science, in particular modern topology, functional

analysis, or quantum physics. It is not inconceivable, especially given his

statements cited here, that they did play a role, however. He also knew

enough mathematics and mathematicians to draw this parallel and to use it.

As I said, he appears to be aware of these epistemological connections, as

some of the statements cited earlier would indicate, even if he did not actu-

ally derive his scheme from the epistemology of complex numbers. There

are other candidates for the sources of this epistemology of the inaccessible

at all levels of signification-the signifier, the signified, the referent-in

more immediate semiotic terms, such as in the work of Saussure and Hjelm-

slef, on the one hand, and Peirce, on the other. In more philosophical terms,

one can see such sources in nonclassical philosophy in the wake of Kant and

Hegel, especially in Nietzsche, although one can, again, also trace some of

these ideas to Plato and the pre-Socratics.58

It is, then, only in the sense of the square root of -1, (L) ' -1 of Lacan's

system, as just delineated, and not of the mathematical system of complex

numbers, that the erectile organ is the square root of -1. This argument

would clearly invalidate the kind of critique that Sokal and Bricmont level

at Lacan, were their critique to survive far lesser levels of scrutiny. Unwit-

tingly, Sokal and Bricmont's own comment in fact says as much: "Even if

[Lacan's] 'algebra' had a meaning, the 'signifier,' 'signified' and 'statement'

that appears within it are obviously not [mathematical] numbers."59

Indeed! This is the point. It is clear even from the most cursory reading that

Lacan never says they are, although, as we have seen, mathematical num-

bers may function similarly to Lacan's in epistemological terms. In the sen-

tence introducing the formula in question and cited by Sokal and Bricmont,

Lacan says: "Thus by calculating that signification according to the alge-

braic method used here."60 That is, Lacan is calculating here according to

his "algebra," not the actual mathematics of complex numbers, which is my

point here. Indeed, Sokal and Bricmont cite other statements by Lacan

where he makes this point even more directly.61

In general, philosophical considerations of the type just outlined are the

main reason why modern mathematics and science enter the work of Lacan

and other so-called postmodernist figures (I use this highly misleading term

154 * The Knowable and the Unknowable

for the sake of convenience here), such as Deleuze, Jean-Frangois Lyotard,

or Derrida. There is nothing new here. The same has been the case through-

out the history of philosophy, beginning with Johannes Kepler, Galileo, and

Newton, or indeed with the mathematics of Plato's time, the irrational num-

bers in particular. Both these "postmodernist" authors and earlier philoso-

phers, beginning with Plato, may and sometimes do get certain things wrong

in specific technical terms or sometimes even in general philosophical terms;

and some of these things are not easy, technically or philosophically. These

problems, however, have nothing to do with the abuse of mathematics and

science by these figures, their desire to parade their familiarity with subjects

they know nothing about, or other motives attributed to "postmodernists"

by Gross and Levitt or Sokal and Bricmont. Lacan's statements and certain

available facts of his intellectual biography clearly suggest that he developed

certain ideas concerning the more complex epistemology of complex num-

bers, primarily, I think, as an extension of the idea of irrational numbers. He

then tried to use these ideas, however loosely. Subsequently he used topol-

ogy in a similar way, that is, one can say (and some often do), loosely

metaphorically. As I indicated earlier, however, the usage of the latter term

here requires further qualifications.

Mediation: Mathematics and Philosophy

Lacan's construction considered here may have been designed primarily for

psychoanalytical purposes, although such purposes in Lacan are a complex

matter. Either way, this construction is accomplished by way of an inven-

tion and construction of philosophical concepts in Deleuze and Guattari's

sense, which activity defines philosophy itself, according to What Is Philos-

ophy? It is in my view this construction that gives us the best sense of

Lacan's usage of mathematics. This point also allows me to close here by

giving a reasonably definite, although not definitive, answer to the question

of what is the place of mathematics in Lacan. Mathematics sometimes func-

tions in Lacan's texts in the more direct fashion of metaphor, illustration,

and the like. For example, on some occasions certain constructions of mod-

ern topology, such as the Moebius strip and the Klein bottle, serve Lacan to

find that which "we [can] propose to intuition in order to show" certain

complex configurations entailed, Lacan argues, by neurosis or psychosis.62

The overall situation is ultimately more complex on these occasions as well,

in particular as concerns the relationships between the visualizable and the

nonvisualizable, algebra and geometry, and so forth; and the role of these

Versions of the Irrational * 155

concepts in Lacan would require a separate discussion, possibly along the

lines of non-Euclidean mathematics considered earlier. I do think that the

primary and most significant usage of mathematical concepts in Lacan is as

components of his own multilayered-irreducibly nonsimple-concepts,

conforming to Deleuze and Guattari's definition (or concept) of the philo-

sophical concept. The presence and role of such concepts in Lacan are, in

my view, unquestionable. One encounters such concepts in virtually any

given sample of Lacan's text. As must be apparent from the passages cited

here, in "The Subversion of the Subject," imaginary numbers are only a por-

tion of a conceptual and metaphorical conglomerate, many components of

which are borrowed from various domains-literature, religion, philoso-

phy, or whatever.

As I said, this view shifts Lacan's usage of mathematics from the psycho-

analytic into the philosophical register, in accord with Deleuze and Guat-

tari's ideas in What Is Philosophy? It is of some interest that in the book-

while examining the difference between philosophy (defined by the

deployment of concepts) and other fields, in particular mathematics, sci-

ence, and art-Deleuze and Guattari omit psychoanalysis from this argu-

ment altogether. The relationships between psychoanalysis and philosophy

have of course been the subject of important recent investigations. One can

think in particular of the ways this question is addressed in Derrida, espe-

cially in The Post Card (where Lacan is the main subject, along with Freud

and Heidegger), or elsewhere in Deleuze and Guattari, especially in Anti-

Oedipus, or in Lacan's essay in question.63 We know from these investiga-

tions that philosophy and psychoanalysis are multiply and perhaps irre-

ducibly entangled, both historically and conceptually. This entanglement,

however, does not diminish the differences, sometimes in turn irreducible,

between them. One of these differences appears to be defined by the differ-

ent (and more fundamental) role that mathematical concepts play in philo-

sophical versus psychoanalytic thought and discourse. Indeed, one may see

Lacan's usage of mathematics as in part an attempt to change this asymme-

try, at least at a certain point, as part of his attempt to make psychoanalysis

more scientific or, with Freud, to affirm its scientific character. In the

process, Lacan did, I think, manage to help us enrich our understanding of

the epistemological nature and complexity of mathematics and science and

their role in our culture. At the very least, he makes apparent the extraordi-

nary conceptual, epistemological, or, of course, metaphorical resources of

modern mathematics, even though he himself does not always takes advan-

tage of the richness of these resources either. The success of his deployment

of mathematics and science in psychoanalysis qua psychoanalysiis is a dif-

156 * The Knowable and the Unknowable

ferent question, in part given the very nature of his thought, work, and

texts. This argument may make mathematics primarily a part of Lacan's

work as an inventor of concepts and, hence, as a philosopher (however one

evaluates him in this role) rather than as a psychoanalyst, to the degree that

these can be distinguished in Lacan's own case, as opposed to psychoanaly-

sis and philosophy in general, as just indicated. At the very least, the role of

mathematics in Lacan is fundamentally philosophically mediated, also in

Hegel's sense of mediation (Vermittlung). That, however-through the

mediation of philosophy-appears to be how mathematics has always func-

tioned outside its own sphere and often within it.

Chapter 4

"But It Is Above All Not True": Derrida,

Relativity, and the "Science Wars"

It is better, and it is always more scientific, to read and to

make a pronouncement on what has been read and

understood.

-JACQUES DERRIDA

Introduction: Tasks of Reading

Derrida's essay "Structure, Sign, and Play in the Discourse of the Human

Sciences" has a central significance for the subsequent history of poststruc-

turalism and related developments and trends, such as deconstruction (more

closely associated with Derrida's work) and various forms of postmod-

ernism, a history extending to the Science Wars and beyond. The essay was

given as a lecture at Johns Hopkins University at a conference entitled "The

Languages of Criticism and the Sciences of Man: The Structuralist Contro-

versy" and published in French in Derrida's Writing and Difference (1967)

and in English in 1969 in the proceedings of the conference, The Languages

of Criticism and the Sciences of Man: The Structuralist Controversy, edited

by Richard Macksey and Eugenio Donato. (The second edition, which I

shall cite here, appeared in 1970.) By then the controversy was on its way to

becoming poststructuralist. During the discussion following the lecture,

another French philosopher, Jean Hyppolite, asked a question concerning

possible parallels between Derrida's ideas and modern mathematics and sci-

ence, specifically Einstein's relativity theory. In his reply, Derrida made the

following statement:

The Einsteinian constant is not a constant, is not a center. It is the very con-

cept of variability-it is, finally, the concept of the game [jeu]. In other

words, it is not the concept of something-of a center starting from which an

observer could master the field-but the very concept of the game which,

after all, I was trying to elaborate.1

This statement has been endlessly circulated in discussions in and around

the Science Wars. Both Gross and Levitt's Higher Superstition and then

158 * The Knowable and the Unknowable

Sokal's hoax article in Social Text commented on the passage and origi-

nated this wider circulation.2 Amid the controversy, there followed the

appearance of Sokal and Bricmont's book and ripple effects in its wake.3 In

the (hopefully) final stages of the Science Wars other authors, such as Lacan

or Latour, moved into the foreground as figures of, and figures for, the

"postmodern" abuse of science (alleged by the authors just mentioned).4

The case of "Derrida and relativity," however, has never disappeared from

the Science Wars radars. It also retains its paradigmatic significance for our

understanding of these confrontations and of the forces that made them

possible and that remain in place and at work, as they were long before the

Science Wars.

The circulation of Derrida's remark and the treatment that it received are

symptoms of a broader problem or set of problems that surrounds nonclas-

sical thinking, affects the current intellectual and cultural landscape, and

shapes the opinions of a significant portion of the scientific community.5

Arguments analogous to the one to be offered here concerning Derrida's

work and its treatment in the Science Wars debates can be made for other

figures prominent in these debates, such as Deleuze, Lyotard, Lacan, Latour,

and still others, and such arguments have been already made here and a few

more will be made later.6 While paradigmatic as far as its treatment is con-

cerned, Derrida's case is remarkable for the extraordinary, and extraordi-

narily misplaced, prominence of the statement just cited and of his work, or

rather name (the work is hardly addressed), in the Science Wars attacks. It

is true that, in addition to the general resistance to nonclassical thought and

the reasons for this resistance, as explained earlier, Derrida has long been an

icon of intellectual controversy (and the subject of much abuse) on the

Anglo-American and Continental cultural scene.7 Still, it is remarkable that,

out of thousands of pages of Derrida's published works, a single extempo-

raneous remark on relativity made in 1966 at a conference in response to a

question was made to stand for all of the deconstructive or even postmod-

ernist (not a term easily, if at all, applicable to Derrida) treatments of sci-

ence. The subsequent "retractions" by some of his scientific critics do not

diminish the significance of this fact but, as I shall explain, instead amplify

this significance and become part of the overall phenomenon.

Derrida has commented more extensively and in more grounded ways on

mathematics and science, and on the philosophical grounding of both. He

also makes use of mathematical and scientific theories, concepts, metaphors,

and so forth (most famously, Godel's concept of undecidability) in his work

and considers their significance in philosophy and elsewhere. His first book

was his introduction to Husserl's "The Origin of Geometry," where philo-

"But It Is Above All Not True" * 159

sophical issues involving and involved in mathematics were considered.8 In

addition, as I have indicated, Derrida's work is fundamentally linked to the

problematic of technology via his investigation of what he calls "writing,"

an investigation that largely defines his work. In his reflections on the rela-

tionships between his work and mathematics and science, and in his actual

claims concerning them, Derrida himself is cautious and circumspect and

offers a number of disclaimers. He emphasizes instead the centrality of his

engagement with philosophical and literary texts for his work.9 One might

argue that mathematics and science play a more significant role in his work

than Derrida is willing to claim or perhaps than he perceives. He acknowl-

edges the possibility and indeed unavoidability of intersections between the

problematics of his work or, more generally, deconstruction, and mathe-

matics and science, and he even argues that "science is absolutely indispens-

able for deconstruction."10 This statement is itself worth attention, at the

very least in relation to the role of mathematics and science for all modern

philosophy, deconstruction included.11 Reciprocally, deconstruction (some-

times virtually in Derrida's sense) may be indispensable for at least some

mathematics and science, and has been used by them, as, I shall argue in the

next chapter, in Heisenberg's analysis of quantum mechanics.

Neither Derrida's more substantive discussions of mathematics and sci-

ence (or, conversely, the relevance of the latter for his work), however, nor

his caution in this respect has been considered by his critics in the scientific

community, either at the earlier or at the later stages of the discussions in

question. These critics instead appear (this has not diminished during the

post-Science Wars debates) to base their views of Derrida's ideas, and those

of other figures just mentioned, on indiscriminately extracted, isolated ref-

erences to science or on snippets of his texts, without placing such state-

ments in the context of his work and often ignoring even the most basic

rules of reading, let alone scholarly reading, of Derrida's and others' texts.12

Such a placement could enable one to give these statements their more

appropriate or at least more productive meaning and significance, while, at

the same time, taking into account the circumstances under which some of

these statements, such as the one on "the Einsteinian constant," were made,

circumstances that account for their somewhat loose and even, on the sur-

face, strange character. In other words, it may only be possible to interpret

such statements as reflecting certain aspects of Derrida's work in relation to

mathematical and scientific theories in question rather than in themselves-

a common but ultimately ineffective approach by either critics or sometimes

defenders of Derrida, especially among the mathematicians and scientists

who commented on them. (By now, one finds some defenders of Derrida

160 * The Knowable and the Unknowable

and other targets of the Science Wars criticism among the latter as well.) In

this case, in particular, in part in view of the circumstances in question, it

may not be possible to say what Derrida's statement on the Einsteinian con-

stant strictly means as a statement concerning relativity or perhaps even

whether it has a strict meaning. It is, however, possible to argue that this

statement reflects a conceptually and epistemologically rigorous analogy,

sensed by Hyppolite, between Derrida's (ultimately nonclassical) argument

presented in "Structure, Sign, and Play" and certain aspects of Einstein's

relativity theory, indeed specifically those aspects that epistemologically link

it to quantum mechanics. This is my argument.

The problems in question may be seen as problems of reading-ulti-

mately involving the most radical limits of this concept, as developed in and

following Derrida's and de Man's work. First of all, however, we are deal-

ing here with the most elementary and traditional norms of reading, such as

those routinely followed by scientists when reading scientific works. These

norms are massively disregarded by most of those scientists as well as by

many nonscientists, who commented, at all stages of the confrontations in

question, on Derrida and others mentioned earlier.13 It is true that, as

indeed Derrida's analysis demonstrates, it is not possible to control (the dis-

semination of) the meaning of Derrida's statements, any more than of any

other statement.14 The overall case considered here offers a powerful, if dis-

tressing, illustration of this point. Nor is it possible to claim that any given

reading (for example, the one offered here) is definitive. This does not mean,

however, that one should not read with utmost care, rigor, and respect the

context within which a given statement is made, or that one cannot argue

about such readings, or that one can simply disregard traditional norms of

interpretation or scholarship-quite the contrary. This view is fully in

accord with deconstructive theory and deconstructive practice, at least the

best theory and the best practice of deconstruction, such as those of Derrida

himself. Derrida's readings and those of other responsible practitioners of

deconstruction scrupulously follow such classical protocols. Deconstruction

does argue, of course, that such protocols, even if scrupulously adhered to,

cannot guarantee determinate results. Deconstruction would aim to explain

what happens in these types of cases and why, and Derrida and others offer

many subtle explanations of them. But this is quite different from endorsing

the kind of practices one finds in the Science Wars.

I shall, therefore, consider here the circumstances, contexts, and mean-

ings of Derrida's remark on relativity in some detail. This consideration

remains pertinent, even though some among these circumstances and con-

texts have been pointed out by a number of commentators and partly recon-

"But It Is Above All Not True" * 161

sidered by some scientists, even by Sokal. This (limited) reconsideration

appears to be primarily responsible for the removal of the planned chapter

on Derrida from Sokal and Bricmont's Fashionable Nonsense, not a bad

idea in itself, and indeed (I could not be more serious) the removal of all

other chapters would further help the book. The authors' comments on the

subject and their overall attitude remain, however, as problematic as ever.

Indeed, a certain, as it were, "virtual," chapter on Derrida remains part of

the book, as Derrida's letter, "Sokal et Bricmont ne sont pas serieux" (Sokal

and Bricmont are not serious), to Le Monde (20 November 1997) would

suggest.15s (Derrida's title, I must confess, seems to me to capture the spirit,

or the lack thereof, of Sokal and Bricmont's book.) The book certainly con-

tributed to the continuing circulation of Derrida's remark, in part since it

was part of Sokal's hoax article, republished in the book. Sokal continued

to use the quote in his lectures, including in his interview with National Pub-

lic Radio, for quite a while after the circumstances of the statement,

reflected in the introduction to Fashionable Nonsense and claimed as rea-

sons for ultimately not having such a chapter, became known to him (8).

Given these facts, the authors' explanation concerning the omission of Der-

rida from the list of alleged "abusers" in the book and in several exchanges

following the publication of the book is, at best, misleading, as Derrida

observes in his letter to Le Monde. Sokal and Bricmont responded (hardly

adequately and, in my view, disingenuously) in their "Response a Jacques

Derrida et Max Dorra" [Response to Jacques Derrida and Max Dorra] to

Le Monde (12 December 1997). Upon the publication of the book, a num-

ber of commentators sympathetic to Sokal and Bricmont's book argued that

such a chapter should have been included after all or, at least, that the book

should have contained more criticism of Derrida. These arguments gener-

ated some further debate in the press. Sokal and Bricmont's "arguments"

concerning Derrida in these exchanges remain part of the massive failures of

the book, already indicated in the previous chapter, but are worth repeating:

the failure to understand the relationships between such work and modern

mathematics and science; the failure to read the relevant texts themselves in

a scholarly and intellectually meaningful way; the failure to discriminate

between good and bad work in the humanities that they target; and, finally,

the failure to handle the key historical and philosophical aspects of mathe-

matics and science and sometimes to adequately represent the latter them-

selves. These problems are even more transparent and more disturbing in

Gross and Levitt's response to their critics in the second edition of their

book.

Furthermore, in question here are serious-and, once unsupported,

162 * The Knowable and the Unknowable

scholarly and ethically unacceptable-accusations (far from restricted to

those against Derrida), with considerable implications. Often it is simply a

matter of the truth of such accusations: hence my chapter title here, which

is a quotation from Derrida. An especially disturbing example is Gross and

Levitt's claim in Higher Superstition, on no basis whatsoever, of "Derrida's

eagerness to claim familiarity with deep scientific matters," of which (this is

clearly their point) he is in fact totally ignorant (79). Beyond being blatantly

untrue, "above all not true," this is hardly an innocent and, if believed,

inconsequential accusation. This is why, while the intellectual significance

of the arguments of Gross and Levitt or Sokal and Bricmont is negligible,

the ethical and political effects of their interventions are far from

insignificant, and the damage done by these books is far from inconsequen-

tial. This damage, I contend here, also includes that done to the positive

impact of mathematics and science on contemporary culture, although these

works are and should not be seen as representing science and scientists or

their views of science and culture. The exposition of mathematics and sci-

ence themselves in these works hardly does much service, and sometimes

does considerable disservice, to both.

Ultimately more significant, however, may be the question of reading

nonscientific texts, such as Derrida's, when these texts engage or relate to

science (or mathematics), especially when they philosophically reflect, and

reflect on, fundamental conceptual conjunctions of scientific and nonsci-

entific fields.16 Accordingly, I shall suggest a reading of Derrida's statement

on relativity that might contribute to the development of more balanced and

productive forms of interaction between mathematics and science and the

work of Derrida and other authors discussed in recent debates. As we have

seen throughout this study, at stake here is at least the possibility of a cer-

tain, nonclassical, philosophical (re)conceptualization in, and of, mathe-

matics and science, and thus also an investigation of a certain conceptual

enclosure, which is at least in part classical, that they share with a broader

philosophical field. As I said, in the present case, this can only be done by

relating Derrida's statement to "Structure, Sign, and Play" and his work in

general and then relating the latter to the epistemology and conceptual

architecture of relativity. As I have argued in this study, however, this type

of work takes place within and is often necessary for mathematics and sci-

ence as well, both in more philosophical texts by mathematicians and scien-

tists and in certain technical texts. Indeed, it sometimes happens, as, again,

in Heisenberg's papers introducing quantum mechanics and uncertainty

relations, that this work reaches its most radical philosophical limits in cer-

tain technical texts and by so doing indeed challenges both our intellectual

"But It Is Above All Not True" * 163

integrity and the philosophical, as well as mathematical and scientific, reach

of our thought. This is why the substantive argument outlined earlier is so

crucial, ultimately more crucial than the ethical problems of mishandling

the work of Derrida (and others) by the scientists in question.

At the same time, however, one cannot in a case like this simply dissoci-

ate the conceptual substance and ethics, and one must consider both and

indeed consider them jointly (and, as will be seen, there is no contradiction

or even ambiguity here). The ethical argument to be offered here is as fol-

lows. If one wants to offer a meaningful argument concerning Derrida's

remark or, to begin with (Derrida's comment is meaningless otherwise), the

Hyppolite-Derrida exchange, one should proceed

(a) by taking into account the particular circumstances of the exchange;

and

(b) by examining the exchange itself and, especially, Derrida's "Struc-

ture, Sign, and Play."

I consider these requirements axiomatic, both as concerns a meaningful

intellectual engagement with the exchange and, especially, as concerns the

ethics of academic and intellectual discussion. This is decidedly not what we

have seen in the discussions in question here (the "retractions" alluded to

earlier included).17 The latter is one of my central points, and part of the

argument in this chapter is framed accordingly. In other words, insofar as

the Hyppolite-Derrida exchange itself is concerned, my argument is more

about how an argument about this exchange should be conducted than an

argument for specific meanings of any particular statement in this exchange

to be suggested here.

This is not to say that I am not about to offer conceptual views, argu-

ments, or claims here, more or less consistent with and supporting my over-

all argument in this study. My point instead is that some of these arguments

are framed in a particular way and that this framing must be taken into

account in considering any claim made here. This refers, in particular, to

possible meanings of the phrase "the Einsteinian constant." The reading I

provide of that phrase may appear to others as plausible or possible, or at

least allowable, or conversely as implausible or even impossible and, hence,

disallowed. My conceptual, let alone ethical, argument, however, is not cen-

tered or anchored in, or even dependent on, any specific interpretation

(there will be several) of "the Einsteinian constant" suggested later, in par-

ticular as the Einsteinian relativistic space-time or the spatio-temporal inter-

val invariant under the Lorentz transformations. (I shall explain both con-

164 * The Knowable and the Unknowable

cepts later.) This argument would remain largely intact if one maintains

other interpretations of this "constant," including the speed of light c (sug-

gested by some)-although the spectrum of such interpretations is, I shall

also argue, not unlimited. Indeed, in view of the fact that this argument is

concerned with the possible relationships between Derrida's work and the

epistemological and conceptual content of relativity, rather than with what-

ever was said (or not said) in the Hyppolite-Derrida exchange, it would

remain intact even if this phrase or related statements were in fact meaning-

less or uninterpretable. I do not think that they are; quite the contrary, I see

them as reflecting, however provisionally or imperfectly (in part, again,

given the circumstances), the relationships in question. Their own meaning

and significance as statements are, however, secondary from my perspective.

It is far more important to show that there exists a view of the relationships

between modern mathematics or science and new philosophy different from

most positions on both sides of the Science Wars. Accordingly, as through-

out this study, I offer a general argument concerning the relationships

between modern science and nonclassical philosophy, here Derrida's. This

argument, in contrast to those concerning particular statements in the Hyp-

polite-Derrida exchange itself, involves more definitive conceptual claims

and support for them. The conceptual content and significance of this

exchange, I argue, lie in the degree to which it reflects the relationships

between the philosophical content of modern mathematics and science, here

in particular relativity, and Derrida's ideas, such as that of "play" (jeu) as

developed in "Structure, Sign, and Play." It is only in relation to these rela-

tionships in Derrida's work, in particular in "Structure, Sign, and Play,"

that the exchange can be meaningfully considered. In other words, the

meaning of the Hyppolite-Derrida exchange is the possibility of such rela-

tionships (suggested by Hyppolite), not in any particular statement of the

exchange itself. The exchange "tells" us that the philosophical content of

Einstein's relativity overlaps with or might be extended into Derrida's idea

of "play" and his conceptuality in general.18

As I have stressed throughout, the question is not whether Derrida's com-

ments on relativity and other areas of mathematics and science, or his work

in general, should be criticized, including by using the philosophical ideas

derived from mathematics and science themselves. The question is at what

level of intellectual engagement, knowledge, and scholarship such criticism

of Derrida and others should take place. Scholars in the humanities should,

of course, exercise due caution as to the claims they make about mathemat-

ics and science and should respect the areas of their specificity, which, as I

have also stressed, has not always been the case. Reciprocally, however, sci-

"But It Is Above All Not True" * 165

entists and other nonhumanist scholars should exercise similar due care and

caution in their characterization of the humanities, especially when they are

dealing with innovative and complex work, such as that of Derrida, all the

more so if they want to be critical about it. Derrida, I would think, would

have been willing to accept any open-minded and meaningful criticism of

his ideas as related to mathematics and science, especially by mathemati-

cians and scientists. However, no such criticism has been offered, at least

not yet. In order for this to happen, reading, in Blanchot's words, must

become a serious task for all of us, scientists and nonscientists alike. On

another occasion (in conjunction with the controversy surrounding his hon-

orary degree from Cambridge), but in responding to the negative sentiments

of certain scientists toward his work (expressed, it appears, without reading

it), Derrida offered the following remarks:

I would be content here with a classical answer, the most faithful to what I

respect the most in the university: it is better, and it is always more scientific,

to read and to make a pronouncement on what has been read and under-

stood. The most competent scientists and those most committed to research,

inventors and discoverers, are in general, on the contrary, very sensitive to

history and to processes which modify the frontiers and established norms of

their own discipline, in this way prompting them to ask other questions,

other types of question. I have never seen scientists reject in advance what

seemed to come from other areas of research or inquiry, other disciplines,

even if that encouraged them to modify their grounds and to question the

fundamental axioms of their discipline. I could quote here the numerous tes-

timonies of scientists in the most diverse disciplines which flatly contradict

what the scientists you mention [in conjunction with the Cambridge incident]

are saying.19

Derrida may here have proved to be more generous to scientists than sub-

sequent events merited-that is, to some scientists, for as I have argued here

the views and discourses of Gross and Levitt, Sokal and Bricmont, and their

like do not and should not be seen as representing the views, work, and

ideas of science and scientists. In any event, one can find many testimonies

of the kind Derrida invokes in the works of Einstein, Bohr, Heisenberg,

Schrodinger, and other founding figures of modern physics or in the works

of many major mathematicians and scientists in other fields. A more serious

engagement with Derrida's and other recent philosophical work on the part

of scientists is possible, too, and we might yet see it. This engagement may

even help them to ask those "other questions, other types of question" that

mathematics and science are likely to bring us in any event and, by so doing,

perhaps even to make this philosophical work obsolete. For these questions

166 * The Knowable and the Unknowable

may well prove to be more radical-more radically classical and more radi-

cally nonclassical-than anything that was hitherto dreamed of in all our

philosophies.

Derrida's "Differantial Topology"

Among the many accusations and complaints of Gross and Levitt's Higher

Superstition, those against Derrida have a special role and significance. This

is hardly in question, even though in their "Supplementary Notes," added to

the second edition of their book, they claim to the contrary (279-80)-

disingenuously, I would argue, and by way of a postfactum readjustment.

Their actual account of Derrida's engagement with mathematics and science

is factually restricted to two isolated instances. One is the Hyppolite-Der-

rida exchange and the other an egregious misrepresentation of Derrida's

statement. By this account, Derrida's engagement with science would have

to be seen as negligible, although Gross and Levitt claim it to be extensive,

without, however, offering any textual support for this claim. Of course,

this type of "reading"-consisting of crude attempts to "catch" direct refer-

ences to scientific terms (and to "catch" Derrida on such references) without

even minimally considering Derrida's text-would be meaningless in any

event. A reading that would make the relationships between Derrida's work

(or that of other nonclassical thinkers) and mathematics and science mean-

ingful is unavailable to the strategies and attitudes, or expertise, of Gross

and Levitt's book, as it is to Sokal and Bricmont's. They do not even com-

ment, critically or otherwise, on Derrida's usage of Godel's theorem,

arguably the most explicit and most famous reference of that type in Der-

rida.20 They only speak of its general abuse by "postmodernists" (78). They

do comment with relish on two references (three, if one counts a sneer at

Derrida's comments on algebra in the Hyppolite-Derrida exchange

[265n.10]). First is the remark on relativity, cited not altogether accurately

and, it appears, from a secondary source (265n.10), but famous ever since:

A further sense of Derrida's eagerness to claim familiarity with deep scientific

matters can be obtained from the following quotation, which also gives one

some sense of how seriously to take such claims: "The Einsteinian constant is

not a constant, [is] not a center. It is the very concept of variability-it is,

finally, the concept of the game. In other words, it is not the concept of

some[thing]-of a center starting from which an observer could master the

field-but the very concept of the game." The "Einsteinian constant" is, of

course, c, the speed of light in vacuo, roughly 300 million meters per second.

"But It Is Above All Not True" * 167

Physicists, we can say with confidence, are not likely to be impressed by such

verbiage, and are hardly apt to revise their thinking about the constancy of c.

Rather, it is more probable that they will develop a certain disdain for schol-

ars, however eminent, who talk this way, and a corresponding disdain for

other scholars who propose to take such stuff seriously. Fortunately for Der-

rida, few scientists trouble to read him, while those academics who do are,

for the most part, so poorly versed in science that they have a hard time

telling the real thing from the sheer bluff. (79)

Since I will discuss Derrida's comment on relativity in detail in the fol-

lowing sections, I shall only say here that nothing can be more misleading

than the assertion of "Derrida's eagerness to claim familiarity with deep

scientific matters," let alone the invocation of "sheer bluff" here. As I

have pointed out, the contrary is in fact true. In truth, all of Gross and

Levitt's assertions (none of them reconsidered in the second edition, in

spite of the fact that the unacceptable looseness of their treatment was

repeatedly pointed out to them) about Derrida's practices and attitudes

are quite simply not true.21 Much else can be said about their "represen-

tation" of Derrida in their book. But it is above all not true ("Mais ce

n'est surtout pas vrai"). Gross and Levitt are self-evidently not among

those "few scientists [who] trouble to read" Derrida. One cannot help but

smile at the naivete of their warning to scholars in the humanities of the

impending danger of disdain and, the second edition adds, "scorn" on the

part of the scientific community (293). On the other hand, as I said, at

stake are serious and, substantively and ethically, unacceptable accusa-

tions with considerable implications. Derrida commented on an earlier,

but, as the earlier discussion would indicate, hardly unrelated, occasion of

criticism of his work without reading it, "this is also extremely funny."

But he added: "The fact that this is also extremely funny doesn't detract

from the seriousness of the symptom."22

I would now like to consider Gross and Levitt's second main example of

Derrida's "abuse" of science. They write:

This [Derrida's remark on relativity] is not, we assure the reader, an isolated

case. In various other Derridean writings there are to be found, for example,

portentous references to mathematical terms such as "differential topology,"

used without definition and without any contextual justification. Clearly, the

intention is to assure readers who recognize vaguely that the language derives

from contemporary science that Derrida is very much at home with its mys-

teries. (79)

Once again, none of these assertions is true. Indeed, their claim of "assur-

ance" (no less!) notwithstanding, they offer no other examples of such "por-

168 * The Knowable and the Unknowable

tentous references," or indeed any references, including in the second edi-

tion.23 At least some familiarity with his work and contemporary (and much

earlier) philosophy would be necessary under all conditions, and this famil-

iarity is manifestly absent (on both counts) in Gross and Levitt's book.

Accordingly, their commentary cannot really be taken seriously as a sub-

stantive argument. On the other hand, using scientific terms "without

definition and without any contextual justification" is something to which

one can respond more seriously. It is of course also the kind of charge that

I am myself making against the usages of Derrida by Gross and Levitt or

Sokal and Bricmont. So, on these grounds (of the ethics of discussion), this

type of accusation requires a response. Gross and Levitt make much of their

observation in a long footnote:

We cannot resist the impulse to point out that in Derrida's usage the word

topology seems to be virtually synonymous with topography-at least the

index regards them as identical. This recollects an experience of one of us

(N.L.) at the age of eighteen. When being interviewed by an insurance exec-

utive for a summer actuarial job he was asked: "What kind of mathematics

are you interested in?" "Topology," he replied. "Well, we don't have too

much interest in topography," said the insurance man. Obviously a decon-

structionist avant la lettre.

Defenders of deconstruction and other poststructuralist critical modali-

ties will no doubt wish to point out that topos (pl.: topoi) is a recognized

term within literary theory for a rhetorical or narrative theme, figure, gesture,

or archetype, and that therefore it is permissible, without asking leave of the

mathematical community, to deploy topology to designate the analysis of

textual topoi. One's suspicions are reignited, however, when the term differ-

ential topology suddenly appears. (In mathematics, differential topology is

used to denote the study of the topological aspects of objects called "differ-

ential (or smooth) manifolds," which are, roughly speaking, higher-dimen-

sional analogues of surfaces in three-dimensional space.) (265-66n.11)

Gross and Levitt's description of differential topology itself is surpris-

ingly (since Levitt is a topologist) uninteresting and will hardly give the

reader unfamiliar with the subject a real sense of this extraordinary branch

of modern mathematics. In fact their statement is not altogether accurate

even mathematically. At least it may confuse a nonmathematician, even

given their "roughly speaking." For one thing, not all two-dimensional sur-

faces are smooth. The relationships between smooth (differential) and not

smooth surfaces or manifolds would need to be explained to the reader.

(Roughly speaking, differential manifolds are the ones on which some form

of differential calculus can be defined.) Second, smooth two-dimensional

surfaces are themselves differential manifolds; so one need not restrict the

"But It Is Above All Not True" * 169

definition to their higher-dimensional analogues. This is, admittedly, a small

matter, except, again, to the degree that this remark and their other com-

ments on science can become a source for a kind of confusion for scholars

in the humanities that should indeed be avoided. I also leave aside the banal

topology versus topography joke, and the overall inappropriate and unpro-

ductive tone of the footnote (unfortunately found throughout much of the

book), or imprecise and even meaningless phrases such as "poststructuralist

critical modalities" that abound in the book.

It is more difficult to leave aside the fact that Acts of Literature consists

of translations of Derrida's various writings on literature, which were

edited, and the index compiled, by someone else. Moreover, Gross and

Levitt clearly did not check their own references to the index. The statement

that "in Derrida's usage the word topology seems to be virtually synony-

mous with topography-at least the index regards them as identical" is a

bizarre non sequitur. The references in the index to Acts of Literature-

"topology (atopology, topography, topoi)" (455)-indicate that these terms

are related or used in similar contexts rather than are identical. Once one

checks the text, one sees that all these terms refer to a general sense of

"topos" as spatiality, which, as even Gross and Levitt admit, need not entail

a direct reference to topology as a mathematical discipline. Even in mathe-

matics the term" topology" carries a double meaning of the discipline and

of the topological structure of particular objects. Here is Derrida's state-

ment itself, from his essay on Kafka, "Before the Law" (Devant la Loi):

This differantial topology [topique differantielle] adjourns, guardian after

guardian, within the polarity of high and low, far and near (fort/da), now

and later. The same topology without its own place, the same atopology

[atopique], the same madness defers the law as the nothing that forbids itself

and the neuter that annuls oppositions [emphasis added].24

The elaboration is part of Derrida's reading of Kafka, and one, obviously,

needs to know and to read both Kafka's and Derrida's texts to make sense

of it. It deals, among other things, with the difficulties of applying any spa-

tial or topological concepts and metaphors to the domain where law oper-

ates as conceived in Kafka's work. This knowledge and a real act of reading

would be necessary, even if Derrida had in fact appealed to mathematical

differential topology here. One can easily see, however, that Derrida says

differantial (differantielle)-and not differential [differentielle]-topology.

That is, he speaks of "topology" relating to his famous neologism or rather

neographism "differance" rather than to differential topology. This differ-

ence is not audible in the spoken French, but it is visible on the page. This

170 * The Knowable and the Unknowable

was one of the reasons why Derrida introduced his neographism. In this

sense Gross and Levitt's mistake is ironic, and it unwittingly proves Der-

rida's point. When Derrida, in fact, uses topography a bit earlier in the same

essay on Kafka, it refers to an "inscription" (in Derrida's sense of writing)

of the "space" or/as "non-space" of the law in Kafka. This is why the edi-

tors list topography in the index. Primarily in question, however, is the con-

ception of "differantial topology" as atopology-a "topology" without its

own place or without the possibility of linking it to a place or a space. This

may be a complex idea, but it entails no claim on Derrida's part concerning

the mathematical discipline of differential topology, although, as will be

seen presently, it is indeed not irrelevant to certain epistemological issues

involved in mathematical topology, as considered in chapter 3. (There is

thus a double irony here.) Derrida does not even say "topology" here,

although he sometimes uses the terms "topology" and "topological" else-

where, including, on another occasion (in "The Law of Genre"), in Acts of

Literature (249). It is true that the English translation says topology-obvi-

ously (it should be clear by now) in the general and perhaps quasi-mathe-

matical, rather than mathematical, sense. However, the French provided in

parenthesis, for that very reason, says "topique differantielle"-a differan-

tial space or place, a certain topos or atopos, or atoposness. The French for

"differential topology" is, of course, "topologie differentielle." Given that

his field is topology and that the French is provided here, it is (or rather it

would be in proper circumstances of reading) inexplicable that Levitt did

not pay attention to or did not bother to check this point. It is also inexcus-

able, given that his aim was to attack publicly Derrida's misuse of scientific

terms. In such circumstances one is obliged to treat with care even what one

perceives as "junk" work or thought, even if in order to check one's own

perception of it as junk.

If there is "topology" in Derrida's reading of Kafka in "The Law of

Genre," it would be something like this. According to Derrida's reading,

Kafka's "Before the Law" is defined by the following allegory, speaking

here in de Man's terms, which, as we have seen, entail the analogous "topol-

ogy" and epistemology. (Analogous allegories are found elsewhere in

Kafka.) Any attempt to configure or even conceive of the "space" of law qua

space is bound to fail. We always fail if we think of the space of law, spatial

or (spatialized as) temporal, as space in relation to which, say, "in front"

("before," devant) of, or at the gate to which (as in Kafka's parable), we can

position ourselves or into which we could in practice or in principle enter.

Moreover, in accord with the (nonclassical) epistemology of this study, we

cannot postulate that law, or more accurately that which produces the effect

"But It Is Above All Not True" * 171

of law, can be ascribed a spatial (or temporal, or any other) nature or struc-

ture, properties, and so forth. All of these, however, may in turn emerge as

effects of the "same" efficacity, that is (since the latter is never strictly the

same), of epistemologically analogous efficacious dynamics. Nor, we recall,

can we postulate that this efficacity of law would exist as inaccessible but

conforming to such a spatial (or any other) ascription; it is both indissocia-

ble from its effects and yet irreducibly inaccessible, as considered earlier. We

can infer the existence of this efficacity on the basis of the sum total of cer-

tain phenomena, which may also have particular topological, spatial effects.

Derrida reads Kafka's parable as allegorizing this situation.

Some among such effects could be quasi-mathematically related to cer-

tain spaces of mathematical topology, analogously (but not equivalently) to

the way Lacan invokes them (on this occasion perhaps in a more conven-

tionally metaphorical and more epistemologically classical way) in his

attempt to describe certain "spatial" mental structures involved in psycho-

analytic situations. Something like the Moebius strip or the Klein bottle or

still stranger objects of modern topology may be specifically invoked in con-

junction with Kafka, who may even have been familiar with them. They

may also be invoked, perhaps more pertinently, in conjunction with Der-

rida's reading of Blanchot (who appears to be familiar with these mathe-

matical ideas) in "The Law of Genre" (Acts of Literature). The situation is

then transferred to the "topology," the space, of the text and our relation to

it (the law of reading, as it were). Or rather (and this is itself a key point of

Derrida's reading) both spaces-that of reading, or of literature, and that of

law-emerge at the same time or "space" and are shown to be irreducibly

entangled, conceptually and historically. This entanglement adds further

loops or entangled spaces and complicates, irreducibly, the overall "topol-

ogy" and epistemology of the case and, hence, also of literature, reading,

and law.

The efficacity itself in question is, within Derrida's framework, more or

less equivalent to the efficacity of differance, with which it is explicitly cor-

related throughout the essay and which bears essentially on other works just

mentioned, as well as "Structure, Sign, and Play," and on the possible con-

nections between the latter essay and relativity.25 This is why Derrida speaks

of "differantial"-differance-effected-topology/atopology (topique) of the

(un)space of law in Kafka. From a nonclassical spatial/topological perspec-

tive, such as that of the nonclassical epistemology of non-Euclidean mathe-

matics considered in the preceding chapter, Derrida's analysis of Plato's

"concept" (if it is in turn a concept in any given sense) of chora in Timaeus

in "Khora" is pertinent here as well.26 As I have indicated earlier, the Greeks

172 * The Knowable and the Unknowable

appear to have a rather deeply developed philosophical topology even

though their mathematical science of space was geometry. By and large,

Plato's chora is read by Derrida as, at least, anticipating and (whether Plato

saw it this way or not) entailing what we may call the differance of space

and spatiality itself. That is, in accordance with the nonclassical view, chora

is that which makes space and spatiality-topology in every conceivable

sense, mathematical topology included-possible and may, in this sense, be

seen as the "site" of the spatial, while not being spatial "itself," or, again,

not "itself" subject to any conception.

Ultimately, differance (it is, again, never the same differance) would be

the efficacity of both spatiality and temporality, or their relationships,

including, for example, spatial-like features of our representations of time

or (these are less common) temporal-like features of our representation of

space. As will be seen in detail later, these efficacious dynamics have con-

siderable implications for the relationships between Derrida's work and rel-

ativity. To anticipate, Derrida's nonclassical epistemology can be, and here

will be, linked, first, to the nonclassical topological epistemology, as just

described and as considered in chapter 3 in the context of complex numbers,

more generally, to non-Euclidean mathematics; second, to the nonclassical

physical, or material, epistemology of quantum mechanics or analogous ele-

ments (specifically experimental technology) of relativity theory; and third,

to the relationships between these fields, discussed in chapter 3. These are

the connections that enable one to link Derrida's "Structure, Sign, and Play"

to relativity and (with qualifications given earlier) give meaning to the Hyp-

polite-Derrida exchange, and, as I shall suggest, enable Derrida to say what

he did concerning "the Einsteinian constant."

For the moment, my point is that, not unlike (but, again, not identically

to) the case of Lacan, Derrida's "topique differantielle" (differantial topol-

ogy) has in fact profound conceptual and epistemological connections to

topology. Indeed, one can say (with more caution and qualification) that it

has profound conceptual connections to differential topology, as it extends

from Leibniz, Gauss, and Riemann to Poincare and then into our own time,

and in fact these connections extend to Einstein's relativity, which itself

extends from the work and ideas of these thinkers. From this perspective,

Derek Attridge's translation of "topique differantielle" and "differantial

topology," while mildly inaccurate, is intriguing and not out of place,

which, obviously, does not diminish the problem of Gross as Levitt's treat-

ment of Derrida here. Thus, on the one hand, there is no "differential topol-

ogy" or even a reference to it in the disciplinary sense of the term, and one

could indeed end (and some commentators did) one's argument against

"But It Is Above All Not True" * 173

Gross and Levitt's "criticism" of Derrida there. On the other hand, how-

ever, a stronger argument is that there are in fact deep conceptual and epis-

temological connections between Derrida's work and topology. Indeed,

these connections, beyond their significance in their own right, may even

help us to illuminate and better understand each and possibly might also

help some mathematicians in their work in mathematical topology. Obvi-

ously, one needs to be a mathematician to be able to do this work; and, as

throughout this study, my argument in no way aims to disregard the disci-

plinary specificity of the scientific fields involved, upon which Derrida

clearly makes no claim.

Gross and Levitt did offer a "clarification" (if such is the word) in the sec-

ond edition of their book. This clarification, however, hardly requires alter-

ing anything in the preceding argument. They write:

It has ... been pointed out, quite correctly, that the term differential topol-

ogy does not appear in Derrida's Acts of Literature; what shows up is differ-

antial topology, as a translation of "topique diff[e]rantielle." Let us assume,

then, that Derrida did not authorize the translator's locution (but remember,

the Wolin affair demonstrated that he can be very finicky about translation

of his writings). The honor of the coinage then goes to translator Derek

Attridge, a well-known American deconstructionist. We leave it to the judg-

ment of the reader to assess the probability that the near-identity of

Attridge's phrase with the mathematical one is merely adventitious. (293)

This is at best disingenuous. The appeal to "the judgment of the reader,"

often used by Sokal, with Bricmont and on his own, as well, is hardly a sub-

stitute for an argument or an excuse for a lack of argument. (The parenthe-

sis on the Wolin affair is irrelevant, if not meaningless.) Gross and Levitt fail

to mention that, as we have seen, the French original is given in the transla-

tion. It is difficult to say whether Attridge had an allusion to differential

topology in mind. Either way it would not help Gross and Levitt's case

much. At stake in the passage is fundamentally differance and a general

nonclassical conceptuality, as just considered, rather than mathematical dif-

ferential manifolds and differential topology in its disciplinary sense. But

even if he had (seriously or playfully) such a metaphorical allusion in mind,

for example, along the lines considered here, it would not make Gross and

Levitt's criticism any more meaningful. In any event, one would have to

read Derrida's text to understand the phrase. To say, therefore, "we think

the point [made in their book] still stands, even if Attridge, rather than Der-

rida, is responsible for the example" (293) is scarcely more compelling than

the original argument.

The statement that bridges the two statements just cited is even less com-

174 * The Knowable and the Unknowable

pelling. Gross and Levitt say: "In any case, there has been an enormous

amount of comment from our critics on this one specific matter, which,

remember, was a casual illustration of an incidental point: literary theorists

sometimes use jargon borrowed from scientific terminology to create the

impression of rigor and congruity with 'cutting edge' science" (293; empha-

sis added). The statements cited there would clearly show (and would make

it difficult to forget) that this "specific matter" was nothing of the kind, the

illustration was not casual, and the point was not incidental. It certainly was

(and still is) a major point for Gross and Levitt, as it is for Sokal and Bric-

mont. Besides, this last claim in the passage just cited is still not true, "above

all not true," as concerns Derrida's writings.

While, thus, Gross and Levitt accuse Derrida of "using" the term "dif-

ferential topology" "without definition and without any contextual

justification" (and, again, of much more), the description appears to be far

more appropriate as a characterization of their own treatment of Derrida's

work. The same can be argued concerning their treatment of the work of

many others whom they criticize in their book, whether these works, as

such, indeed deserve criticism along these lines (as some do) or not. Schol-

arly problems of monumental proportions are, to use the language of topol-

ogy, found in the immediate vicinity of just about every point of Higher

Superstition. It is not so much embarrassing errors, such as misreading

"topique differantielle" as "differential topology," that are most crucial.

We all make mistakes and sometimes make absurd mistakes. Most crucial

here are the intellectually and scholarly inadmissible practices and attitudes

that pervade this irresponsible book. Gross and Levitt's warning concerning

"threats to the essential grace and comity of scholarship and the academic

life" (ix) becomes, in one of many bizarre ironies of this bizarre book, its

self-description. It would be equally difficult not to think of Sokal and Bric-

mont's title phrase, "fashionable nonsense," as a more fitting description of

their book itself than of most of the works and ideas they criticize. (I have

not encountered this observation but would be surprised if others had not

made it as well.)

To be sure, and one must acknowledge this without false pretense or hes-

itation, some of the "postmodernist" work on science is indeed bad. Some

commentaries on mathematics and science cited by Gross and Levitt or

Sokal and Bricmont are at best embarrassing, as are, one might add, many

commentaries by the humanists on the major "postmodernist" figures in

question. They offer the editors of scholarly journals and university presses

no more excuse for publishing them than Higher Superstition, The House

Built on Sand, or Fashionable Nonsense. It would be difficult to defend the

"But It Is Above All Not True" * 175

decision of the editors of Social Text to publish Sokal's hoax article,

although one can think of some good "excuses," not excluding trusting in

the ability of a good scientist to understand complex nonscientific texts. It is

the opinion of the present author, however, that Sokal's hoax article would

have probably been published somewhere in any event. The reasons for this

are indeed the lack of adequate knowledge and misconceptions concerning

both modern mathematics and science, as well as concerning the humanist

discourses in question on the part of the possible editors and reviewers in

the humanities. One need not know any mathematics and science to reject

Sokal's hoax article for publication; a fairly rudimentary knowledge of Der-

rida and other authors it comments on, and of course a minimally careful

reading (both found lacking on this occasion), would be enough to send it

back to the author as unacceptably incompetent, even if one did not per-

ceive the hoax nature of the piece. If this is what Sokal's hoax aimed to

prove, then it has achieved its goal. Hardly a revelation, however. There is

always some bad work in any field, including mathematics and science. The

problem is that, as I said, Gross and Levitt or Sokal and Bricmont are (by

virtue of their lack of expertise, or even superficial familiarity with the texts

in question, and unacceptable reading practices) in no position to discrimi-

nate between good and bad work.27 If they were, very different books

would ensue. The comedy of these books is that they say the worst things

about some of the best work and accept and sometimes praise-and draw

on-some of the worst. The tragedy is that some scientists, including some

among the best scientists, have taken it seriously and accepted its arguments

and have even adopted some of its unacceptable attitudes.

Thus, in the case of the major thinkers in question, such as Derrida,

Lacan, or Deleuze, where Gross and Levitt or Sokal and Bricmont see abuse

of science, there is no science. When there is science or a role for it to play,

there is no real abuse, even though one cannot deny that errors do occur.

This science and this role, however, the critics in question do not see, in

part, again, because they are entirely unequipped intellectually and philo-

sophically to read such texts in general. For, as I argue here, one can, in

principle, decouple mathematics and science from such authors, say Godel's

undecidability from Derrida's, complex numbers from Lacan's psychoana-

lytic "algebra," or differential topology (of Riemannian manifolds) from

Deleuze's "topology," or, again, from Derrida's differance. The reverse,

however, cannot be done. One cannot decouple Deleuze, Derrida, or Lacan

(their ideas, idiom, rhetorical and textual strategies and moves, and so

forth) from mathematics or science in their texts, especially in the manifestly

crude way of Gross and Levitt or Sokal and Bricmont. Derrida's, Deleuze's

176 * The Knowable and the Unknowable

or Lacan's engagement with these mathematical or, again, quasi-mathemat-

ical ideas is not mathematics in its disciplinary or technical sense and cannot

be treated as such, not without massive, or at least some, engagement with

other conceptual, metaphorical, and textual elements of their work. From

this perspective, the whole criticism of these authors by Gross and Levitt or

Sokal and Bricmont and the like is inapplicable altogether.

Given the egregious nature of some of Gross and Levitt's errors it is sur-

prising that they were not discussed by reviewers immediately upon the pub-

lication of the book or indeed were not noticed before the book was pub-

lished. It is also singularly unfortunate, since it could have diminished the

extraordinary harm done by the book, that is, assuming that it should have

been published to begin with, given its flaws, which are unredeemable

regardless of the problems one might have with the authors discussed there.

The decision, even more unfortunate, to issue the second edition poses fur-

ther questions in this respect. By now some of these problems have been

pointed out by reviewers and commentators.28 However, in contrast to my

argument here, often critics and even defenders of, say, Derrida still think

that Derrida's (or Hyppolite's) comments should at best be discounted (at

worst they are seen as inept or senseless) rather than understood in the con-

text of the relationships between philosophy and science.29 The same (with

suitable modifications) goes for other figures. The very critique of Gross and

Levitt's book often amounts to a "yes, but... ," such as "they got a few (or

even not so few) things wrong, but..."-not the kind of change of attitude

that is, I think, necessary here.

Nor, in my view, have these books, or Sokal's hoax, had as much value

in terms of provoking debate as some have contended. To some degree they

have done so, of course. There are, however, better, much better, ways to

engender debate-and better, much more productive, debates. They are

such by virtue of confronting the fundamental questions at stake in the rela-

tionships between mathematics and science and the humanities (or the

social sciences), again with a very different set of intellectual attitudes and

in a very different spirit of intellectual discussion. As I have suggested ear-

lier, one would do well to argue that these works are taken far too seriously,

that is, again, as works rather than symptoms, are given far too much space,

and are trusted by too many scientists (or indeed nonscientists). Once one

indeed provides the proper "definition" and "contextual justification" for

Derrida's terms, or those of other authors under attack, there begins to

emerge a very different sense of their work and of the relationships between

this work and mathematics and science. The true richness and complexity of

these relationships and the true possibility of reading these texts, including

"But It Is Above All Not True" * 177

of course reading them critically, and of exploring these relationships in

them begin to emerge.

"The Einsteinian Constant"

To (re)cite Derrida's comment one more time:

The Einsteinian constant is not a constant, is not a center. It is the very con-

cept of variability-it is, finally, the concept of the game [jeu]. In other

words, it is not the concept of something-of a center starting from which an

observer could master the field-but the very concept of the game which,

after all, I was trying to elaborate.

To begin with, Derrida's last sentence, "which, after all, I was trying to elab-

orate [in the lecture]," was omitted by most-it appears, by all-Science

Wars commentators on this statement. This clause, however, is crucial

because it indicates that the term "game" or "play" (in this context a better

translation of the French jeu, which carries both meanings) has a very

specific meaning, elaborated in Derrida's essay. I shall spare the reader

proper "editing" (of the type I suggested for Lacan in chapter 3) of Derrida's

or Hyppolite's remarks, which would "deglitch" them without, I think,

deviating too much from their meaning (at least a possible and even plausi-

ble meaning), albeit at the expense of their great value for the Science Wars

type of exchanges. I am, however, reasonably certain that if the word

"game" did not appear here, and Derrida had merely said "variability," the

statement would never have surfaced in the Science Wars debates, which

(alas!) says something in itself. Of course, the appearance of the term is

defined by Derrida's essay, where it is motivated by the history of the con-

cept in the human sciences and its deconstructive potential, in view of this

history. "Play" in Derrida's sense designates a certain "decentering" (desta-

bilization) of the organization of elements within a given system, presumed

to be stabilized by a "center," understood as a particular privileged

configuration within this system, or a principle governing its organization.

Indeed, one may speak of decentering effects of nonclassical efficacity desig-

nated by Derrida's differance. This meaning, I shall suggest, may be argued

to be consistent with the philosophical content of relativity, which possibil-

ity is at the core of the Hyppolite-Derrida exchange. In other words, rela-

tivity allows for understanding a substantive portion of its philosophical

content in terms of (decentered) "play" in Derrida's sense of the term and of

differance (as well as related elements of Derrida's matrix). At the very least,

178 * The Knowable and the Unknowable

it is unquestionable that the concept of play is central to Derrida's "Struc-

ture, Sign, and Play," an oral presentation of which occasioned the

exchange.30 Accordingly, understanding how Derrida uses the term "play"

in his essay and understanding what Hyppolite and Derrida mean by "the

Einsteinian constant" are both essential for a meaningful reading of Der-

rida's statement, whether the latter statement itself is meaningful or not as

far as relativity is concerned. While, however, nearly everyone involved

appears to have commented on "the Einsteinian constant," most scientists

who commented in print on this statement have not carefully considered

Derrida's concept of play.31

The accuracy of quotations from Derrida and other figures by their crit-

ics has been often stressed by scientists and others who welcomed and sym-

pathetically commented on this criticism. As we have seen, not all of these

quotations proved to be as accurate as these commentators believed. How-

ever, even when they are accurate, their literal accuracy is meaningless if the

reader is not provided with the meanings of the terms involved (such as

"play/game" or "the Einsteinian constant"); is deprived of the possibility of

establishing such meanings; or is free to construe them on the basis of, say,

one's own or "common" ideas associated with a given term ("play/game")

or one's general knowledge of physics ("the Einsteinian constant"), as

opposed to the meaning given to these terms by Derrida's essay or by Hyp-

polite's question. How meaningful can the accuracy of one's quotation be if

"the Einsteinian constant" is made to be the gravitational constant as it

figures in general relativity (as suggested in Sokal's hoax), or even the

famous c, the speed of light in a vacuum (as Gross and Levitt claim), if Hyp-

polite meant something else by it?32 It is true that Hyppolite's meaning may

in fact be seen as correlative, if not identical, to the constancy of c in a vac-

uum, which is not surprising, given that this constancy defines (special) rel-

ativity. Even so, however, Derrida gives it a special meaning, related to or

correlated with his concepts of variability and play, the concepts specifically

considered in his lecture. The very word "concept," persistently repeated by

Derrida in his remark (a fact ignored by the Science Wars critics), indicates

that a certain particular conceptuality is in question, rather than some allu-

sion to some "game" played by somebody somewhere. The latter reading

could make this comment laughable or nonsensical. Derrida's reference

(given that he refers to his lecture) is nothing of the kind. I shall return to the

relationships between variability and play in Derrida's sense and concepts of

relativity, in particular the constancy of c. My main point at the moment is

that Derrida's statement cannot be read without a further explanation of the

"But It Is Above All Not True" * 179

terms of his essay, which the readings or unreadings in question lack, even

though there are, again, clear indications in his comment that such an expla-

nation is necessary. If it is so read, one cannot be surprised at a reaction of

Steven Weinberg's in "Sokal's Hoax," "I have no idea what this is intended

to mean" (11), or any number of similar responses encountered in the

debates in question.

In his contribution to the exchange on his article, Weinberg, to his credit,

acknowledges that he did not initially pay much attention to the meaning of

Derrida's key terms and gives some consideration to the context of Der-

rida's statement, specifically to Hyppolite's remarks. He says, in "Steven

Weinberg Replies," in particular that in his initial reaction to Derrida's

comment in "Sokal's Hoax," he "was bothered not so much by the obscu-

rity of Derrida's terms 'center' and 'game,'" and that he "was willing to

suppose that these were terms of art, defined elsewhere by Derrida." He

then says, "What bothered me was his phrase 'the Einsteinian constant,'

which I have never met in my work as a physicist" (56). It would seem nat-

ural to check what Derrida's "terms of art" could have meant, especially

given the weight of a commentary by someone in Weinberg's position and

of his stature. Be that as it may, Weinberg proceeds, first, to suggest a pos-

sible meaning for the phrase and then to offer some comments on Derrida's

essay and the Hyppolite-Derrida exchange. He does not consider Hyppo-

lite's own description of the (Einsteinian) "constant." Nor does he offer any

meaningful commentary on or interpretation of the concept of play. Such a

commentary, however critical, would be of considerable interest. I would

even risk a claim that some of Weinberg's own work on quantum field the-

ory (which brought him his Nobel Prize) carries in it certain philosophical

implications that are not that far away from the (nonclassical) epistemology

that underlies Derrida's concept of play. Weinberg's quotation from Der-

rida's essay on the term "center" is hardly sufficient to explain Derrida's

idea of decentering and play, and it is not surprising that this quotation was

"not much help" to Weinberg (56). The passage that Weinberg cites occurs

in the introductory portion of the essay, as part of the discussion of the joint

historical functioning of the concepts of "structure" and "center." Derrida

says: "Nevertheless . . . structure-or rather, the structurality of structure-

although it has always been involved, has always been neutralized or

reduced, and this by a process of giving it a center or referring it to a point

of presence, a fixed origin." Derrida's phrase, omitted by Weinberg, "up to

the event which I wish to mark and to define," indicates that Derrida is

making an introductory historical point here. His concept of "decentered

180 * The Knowable and the Unknowable

play" emerges later in the essay, although a few sentences following the one

cited by Weinberg already give one a better sense of Derrida's ideas con-

cerning "structure," "center," and "play." Derrida writes:

The function of this center was not only to orient, balance, and organize the

structure-one cannot in fact conceive of unorganized structure-but above

all to make sure that the organizing principle of the structure would limit

what we might call the play of the structure. By orienting and organizing the

coherence of the system, the center of a structure permits the play of its ele-

ments inside the total form. And even today the notion of structure lacking

any center represents the unthinkable itself.33

In short, those unfamiliar with Derrida's ideas would need a more exten-

sive reading of Derrida's essay and a more comprehensive explication of its

terms. Therefore, more patience and caution may also be necessary before

one is ready to agree, or disagree, with Weinberg's conclusion: "It seemed to

me Derrida in context is even worse than Derrida out of context."34 The con-

texts and concepts at issue may well not be sufficiently familiar to most sci-

entists for them to be able to offer the kind of reading of Derrida's statement

to be suggested here. Nor should they be expected to be familiar with these

ideas and contexts or have any obligation to engage them in any way. It is

not a question of blaming Weinberg-a great physicist and (which is not

irrelevant) one of the most open to radical and innovative theories in physics

itself-or many other physicists who commented on the subject. One might

regret a certain lack of intellectual curiosity on the part of those scientists or

their unwillingness to consult the experts on Derrida, or indeed-why not?-

Derrida himself, something that, in more general terms, Weinberg appears to

endorse as well.35 Reciprocally, scientists can be exceptionally helpful to

scholars in the humanities in clarifying both science itself and philosophical

concepts emerging in science, and they, (including on other occasions Wein-

berg himself) have been throughout intellectual history.

Throughout the debates in question Hyppolite's and Derrida's critics in

the scientific community not only cite their comments out of context but vir-

tually disregard the minimal relevant norms of intellectual and, especially,

scholarly exchange. To restate a few basic facts, Derrida's statement

appears in the transcript of an improvised response to Hyppolite's question

following an oral presentation of his essay. The essay does not mention rel-

ativity, and the statement itself makes no substantive scientific claims. Rela-

tivity and "the [Einsteinian] constant" are brought in by Hyppolite, not

Derrida, who responds to Hyppolite extemporaneously, in the context of

his just-delivered paper. Given these circumstances, a responsible commen-

tator would be hesitant to judge Derrida's statement without undertaking a

"But It Is Above All Not True" * 181

further investigation of his work, beginning with "Structure, Sign, and

Play." The conclusions may of course be different from those reached by the

present analysis, but no conclusion would be ethically, intellectually, or

scholarly responsible short of such an investigation.

There is nothing exceptional in the circumstances themselves. Such com-

plexities of improvisation, transcription, translation, and interpretation

often arise at conferences, including on mathematics and science. The cir-

cumstances that lead to them remain significant when such exchanges are

subsequently reproduced in conference volumes, as is the case here and as is

made clear by the editors of the volume.36 It is true that such statements are

sometimes edited by the authors before publication and technically require

their permission to be reproduced. Such is not always the case, however,

and it is doubtful that it was done here; indeed it is virtually certain that it

was not. Hyppolite died before the volume (eventually dedicated to his

memory) went into production and did not even have a chance to edit his

own contribution, let alone his exchange with Derrida. One might, accord-

ingly, argue (and some have) that such improvised statements should be dis-

counted or, at least, not considered as significant as far as key philosophical

ideas or positions are at stake. However, in spite, and sometimes because, of

the interpretive problems that they pose, such statements and exchanges are

significant, historically and conceptually. My argument, therefore, is differ-

ent. I argue that the circumstances of these statements must be given special

consideration in interpreting and evaluating them, rather than serving as a

reason for dismissing them.

Some scientists, such as Weinberg, considered earlier, have admitted,

grudgingly, that the circumstances of Derrida's remark may require addi-

tional consideration. Such admissions in themselves are hardly sufficient,

however. First of all, they were far "too little, too late"-after several years

of relentless abuse, beginning with Gross and Levitt's book, and, as I have

already argued, some of this abuse continues by only slightly different

means. Second, they do not appear to signal much change in the overall hos-

tile and unprofessional-and, one might even say, unscientific-attitude

toward the work of Derrida and other figures on the part of the scientists

involved in these debates. (Some more encouraging signs have begun to

appear.)

I now turn to Hyppolite's remark, introducing the famous "constant."

Hyppolite said, according to the transcript of the exchange:

With Einstein, for example, we see the end of a kind of privilege of empiric

evidence. And in that connection we see a constant appears, a constant which

is a combination of space-time, which does not belong to any of the experi-

182 * The Knowable and the Unknowable

menters who live the experience, but which, in a way, dominates the whole

construct; and this notion of the constant-is this the center [i.e., would it be,

according to Derrida's argument] [emphasis added] ?37

Hyppolite's first sentence is somewhat obscure, which is not surprising,

given the improvised, tentative, and probing nature of his comments. It can

be read as compatible with special relativity, in particular the idea that the

distinction between space and time depends on the observer. Certain state-

ments, which would have objective (universal) "empirical" value according

to classical physics-say, as concerns a sequence of two given events (A

before B)-can no longer be seen as universally valid. Instead they depend

on a specific reference frame, since the sequence can be reversed if seen from

the perspective of another frame (in which B will precede A).38

Hyppolite does not actually use the phrase "the Einsteinian constant" as

such, which is introduced by Derrida. The phrase clearly refers to Hyppo-

lite's remark, rather than to any accepted scientific term, and is, in this

sense, a local contextual reference. As used by Hyppolite, the "constant"

here may not mean-and to the present author does not appear to mean

a numerical constant, as virtually all the physicists who commented on it

appear to assume. Instead it appears to me to mean the Einsteinian (or Ein-

steinian-Minkowskian) concept of space-time itself, since Hyppolite speaks

of "a constant which is a combination of space-time" (emphasis added), or

the so-called spatio-temporal interval, invariant ("constant") under Lorentz

transformations of special relativity. This interval is also both "a combina-

tion of space [and] time" and something that "does not belong to any of the

experimenters who live the experience," and it can be seen as "dominat[ing]

the whole construct." ("The whole construct" here refers to the conceptual

framework of relativity in this Minkowskian formulation, on which I shall

comment in detail later). Given the text, either of these interpretations is

more plausible than seeing the phrase as referring to a numerical constant.

It is also possible that Hyppolite has in mind this latter interpretation, while

Derrida understood the "constant" as referring to the Einsteinian concept of

space-time itself. Or perhaps, in accordance with the analysis given earlier,

he means that-it may not be given a name!-efficacious dynamics or, let

me risk it, efficacious play, jeu, a protospatial differance of space-time, in

and through which the space-times of relativity (or in mathematics, Rie-

mannian spaces) emerge along with their difference from classical spaces,

Euclidean, Newtonian, and so forth. This difference is ultimately not that

crucial, since both these notions are correlative, and both are correlative to

"But It Is Above All Not True" * 183

the constancy of the speed of light c in a vacuum and its independence of the

state of motion of the source. Also, both "constants" equally reflect key fea-

tures-decentering, variability, play (in Derrida's sense), and so forth-at

stake in Hyppolite's and Derrida's statements and Derrida's essay. These

features, found in both Derrida's matrix and (with due qualifications, to be

offered later) relativity, appear to make Derrida see the "constant" in ques-

tion as "not a constant," but as (that is, as reflecting) "the very concept of

variability" and, "finally, the concept of the [play]" (in Derrida's sense).

These features are most fundamentally at stake in Derrida's essay and led

Hyppolite to link, albeit in a cumbersome way, Derrida's ideas to relativity.

It follows that one can read c in this way as well (all three readings are cor-

relative insofar as they entail the relativistic "variability" in question). In my

view the two conceptual readings of the Einsteinian constant just indicated

appear to be suggested more strongly by Hyppolite's statement (in the avail-

able transcript) and to suggest the variability in question more immediately.

In any event, in view of those aspects of Hyppolite's and Derrida's meanings

that can be established with more certainty from broader contexts (such as

that of Derrida's essay), these interpretations of "the Einsteinian constant"

are at least allowable by their statements.

None of these interpretations is definitive, and no definitive interpreta-

tion may be possible, given the status of the text as considered earlier. At the

same time, interpretations of these statements are possible and may be nec-

essary, at least in the broader sense of relating them to the content of Der-

rida's essay. These statements have been interpreted without any considera-

tion of these complexities or any serious attempt to make sense of them. It

is more productive, however, to take these complexities into account, to sort

them out to the degree possible, and to give these statements the most sensi-

ble rather than the most senseless interpretation.

Derrida's statement acquires more congruence with relativity once one

understands the term "play/game" as in this context (it is a richer concept

overall) connoting the impossibility within Einstein's framework of space-

time of a uniquely privileged frame of reference-a center from which an

observer could master the field, that is, the whole of space-time (if such a

thing as the whole of space-time is indeed possible). It is in part the under-

standing, arising from a reading of Derrida's essay (such a reading is, again,

imperative for a meaningful examination of the exchange), that leads to the

considerations just mentioned. While my readings of "the Einsteinian con-

stant" are more conjectural, the meaning I suggest for Derrida's term "play"

(jeu) is easily supportable on the basis of his essay and related works. So is,

184 * The Knowable and the Unknowable

it follows, the understanding of this concept as congruent with certain

philosophical ideas and implications of relativity, and it may be in part

indebted to these ideas, however indirectly.

Keeping this broader conceptual field as a background (I shall return to

it presently), one might see Derrida's statement as correlative to the follow-

ing aspect of relativity theory. In contrast to classical-Newtonian-

physics, the space-time of special, and even more so of general, relativity dis-

allows a Newtonian universal background with its (separate) absolute space

and absolute time or a uniquely privileged frame of reference for physical

events. These two notions are, of course, not the same, and my "or" also

means "and" here. Both are disallowed, but the difference between them is

of some significance, in particular in the context of the question of whether

the Newtonian classical framework is in fact decentered as well. I shall

return to this question later. My point at the moment is that the broader

matrix of relativity just indicated is philosophically and epistemologically

correlative to this more general conceptual field of "decentering." The idea

can be traced to Galileo (insofar as the relativity of spatial description is

concerned) and, in its more radical implications, to Leibniz and his critique

of Newton, both of which are among Einstein's sources. The Einsteinian or

Einsteinian-Minkowskian concept of space-time may be seen as correlative

to the assumption that the speed of light is independent of the state of

motion of either the source or the observer. The "constancy," that is, invari-

ance, in special relativity, of the spatio-temporal interval under Lorentz

transformations arises from the same considerations and was introduced in

this form by Minkowski. These considerations also led him to the concept

of space-time, which in turn helped Einstein in developing the general rela-

tivity theory, his theory of gravity. (I bypass the explanation of these more

technical terms, since they are not essential for my main point-the decen-

tered structure of the space-time of relativity.)39 As I said, it is plausible that

Hyppolite had in mind precisely this concept. The Einsteinian (concept of)

space-time, however, can be more immediately linked to Derrida's concepts

of decentering, variability, and play, and this is why, as I suggested earlier,

it is possible that Derrida and Hyppolite have two different "constants" in

mind here. Hyppolite, of course, might have had space-time itself in mind as

well, in spite of his somewhat odd expression "a combination of space-

time." (The special circumstances of both the exchange itself and of its tran-

scription, translation, and editing appear to have played a role here, how-

ever.) Both "constants," or c, are inherent in the same matrix of Einstein's

(special) relativity and may, again, be seen as conceptually equivalent inso-

far as they reflect a certain conceptuality of "variability" found in this the-

"But It Is Above All Not True" * 185

ory. Or, more accurately, "two Einsteinian constants," space-time itself and

the Lorentz distance, while both correlative to the constancy of the speed of

light and to each other, allow for two conceptually equivalent frameworks

of relativity.

Thus, while I do think that the conceptual (rather than numerical) inter-

pretations that I suggest are "more plausible," I do not think that reading

"the Einsteinian constant" as c undermines my overall argument here. In

criticizing those who used c in the Science Wars debates, I do not mean to

suggest that this interpretation is impossible. I mean instead to emphasize,

first, that other interpretations are also possible and perhaps more likely, a

decision, if one wants to makes such a decision more determinately, that

requires a more careful examination of the text. Second, my argument is

that the treatment of the Hyppolite-Derrida exchange by some of those who

used this interpretation is unacceptable on ethical and intellectual grounds.

My overall interpretation of the Hyppolite-Derrida exchange on relativity

allows for reading "the Einsteinian constant" as c, since it is primarily about

relativity and/as a decentered play, a concept that easily tolerates this read-

ing. Accordingly, if some, such as Richard Crew (in his exchange with the

present author), are right in arguing that c makes more or even most sense

for "the Einsteinian constant" (and this argument is not uncompelling), it

would not change my overall argument.40 As I have stressed from the out-

set, my overall conceptual, let alone ethical, argument does not fundamen-

tally depend on two particular interpretations of the phrase "the Einsteinian

constant." Certain reservations, expressed by Crew, concerning the differ-

ence between the terms "constant" and "invariant" are justified, especially

as concerns "space-time," and I pondered that point when I encountered the

Hyppolite-Derrida exchange. I do think, however, that the space-time inter-

val, invariant under the Lorentz transformations, could have been assimi-

lated by Hyppolite into his "constant" as referring to something invariant in

the sense of remaining constant.

Either way, in philosophical terms, the overall conceptual framework

outlined here can be seen in terms of the conceptuality of "play" in Der-

rida's sense. Derrida sometimes speaks, via Nietzsche and Heidegger, of

"the play of the world," as opposed to play in the world. This shift indicates

a complex nonclassical dynamic. In this dynamic, space or space-time, or

even "the world," however conceived, is not given as a background in

which whatever "play" of events takes place. Instead we deal with dif-

ferance and play in which such effects as space and time, or space-time, or

even whatever appears (including phenomenologically) to us as worlds,

emerge in a nonclassical manner, while their ultimate efficacity remains

186 * The Knowable and the Unknowable

inaccessible. From this perspective, "the Einsteinian constant" could indeed

be seen as correlative to "the very concept of variability" and, at the limit,

as the concept of play/differance developed by Derrida. In other words, Der-

rida posits a certain irreducible variability of the world itself and/as our con-

struction of it, as opposed to the concept of the world as a ("flat") back-

ground of events given once and for all, such as Newton's absolute space

(with absolute time) in classical physics. This idea was questioned already

by Newton's contemporaries, in particular, Leibniz.41 Leibniz was philo-

sophically (his physics is another matter) not that far from Einstein's or

from Gauss's and Riemann's, and subsequent geometrical/topological, con-

ceptions, as considered earlier.

The latter were in turn crucial to Einstein's framework of geometrization

of the space of gravitation itself, as curvature of the space, in general rela-

tivity. The connections in question would indeed be more pronounced in

general relativity, which links gravitation to the geometry, here non-Euclid-

ean (Riemannian) geometry and topology, of space-time. In this case, the

Lorentz invariance can no longer be maintained globally but only locally,

correlatively to the fact that in general relativity space can be seen as flat-

Euclidean or, more accurately, Lorentzian-only locally. Globally space is

curved. Gravity measures the curvature of the space, or (these indeed

become equivalent) the curvature of the space measures gravity. The vari-

ability and "the play of the world" (in the present sense) are, however, not

only retained but enhanced as a result. One can see such spaces, manifolds,

in terms of infinite possibilities of maps or atlases (these are indeed used as

mathematical terms, essentially developed already by Riemann), mapping

the neighborhood of each point and transitions from one such neighbor-

hood to another, as discussed in the preceding chapter. The variability

becomes all the more radical if one connects it to the geometrical/topologi-

cal considerations suggested, in various contexts, in the preceding chapters.

That is, we deal here with nonclassical efficacity of the conceptuality or con-

ceivability-pictorial, representational, intuitive (anschaulich), and so

forth-of the spatial (in mathematics and physics) or (in physics) also the

temporal or of their interaction. As I have indicated, Derrida's subsequent

investigations of, among others, Blanchot's, Kafka's, and especially Plato's

concept of chora proceed along these lines as well. This is, then, how dif-

ferance and its differantial (a)topology may be read, let us say, again, quasi-

mathematically, in this context.

As we have seen, however, there are also quasi-physical, material aspects

of differance, which are germane to the epistemological connections

between Derrida's framework and quantum mechanics. These connections

"But It Is Above All Not True" * 187

can be extended to relativity, by virtue of the elements the latter itself shares

with quantum mechanics in view of the following key point of Einstein's

theory. According to the latter, space and time are not given independently

of observation and our instruments of observation and, then, being repre-

sented by means of these instruments, such as rulers and clocks, or (in a

more complex way) theories. Instead, space and time, in any way we can

observe or conceive of them, are effects of instruments-technology-of

observation (and, again, in more complex way, of our theories) and indeed

represent or embody experimental and theoretical practices.42 One of the

key subjects of Derrida's essay (especially if read with the companion analy-

sis in Of Grammatology) is the analysis of Claude Levi-Strauss and struc-

tural anthropology from this perspective of the reciprocal relationships

between that which is observed and that by means of which one observes.43

The questions of "structure," "sign," and "play" (or center, decentering,

and so forth), or those of "the discourse of the human sciences," are posi-

tioned accordingly, and are all linked, at several levels, to (the concepts of)

technology (instruments, tools, and so forth) and/as writing. The analysis of

the question of writing is especially crucial to Writing and Difference (where

the essay is included), along with Of Grammatology, where Derrida con-

siders Levi-Strauss in the same set of contexts. This dynamic (in general and

in the more specific context of spatiality and temporality) acquires an even

more complex and radical form in quantum physics and its epistemology. In

this respect, as Bohr understood well, Einstein's relativity is a transitional

stage from classical to quantum-mechanical epistemology, and in the case of

general relativity the epistemological proximities become even more pro-

nounced, once one deals with more exotic gravitational phenomena, such as

neutron stars or black holes. Also, as the preceding analysis makes clear,

Heisenberg's original work and his very invention of quantum mechanics,

which were decisive for all of Bohr's epistemological ideas, were crucially

indebted to and inspired by Einstein's physics and some of his philosophical

ideas involved here, specifically the role (ultimately irreducible) of measur-

ing instrument in defining the property of objects under consideration. By

coupling it to differance, Derrida also conceived of "play" (the play of the

world) as part, or (this may depend on one's interpretation) itself an inter-

mediary effect of this machinery, of the efficacity producing the effects of

the spatiality of space and the temporality of time-as that which makes

space conceptually spatial and time temporal. In addition, under certain

conditions, differance and "play" also produce certain spatial aspects of

time and temporal aspects of space-that is, they enable us to see time in

terms of space and, conversely, space in terms of time or to combine both

188 * The Knowable and the Unknowable

within one or another concept of space-time. Now, however, we can also

see the situation from a material-technological, rather than phenomenal,

nonclassical perspective and indeed link both. As I have suggested from the

outset of this study, we may define as "nonclassical" those situations in

which the role of technology is irreducible. The irreducible technology of

measurement in quantum physics, even if not to the same degree in relativ-

ity, or of tekhne of "writing" in Derrida's sense, makes each theory non-

classical. With the accompanying concept of phenomenon in Bohr's sense

and its extension to the work of other authors discussed here, we may speak

of nonclassical thought as techno-phenomenology. By contrast, in "classi-

cal" situations the role of technology is, at least in principle, reducible, as

for example in the case of measurement in classical physics, since measuring

instruments play only an auxiliary role there so as to allow us to speak of

the independent properties and behavior of classical objects.44 At stake is a

rigorous deconstruction of, among others, the concepts of space and time

(or space-time) in the original sense of Derrida's idea and practice-as an

analytical dismantlement and, then (or simultaneously), reassembling of the

fundamental architecture of these concepts.

Thus, Derrida's nonclassical framework has deep philosophical connec-

tions to both modern mathematics (including differential geometry, differ-

ential topology, and related fields), extending from Gauss and Riemann to

our own time, and modern physics, such as relativity, especially general the-

ory, and quantum theory, or indeed to the possible parallels, connections,

and interactions between both, as considered in the preceding chapter. From

this perspective we may even call, epistemologically and philosophically rig-

orously, this topology and perhaps (again, if considered epistemologically)

even mathematical differential topology "differantial (a)topology." It is not

possible to consider networks just indicated in detail here. My point is that

Einstein's relativity and other key areas of modern physics, such as quantum

mechanics, and other sciences or mathematics are, philosophically and his-

torically, bound to be entangled in these networks within a richly interactive

nonclassical matrix. This is what Hyppolite's comments reflect, even

though, I would think, short of the nonclassical limits here in question. On

the other hand, it appears that it was at least intimations or intuitions of

these nonclassical features of relativity, along with their being clearly at

work in Derrida's essay itself, that shaped Derrida's statement.

One might, thus, see Hyppolite's and Derrida's remarks as relating to

certain, most of them reasonably standard, features of Einstein's relativity-

presented, admittedly, in a somewhat nonstandard idiom. At the very least,

these remarks can be read as consistent or, again, congruent with the philo-

"But It Is Above All Not True" * 189

sophical ideas and implications of relativity, as they have been elaborated in

the scientific and philosophical literature on the subject. What Hyppolite

appears to have sensed and suggested here is that part of the conceptual con-

tent of Einstein's relativity with its space-time may serve as a kind of model

for the Derridean concept of (de)centered play and related ideas. This sug-

gestion is neither surprising nor especially difficult for anyone who has read

Derrida's essay and has some knowledge of certain key ideas of relativity.

Derrida responds more or less positively but suggests that one needs a more

decentered and more nonclassical view of "the Einsteinian constant"-

which is to say of the physical world according to Einstein's relativity or, as

will be seen, of scientific theories themselves-than Hyppolite appears (to

Derrida) to suggest.

As I have stressed throughout, at stake are primarily philosophical ques-

tions, rather than questions of physics, and both Hyppolite's and Derrida's

remarks must be read and evaluated accordingly. It is worth citing Deleuze

here:

Of course, we realize the dangers of citing scientific propositions outside

their own sphere. It is the danger of arbitrary metaphor or of forced applica-

tion. But perhaps these dangers are averted if we restrict ourself to taking

from scientific operators a particular conceptualizable character which itself

refers to non-scientific areas, and converges with science without applying it

or making it [simply] a metaphor [emphasis added].45

This is not a quotation one is likely to find in recent critics of Deleuze in the

scientific community. I do not think, however, one can read Deleuze on sci-

ence without keeping this statement in mind (it is of course congruent with

and indeed correlative to his other statements on science cited here). Such

reading, indeed this statement itself, would immediately invalidate most of

what these critics, and in particular Sokal and Bricmont, say about Deleuze,

whose ideas concerning science must, too, be considered, positively or criti-

cally, from the same perspective of philosophical questions at stake in mod-

ern science, or in terms of what I call here quasi-mathematical concepts. It

could be easily shown that the same cautious attitude toward and utmost

respect for mathematics and science manifestly defines most of the "targets"

of the criticism in question-certainly Derrida, Michel Serres, and Lyotard,

or, self-evidently, Hyppolite.

Philosophical questions and their implications are significant, however,

and their significance is in no way diminished by the circumstances of the

exchange, such as the improvised nature of these remarks. This exchange

reflects concepts, including those of Hyppolite and Derrida, which are any-

190 * The Knowable and the Unknowable

thing but improvised-quite the contrary, these concepts, such as "play,"

are thought through by Derrida in the most rigorous way. Even more

significantly, they reflect and, at least in part, derive from the philosophical

questions arising in modern science itself. Here I would think that one is

obliged to go (perhaps) beyond Deleuze, if not in the sense of philosophical

disciplinarity (defined by the practice of construction of concepts in his

sense), then in the sense of epistemological necessity. This is in part why, in

introducing his question, Hyppolite says that "we have a great deal to learn

from modern science."46 And perhaps science has a few things to learn from

modern or even postmodern philosophy as well, thus continuing the recip-

rocal relationships that have defined Western intellectual history since

Galileo and Descartes, or indeed since Plato.

The very question of how casual such "casual" remarks are, or can be,

would have to be reconsidered from this perspective. Some of the most

significant ideas in science and philosophy alike were introduced by way of

or developed from "casual" remarks, footnotes, and so forth. This is why I

would be hesitant to treat these remarks as merely "casual," "offhand," and

so forth and dismiss them on these grounds. As I have indicated, this type of

argument was advanced by some in defending Derrida and has even been to

a degree accepted by some critics as well. However, if one wants to defend

Derrida here, this is a weak defense, that, I think, does not pay sufficient

attention to Derrida's thought, as reflected in his exchange with Hyppolite,

which gives significance to the exchange. As I have stressed throughout, at

stake in my argument is not a defense of Derrida on relativity (or of his work

in general), anymore than that of Lacan on mathematics, or other figures

considered here. Indeed, my position is that a meaningful critique of the

work of these figures, whether specifically on science (or as concerns the rela-

tionships between their work and science) or in general, would be more inter-

esting and productive than any defense. The best position for making such a

critique effective is clearly one established through a substantive understand-

ing of the relationships between modern mathematics and science and the

philosophical work, the architecture of the concepts, in question. Insofar as

such new philosophical building blocks, which can be provided by mathe-

matics and science, are delivered to their projects via other trajectories (direct

or implicit), the degree of Hyppolite's or Derrida's specific knowledge of rel-

ativity or other areas of mathematics and science, and even their misrepre-

sentation of them, as mathematics and science, matters little. Besides, there is

no significant misrepresentation in this particular case in any event. What

matters is their (and others') philosophical thought, however shaped, in con-

nection to the philosophical content and implications (more specialized or

"But It Is Above All Not True" * 191

broad) of modern mathematics and science, rigorously and carefully consid-

ered, even when the thinkers in question do not offer such a consideration

themselves. It hardly needs to be repeated that these connections are recipro-

cal, and that the trajectories involved are multiply entangled, although they

are also heterogeneous enough to disallow a classical, such as dialectical,

synthesis or unification within a single system.47

The possibility of such an argument in the present case should not be sur-

prising, either conceptually or historically. Neither Hyppolite nor Derrida

pretends or claims to have expertise or competence in physics itself. How-

ever, leaving aside their general erudition, both have been the readers and

(especially Hyppolite) colleagues of such world-famous philosophers and

scholars of science as Alexandre Koyre, Gaston Bachelard, Jean Cavailles,

Albert Lautman, and others and of a number of major mathematicians and

scientists, including those working in related areas. These authors, including

mathematicians and scientists, commented extensively on the philosophical

issues in and implications of relativity and are cited by many experts in the

history and philosophy of science. Many discussions of the Leibniz-Clarke

debate in philosophical literature, known to Hyppolite (or Derrida), con-

sider Einstein's relativity, both specific and general theory, as a culmination

or at least a crucial point in the history opened by this debate, which can in

turn be multiply linked to Derrida's philosophy. As I have indicated, Leib-

niz's ideas on these subjects are not that far from those of Einstein (and one

might even argue that philosophically they go further). This may be espe-

cially said about Leibniz's critique of Newton's absolute space and absolute

time. Equally crucial is the accompanying idea that space and time (absolute

space and absolute time being no longer possible), rather than being a fixed

background of physical events, are relational and indeed relative quantities

(or even concepts) that depend on a particular configuration and relation

between such events. As we have seen, this idea opens a history of major

transformations of our thinking in mathematics, science, and philosophy.

Obviously, Leibniz's (and subsequently Einstein's) ideas concerning space

and time have crucial connections to both Kant's and Hegel's philosophy,

the main subject of Hyppolite's own investigation, as well as connections to

earlier philosophical thought from and before Plato on. Hyppolite's own

presentation at the same conference invokes Leibniz in a context that is rel-

evant to some of his comments in his exchange with Derrida. Derrida's ideas

in "Structure, Sign, and Play" have a number of connections to Leibniz's

philosophy, as, it can be added, do the ideas of Deleuze, in whose philoso-

phy one finds even more pronounced links between Leibniz and relativity,

especially general theory, via Gauss and Riemann, as indicated earlier. In

192 * The Knowable and the Unknowable

addition, Derrida's investigation of temporality in Aristotle, Hegel, Husserl,

and Heidegger can hardly be disconnected from this problematic. As the

director of the Ecole Normale, the center of French philosophy, mathemat-

ics and science, which he headed for ten years (1954-63), Hyppolite had

access to the most sophisticated scientific and philosophical information on

the subject. He was previously a chair at the Sorbonne and a professor at the

College de France thereafter, where he also had ample opportunities to dis-

cuss modern mathematics and science, in which he had considerable inter-

est throughout his life. It is worth mentioning in this context that Hyppolite

was granted admission to the Ecole Normale on the basis of his ability in

philosophy and mathematics. Derrida, too, spent years of his career at the

Ecole Normale, first as a student (of, among others, Hyppolite) and then as

a professor, and had similar access to key ideas of modern mathematics and

science, although his background and interests in science are less extensive

than those of Hyppolite.

It cannot therefore be surprising that both Hyppolite and Derrida would

know enough about relativity or what is philosophically at stake there to

make philosophically sensible or even suggestive remarks about it. More-

over, as I have argued here, there are considerable independent philosoph-

ical affinities between relativity and Derrida's ideas in "Structure, Sign, and

Play" and beyond-if, once again, these affinities are indeed independent,

given the history just indicated. Einstein's relativity dates to 1905 and has

earlier genealogies overlapping with genealogies of Derrida's ideas, in par-

ticular via the question of temporality, specifically in conjunction with Hei-

degger's work. Hyppolite's invocation of relativity (or other areas of math-

ematics and science) in relation to Derrida's essay is, then, not only

justifiable, conceptually and historically, but is to the point. His point,

again, however imperfectly made, has significant implications insofar as

the relationships between modern science and contemporary philosophy

are concerned. In spite of their improvised nature (again, not to be dis-

counted), Hyppolite's remarks and the Hyppolite-Derrida exchange reflect

and arise from, and are given their significance by, an extraordinarily rich

network of mathematical, scientific, and philosophical ideas. The same net-

work shapes the relationships between the work of Derrida (or that of

Deleuze and other authors mentioned here) and modern mathematics and

science. All this is entirely missed by their recent critics in the scientific

community or in the philosophical community, including in the philosophy

of science, where a similar criticism is, sadly, equally widespread and

indeed has a longer history.

It is, then, hardly accidental either that Hyppolite, whose thought was

"But It Is Above All Not True" * 193

shaped by this type of understanding, invokes next still more radical con-

ceptual possibilities suggested by modern science. He refers, first, implicitly

(at least it can be read in this way), to quantum physics and then, overtly, to

biology. These references, their connections to Derrida's ideas, and Der-

rida's response to them require a separate analysis, although the preceding

arguments of this study offer us a good sense of these connections. As I have

suggested earlier, the Derridean epistemology may indeed be even closer to

that of quantum mechanics than that of relativity, even though Derrida

appears to have arrived at it through different, primarily literary and philo-

sophical, trajectories. The remainder of the exchange raises questions con-

cerning the relationships between (post)structuralism and the philosophical

aspects of mathematics, in particular algebra or, more accurately, "alge-

braization," and science. Some of these questions have an interesting history

in the context of the "structuralist controversy" (and of course a still longer

history, extending to/from Greek and Babylonian mathematics). Serres even

argues that we might yet need to rethink structuralism from the perspective

of its connections to twentieth-century mathematics, specifically the Bour-

baki project.48 Andre Weil, one of the great mathematicians of this century

and a founding member of the Bourbaki group (and the brother of Simone

Weil), wrote an appendix to Levi-Strauss's The Elementary Structures of

Kinship.49 Both Derrida and Hyppolite (or Serres) must have been aware of

Weil's article and might have been familiar with it. Derrida's "Structure,

Sign, and Play" is primarily an analysis and a deconstruction of Claude

Levi-Strauss and structuralism.s50 The essay does not consider this mathe-

matical or, again, mathematical-philosophical problematic and its relation-

ships with structuralism. However, it can hardly be simply disconnected

from them, and some of these connections emerge more explicitly in other

essays in Writing and Difference and elsewhere in Derrida. At the very least,

Derrida's philosophical ideas can be meaningfully engaged in exploring

these relationships, as Hyppolite suggests. As we have seen in the preceding

chapters, crucial philosophical and epistemological questions are at stake,

and the poststructuralism-structuralism axis, where Lacan's and Derrida's

work belongs as well, is fundamentally linked to these questions.

There are further nuances concerning relativity as well, especially those

relating to the difference between the centering of "the whole [theoretical]

construct"-that is, the overall conceptual framework of relativity-around

the concept of space-time and the centering of the space-time of special or

general relativity itself.51s It would be difficult from the Derridean perspective

to claim any central or unique concept defining the Einsteinian framework.

It is, therefore, possible and even likely that Derrida has this point in mind;

194 * The Knowable and the Unknowable

and it is certainly a major question posed by "Structure, Sign, and Play," and

by Derrida's work in general, whether he specifically had it in mind in rela-

tion to relativity on that occasion or not. Invariance or stability of a concep-

tual center (if any) of a theoretical structure, such as that of the relativity the-

ory, is, of course, quite different from invariance of a physical constant. One

might suggest, however, that in the Hyppolite-Derrida exchange a certain

concept of decentering defining the space-time of relativity coincides with, or

even becomes a model for, the idea of decentering the overall conceptual

structure of the theory itself. We here keep further nonclassical aspects of

Derrida's framework in mind, specifically the inaccessible efficacity of both

centering (centers are seen as local and multiple, but are not eliminated) and

decentering effects. Accordingly, no concept belonging to the theory, not

even that of the decentered space-time, may be seen as an absolute center of

relativity theory-a center invariant under all theoretical and historical

transformations of this theory. That is, such conceptual centering may

change from one version of relativity to another (this centering is relative in

this sense), and some forms of relativity may be constructed as conceptually

decentered in themselves, and perhaps all of them in fact must ultimately be

reconstructed as such. There have certainly been considerable debates among

historians of science as to the relative centrality of key experimental facts and

theoretical ideas of special relativity, either as originally introduced by Ein-

stein or in its subsequent, such as Minkowskian, forms.

Given Derrida's argument in "Structure, Sign, and Play" and the Der-

ridean perspective on the structure of theoretical argument in general, such

considerations concerning a possibly decentered conceptual framework of

relativity could have primarily been, and would naturally have been, on

Derrida's mind in replying to Hyppolite's statement. This meaning also

allows for several meanings of "the Einsteinian constant" itself, including

the speed of light c, which may indeed have been what Derrida had in mind

after all. Hyppolite might have also thought of the center in the sense of

conceptually centering the framework of relativity around a certain "con-

stant" (however the latter itself is interpreted), when he spoke of the con-

stant as "dominat[ing] the whole construct." A certain conflation of both

these levels of "(de)centering" is possible, and these levels may mirror each

other, if one should speak of conflation here. Given that at stake is a gen-

eral question of thinking "structure without center" and that ultimately

every conceptual structure may need to be seen as decentered, certain

aspects of relativistic space-times would serve as models or "allegories" of

the Derridean decentering or, better, "play." Both relativity and "play" in

Derrida's sense entail the great complexity of the relationships between cen-

"But It Is Above All Not True" * 195

tering and decentering, ultimately within a nonclassical epistemological

framework. The nonclassical view only suspends a center that is (perceived

to be) unique, absolute, "central," and so forth, not local or conditional

centralization.

Is the space-time itself of relativity decentered and decentered more radi-

cally than that of Newtonian physics (since there is a certain decentering

there as well, via the Galilean relativity), or is it decentered radically

enough, given other physical data or theoretical arguments now available?52

Is relativity a centered or decentered, or un-centered (de-centralized), con-

ceptual or theoretical framework? To what degree can the latter be concep-

tualized via or metaphorized by the former? These are some among the

questions that are, if not in fact asked, provoked by Hyppolite's reaction to

Derrida's concept of play and other ideas in "Structure, Sign, and Play."

Questions of this type naturally arise in the philosophy or history of relativ-

ity, even if the philosophers of science may not address them in the form

that Derrida's work would suggest. On occasion, however, one does find

formulations in their work that are close to those that would emerge were

one to consider relativity from a Derridean perspective. Even as he criticizes

the exchange and "Structure, Sign, and Play," Richard Crew (in the

exchange with the present author) in effect demonstrates the potential

significance of these questions for our understanding of both relativity and

Derrida's framework in his commentary, cited earlier.53 While these ques-

tions are independent of whatever was said or meant in the Hyppolite-Der-

rida exchange, they are intimately related to the Derridean problematic.

Crew's discussion may, even if, again, against his aims, also be seen as

supporting rather than diminishing Derrida's historical point, in "Structure,

Sign, and Play," concerning the relationships between continuity and rup-

ture in the reciprocal history of the concepts of structure and play. While, as

Crew correctly points out (par. 11), Derrida does speak of and consider a

certain "rupture" in "Structure, Sign, and Play," this "rupture" cannot be

seen as absolute. The concept of absolute rupture is itself subject to Der-

rida's critique throughout his work, along with absolute origins, ends, and

so forth. This is suggested even by the way in which the very term "rupture"

is introduced in Derrida in conjunction with the "event" at issue in "the his-

tory of the concept of structure." ("Event" is already used in quotation

marks in Derrida's opening paragraph.) Derrida writes: "What would this

event be then? Its exterior form would be that of a rupture and redoubling"

(emphasis added).54 This would hardly seem to suggest a simple or unequiv-

ocal rupture, at most something whose exterior form appears as and has the

effect of a rupture, and this effect is important. Overall, however, a much

196 * The Knowable and the Unknowable

more complex historical model is at stake. Indeed, the very formulation is

suggestive of a kind of nonclassical epistemology of history, whereby one

finds certain effects of rupture or, conversely, continuity, without ascribing

either, or again, anything to the efficacity of these effects.55 A very long his-

tory of continuities and ruptures around the idea of structure (or again,

play) is at stake, as the second paragraph of "Structure, Sign, and Play" sug-

gests.56 From this perspective, one should expect some intimations, antici-

pations, and so forth of the "event" at issue, sometimes radical and striking,

as in Leibniz or indeed Galileo (who ruptured the history of classical physics

even before Newton), and there are much earlier cases, for example, in Plato

and his concept of chora. "Derrida's intention[s]" in "Structure, Sign, and

Play" were likely of that nature.57 And yet, at the same time, the kind of

("decentered") understanding of even the Newtonian physics that Crew

suggests does not come into the foreground in any significant measure

before precisely the historical period suggested by Derrida, in particular

around Nietzsche's time.58 Derrida's key concept of "the play of the world,"

as considered earlier, is equally associated with Nietzsche.59 Nietzsche,

however, was a contemporary of Maxwell, and his thought belongs to the

period of Maxwell's physics and other developments, some of which even-

tually culminated in relativity. It is indeed true that, as Crew says, "this is

not the heyday of classical mechanics" (par. 9). The point is, however, that

a decentered understanding of classical mechanics, where this understand-

ing is viable, is also-and decidedly-not what governed the heyday of clas-

sical mechanics. Crew seems to locate the decentering possibilities of inter-

preting classical mechanics, ahistorically, in classical mechanics itself (par.

10). (Let us assume for the moment that these possibilities are viable even if

taken in Derrida's radical sense, which is a complex question of interpreta-

tion, along the lines discussed in chapter 1.) As I have indicated, there are

indeed reasons for doing so, since already with Galileo and then Leibniz cer-

tain key features necessary here were in place. However, the history of clas-

sical mechanics in this respect is for the most part a history of physically

centered theory, prior to relativity or developments that led to it, from

Maxwell on. So Derrida's genealogy is on the mark and not at all in conflict

with the decentering possibilities of interpreting classical physics, to the

degree that such possibilities are available (par. 10).

One can, thus, read Derrida's comment in the following way (which, as

I shall explain presently, is not the same as finding out what Derrida meant

or even what he might have meant by his statement at the time it was

made). Whatever is a constant-numerical, conceptual (an invariant, for

example), or other-in Einstein's relativity is likely to be a correlative of a

"But It Is Above All Not True" * 197

Derridean (or analogous) decentering, variability, play (jeu), the nonclassi-

cal efficacity of differance, and so forth. Such "constants" may be found

either at the level of the structure of the Einsteinian space-time or at the

level of the conceptual structure of Einstein's theory itself. Both may be seen

as mirroring each other in this respect, with the qualifications indicated ear-

lier and keeping in mind the rigorous specificity of the mathematics and

physics of relativity. In this reading the meaning of "the Einsteinian con-

stant" itself is left open, although it cannot be seen as arbitrary (and the

spectrum of possibilities is in fact not that large). The "inconstancy" of "the

Einsteinian constant" will refer to the relationships between many conceiv-

able Einsteinian "constants"-numerical, conceptual, invariants, and so

forth-and the irreducible decentering, variability, and play in Derrida's

sense, as the effects of differance. The former signals the latter: that is, the

Einsteinian constants signal the Einsteinian variability as the Derridean or

quasi-Derridean play.

This might or would have been an intriguing insight or a lucky guess (or

both) on Derrida's part, if due primarily to his general ideas, in relation to a

broader nonclassical field outlined earlier, rather than his knowledge of rel-

ativity or related mathematics, from Gauss and Riemann on. This is what

Derrida's remark in fact or in effect performs. That is, it might be that, pro-

voked by Hyppolite's remarks, Derrida guessed that something of that

type-that "constants" or "invariants," numerical or other, of the theory

signal decentering, variability, play-must take place in relativity and else-

where in mathematics and physics. If so, it was a rather brilliant guess. If

not-that is, if this is not what he meant, or even could have possibly

meant-this is still how this statement may be read, in view of the nonclas-

sical features of both his argument and mathematical or scientific theories in

question. One might, thus, see Derrida's remark as a kind of philosophical,

as opposed to strictly scientific, proposition, assembled, built, out of freely

floating terms and ideas, which may be read in the way just described. Even

if the statement had a different meaning when produced, once produced,

this statement or, more accurately, the field of interpretive possibilities

thereby established allows one to interpret it in the way suggested here. This

reading, however, is not arbitrary and, I would argue, philosophically rig-

orous-that is, the statement can be given a rigorous meaning on the basis

of both Derrida's text, specifically "Structure, Sign, and Play," and relativ-

ity. As I said, if this was what Derrida had in mind, it was likely only a lucky

guess, although such guesses are never purely a matter of luck, a dynamics

itself considered by Derrida on many occasions and in a certain sense

throughout his work. At the very least, the history of Derrida's ideas has

198 * The Knowable and the Unknowable

some connections to relativity, as well as quantum physics, topology, post-

Godelian mathematical logic, and other areas of modern mathematics, as

well as certain areas of modern biology and genetics, especially in their non-

classical aspects. As I argue here, in all of these theories one finds the philo-

sophical conceptuality of the kind found in Derrida and thinkers, such as

Nietzsche, Bataille, Blanchot, Lacan, and Levinas, who influenced him or

whom he confronted in his work. One also finds related problematics else-

where in philosophy itself or in literature, both of which are more likely to

be among Derrida's sources, some of which have in turn been influenced by

mathematics and science.

What I have tried to do here is to suggest such general connections,

whether perceived or not by Derrida himself, between Derrida's ideas and

the philosophy of relativity. One must take responsibility for this reading

and claims, be they ultimately right and wrong, since Derrida makes none of

them. I am perfectly happy to take this responsibility and a degree of credit,

if it is due, for establishing or constructing these connections. Indeed, as I

have argued here, more radical ideas than those found anywhere in contem-

porary philosophy may ultimately be at stake in the problematics of modern

mathematics and science, in quantum physics in particular. If I am right,

part of the credit will still have to go to Derrida. If I am wrong, the fault, as

the saying goes, is all mine.

Derrida is in a rather enviable position here. On the one hand, he does

not mention relativity in his essay (nor to my knowledge anywhere in his

work in any substantive way) and gives only a brief improvised response to

a somewhat muddled question by Hyppolite. He might have had something

in mind concerning relativity and his concept of play, quite possibly some-

thing vague and induced by Hyppolite's question. What that was is not eas-

ily, if at all, recoverable after thirty years. In any event, while, as some, such

as Crew or indeed Derrida himself, have pointed out, Derrida might not

have said enough or was too vague to be definitive or definitively right, or

even meaningful, about relativity, he did not say enough to be (definitively

provable) wrong about it either. Even if he were proved to be wrong, it

would be unreasonable to make too much of it, given the circumstances, as

indeed nearly everyone, on both sides of the Science Wars, admits at this

point, including (however lamely) Sokal, although not Gross and Levitt. In

short, there is little ground for any meaningful criticism, and certainly not

for the kind of attack to which Derrida was subjected during the Science

Wars. At most, if one is inclined to criticize, one can see Derrida's remarks

as irrelevant as far as relativity is concerned. In this latter case, one has from

the ethical standpoint no choice but to acknowledge that the circumstances

"But It Is Above All Not True" * 199

contributed to the questionable character of his statement, especially given

that Derrida is customarily cautious in his statements concerning mathe-

matics and science, and the relationships between them and his own work.60

If, however, one does want to suggest connections between Derrida's work

and relativity or to establish them rigorously, one is free to do so. One has

to extract and indeed to construct (including in Deleuze's sense of concep-

tual construction) a "Derridean" conceptual matrix for relativity out of

Derridean and related philosophical materials, as suggested earlier. It is out

of a combination of Derrida's conceptuality and that of (the philosophy of)

relativity that the propositions mentioned earlier, such as "constants" of

Einstein's relativity are manifestations of variability, decentering, play (in

Derrida's sense), and ultimately the nonclassical efficacity of differance, are

produced. Is a claim of that nature-that is, that Einstein's relativity is, at

least in part, a Derridean theory in this sense-supportable? I think so. At

the least, this engagement of Derrida's and related nonclassical work

appears to lead to important philosophical questions concerning both. At

the very least, this suggestion can be put on the table, and it would, I think,

be difficult to criticize anyone for doing so.



Chapter 5

Deconstructions

After a critique of the wave concept has been added to that of

the particle concept all contradictions between the two

disappear-provided only that due regard is paid to the limit

of applicability of the two pictures.

-Werner Heisenberg

As must be apparent from the preceding chapters and as this chapter will

further suggest, it is not always easy to disentangle different strata of non-

classical concepts or, as in the case of Derrida's "differance," neither terms

nor concepts. Nor is it always easy to relate effectively some among these

strata to the relevant mathematical and scientific ideas. This is in part why

negotiations, let alone building bridges, between the fields and cultures

involved may be difficult and may be bound to encounter unsurpassable

limits at certain points, at which the difference between the sides involved

(there may be and, in a certain sense, always are more than two) becomes

irreducible. Could one argue for a more symmetrical situation, whereby a

more technical expertise in the humanities could also be argued to be neces-

sary for a meaningful commentary? Some humanists do, and there is some

ground for this argument. As I have suggested from the outset of this study

in more general terms, I would not see the situation as altogether symmetri-

cal in this respect. Philosophy (speaking now of Continental, rather than,

say, analytic, philosophy) appears to occupy a space somewhere between lit-

erature (or art) and mathematics and science. There are also significant dif-

ferences between the latter as well, which may, however, be less germane for

the moment).1 It is true that a number of major mathematicians and scien-

tists had a considerable knowledge, if not strictly expertise, of philosophy

and were certainly able to read philosophy, and the work of some of them,

or in any event some of their work, can be read as philosophy, and is so read

in this study. It would be more difficult to find the reverse cases. Leaving the

Greeks aside, both Descartes and Leibniz, or differently, Galileo (but not

many others), pose complex questions in this respect, in particular as to

whether they may be seen primarily as philosophers or as scientists. Leibniz

appears to the present author to be primarily the former, but one would

202 * The Knowable and the Unknowable

want to be cautious here. It may be added that the problem is not only and

not so much the lack of the necessary expertise in, say, Deleuze or Derrida,

on the part of their scientific critics, but their lack of even a minimal engage-

ment with the texts they criticize. The latter would easily enable them to

avoid the more embarrassing errors and ethical mistreatments of the works

in question. Some expertise in this work would, however, be necessary in

order to perceive the proper character of the relationships between these

works and modern mathematics and science. My argument in this book is

that nonclassical thought can help us in these negotiations and bridge build-

ing, or indeed in developing the spaces themselves that we want to bring

together rather than hindering such efforts, as is often argued by its oppo-

nents. It does so at the very least in view of the nonclassical features found

in modern mathematics and science themselves. To these features nonclassi-

cal thinkers in mathematics and science and elsewhere are especially sensi-

tive, while their scientific critics often miss or refuse to confront the non-

classical for the reasons that I have discussed here beginning with the

preface to the book.2

I further argue that, even if and to the degree that nonclassical authors

(speaking of those of them who are not mathematicians and scientists) are

technically wrong sometimes, their work does tell us something philosoph-

ically significant and deep about mathematics and science, without, as I

have stressed throughout, infringing on the disciplinary specificity of the

latter. At the very least, sometimes through deliberate effort (as in Deleuze),

sometimes despite their reticence to engage mathematics and science

directly (as in Derrida), this work is an extraordinary manifestation of the

presence and power of mathematics and science in philosophical imagina-

tion or of course in literature and art. But then, mathematics and science

cannot be reduced to their disciplinary sense either, even assuming that we

can ever strictly delimit the latter (not an easy assumption, if possible at all).

These complexities, however, need not erase the disciplinary specificity of

either, which, I argue, must be rigorously respected, especially in all claims

we make concerning their findings. In other words, while Deleuze and Der-

rida do not do mathematics or science, they do sometimes tell us something

about mathematics and science. Bohr, of course, both does physics, experi-

mental theoretical physics, "that new type of theoretical physics . . . which

was more experimental than mathematical," as Heisenberg, who does

physics, too, aptly called it, and also tells us a great deal about it, as does

Heisenberg.3

By contrast, much of the recent (Science Wars) criticism of these authors

by scientists tells us virtually next to nothing either about the nature of

Deconstructions * 203

mathematics and science or about their role in intellectual history and cul-

ture. Naturally, I am not saying that we have nothing to learn, in philo-

sophical and cultural terms, from mathematicians and scientists in their

popular or otherwise more accessible accounts, or indeed from their techni-

cal works. (The humanists might want to and, I would urge, should some-

times consult the latter as well, which is not as impossible as it may seem.)

As I have stressed throughout, the opposite is true, as many works by math-

ematicians and scientists cited in this study would demonstrate. One cer-

tainly finds an immense power of philosophical thought, classical and non-

classical, in mathematics and science, as in the cases of Bohr or Heisenberg

or, on the (more) classical side, Einstein and Schrodinger, which are espe-

cially crucial for this study. A list of such examples would be long and

would include a great many of the major figures whose work and ideas

shaped the history of mathematics and science, and sometimes philosophy

and literature and art, and an even greater number of lesser-known figures.4

These works, however, are something altogether different from what one

finds in Gross and Levitt, Sokal and Bricmont, and several related writings.

I shall, in this chapter, argue that the representation of mathematics and

science in the work of the Science Wars figures in question suffers from

problems that are not altogether unlike, or complicit with, those they

bemoan in the works in the humanities that they criticize. I do speak here of

their treatment of mathematics and science, rather than of the humanities,

where the problematic character of their treatment and the symmetry just

indicated are much more pronounced and have been extensively commented

upon (while, by contrast, their treatments of mathematics and science have

not been much, if at all, considered). By "complicit" I mean that the prob-

lems in the exposition of science that one finds in Gross and Levitt (and

other popular, and even some technical, accounts by mathematicians and

scientists) contribute to the problematic nature of the treatment of mathe-

matics and science by the humanists. This exposition also contributes to the

problematic and, at their worst, damaging impact of their interventions,

especially on those humanists or intellectuals (there are some) who are, for

various reasons, benign or problematic, sympathetic to their cause. This

exposition is certainly far from explaining mathematical and scientific ideas

with the lucidity and precision necessary for nonspecialists, who want to

understand the mathematics and science involved; and it is more likely to

increase some of the "postmodern" confusion than to help clear it up. We

have already encountered these problems in Sokal and Bricmont's book,

and I shall discuss others, specifically dealing with quantum mechanics, in

this chapter. Gross and Levitt's discussion of quantum mechanics in Higher

204 * The Knowable and the Unknowable

Superstition is, however, a particularly revealing and significant case here.

This discussion is also philosophically central to their book. Far from being

lucid, precise, and informative, their long footnote on the subject is a

significant misrepresentation of the current states both of quantum theory

itself and of the debates concerning it, and I shall consider it in some detail

here. This discussion comprises the first section of this chapter.

The second section, the final section of this study (a short conclusion is to

follow this chapter), returns to my argument concerning the significance of

nonclassical thinking, specifically Derrida's, for our understanding of mod-

ern mathematics and science. In this case, I shall be concerned with the rela-

tionships between Derrida's work and quantum mechanics, although the

key aspects of my argument here extend those of my argument concerning

Derrida's work and relativity in the preceding chapter. I shall, however, pur-

sue this argument via Heisenberg's, rather than Bohr's, work and via

Heisenberg's, rather than strictly Derrida's, "deconstruction." But then, I

shall also argue that this deconstruction is in fact, or in effect, essentially the

same as Derrida's, which allows one to remove the quotation marks around

deconstruction here. Indeed, for those familiar with Derrida's work it is

difficult to avoid the impression that Heisenberg, writing in 1929, has in

fact read Derrida, which is factually impossible, of course. Or, perhaps, it is

Derrida who read Heisenberg's 1929 text, which is factually unlikely but

culturally not impossible, however indirect and multiply mediated such a

reading may have been. Even though Derrida does not appear ever to refer

to quantum mechanics, it is almost as impossible for him not to have "read"

the philosophy of quantum mechanics, anymore than that of relativity, from

one source or another, in one way or another, through one mediation or

another, for example, via, among others, such figures as Bataille (for whom,

in particular, quantum mechanics was a more or less direct epistemological

source), Blanchot, and Lacan, with whose work he has engaged extensively.

Quantum Mechanics and the Science Wars

While germane to the key philosophical and epistemological questions at

stake, the discussions of quantum mechanics, by both sides of the Science

Wars, rarely bring additional clarity and often increase the confusion. The

complexity of the subject and of the debates surrounding it is in part respon-

sible, but is hardly sufficient, to justify the poor quality of these discussions,

especially on the part of the scientists involved. Consider Sokal and Bric-

mont's discussion of "linearity," in general a rather heated subject in the

Deconstructions * 205

Science Wars debates. I leave aside their hazy sense of the philosophical idea

of linearity (or of the philosophical aspects of the idea of the line, mathe-

matics included) from Aristotle on. It is crucial to its usage in philosophical

discourse, specifically (for example, in considering the question of tempo-

rality) in Heidegger's and Derrida's work, which significantly shaped cur-

rent discussions of linearity. More unfortunately, their mathematics and

physics are confusing and misleading as well. Thus, they say, "one may

speak of a linear function (or equation): for example, the functions f(x) = 2x

and f(x) = -17x are linear, while the functions f(x) = x2 or f(x) = sin x are

nonlinear."'5 There is no problem so far. But here is an (unfortunately,

habitual) jump: "Likewise, [while] quantum mechanics is often cited as the

quintessential example of a 'postmodern [nonlinear] science,' the funda-

mental equation

of quantum

mechanics-Schrbdinger's equation-is

absolutely linear" (145). This is careless. First, while Schrbdinger's equation

is indeed linear (and this fact is indeed crucial to quantum mechanics and its

epistemology), it is a (linear) differential equation; and it would be mislead-

ing to think about it, let alone of the so-called v-function that satisfies it, in

the terms just indicated by Sokal and Bricmont. This function, which bears

significantly on the nature of quantum theory, is itself not linear. The lin-

earity of Schrbdinger's equation manifests itself in the fact that it is linear

with respect to the functions to which it applies. That is, if you apply an

operator (a mathematical entity different from function) H to the sum of

two functions satisfying Schrbdinger's equation, w and p, H(W + (p)= H(W) +

H((p), and if a is a number H(aW) = aH(W). In this case, moreover, we are

dealing with vectors in a Hilbert space, which is associated with the quan-

tum system in question and which is, in general, infinite-dimensional. So,

unless one has a substantial (at least for a non-physicist) substantive knowl-

edge of the mathematical formalism of quantum theory, Sokal and Bric-

mont's commentary will not be especially enlightening or helpful. It is true

that ordinary linear functions, such as those mentioned by Sokal and Bric-

mont, behave in the same way (obey the same linearity rules) with respect to

their variables. It would, however, be beneficial and indeed essential to

explain the differences just indicated for nonspecialists, in part because

(similarly to causality) the linearity of quantum theory is itself a complex

issue. The mathematical equations (or the Hilbert-space formalism) that

account for the experimental data of quantum mechanics are linear. The

processes themselves, however, considered in quantum mechanics can

hardly be characterized as "linear."

Steven Weinberg, too, calls "an English Professor Robert Markley" to

task for "calling quantum theory nonlinear, though it is the only known

206 * The Knowable and the Unknowable

example of a precisely linear theory."6 I assume that Weinberg means "in

physics," for otherwise the statement is simply incorrect; it is actually not

altogether correct even in physics, but a small matter here. Markley may or

may not be confused as concerns some of these questions. If he is, his con-

fusion, while unfortunate, is understandable, given that quantum theory is

concerned with processes that are themselves nonlinear, or conceivably

indescribable by any mathematical, or other, means, linear or nonlinear.

This is still leaving aside that some quantum field theories (for example, the

so-called Yang-Mills theories, to which Weinberg himself made major con-

tributions) are in fact nonlinear. Common claims concerning the determin-

istic nature of quantum theory, specifically of Schrodinger's equation, or

concerning the nature of quantum reality involve similar complexities and

require much explanation and qualification, both substantively and as con-

cerns different views of these matters even among physicists. These facts,

once not properly sorted (as they are often not in popular or even technical

literature), could easily contribute to the confusion of those in the humani-

ties. But Weinberg does not much clarify these matters either, as he could

and, I would argue, should have. Indeed, to some degree he even adds to the

confusion. However, Weinberg at least makes well-taken and important

points, specifically as concerns similarities between Newtonian and quan-

tum physics, in commenting on Andrew Ross's formulation, which is

indeed not careful or well informed.7 As we have seen, however, quantum

mechanics may not account for the behavior of its objects in the same way

classical physics does, even though, as we have seen earlier and as Weinberg

correctly argues (against Ross), in its disciplinary practice as physics it is as

quantitative and rational as is classical physics.8 Weinberg himself makes

this difference clear on this very occasion, albeit in part against the grain of

his argument.

In their long note on quantum mechanics, then, Gross and Levitt offer

the following commentary:

The assertion that deterministic causality is still viable within the phenome-

nal world of quantum mechanics may come as a surprise to people far better

informed than [Stanley] Aronowitz, but recent work in the mathematical

foundations of the subject seems to support it strongly. A beautiful result by

Diirr, Goldstein, and Zanghi [hereafter DGZ] shows that a large part of clas-

sical quantum theory emerges naturally from a dynamical model which is

deterministic through and through, and in which the only "hidden" variables

are thoroughly classical variables like position and velocity ....

This work amplifies and makes rigorous rather old ideas of the philoso-

pher/physicist David Bohm, vindicating him and bluntly contradicting the

Deconstructions * 207

"Copenhagen interpretation" of Bohr and Heisenberg. Aronowitz mentions

Bohm briefly and skeptically.., but seems to be unaware that Bohm's pro-

gram is not just philosophical but involves a specific strategy, now seen to be

quite fruitful, for doing mathematical physics.

We emphasize, of course, that this particular work only rederives the clas-

sical (i.e. nonrelativistic) quantum mechanics of a collection of point-masses

in a potential field. The philosophical point is strongly clear, however, since

the standard debates use this case as a touchstone. The moral, if one must be

whimsical, is that Occam's Razor may now cut through the leash that hereto-

fore bound us to Schrodinger's cat, that half-and-half beast weirder by far

than Centaur, Sphinx, and Hippogriff. In other words: (a) Einstein was right:

God does not play dice with the cosmos. (b) On the other hand, since indi-

vidually and collectively, we are not God, nor Laplace's Demon, nor any

other demiurge of comparable intellectual power, we must inevitably regard

the universe as something of a crap game. As far as (b) is concerned, note well

that even if the universe were purely Newtonian in the best eighteenth-cen-

tury tradition, and the perplexities of quantum mechanics entirely avoided,

the crap game would still be inevitable. This follows from the work of Poin-

care on classical mechanics and from that of his latter-day disciples, which

goes under the fashionable name "chaos theory."9

I omit, for the moment, their technical summary of DGZ's model, which

is not necessary for my argument, since I am less concerned at the moment

with mathematics and physics as such than with Gross and Levitt's claims.

I shall, however, return to some aspects of this summary presently. Their

exposition of this model would not be accessible to nonspecialists in any

event. They do see their summary as one "for readers with a little knowl-

edge of mathematics and science" (261; emphasis added). It depends of

course on how one defines "a little." In order for this summary to be useful

it would require at least two years of college mathematics and physics,

maybe a year if one were a mathematics or physics major in a better college

and had very good teachers. Indeed, one would need a good (technical)

introductory course on standard quantum mechanics and some differential

geometry (not, incidentally, the same as differential topology). How would

one otherwise follow statements such as "the Laplacian [in Schrodinger's

{partial differential} equation] ... and gradient [involved in the ordinary dif-

ferential equation added in Bohm's or DGZ's version of quantum theory]

are given in terms of a Riemannian metric scaled by the masses of the parti-

cles" (262 n. 9)? Even if one were, justly, to lament the dismal state of math-

ematical and scientific education in this country, this statement would seem

to put a rather excessive demand on a lay reader. Understanding it would

require something close to a graduate-studies level of knowledge of mathe-

208 * The Knowable and the Unknowable

matics and physics. I am not saying that some of these things cannot be

explained otherwise; they can. They are just not so explained by Gross and

Levitt, who could have presented the key ideas involved in a more accessi-

ble form, and indeed much better and more lucidly and, most of all, more

accurately than they do here. (The phrase "the phenomenal world of quan-

tum mechanics" in their first sentence is hardly self-evident, if meaningful,

and certainly requires further explanation, but we may let it pass.) If they

did, their energy would be far better spent than in their incompetent treat-

ment of the humanities, although it appears that they developed a kind of

taste for public discussions of the latter. Perhaps they will learn to do it right

eventually.10

As the text stands, even readers with a little more than a little knowledge

of mathematics would only have Gross and Levitt's authority to assume, for

example, the "modest and unproblematic" nature of the "assumptions [in

DGZ's theory]" that make, "statistically speaking, the standard quantum

mechanical formalism appl[y]," while, in contrast to the overall standard

theory in the standard, noncausal interpretation, the underlying dynamics is

Bohmian and hence causal (262n.9). Well, these readers may not take this

authority for granted, and they should not. The assumptions in question

may be (may be!) modest and unproblematic in terms of the mathematics

involved. Their physical justification and the overall status of DGZ's theory,

and hence of how "viable" deterministic causality is in quantum theory, are,

as will be seen momentarily, quite another story.

More generally, some (less informed) readers may get an impression here

that one is dealing with a reasonably established theory offering an alterna-

tive to the standard (nonrelativistic) quantum mechanics. Such is not the

case. At best, as far as physics is concerned, even in the nonrelativistic case

in question here we are dealing with a hypothetical proposal, by no means

widely entertained even as such, especially outside a rather contained and,

by and large, marginal community of adherents to Bohm's views. The latter

is of course not in itself an argument against Bohmian theories themselves

but against Gross and Levitt's presentation of the situation. DGZ them-

selves do see considerable promise in their theory, or in the Bohmian

approach in general. Goldstein is one of the staunchest advocates of Bohm

and opponents of Bohr, as is also Bricmont, which may not be coincidental,

given that Bohm's is a classical-causal and realist-theory, and which I

shall discuss presently. Their actual claims for it are more circumspect than

those made for them by Gross and Levitt. One would easily see from, for

example, an account given by James Cushing (no friend of the Copenhagen

interpretation and especially of Bohr, and a sympathetic reader of DGZ)

Deconstructions * 209

that DGZ's theory is fundamentally hypothetical. In many respects it is ten-

tative even as a hypothesis, including as concerns the assumptions in ques-

tion. The latter involve no less than the initial configuration of our universe,

obviously a wide open question. Cushing says: "The crucial question here is

how convincing or applicable one finds probability arguments applied to a

singular event (i.e., the actual and only initial configuration of our uni-

verse)."11 Besides, DGZ's theory is only one of several competing attempts

to develop Bohm's views so as to give them more viability. There have been

several versions of the theory proposed by Bohm himself over several

decades.12 These attempts are of course legitimate, and sometimes interest-

ing, even though the attitude toward Bohmian theories in the physics com-

munity remains skeptical, for several reasons.

As I have noted earlier, some among Bohmian theories are different from

quantum mechanics also in terms of their mathematical formalism rather

than only in terms of the physics they imply, such as causality or nonlocal-

ity. Hence, these theories must be seen as different physical theories

accounting for the same data, rather than different interpretations of the

standard quantum mechanics. As Gross and Levitt point out, in view of

additional assumptions (and this is in part why these assumptions are intro-

duced) in DGZ's version of Bohmian mechanics, the standard quantum-

mechanical formalism itself would apply in statistical terms. However, in

contrast to the standard quantum mechanics, the underlying (causal)

dynamics is described by a different formalism, since, unlike in the standard

quantum mechanics, well-defined trajectories are assigned to particles.13

Another attractive feature of DGZ's theory is that it uses as its "hidden"

variables standard physical variables rather than "hidden" parameters of an

unknown and perhaps unknowable nature, as do other versions of Bohm's

theory, beginning with Bohm's original (1952) version. (This is why these

theories are called "hidden variables" or "hidden parameters" theories.)

However, beyond the extraordinary practical success of the standard ver-

sion and its extensions (in particular to quantum field theories), Bohmian

theories possess certain (for most physicists) undesirable features that make

their status problematic.

Gross and Levitt fail to mention that, in contrast to quantum physics,

Bohmian theories, DGZ's included, definitionally entail the violation of

locality and hence of the standard requirements of relativity theory, even

though in some versions of these theories such violations cannot in fact be

observed. Even in theses cases, however, this violation, and hence a conflict

with relativity, is a built-in feature of the theory, which is not the case in the

standard quantum mechanics, at least by a large majority of accounts of the

210 * The Knowable and the Unknowable

latter. This is arguably the main reason why these theories are suspect to

many practicing physicists. At earlier stages of these developments, one of

the skeptics was Einstein, who was on this (and other) grounds rather nega-

tively inclined toward Bohm's original version of his mechanics, published in

1952.14 As we have seen, the question of the nonlocality of quantum theory,

initiated by Einstein and brought into the foreground in the wake of Bell's

theorem, involves subtle arguments. Gross and Levitt claim that "Bell's the-

orem . . . demonstrates nonlocality of the quantum universe" (presumably

regardless of a particular version or interpretation of the quantum data)

(278n. 21). This statement is inaccurate, if not outright incorrect, at least in

this form and without much further qualification, beginning with what they

mean by "nonlocality." There are significantly different concepts of nonlo-

cality (related, but often different, concepts include nonseparability and

entanglement) differently used in the context of Bell's theorem or the class of

corresponding theorems. Bell's theorem demonstrates the nonlocality of par-

ticular quantum theories, similar to Bohm's in their assumptions. There are,

it is true, those who argue for a general nonlocality of the quantum world.

As I have explained in chapter 2, however, such arguments, if tenable at all,

are by no means definitive or widely accepted in the physics community. In

any event, the existence and the nature of these arguments would confirm my

overall point here, as concerns the presentation of the state of quantum the-

ory and the debates concerning it. Certainly, nonlocality, manifest or not, is

seen as an undesirable feature of any physical theory by most physicists, and

Bohmian theories are demonstrably nonlocal.

Specifically on this subject and in more general terms, Goldstein's

account-mostly an apologia for Bohm's work-of the current state of the

interpretation of quantum mechanics in Physics Today is misleading as well.

This may be said, even if one leaves aside the problems of his exposition of

the Copenhagen interpretation itself, which hardly does justice to the com-

plexity of, specifically, Bohr's thought.15s This is in a rather sharp contrast to

Bohm's own account and views. Bohm treats Bohr's arguments and his

thought with the greatest respect and sees it as crucial for his own views.

Specifically, on nonlocality, Goldstein says that "the implications of [Bell's]

work have been widely misunderstood as demonstrating the impossibility of

hidden variables rather than the inevitability of nonlocality" (emphasis

added).16 If this is the case, among those who misunderstood Bell's work

was certainly Bell himself. I was unable to find in any of Bell's articles, for

example, those assembled in The Speakable and the Unspeakable in Quan-

tum Mechanics, an argument for the inevitability of nonlocality, and I do

not think that one can find such an argument there. If anything, Bell sees the

Deconstructions * 211

latter as an undesirable feature, even if a hypothetical, but only hypotheti-

cal, possibility (just as that of hidden variables). According to Bell, it is no

more, and perhaps less, likely than Bohr's interpretation, however unap-

pealing the latter was to Bell.17 In any event, nowhere does he say that the

available data of quantum mechanics entails nonlocality.

The nonlocality of quantum mechanics is, at best, a controversial

hypothesis. However, a number of recent prorealist, and specifically pro-

Bohmian, arguments concerning quantum mechanics and its epistemologi-

cal implications, including for the contemporary cultural debates, such as

the Science Wars, depend on it and present it as a scientific fact rather than

a possibility. For example, Mara Beller (a pro-Bohmian historian of science,

mentioned earlier) significantly relies on it in her "The Sokal Hoax: At

Whom Are We Laughing?" in Physics Today.18 Beller argues that the

"philosophical pronouncements" of some among the founders of quantum

mechanics are comparable to some postmodernist arguments, in which she

follows Goldstein's article, "Quantum Philosophy: The Flight from Reason

in Science," published in Gross, Levitt, and Lewis, The Flight from Science

and Reason, cited earlier. This point itself is not wrong, or rather it can be

properly made. It is not, however, so made either by Beller or Goldstein.

Goldstein's article lumps together "postmodernist" authors (the present

author among them), who not only have little in common but whose argu-

ments are also clearly in conflict with each other, which makes one doubt

whether Goldstein had actually read the works in question. Be that as it

may, the main point at the moment is the representation of quantum

mechanics and the thought of the key founding figures, both in general and

as grounded in the argument for nonlocality. Thus Beller attributes to Bohr

a "positivistic dictum that quantum theory is merely a predictive and

descriptive calculational tool," to which, as we have seen, Bohr does not

subscribe, and indeed, as we have also seen, he would not see quantum

mechanics as a descriptive tool, perhaps more an undescriptive tool.19 It is

of course a predictive tool, but it is also much more than that. His position

is not positivist in general, as should be clear from the earlier discussion in

this study, and he makes it clear throughout his writings. Beller also appears

to misrepresent John Archibald Wheeler's position and his view of Bohr.20

The responses to Goldstein's and Beller's articles in Physics Today show

a very different picture of the state of the interpretation of quantum

mechanics, the status of experimental and theoretical findings involved, and

the opinion of the physics community than that offered by Goldstein and

Beller, or, again, Gross and Levitt. As Murray Gell-Mann and John Hartle's

and Robert R. Griffiths's letters suggest, Goldstein's account of the so-called

212 * The Knowable and the Unknowable

histories interpretations of quantum mechanics suffers from several prob-

lems, physical and philosophical, similar to those found in his account of

Bohr.21 Let me reiterate that it is not a question of objecting to a search for

alternatives to, say, Bohr's interpretation but of the nature of the claims

concerning it, or the nature and state of quantum physics.

It becomes clear that Gross and Levitt's "moral" in no way corresponds

to either the actual state of quantum theory or the debates about it, either

among physicists themselves or among the philosophers of quantum

physics. The support for the deterministic causality of the quantum world

does not seem (at least they say "seem") as strong as Gross and Levitt's

account may suggest, and while their "philosophical point" is indeed

"strongly clear," it is by no means a strong point. Certainly, if one wants to

invoke Occam's Razor (which is not easy to apply in this situation), it

appears to cut pretty much the way it has for a while now, including as con-

cerns DGZ's "modest and unproblematic assumptions." That is, again, not

to say that there is no debate concerning various aspects of quantum

mechanics; quite the contrary, as we have seen throughout this study, this

debate continues. Indeed, it is this debate that makes the status of DGZ's

theory very different from what Gross and Levitt's account suggests. What

makes the invocation of Occam's Razor difficult in this case is the wide

diversity of versions of quantum theory, of interpretations of each such ver-

sion, and of general philosophical views (often different even within each

version or interpretation), as discussed earlier. DGZ's theory would have to

be something different in its assumptions and conclusions and have a differ-

ent status in the physics community in order for Gross and Levitt's claim to

correspond to the current state of our knowledge concerning the physical

world and of our debates concerning this knowledge.

In general, while perhaps not quite as confused as the more problematic

expositions of these physical theories in the humanities, Gross and Levitt's

exposition can easily serve as a source of such confusion. I am perfectly will-

ing to see these problems as a result of sloppiness rather than misconcep-

tion, although the latter case may be made as well. This sloppiness is far

from unproblematic in all circumstances but is especially disturbing given

their belligerently unforgiving critical stance. Indeed, this sloppiness is

sometimes reminiscent of the kind of irresponsible or, in Gross and Levitt's

language, "idle" use of technical "phrases," against which they protest

when mathematical and scientific ideas are used by humanists, and which is

more troubling when found in the commentary by scientists. On the other

hand, in fairness, but also as a stronger point, it reflects the complexity of

contemporary mathematics and science and the fact that mathematicians

Deconstructions * 213

and scientists do not always have proper expertise in areas outside their

own. This is, obviously, no sin; and nobody (naturally, the present author

included) is guaranteed from getting certain things not quite right or even

wrong altogether. But the situation does requires a very different type of

attitude toward all discourses involved from that of Gross and Levitt, Sokal

and Brictmont, and most of their fellow critics and supporters, or of course

from many of those on the opposite side of the Science Wars.

It would also be nice if Gross and Levitt explained what Laplace's

Demon and Schrbdinger's cat are. Both ideas are pertinent to the argument

in question and the reader's ability to follow and access it. Laplace's demon

is a superhuman intelligence that would know all the positions of all the

particles (i.e., all the ultimate constituent atoms of the universe) and all the

forces that act upon them. Accordingly it would be able to predict strictly

(rather than in terms of probabilities) the future state of the universe at any

given point, by using the laws of the Newtonian mechanics or some version

thereof. This "program," however, is bound to fail even in classical physics,

at least given the actual laws of classical physics, as we know them, laws

that are, of course, demonstrably incorrect beyond certain limits.22 All cor-

responding theories (gravity, electromagnetism, and so forth) are classical

and specifically causal and realist. They do not, however, appear to be

unifiable, or even reconcilable with each other, even within classical physics

and leaving aside the limits, sometimes stringent, within which each of these

theories are applicable. Nor, in view of quantum theory and indeed already

relativity, does it appear, at least for the moment, that there is a classical-

like set of alternatives for different forces that will enable such reconcilia-

tion. (Obviously, the role of interpretation, as considered in chapter 1, must

be kept in mind throughout, whatever physical theories one considers, clas-

sical or not.) The so-called quantum gravity and other unifying programs

(such those emerging from string and, by now, "brane" theories) aim to do

so at least in principle. Whether the outcome will ultimately be classical-like

or nonclassical, or whether any such unification is in fact possible, is unclear

at present. Various Bohmian programs are designed in part in order to make

ultimately a classical-like view of the physical world possible, at least in

principle. If they succeed, however, it may be, in its rigorous specificity, nei-

ther Newtonian nor Laplacian, since Bohmian mechanics is not Newtonian.

Both, however, are, in terms introduced earlier, classical configurations of

chance, in contrast to that of quantum mechanics, where one encounters

Schrbdinger's cat.

Schrbdinger's cat is no particularly "weird or special beast" (I realize, of

course, that this characterization may be applied by Gross and Levitt to the

214 * The Knowable and the Unknowable

nature of the phenomenon in question) but a regular, if hypothetical, cat. It

could be replaced by any other animal, human beings included.23 In the

famous thought-experiment invented by Schrodinger, this cat is (hypotheti-

cally) locked in a box inside which a certain quantum, and hence irreducibly

probabilistic, event may or may not take place. If it does, it will trigger a

device (also installed in the box) that will kill the cat. We would not know

the outcome before opening the box, which we can do long after the event

did or did not take place inside the box, although this time, too, is subject to

uncertainty.24 (The box-with the subsystem capable of generating the

quantum event in question, the triggering device, the cat and all-is consid-

ered as a quantum-mechanical system.) The reason for this is that such a

quantum event has only a given probability (say, 50 percent) of occurring;

and this probability is not subject to a classical account, according to the

standard quantum mechanics in the standard interpretation(s). On the other

hand, it is indeed difficult to assume that the cat would not be definitively

dead or alive inside the box immediately after the quantum event in question

took place inside the box. The "paradox" (not everyone sees it as such)

requires a subtle analysis and remains a part of the debate concerning quan-

tum mechanics, a subtlety that need not be followed here. It may be observed

briefly that this subtlety has mainly to do with the relationships between the

quantum microworld and the classical macroworld, and the character of the

ultimate constitution of the latter out of quantum micro-objects (an open

question in modern physics), as considered earlier.25 Hence it has to do with

the question of to what degree the whole arrangement can be considered as

a quantum system. For example (this would be more or less Bohr's view),

one can see the (classical) cat as itself a macro-"observer" of a quantum

event inside the box, and hence as dead or alive, depending upon whether the

event did or did not take place. In this view, there is no paradox.

Thus, it is indeed true that chance appears to be in practice irreducible

from any account of the physical world. This would be the case whether we

subscribe to radical quantum-mechanical chance, chance that is irreducible

both in practice and (in contrast to the classical chance) in principle, or clas-

sical chance, irreducible only in practice but leaving the causality in princi-

ple intact. The latter would be the case in classical mechanics, classical sta-

tistical physics, chaos theory, or Bohm's version of quantum mechanics.

These are all different theories, physically and mathematically. In particular,

as I said, Bohmian mechanics, while causal and otherwise classical, is not

Newtonian.26 Accordingly, the parallel with other classical theories would

have to be significantly qualified. As stated by Gross and Levitt, this paral-

lel is far too loose. More generally, the status and roles of these theories in

Deconstructions * 215

our description of the physical world on different (macro and micro) levels

are quite different, even if one maintains the underlying causality in all

cases. These differences and nuances, blurred by Gross and Levitt, are

important, in particular if one wants humanists to avoid the kind of confu-

sion about science that bothers Gross and Levitt and their fellow critics.

Poincare's extraordinary findings, which are indeed one of the early inti-

mations of chaos theory, concern the behavior of classical macro-objects in

a gravitation field. They show the general impossibility of predicting, even

in principle and even in idealized situations, this behavior (once the number

of such bodies is three or more), even though we know the (Newtonian,

causal) laws of such behavior, specifically his law of gravitation, and can

write corresponding equations for it. In other words, the behavior of such

systems, in an ideal case, is in principle causal, but (deterministic) predic-

tions of such behavior are not possible even in principle, beyond certain lim-

its. This behavior is too sensitive in its dependence on initial conditions.

That is, a small change in such conditions can lead to a big change in the

behavior of the system. In other words, in the case of the theories assembled

under the rubric of chaos theory, indeterminism arises from the nature of

the nonlinear equations involved describing their objects. The underlying

dynamics may be assumed to be causal, but the (causal) fluctuations in

behavior involved are well beyond any meaningful deterministic prediction.

In any event, the equations are assumed to map the behavior of these objects

as such, which may or may not be constituted by more or less large multi-

plicities of more elementary objects. In the case of classical gravitation,

already three bodies would do; in the case of (long-term) weather predic-

tions, or the impossibility thereof (defined by systems whose behavior obeys

chaos theory), we deal with much greater multiplicities of objects. As I have

indicated, actual systems are often chaotic. The point of chaos theory is that

deterministic predictions are, in general, impossible even in idealized situa-

tions. Conversely, it is possible that a particular actual system may ulti-

mately prove to be nonchaotic. For example, the question of whether the

solar system, widely expected to be chaotic, actually is chaotic is a matter of

some controversy.

In contrast to chaos theory, in classical statistical physics the origins of

the statistical nature of our predictions are specifically the multiplicity of

objects involved, such as molecules of a gas, each behaving according to the

(nonchaotic or not necessarily chaotic) equations of Newtonian mechanics.

Gravity is disregarded in this case. Once it is brought in, even as Newton-

ian, let alone as that of Einstein's general relativity, the complexity becomes

immense and perhaps unsurmountable in principle. It may, for example,

216 * The Knowable and the Unknowable

involve certain nonlinear dynamics, akin to those of chaos theory, in addi-

tion to the standard statistical considerations. Accordingly, the formulas of

statistical physics, enabling our statistical predictions, do not describe the

(causal) behavior of these objects as such, even as they generally presuppose

such behavior, again, in contrast to standard Newtonian mechanics, even in

idealized situations. Thus, in contrast to chaos theory, in this case we have

in principle two sets of descriptions: one maps the actual behavior of objects

involved, and the other does not, while it depends on this behavior in estab-

lishing the counting procedures that enable statistical predictions of the the-

ory. In certain cases, dealing with greater multiplicities of objects, in chaos

theory the classical laws of Newtonian mechanics may be involved as well,

for example, in our assumptions concerning the behavior of certain compo-

nents or elements of chaotic systems. This would, however, not eliminate

the difference between the two types of systems just described. One of the

main differences is that, just as does quantum mechanics (which, however,

is neither causal nor realist, at least, again, in Bohr's interpretation), classi-

cal statistical physics makes excellent statistical predictions of the behavior

of the systems it considers, while the behavior of chaotic systems (described

by chaos theory) is, beyond certain limits, unpredictable. In this sense, the

equations of chaos theory "predict" the unpredictability of the physical

behavior of the systems in question, while the formulas of classical statisti-

cal physics make statistical predictions. The theory does show certain com-

plex patterns or forms of order. One can see these patterns in, by now

famous, pictures and computer simulations. Indeed, a better name for it

would be the "order theory," the theory of certain complex and unpre-

dictable forms of order.

In quantum theory (in the standard, rather than Bohmian, version and in

Bohr's interpretation) the ultimate constituents of matter, sometimes called

"elementary particles" (a term applicable at best only provisionally in

Bohr's interpretation) are not and cannot be assumed, even in idealized

models, to exhibit causal (or indeed any describable) behavior; and individ-

ual quantum events are, in general, not subject to physical law. (Further

qualifications concerning the nature of this individuality, as considered ear-

lier, must be kept in mind.) Nor, in dealing with quantum statistical multi-

plicities, can "particles" be seen as individually distinguishable, as they can

and must be in the case of classical statistical physics (that is, we cannot

assume that we can mark, "flag," each and trace, even in principle, the indi-

vidual trajectory of each). On the other hand, as in classical statistical

physics and in contrast to chaos theory, excellent statistical predictions are

possible and are made possible by quantum theory. This theory, accord-

Deconstructions * 217

ingly, is a theory of (irreducibly) statistical predictions, correlations between

macroscopically observed experimental events, and so forth, rather than a

theory describing individual objects and their behavior in the way classical

physics does. Indeed, quantum mechanics (again, at least in Bohr's and

related interpretations) tells us that such a classical-like mechanics of indi-

vidual quantum objects is in principle impossible. Bohmian (causal) theo-

ries, such as DGZ's, are yet another story, and their relationships to other

theories just mentioned, in particular chaos theory, are a complex matter

and the subject of ongoing investigations. As I said, considerable further

epistemological complexities, as yet barely explored, in our understanding

(or the lack thereof) of quantum physics are introduced by quantum field

theories, the currently standard way of theorizing what we see as the ele-

mentary (quantum) level of matter.

One might say that the scientific and philosophical expectation, concern-

ing either exact physical description or approximation and statistical pre-

diction, built jointly by classical Newtonian mechanics and classical statisti-

cal physics were defeated from two sides, that of chaos theory and that of

quantum mechanics. The first made apparent that the behavior of certain

systems is not predictable even statistically and even in principle, although

the underlying dynamics is ultimately causal. The second (again, at least in

certain interpretations) showed that while, in the case of quantum systems,

excellent statistical predictions are possible, the individual behavior of the

ultimate constituents of the quantum world is irreducibly noncausal.

Indeed, in this type of interpretation, it is not subject to any realist descrip-

tion (in the sense defined earlier), since individual quantum events are seen

in general as not subject to physical law. Bohmian mechanics, such as

DGZ's theory, is aimed at "correcting" these "deficiencies" of quantum

mechanics, or what are perceived as deficiencies by some.

These are well-known considerations, and I repeat them here for the fol-

lowing reasons. First, it is helpful to have a more clear and comprehensive

picture of what is at stake here. Second, this picture further exposes the

problems, rather than merely the incompleteness, of Gross and Levitt's

statements cited earlier. One may say, as Gross and Levitt do, that "even if

the universe were purely Newtonian in the best eighteenth-century tradi-

tion, and the perplexities of quantum mechanics entirely avoided, the crap

game would still be unavoidable." It is worth keeping in mind, however,

first, the significant differences between different classical crap games, such

as those of chaos theory versus classical statistical physics, and, second, the

fundamental and irreducible difference between the two "crap games," clas-

sical and nonclassical. This latter difference is of immense significance even

218 * The Knowable and the Unknowable

technically (very different formalisms are required in each case), let alone

epistemologically. To say, however, as Gross and Levitt do, that "this [i.e.,

the unavoidability of the crap game in the classical case] follows from the

work of Poincare on classical mechanics and from that of his latter-day dis-

ciples, which goes under the fashionable name 'chaos theory,"'" is hardly

accurate. This statement certainly requires much further qualification. As it

is, the statement appears to be very much in the "classical" tradition of con-

fusion, indeed common enough among humanists, about these matters; and

here again a statement like this is more likely to cause or increase this con-

fusion rather than help to avoid it. There are different "Newtonian uni-

verses," different levels of classical description. If we speak of our predic-

tions of the behavior of celestial bodies (be they planets of the solar system,

stars in a galaxy, or galaxies in the universe) in a gravitational field, Gross

and Levitt's statement is more or less true. If we speak of the universe as a

collection of atomic objects (suspending their gravitational force), it is not

true, since the crap game would then be that of statistical physics, possibly

with that of chaos theory. If such, or more elementary, objects are subject to

a Bohmian mechanics, such as DGZ's theory (which does not take into

account gravity or even electromagnetism), this statement is not true either.

Nor is this statement true, if we take, in a Laplace-like manner, all the

forces of nature into account, even if we assume them to be classical-like.

We have very little sense at present how such different classical ways of

description can come together even in idealized cases (and hence leaving

aside whether and to what degree classical theories correspond to actual

physical data). The history of modern physics is the history of dealing with

the complexities of both types-those of bringing classical physics together

and those of its correspondence with the available data, including the data

that eventually led to relativity and quantum physics. The success has been

limited on both scores. For a program of the Laplacian type to succeed one

would need to reconcile too many different classical configurations and

laws, which in the end have proven to be irreconcilable, even in principle.

One might argue that the preceding commentary is merely nitpicking,

since Gross and Levitt's elaboration is, after all, a ("casual"?) footnote. The

note and the argument, however, are hardly casual, even leaving aside that

Gross and Levitt's sloppiness is out of place, given their critical stance as

concerns the presentation of science in the humanities. It is by far the longest

footnote in their book, and for obvious reasons. First of all, the subject of

deterministic causality, or the lack thereof, in the physical world (or, again,

the possibility of a realist physical description), or the current state of

research in and debate on the subject, is hardly a casual matter. I am not

Deconstructions * 219

saying that the outcome of this debate, whatever it is, has a direct bearing

on how one sees the world elsewhere, for example, in the humanities and

the social sciences. One does not need quantum physics to argue the non-

classical epistemology to be applicable there, although the parallels are

instructive. On the other hand, these are not some "postmodernists," but

classical philosophers and scientists, who have, since Newton at least, used

the classical behavior of nature both as a model of and as support for their

arguments concerning human nature, individual or collective. Gross and

Levitt or Sokal and Bricmont do just the same, albeit in a much cruder way.

As I have argued throughout, it is one of the remarkable features of non-

classical theories, and possibly of the greatest anxieties of their opponents,

that these theories both incorporate classical theories and enable, better that

classical theories could in these circumstances, the disciplinarity and the dis-

cipline of the fields and projects that sustain, and are sustained by, classical

theories.

Nor is Gross and Levitt's (or, as we have seen, Sokal and Bricmont's)

deficient presentation of mathematics and science a casual matter. For one

thing, how are the humanists, in particular students, to correct their views

and improve their knowledge of mathematics and science, as Gross and

Levitt or Sokal and Bricmont appear to want them to do, if their own pre-

sentations suffer significantly in this respect? Of course, as I said, the

humanists should be aware that mathematicians and scientists do not

always have a sufficient knowledge or expertise in all areas of mathematics

and science on which they comment. My point is the profoundly problem-

atic nature of Gross and Levitt's (or Sokal and Bricmont's) book and related

recent publications. These problems concern not only (self-evidently) their

critique of virtually anything they criticize (whether the subject is worthy of

criticism or not) but also the nature and complexity, and sometimes strictly

technical accuracy, of the mathematics and science themselves, let alone

their relation to culture. The authors may say that it was not their goal to

discuss the latter subject. The point, however-and this may well be the

broadest and the most important critical point one can make here-is that

no meaningful criticism of what they call "postmodernism" on science is

possible otherwise.

Heisenberg's Deconstruction

My argument throughout this study has been that, in contrast to their Sci-

ence Wars critics, the best nonclassical thinkers do tell us something

220 * The Knowable and the Unknowable

significant, and sometimes deep, about the nature of mathematics and sci-

ence. The point is more or less self-evident when one encounters such sci-

entific nonclassical thinkers as Bohr and Heisenberg, which is not to say that

this point is insignificant even in this case in view of the fact that in question

is still what nonclassical philosophical thought tells us about mathematics

and science. We do, however, learn something about the philosophical and

epistemological nature of quantum mechanics from Derrida and de Man,

about complex numbers from Lacan, and about topology and relativity

from Deleuze and Derrida. Reciprocally, we also learn a great deal about

nonclassical epistemology from mathematics and science, even when we do

not have the benefit of Bohr's and Heisenberg's arguments. Certainly quan-

tum mechanics appears to offer us, to return to Bohr's locution, a tremen-

dous "opportunity of testing the foundation and scope of some of our most

elementary concepts." As we have seen in this study, however, we are deal-

ing with a spectrum of nonclassical mathematics and science, extending

from complex numbers to quantum mechanics, itself, mathematically, a the-

ory irreducibly involving complex numbers, arguably the first manifestation

of non-Euclidean mathematics in the broad sense of this study.

We are far from finished with this nonclassical testing, and perhaps new,

more severe and more unimaginable, tests and trials are in the offing. In

physics, while some of the currently available theories, such as much of clas-

sical physics or (nonrelativistic) quantum mechanics, may be seen as com-

plete within their limits, overall our theories remain manifestly incomplete

as regards our knowledge concerning the physical world. We have at the

moment classical physics, relativity, chaos theory, quantum mechanics,

quantum electrodynamics, quantum field theories, or still other theories,

such as string theories, by now extended into "branes" theories; and each of

these denominations appears to branch out nearly interminably. These the-

ories describe various macro and micro aspects of the physical world and of

our interaction with it by means of experimental technology. They do so

sometimes in classical-like ways, sometimes in quantum-mechanical-like

ways, sometimes by combining both. The epistemological status of many of

these theories is far from established, and some of them are highly specula-

tive. Quantum mechanics (at least in the interpretation considered here)

and, by implication, its extensions rigorously suspend the possibility of

physical description at the level of the ultimate constituents of matter them-

selves. This may or may not continue to be the case, once new theories or

new interpretations of them (or of existing theories, such as quantum

mechanics) take shape. Still more radical epistemological configurations are

not inconceivable either. For the moment we can at best correlate some

Deconstructions * 221

among the available physical descriptions and try to maintain their consis-

tency with experimental data within sufficiently workable limits. (Some of

these theories are manifestly inconsistent with each other.) It remains an

open question whether physics can ever be reasonably brought together. It

is conceivable that, as Einstein hoped, future theories, or new data, will

transform physics, and will do so by means of a more homogeneous single

theory, or at least a set of more homogeneous theories, which will, in par-

ticular, be no longer irreducibly nonclassical. It is also conceivable that, as

Bohr thought, future developments will preserve the joint significance of

classical-like and quantum-like physical, or philosophical, theories in

describing, or making it impossible to describe, the ultimate constituents of

matter. This conjunction of classical and nonclassical theories has defined

the century of physics that began with Planck's discovery of the quantum of

action in 1900. It is also possible that the future will produce not only as yet

unencountered but as yet inconceivable configurations of nature and sci-

ence, of matter and mind, or of both or of neither, of something else alto-

gether, and of entirely other questions (if this, for now seemingly

inescapable, notion will be retained), and ever more radical (re)conceptual-

izations of the conceivable and the inconceivable themselves.

It would be difficult to imagine anything of a lesser complexity in math-

ematics, biology, or computer technology, as all these fields invade and fuse

with each other ever more aggressively, without, it appears, worrying too

much about interdisciplinary hurdles (mathematics and science do not

appear to have ever been proceeding otherwise than by means of interdisci-

plinary projects). Will these developments give an ever-increasing role to

nonclassical epistemology? The answers, it appears, would be the same as

the one just given for physics. We do not know at the moment, but the

significance and even expansion of the nonclassical character of our knowl-

edge in these fields are at least as likely as any return to classical knowledge

at the ultimate level. It is worth reiterating that classical knowledge retains

its significance both in nonclassical knowledge and in its own right in many

areas of modern mathematics and science or, of course, elsewhere.

Will nonclassical epistemology continue to help us to learn new things in

mathematics and science, assuming, as I do, that it has done so in the past?

It would appear that at least in the near future it would; at least for the

moment it does. A few examples, indeed a few chapters, on mathematics

and science in nonclassical works (by nonscientists) could be added to what

has already been considered here from this perspective. I would like to close

this study, however, with a reverse or-reciprocity has been my theme

through this study-reciprocal example of Heisenberg, as it were, "reading"

222 * The Knowable and the Unknowable

Derrida, in his extraordinary 1929 lectures, The Physical Principles of

Quantum Theory, already cited in this study. It is tempting to think here in

terms of what Harold Bloom, in the context of the anxiety of influence in

poetry, calls "apophrades." He writes, "the uncanny effect is that the new

poem's achievement makes it seem to us, not as though the precursor were

writing it, but as though the latter poet himself has written the precursor's

characteristic works."27 Thus, it might seem to us, as readers, that, say,

Percy Bysshe Shelley, William Wordsworth, William Blake (who himself

makes this type of point in Milton), or Wallace Stevens has written certain

passages in Milton, even if not some of his poems. In the present case we

are, it is true, not dealing with poetry or a single philosophical tradition, or

with the anxiety of influence on Derrida's part as concerns Heisenberg or

quantum mechanics. As Bloom recognized, however, these relationships

need not be strictly personalized or need not, and indeed cannot, be con-

tained within a single field either. As we have seen, the anxieties concerning

nonclassical theory are many and diverse. There are those pertaining to the

possibility that such theories are right after all, a difficult circumstance for

those who do not want them to be right or (it is rarely a question of choice)

those who cannot accept them as possibly ultimately right. There are also

anxieties that relate to the possibilities (often more important than fact

when anxieties are concerned) that somebody was there already, a while

ago. "Poets were here before me," Freud liked to say. To the degree that this

is true (not a simple question, but secondary for the moment), in our cul-

ture, in our "two cultures," it appears easier both for poets to be there first

and for others to confront poets who are ahead of them. "The shadows of

gigantic mirrors which futurity casts upon the present" and "the unac-

knowledged legislators of the world," Shelley famously called poets.28 Nor

do poets seem to be targets of the Science Wars critics, although many of

them easily could be on the same grounds that philosophers like Deleuze

and Derrida are. But then, as we have seen throughout this study, nonclas-

sical thinkers are rather willing to give mathematics and science, and math-

ematicians and scientists, the place Freud gives to poets and often place

mathematicians and scientists alongside poets, for example, in going non-

classical or, as, among others, Lyotard suggests, indeed postmodern, first.

Shelley certainly would, and did, apply his view of poetry, as always the

poetry of the future, always coming from the future, to mathematic and sci-

entific discoveries. For him physics, chemistry, and (then emerging) biology,

too, were "the shadows of gigantic mirrors which futurity [casts] upon the

present," and he proved to be right about them.

Heisenberg's (1929) lectures were significantly influenced by, and paid a

Deconstructions * 223

lavish tribute to, Bohr's preceding work on complementarity as physics and

as epistemology, although as I have indicated he in fact manages to avoid

some of the problems of Bohr's Como version, in part through his more

deconstructive (than Bohr's) approach. I shall comment on this influence

and its significance for Heisenberg's "deconstructive" argument presently.

(Schrodinger's 1926 wave quantum mechanics played a role as well.) Ironi-

cally, however, a number of key features of Heisenberg's argument itself are

closer to Bohr's later (post-EPR) work on complementarity, as considered

here-ironically, but not surprisingly. For Heisenberg's own preceding

work (before Schrodinger's equation and Bohr's Como version of comple-

mentarity) on quantum mechanics and uncertainty relations, which is the

physical basis of the lectures, is closer to Bohr's later views, the views in turn

indebted or proximate to Heisenberg's early work. (It is difficult to disen-

tangle or unambiguously sequentialize influences here.) This proximity is

especially apparent in Heisenberg's extraordinary appendix (expanded for

the English publication of the lectures), "The Mathematical Apparatus of

the Quantum Theory." This appendix is much more than an appendix and

much more than an exposition of "the mathematical apparatus of the quan-

tum theory," unless we understand, as Heisenberg perhaps did, this appara-

tus as irreducible linked to physics, specifically the new kinematics of quan-

tum theory in terms of manifest physical effects of measurement, as

discussed earlier. Heisenberg's understanding of the mathematics of quan-

tum theory is, epistemologically, close to the view adopted by this study, in

particular the argument that relates the nonclassical epistemology of quan-

tum theory to that of non-Euclidean mathematics and specifically complex

numbers, themselves crucial for quantum mechanics. As he writes:

We shall now sketch the deduction of the fundamental equations of the new

quantum mechanics, following the program outlined above. It should be dis-

tinctly understood, however, that this cannot be a deduction in the mathe-

matical sense of the word, since the equations to be obtained form themselves

the postulates of the theory. Although made highly plausible by the follow-

ing considerations, their ultimate justification lies in the agreement of their

predictions with experiment. (The Physical Principles of Quantum Theory,

108).

A qualification may be in order concerning Heisenberg's famous empha-

sis in his first paper on quantum mechanics on the "magnitudes, which in

principle are observable," in other words, more or less individual quantum

effects in the above sense, which, rather than properties of quantum objects

and of their behavior, become subject to his "new kinematics." As is well

known and well documented, however, it was Einstein's argument that "the

224 * The Knowable and the Unknowable

theory decides what can be observed" that was guiding Heisenberg in his

work leading to his invention of quantum mechanics. His theory was not

founded on such magnitudes. Instead, at stake is a much greater complexity

of the very processes of observation and theory production alike, which

involves their irreducible mutual reciprocity, which found its way and

indeed shaped "phenomenon" and related concepts of Bohr's complemen-

tarity. This complexity, however, transpires already in Heisenberg's

famous, but not always carefully read, opening statement: "The present

paper seeks to establish a basis for theoretical quantum mechanics founded

exclusively upon relationships between quantities [magnitudes] which are in

principle observable" (van der Waerden, Sources in Quantum Mechanics,

261; emphasis added). "Relationships" is the key word here, and the title of

the paper was, we recall, "On Quantum-Theoretical Re-Interpretation of

Kinematic and Mechanical Relations" (emphasis added). "In principle" is

crucial too, for, no matter how theory-laden and how complicated the

processes of observation, the magnitudes in question could, in principle, be

observed and "kinematized" in the sense of the "new kinematics," as out-

lined earlier, while one cannot mathematically, or even conceptually, relate

in this or any other way quantum objects and their behavior beyond the

effects in question.

Thus, even leaving aside for the moment the theory-laden character of all

conceivable data (including that of classical physics), dealing with such, "in

principle observable," magnitudes is not the same as founding the theory on

them, and Heisenberg's paper does not conform to the latter conception.

Elsewhere he called his theory "a calculus of observable quantities" (The

Physical Principles, 109), by, undoubtedly, a deliberate and deep-going

analogy with Newton's calculus, which was both born from classical

physics and through which classical physics found its ultimate representa-

tion. Equally undoubtedly, however, Heisenberg was aware of and had in

mind in speaking in these terms, the radical nature of the revolution in

physics his calculus brought about. While working with the available data

of quantum physics (such as the Rydberg-Ritz formulas and the Bohr fre-

quency relations), his theory qua theory was founded above all on Bohr's

correspondence principle, which he used with extraordinary effectiveness,

as Bohr immediately grasped. The correspondence principle was used to

argue that for large quantum numbers the data becomes the same as it

would be in a classical case, at least as far as predictions are concerned (the

description could, in all rigor, no longer be the same). The principle was also

used to argue, correlatively, that the equations should be formally the same

as those of classical mechanics, the Hamiltonian equations. For large quan-

Deconstructions * 225

turn numbers these of course would give correct predictions classically.

From this viewpoint, physics-wise, Bohr's "correspondence principle" may

have turned out to be his greatest and (it is uniquely his) most original con-

tribution to quantum theory. Complementarity might be seen as a variation,

quite original but a variation nonetheless, on Heisenberg's themes, in turn,

however, depending on (but not merely a variation on) Bohr's correspon-

dence principle. Philosophical conceptuality is another matter. In any event,

this (as far as classical physics is concerned, "lethal") combination of the

data and the correspondence principle leads to the remarkable features of

quantum mechanics, such as Born's probability rules, uncertainty relations,

and so forth, and mathematically to the irreducibility of complex numbers

and replacing functions with operators as the kinematical and dynamical

variables of the theory. Both Dirac's and von Neumann's schemes are more

or less automatic translations of Heisenberg's matrix mechanism. Heisen-

berg's stroke of genius of finding his matrices (not altogether unprepared by,

among others, Bohr, but a stroke of genius nevertheless) was itself a found-

ing theoretical move. That is, this arrangement of the relationships between

observable magnitudes, made moreover into complex, rather than real,

quantities (never observable), and into infinite matrices is already a theory,

not an observation of nature, which does not arrange anything in this way.

(One might also recall that these matrices must be infinite in order to derive

uncertainty relations.) It has been sometimes pointed out that Heisenberg's

matrices are not observable quantities. The latter is indeed a correct point,

which, however, cannot be used as an argument against Heisenberg. He

never claimed they were. They were, however, linked to observable quanti-

ties and related the latter in terms of probabilities.

Heisenberg's philosophical attitude in the lectures is defined primarily by

that part of Bohr's philosophy, on which Heisenberg was later to comment

in the statement cited earlier in this study that, while "Bohr was primarily a

philosopher, not a physicist, ... he understood that natural philosophy in

our day and age carries weight only if its every detail can be subjected to the

inexorable test of experiment." Heisenberg's own argument-indeed, his

lectures could also be entitled "the philosophical principles of quantum the-

ory"-proceeds in philosophical terms of the critique of concepts, closer to

the Kantian sense of critical philosophy, and in effect establishes a certain

Kant-Derrida axis (as suggested in chapter 1) in the epistemology of modern

physics, and to some degree even in physics itself. Heisenberg starts with

special relativity, along the lines indicated in the preceding chapter, as an

experimentally grounded (but, in general, more broadly based) critique of

classical concepts. He writes:

226 * The Knowable and the Unknowable

Thus it was characteristic of the special theory of relativity that the concepts

"measuring rod" and "clock" were subject to searching criticism in the light

of experiment; it appeared that these ordinary concepts involved the tacit

assumption that there exists (in principle, at least) signals that are propagated

with an infinite velocity. When it became evident that such signals were not

to be found in nature, the task of eliminating this tacit assumption from all

logical deductions was undertaken, with the result that a consistent interpre-

tation was found for facts that had seemed irreconcilable. A much more rad-

ical departure from the classical conception of the world was brought about

by the general theory of relativity, in which only the concept of coincidence

in space-time was accepted uncritically. According to this theory, ordinary

language (i.e. classical [physical] concepts) is applicable only to the descrip-

tion of experiments in which both the gravitational constant and the recipro-

cal of the velocity of light may be regarded as negligibly small. (The Physical

Principles of Quantum Theory, 2; translation slightly modified)

Heisenberg's parenthesis is crucial in indicating some of the key com-

plexities of the situation, as considered earlier: first, those concerning the

relationship between ordinary (everyday) language and classical physics,

including as ultimately inapplicable to the ultimate objects of nonclassical

physics; and second, those concerning the unknowable and inconceivable

nonclassical efficacity of certain observable effects. The latter point becomes

especially significant once Heisenberg moves to quantum mechanics. It can

be shown (the preceding chapter would indicate how such an argument

would proceed) that the outcome of this critique epistemologically ulti-

mately amounts to the configuration analogous to Derrida's decentered

"play" and/as the inaccessible efficacity of differance, and, correlatively, or

indeed correlative to, the irreducible role of technology and "writing" in

Derrida's extended sense of the term. This critique may also be seen in (par-

allel) terms of Bohr's epistemology, as considered here, of phenomena as

effects of quantum-mechanical efficacity, although, as will be seen presently,

these terms to some degree allow one to proceed otherwise than by means

of (Derrida's) deconstruction. Both, however, radicalize this Einsteinian

scheme, to a nonclassical level, to which Derrida's matrix ultimately belongs

as well. This radicalization, as a form of critique and ultimately deconstruc-

tion, is enacted by Heisenberg in his next two chapters, "Critique of the

Physical Concepts of the Corpuscular Theory of Matter" and "Critique of

the Physical Concepts of the Wave Theory." (Those familiar with decon-

struction will recognize some deconstructive moves in the passage just

cited.) Heisenberg's analysis is extraordinary, and it is regrettable that I only

sketch it here. Indeed, I do so only because the nonclassical epistemology

that he ultimately arrives at has been already established in this study by

Deconstructions * 227

means of a somewhat different (technically less deconstructive) argument,

proceeding primarily via Bohr, although with Heisenberg's help through-

out. I shall return to this point presently. Now, I shall address the two cri-

tiques in question and argue that at stake is in fact a mutual deconstruction

of the classical particle and wave theories, virtually in the strict sense of Der-

rida's deconstruction, especially in his early works, such as "Structure, Sign,

and Play." According to Heisenberg:

In the foregoing chapter ["Critique of the Physical Concepts of the Corpus-

cular Theory of Matter"] the simplest concepts of the wave theory, which are

well established by experiment, were assumed without question to be "cor-

rect." They were taken as the basis of a critique of the corpuscular picture,

and it appeared that this picture is only applicable within certain limits,

which were determined. The wave theory, as well, is only applicable with cer-

tain limitations, which will now be determined. Just as in the case of particles

the limitations of a wave representation were not originally taken into

account, so that historically we first encounter attempts to develop three-

dimensional wave theories that could be readily visualized (Maxwell and de

Broglie waves). For these theories the term "classical wave theories" will be

used; they are related to the quantum theory of waves in the same way as

classical mechanics [of particles] to quantum mechanics. . . . (The reader

must be warned against an unwarrantable confusion of classical wave theory

with the Schrodinger's [quantum] theory of waves in a phase space.) After a

critique of the wave theory concept has been added to that of the particle

concept all contradictions between the two disappear-provided only that

due regard is paid to the limit of applicability of the two pictures. (47)

I am aware that I have not properly introduced (Derrida's) deconstruc-

tion in order to argue my point. But, in fact-this is in a way my point!-I

need not. Heisenberg virtually does it for me! The very essence of decon-

struction, both in more general terms and in a particular form of mutual

deconstruction employing conflicting classical theories against each other, is

defined here, as well as, and indeed correlatively to, certain key aspects of

Bohr's complementarity (visualization; the disappearance of contradictions,

replaced with complementary features of description; and so forth).29

Heisenberg, however, is closer here to Derrida than Bohr's post-EPR ver-

sion, or Heisenberg's own earlier views, ultimately, since, while there are

quasi-deconstructive elements in Bohr's Como approach or later writings, in

some respects complementarity itself may be seen as an alternative (to

deconstruction) way of handling the situation, which becomes especially

effective in Bohr's later work. On the other hand, it may also be argued that

this proximity to Derrida and this more deconstructive approach ultimately

allow one to avoid some of the problems of Bohr's Como argument, as con-

228 * The Knowable and the Unknowable

sidered earlier. In a way, Heisenberg's lectures may be seen as refining

Bohr's Como argument, a task that Bohr himself accomplishes later rather

differently and, again, closer to Heisenberg's own, in turn, less deconstruc-

tive, argument in his earlier work. Here, however, Heisenberg proceeds

deconstructively. In his earlier texts, "Structure, Sign, and Play," among

them, Derrida applies a similar procedure to certain conceptual frame-

works, for example, the (post-Husserlian) phenomenological one and the

structuralist one. Heisenberg's last qualification is of course crucial, and the

proper understanding of the limits of applicability of conceptual frame-

works under deconstruction is equally crucial for Derrida. A bit earlier in

his analysis, Heisenberg makes a decisive, physically and philosophically,

point (ultimately correlative to uncertainty relations): "As a matter of fact,

it is experimentally certain only that light sometimes behaves as if it pos-

sessed some of the attributes of a particle, but there is no experiment which

proves that it possesses all the properties of a particle; similar statements

hold for matter and wave motion. The solution of the difficulty is that the

two mental pictures which experiments lead us to form-the one of parti-

cles, the other of waves-are both incomplete and have only the validity of

analogies which are accurate only in limiting cases [where quantum and

classical physics give the same results]" (10). The main point, however,

remains that in question in this (or Derrida's) deconstructive procedure is,

first, an investigation of the proper limits of classical concepts and theories

under deconstruction and, second, their workings, against each other within

and in order to establish the nonclassical framework, which the phenomena

in question require.

There are, it is true, significant differences as well, even leaving aside the

disciplinary specificity of quantum physics (or, conversely, deconstruction).

Thus, one of Derrida's deconstructive aims in his earlier work is to establish

the common metaphysical or, one might also say, classical base for views

and theories, such as phenomenology and structuralism, which seem to be

in opposition to each other, and to use their mutual deconstructions in order

to do so. One can show a similar common philosophical appurtenance of

the classical concepts of particle and wave, although of course not the com-

monality of the physical features of each, which remain irreducibly mutually

exclusive and, hence, strictly complementary in Bohr's sense. This feature is

not essential in the same way in Derrida's deconstruction. The differences,

however, in no way diminish the significance of affinities, especially insofar

as the latter entail the ultimate necessity of deploying nonclassical theory in

approaching the phenomena in question in either case.

How bad, then, can Derrida's "commentary on physics" be, even though

Deconstructions * 229

and possibly because it is in fact something other than physics that is his

subject?

From another perspective, however, Heisenberg's lectures were delivered

in the wake and following Bohr's 1927 (Como) version of complementarity,

which was, in contrast to his post-EPR version(s), significantly conditioned

by Schrbdinger's wave version of quantum mechanics. Initially, for both

well-taken (in view of some of Schrbdinger's own arguments) and weakly

justified reasons, Heisenberg was critical of Schrbdinger's theory, while

Bohr welcomed it from the outset because he detected in it something very

different and, as it were, more quantum-mechanical than what Schrodinger

(more classically) expected from it. It would be difficult to discuss the rela-

tionships between the two versions of quantum mechanics in detail, and it is

not necessary for my purposes at the moment. It is worth it to recapitulate

briefly a few key facts especially relevant to my argument.

As I have indicated earlier, while equivalent (i.e., derivable from one

another) mathematically, these two versions are different epistemologi-

cally insofar as Heisenberg leads us more directly to the nonclassical epis-

temology of quantum theory, and indeed almost immediately to Bohr's

post-EPR version of it. Schrbdinger's version requires more interpretive

work to show that it requires or (keeping in mind the analysis given ear-

lier) at least permits a nonclassical interpretation. Schrbdinger's equation

simplified doing theoretical physics (making theoretical predictions con-

cerning the outcomes of quantum-mechanical experiments). This was

important at the time, since Heisenberg's matrix mechanics was almost

prohibitively difficult to handle mathematically. On the other hand, it also

appeared (and, as we have seen, it still does) more natural to think of

Schrbdinger's equation in terms analogous to the equations of classical

physics. This view, however, created new physical and epistemological

complexities and paradoxes, quickly realized by Schrodinger himself, as

we have seen in chapter 2. (In this respect, his initial hopes for his equa-

tion did not materialize.) In his Como version of complementarity,

through juggling both wave and particle pictures and complementarity (in

that version), Bohr managed to use Schrbdinger's equation to find a way

out of the paradoxes, at least to make some key moves in this direction. As

I said, he appears to have sensed these possibilities immediately in the

wake of the introduction of Schrbdinger's mechanics, which is why it

appealed to him from the outset. Bohr's argument ultimately proved to be

in need of further refinement and even some corrections. Heisenberg's

Chicago lectures were, in my view, more successful, in part through a

more effective use of the mathematics of quantum theory and in part,

230 * The Knowable and the Unknowable

which was my main point earlier, by using a more deconstructive type of

argument, as just considered.

Eventually, in the wake of the EPR argument, Bohr arrived at a more rad-

ical interpretation, as considered in this study, to which, however, Heisen-

berg's work prior to Bohr's early version of complementarity leads virtually

directly, although that was not so apparent at the time.30 This interpretation

does not depend on either wave or particle theories or properties, not even in

partial terms (as indicated by Heisenberg's statement earlier) in "describing"

quantum objects, to whose characterization and behavior neither, or indeed

any, description nor theory is applicable. At most, some properties of either

theory are retained at the level of the effects of the quantum (and hence ulti-

mately in turn indescribable) interaction between quantum objects and mea-

suring instruments upon the latter. Nor, accordingly, would one need to

depend on a mutual deconstruction of the type just described, or it appears

any deconstruction, in developing this interpretation. One might proceed in

this way, too, as Heisenberg does in his Chicago lectures, but one need not;

and in a way Bohr might have been (it is difficult to be certain) better off fol-

lowing Heisenberg all along rather than taking a Schrodingerian detour.

I cannot consider the subject in proper detail here, but it may be argued

that the introduction of new mechanics in Heisenberg's great first paper on

the subject, published in 1925, may be seen as arising from, or at least as

linked to, an extraordinary form of "vision" of the material constitution

and, with respect to the viewpoint of classical physics, deconstitution of the

data in question.31 I am now speaking of the vision relating to the material

marks/traces constituting the quantum data rather than to the theoretical

conceptualization of the quantum-mechanical situation as a whole, such as

complementarity, which, however-this is my argument-is ultimately

made possible by this vision. (I continue to remain here within the linea-

ments of complementarity as an interpretation of quantum mechanics, for

otherwise my argument at the moment may not apply.) This vision itself

may be seen as material, insofar as it relates to certain material marks, as just

explained. It may, however, not be seen as configurative, unless in terms of

(if one may use such an expression) a radically deformalized form-that is,

if we can, phenomenally, and especially geometrically, "see" anything in this

way at all. It may not be humanly possible to do so, even though, in contrast

to the ultimate constituents of matter or the ultimate efficacity of the data in

question, the (material) elements constituting these data are available to phe-

nomenological apprehension. We do, however, now treat these marks, even

their collectivities, purely "formally" (without "form") rather than in any

way configuratively, and indeed deal with, as de Man would have it, "the

Deconstructions * 231

material disarticulation . . . of nature," or, more accurately, with the disar-

ticulation of material nature, of matter.32 In particular, we divest them of

their classical and hence configurable appearance (in either sense), of any-

thing that could possibly be mapped by a classical model, even though they

do form configurations, or what can be so seen in certain circumstances-

say, wavelike patterns, or a trace of a particle in a cloud chamber. These

marks must be divested of the possibility of being explained in classical

terms and hence of their manifest classical configurativity. For example, they

should not be seen either as points resulting from classically conceived colli-

sions between "particles" and the screen or as forming a classically con-

ceived wave pattern. Neither "picture" corresponds to what in fact occurs.

At this stage, even the radical (Derridean) tracelike character of these marks

is suspended, although this type of character will have to be given to these

data in order to explain them in quantum-theoretical terms.

This suspension is necessary, and the vision that results is possible, to the

degree it is possible as a vision, for the following reasons. (Once again, the

impossibility of vision is here of a phenomenal nature, the impossibility fully

to divest the phenomena, the phenomenal effects, in question of any organi-

zation, rather than that relating to the nonphenomenalizable efficacity of

these effects.) As we have seen, the mathematical formalism of quantum

mechanics, viewed nonclassically, does not formalize or otherwise describe

the (emergence of) configurations, individual or collective, of these traces as

such, or for that matter any material physical process in the way classical

physics would. Instead, it enables (excellent) statistical predictions concern-

ing certain, especially collective, configurations or correlations of material

marks in certain circumstances.33 Accordingly, in order for a theoretical for-

malization and interpretation of quantum physics to take place, these

marks, while "visible," have to be divested of any form of mathematical and

specifically geometrical representation as concerns their emergence or possi-

bility. Classical physics is largely defined by the possibility of such represen-

tation or phenomenalization (in the usual sense) or conceivability of the sit-

uation, making it available to human intuition (Anschaulichkeit), as

considered earlier. In quantum mechanics, in a nonclassical interpretation,

neither is possible any longer. This impossibility is reflected in the tracelike

character of the marks in question and the nonclassical nature of the inter-

pretation of these traces or of the mathematical formalism of quantum

mechanics. The efficacious processes themselves will, again, be far beyond

the reach of any pictorial visualization, intuition, phenomenalization, repre-

sentation, conception, and so forth. Now, however, indeed "beforehand,"

the emergence of the manifest effects, the visible marks, involved, must also

232 * The Knowable and the Unknowable

be divested of any geometrical structure consistent with classical physics.

The emergence of these traces, either individually or collectively, must be

seen as allowing for no classical physical description.

Heisenberg's first paper on quantum mechanics appears to reflect the

process just described. First, it suspends the application of classical physics

to quantum data and the very possibility of configuring these data accord-

ingly. Instead it treats them formally, as material and phenomenal effects,

divorced from all mechanical configurativity. His introductory elabora-

tions in the paper itself would suggest nearly as much. Heisenberg does not

philosophically explore the epistemological consequences of the situation,

of which he was only vaguely aware at the time. His main concern was to

offer a mathematical formalism that would enable theoretical predictions

in the situations where all previous attempts had failed. These conse-

quences emerged in subsequent developments, both in Heisenberg's own

work cited earlier, specifically his paper introducing the uncertainty rela-

tions, and in Bohr's work. Heisenberg's invention of quantum mechanics,

however, appears to have been partly enabled by the deconfigurative phe-

nomenology just discussed. From this perspective, too, Heisenberg, rather

than only Bohr, may, as I suggested earlier, be seen as the discoverer of the

nonclassical epistemology of quantum mechanics, even though at this point

there are no complementary features of description, which came later cour-

tesy of Bohr, who also gave this epistemology a firmer philosophical base.

It is indeed a miracle that Heisenberg proceeded from the disassemblage,

just considered, of experimental data (in this case, spectra) to assembling

or, as against the classical view, reassembling this data through his matri-

ces, infinite square tables of complex-number quantities related to these

data in terms of probabilities. This arrangement itself was a key founding

theoretical element of the new mechanics if indeed the very denomination

mechanics may still apply. He was not even aware at the time that the cor-

responding mathematical theories (matrix algebra) had already existed at

the time, which was realized by Max Born, his teacher, upon reading the

paper. Heisenberg reinvented it. This new approach to the data is what

enabled him and others (this took a few months) to formalize these effects

of the interaction between quantum objects and measuring instruments

upon the latter in a new way, ever since called quantum-mechanical for-

malization. The proper manipulation of these tables became quantum

mechanics, in its matrix version (Schrodinger's equation came later). As

explained earlier, the new (as opposed to classical physics) kinematics was

defined by the data arranged into these tables, now used as variables of

Deconstructions * 233

equations borrowed from classical mechanics and applied there to standard

physical variables. Of course, once one studies how Heisenberg arrived at

his ideas, that is, on the basis of what principles and data (for example, for-

mally taking over the equations of classical mechanics and substituting new

mathematical objects for classical variables, or using previous theoretical

arguments), the situation of his invention does not look quite so dramatic.

It never does. His invention was quite miraculous nevertheless, and in a

way poetic, for this, too, is "the highest form of musicality in the sphere of

thought."

However one proceeds, the outcome is the nonclassical epistemology of

the type Derrida reaches (more deconstructively in his earlier work, more

directly in his later works) in and through his analysis of writing. This argu-

ment poses further questions as concerns Derrida's own nonclassical episte-

mology and its relation to various, let us say, more technical deconstructive

procedures, such as those used in his early, more technically deconstructive

works. In other words, the question is this. Do we need this type of techni-

cal deconstruction, used in Derrida's early works, in order to develop non-

classical epistemology, for example, that entailed by and entailing such Der-

ridean structures as differance, trace, writing, and so forth (we recall that,

while finite, this list does not permit termination)? Conversely, can we pro-

ceed to this type of epistemology more directly, as Nietzsche or Derrida in

his later work appears to have done? Or how is deconstruction (in this

sense) different from complementarity, especially when the latter is seen as

a generalized epistemological and conceptual framework, which, as I said,

may also define the difference between Heisenberg's approach in his

Chicago lectures and Bohr's Como approach, while earlier work of Heisen-

berg and Bohr's post-EPR work link nonclassical epistemology and comple-

mentarity more immediately? There may be no simple answer to this ques-

tion and no simple dichotomy either, but rather a complex reciprocity,

which, however, may make these questions all the more significant.34 In any

event, in considering the quantum data, we deal with nothing less than writ-

ten traces in Derrida's sense and perhaps, as we have seen, with something

more radical. The "traces" one encounters in quantum physics may or may

not be traces (the term provisionally deployed by Derrida as well) of some-

thing, or something (material marks) left by something interacting with our

technology, or indeed with the same type of something in our technology.

But they may not be written, nor be traces, in Derrida's sense.

Then, the question of how bad Derrida can be on physics (or in general)

indeed acquires a meaning rather different from the one it received in most

234 * The Knowable and the Unknowable

recent (and some no longer recent) criticism of Derrida's work. Rather than

being bad, as and in the way these critics contend, he may only not be good

enough.

Physics itself is yet another question. It does appear, however, that one

needs to be at least as good as Bohr or Heisenberg, or, in spite of himself,

Einstein, to enable physics to remain physics, and perhaps to become

physics more than it has even been classically, under these nonclassical con-

ditions. This task may, as Einstein said, indeed require "the highest musi-

cality in the sphere of thought." It may even require the highest musicality

in the highest sphere of thought, where musicality and thought become one

and the same. But then perhaps this is what thought is, wherever we

encounter it, assuming, somewhat improbably, impossibly, that we can call

it thought or indeed anything, or can even (classically) think, conceive, of it

in any way. It appears likely that this thought, just as modern mathematics

and science, can only be thought of, unthought of, nonclassically-if then

just as quantum objects. We may need to think of them in the same unthink-

able way.

Conclusion

In his 1927 Como lecture, Bohr speaks of the "inherent 'irrationality' "

(Bohr's quotation marks) of "the quantum postulate" at the core of quan-

tum mechanics (PWNB 1:54). He eventually abandons this way of speaking

of the situation, in view of a misunderstanding, indeed, I would argue, a

massive misunderstanding, of his point by his audience, critics, and advo-

cates alike, including by ignoring the quotation marks around "irrational-

ity," as well as his stress on the rational character of complementarity itself.

The characterization "rational," as defining complementarity, especially as

a "rational generalization" of classical physics, is persistent throughout

Bohr's writing on quantum theory. The idea itself of the quantum irra-

tionality, conceived along the nonclassical lines of complementarity, contin-

ued to inform his understanding, his rational understanding, of quantum

mechanics and complementarity. Indeed, Bohr's nonclassical epistemology

as considered here-the epistemology of knowable and quantum-mechani-

cally theorizable (phenomenal) effects and of the irreducibly unknowable

efficacities of these effects-is a rational version of the irrational in this

sense of the irreducibly unknowable. It is more than merely likely, more-

over, that in using this term Bohr in the Como lecture had in mind the way

it has functioned in mathematics, as discussed in chapter 3. He stresses that

both Schrodinger's formulation and Heisenberg's matrix mechanics,

"[depend] essentially upon the use of imaginary arithmetic quantities,"

which cannot correspond to anything that actually manifests itself in physi-

cal space and time (PWNB 1:76). It is true that he actually spoke of this

dependence in relation to the "simplicity" of both schemes, while he main-

tained the last point itself and hence the symbolic character of both methods

on other grounds, which are indeed many, throughout the articles, includ-

ing at the juncture just cited. The essential dependence on, the irreducibility

of, the role of complex numbers in quantum mechanics became clear shortly

thereafter and is properly reflected in Bohr's later writings. He also appears

to have been thinking of complex numbers as irrational in this epistemolog-

ical sense, as they are in their mathematical sense, thus, I would argue, pur-

236 * The Knowable and the Unknowable

posely extending the Greeks' conception of X/2 to its nonclassical limit. In

other words, he thought of this irrationality as the impossibility of repre-

senting or conceiving of something in terms that are available to us, in this

case taken to a nonclassical limit. At this limit, no representation or con-

ception of the unknowable is possible by any means-arithmetical, geomet-

rical, algebraic, topological, or other. Nor (this, as I have explained in chap-

ter 1, is a key nonclassical part of this view) will it ever be available to us.

We recall that, in Bohr's interpretation, the "algebra" of quantum mechan-

ics, its mathematical formalism, which might, and to some does, take on the

function of representation in quantum theory, does not describe quantum

objects but only enables us to predict the effects of their interactions with

measuring instruments upon the latter. This formalism, we also recall, irre-

ducibly depends on complex numbers.

Remarkably, all these elements appear and come together in the famous

picture that Bohr drew in 1962, one day before his death, on the blackboard

in his house (BCW 7:286). At the top there is the general formula for the

functions of complex variables and a sketch of two separate complex num-

ber sheets. Riemann introduced this idea for dealing with such functions in

order to be able to have a single rather than double (and hence, ambiguous)

meaning corresponding to each value of the argument, which, in a proto-

complementarity fashion, makes it possible to treat them as regular func-

tions (defined in mathematics so that only a single value of the function cor-

responds to each value of the variable). Bohr tried to use Riemann's idea

before he became a physicist, when he was interested in certain problems of

psychological epistemology, in part thanks to the work of his brother Har-

ald Bohr, who was to become a great mathematician in his own right and

whose doctoral dissertation was on Riemann. These ideas were to resurface

later in his work on complementarity. In between, in a 1910 letter to Har-

ald, Bohr described his emotional state during his early work on electron

theory: "Emotions, like cognitions, must be arranged in planes that cannot

be compared" (BCW 1:513). Whether or not the idea works as a good

description of an emotional state, it will work for electrons themselves, that

is, for their effects upon measuring instruments. Of course, Bohr's sketch on

his blackboard refers to his pre-quantum-mechanical ideas. By that point,

however, he could hardly avoid thinking of the role of complex numbers in

quantum mechanics, on whose symbols he was later to comment: "These

symbols themselves, as is indicated already by the use of imaginary num-

bers, are not susceptible to pictorial interpretation" (BCW 7:314). Now,

below on the blackboard is a sketch of "Einstein's box" involved in one of

Einstein's arguments, epistemologically, ultimately, of the same type as the

Figure 3.

238 * The Knowable and the Unknowable

EPR argument, discussed by Bohr in "Discussion with Einstein" (PWNB

2:54-57). Thus, as has been often observed, Bohr's sketch represents the

beginning and the end or, rather, the last key stage of his work on comple-

mentarity. What does not appear to have been commented upon, however,

is that right underneath the formula for the functions of complex variables,

Bohr in fact writes J 2, the first irrational number! One might argue that it

could also be z, rather than 2, an often used symbol for complex variables

(x + iy). However, Bohr does have a "z" in his sketch. It is used as the des-

ignation just explained above his formula, and it looks very different in his

handwriting. So it has to be a J 2, and it is used, it appears, as the symbol of

"irrationality" in the sense of the unknowable, as a bridge over the cen-

turies, indeed millennia, from perhaps the first scientific encounter with the

rigorously irrational to complex numbers and then to quantum mechanics,

the key nonclassical scientific junctures of this study.

In the twentieth and by now the twenty-first century we have moved into

extraordinarily complex territories and, as Deleuze would have it, deterrito-

rializations of both knowledge and the unknown, or, ultimately, the

unknowable, in mathematics and science. A similar progression (one may

even speak of progress here, if only against some recent critics) can be traced

in other fields, such as those traversed here, and in the interactions among

these fields or between them and mathematics and science, and hence in the

confrontations between the two cultures. This confrontation apparently

refuses to go away, or continues to return or, as Nietzsche would have it,

eternally return, even though and perhaps also because, as I have argued

here, there are more than two cultures involved in these interactions and

confrontations, more that two and less than one. Something on the order of

the "two cultures" continues to invade this multiplicity and this less than

unity, and in their conceivably irreducible differences, these "two cultures"

still appear to one another as ghostly strangers never heeding Hamlet's

advice to Horatio upon the appearance of his father's ghost: "This is won-

drous strange! / And therefore as a stranger give it welcome. / There are

more things in heaven and earth, Horatio / Than are dreamt of in your phi-

losophy." (Hamlet, Scene V). Perhaps, even a certain nonclassical way of

welcoming these strange things is in order. For now, we confront each other

more in the way Levinas and Blanchot describe in speaking of ethical rela-

tionships. In Blanchot's words: "The Stranger comes from elsewhere and is

always somewhere other than we are, not belonging to our horizon and not

inscribing himself upon any representable horizon whatsoever, so that his

'place' would be invisible."1

Remarkably, it is also modern mathematics and science-specifically, via

Conclusion * 239

Paul Valery, Riemann's theory of complex functions, which inspired Bohr's

thinking on complementarity, post-Einsteinian relativistic cosmology, and,

implicitly, quantum theory-that guide Blanchot's thought alongside, and

indeed reciprocally with, Levinas's ethical conception of alterity. The latter,

as we have seen, may in turn be argued to have complex reciprocities with

twentieth-century science and its epistemology. Elsewhere in The Infinite

Conversation (which I cite here), Blanchot invokes his extraordinary con-

cept of the unfigurable Universe, correlative to the concept of the unknow-

able in question in this study. Blanchot writes:

It is nearly understood that the Universe is curved, and it has often been sup-

posed that this curvature has to be positive: hence the image of a finite and

limited sphere. But nothing permits one to exclude the hypothesis of an

unfigurable Universe (a term henceforth deceptive); a Universe escaping every

optical exigency and also escaping consideration of the whole-essentially

non-finite, disunified, discontinuous. What about such a Universe? ... But

will [man] ever be ready to receive such a thought, a thought that, freeing

him from fascination with unity, for the first time risks summoning him to

take the measure of an exteriority that is not divine, of a space entirely in

question, and even excluding the possibility of an answer, since every

response would necessarily fall anew under the jurisdiction of the figure of

figures? This amounts perhaps to asking ourselves: is man capable of a radi-

cal interrogation? (350)2

Riemann's ideas are used in a similar way (albeit somewhat more loosely)

and indeed via formulations echoing the ones just cited, specifically lan-

guage that is "without reference to unity" (78). An analysis of Blanchot's

unfigurable Universe, which is also the universe, or more accurately (given

the misleading nature of the latter concept), un-universe, of the unfigurable,

would show that at stake here is a nonclassical epistemology. Indeed, the

point is readily apparent, given the argument presented in this study. As

such, it suggests, along the lines indicated earlier in this study, the poten-

tially nonclassical nature of Einstein's general relativity, the basis of the cos-

mology Blanchot invokes, in order, it must be added, also to link it to the

question of literature. For the question of whether we are capable of radical

interrogation is, for Blanchot, also, finally, the question of whether we are

"capable of literature" under nonclassical conditions, anymore than we are

capable of mathematics and science. Thus, the question of the "two cul-

tures" is also the question of the nonclassical culture or, irreducibly, cul-

tures, always more than two and less than one, within which we must think

differently the work, old and new, of both literature, and mathematics and

science, and their relationships. It is indeed not easy to confront such a task

240 * The Knowable and the Unknowable

either in mathematics and science, defined by Blanchot's last question

throughout their history, or in our (in the broadest sense) ethical encounters

with strangers or with what is strange, which we often find in those whom

we know or thought we knew. Nor is it easy for us, for example, as

strangers coming from the "two cultures," to hear in Levinas's and Blan-

chot's words what they want us to hear: that invisible that "turns away

from everything visible and everything invisible," that unknowable that

turns away from everything knowable and everything unknowable (52).

This unknowable defines the limit of radical interrogation, which is invoked

by Blanchot and which is the subject of this study. At this limit what we are

capable of in science and human relations, or how we conceive of them, may

indeed be put to its ultimate test. We tend to steer away from this limit or

even not to see it, especially as far as concerns human relations-ethical,

cultural, political, or other-such as those between the "two cultures."

Mathematics and science themselves often show more boldness, and, as I

have argued here, with Nietzsche, they test our ethics, too, at least our self-

ethics, our integrity (Redlichkeit). Elsewhere we may need to catch up.

Instead, we tend to prefer more convenient and more comfortable forms of

blindness, often under an illusion of sight, or even of insight, especially

when we approach the strangeness of, and strangers from, other fields or

strangers from far afield, such as modern mathematics and science, on one

side, and French philosophical thought, on the other. It is not that we can

ever completely avoid blindness or illusion here, or elsewhere. The

significance of Blanchot's insight could not be more profoundly, if again

conveniently, misunderstood. The question is, which illusions do we offer to

others-to our fellow countrymen, to our fellow culture-men, or to

strangers from other countries and cultures?

Perhaps the ultimate ethics or (since the ultimate ethics may not be pos-

sible, classically, in practice and, nonclassically, in principle) at least a good

ethics of intellectual inquiry, or of cultural interaction, is the following:

being strangers ourselves, to offer other strangers, strangers in our own or

in other cultures, those ideas that bring our own culture-say, science, on

one side, and the humanities, on the other-to the limits of both what is

known and unknown, or unknowable, to them. To have an expertise is to

reach the limits of both what is knowable and what is unknowable in one's

field; and to be ethical in intellectual exchange is to offer others the sense of

both of these limits, to tell the other culture or field both what we know and

what we do not know ourselves, and what is knowable and what is

unknowable, in our own field or culture.

Conclusion * 241

Naturally, we can never be fully certain how our offerings will be received,

not even within our own culture, since we are often strangers there, too. Nor

can we be quite certain as to what we know and what we don't know. Nei-

ther uncertainty can be circumvented. We might, however, want to give both

of them welcome in turn. For these uncertainties, too, can be wondrous

strange, as they turn us away from everything that is definitively certain or

uncertain, just as the invisible of quantum mechanics turns us away from

everything that is definitively visible or invisible and as its unknowable turns

us away from everything definitively knowable or unknowable.



Notes

Chapter 1

1. Niels Bohr, The Philosophical Writings of Niels Bohr (hereafter PWNB), 3

vols. (Woodbridge, CT: Ox Bow Press, 1987), 2:67-69; 3:1; Werner Heisenberg,

The Physical Principles of the Quantum Theory, trans. Carl Eckart and Frank C.

Hoyt (Chicago: University of Chicago Press, 1930), 1-2, 13.

2. As early as the 1930s, Bohr (in collaboration with Leon Rosenfeld) extended

key aspects of complementarity to relativistic quantum theory in his influential arti-

cles, "On the Question of the Measurability of Electromagnetic Field Quantities"

(1933) and "Field and Charge Measurements in Quantum Electrodynamics" (1950),

both reprinted in volume 7 of Niels Bohr: Collected Works (hereafter BCW), 10

vols. (Amsterdam: Elsevier, 1972-1996), and in Quantum Theory and Measurement

(hereafter QTM), ed. John Archibald Wheeler and Wojciech Hubert Zurek (Prince-

ton, NJ: Princeton University Press, 1983), 479-522, 523-38.

3. The statement is reported in Q TM, 8.

4. Georges Bataille, "Conferences sur le Non-Savoir," Tel Quel 10 (1962): 5.

This sentence is omitted from the text published in "Conferences 1951-1953," in

vol. 8 of Georges Bataille, Oeuvres completes, 12 vols. (Paris: Gallimard, 1970-88).

5. Georges Bataille, "Conferences sur le Non-Savoir," 5; Oeuvres completes

8:219.

6. This confluence may not be coincidental, even beyond the more general

influence of quantum physics on Bataille, who refers to it on several occasions. One

of the figures involved in the discussion of Bataille's ideas in "Conferences

1951-1953" was Georges Ambrosino, an atomic physicist, whom Bataille even

credits with a partial coauthorship of the contemporary Accursed Share (The

Accursed Share: Volume 1, trans. Robert Hurley [New York: Zone, 1988], 191 n.

2). Indeed, a chapter on Bataille could have been easily included in this study. I have,

however, extensively written on Bataille previously in a similar set of contexts, espe-

cially in Complementarity: Anti-Epistemology after Bohr and Derrida (Durham,

NC: Duke University Press, 1994) and more recently in "The Effects of the Unknow-

able: Materiality, Epistemology, and the General Economy of the Body in Bataille,"

in Parallax 18 (January 2001): 16-28.

7. I have discussed this point in a related article, "Reading Bohr: Complemen-

tarity, Epistemology, Entanglement, and Decoherence," in Decoherence and Its

244 * Notes to Pages 11-13

Implications in Quantum Computation and Information Transfer, ed. Tony Gonis

and Patrice E. A. Turchi (Amsterdam: IOS Press, 2001), 3-37.

8. Bohmian theories are different from the standard quantum mechanics, since,

in contrast to the latter, they expressly assign trajectories to particles, which the stan-

dard version does not do (Bohr's interpretation of course expressly forbids such an

assignment) and, in some versions, expressly modify the mathematical formalism of

the standard version. Accordingly, I shall collectively refer to Bohmian theories here

as Bohmian mechanics or theory, in contradistinction from the standard version,

which, and only which, is in question in complementarity in the sense of Bohr's

interpretation. Bohmian mechanics does contain certain complementary or quasi-

complementary features, such as those involved in uncertainty relations, which it

preserves as well, but to which it gives an epistemologically classical interpretation.

The latter is made possible by virtue of assuming certain additional physical quanti-

ties ("variables") that are, at least at this stage, unavailable to experimental detec-

tion ("hidden") but that reinstate a classical-like physical picture (trajectories,

causality, and so forth) at the level of quantum objects and processes.

9. It would be difficult to survey these alternatives or the spectrum of interpre-

tations of complementarity here, which often involve complex negotiations of clas-

sical and nonclassical features, as one can easily see from the introductory materials

in volumes 6 and 7 of BCW. For a representative example, see Henry Folse, "Niels

Bohr's Concept of Reality," in Symposium on the Foundations of Modern Physics

1987, ed. Pekka Lahti and Peter Mittelstaedt (Singapore: World Scientific, 1987),

161-80, on which I shall comment later. I have considered some among the promi-

nent studies of Bohr in Complementarity, 78-85 (and references there).

10. Ultimately, there may be no physical theory that is fully free of interpreta-

tion, to the degree that we can unequivocally distinguish between theory and inter-

pretation, even if by theory we mean only the mathematical formalism of a physical

theory, such as quantum mechanics. Some do see quantum theory, or any physical

theory, as independent of interpretation, and some see a working out of a mathe-

matically consistent scheme of the theory as a prerequisite for any meaningful inter-

pretations. For example, Bas C. van Fraassen takes this view in his exposition of his

version of the so-called modal interpretation in Quantum Mechanics: An Empiricist

View (Oxford: Clarendon, 1991). Van Fraassen, accordingly, does not see most of

Bohr's or Heisenberg's work as a full-scale interpretation, which he argues only to

be possible after John Von Neumann's rigorous mathematization of quantum theory

in terms of Hilbert-space formalism in the 1930s, on which I shall comment in the

next chapter. It appears difficult and perhaps impossible to sustain so restricted a

view, and it may be preferable to speak in terms of certain mathematizable models,

such as those used in classical or in quantum physics, as distinguished from full-scale

interpretations. It may also be observed that, while my exposition here is closer to

Bohr, all my epistemological arguments can be properly correlated with Von Neu-

mann's version of the quantum-mechanical mathematical formalism, which uses

more elegant mathematics. Indeed, using the latter would make some of my key

arguments easier to make. However, the highly abstract nature of this formalism

would make my exposition far less accessible for the general reader.

Notes to Pages 13-16 * 245

11. Predictably, some among such arguments appeal to the limited scope of most

current theories, specifically quantum mechanics, and contemplate and pursue the

alternatives (different also from more standard extensions of quantum mechanics,

such as quantum field theories) that take into account higher-level quantum effects

or gravity. Among the attempts of this type are those by Roger Penrose, David

Deutsch, Giancarlo Ghirardi, Alberto Rimini, Tulio Weber, and several others. Most

of these proposals contain unappealing physical features, such as nonlocality, which

they share with Bohmian mechanics (which inspired many of these proposals). See,

for example, Roger Penrose, Shadows of the Mind: A Search for the Missing Science

of Consciousness (Oxford: Oxford University Press, 1994), 331-46, which also sur-

veys some of these attempts, and Giancarlo Ghirardi, "Beyond Conventional Quan-

tum Mechanics," in Quantum Reflections, ed. John Ellis and Daniele Amati (Cam-

bridge: Cambridge University Press, 2000), 79-118.

12. It is worth keeping in mind that Foucault's "modern," juxtaposed by him to

"classical," for example, in The Order of Things: An Archeology of Human Sciences

(New York: Random House, 1994), often refers to views that are classical in the pre-

sent sense, as are also some among the views associated with such denominations as

poststructuralist, deconstructive, and postmodernist. It is also worth noting that the

terms "mathematics" and "science" are not always as self-explanatory as they might

appear either and sometimes need to be qualified in turn.

13. See, for example, Jacques Derrida, "Differance," in Margins of Philosophy,

trans. Alan Bass (Chicago: University of Chicago Press, 1982), 17.

14. Friedrich Nietzsche, The Gay Science, trans. Walter Kaufmann (New York:

Vintage, 1974), 305.

15. I commented on these connections in In the Shadow of Hegel: Complemen-

tarity, History, and the Unconscious (Gainesville, FL: University of Florida Press,

1993), 54-74.

16. Ludwig Wittgenstein, Tractatus Logico-Philosophicus, trans. C. K. Ogden

(London: Routledge, 1985), 179.

17. This circumstance poses the question of whether the interpretations of clas-

sical theories pursued by such investigations are in fact nonclassical, rather than still

classical, readings of both (more appropriately) classical and (more problematically)

nonclassical theories they consider. If (and when) the latter is the case, the paradox

disappears. On the other hand, as has been pointed out (by, among others, Kant),

there will always be "savants" who would find anything in one or another of the

pre-Socratic philosophers, provided they are told what to look for. I am saying this

not in order to dismiss all such rereadings of old texts via new theories but to sug-

gest that new theories, whose interpretations (such as those offered by this study)

may of course be debated in turn, entail a complex balance of (re)reading both the

"old" and the "new."

18. For Bell's view of quantum mechanics see his The Speakable and the

Unspeakable in Quantum Mechanics (Cambridge: Cambridge University Press,

1987). The title of this study may be seen as an implicit argument with Bell's views

of quantum mechanics and of Bohr, which I have considered in Complementarity,

176-90.

246 * Notes to Pages 17-31

19. Nietzsche, The Gay Science, 356.

20. Werner Heisenberg, "Quantum Theory and Its Interpretation," in Niels

Bohr: His Life and Work as Seen by His Friends and Colleagues, ed. Stephan S.

Rozental (Amsterdam: North-Holland, 1967), 95.

21. Friedrich Nietzsche, Daybreak, trans. R. J. Hollingdale (Cambridge: Cam-

bridge University Press, 1982), 60.

22. Werner Heisenberg, Philosophical Problems of Quantum Physics (Wood-

bridge, CT: Ox Bow Press, 1979), 13.

23. I have addressed the latter subject in In the Shadow of Hegel, 54-83.

24. Friedrich Nietzsche, The Birth of Tragedy and the Case of Wagner, trans.

Walter Kaufmann (New York: Vintage, 1966), 97-98.

25. John Archibald Wheeler, "Law without Law" (QTM, 189).

26. Gilles Deleuze and Felix Guattari, What Is Philosophy? trans. Hugh Tom-

linson and Graham Burchell (New York: Columbia University Press, 1993), 11-12,

24.

27. One can mention such recent endeavors as algebraic topology and its exten-

sion in Alexandre Grothendieck's program in algebraic geometry, and infinite-

dimensional group representations in Robert Langlands's program, both of which

have implications in quantum physics. I shall further comment on these develop-

ments in chapters 2 and 3.

Chapter 2

1. I cannot consider here Planck's law and its history or key events following

Planck's discovery and leading to quantum mechanics and its interpretation. See

Thomas S. Kuhn's Black-Body Theory and the Quantum Discontinuity, 1894-1912

(Oxford and New York: Oxford University Press, 1978) for arguably the best

account of the early stages of this history. Bohr gives an excellent and conceptually

important account in "Discussion with Einstein on Epistemological Problems in

Atomic Physics" (PWNB 3:32-36). Some of Heisenberg's essays, in particular

"Development of Concepts in the History of Quantum Mechanics" and "The Begin-

nings of Quantum Mechanics in Gottingen," both in Encounters with Einstein

(Princeton, NJ: Princeton University Press, 1983), are exceptionally useful as well. In

the context of Bohr's work, see also introductory and historical material in volumes

5,6, and 7 of BCW.

2. This may be said, even if one were to leave aside the fact that the conven-

tional (nonrelativistic) quantum mechanics has only a limited scope in our, as yet

incomplete, understanding of nature at the level of its ultimate constitution and that

the epistemology of the higher-level quantum field theories remains as yet a largely

unexplored subject. Paul Taylor's An Interpretive Introduction to Quantum Field

Theories (Princeton, NJ: Princeton University Press, 1995) is among a few recent

attempts and also contains useful references. By contrast, the literature (technical,

philosophical, historical, or popular) dealing with quantum mechanics is immense,

and even a list of classic works is long and not easily claimed to be definitive in view

Notes to Page 31 * 247

of the diversity of the views concerning virtually every aspect of quantum mechan-

ics. This immensity is matched by the number of interpretations of quantum

mechanics itself. Even within the cluster of the standard or Copenhagen (or, as it is

also called, orthodox) interpretations, to which Bohr's interpretation belongs, the

range is formidable, even if one restricts oneself to such founding figures as (in addi-

tion to Bohr) Heisenberg, Born, Pauli, Dirac, Von Neumann, and Wigner. The two

main lines of thought within the Copenhagen cluster are defined by the argument of

whether or not the formalism of quantum mechanics describes (even if noncausally)

the behavior of quantum objects themselves. The first line often follows Dirac's and

Von Neumann's versions of the mathematical formalism of quantum mechanics. For

Dirac's and Von Neumann's versions see their seminal works: Paul A. M. Dirac, The

Principles of Quantum Mechanics (Oxford: Clarendon, 1995), and John Von Neu-

mann, Mathematical Foundations of Quantum Mechanics, trans. Robert T. Beyer

(Princeton, NJ: Princeton University Press, 1983). Their own positions on the sub-

ject are actually more complex and are closer to Bohr's view, especially in Dirac's

case (The Principles of Quantum Mechanics, 10-14). Indeed, this is not surprising,

even from a mathematical standpoint, since Dirac's and Von Neumann's versions

are closer to Heisenberg's matrix mechanics, which is more conducive to Bohr's

view. Schrodinger's version is epistemologically more conducive to (which is not to

say it entails) the other view. Richard Feynman's version is still another story and

will not be discussed here. The profusion of new interpretations during recent

decades was in part motivated by the EPR argument. It may, however, be seen as

triggered by David Bohm's reformulation of the EPR argument in terms of spin and

then his hidden-variables interpretation, introduced in 1952. It received a further

impetus from Bell's theorem (1966) and related findings, especially the so-called

Kochen-Specker theorem and then from Alan Aspect's experiments (around 1980)

confirming these findings. Bell's theorem states, roughly, that any classical-like the-

ory (similar to Bohm's) consistent with the statistical data in question in quantum

mechanics is bound to involve an instantaneous action-at-a-distance and, hence, vio-

late relativity theory. Bohm's theory does so explicitly, in contrast to the standard

quantum mechanics, which does not. As will be seen, these developments recentered

the debate concerning quantum mechanics around the question of nonlocality.

Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, ed.

James T. Cushing and Ernan McMullin (Notre Dame, IN: Notre Dame University

Press, 1989), offers a fairly comprehensive sample, although it requires some updat-

ing. See also Ellis and Amati, eds., Quantum Reflections. David Mermin's essays on

the subject of quantum mechanics in Boojums All the Way Through (Cambridge:

Cambridge University Press, 1990) offer one of the better nontechnical, although

demanding, expositions of some of these subjects. By now, dealing only with non-

relativistic quantum mechanics (considered here), the list of interpretations includes,

among others (and with variations within each denomination), the hidden-variables

interpretation; the many-worlds interpretation; the modal interpretation; the histo-

ries interpretation; the Ghirardi-Rimini-Weber and related interpretations (these

incorporate higher-level theories, specifically gravity); and the relational interpreta-

tion. Among the most recent additions is Mermin's provocative proposal for what he

248 * Notes to Page 31

calls "the Ithaca interpretation," which maintains that only statistical correlations

between quantum events, not events (correlata) themselves, can be meaningfully

considered by quantum theory. While, on the one hand, Mermin's approach aims

to dispense with the irreducible role of measurement, advocated by Bohr, it aims

to ascribe physical reality to correlations alone, not correlata, as would be the case

in mentioned approaches. See David Mermin, "What Is Quantum Mechanics Try-

ing to Tell Us?" American Journal of Physics 66, no. 9 (1998): 753-67, and refer-

ences there. Mermin credits such theorists as Arthur Fine, Paul Taylor, Carlo Rov-

elli, and Lee Smolin (the latter two contributed significantly to both Bohmian

mechanics and relational interpretation). My argument here is that Bohr's inter-

pretation is at the very least as consistent and comprehensive as any available,

albeit to some unsatisfactory by virtue of its radical epistemological features, lead-

ing one to nonclassical epistemology. The present argument develops and some-

times significantly refines several related works by the present author, in particular

"Reading Bohr"; "Techno-Atoms: The Ultimate Constituents of Matter and the

Technological Constitution of Phenomena in Quantum Physics," Tekhnema: Jour-

nal of Philosophy and Technology 5 (1999): 36-95; "Landscapes of Sibylline

Strangeness: Complementarity, Quantum Measurement, and Classical Physics," in

Metadebates, ed. G. C. Cornelis, J. P. Van Bendegem, and D. Aerts (Dordrecht:

Kluwer, 1998); "Complementarity, Idealization, and the Limits of Classical Con-

ceptions of Reality," in Mathematics, Science, and Postclassical Theory, ed. Bar-

bara H. Smith and Arkady Plotnitsky (Durham, NC: Duke University Press, 1997);

and Complementarity.

3. It is worth keeping in mind that quantum phenomena are not restricted to

those associated with what happens on the microscale, established by Planck's law,

and can be extended to macro-objects, including the Universe, which may (there are

significant complications involved) be considered as an ultimately quantum object.

Their quantum "behavior" is, however, due to their ultimate quantum microconsti-

tution (regardless of how one interprets the latter), at least the way we understand

the situation now, an important qualifier to be applied to all claims made here con-

cerning physics and its epistemology. These complexities have largely to do with the

possibility of placing an observer outside the Universe, a fact that is sometimes used

in arguments that quantum mechanics, at least in any observation-based interpreta-

tion, such as Bohr, has to be deficient on that scale. See, for example, Lee Smolin,

The Life of the Cosmos (New York and Oxford: Oxford University Press, 1997),

and, for a counterargument, a review by the present author, "The Cosmic Internet:

A Review of Lee Smolin's The Life of the Cosmos," Postmodern Culture 8.3 (May

1998) (published electronically). Smolin's more recent book, Three Roads to Quan-

tum Gravity (New York: Basic Books, 2001), avoids some of the problems of The

Life of the Cosmos. In particular, the book offers a useful argument for the observer

dependent logic, linked to Grothendieck's topos theory, mentioned earlier. Accord-

ing to this argument, while all observers relate to the universe as a whole, each

observer must be excluded from those parts of the universe this particular observer

can describe (48). The latter point is consistent with Bohr's argument concerning the

difference between quantum objects and measuring instruments. The concept of the

Notes to Pages 31-35 * 249

universe as a whole (i.e. as a wholeness), retained from The Life of the Cosmos, is

a more complex and problematic part of Smolin's argumentation, which, I would

argue, in fact undermines the argument for the observer dependent logic. I

addressed this latter issue in "The Cosmic Internet." These considerations do not

affect the present argument. It may be observed, however, that it is not inconceiv-

able that, in view of its possible ultimate constitution (quantum or not), the "Uni-

verse" on this scale may not allow for a conception in terms of any attributes or,

especially, wholeness. It may prove to be nonclassically "unfigurable," as Maurice

Blanchot said in his The Infinite Conversation (trans. Susan Hanson [Minneapolis:

University of Minnesota Press, 1993], 52), which, as I said, may even be the case

already at the level of Einstein's general relativity at the cosmological scale (i.e.,

even short of considering its quantum constitution). I shall comment on Blanchot's

conception of "the unfigurable Universe" in the conclusion to this study. The con-

cept itself of the unfigurable universe, as just explained, would be consistent with

Bohr's scheme, without at the same time disabling our investigation of its physical

nature. Heisenberg anticipates these arguments in The Physical Principles of the

Quantum Theory, 58.

4. Such constituents, including those associated with the particles of matter and

radiation, such as electrons and photons, remain different in other respects. In addi-

tion, as will be seen, only some particle-like or wave-like properties could apply in

any event.

5. Most key (technical) papers on the subject are assembled in Sources of Quan-

tum Mechanics, ed. B. L. van der Waerden (Toronto: Dover, 1968), which contains

a useful, if, again, technical, historical introduction and commentary.

6. For a useful discussion of and references on Einstein's work on this subject

see Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein

(Oxford: Oxford University Press, 1982), 402-14, and Niels Bohr's Times, in

Physics, Philosophy, and Polity (Oxford: Clarendon, 1991), 191-92.

7. For a discussion see Kuhn, Black-Body Theory, 185.

8. This point echoes Mermin's (different) argument in "What Is Quantum

Mechanics Trying to Tell Us?"

9. See Heisenberg, The Physical Principles of the Quantum Theory, 108-9.

Heisenberg considers spectra, but this does not change the epistemological consider-

ations in question. Indeed, this fact makes these considerations more pertinent, since

it makes the very term kinematical (if understood as something representing motion)

inapplicable. His argument can however be adjusted so as to refer to traditional vari-

ables, such as "position" and "momentum," pertaining to certain parts of measur-

ing instruments, and thus translated Bohr's argument, which Bohr in effect has done.

Much of this work summarizes more complex arguments of his great earlier papers,

"Quantum-Theoretical Re-Interpretation of Kinematical and Mechanical Rela-

tions" (in B. L. van der Wareden, ed., Sources of Quantum Mechanics, 261-77) and

"The Physical Content of Quantum Kinematics and Mechanics"(Uber den

anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik)" (in

Wheeler and Zurek, eds., Quantum Theory and Measurement, 62-86), introducing,

respectively, quantum mechanics and uncertainty relations. The English translation

250 * Notes to Pages 36-40

of the second title is misleading and should read instead "on the representable (intu-

itable) content of quantum-theoretical kinematics and mechanics." I shall return to

the role of the term "anschaulich," crucial for both Bohr and Heisenberg, later. The

Physical Principles of the Quantum Theory gives Heisenberg's arguments a more

Bohrian twist, following Bohr's Como lecture, which introduced complementarity,

while earlier papers are in fact closer to Bohr's post-EPR argument, on which point

I shall comment later in this chapter and in chapter 5.

10. Niels Bohr, interview by Thomas Kuhn, Aage Petersen, and Eric Riidinger,

November 17, 1962, Niels Bohr Archive, Copenhagen.

11. The concept of phenomenon was introduced by Bohr in his Warsaw lecture

of 1938, "The Causality Problem in Atomic Physics" (BCW 7:303-22).

12. It may appear that, following Kant, one should more properly speak of

"noumena" here. Bohr is, however, right not to do so, since "noumena" would more

properly refer to the ultimate constitution of the overall experimental arrangement,

which we cannot observe.

13. It is worth keeping in mind that the latter concepts are not independent of

other conventional, more or less idealized, physical or mathematical attributes. In

the case of particles, for example, such attributes would include "position" or

"momentum," or "trajectories of motion" in classical physics (classical-like trajec-

tories are more or less immediately prohibited in quantum mechanics). In the case of

elementary particles of modern quantum physics, such attributes would also include

their mathematically pointlike (structureless) character, which is the standard and, it

appears, irreducible idealization in quantum physics. Short of string theories, it

appears impossible to theoretically treat elementary particles otherwise, even though

such "objects" are not seen as likely to exist in nature. As we have seen, the objects

of classical physics, too, are sometimes idealized, for example, as (massive) dimen-

sionless material points. There, however, this idealization is necessary because of

practical complexities (sometimes immense), rather than, as in quantum physics,

because of the possible conceptual contradiction in the ensuing theoretical model

itself. It is also worth observing that the "size" of the electron was a problem for the

classical electrodynamics (the theory of bodies moving in an electromagnetic field) as

well.

14. Heisenberg, The Physical Principles of the Quantum Theory, 10.

15. It may appear that some interpretations of quantum mechanics allow us at

least to directly refer to, if not quite to describe, quantum processes themselves and

their key features, while bypassing measurement as an (irreducibly) constitutive

component of quantum physics. Measurement would in this case be seen as auxiliary

as it is in classical physics. Hence, such interpretations may be seen, and are often

offered, as realist, and, accordingly, as epistemologically different from Bohr's.

Thus, we would have at least competing and, to many, epistemologically more palat-

able alternatives to, if not counterarguments against, Bohr's interpretation. I am not

referring to the hidden-variables theories and related nonlocal theories (these theo-

ries are expressly different from quantum mechanics) or the many-worlds interpre-

tation of quantum mechanics, as the latter do not retain the key features of quantum

mechanics here under discussion. I have in mind (in addition to some earlier ver-

Notes to Pages 41-43 * 251

sions, such as Richard Feynman's) certain versions of the modal interpretation

(specifically that of Bas C. van Fraassen); Robert Griffiths's histories interpretation;

Roland Omnes's logical interpretation (a version of the histories interpretation), as

presented in The Interpretation of Quantum Mechanics (Princeton, NJ: Princeton

University Press, 1994); and possibly (this case is more complex) Mermin's "Ithaca

interpretation." These are, arguably, among the closest to Bohr's among such inter-

pretations. Von Neumann's and related interpretations, such as Peter Mittelstaedt's

minimal interpretation, as presented for example, in The Interpretation of Quantum

Mechanics and the Measurement Process (Cambridge: Cambridge University Press,

1998), are more grounded in the concept of measurement. Both van Fraassen and

Omnes associate their interpretation with the Copenhagen interpretation; and Mer-

min's indicates further proximities between Bohr's and the histories interpretations

of Griffiths and Omnes (Mermin, "What Is Quantum Mechanics Trying to Tell Us?"

763). I shall refrain from an ultimate judgment concerning such claims and offer

only the following observation, related to what Bohr's or a Bohrian response would

be as concerns the possibility of such claims in principle, referring the reader to the

more extended analysis in Plotnitsky, "Reading Bohr." Bohr's argument appears to

imply the following for the interpretations in question, given that the mathematical

formalism involved is that of standard quantum mechanics, that is, a highly sym-

bolic scheme, involving infinite-dimensional mathematical objects. Once their argu-

ment is related to anything that can actually occur in space-time, Bohr's phenomena

and, hence, measurement may have to be reinstated as constitutive rather than

merely auxiliary part of the overall situation.

16. This view of the ultimate nature of the physical world was recently advo-

cated by Julian B. Barbour in The End of Time: The Next Revolution in Physics

(Oxford: Oxford University Press, 1999). Barbour is right to argue that motion (or

change, differentiation, and so forth) may not apply at the ultimate level of the con-

stitution of nature, which is the present view as well. The present argument also sug-

gests, however, that what Barbour calls "Platonia" (the state of nature without

change and motion) need not follow (indeed its attribution is no less questionable

than that of its opposite), unless, of course, strong further evidence for such a view

is given, which Barbour does not appear to offer even at the level of conjecture.

Overall, his argument clearly and perhaps deliberately echoes that of Parmenides

specifically as it is spelled out in Plato's Parmenides.

17. Abraham Pais, "Einstein and the Quantum Theory," Reviews of Modern

Physics 51 (1979): 907. Cf. also David Mermin's commentary in Boojums All the

Way Through, 81-82.

18. Marcel Proust, "The Guermantes Way," The Remembrance of Things Past,

trans. C. K. Scott Moncrieff and Terence Kilmartin (New York: Vintage, 1981),

3:64; "Le C6te de Guermantes," A la recherche du temps perdu (Paris: Gallimard,

1988), 2:366.

19. Erwin Schrodinger, "The Present Situation in Quantum Mechanics," in

Wheeler and Zurek, eds., Quantum Theory and Measurement, 154.

20. The potential inapplicability of these terms and concepts (rather than the

nonexistence of that which we so characterize) appears to be what led Bohr to say

252 * Notes to Page 43

in a famous (reported) statement: "There is no quantum world." The statement is

reported in Aage Petersen, "The Philosophy of Niels Bohr," Niels Bohr: A Cente-

nary Volume, ed. A. P. French and P. J. Kennedy (Cambridge, MA: Harvard Uni-

versity Press, 1985), 305. One must, of course, exercise caution in considering such

reported statements. It would, however, be very difficult to conclude on the basis of

Bohr's works that he denies the existence of that to which "the quantum world"

would refer or, again, from (the effects of) which Bohr's "quantum objects" would

be, nonclassically, idealized as that to which no such attributes can be assigned. The

statement may instead be read, especially given the context (the question of whether

quantum mechanics actually represents the quantum world), by putting the empha-

sis on "quantum," as without in any way indicating the nonexistence of material

efficacity of the effects in question. Instead, it indicates the inapplicability to the lat-

ter of conventional "quantum" attributes-such as discontinuity (of radiation),

indivisibility (of quanta themselves), and so forth, or any other physical attributes,

even "objects," "constituents," and so forth, including, ultimately, individuality of

quantum objects ("particles") as such, or the wavelike character of quantum

processes-to the "quantum objects" of Bohr's interpretation. Beyond this level (of

Bohr's idealization), even this latter claim may, as I said, be inapplicable. At the

level of phenomena, certain individuality remains, and the existence of the

microlevel (that of the ultimate constituents of matter) efficacities of such individ-

ual phenomenal effects is essential. Both have been at stake in quantum physics

from Planck on. In fact, Planck's law is incompatible with assigning identities to

individual particles (of distinguishing them) within quantum-physical multiplicities,

which is what mathematically distinguishes Planck's and other statistical counting

procedures of quantum physics from classical statistical physics. This statistical

configuration is already phenomenologically inconceivable or, in Bohr's language,

beyond pictorial visualization or intuition (Anschaulichkeit), an impossibility that

defines quantum physics for Bohr, primarily in view of the nonclassical considera-

tions here in question. This fact has far-reaching consequences in quantum physics,

from Planck's law on. On the one hand, "identity" in the sense of the interchange-

ability of all particles of a given type (photons, electrons, and so forth) is crucial; on

the other, this identity also peculiarly manifests itself in the impossibility of assign-

ing particles individual identity in certain situations, or perhaps ultimately ever (for

the reasons to be explained presently). In quantum field theories, such as quantum

electrodynamics (QED), beyond the impossibility of distinguishing individual par-

ticles, one can no longer quite speak of the particles of the same type. An investiga-

tion of a particular type of quantum object (say, electrons) irreducibly involves

other types of particles, conceivably all existing types of particles. This is the main

reason why Heisenberg saw Dirac's discovery of antiparticles in 1928-32 (the

process was somewhat prolonged), which entails this situation, as one of the great-

est discoveries of modern physics, "perhaps the biggest of all the big changes in

physics of our century" (Heisenberg, Encounters with Einstein, 31). According to

Heisenberg, quantum field theories push the complexities in question to their

arguably most radical available limits, even beyond those of the standard quantum

mechanics of Heisenberg and Schrodinger (31-35). The latter is a complex and lit-

Notes to Pages 45-56 * 253

tie developed subject, which cannot be addressed here without expanding the scope

of this discussion beyond its limits. The circumstances themselves in question, how-

ever, are not only fully consistent with but would reinforce the present epistemo-

logical argument.

21. I shall not address the question of models in physics, a major subject of inves-

tigation in its own right, which would require a separate treatment. The present dis-

cussion is self-sufficient, although limited, as concerns this subject. These limitations,

however, do not affect my main epistemological argument.

22. This is a clear implication of Stephen Hawking's argument in his debate with

Penrose in Stephen Hawking and Roger Penrose, The Nature of Space and Time

(Princeton, NJ: Princeton University Press, 1996).

23. I consider this subject in "Reading Bohr."

24. This argument is indebted to, although different from, Arthur Fine's point in

"Do Correlations Need to Be Explained?" (in Cushing and McMullin, eds., Philo-

sophical Consequences of Quantum Theory, 180).

25. Mittelstaedt offers important commentaries on this impossibility from a

more formal viewpoint of quantum logic, most recently in his The Interpretation of

Quantum Mechanics and the Measurement Process.

26. Bohr's sense of the situation is of some interest. He opens his remarks with a

statement, cited earlier, on "Galileo's programme" of basing "the description of

physical phenomena on measurable quantities," applicable to quantum mechanics

as well, provided one redefines, as Bohr does, the phenomena in question accord-

ingly and, correlatively, rethinks what is in fact described and how by quantum

mechanics. However, he traces the emergence of the classical model just described

and the program it entails more specifically to Newtonian mechanics and subsequent

developments (PWNB 3:1). This view is astute in presenting this model both as a

version of Galileo's more general program and as departing from Galileo's own view

and practice.

27. For a relevant commentary see Lawrence Sklar, Physics and Chance: Philo-

sophical Issues in the Foundations of Statistical Mechanics (Cambridge: Cambridge

University Press, 1998), and references there. Indeed chaos-theoretical considera-

tions are often pertinent in situations treated by means of classical statistical physics

alone and complicate such situations considerably, even though chaos theory itself

is classical.

28. Heisenberg, The Physical Principles of the Quantum Theory, 64.

29. See also Pauli's elegant commentary (whose essential points he attributes to

Bohr) in Max Born, The Born-Einstein Letters, trans. Irene Born (New York:

Walker, 1971), 224-25.

30. There exists the quantum-theoretical concept of state defined via the formal-

ism of quantum theory and specifically the so-called state-vector, a concept bound

by the uncertainty relations. This concept is more significant when the formalism of

quantum theory is seen as describing the behavior of quantum objects themselves.

Realism appears to remain the main aspiration of most such interpretations. Bohr's

interpretation does not assign, and does not allow one to assign, physical reality to

the state-vector.

254 * Notes to Pages 57-58

31. Arguably the best definitions are those of Bohr (PWNB 1:8) and Heisenberg

(The Physical Principles of the Quantum Theory, 105).

32. Paul Arthur Schilpp, Albert Einstein: Philosopher-Scientist (New York:

Tudor, 1949).

33. It is epistemologically much closer to, indeed virtually the same as (the con-

cept of "phenomena" as such was to come later, but the epistemology is in place) the

version presented in "Discussion with Einstein" than to the Como lecture. Some of

Bohr's critics did not fail to point out these inconsistencies, and some argue that they

undermine Bohr's project, the logic of complementarity, and even that of quantum

mechanics itself, for example, versus Bohm's theory. This type of argument is

advanced, for example, by James T. Cushing, Quantum Mechanics, Historical Con-

tingency, and the Copenhagen Hegemony (Chicago: University of Chicago Press,

1994), and, in part following Cushing, to whom and to whose work she acknowl-

edges a special debt, by Mara Beller as part of their defense of Bohm's theory, or (as

she might argue) in Beller's case, her advocacy of Bohm's theory versus Bohr and the

dominance of the Copenhagen approach(es), in Quantum Dialogue: The Making of

a Revolution (Chicago: University of Chicago Press, 1999). Cushing's discussion of

Bohr is at best a gloss, as Cushing nearly admits himself, and cannot, accordingly, be

of much help, even if one were to accept this gloss qua gloss, which would not be

easy for most who read Bohr (26). Beller's treatment of Bohr is, by contrast, exten-

sive, indeed it takes most of the book. In this author's view (and given the analysis

offered in this study), Beller's analysis of Bohr's key concepts lacks sufficient preci-

sion and discrimination of crucial nuances, which are all the more necessary in view

of the complexities involved in different versions of complementarity, a difference

that grounds much of the argument here. (I leave aside Beller's poorly supported

commentaries on Bohr's personality, which are in obvious conflict with most known

accounts of Bohr, by critics and admirers alike, beginning with Einstein, who was

both.)

Beller is not wrong in arguing that there is no single framework of complemen-

tarity (or that there is a conflict between Bohr's earlier and later arguments, as just

indicated), or that Bohr's argument for it (in whatever version) is not inevitable, as

some argue, but rather only consistent with the available data and the formalism of

quantum mechanics. (Surprisingly, Beller devotes little, if any, attention to different

versions of Bohm's theory, which changed considerably over the years, even in

Bohm's own work.) As I said, it is true that Bohr does not always help either, espe-

cially on the first question. He is, I would argue, more cautious as concerns the

inevitability of complementarity than Beller argues. Given immense difficulties

involved (we are not quite finished with them even now), it is hardly surprising that

Bohr had to refine or even correct his argument. Glitches are found virtually until

"Discussion with Einstein," including in his reply to EPR, and I shall indicate some

of them. It is difficult to expect one to fully control such a text, a point often made

by Bohr himself, who famously never stopped revising his work and was reluctant to

publish at almost any stage. The "conflict" between the Como version and the post-

EPR one is that the first, while an extraordinary step, ultimately did not quite work

and had to be refined or even reworked, as just explained. The second ultimately

Notes to Page 59 * 255

does, which is quite enough, even if only as a local interpretation, which is more than

most and perhaps all others are able to offer; certainly it is something that Bohm's

theory fails to offer. This is an immense labor. Beller is right to argue for its multifar-

ious complexity, and she summons much valuable historical evidence and documen-

tation. Her interpretation of this process or, again, of Bohr's work and complemen-

tarity itself, is far less compelling. Admittedly, more needs to be said in order to

ground appropriately this view of her book and make it more than a view of this par-

ticular reader, which cannot be done here. So a view it must remain-on that score.

On the other hand, one can argue much more definitively that Beller's book con-

tains significant misconceptions concerning quantum mechanics and Bell's theorem

and related finding, which are, in any event, never properly, or even minimally,

explained in the book but are merely (and barely) referred to (along with some liter-

ature), while they are themselves subject to a considerable interpretive complexity.

Beller, problematically, sees these findings as demonstrating the nonlocality of quan-

tum mechanics (or even the data in question themselves) and, hence, as erasing the

main problem of Bohm's theory versus the standard quantum mechanics, which is

local or may be interpreted as local. This view is among the factors that lead her to

significant misconceptions concerning Bohr's conception of the wholeness of phe-

nomena, which I shall consider in detail later (Quantum Dialogue, 252-59). Beller

reads this concept through the prism of Bohm's nonlocal quantum wholeness, which

she sees as better responding to "the post-Bell notions of 'inseparability' [quantum

entanglement] and 'nonlocality'" (253). This may be true, if one sees nonlocality (as

the conflict with relativity theory, found in Bohmian mechanics) as revealed by the

experimental data or as an acceptable feature of a theory, which, however, is not the

case so far. Indeed, Beller admits that Bohr's "reasoning makes sense in an anti-real-

ist approach, where it is meaningless to discuss behavior and properties indepen-

dently of measurement" (254). This is a curious admission of Bohr's rigor on Beller's

part, something that she persistently (and I would argue, mostly mistakenly) denies

to Bohr; and indeed this point would be enough to justify my claims in this study or

to diminish much of Beller's argument against Bohr. It is not difficult to perceive

Beller's strong realist agenda in the book; and, however much she tries to separate

herself from naive realism, it remains realism, nevertheless, which, as I said, is per-

haps always naive. She also (correctly) sees Bohr's argument as rejecting nonlocality.

On the other hand, she sees Bohm's nonlocal theory as an equally viable, and (it

appears) even better, epistemological alternative in view of Bell's theorems and

related results. That, however, could only be argued were nonlocality (in the sense

of Bohm's theory, and hence incompatibility with relativity) an established fact of

nature or a rigorous consequence of quantum mechanics, which it is not, in part pre-

cisely in view of Bohr's antirealist interpretation. As will be seen later in this chap-

ter, locality and realism may indeed be incompatible. In any event, nonlocality is not

seen as an acceptable feature of physical description by an overwhelming majority of

physicists, beginning with Einstein, who, incidentally, did not care for Bohm's the-

ory, in part for this reason. There may be no alternative on that score, at least for

now.

34. In his The Physical Principles of the Quantum Theory (13, 63-65), Heisen-

256 * Notes to Pages 65-71

berg was rather closer to the mark and to Bohr's later argument in both his great

early papers, introducing, respectively, his matrix quantum mechanics and uncer-

tainty relations. To some degree he was also able to handle more successfully the

argumentations of the type of the Como lecture in his 1929 Chicago lectures, just

cited, which were clearly influenced by Bohr's Como argument.

35. Anthony J. Leggett, "Experimental Approaches to the Quantum Measure-

ment Paradox," Foundations of Physics, 18, no. 9 (1988): 940-41. There are a vari-

ety of approaches to quantum probability, in part because, in addition to the

specificity of quantum probability, there are a variety of approaches to probability

as such. These topics would require a separate discussion. Among the most interest-

ing approaches to these questions from the present perspective are those along the

lines of quantum information theory, further coupled to the so-called Bayesian ver-

sion of probability. Most literature on these subjects is technical. For an excellent, if

demanding, introduction to these subjects and to quantum information theory in

general, and further references, see Christopher A. Fuchs, "Quantum Foundations in

the Light of Quantum Information," in Decoherence and Its Implications in Quan-

tum Computation and Information Transfer, ed. Tony Gonis and Patrice E. A.

Turchi (Amsterdam: IOS Press, 2001), 38-82.

36. Henry Stapp is one of the most persistent advocates of this view, which, he

argues, is a consequence of quantum mechanics. See, for example, Henry P. Stapp,

"Quantum Nonlocality and the Description of Nature," in Philosophical Conse-

quences of Quantum Theory, ed. Cushing and McMullin, 154-74, and Henry P.

Stapp, "Nonlocal Character of Quantum Theory," American Journal of Physics 65:

(1997): 300-304. For an effective counterargument, via Bohr's reply to EPR, see

David Mermin, "Nonlocal Character of Quantum Theory?" American Journal of

Physics 66 (1998): 920-24.

37. Bohr illustrates this argument by his famous drawing showing only heavy,

immovable instruments and, sometimes, traces left by the experiments performed,

without showing, even diagrammatically, anything actually happening at the quan-

tum level.

38. The assessment of the relative merits of both interpretations could change

were nonlocality to follow from the formalism of quantum theory or data them-

selves, as some argue. I would contend that such arguments are unconvincing so far

(which, of course, may change, too), as the Mermin-Stapp exchange cited earlier

(note 36) suggests.

39. See Bohr's comments in PWNB 3:5-6 and "On the Notion of Causality and

Complementarity" (BCW 7:335). Along these lines (there are considerable differ-

ences otherwise), Bohr's interpretation may be related to Roland Omnes's "logical

interpretation of quantum mechanics," in which, roughly, only what is logically pos-

sible is physically possible in quantum mechanics. See Roland Omnes, The Interpre-

tation of Quantum Mechanics and Understanding Quantum Mechanics (Princeton,

NJ: Princeton University Press, 1999).

40. This concept, accordingly, has nothing to do, as some argue, with David

Bohm's concept of "wholeness" as the nonlocality of the quantum world, indebted

as the latter may be to Bohr's ideas. Complementarity is a local interpretation.

Notes to Pages 73-80 * 257

41. On quantum "techno-atomicity," I permit myself to refer to my "Techno-

Atoms." Among very few authors who commented on the significance of Bohr's spe-

cial conception of atomicity is Henry Folse in, among other works, "Niels Bohr's

Concept of Reality" (Lahti and Mittelstaedt, Symposium of the Foundations of

Modern Physics 1987), cited above. Folse's interpretation of this conception is dif-

ferent from the one offered here. In particular he argues that Bohr, in fact, did not

carry his program far enough insofar as he failed to provide an account of individ-

ual quantum objects themselves. According to the present reading, Bohr instead

argued that his conception of atomicity at the level of phenomena makes such an

account or indeed such a conception rigorously impossible. On this view, while, as I

said, crucial to all of quantum physics, the "identity" of individual quantum objects,

say, all electrons, is in all rigor only a shorthand for properly correlated individual

phenomena in Bohr's sense, the same class of equivalent measurements. Among

closer arguments is that offered in an excellent recent article of Ole Ulfbeck and

Aage Bohr, "Genuine Fortuitousness: Where Did That Click Come From?" Founda-

tions of Physics 31, no. 5 (2001): 757-74. In this case while the authors' handling,

in (Niels) Bohr's language, of "the peculiar individuality of quantum effects" or, in

their language, "clicks," is close to that of the present study, their argument, by con-

trast, would be different from the present one as concerns the role given to quantum

aspects of the efficacity of these effects. These aspects, I argue here, are crucial (espe-

cially in the argument concerning the locality of quantum mechanics), even though

and in part because this efficacity must be seen as nonclassically unknowable in the

present sense. In this sense Ulfbeck and Bohr's view appears to differ from Bohr's, as

here read (their article itself is not a reading of Bohr, but an independent argument

for an interpretation of quantum mechanics). In certain respects their view is closer

to some of Heisenberg's arguments, especially in their emphasis on the mathemati-

cal, rather than spatio-temporal, nature of the key elementary concepts, such as par-

ticles or atoms, especially from the point of view of symmetry and group theory.

This argumentation would also contrast with the present one, even though both

share the view that an attribution of such concepts (or, again, any conceivable con-

cepts) is, in all rigor, impossible at the quantum level. I am grateful to the authors for

the opportunity to see the article before it was published.

42. Hawking and Penrose, The Nature of Space and Time, 60.

43. Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World

(Oxford: Oxford University Press, 1986), 262.

44. Cf. also PWNB 2:73.

45. Most of the better treatments are, unfortunately, technical. See, however,

Roger Penrose, The Emperor's New Mind: Concerning Computers, Minds, and the

Laws of Physics (Oxford: Oxford University Press, 1989), 236-42. The exposition

may require effort, but it is elegant and, in principle, does not require any specialized

knowledge, as all the necessary background is provided in the book itself.

46. See, again, his commentary in The Principles of Quantum Mechanics

(10-14), which, while technical, would convey the point even to a nonspecialist.

47. These considerations are significant even in the case of the so-called spin (a

particular quantum-mechanical element, a kind of momentum, associated to each

258 * Notes to Pages 86-89

particle, which has no equivalent or even strict analogue in classical physics), where

we deal with finite-dimensional Hilbert spaces, but especially in the case of standard

kinematical and dynamical variables and the infinite-dimensional Hilbert spaces.

There are, however, alternative views as concerns the physical reality of this formal-

ism, found, for example, in Penrose's works cited here, as well as in his more acces-

sible summary in The Large, the Small, and the Human Mind (Cambridge: Cam-

bridge University Press, 1997). On this point, I permit myself to refer to my earlier

discussions in "Complementarity, Idealization, and the Limits of Classical Concep-

tions of Reality" (161-67) and "Penrose's Triangles: A Review of Roger Penrose's

The Large, the Small, and the Human Mind, with a Glance Back at The Emperor's

New Mind, The Shadows of the Mind, and The Nature of Space and Time," Post-

modern Culture 7, no. 3 (May 1997) (published electronically).

48. Einstein, "Reply to Criticisms," in Schilpp, Albert Einstein: Philosopher-

Scientist, 681-82.

49. As I said, Einstein did not much care for Bohm's theory either, perhaps pri-

marily in view of its nonlocality.

50. See an interesting remark by Einstein reported by Pauli in The Born-Einstein

Letters: "Although the description of physical systems by quantum mechanics is

incomplete [by Einstein's criteria?], there would be no point of completing it, as a

complete description would not agree with the laws of nature" (226). This view

echoes Schrodinger's sentiment that classical ideals and models "cannot do justice to

nature," according to the quantum-mechanical "doctrine" (QTM, 153).

51. The literature on the subject is massive, and it would not be possible to

address it here. See especially the commentary by John Bell, The Speakable and the

Unspeakable in Quantum Mechanics, 155-56, 189-90, and the essays assembled in

Cushing and McMullin, eds., Philosophical Consequences of Quantum Theory, and

in Ellis and Amati, eds., Quantum Reflections. As several essays in these collections

suggest, Bohr's views and writings encounter considerable resistance among the pro-

ponents of Bohm's approach, who find their inspiration in Einstein's and Bell's cri-

tiques, as, to give two recent examples cited earlier, in Beller's Quantum Dialogue

and Cushing's Quantum Mechanics, Historical Contingency, and the Copenhagen

Hegemony. I shall comment on this resistance from a different perspective in chap-

ter 5. As I have indicated earlier a propos Beller's book, to paraphrase Bohr on EPR's

argument, the trends of these arguments do not seem to me adequate to meet the

case that Bohr's writings present to us. Bohm himself had a more balanced view and

cogent sense of Bohr's argument and work, which he admired, as is clear from most

of his writings, for example, Wholeness and Implicate Order (London: Routledge,

1995). Bohm is, of course, also the author of the classic exposition of the standard

quantum mechanics, Quantum Theory (New York: Dover, 1979), originally pub-

lished in 1951 (shortly before he discovered his hidden-variables version of quantum

theory), where, however, Bohm's epistemological argument is quite different from

Bohr's, as presented here. For arguably the final version of Bohm's theory itself see

David Bohm and Basil J. Hiley, The Undivided Universe: An Ontological Interpre-

tation of Quantum Mechanics (London: Routledge, 1993). I have considered Bell's

and Einstein's commentaries on Bohr in Complementarity, 160-63, 176-90. As I

Notes to Pages 93-97 * 259

have indicated, while they have attracted some public attention recently, the

Bohmian approach represents a minority view among the physics community (they

are somewhat more widespread among the philosophers of quantum mechanics).

Naturally, this fact is not an argument against such views (the nonlocality of

Bohmian mechanics is, in my view) or against this or any criticism of Bohr. For dif-

ferent views of these questions, closer to the present position, see essays by Abner

Shimony, "New Aspects of Bell's Theorem," and Kurt Gottfried, "Does Quantum

Mechanics Carry the Seeds of Its Own Destruction?" in Ellis and Amati, eds., Quan-

tum Reflections, as well as several essays in Cushing and McMullin, eds., Philo-

sophical Consequences of Quantum Theory, especially by Shimony, Mermin, van

Fraassen, Fine, and Folse ("Bohr on Bell," which, however, offers a reading of Bohr

that is rather different from the one offered here). See also Mermin's discussion of

these questions in Boojums All the Way Through, 81-185. Gottfried appears to

express the majority view in arguing that relativity in fact provides the experimental

evidence ("test") against Bohmian mechanics and that Bell's theorem(s) and related

theoretical findings further support this evidence in his "Inferring the Statistical

Interpretation of Quantum from the Classical Limits," Nature 405 (2000), 533-36,

a point that extends the argument of his "Does Quantum Mechanics Carry the Seeds

of Its Own Destruction?"

52. See, again, Mermin's essays on the subject in Boojums All the Way Through,

81-185.

53. The arguments for it, such as those by Stapp, mentioned earlier, appeared to

be unconvincing at best.

54. Stapp argues this on several occasions, for example, again, in "Quantum

Nonlocality and the Description of Nature," clearly attributing to Bohr terms and

concepts, such as "epistemological disturbance," never found in Bohr's writing (in

Cushing and McMullin, eds., Philosophical Consequences of Quantum Theory,

162). As my elaborations to be offered presently suggest, it is difficult, indeed impos-

sible, to sustain such an argument. Beller, too, appears to misconstrue Bohr's depen-

dence on the concept of disturbance, specifically in his reply to EPR (her argument is

different from Stapp's), which is less significant than she and others argue even in his

earlier writing (Beller, Quantum Dialogues, 155-60).

55. I omit Bohr's intermediate propositions concerning the interactions between

quantum objects and measuring instruments. While they are important in explaining

the reasons for Bohr's argument, they are fully consistent with the preceding analy-

sis, presented in terms of "Discussion with Einstein."

56. See Fine, "Do Correlations Need to Be Explained?" (in Cushing and

McMullin, eds., Philosophical Consequences of Quantum Theory) and Mermin,

"What Is Quantum Mechanics Trying to Tell Us?"

57. See Mermin's comments in Boojums All the Way Through, 108-9.

58. Einstein, The Born-Einstein Letters, 209; see Mermin's discussion in Boo-

jums All the Way Through, 171-74.

59. For a preliminary approach see an article by the present author, "Landscapes

of Sibylline Strangeness."

60. See Linda Wessels, "The Way the World Isn't: What the Bell Theorems Force

260 * Notes to Pages 98-106

Us to Give Up," in Cushing and McMullin, eds., Philosophical Consequences of

Quantum Theory, 80-96, for a useful discussion of possible "options" given Bell's

and related theorems, which, however, does not discuss a "nonclassical" option in

the present sense.

61. See, for example, Shimony's "New Aspects of Bell's Theorem" in Ellis and

Amati, eds., Quantum Reflections (136-64), and Mermin's "What Quantum

Mechanics Is Trying to Tell Us," and references there. Arguments concerning the

possible inapplicability of counterfactual reasoning in quantum mechanics have

been made by others as well, including, interestingly, by Henry Stapp in his earlier

works, before he became a strong proponent of quantum nonlocality.

62. See Mermin, "What Quantum Mechanics Is Trying to Tell Us," 765 n. 31.

63. Hans Reichenbach, The Direction of Time (Los Angeles: University of Cali-

fornia Press, 1956), 159.

64. While the earlier qualifications, especially those concerning nonlocality, must

be kept in mind they would not affect the points made at the moment.

65. Werner Heisenberg, "Quantum Theory and Its Interpretation," 95.

66. I refer primarily to the work of Thomas Kuhn, Imre Lakatos, and Paul Fey-

erabend, and following them the work in, as they became known, the social studies

of science (for example, in the work of David Bloor and his school) and related

developments, in particular the work of Bruno Latour, one of the primary subjects

of the Science Wars debates. I shall comment on these developments in this latter

context in chapter 4.

67. Obviously, an appeal to "convincing arguments" would indicate the com-

plexities mentioned earlier. However, the core of the present argument concerning

the disciplinarity of physics under the radical epistemological conditions in question

would be maintained, indeed, I would argue, all the more so once these complexities

are taken into account.

68. On these questions I permit myself to refer to an article by David Reed and

the present author, "Discourse, Mathematics, Demonstration and Science in

Galileo's Discourses Concerning Two New Sciences" Configurations (winter 2001):

65-92.

69. Martin Heidegger, What Is a Thing? trans. W. B. Barton Jr. and Vera

Deutsch (South Bend, IN: Gateway, 1967), 93. Quantum mechanics and comple-

mentarity, as considered here, obviously give new and more radical dimensions to

Heidegger's point, made by him, fittingly, in the context of Galileo's project, but not

inconceivably and equally fittingly with Heisenberg in mind.

70. As Sylvan S. Schweber argues in his QED and the Men Who Made It: Dyson,

Feynman, Schwinger, and Tomonaga (Princeton, NJ: Princeton University Press,

1994), in the case of quantum electrodynamics (QED), at a certain point in the his-

tory of quantum physics, it was the persistence in keeping the existing framework,

with incremental modifications, rather than attempts at radically transforming it

that paid off. In the case of QED it was, ironically, Dirac, its founder, who gave up

on his creation and believed that yet another radical transformation, similar to that

of the original quantum mechanics in relation to classical physics, would be neces-

sary. Schweber speaks of the "extreme conservatism" of the figures mentioned in his

Notes to Pages 106-10 * 261

title in this context and in this sense. From the present perspective, the extreme con-

servatism may apply even when a radical transformation is ultimately at stake. On

the other hand, it cannot be seen as necessary in all conditions or at all points, even

in science, although virtually all the founders of quantum mechanics appeared to

conform to this view at the time of its emergence. We can never be certain what will

ultimately pay off. In some respects the creation of modern QED was quite radical

as well, in particular in employing rather unorthodox, and indeed mathematically

strictly forbidden, techniques in the so-called renormalization procedure, the center-

piece of quantum field theory ever since. So the creators or (it was mostly founded

by Dirac and several others earlier) perhaps "saviors" of modern QED, too, were

both extreme conservatives and extreme radicals, just as were the founders of quan-

tum mechanics earlier.

71. Albert Einstein, "Autobiographical Notes," in Schilpp, Albert Einstein:

Philosopher-Scientist, 45-47. It is true that Einstein here refers to Bohr's 1913 the-

ory of the atom, which appeared at the time to hold some promise for a classical res-

olution, even to Bohr then. As we have seen, however, certain nonclassical features

were in place in Bohr's work even then. Einstein's statement refers precisely to

Bohr's ability to do physics under these conditions of extremely uncertain founda-

tions, of which Bohr was himself acutely aware at the time as well. This is not unlike

Keats's famous "negative capability."

72. Obviously, Milton's conception has its sources as well, especially Lucretius,

who is sometimes also argued, for example and in particular, by Michel Serres, to be

one of the precursors of chaos-theoretical, as well as quantum-mechanical, philo-

sophical ideas.

Chapter 3

1. Paul R. Gross and Norman Levitt, Higher Superstition: The Academic Left

and Its Quarrels with Science (Baltimore: Johns Hopkins University Press, 1994)

(the second edition with supplementary notes appeared in 1998); Alan D. Sokal,

"Transgressing the Boundaries-Towards a Transformative Hermeneutics of Quan-

tum Gravity," Social Text (spring/summer 1996): 217-52; Alan Sokal and Jean Bric-

mont, Impostures intellectuelles (Paris: Odile Jacob, 1997), published in English as

Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science (New York: Pic-

ador USA, 1998).

2. Jacques Lacan, Ecrits (Paris: Seuil, 1966) and Ecrits: A Selection, trans. Alan

Sheridan (New York: W. W. Norton, 1977). Unless indicated otherwise, I refer here

and throughout this chapter to the English edition, Ecrits: A Selection. The follow-

ing other works by Lacan will be considered: "Desire and the Interpretation of

Desire in Hamlet," trans. James Hulbert, Yale French Studies 55/56:11-52; and "Of

Structure as an Inmixing of an Otherness Prerequisite to Any Subject Whatever," in

The Languages of Criticism and the Sciences of Man: The Structuralist Controversy,

ed. Richard Macksey and Eugenio Donato (Baltimore and London: Johns Hopkins

University Press, 1970), 186-200.

262 * Notes to Pages 111-16

3. I have commented on these questions in "Complementarity, Idealization, and

the Limits of Classical Conceptions of Reality," 161-67.

4. Accordingly, the more strictly Lacanian part of the present essay may be

more difficult for the general reader than ideally desirable. This may be harder to

avoid than in the case of other figures discussed here, complex as their work may be.

It is conceivable that no presentation of Lacan available to the general reader is pos-

sible without major distortions, thus defeating any attempt to give a nonspecialized

version of Lacan's arguments. In this sense Lacan's text may be irreducibly special-

ized-to Lacan. This quality would not be inconsistent with Deleuze and Guattari's

view of philosophical discourse, to be partially adopted here in regard to Lacan's

work, and applicable to other figures considered here, and this view is adopted here

in relation to their work as well. Lacan's case appears to be especially cumbersome,

however, in part for the reasons just explained. All one can do is to try to make one's

reading of extracted chains of Lacan's discourse consistent with possible readings of

his ambient discourse. This is what I shall attempt here. On the other hand, my argu-

ment concerning how Lacan uses mathematics is independent of these complexities

and is, thus, fully available to the general reader. I am grateful to Guy Le Gaufey for

productive exchanges and for permission to read his yet unpublished essay on Lacan,

"Comment l'objet a et pourquoi?" He is also the author of several works dealing

with the relationships between Lacanian problematics and mathematics and science,

especially L'incompletude du symbolique: De Rene Descartes a Jacques Lacan

(Paris: E.P.E.L., 1991) and L'Eviction de l'origine (Paris: E.P.E.L., 1994).

5. A number of works, including by scientists, supporting these points, and

more direct defenses of Lacan, have appeared since the publications of these books,

and I shall comment on some of them later and in the next chapter. For example,

Henry Krips's "Review of Intellectual Impostures" (Metascience 9, no. 3 [Novem-

ber 2000: 352-57]) argues that Lacan's use of topology is not as bad or nonsensical

as Sokal and Bricmont or Gross and Levitt make it out to be. Krips sees this use in

more conventional metaphorical terms rather than in terms of epistemological par-

allels considered in this study, which may indeed be more appropriate in the case of

Lacan's use of topology, in contrast to complex numbers. Most readers of these

books, however, and even their critics still tend to submit to their authority on math-

ematics and science, in my view, too readily, as I shall argue here and in the next two

chapters.

6. Friedrich Nietzsche, On the Genealogy of Morals and Ecce Homo, trans.

Walter Kaufmann (New York: Vintage, 1967), 18.

7. The secondary literature on the subject is vast (although more limited when

dealing with the connections between Lacan and mathematics and science) and can

only be minimally addressed in this essay, which centers on Lacan's own key quasi-

mathematical concepts. It may be observed that this literature is extremely uneven,

which makes one's task of navigating among both Lacan's own ideas and commen-

taries on them all the more difficult.

8. Just in case, I shall give here a brief explanation of the Moebius strip as a

mathematical object. The model of the Moebius strip is a strip of paper with its

edges glued together by turning one of them upside down. Locally, in the vicinity of

Notes to Pages 116-17 * 263

any given point, the Moebius strip is, thus, indistinguishable from a regular (cylin-

der) strip, which one obtains by gluing a strip of paper, while keeping both edges on

the same side. Each point is an intersection of a vertical interval (think of each such

interval as a stalk to be added to a sheaf) and a horizontal interval common to all,

drawn in the middle of the strip (think of it as a single long stalk to be tied to the

sheaf). We can now glue the end intervals of the strip either regularly or twist one of

them into a Moebius strip or tie our stalk accordingly (hence "sheaf" terminology).

No single vertical stalk, or its intersection with a long stalk, is affected in itself, and

the (tied) horizontal stalk is a circle in both cases. The two global objects are, how-

ever, different topologically. If you begin to paint on one side of the Moebius strip,

say, in blue, you will end up painting "both" sides the same color-that is, it only

has one side. Its boundary is topologically a single circle, while the boundary or,

rather, boundaries of a regular (glued) strip are two separated circles. (One can paint

its two sides in two different colors, say, red and blue.) A regular strip is also a sheaf,

but an untwisted one, a trivial sheaf, as mathematicians call it. In general, a circle

and a line interval are replaced by other, usually more complex, mathematical

objects. A horizontal circle or its equivalent is called the "base" of a sheaf, while ver-

tical intervals or their equivalents are called "fibers." The space as a whole is a bun-

dle of "fibers" (hence the "fiber bundle" terminology). Technically, the terms

"sheaves" and "fiber bundles" serve to designate somewhat different mathematical

objects, but these differences are not essential for the present purposes. The essence

is that one can glue a multiplicity of objects into a single object so that a global

"twist" may result, or conversely one can decompose an object such as the Moebius

strip into this type of configuration in order to distinguish it from the untwisted

ones. Leibniz appears to have had the earliest intuitions concerning such objects.

These objects play a major role in contemporary physics, and indeed most field the-

ories currently in use (either in relativity or in quantum physics) may be translated

into these terms. Their significance in twentieth-century mathematics is truly

momentous.

9. As this reference to Hegel indicates, the question of the relationships between

Lacan and philosophy, and the philosophers, beginning with Hegel, is a broad and

complex subject, which is, in its full scope, well beyond the limits of the present

analysis. The reader may be referred, for example, to the collection Lacan avec les

philosophes, ed. N. Avtonomova (Paris: Albin Michel, 1990), and the proceedings of

the UNESCO colloquium under the same title. I would argue, however, that the pre-

sent subject (the architecture of philosophical concepts in Deleuze and Guattari's

sense) is crucial to understanding the relationships between Lacan's work and

philosophy.

10. The discussion to follow is indebted to a number of technical and semitech-

nical accounts, in particular, Elie Cartan, "Nombres complexes. Expose, d'apres

l'article allemand de E. Study (Bonn)," Oeuvres Completes (Paris: Editions du cen-

tre national de la recherche scientifique, 1984), 2:1, 107-247 (which, along with

Study's article itself appears to shape most accounts of the subject by mathemati-

cians); Reinhold Remmert's chapter, "Complex Numbers," in Heinz-Dieter Ebbing-

haus et al., Numbers (New York: Springer-Verlag, 1990), which follows Cartan

264 * Notes to Pages 118-25

closely; and David Reed, Figures of Thought: Mathematics and Mathematical Texts

(London: Routledge, 1995). I am also grateful to Barry Mazur, David Mermin, and

David Reed for productive exchanges on these subjects.

11. The method used in the proof-an argument based on the so-called excluded

middle (still the most common and effective form of mathematical proof)-was orig-

inally used by Parmenides and Zeno and formed one of the foundations of dialectic,

the invention of which is credited to them.

12. Maurice Blanchot, The Writing of the Disaster, trans. Ann Smock (Lincoln:

University of Nebraska Press, 1986), 103; emphasis added. Blanchot probably has in

mind Hippassus, who is often mentioned as the protagonist of the legend.

13. I discuss de Man's allegory in this context (and in connection with Bohr's

epistemology of quantum mechanics) in "Algebra and Allegory: Nonclassical Epis-

temology, Quantum Theory, and the Work of Paul de Man," in Material Events:

Paul de Man and the Afterlife of Theory, ed. Barbara Cohen, Thomas Cohen, J.

Hillis Miller, and Andrzej Warminski (Minneapolis: University of Minnesota Press,

2000), 49-92. For other aspects of the legend and relevant epistemological links,

especially in the context (highly relevant here) of Descartes's work, see Claudia

Brodsky Lacour, Lines of Thought: Discourse, Architectonics, and the Origins of

Modern Philosophy (Durham, NC: Duke University Press, 1996), 55-60.

14. The relevant aspects of Godel's work and related areas of mathematical logic

and the philosophy of mathematics, and their connections to nonclassical thought

(or to the epistemology of quantum mechanics), would require a separate consider-

ation. The standard collection containing key major works is Philosophy of Mathe-

matics, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University

Press, 1991). I have considered some of these questions in Complementarity

(193-224).

15. It is worth observing that the mathematical legitimacy of negative numbers,

too, was a matter of long debate, extending to the eighteenth century.

16. This statement involves further complexities, which have a long history of

their own, extending to Richard Dedekind's and Georg Cantor's work, and then to

the twentieth-century work on the foundations of mathematics, extending to

Godel's investigations and beyond.

17. One can define two other operations in vector spaces, the so-called scalar and

vector multiplication. The first, however, always gives us numbers, not vectors; the

second does give us vectors, but ones no longer belonging to the original vector

space, here the real plane. The vector product of two plane vectors gives us vector

configurations in three-dimensional space.

18. To recapitulate, in the currently standard view the equations of quantum

mechanics involve vectors in certain vector spaces, the infinite-dimensional Hilbert

spaces, over the field of complex numbers and the so-called (probability) amplitudes,

which are complex numbers. In order, however, to establish correlations with mea-

surements in terms of actual probabilities (which define quantum-mechanical pre-

dictions) one needs to use the squared moduli, which are always positive real num-

bers. Quantum mechanics provides definite rules for how to do this, such as Born's

rule. Heisenberg's uncertainty relations can be shown to be direct mathematical con-

Notes to Pages 127-29 * 265

sequences of this scheme. It is also worth recalling that, while it may deal with that

which cannot be visualized, quantum mechanics, too, uses visualization, either in the

more direct sense of this word or in the sense of intuitive pictures or representations,

closer to German Anschaulichkeit. The "pictures" of that type may be either partial

and specifically complementary (mutually exclusive and, hence, un-unifiable within

a single picture or configuration) such as wave- and particle-pictures (keeping in

mind qualifications given in chapter 2), or diagrammatic, such as the so-called Feyn-

man diagrams, primarily used as heuristic tools in quantum field theories.

19. We can do so for the Riemannian geometries, which have positive curva-

tures. There is only one (flat) Euclidean geometry, but there are many non-Euclidean

geometries. Topologically, the situation is still more complex.

20. Let me reiterate that this is an epistemological rather than a mathematical

question, a difference that is sometimes missed.

21. These are much more entangled, more intricately mixed, than is commonly

acknowledged or realized. For relatively rare arguments concerning this complexity,

see Lacour's Lines of Thought (49-67) for a postdeconstructive treatment and

Reed's Figures of Thought (21-48) for a more mathematical-historical-philosophical

treatment. In many respects, Reed's analysis may be seen as postdeconstructive as

well. The confluence of these two titles themselves is of some interest here, and, in

addition, both books deal with the question of textuality in Descartes and beyond.

Both also contain useful further references.

22. I leave aside the questions of geometrical or topological definition and of our

spatial intuition concerning the idea of dimension itself. It was a subject of profound

investigations, first in the wake of Georg Cantor's introduction of the idea of sets

and, then, in early topology, in particular in the work of Henri Poincare and Luitzen

E. J. Brouwer. Brouwer was also one of the creators of mathematical intuitionism, a

program that is relevant to the problematic considered here. Lacan refers to

Brouwer's work on several occasions.

23. Richard Feynman made interesting comments on this point in describing his

own visual intuition in thinking about quantum objects. See Sylvan Schweber, QED

and the Men Who Made It, 465-66. The present argument obviously does not take

into account the relevant research in psychology and physiology (and to some

degree, sociology) of perception and cognition, which may complicate the situation

even as concerns the very constitution (cognitive-psychological or cultural) of two-

dimensional (or for that matter, one-dimensional), spatial representations to which

I refer here as Euclidean. It is not inconceivable that we are capable of constructing,

and even actually, but "imperceptibly," deploy, more complex perceptual or phe-

nomenological configurations, which are, as it were, "repressed" by our Euclidean

machinery of (spatial) perception and cognition, however this machinery came

about. Conversely, we might not even "see"-that is, may not really be capable of

visualizing, intuiting, imagining and so forth-elements of the two-dimensional

plane. Even though there is a considerable accumulation of literature on these sub-

jects, the research concerning them remains at a relatively early stage and is often

inconclusive at best, and thus is difficult to utilize in the present context. It does not

appear to be in conflict with the present argument. Indeed, it is conceivable that this

266 * Notes to Page 129

research can eventually help to support this argument and even radicalize it. We may

well be ultimately less likely to visualize even two-dimensional real objects than to

be able to improve on our intuition concerning the actual geometry of the complex

plane, let alone on our intuition concerning the behavior of quantum objects. I shall

further comment on some of the complexities concerning real numbers and their

geometrical representation presently.

24. I am indebted to Barry Mazur for this particular point and for productive

exchanges on these matters. Naturally, I am not saying that "geometry" or "topol-

ogy" (mathematical or metaphorical) has no role to play in nonclassical thought;

quite the contrary, in mathematics and science or elsewhere it was crucial to various

aspects of nonclassical thought all along, beginning with the pre-Socratics. Nor, con-

versely, am I suggesting that "algebra" cannot be used analogously to geometry,

either classically (say, symbolically or as a direct representation of something) or

nonclassically. Geometrical configurations (such as those derived from classical

physics) and language, both used nonclassically, play significant roles in quantum

physics as well. My point is that nonclassical thought, at the limit, always irreducibly

suspends a geometrical and ultimately any representation of its "object(s)," or,

again, more accurately, what it relates to. This is why I associate nonclassical

thought here with "algebra" rather than "geometry," even though algebra, too, can

serve classical epistemology, as it appears to have done in Leibniz's work. Leibniz is

a crucial figure in this context (of the relationships between algebra, geometry,

analysis, and even topology), and his work would require a separate analysis,

specifically via Deleuze's work on Leibniz and the Baroque in The Fold: Leibniz and

the Baroque, trans. Tom Conley (Minneapolis: University of Minnesota Press,

1993). On the other hand, on the particular point in question at the moment, see

Derrida's discussion of Leibniz in the section "Algebra: Arcanum and Trans-

parence" in Of Grammatology, trans. Gayatri C. Spivak (Baltimore: Johns Hopkins

University Press, 1974), 75-81, and of "the linearity of the symbol" and its invasion

by nonlinear and, hence, nonclassical "writing" in Derrida's sense a bit further in the

book (85-87). It may, however, be argued that, while both algebra and geometry in

mathematics or physics may be used either classically or nonclassically, geometrical

representation qua geometrical representation is always ultimately classical. This has

significant implications for the very understanding of what would count as rigor-

ously "geometrical" representations in classical mathematics and science, from

Kepler and Galileo on, or elsewhere. At the same time, it would be tempting to con-

sider the parallel histories of algebra, geometry, and representation (including alle-

gorical aspects of the latter, ultimately in de Man's nonclassical sense) from their

Medieval roots; to their role in the Renaissance, and the role of algebra (rather than

only geometry) in Galileo; to the invention of calculus in the late Renaissance and

the early Baroque by Newton and Leibniz; to the emergence of modern algebra (in

particular complex numbers and the theory of algebraic equations) in the work of

such figures as Gauss, Niels Henrik Abel, and Evariste Galois, contemporary with

Idealism and Romanticism; to their more or less immediate extensions in nineteenth-

century mathematics; to finally, twentieth-century mathematics and its role in

physics, specifically quantum theory. Several works and analyses more directly rele-

Notes to Page 129 * 267

vant to the present argument may be considered from this perspective as well. They

include Derrida's analysis just mentioned, as well as Deleuze's discussion in Differ-

ence and Repetition, trans. Paul Patton (New York: Columbia University Press,

1993), and Michel Foucault's interesting appeal to algebra or (a related considera-

tion) to "the Leibnizian project of establishing a mathematics of qualitative orders as

situated at the very heart of Classical thought" (in Foucault's sense) in The Order of

Things, 57, 83. I have addressed some of these issues in the context of de Man's

work in "Algebra and Allegory."

This is of course an immense program, which can only be indicated here as a pos-

sibility (perhaps yet another insurmountable possibility). To give a sense of what is

at stake, one can specifically mention investigations, in the wake of Kant, into the

nature of algebra as "the science of pure time." I think, in particular, of the work of

Sir Rowan Hamilton, who was closely aligned with Coleridge and his circle, whose

members not only had strong connections to German Idealism (an obvious point)

but were also specifically interested in questions of the type considered here. Hamil-

ton was especially interested in ideas of symbol and symbolic representation, both in

a more general sense and, I would argue, specifically in Coleridge's sense of the term.

Conversely, it is hardly conceivable that Coleridge's ideas or those of his fellow

Romantics, such as Kleist, were not, reciprocally, influenced by these mathematical

connections. Hamilton eventually, in 1843, became the inventor of the so-called

quaternions. For a useful historical account of this invention (from a different per-

spective bypassing the epistemological questions considered here) see Andrew Pick-

ering, "Concepts and the Mangle of Practice: Constructing Quaternious" (in Smith

and Plotnitsky, eds., Mathematics, Science, and Postclassical Theory, 40-82). This

invention can be linked to the philosophical ideas just mentioned, even though some

of Hamilton's philosophical positions had changed by that time. Quaternions (con-

sidered, analogously to real or complex numbers, as a single mathematical structure,

a kind of "numbers") was the first example of, topologically, a mathematical object

of more than three (in this case, four) dimensions. As such, it in many ways

prefigured and indeed initiated both twentieth-century abstract algebra and such

mathematical objects as Hilbert spaces (which may have an infinite number of

dimensions), used in quantum mechanics. (Hilbert spaces are usually associated with

the so-called functional analysis.) David Hilbert's own extraordinary mathematical

work and philosophical ideas were in turn crucial for twentieth-century mathemat-

ics and physics, including, again, in the present context. At the core of Hamilton's

work on quaternions are questions of the relationships between algebraic and geo-

metrical objects and their mutual representation. At the same time, fundamental

connections to physics and the question of mathematical representation in physics,

on the one hand, and the question of reality in more general philosophical terms, on

the other, were central to Hamilton's thinking on quaternions. The latter themselves

introduced new epistemological dimensions to these questions, extending, once

again, to twentieth-century mathematics and physics (or the mathematization of

physics), specifically to quantum mechanics. Earlier (in 1834), Hamilton also intro-

duced modern or Hamiltonian, as it has been known ever since, formalism of classi-

cal mechanics, by exploring the analogy between mechanics and optics. It was, as we

268 * Notes to Pages 129-31

have seen, this formalism that served as the primary model for the mathematical for-

malism of quantum mechanics. As I said, Hamilton's mechanical and mathematical

ideas were instrumental to the thought of Bohr, Heisenberg, Dirac, and other

founders of quantum mechanics, including, in Bohr's case, specifically in shaping his

nonclassical interpretation of quantum mechanics. As I also argue here, the history

of imaginary and complex numbers, immediately preceding and prompting Hamil-

ton's work on quaternions, already appears to pose some of the epistemological

questions at stake, and Gauss's and then Riemann's work, and their connections to

Kant and German Idealism, may well be even more fertile cases here than that of

Hamilton.

25. See Robert P. Langlands, "Representation Theory," in Proceedings of the

Gibbs Symposium, Yale University, 1989, ed. G. G. Caldi and George D. Mostow

(College Park, MD: American Mathematical Society Publications, 1990), 209.

26. This is yet another question, also having to do with the difference between

curved lines and surfaces in space and the curvature of space itself (with some con-

nections to the epistemology of Einstein's general relativity), which I shall bypass

here. It is of some interest that the complex numbers allow one to construct more

naturally models for non-Euclidean geometries, specifically those of negative curva-

ture (hyperbolic geometries). It can also be added that, according to this view and

contrary to a common claim, Kant's argument concerning the Euclidean nature of

the phenomenal world is not disproved merely by virtue of the discovery of non-

Euclidean geometries. One would need to mount a far more complex argument

against Kant, if one is to refute his view.

27. These considerations obviously relate to the question of what is known as

mathematical Platonism (which postulates, in one way or another, with one degree

of complexity and indirectness or another, the reality of mathematical objects), and

related questions in the philosophy of mathematics. This is an immense subject in its

own right, which cannot be addressed here, beyond the implications of the present

analysis. The latter itself is reasonably conditional in the sense that, similarly to the

case of complementarity but more provisionally, it only suggests a certain interpre-

tation of the situation rather than offers an argument against the inevitable nature of

this interpretation, which is not required for my purposes here. As follows from the

preceding analysis, a nonclassical view, such as the one suggested here, cannot be

realist by definition. It may also be pointed out that the Platonist position, however

defined, is not anymore universally accepted in mathematics than realist is in

physics. I have addressed some of these questions in "Complementarity, Idealiza-

tion, and the Limits of Classical Conceptions of Reality," 161-67.

28. This area has, however, been a subject of intense investigation in the wake of

Penrose's proposal, advanced in his recent works cited earlier. It may be noted that

some of the claims advanced along these lines heavily depend on the arguments for

the nonlocality of quantum theory (either in standard or Bohmian versions), often

missing the conjectural nature of these arguments and the lack of experimental evi-

dence supporting them, as considered earlier and as will be further discussed in chap-

ter 5. I have discussed Penrose's ideas in "Penrose's Triangles."

29. Gerald Edelman's work is perhaps best known and is, predictably, the sub-

Notes to Pages 131-42 * 269

ject of much controversy and counterargument. His Bright Air, Brilliant Fire: On the

Matter of the Mind (New York: Basic Books, 1992) is arguably his best popular

exposition that also contains key references to philosophical and (some) technical lit-

erature. It also takes issue with Penrose's proposal for the quantum nature of con-

sciousness in The Emperor's New Mind, as discussed earlier (216-18). For the most

recent exposition, see Gerald Edelman and Giulio Tononi, The Universe of Con-

sciousness: How Matter Becomes Imagination (New York: Basic Books, 2000), and

for the original (more) technical presentation, see Gerald Edelman, The Remem-

bered Present: A Biological Theory of Consciousness (New York: Basic Books,

1990).

30. Thus, Penrose in the works cited earlier conjectures (and hopes) that such

would be the case.

31. Detlef Laugwitz, Bernhard Riemann: Turning Points in the Conception of

Mathematics, trans. Abe Shenitzer (Boston: Birkhaiuser, 1999), 295.

32. Misha Gromov, "Local and Global in Geometry," preprint, October 29, 1999.

33. Laugwitz, Bernhard Riemann, 299.

34. Laugwitz, Bernhard Riemann, 225-26. Gauss's paper itself, Beitrdige zur

Theorie der algebraischen Gleichungen (Contribution to the theory of algebraic

equations), is found in volume 3 of Gauss's collected works, Werke . . . Her-

ausgegeben von der K. Gesellschaft der Wissenschaften zu Gottingen (Gottin-

gen: Gedruckt in der Deiterichschen Universititsdruckerei [W. F. Kaestner],

1863-).

35. Laugwitz, Bernhard Riemann, 226.

36. Euclid, The Thirteen Books of Euclid's Elements, ed. Thomas L. Heath, 3

vols. (New York: Dover, 1989), 2:278, 316-17.

37. Lacan appears to have been aware of some of these complexities, as is sug-

gested by his comments on the foundations of mathematics, including some of those

cited by Sokal and Bricmont in Fashionable Nonsense (27-36). These passages cause

Sokal and Bricmont much aggravation. In truth, however, they are at worst harm-

less, and often there is nothing especially wrong with them-especially, again, if one

tries to understand Lacan's actual argument where these passages are used. On the

other hand, Lacan and other radical thinkers under criticism in the Science Wars

appear to be more aware of both the philosophical dimensions of the mathematical

concepts in question and of their history and of the philosophical thought of the key

mathematical and scientific figures involved than are their recent critics in the sci-

entific community. In Lacan's works one finds numerous references to the history of

mathematics and, specifically, algebra and the theory of equations, which indicates

this awareness, however one evaluates his deployment of these ideas themselves.

38. See Remmert's commentary in Ebbinghaus et al., Numbers, 62-63.

39. This assessment is cited by Friedrich Waismann in Introduction to Mathe-

matical Thinking, trans. Theodore J. Benac (New York: Frederick Ungar, 1951),

107.

40. Here and later the term "analogon" may also be understood in its Greek

sense, as connoting a parallel or "proportionate" relation, rather than identity, of

one logos (here as "discourse") to another.

270 * Notes to Pages 142-43

41. See Derrida's chapter "Linguistics and Grammatology" in Of Grammatol-

ogy, 27-73, which is one of the defining texts in the history of deconstruction.

42. Two recent commentaries address the relationships between Lacan and

quantum mechanics. The first is Slavoj Zizek's "Lacan with Quantum Physics," in

Futurenatural: Nature, Science, Culture (Futures, New Perspectives for Cultural

Analysis), ed. George Robertson et al. (London: Routledge, 1996). The second is

Henry Krips, "Catachresis, Quantum Mechanics, and the Letter of Lacan,"

Configurations 7, no. 1 (winter 1999): 43-60. Krips follows Zifek's argument but

pursues a different subject, that of the history of the interpretation of quantum

mechanics in terms of a confrontation with the Lacanian Real. Ultimately, the argu-

ments of both articles turn on how one reads Lacan's Real, which is pertinent. For

Lacan's Real is indeed crucial to Lacan's thought and to the relationships between it

and quantum mechanics. On the present view, both of these reading are ultimately

classical as concerns both Lacan and Bohr, and Krips in fact offers more a reading

of Zizek rather than Lacan. Krips's essay contains significant misunderstandings of

Bohr's views, which are not sufficiently elaborated by Krips in any event, as well as

some among the views of Heisenberg (or of the exchange between the latter and

Bohr), or indeed some of the key elements of quantum mechanics. I further think

that the article equally misconceived the transition from the earlier stages of quan-

tum mechanics and Bohr's interpretation to Von Neumann's (Hilbert-space) version

of quantum mechanics. First of all, some key aspects and refinements of Bohr's inter-

pretation, as considered earlier, emerged following this version, a point entirely dis-

regarded by Krips. Most of the Bohr-Einstein debate and arguments concerning the

EPR argument and nonlocality took place after it as well. Indeed, as I have indicated,

all of the key features of Bohr's interpretation can be reformulated in terms of Von

Neumann's Hilbert-space scheme, as Bohr was well aware, as are indeed most

authors who write on the subject. In fact, this reformulation helps Bohr's post-EPR

version of complementarity since Von Neumann's version is epistemologically close

to Heisenberg's scheme. Heisenberg's matrices are the "observables" (defined as

operators in Hilbert spaces) of Von Neumann's scheme. Thus, there is no break here

of the kind for which Krips argues, especially if one considers Bohr's post-EPR ver-

sion of complementarity (the difference by and large ignored by Krips). There seems

to be a degree of confusion in Krips's article between Von Neumann's (different)

mathematical formalism of quantum mechanics and the interpretation of this for-

malism. Krips's claim that complementarity sees quantum mechanics as merely pro-

viding the rule for predicting the outcome of experiments is also mistaken, as this

study would suggest. It does that, of course, but this is hardly sufficient to charac-

terize Bohr's interpretation. Indeed, I was unable to locate the statement "the math-

ematical formalism of quantum mechanics . . . merely offers rules of calculation [as

opposed to meaningful truth claims]," which Krips attributes to Bohr either on page

60 of PWNB 2 (Krips refers to the original edition of which this is a reprint) or else-

where in Bohr. Perhaps the ellipses prevented me from locating it. In reading the arti-

cle one gets an impression that the Bohr quotations come from secondary sources,

since they do not always exactly correspond to the original, and passages from the

same essays, specifically "Discussion with Einstein," are cited from different publi-

Notes to Pages 143-47 * 271

cations (notes 6 and 7, pp. 44, 45). As stated, the above formulation is closer to Ein-

stein's view of the mathematical formalism of quantum mechanics as commented

upon by Bohr there (PWNB 2:61). Phenomena in Bohr's sense are unsustainably

seen by Krips as referring to the attributes of quantum objects, and the role of mea-

suring instruments, decisive for Bohr, is ignored altogether, while it is, as we have

seen, crucial to the question of the relationships between complementarity and post-

Von Neumann's versions of quantum mechanics. In short, most key aspects of

Bohr's argumentation are disregarded. Krips's attempt to explain the persistence of

certain aspects of Bohr's interpretations, despite its decline in theoretical and con-

ceptual terms (alleged by Krips) in, at least as presented in the article, naively psy-

choanalytic terms, is especially unfortunate, and it misses most key aspects of the

history of Bohr's ideas and of their impact. Indeed, while Krips offers this situation

as an example of a confrontation with the Lacanian Real, this argument hardly

amounts to a meaningful deployment of Lacan's ideas. At most, Krips's suggestion

may be treated as a conjecture, rather than an argument, which yet needs to be

made. Thus, if there is little, if any, Bohr in the article, there is hardly much more of

Lacan. Zizek's argument concerning Bohr is rather looser and, in essence, claims far

less in term of the specifics of Bohr's interpretations. It is, again, another question

how close this argument is to Lacan's.

43. See, in particular, Paul de Man, "The Rhetoric of Temporality," Blindness

and Insight: Essays in the Rhetoric of Contemporary Criticism (Minneapolis: Uni-

versity of Minnesota Press, 1983), 208-28.

44. One must keep in mind the difference between complex numbers, or indeed

any mathematical object, and the Lacanian system in question as concerns their

respective relationships with materiality (whether one sees the latter in terms of real-

ity in the classical sense or otherwise). The situation is in some respects analogous,

but not identical, to that of the difference between mathematics and quantum

mechanics, as considered earlier. In the case of the Lacanian system, the relation-

ships between the symbolic and the material are more immediately germane, some-

what similarly (although not identically) to the way mathematical models function

in physics. In the case of mathematics, the symbolic systems that mathematics uses

may be seen as more or less independent of material objects-such as those consid-

ered in physics. Other forms of materiality are, however, found in mathematics as

well, and, as mentioned earlier, there is of course still the question of nonmaterial

mathematical (or for that matter Lacanian) reality.

45. Gottfried Wilhelm Leibniz, Mathematische Schriften, ed. C. I. Gerhard

(Hildesheim: G. Olms, 1962), 5:357. Cited by Remmert in Ebbinghaus et al., Num-

bers, 58.

46. See Cartan, "Nombres complexes," 330n. 3.

47. The statement occurs in "La geometrie," published in 1637; it is cited by

Remmert in Ebbinghaus, et al., Numbers, 58.

48. Cited by Remmert in Ebbinghaus et al., Numbers, 55.

49. Sokal and Bricmont, Impostures intellectuelles, 31; Fashionable Nonsense,

25.

50. To the extent that one can speak of a metaphorical parallel, it operates at the

272 * Notes to Pages 149-51

level of the two systems themselves. This is a classic Lacanian move, and it is often

found elsewhere as well. For example, Poe's "The Purloined Letter" is read by Lacan

as textualizing the scene and indeed the field of psychoanalysis ("Seminar on 'The

Purloined Letter,'" Ecrits [French edition]). It is reread by Derrida as the scene of

writing in Derrida's sense in "Le facteur de la verite" (The Post Card: From Socrates

to Freud and Beyond, trans. Alan Bass [Chicago: University of Chicago Press,

1987]), as part of his deconstruction of Lacan. In the sense just explained, however,

one can also speak of a certain "repetition" of Lacan on Derrida's part, albeit a rep-

etition in the sense of Derrida's differance as the interplay of differences and simi-

larities, distances and proximities, and so forth. Sokal and Bricmont may be seen as,

at best, confusing the systemic metaphor, as just described, with a direct metaphor

applied to a single element of, or, more accurately, extracted from, the Lacanian sys-

tem. That is, they see Lacan's concept of the erectile organ as defined, metaphori-

cally, through the mathematical J -1, while missing or ignoring the Lacanian system

within which his concept is inscribed (as "the square root of J -1," (L) -1, of his

system) and, hence, also the systemic metaphor in question. They even profess a total

lack of familiarity with the Lacanian system. This indeed did not need to be a factor,

or at least as much of a factor, were their view of Lacan's "metaphor" correct, which

it is not.

51. See also the discussion in "Of Structure as an Inmixing," 193-94.

52. It might be possible to speculate as to whether Lacan's "symbols" here-

"-1" and "the square root of -1"-can be read as "operations" upon the elements

of the "space" of the subject (thus giving the set of these operations itself a certain

structure) rather than only these elements.

53. See, however, the qualifications offered earlier. The question of "negativity"

in psychoanalytical or (via Hegel) philosophical terms in Lacan would require a sep-

arate discussion.

54. One can speak of "distancing" here only with caution. Although an

efficacious materiality of the Lacanian Real can be seen as, in a certain sense, more

"remote," it cannot be postulated as existing in any conceivable (specific) form by

itself and in itself, as absolutely anterior, prior to or otherwise independent of

signification. Hence, it cannot be seen as something from which the distance of

signifiers can be unequivocally "measured." One can speak in these terms only pro-

visionally. Nor, as was suggested earlier, can the overall efficacity of the Lacanian

signification be contained by this materiality: this efficacity, along with its effects,

such as signifiers, is fundamentally reciprocal in nature, including as concerns the

relationships between materiality and phenomenality. In this sense, the expression

"the image of the image of the penis," discussed earlier, must be seen as designating

the "site" of the multidirectional and ultimately interminable reciprocal network of

the "material" and the "phenomenal." However, indeed by the same token, other

terms will be necessary in order to follow and to consider this network.

55. Lacan, Ecrits, 319. In order to read this passage one would also need to con-

sider the question of sacrifice in Lacan, via Hegel, in particular in the Phenomenol-

ogy (Hegel's Phenomenology of Spirit, trans. A. V. Miller [Oxford: Oxford Univer-

Notes to Pages 151-58 * 273

sity Press, 1977]), and, then, via Alexandre Kojeve and Georges Bataille, both of

whom appear to be on Lacan's mind here.

56. The difference between the erectile organ and the phallus would be inscribed

accordingly, as indicated earlier.

57. It is significant that Lacan says "equivalent" and not "identical," which,

again, suggests the difference between Lacan's "algebra" and that of the actual

mathematical complex numbers rather than a claim of their identity on Lacan's part.

58. It is worth noting, however, that certain key ideas, including key epistemo-

logical ideas, of the figures just mentioned-in particular, Saussure, Hjelmslef, and

Peirce-were influenced by mathematics and its epistemology.

59. Sokal and Bricmont, Impostures intellectuelles, 32; Fashionable Nonsense,

26. The reader may be spared the rest of Sokal and Bricmont's sentence, equally

ironic in its confirmation of my point here and equally remarkable in its naivete and

blindness.

60. Lacan, Ecrits, 317; emphasis added.

61. Sokal and Bricmont, Impostures intellectuelles, 32; Fashionable Nonsense,

26.

62. Lacan, "Of Structure as an Inmixing," 192.

63. Gilles Deleuze and Felix Guattari, Anti-Oedipus: Capitalism and Schizo-

phrenia, trans. Robert Hurley, Mark Seem, and Helen R. Lane (Minneapolis: Uni-

versity of Minnesota Press, 1983).

Chapter 4

1. Macksey and Donato, Languages of Criticism, 267.

2. Gross and Levitt, Higher Superstition; Sokal, "Transgressing the Bound-

aries," 217-52. There is an earlier reference to Derrida's remark in Ernest Gallo's

"Nature Faking in the Humanities," Skeptical Inquirer 15, no. 4 (1991): 371-75. It

offers a criticism similar to that of Gross and Levitt, who borrow the passage itself

from this source, rather than from the original, with which they do not appear to

have been familiar at the time.

3. The proliferation of commentaries and discussions in scholarly or public

domains, including the popular press in the United States and Europe, on and

around Sokal's hoax has been staggering, even leaving the innumerable exchanges

on the Internet aside. Several Web sites, however, both serve as good bibliographical

sources and contain the texts of articles and exchanges themselves involved. A num-

ber of these sites are listed on Alan Sokal's Web page, <www.physics.nyu.edu/

faculty/sokal> itself a reasonable, if somewhat selective, source. Derrida's comment

has figured nearly uniquely throughout these discussions, even though, more

recently, Derrida has overtly moved into the background and other figures have

become main targets. Of particular interest is the commentary by Steven Weinberg,

"Sokal's Hoax," New York Review of Books, August 8, 1996, 11-15, and Steven

Weinberg et al., "Sokal's Hoax: An Exchange," New York Review of Books, Octo-

274 * Notes to Page 158

ber 3, 1996, 54-56, on which I shall comment later. Subsequently Weinberg pub-

lished an article on Kuhn, "The Revolution That Didn't Happen," New York

Review of Books, October 8, 1998, which is a more general response to the current

stage of the drama, and sometimes comedy, of "the two cultures." The article is con-

cerned primarily with the nature of scientific knowledge rather than with the

"abuse" of mathematics and science by radical philosophers or literary scholars.

Two other collections, inspired by Gross and Levitt's book and Sokal's hoax, and

containing contributions by them, are The Flight from Science and Reason, ed. Paul

R. Gross, Norman Levitt, and Martin W. Lewis (Baltimore: Johns Hopkins Univer-

sity Press, 1997), and A House Built on Sand: Exposing Postmodernist Myths about

Science, ed. Noretta Koertge (Oxford: Oxford University Press, 1998). Most of the

articles contained in these volumes carry the problems of Gross and Levitt's and

Sokal and Bricmont's books (exceptions are few and far between). See also The

Sokal Hoax: The Sham that Shook the Academy, ed. Lingua Franca (Lincoln: Uni-

versity of Nebraska Press, 2000), a collection of commentaries on and around the

Sokal affair, assembled by the editors of Lingua Franca, where Sokal's hoax was

originally disclosed. The characterization "Science Wars" has been objected to from

both sides on several occasions. I shall continue to use it here, even if only for con-

venience's sake. It is, however, not altogether out of place, and Sokal protests too

much on that score. His own commentaries have never managed to transcend a cer-

tain militancy, if not belligerence, even and sometimes especially when reasonable-

ness and moderations are professed, and appealed to, by him.

4. Latour's case has a special significance in this context, both on its own terms

and in the broader context of (constructivist) science studies, to which the debate in

question has shifted more recently. These questions would, however, require a sepa-

rate treatment. It would not so much concern commentaries by Gross and Levitt or

Sokal and Bricmont (or, for the most part, those in the partisan collections, cited ear-

lier). These do not have much interesting to say on these subjects, nor would they

require a significant modification of the main ethical points made here. I have in

mind instead more constructive discussions elsewhere, which also productively

engage, including in commentaries by scientists, with what is indeed problematic in

these areas of the history and philosophy of science. Most of these discussions have

been conducted by prominent mathematicians and scientists, on the one hand (such

as Kurt Gottfried, Michael Harris, Jay Labinger, David Mermin, Stephen Weininger,

and Kenneth Wilson), and, on the other, leading representatives of the constructivist

school in science studies (such as, in addition to Latour, David Bloor, and several

prominent representatives of his school, such as Harry Collins, Andrew Pickering,

and Simon Schaffer) in scientific journals, such as Nature and Physics Today, and in

Social Studies of Science. See also The One Culture?: A Conversation about Science,

ed. Jay A. Labinger and Harry Collins (Chicago: University of Chicago Press, 2001).

I would especially like to mention here David Mermin's commentary on Latour's

case, "What's Wrong with This Reading?" Physics Today (October 1997): 11-13,

and the subsequent exchange, with some predictable reaction from the prominent

"Science Warriors," and his "Review of Fashionable Nonsense" Physics Today,

April 1999: 70-71. I also refer to John Huth's "Latour's Relativity" (in Koertge, ed.,

Notes to Page 158 * 275

The House Built on Sand, 181-92), which, while (mostly pertinently and interest-

ingly) critical of Latour, is one of the few, if not the single, exceptions in Koertge's

collection. Otherwise the latter exemplifies some of the worst aspects of the Science

Wars and its aftermath, which is of course not surprising given the list of contribu-

tors. I would also like to mention Michael Harris's letter, "Science Wars or Wars of

Derision?" Notices of the American Mathematical Society 5 (May 1997): 543-44,

and his unpublished essay on these subjects, "I Know What You Mean." See also

Latour's own recent Pandora's Hope: Essays on the Reality of Science Studies (Cam-

bridge, MA: Harvard University Press, 1999). A number of other prominent histori-

ans and philosophers of science contributed to these exchanges on both or, again, all

sides. Ian Hacking's recent The Social Construction of What? (Cambridge, MA:

Harvard University Press, 2000) discusses the Science Wars and the question of

social constructivism as well. Two volumes published in France are of considerable

interest as well and, while uneven, are of much higher quality than the Science Wars

collections cited earlier: Yves Jeanneret, L'affair Sokal ou la querelle des impostures

(Paris: Presses Universitaires de France, 1998), and Baudouin Jurdant, ed., Impos-

tures scientifiques: Les malentendus de l'affair Sokal (Alliage #35-36, 1998, coedi-

tion with La Decouverte).

5. As I indicated at the outset, these problems have a long history. It is not

inconceivable that mathematicians in Plato's time had objected to some of Plato's

uses of his contemporary mathematics as confused and misapplied. On the basis of

the major available works, historical documents, and scholarly commentaries, it

appears unlikely that some form of the problem of the "two cultures" did not exist

in Plato's time.

6. Obviously we must allow for the differences between these figures and

specific critical points concerning them; hence, I say analogous here. I shall indicate

some of these arguments later.

7. Certain earlier events, such as Derrida's exchange with John Searle, are rele-

vant here. On the exchange with Searle, see Derrida, Limited Inc abc. .. (Evanston,

IL: Northwestern University Press, 1988). See also Derrida's discussion in

Points... : Interviews, 1974-1994, trans. Peggy Kamuf (Stanford, CA: Stanford

University Press, 1995), 399-456. Mathematics and science do bring new dimen-

sions to these debates and contribute to their public reverberations, in part in view

of the particular role science and scientists play in modern society. The latter is a

broader question, which would require a separate analysis. Substantively, one does

not find much new or enlightening in most of the arguments by the Science Wars sci-

entists (to some degree, in contrast to the discussions of the Science Wars them-

selves). In terms of their substance, many of these interventions (certainly, Gross and

Levitt's or Sokal and Bricmont's) are irrelevant. In cultural, ethical, and political

terms the case is, again, different, given the unfortunate implications of these inter-

ventions. There is no contradiction between this contention and the argument, given

in the main text, concerning the inseparability, within the present discussion, of

these and conceptual dimensions of the questions at issue. First, this inseparability is

part of a particular argument offered here, an argument involving both ethical and

conceptual issues (in contrast, say, to the previous chapters of this study, which are

276 * Notes to Page 159

more conceptually oriented). Second, while ethical (or political) and conceptual sub-

jects are linked and, perhaps, ultimately indissociable, they are not the same, and at

certain points their analytical separation is just as necessary as their conjunction. In

particular, it is crucial to stress the intellectual impoverishment of "critiques," such

as those offered by Gross and Levitt or Sokal and Bricmont. In this case we need to

consider both aspects, ethical and intellectual, even though the intellectual contribu-

tion of these authors is negligible, including as concerns their presentation of math-

ematics and science themselves.

8. Jacques Derrida, Edmund Husserl's Origin of Geometry: An Introduction,

trans. John P. Leavey (Stony Brook, NY: Nicolas Hays, 1978). Beyond commen-

taries throughout his works, one can especially mention here the as yet unpublished

seminar "La vie la mort," concerned, in Derrida's words, "with a 'modern' prob-

lematic of biology, genetics, epistemology, or the history of life sciences (reading of

Jacob, Canguilhem, etc.)"'" (Derrida, The Post Card, 259n. 1). A number of com-

mentaries on the relationships between Derrida's work and modern mathematics

and science have been published, including Complementarity by the present author.

It is true that Derrida does not, for the most part, engage mathematics and science in

overt ways, as, for example, Deleuze does. And yet, deeper and more meaningful

connections to mathematics and science are found in Derrida's work. The connec-

tions emerge, for example, by virtue of Derrida's more specific engagements with

mathematics and science such as that leading to Derrida's quasi-Godelian "undecid-

ability," or in his discussions of the common philosophical grounding shared by the

humanist and scientific disciplines. They may also be implicit and unperceived by

Derrida himself, such as, conceivably, certain connections to relativity, quantum

physics, or algebra suggested by Hyppolite in his exchange with Derrida. Such con-

nections are inevitable and irreducible within what may be called the culture of sci-

entific modernity (it may, again, not be possible to think modernity otherwise), from

Kepler and Galileo (perhaps, again, from Plato and Aristotle, or even the pre-Socrat-

ics) on. As I stressed from the outset, philosophy is multiply interlinked with mathe-

matics and science within this culture or these cultures, and these links proceed from

both sides. Whether mathematicians and scientists aim or want to do so or not,

mathematics and science provide concepts, ideas, and ways of thinking to philoso-

phy (and much else). That sometimes includes ideas of a fundamentally philosophi-

cal nature, as, for example, according to Deleuze, those that Abel and Galois intro-

duced by inaugurating a new (non-Kantian) critique of pure reason (Deleuze,

Difference and Repetition, 177-80). Deleuze's mathematical references may require

some finessing. (He should, at this juncture, have also mentioned Lagrange, whose

work was crucial for both Abel and Galois.) His point, however, not only is funda-

mentally correct and well taken but is indeed of crucial importance. Conversely,

from the invention of the proof from contradiction by Parmenides and his school,

philosophy and other nonscientific (if they even are or have ever been strictly non-

scientific) disciplines, or literature, do the same for mathematics and science. These

relationships are and have always been reciprocal, which is, again, not to say that the

disciplinary boundaries should not be respected.

9. See his remarks in Florian Rotzer, Conversations with French Philosophers,

Notes to Page 159 * 277

trans. Gary E. Aylesworth (Atlantic Highlands, NJ: Humanities Press, 1995), 52.

10. See, again, Conversations with French Philosophers, 52, and "As if it were

Possible: 'within such limits,'" Questioning Derrida: With His Replies on Philoso-

phy, ed. Michel Meyer (London: Ashgate, 2000), 113-15.

11. Specifically on the critical and deconstructive epistemological potential of

mathematics, see Derrida's comments in Of Grammatology, 75-81. I have

addressed this question in Complementarity and "Complementarity, Idealization,

and the Limits of Classical Conceptions of Reality," 161-67.

12. As I said, Derrida's, while a more special case, is no exception. Consider, as

a typical example, Sokal's comment on Deleuze and Guattari "holding forth on

chaos theory" in his chapter "What the Social Text Affair Does and Does Not

Prove" (in Koertge, ed., A House Built on Sand, 9-22). I would welcome here an

opportunity to take up Professor Sokal's invitation in his essay to examine whether

Deleuze and Guattari are better or "worse in context than out of context," when

they speak about chaos in What Is Philosophy? which incidentally is an exact quo-

tation, unperceived as such or unacknowledged by Sokal, from Steven Weinberg's

comment on Derrida, on which I shall comment later (Koertge, A House Built on

Sand, 12). Note that, unlike Sokal, I say chaos, not chaos theory, and I have good

reasons to do so. Here is the quotation:

To slow down is to set a limit of chaos to which all speeds are subject, so that they

form a variable determined as abscissa, at the same time as the limit forms a uni-

versal constant that cannot be gone beyond (for example, a maximal degree of

contraction). The first functives are therefore the limit and the variable, and ref-

erence is a relationship between values of the variable or, more profoundly, the

relationship of the variable, as abscissa of speeds, with the limit. (What Is Philos-

ophy? 118-19)

This may be a difficult passage to understand without reading the two-page elab-

oration of which it is the conclusion and without perhaps spending some time with

the book as a whole, and it may appear as nonsense to those who, having not done

so, do not understand it and in part because they do not understand it. Formidable

as this passage may be, however, even a cursory examination of the paragraph from

which this quotation is extracted by Sokal would show that he is in error in think-

ing this passage to be on chaos theory. The paragraph, including the part cited by

him, and much of the book is indeed about chaos, but it is decidedly not about chaos

theory. It is about a particular idea of chaos and how it functions differently in phi-

losophy and science-science in general, not chaos theory-and how this difference

helps us to understand the difference between philosophy and science in general. The

first sentence of that paragraph says: "The primary difference between science [sci-

ence in general] and philosophy [philosophy in general] is their respective [different]

attitudes toward chaos" (117). The next sentence expressly defines chaos in terms

that have self-evidently little, if anything, to do with "chaos theory." It defines chaos

in terms of the speed of emergence and disappearance of configurations, shapes,

forms, and so forth, which "spring up only to disappear immediately.... Chaos is

an infinite speed of birth and disappearance [of forms]" (What Is Philosophy? 119).

278 * Notes to Page 160

Obviously, the passage would make no sense if one does not explain the key

terms used in it, such as chaos or "functives." The latter are defined in the preceding

page quite simply (nothing obscure or "postmodern" here) as the elements of which

functions are composed-"variables," "values," "limits," and so forth (117). And

the passage makes no sense as a comment on chaos theory because it is not a com-

ment on chaos theory. It would take a considerable amount of space (which I do not

have here) to explain Deleuze and Guattari's ideas and the elaboration itself, which,

as all good philosophy, requires fast thinking but slow reading. My point is that the

passage cited by Sokal is unquestionably not about "chaos theory," to which the

book briefly refers much later in a different context (206). However undeliberately,

Sokal's reading not only takes the quotation out of context, but it deprives it of its

content. There is nothing in the quotation itself that suggests chaos theory, except

the word "chaos," used once!

The point seems so obvious that it is difficult for me to conceive how Sokal could

make an error here. Naturally, I would not presume that he himself has as hazy a

notion of what "chaos theory" actually is as, regrettably, do many humanists who

have commented on it. As I have indicated, chaos theory is, for the most part, a the-

ory of nonlinear dynamically evolving systems, the behavior of which is marked by

extremely sensitive dependence on their precise initial state. As a result, although

most such systems are in fact causal, they can, in practice, appear random-

chaotic-because deterministic predictions of their behavior are not possible. In this

sense, catchy as it is, the term "chaos theory" is somewhat of a misnomer. It might

have been better called "order theory" or "complex-order theory," and it is indeed

close to the so-called complexity theory, another fashionable and much misused item

in current academic discussions in the humanities. Deleuze and Guattari, by con-

trast, are interested precisely in chaos and offer a specific new concept of it. There is,

of course, no single concept of chaos, assuming that we can conceive of it, to begin

with.

It would only take a little more argument to show similar problems throughout

the chapter on Deleuze in Fashionable Nonsense and throughout the book on most

other authors, and many of these problems were indeed made apparent by a number

of commentators and reviewers since the book appeared. Finally, I ought to mention

that the argument just offered was presented at the conference entitled "Debate on

Science, Science Studies, and Their Critics," at the University of California, Santa

Cruz, in May 1997, where the present author was paired with Sokal, who presented

the identical version of his comments just discussed. This, however, obviously did

not make him change these comments in the subsequent publication of his article.

13. On these issues in a more general context, see Derrida's analysis in "Limited

Inc a b c . . ." and, especially, "Afterword: Toward an Ethic of Discussion," both in

Limited Inc. A number of reviews of Higher Superstition and Fashionable Nonsense,

such as, and in particular, by Richard Dawkins ("Postmodernism Disrobed," Nature

394 [1998]: 141-43) and Thomas Nagel ("The Sleep of Reason: Review of Alan

Sokal and Jean Bricmont, Fashionable Nonsense," The New Republic, October 12,

1998), contributed to and exacerbated the situation by uncritically accepting Sokal

and Bricmont's commentaries on the quotations they used as definitive, without

Notes to Pages 160-62 * 279

checking the original texts with any care (if at all) and without stopping to think

even minimally about the content of the quotations. These quotations are often

clearly mistreated-"abused"-by Sokal and Bricmont. Nagel, whose unfamiliarity

with the texts is pronounced, goes so far as to call Sokal and Bricmont's commen-

taries "patient explanations" ("why [the postmodernist work in question] is gibber-

ish"). These commentaries are clearly nothing of the kind, regardless of whether or

not some of these postmodernist works are gibberish. Patience is what these authors

and these reviewers lack above all. Such reviews, just as the books under review,

depend on and perpetuate the ignorance of their readers and can hardly be of much

service to the latter or contribute to any productive debates, regardless of the prob-

lems of some among the postmodernist figures under discussion. To encounter this

from prominent scientists, philosophers, and scholars is sad but, unfortunately, not

surprising. This is not altogether new either, and earlier examples are not hard to

come by. There have been more balanced assessments of these books or, as I said, of

the situation under discussion in general by a number of mathematicians and scien-

tists, without necessarily defending the "targets" of these books, which makes these

assessments all the more valuable in this context. In general, as I said, in question

here is not a defense of these targets as such, but a defense of the ethically and intel-

lectually proper conduct of the discussion, as well as an argument for more substan-

tive arguments concerning the key conceptual issues involved. As I have stressed

throughout, meaningful critique should not only be welcomed but is essential. It is

essential even if (by virtue of not being always sufficiently familiar with modern

mathematics and science) the "postmodernist" works miss some of the philosophi-

cal content and certain implications of mathematics and science. Such a critique

should of course also be responded to whenever necessary. But-this is the point

here-a very different form of exchange would then emerge. A characteristic exam-

ple is Margaret C. Jacob's commentary on Latour's postmodernism in "Reflections

on Bruno Latour's Versions of the Seventeenth Century" (in Koertge, ed., House

Built on Sand, 240-54). Jacob not only misses that, in Latour's We Have Never Been

Modern, trans. Catherine Porter (Cambridge, MA: Harvard University Press, 1993),

which she considers (she gives the wrong publication data [252n. 2]), Latour

expressly dissociates his framework from "postmodernism," as indeed his title indi-

cates, but she also misses that he criticizes Steven Shapin and Simon Schaffer's

Leviathan and the Air Pump: Hobbes, Boyle and the Experimental Life (Princeton,

NJ: Princeton University Press, 1985). This can hardly give one much confidence in

Jacob's criticism of Latour.

14. On Derrida's concept of dissemination, see Dissemination, trans. Barbara

Johnson (Chicago: University of Chicago Press, 1981).

15. See also his response to the articles, "As if it were Possible, 'within such lim-

its,"'" in Questioning Derrida: With His Replies on Philosophy 96-119.

16. These relationships may be, and often are, seen in terms of metaphor rather

than, as here, in terms of conceptual (in Deleuze and Guattari's sense) junctures.

Some among recent treatments along these lines tend to suppress the complexity

involved in the nature of metaphorization and the very concept or "definition"

(assuming that it is possible) of metaphor. At times, a reference to metaphor func-

280 * Notes to Pages 163-66

tions as an illusion of explanation: as if it were enough to explain either the specific

usage of metaphors or the nature of metaphorization, to say that, for example,

Deleuze uses a metaphor of Riemannian spaces or Derrida a metaphor of undecid-

ability. In fact, in both cases at stake is a specific concept that is quite different from

its mathematical counterpart, while transferring some (philosophical) aspects of the

latter. On these subjects, the reader may be referred to well-known works by Der-

rida (such as "White Mythology," Margins of Philosophy) and de Man ("The Epis-

temology of Metaphor," Aesthetic Ideology [Minneapolis: University of Minnesota

Press, 1997]), and related discussions elsewhere in their writings, or of course to

Nietzsche's great pioneering work, "On Truth and Lying in an Extra-Moral Sense."

Related commentaries are found in Rodolphe Gasche in The Tain of the Mirror

(Cambridge, MA: Harvard University Press, 1986) and The Wild Card of Reading:

On Paul de Man (Cambridge, MA: Harvard University Press, 1998) and by the pres-

ent author in Complementarity and earlier in this study.

17. I am, again, not saying that there have not been different discussions, such as

those I mentioned earlier, which ask legitimate and important questions concerning

the usage and treatment of mathematics and science in the humanities and the social

sciences or which offer discriminating criticisms of the latter.

18. Accordingly, all that I claim with any definitiveness is the existence of such

philosophical relationships between Derrida's ideas and relativity-and that the pos-

sibility of these relationships is reflected in the Hyppolite-Derrida exchange. The

nature of my argument was entirely missed both by Sokal and Bricmont (Fashion-

able Nonsense, 263n.113) and by Gross and Levitt (Higher Superstition [2d ed.],

293) in their comments on the earlier version of the present article, published in

Postmodern Culture 7, no. 2 (1997) (published electronically). These comments are

themselves reflective, sometimes comically, of the Science Wars situation, but I shall

put them aside here.

19. Derrida, Points ..., 414.

20. This subject requires a separate analysis. I have considered it in Complemen-

tarity, where, however, I have concentrated on Derrida's practice and the philo-

sophical (conceptual and metaphorical) aspects of undecidability rather than on

mathematical logic. It may be suggested, following Derrida's remark cited earlier,

that one encounters here only a loosely metaphorical analogy rather than a rigorous

or comprehensively conceptual proximity. That is, only certain aspects of Gbdel's

undecidability, rather than a broader framework emerging from Gbdel's incom-

pleteness theorems, are used by Derrida. Many key features of the Gbdelian

matrix-such as incompleteness, consistency, the relationships between systems and

metasystems, and so forth-are suspended in the process. By contrast, Complemen-

tarity argues for a more comprehensive philosophical parallel between Bohr's com-

plementarity as a philosophical framework and the overall framework of radical

epistemology, which is in fact closer (although not identical) to Derrida's. Accord-

ingly that study and a companion work, "Complementarity, Idealization, and the

Limits of the Philosophical Conceptions of Reality," explore the connections

between Derrida's and quantum-mechanical epistemology

significance.

and stress their

Notes to Pages 167-69 * 281

21. Gross and Levitt neither offer any support nor retract any of these accusa-

tions in responding to the criticism of their work, including in their response to the

original version of the present essay (Higher Superstition, 293).

22. Derrida, Points ..., 404.

23. All they manage to come up with in reply to this criticism in the second edi-

tion is a quotation on topology not by Derrida but by John Law, "an ally of Bruno

Latour" (which is not altogether accurate either). Law's comment is, it is true, not

altogether coherent but is basically benign. This quotation, moreover, is taken from

an unpublished preprint, marked as a "draft" (Higher Superstition, 293-94). Had it

taken place, an "abuse" or misrepresentation of differential topology would, of

course, be unfortunate. It is an extraordinary discipline, a grand achievement of the

human mind. I studied differential topology at the University of Leningrad with two

extraordinary mathematicians, Vladimir Rokhlin and Misha Gromov. Mathemati-

cians would know these names and those of other figures just mentioned, and it is a

pity that nonmathematicians do not know them-a subject that would require a sep-

arate consideration. The contribution of the French mathematicians to the founding

and development of this field was extraordinary, from such founding figures as

Henri Poincare to the extraordinary contributions, throughout the first half of this

century, of such figures as Elie Cartan, Jean Leray, Henri Cartan, Jean-Pierre Serre,

Rene Thom, and many others, and then by their younger followers up to the present.

I mention the French names (mathematicians from other countries also made major

contributions) because key developments to which they contributed took place when

Hyppolite, a key figure for my discussion, was first a student at the Ecole Normale

and then a professor at the Sorbonne, the Ecole Normale, and the College de France,

where many of these figures were Hyppolite's fellow students and then colleagues.

Derrida was a student at the Ecole Normale (where he studied with Hyppolite)

around the time of major breakthroughs in the field, which were widely discussed in

the intellectual community to which he, Hyppolite, and other philosophical figures

mentioned here, such as Michel Serres, belonged. This community also included

major historians and philosophers of science. Some of these figures (it is a well-doc-

umented fact) were also interested in contemporary philosophy and other disci-

plines, including Althusser's and Lacan's work, and discussed mathematics and sci-

ence with the latter figures as well. Michel Serres, for example, was not only an avid

reader of Jacob's and Monod's work (both Nobel Prize laureates in biology), but he

knew Monod well and discussed science extensively with him. This is leaving aside

that Serres was educated in mathematics, science, and (naval) engineering and that

at some point of his life taught mathematical logic at the Sorbonne. This is immea-

surably more than the likes of Gross and Levitt or Sokal and Bricmont can claim as

credentials in any field in the humanities, let alone what they refer to as "postmod-

ernism," if not in mathematical logic. The irresponsible attitude on Derrida's part

imagined or fantasized (with no basis whatsoever) by Gross and Levitt is inconceiv-

able for anyone even remotely familiar with the intellectual environment just indi-

cated and with Hyppolite's and Derrida's work and attitudes.

24. Derrida, Acts of Literature, ed. Derek Attridge (New York: Routledge, 1992),

208-9.

282 * Notes to Pages 171-78

25. See most specifically his "Differance" (in Margins of Philosophy, 1-28) as

well as related earlier works, Dissemination, Of Grammatology, Positions, trans.

Alan Bass (Chicago: University of Chicago Press, 1980), and Writing and Difference,

trans. Alan Bass (Chicago: University of Chicago Press, 1978).

26. Jacques Derrida, "Khora," in On the Name, ed. Thomas Dutoit (Stanford,

CA: Stanford University Press, 1995).

27. Accordingly, the fact that some of the passages they cite are at best "junk"

thought is irrelevant. One could argue (and Sokal and Bricmont did at some point)

that one would need to argue against (or for) what they have to say, author by

author, point by point. I do not think, for the reasons just explained, that they, or

Gross and Levitt, have earned the right to expect or demand this, even if some of

their targets are worthy of criticism or even laughter (some are). Nothing in their

subsequent interventions, responses to criticism, and so forth appears to contradict

this assessment, and in fact some of the collections cited here are diminished by the

inclusion of their commentaries, however commendable the intent of so doing may

be. What was already cited and referred to here is, I think, quite enough to prove my

point. I would, accordingly, stand by my assessments of these authors. The reader is,

instead, invited to read this criticism as an appeal to a more serious engagement,

which may, again, be critical with the work of Derrida and other nonclassical

thinkers under attack.

28. An extensive survey of such problems has been published by Roger Hart in

"The Flight from Reason: Higher Superstition and the Refutation of Science Stud-

ies," in Science Wars, ed. Andrew Ross (Durham, NC: Duke University Press, 1996),

259-92.

29. This attitude is found, for example, in Hart's article, just cited, which is

sometimes also problematic in its attempt to defend the authors, whose work on sci-

ence is, in my view, indefensible. Hart is of course right to argue that many of these

authors were mistreated by Gross and Levitt.

30. Both the essay and the discussion are in Languages of Criticism. The essay,

as I said, is also included in Derrida's Writing and Difference, 278-94.

31. This point of the necessity of understanding both terms is clearly brought

into the foreground by Steven Weinberg in "Steven Weinberg Replies" (New York

Review of Books, October 3, 1996, 56). Weinberg there also qualifies his original

remarks on Derrida somewhat (without changing his view) and comments on the

context of Derrida's statement in response to the letters published in "Sokal's Hoax:

An Exchange." As will be seen, these qualifications are hardly sufficient to change

my argument here. I also leave aside for the moment the problem of translation, even

though it is significant. Thus, the translation of Derrida's essay published in the con-

ference volume has several problems, and one is better off reading the version pub-

lished in Writing and Difference. In particular, the version in the conference volume

translates Derrida's jeu as "freeplay"-which may lead to a misunderstanding of

Derrida's idea of play. Translation is a crucial concern in considering the circum-

stantial context of the statements at issue. This context may make any claim con-

cerning these statements, including any claim to be offered here, irreducibly tenta-

tive. On the circumstances themselves, see Languages of Criticism, xi-xiii.

Notes to Pages 178-84 * 283

32. The very disagreement between Sokal's and Gross and Levitt's interpreta-

tions suggests that a more careful reading may be necessary. Of course, Sokal's arti-

cle, being a hoax, cannot be considered as offering a meaningful interpretation of

anything, and it can be shown that it misrepresents (deliberately or not) virtually all

the significant ideas that it invokes, certainly Derrida's. Sokal's interpretation of

Derrida's remark makes no sense whatsoever given Hyppolite's question and Der-

rida's essay. It is strange that several scientists appear to have accepted this interpre-

tation on the basis of an admitted hoax, especially since, as Weinberg points out, this

is not a standard term in physics. This makes Weinberg (in this case more under-

standably) puzzled about the phrase and makes him suggest the meaning of the

phrase as, again, referring to a numerical constant, that of Newton's constant figur-

ing in Einstein's theory ("Steven Weinberg Replies," 56). He does not appear to

attribute this meaning to Derrida, which indeed would not make any sense. Yet

another reading proposed by some scientists, that of the so-called cosmological con-

stant appearing in certain versions of relativistic cosmology, makes even less sense,

historically or conceptually. Such a constant was indeed introduced by Einstein in

his early cosmological investigations but was quickly abandoned by him. (He even

spoke of its introduction as the greatest scientific mistake of his life.) It was resur-

rected by recent cosmological theories and has had considerable prominence in

recent discussions of these theories. It would, however, be very unlikely for it to be

invoked at the time of the Hyppolite-Derrida exchange in 1966. Nor does it appear

to make much sense as Hyppolite's reference, given what he says here or given Der-

rida's discussion in "Structure, Sign, and Play" itself.

33. Derrida, Writing and Difference, 278-79.

34. Weinberg, "Steven Weinberg Replies," 56.

35. Weinberg, "Sokal's Hoax," 14.

36. Macksey and Donato, Languages of Criticism, xi-xiii.

37. Ibid., 266.

38. I am grateful to Joshua Socolar for his suggestions in clarifying this particu-

lar point. It must be kept in mind that, in Bohr's formulation, "the space-time coor-

dination of different observers never implies reversal of what may be termed the

causal sequence of events" (PWNB 3:2). As was indicated earlier, in contrast to

quantum physics, Einstein's relativity remains a causal and otherwise classical phys-

ical theory, at least special relativity does, since these questions-causality, reality,

and so forth-become more complex in the case of Einstein's general relativity, his

theory of gravitation. This point is intimated by Hyppolite in his remarks, when he

invokes a more radical dislocation of classical thinking emerging in modern science

(Languages of Criticism, 266).

39. According to Einstein's original paper on relativity, "Zur Elektrodynamik

bewegter Korper" (On the electrodynamics of moving bodies), one may firmly con-

jecture the following on the basis of the available experimental evidence: "[T]he

same laws of electrodynamics and optics will be valid for all frames of reference for

which the equations of mechanics hold good. We will raise this conjecture (the pur-

port of which will thereafter be called the 'Principle of Relativity') to the status of a

postulate, and also introduce another postulate, which is only apparently irreconcil-

284 * Notes to Pages 185-88

able with the former, namely that light is always propagated in empty space with a

definite speed c which is independent of the state of motion of the emitting body"

(Einstein: A Centenary Volume, ed. E. P. French [Cambridge, MA: Harvard Univer-

sity Press, 1979], 281-82). Einstein's "reconciliation" of these two apparently irrec-

oncilable postulates within the framework of special relativity was his great achieve-

ment.

40. See the exchange between Richard Crew and the present author, "An

Exchange: Richard Crew and Arkady Plotnitsky," Postmodern Culture 8, no. 2

(January 1998) (published electronically). I am grateful to Richard Crew for his

comments, which, while critical of Derrida and the present author alike, are in sharp

contrast to the standard Science Wars criticism (and some defenses) of Derrida. Sub-

stantively my argument is adjusted minimally in the present version. It takes Crew's

comments into account in clarifying and amplifying several key points.

41. There are in turn complex reasons, physical and philosophical, that com-

pelled Newton to introduce his notions of absolute space and absolute time.

42. For a comparison on this point, see Mermin's commentary on Latour,

"What's Wrong with This Reading?"

43. These questions, as well as Derrida's early work, have, again, significant con-

nections to Latour's work.

44. A (perhaps) more technically phenomenological example of this situation is

Emmanuel Levinas's phenomenology of ethical otherness, of the relationships to

what he calls the Other (Autrui), to which I shall return, via Blanchot (Levinas has

been crucial for Derrida as well), in the conclusion and which has been the subject

of many recent discussions. In a rough outline, following primarily his 1961 Total-

ity and Infinity, trans. Alphonso Lingis (Pittsburgh, PA: Duquesne University Press,

1990), arguably his most influential work, Levinas's phenomenology and epistemol-

ogy of the radical alterity of the Other may be viewed as follows. This radical alter-

ity, the emergence of the (effect) of the Other in, or even on the horizon of, the phe-

nomenal, of the radar of our thought, radically restructures the latter by the power

of its effects, especially in the ethical context. Indeed one may see the effects in ques-

tion as the opening of ethics for Levinas. The ultimate locus of the Levinasian other-

ness itself (if such or, again, any terms may ultimately apply) may remain indetermi-

nate rather than, for example, merely identified, as is commonly (mis)understood,

with others (other individuals, cultures, and so forth), even though the role of the lat-

ter may be decisive in the emergence of these effects. One may speak of a certain

materiality in this respect, by analogy with quantum mechanics, which, for example,

originally made physicists rethink the nature of light along these lines. To some

degree this is true even as concerns (more classical) relativity, which alienated light

from the classical theory, or indeed from the (classical thought of the) Enlighten-

ment. Indeed this "alienation" of light in physics was happening just around the time

when Levinas, for whom light is a crucial figure (including in de Man's allegorical

sense), began to develop his ideas, originally through his encounters with Husserl's

phenomenology and Heidegger's work. See especially the section "The Violence of

Light" of Derrida's early deconstructive reading of Levinas in "Violence and Meta-

physics: An Essay on the Thought of Emmanuel Levinas," in Writing and Differ-

Notes to Page 188 * 285

ence, 84-92. However, ultimately it is the corresponding restructuring of phenome-

nological effects that is more crucial, while the nature of the efficacity, even if ulti-

mately material (which is not always clear in this context), is best understood along

(or closer to) the lines of the nonclassical epistemology of effects as considered here.

Levinas's infinity entails apparently (and from the classical standpoint inevitably)

paradoxical imperatives. Most crucial of them is that of placing the radical, irre-

ducible alterity of the Other (Autrui) (that which is irreducibly unavailable to sub-

jectivity) in the position of something that is irreducibly linked to and indeed is the

efficacity of subjectivity rather than something that is absolutely excluded from it, is

an absolute other of it. (Indeed, subjectivity itself is (re)defined by Levinas accord-

ingly.) In short, the situation is epistemologically equivalent to Bohr's atomicity of

(techno) phenomena, albeit, in Levinas, under the opposing name of infinity (vs.

quantum finitude). Accordingly, it requires nonclassical epistemology and shares

with Bohr's scheme other features of "atomicity"-individuality and/as singularity,

indivisibility of phenomena involved, and irreducible discontinuity with respect to

the efficacity of effects in question. This argument opens or enters an immense field

of questioning. Such questions would, for example, concern how nonclassical Lev-

inas's thought ultimately is, that is, whether his particular concept of radical alterity

(a term used by others, specifically Derrida, via Levinas but in more determinately

nonclassical arguments) in fact reinstates the instance of classical thought, similarly

to the way early Heidegger does, according to Derrida (Of Grammatology, 19-20).

(The answer, or rather an interpretation, may also depend on particular works by

Levinas, especially those following Totality and Infinity, where he specifically

responds to nonclassical theories and encounters with his works, such as Derrida's.)

Another central, and to some degree correlative, question would be how strictly phe-

nomenological, as opposed to more linguistic or/as technological, Levinas's scheme

just outlined is or can be, that is, whether it is deconstructible into an irreducibly

techno-textual play of forces. The question of visualization (central for Levinas) is

also crucial in this context. Finally, the connections between Levinas's ideas and new

science, such as relativity and quantum physics, while, at some level, inescapable,

require a careful analysis. These questions cannot be addressed here. Nor can I con-

sider here the, by now, massive literature on Levinas. Among more recent works,

Derrida's commentaries, especially his Adieu to Emmanuel Levinas, trans. Michael

Naas and Pascale-Anne Brault (Stanford, CA: Stanford University Press, 1999),

remains, arguably, the most cogent and effective, at least in the present context. My

point here is the efficacious dynamics of Levinas's phenomenology of radical alterity

and its nonclassical aspects and potential. From this point of view, it may well be

that it is as a form of phenomenology that Levinas's work is especially interesting,

even though and because it may also be the (self-)deconstruction of phenomenology,

just as it happens (more overtly) in Heidegger, even if not in Husserl, that Levinas's

investigation ultimately arrives at, even if against himself, at least as he saw his pro-

ject initially or even by the time of Totality and Infinity. (It is more pronounced in

his later works.) On the other hand, Derrida's work offers us cases of more deliber-

ately techno-textual nonclassical theory, and, as I argue here, in all rigor, no theory

can be nonclassical otherwise.

286 * Notes to Pages 189-202

45. Gilles Deleuze, Cinema 2: The Time-Image, trans. Hugh Tomlinson and

Robert Galeta (Minneapolis: University of Minnesota Press, 1995), 129.

46. Macksey and Donato, Languages of Criticism, 266.

47. This type of dialectical synthesis is sometimes (and not always accurately)

associated with Hegel, a subject that would, however, require a separate analysis.

48. Michel Serres and Bruno Latour, Conversations on Science, Culture, and

Time, trans. Roxanne Lapidus (Ann Arbor: University of Michigan Press, 1995), 35.

49. Claude Levi-Strauss, The Elementary Structures of Kinship, trans. James H.

Bell, John R. von Sturmer, and Rodney Needham (Boston: Beacon Press, 1969),

221-27. I am grateful to David Reed for reminding me about this fact and, again, for

most helpful discussions of several subjects considered here.

50. Here I use this, by now complicated, term "deconstruction" in the sense of

the analytical practice of Derrida's own (mostly earlier) work, such as "Structure,

Sign, and Play." This is not the place to consider the "continuities" and "disconti-

nuities" in Derrida's work over the last thirty years nor the differences in the ways

this work is received on different sides of the Atlantic. These factors are relevant to

recent debates, but they would not affect my argument.

51. Differences of that type-those between the centering of a given theoretical

framework, say, around a given concept, and the centering of the structure(s) con-

structed or investigated within this framework-appear to be on Hyppolite's mind

throughout his remarks, beginning with his invocation of algebra.

52. These questions of course also open the general problematic, introduced ear-

lier in this study, of how we interpret classical physics, which would require a sepa-

rate treatment.

53. See especially paragraphs 5-11 of Crew's response.

54. Derrida, Writing and Difference, 278.

55. On these points I permit myself to refer to my book In the Shadow of Hegel,

84-95, 380-88.

56. Derrida, Writing and Difference, 278.

57. In any event, they seem to me more likely than those that Crew suggests (par.

10).

58. Derrida, Writing and Difference, 292.

59. Derrida, Of Grammatology, 50.

60. Cf., again, Derrida's comments in "Sokal et Bricmont ne sont pas serieux."

Chapter 5

1. These differences are one of the main subjects of Deleuze and Guattari's

What Is Philosophy? considered primarily from the perspective of their particular

understanding of philosophy as the creation of concepts. This perspective, however,

is helpful, at least as a good starting point.

2. The significance of such features is part of Jean-Francois Lyotard's (often

misunderstood) argument in The Postmodern Condition: Report of Knowledge,

trans. Geoffrey Bennington and Brian Massumi (Minneapolis: University of Min-

Notes to Pages 202-9 * 287

nesota Press, 1984). He argues that, if one wants to follow mathematics and science

or what they tell us about nature and mind, in the way, say, the Enlightenment fol-

lows classical mathematics and physics, one might at least ask and examine carefully

what modern mathematics and science-relativity, quantum physics, chaos theory,

modern biology, post-Godelian mathematical logic, and so forth-tell us. Naturally,

what they are "telling us" itself becomes a matter of interpretation and debate, such

as those involving the "two cultures" or those of the Science Wars. The stakes are

obviously immense, given the irreducibly mathematical-scientific nature of moder-

nity and postmodernity, or by now perhaps post-postmodernity, hence in shaping

our conditions of knowledge.

3. Werner Heisenberg, interviewed by Thomas Kuhn, November 30, 1962,

Niels Bohr Archive, cited by Pais in Niels Bohr's Times, 263.

4. I am thinking in particular of Lagrange's, Abel's, and Galois's philosophical

revolution, stressed by Deleuze, and Bohr's complementarity (a massive new con-

ceptual architecture). One can, however, also think of Parmenides's invention, men-

tioned earlier, of the proof from contradiction as a great reciprocal example of both.

5. Sokal and Bricmont, Fashionable Nonsense, 143.

6. Weinberg, "Sokal's Hoax," 12.

7. Weinberg, "Sokal's Hoax," 13.

8. On the other hand, Weinberg's strong claim, "And both the philosopher

Michael [sic!-Michel] Serres (a member of the Academie Frangais) and archpost-

modernist Jean-Francois Lyotard grossly misrepresent the view of time in modern

physics" (12), does not appear to be based on familiarity with their usage of the term

"time." In any event this claim is not supported by textual evidence. Indeed, it would

be naive to think that there is a simple or single view of time in modern physics itself,

again not specified by Weinberg.

9. Gross and Levitt, Higher Superstition, 261-62n. 9.

10. If one judges by their more recent publications, however, such as Norman

Levitt's recent book Prometheus Bedeviled: Science and the Contradictions of Con-

temporary Culture (New Brunswick, NJ: Rutgers University Press,1999), that goal is

hardly in sight.

11. James T. Cushing, Quantum Mechanics, Historical Contingency, and the

Copenhagen Hegemony, 168. As I indicated in chapter 2, the treatment of Bohr's

work and text in this book is extremely fragmentary and problematic.

12. Still more recent work in these areas, including by DGZ, does not change

anything in the present argument, either in substance or, especially, as concerns

Gross and Levitt's treatment of the subject (and the latter treatment does not take

these subsequent developments into account in any event). For more recent refer-

ences, see Sheldon Goldstein, "Quantum Theory without Observers" (parts 1 and

2), Physics Today, March 1998, 42-46, and Physics Today, April 1998, 38-42, and

for the subsequent exchange see Physics Today, February 1999, on which I shall

comment later.

13. As I have pointed out, some argue that Schrodinger's equation is determinis-

tic. For the reasons explained already, such a view does not appear to me sustain-

able, at least not without much qualification, usually not provided. It can, perhaps,

288 * Notes to Pages 210-12

be argued that Schrbdinger's equation itself is mathematically deterministic insofar

as it unambiguously determines the (mathematical) value of v-function at any given

point, once the initial conditions are given. This is important and rightly stressed, for

example, by Weinberg. It is, however, a very different question as to what, if any-

thing, in nature it is deterministic about. It may well be deterministic only about

indeterminism or acausality, insofar as it gauges the distribution of the probabilities

in registered quantum-mechanical effects, which, we recall, manifest themselves only

at the macrolevel of measuring instruments. One cannot, however, definitively infer

from this fact that the behavior of quantum objects themselves is causal, in all rigor,

even if one assumes that the mathematical formalism of quantum mechanics, such as

Schrbdinger's equation, describes this behavior. Either one of these two assumptions

is a metaphysical assumptions and not an inevitable logical inference on the basis of

the available data of quantum mechanics. Probabilities can be gauged in a reason-

ably predictable manner, for example, by using Schrbdinger's equation. The out-

comes of the experiment are never certain but are constrained by uncertainty rela-

tions that are inherent in Schrbdinger's equation. Obviously, one must keep in mind

qualifications, given earlier, concerning models, interpretations, and so forth; but

these qualifications would not change the present argument, which only concerns the

fact that the claim of the deterministic character of Schrbdinger's equation requires

significant qualifications not given by the scientists considered here.

14. I have discussed these early theories of Bohm in Complementarity, 168-76.

15. This last point may also be made about Jean Bricmont's commentaries on

Bohr, cited by Sokal and Bricmont on several occasions as an example of a rigorous

critique of Bohr's interpretation, which, unfortunately, they do not appear to be. It

may, again, be noted that it appears that, among just about all of the adherents of

Bohmian mechanics, Bohm himself is the one who does justice to Bohr's views. I

leave aside the question of whether Goldstein's exposition may be seen as a definitive

representation of either Bohmian mechanics or of Bohm's views, goals, and argu-

ments. It is quite clear, at the very least, that not all Bohmian physicists or scholars

share Goldstein's views concerning these matters.

16. Physics Today, April 1998, 40.

17. See, specifically, his commentary in "Bertlmann's Socks and the Nature of

Reality" (in The Speakable and the Unspeakable, 154-55), arguably his most

definitive assessment of the situation.

18. Mara Beller, "The Sokal Hoax: At Whom Are We Laughing?" (Physics

Today, September 1998, 29-34). The article is followed by the exchange in Physics

Today, January 1999.

19. Physics Today, January 1999, 96.

20. Ibid. I also permit myself to refer to Complementarity, where this point is

argued on several occasions (84-85, 101-2, 115-16). As I have indicated in chapter

2, Beller's Quantum Dialogue contains similar problems as well, especially as con-

cerns the question of nonlocality.

21. Robert Griffiths and Roland Omnes, "Consistent Histories and Quantum

Measurements," Physics Today, August 1999, 26-31. On "histories," the reader

might want to consult Roland Omnes's books cited earlier, as well as Griffiths and

Notes to Pages 213-39 * 289

Omnes's article itself, technical but still useful to a nonspecialist. It gives a far more

balanced account of its subject and of the state of quantum theory.

22. It is worth stressing that Laplace was, at most, skeptical as concerns the pos-

sibility for human knowledge to come anywhere close to developing an adequate

understanding of the universal "machine" in question.

23. There is a related thought experiment designed by Eugene Wigner, where

Wigner replaces the cat with his hypothetical friend, known as "Wigner's friend"

ever since.

24. See Carlo Rovelli's article, "'Incerto Tempore, Incertisque Loci': Can We

Compute the Exact Time at Which a Quantum Measurement Happens?" Founda-

tions of Physics 28, no. 7 (1998): 1031-43.

25. On some of these arguments see, for example, the works by Leggett,

Griffiths, and Omnes, cited earlier.

26. See Cushing's discussion in Quantum Mechanics, 166.

27. Harold Bloom, The Anxiety of Influence, 16.

28. Percy Bysshe Shelley, Shelley's Poetry and Prose, ed. Donald Reiman and

Sharon B. Powers (New York: W. W. Norton, 1977), 508.

29. I cannot discuss these works here, but if one wants to look for other exam-

ples (I am not saying anticipations) of deconstructive approaches (of the Derridean

type) outside deconstruction, one can think of, among unexpected places, Godel's

remarkable reading of Russell in "Russell's Mathematical Logic" (in Benacerraf and

Putnam, eds., Philosophy of Mathematics, 447-69), or, where it could be expected

more readily in Imre Lakatos's works, especially Proofs and Refutations: The Logic

of Mathematical Discovery (Cambridge: Cambridge University Press, 1976) and his

other major essays on the philosophy of mathematics.

30. Indeed, the point is rarely, if ever, made, including by Heisenberg himself,

who in his subsequent works, especially those philosophically oriented, moves in yet

other directions.

31. Van der Waerden, Sources of Quantum Mechanics, 261-76.

32. De Man, Aesthetic Ideology. I have discussed de Man's work in this context

in "Algebra and Allegory."

33. We recall that the nature of quantum probability is in turn nonclassical and

is not defined, as in classical physics, by, in practice, insufficient information con-

cerning the systems that, in principle, behave classically.

34. I have discussed these questions in Complementarity, 191-269.

Conclusion

1. Blanchot, The Infinite Conversation, 52.

2. The omitted part of the quotation would slightly modify but not undermine,

indeed amplify, my argument.



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Index

Note: Certain terms, such as "epistemology," "mathematics," "science,"

"quantum mechanics," and "nonclassical theory," are used throughout the

book and are indexed only via "key discussions" pertaining to them, or via

subheadings.

Abel, Niels Henrik, 266n. 24, 276n. 8,

287n. 4

Adorno, Theodor W., 25

Agencies of observation. See Measuring

instruments in quantum theory

Algebra, 25, 122, 124, 127-29, 133,

135, 139, 146, 192, 266-68n. 24,

269n. 27, 276n. 8, 286n. 51; of com-

plex numbers, 127; and geometry,

128, 154, 266-68n. 24; main theo-

rem of, 122; and nonclassical

thought, 127-29, 266-68n. 24

Algebraic geometry, 25, 129, 246n. 27;

as metaphor, 129

Allegory, 119, 194, 284n. 44; in de

Man's sense, 119, 127, 144, 170,

264n. 13, 284n. 44

Alterity (otherness), radical, 284n. 44

Althusser, Louis, 281n. 23

Ambiguity. See Bohr, Niels, on ambigu-

ity of assignment of properties to

quantum objects

Ambrosino, Georges, 243n. 6

Amplitudes. See Probability and statis-

tics

Analysis (mathematical), 135, 266n. 24

Anaximander, 120

Anthropology, structural, 187, 193

Argand plane. See Gauss-Argand plane

Aristotle, 55, 192, 205, 276n. 8;

Physics, 55

Arithmetic, 118, 127, 135, 139; arith-

metical representation (of irrational

numbers), 118; of complex numbers,

12; and geometry, 18-19

Aronowitz, Stanley, 206-7

Arts, xv, 25

Aspect, Alain, 89

Astronomy, xv

Atomicity (see also Bohr, Niels, atomic-

ity), 41, 71-73; classical vs. Bohr's,

72-73

Atoms (see also Bohr, Niels, atomicity),

5, 48, 71-73

Attributes. See Properties

Attridge, Derek, 172-73

Bach, Johann S., 106

Bachelard, Gaston, 191

Barbour, Julian, 25 1n. 16

Baroque, 266n. 24

Basic principles of science, 30, 44,

99-107; consistency, 103; experimen-

tal rigor, 104; mathematical charac-

ter of modern physics, 103; unam-

biguous communication, 103-4

Bataille, Georges, xiv, 9, 19, 25-26,

105, 120, 204, 243nn. 4, 5, 6, 272n.

54; and quantum physics, 243n. 6;

and unknowledge [nonsavoir], 9

Beckett, Samuel, 25

Beethoven, Ludwig van, 106

302 * Index

Bell, John S., xix, 16, 210, 245n. 17,

247n. 2, 258n. 51; on Bohr, 245n.

17, 247n. 2; on quantum mechanics,

245n. 17; theorem, xix, 16, 88-89,

96, 98, 210, 255n. 33,259n. 51,

260n. 60

Beller, Mara, 211,254-55n. 33, 258n.

51,259n. 54, 288nn. 18, 20

Benjamin, Walter, 25

Binary oppositions, 13-14

Biology (and genetics), 14, 15, 193,

198,221,276n. 8,287n. 2

Black holes, 47, 73-74, 82-83, 187; as

quantum objects, 73-74

Blake, William, 222

Blanchot, Maurice, xiv, 15, 25-26,

105, 118-20, 165, 171, 186, 204,

238-40, 249n. 3,264n. 12, 289nn.

1, 2; on irrationality and irrational

numbers, 118; on literature, 239-40;

on radical interrogation, 240; on the

unfigurable Universe, 239

Bloom, Harold, 222, 289n. 27; anxiety

of influence, 222; "apophrades," 222

Bloor, David, 260n. 66, 274n. 4

Body, materiality of, 131

Bohm, David, 11, 13, 67, 89, 206-14,

247n. 2, 254-55n. 33,258nn. 49,

51, 288nn. 14, 15; on Bohr, 210,

258n. 51,288n. 15; on wholeness

(nonlocality), 255n. 33, 256n. 40

Bohmian mechanics (also Bohmian the-

ories and views; hidden variables and

hidden-variables theories), 11,

49-50, 63, 66-67, 81-82, 88-89, 96,

206-18,244n. 8; 247n. 2, 250n. 15,

254-55n. 33,258n. 49, 259n. 51,

268n. 28,288nn. 14, 15; vs. Bohr's

and Copenhagen interpretations,

210-11,216-17, 254-55n. 33; com-

plementary features in, 244n. 8; dif-

ferent versions of, 254n. 33; nonlo-

cality of, 11, 88, 209-11, 247n. 2,

255n. 33,259n. 51,268n. 28; non-

Newtonian character of, 213-14;

uncertainty relations in, 81, 88,

244n. 8; wholeness (nonlocality),

255n. 33,256n. 40

Bohr, Aage, 257n. 41

Bohr, Harald, 236

Bohr, Niels: key discussion, 29-107;

xiii-xiv, xvi, xviii, 2-5, 7-12, 14-17,

19-20, 109, 111- 12, 120, 126, 136,

165, 187, 201-4, 207-8,210-12,

216, 220, 223-30, 233-39, 243nn. 1,

2, 244nn. 8, 9, 10, 245n. 17, 246n.

1,247-48n. 2, 248n. 3, 249n. 9,

250nn. 9, 10, 11, 12, 15, 251n. 15,

252n. 20, 253nn. 26, 30, 254n. 31,

254-55n. 33,256nn. 34, 37, 39, 40,

257n. 41,258-59nn. 51, 54, 55,

261n. 7, 264n. 13,268n. 24,

270-71n. 42, 285n. 44, 287nn. 4,

11, 288n. 15; on ambiguity of assign-

ment of properties to quantum

objects ("essential ambiguity"), 57,

60, 66-67, 71, 76, 80, 92, 95,

98-101, 104; atomic theory of 1913,

33, 52, 73, 261n. 71; atomicity (see

Index * 303

non (see Phenomenon, in Bohr's

sense); on probability in quantum

mechanics, 24, 83-85; on the "quan-

tum world," 9, 43, 98, 252n. 20; on

the rationality of quantum mechan-

ics, 59, 235; on reality, 29, 44; on

visualization (Anschaulichkeit), 126;

Warsaw lecture, 58, 250n. 11

Bohr's frequency relations, 223

Born, Max, 32, 80, 232, 247n. 2,

253n. 29; probability rule, 51, 80,

225

Bourbaki (group), 193

Brain, quantum aspects of, 130

"Brane" theory, 220

Bricmont, Jean (see also Sokal, Alan,

and Jean Bricmont), 208, 288n. 15

Brouwer, Luitzen E. J., 265n. 22

c (the speed of light), 164, 166, 177,

183-85, 226, 284n. 39

Calculus, 48, 136, 224

Canguilhem, Georges, 276n. 8

Cantor, Georg, 264n. 16, 265n. 22

Cartan, Elie, 263n. 10, 271n. 46, 281n.

23

Cartan, Henri, 281n. 23

Cauchy, Augustin-Louis, 127, 133,

139; on complex numbers, 139

Causality (and noncausality), 1-4, 6-7,

15, 24, 29, 42-49, 52-53, 56-57, 59,

74, 83, 94, 101-3, 206-9, 212,

215-17, 244n. 8, 278n. 12; and con-

tinuity, 4; vs. determinism, 2, 47-48,

52-53, 74, 206, 215-16, 278n. 12;

and reality, 15, 43, 56, 215-16

Cavailles, Jean, 191

Chance (see also Probability and statis-

tics; Quantum mechanics, probability

and statistical predictions in): in clas-

sical vs. quantum theory, 24, 83-86,

214-18

Change, 41, 43, 251n. 16; vs. perma-

nence, 41, 43, 251n. 15

Chaos, 106-7, 277-78n. 12

Chaos theory, 1, 47, 53, 81, 107, 207,

215-18, 220, 253n. 27, 261n. 72,

277-78n. 12, 287n. 2; vs. classical

statistical physics, 217; as classical

theory, 47, 215-17; nonlinearity of,

278n. 12; order in, 278n. 12

Classical ideal in physics, 45

Classical mechanics (also Newtonian

mechanics), xiv, 1, 33, 44-47, 52-53,

57, 78, 81-85, 196, 207, 213-18,

283n. 38; conceptual framework of

(centered and decentered), 196;

mathematical formalism of, 2, 47,

57, 267n. 24; and optics, 267n. 24

Classical physics: key discussions,

44-48, 52-57, 213-20; ix, 1-3, 6,

19, 29, 34, 36, 38-41, 50, 59, 71,

73-79, 81-85, 88, 94, 126, 182, 184,

196, 206, 226, 231, 244n. 10, 250n.

15, 286n. 52, 289n. 33; as a classical

(causal and realist) theory, 45,

52-53; conceptual framework of

(centered and decentered) 196; exper-

imental data, 47; idealization in (see

Idealization, in classical physics);

304 * Index

Collins, Harry, 274n. 4

Common cause principle, 98

Complementarity: key discussions,

1-27, 29-107; 142, 244n. 8,

254-55n. 33, 256n. 40, 260n. 69,

268n. 27, 270-71n. 42; vs. alterna-

tive interpretations, 51; and com-

pleteness, 12, 49, 87-101 passim;

and consistency, 12, 49, 67; of coor-

dination and causality, 59; different

versions of, 58-59, 223, 227-29,

254-55n. 33; of effects, 71; of

efficacities, 71; as a generalization of

causality, 59; as interpretation,

49-51; interpretations of, 244n. 9;

locality of, 255n. 33, 256n. 40; as a

model, 49-50; of mutually exclusive

phenomena, 59, 67-70, 74-77; of

mutually exclusive experimental

arrangements (or procedures), 70,

74-77, 99; as mutual exclusivity, 32,

36, 39-41, 56, 58-59, 67-69, 74-77,

99; of observation and definition, 59;

of position and momentum, 75; of

space-time concepts and application

of conservation laws, 59, 77; of time

and energy, 75; wave-particle, 39

Complex and imaginary numbers: key

discussion, 117-40; xx-xxi, 57, 80,

144-45, 150-54, 175, 220, 223, 225,

235-38, 262n. 5, 263n. 10, 266n.

23, 267n. 24, 271n. 44, 272n. 50,

273n. 57; algebraic representation,

124; complex vs. imaginary numbers,

122; epistemology of, 110-11,

116-40 passim, 144-45, 153-54;

geometrical representation of,

122-26, 135-40, 144-45, 266n. 23,

267n. 24; i (the square root of -1),

xxi, 57, 110, 112-13, 117, 124, 133,

139-42, 144-47, 150-53, 272n. 50;

as irrational numbers, 120, 145-46,

235-38; in Lacan (see Lacan,

Jacques, and complex and imaginary

numbers); legitimacy of, 121; moduli

of, 125; and quantum mechanics

(epistemological parallels), xxi,

109-11, 146; in quantum mechanics,

57, 80, 109, 220, 223, 225; reality

of, 121

Complexity theory, 278n. 12

Compton, Arthur, 31

Computer technology, 221

Concepts, 22-23, 46, 48, 115, 155-56,

190, 201, 286n. 1; classical, in

physics, 46, 48, 225-27; classical

(physical), in quantum mechanics,

24, 225-27; in Deleuze and Guat-

tari's sense, xxi, 22-23, 115-16,

155-56, 190, 286n. 1; and language,

226; vs. metaphors, 279-80n. 16;

mathematical, 48; nonclassical, 201

Consciousness, 269n. 29; quantum

nature of, 269n. 29

Conservation laws, 55, 77

Continuity, 4, 70; vs. revolution (in the

history of science), 19

Continuum, 136

Coordinates (Cartesian), 124

Index * 305

221-28 passim, 233,245n. 12,

270n. 41, 284-85n. 44, 286n. 50,

289n. 29; and ethics, 160-61; and

mathematics and science, 159, 165;

mutual, 227-28; of phenomenology

and structuralism, 228; and quan-

tum mechanics, xxiii, 159, 221-28

passim

Dedekind, Richard, 264n. 16

Deleuze, Gilles, and Felix Guattari,

xiv-xv, xxi, 19, 22, 26, 105, 146,

154-55, 158, 175, 189, 202, 220,

222, 238,246n. 26, 262n. 4, 263n.

9,266-67n. 24, 273n. 63,276n. 8,

277-78n. 12. 279-80n. 16, 286n. 45,

286n. 1,287n. 4; on chaos, 277n.

12; on concepts and philosophy, 22,

155, 262n. 4, 286n. 1; on Leibniz,

266n. 24; and mathematics and sci-

ence, 202, 276n. 8; on rhizome, 26;

and Riemann, 26, 175, 280n. 16; on

scientific concepts in philosophy,

189; and topology, 175

De Man, Paul, 26,105,119-20,127,

143-44,160,170,220,230,264n.

13, 284n. 44; allegory, 119, 127,

144, 264n. 13,266-67n. 24, 271n.

43, 284n. 44; on irony, 143; on read-

ing, 160

Democritus, 72-73

Derrida, Jacques (see also Deconstruc-

tion): key discussion, 157-99; xiv,

xxii-xxiii, 9, 13, 19-20, 22, 26, 105,

112, 120, 134, 201-2, 204-5,220,

222,225-28,233-34,245n.13,

266-67n. 24, 270n. 41,272n. 50,

273n. 3,275n. 7, 276n. 8,277nn.

11, 12, 278n. 13,279n. 14, 280nn.

16, 18, 19, 20, 281nn. 22, 23, 24,

282nn. 26, 27, 30, 31,283n. 32,

284nn. 40, 43, 44, 285n. 44, 286nn.

50, 54, 56, 58, 59, 60; and biology

and genetics, 198, 276n. 8; on center

and decentering, 157, 166, 177-80,

184-85, 193-97, 199, 226; and dif-

ferance, 9, 169-75, 182, 185-87,

197, 199, 201, 226, 272n. 50; and

differance as the efficacity of space

and time, 182-83, 185-87; and dif-

ferantial topology [topique dif-

ferantielle], 167, 169-75; and dissem-

ination, 279n. 14; and Heisenberg,

204; and Hyppolite (see Hyppolite-

Derrida exchange); on Kafka,

169-71; on Lacan, 272n. 50; on law,

169-71; on Leibniz, 266n. 24; on

Levinas, 284-85n. 44; and literature,

159, 171; and mathematics and sci-

ence, xxii, 158-59, 162, 165-66,

157-99 passim; on mathematics and

science, 158-59, 188, 199, 202,

278n. 8; and metaphor, 280n. 16; on

phenomenology (deconstruction of),

228; and philosophy, 159; and

play/game [jeu] (also decentered

play), 157, 166, 177-80, 183,

185-87, 190, 194, 196-99, 226,

282n. 31; and "play of the world,"

185, 196; and quantum mechanics,

306 * Index

DGZ (Diirr, Goldstein, and Zanghi)

hidden-variables theory, 206-9,212,

217-18,287n.12

Diagonal (of a square), 117-20; irra-

tionality of, 117-18

Dialectic, 115-16, 264n. 11, 286n. 47;

of desire, 116; synthesis, 191, 286n.

47

Diaphantus, 118

Differance. See Derrida, Jacques, dif-

ferance

Differential topology. See Topology,

differential

Dirac, Paul, 32, 35, 80, 225,247n. 2,

252n. 20, 260-61n. 70, 268n. 24

Disciplinarity, xx, 18-20, 23, 26, 30,

43-44, 99-107 passim, 112, 201,

206, 219, 260n. 67; disciplinary

specificity, 112, 201; and interdisci-

plinarity, 19, 23, 112; and nonclassi-

cal theory, xx, 18-19, 105-6; and

physics, xx, 99-107, 206; and radi-

cality, 20, 30, 260n. 67; and rigor,

18-20, 44

Disciplinary conservatism, 18-20, 106,

260-61n. 70

Discontinuity, 4, 30-31, 41, 54-55, 61,

70, 72-73, 285n. 44; and atomicity,

72-73

Disturbance (by measurement in quan-

tum mechanics), 39, 90, 100, 259n.

54

Donato, Eugenio, 157, 273n. 1, 283n.

36, 286n. 46

Double-slit experiment, 61-69, 72, 84,

95-99; and statistical nature of quan-

tum mechanics, 64-65

Duchamp, Marcel, 25-26

Edelman, Girard, 268-69n. 29

Effects, xiv, 3-4, 7, 9-11, 36-41, 43,

49-52, 60, 63, 68, 70-72, 74-80,

82-83, 86, 93-94, 98, 100, 130-31,

142-43, 170-71,187, 196-97, 223,

226, 231, 235, 252n. 20, 257n. 41,

272n. 54, 285n. 44, 288n. 13; of dif-

ferance, 197, 226; and efficacity, 3-4,

7, 9-11, 37-40, 43, 51-52, 70-72,

75, 79, 82-83, 86, 94, 98, 100,

130-31,142-43, 170-71,186-87,

196, 199, 226, 231,235, 252n. 20,

257n. 41,272n. 54, 285n. 44; indi-

vidual, 39, 60, 68, 72, 78, 252n. 20,

257n. 41; individual vs. collective,

39; of the interaction between quan-

tum objects and measuring instru-

ments (see Measuring instruments in

quantum theory, interaction with

quantum objects); irreversible

amplification, 63; of law (in Derrida),

170-71; macroscopic or classical, 37,

50; mathematical, 130-31; quantum-

mechanical, 9-11, 36-39, 50, 60

Efficacity (nonclassical) of effects (see

Effects, and efficacity); of differance

(in Derrida), 171, 199; individuality

of, 39, 70-72; of law (in Derrida),

171; material, 130, 142, 272n. 54; in

mathematics, 130-31

Einstein, Albert, xvi, xix-xx, 7, 11, 13,

Index * 307

Electrodynamics, 1, 17, 44, 48, 53, 73,

218, 250n. 13, 283n. 38

Electromagnetism. See Electrodynamics

Electrons, 31-32, 48, 73, 98, 236,

249n. 4, 250n. 13; in the atom, 48;

wavelike aspects of, 31, 249n. 4

Elementary particles. See Particles, ele-

mentary

Enlightenment and classical mathemat-

ics and physics, 287n. 2

Entanglement (in quantum mechanics),

87-89, 92-93, 98, 255n. 33; and

locality, 89, 255n. 33; quantum

objects and measuring instruments,

89-90, 94

Epicurus, 73

Epistemology, xix, 35; classical (see

also Classical theory); of complex

and imaginary numbers (see Complex

and imaginary numbers, epistemol-

ogy of); Derrida's, 157-99 passim; in

mathematics, 126-32; Lacan's,

110-16, 141-56 passim; nonclassical

(see also Nonclassical theory), xx,

xxiii, 6, 10, 13, 27, 35-36, 132, 219;

of quantum mechanics (see also Bohr,

Niels, epistemology of quantum

mechanics), xix, xxiii, 9, 27, 110

EPR (Einstein, Podolsky, and Rosen):

key discussion, 87-101; xix, 29-30,

36, 42, 44, 58, 60, 66-67, 70, 223,

227-30, 233, 238, 247n. 2, 259n.

54, 270n. 42; argument, xvi, 29-30,

58, 66, 70, 75-76, 80, 84, 87-101,

230, 238, 247n. 2, 254n. 33, 270n.

42; "criterion of reality," 89-91,

102; experiment, 30, 66, 87-91, 93

Equations: of classical physics (see also

Hamiltonian equations), 48, 57,

283n. 32; differential, 48

"Essential ambiguity." See Bohr, Niels,

on ambiguity of assignment of prop-

erties to quantum objects

Ethics of discussion, xxii-xxiii, 114,

160-65, 240-41, 275-76n. 7; and

argument, 163-64, 275-76n. 7; and

"two cultures," 240-41

Euclid, 119, 124, 126, 138, 269n. 36;

on arithmetical operations in, 138;

Elements, 119, 126, 138; on multipli-

cation, 138

Experiment, 17, 22, 38-40, 102

Experimental arrangements. See Mea-

suring instruments in quantum theory

Experimental data. See Classical

physics, experimental data; Quantum

mechanics, experimental data; Mea-

surement

Feyerabend, Paul, 25, 260n. 66

Feynman, Richard, 247n. 3, 251n. 15,

265n. 23; diagrams, 265n. 18

Fiber bundles (in topology), 116, 263n.

8

Fields (in algebra), 122, 124-26; alge-

braically closed, 122; vs. vector

spaces, 125-26

Field theories (in physics), 263n. 8

Fine, Arthur, 96, 248n. 2, 253n. 24,

259nn. 51, 56

308 * Index

Gauss, Karl Friedrich, xiv, 25, 123,

126-27, 133-37, 139, 145, 172, 186,

188, 191,197, 266n. 24, 268n. 24,

269n. 34; on complex numbers, 135,

139, 145; on intuition (Anschau-

lichkeit), 135-36, 139

Gauss-Argand plane (also Argand

plane), 123-27, 136-37; as a dia-

gram, 123-27, 137, 139-40; vs. real

plane, 124-27

Gell-Mann, Murray, 211

Geometrical representation (see also

Complex and imaginary numbers,

geometrical representation of; Num-

bers, geometrical representation of;

Visualization), 118, 123-27, 136

Geometry, 118, 124, 126-29, 132-37,

145, 154, 172, 266-68n. 24; and

algebra (see Algebra, and geometry);

analytic, 128; and classical physics,

266n. 24; of complex numbers, 109;

differential, 188,207; Euclidean,

126, 265n. 19; as mathematics of

space, 136; of negative curvature or

hyperbolic (also Gauss-B61yai-

Lobachevsky's geometry), 127, 265n.

19, 268n. 26; and nonclassical

thought, 127-29, 266-68n. 24; non-

Euclidean, xxi, 26, 123, 132-34,

136, 186, 265n. 19, 268n. 26; as

non-Euclidean mathematics, 127; of

positive curvature, Riemannian, 186,

265n. 19; and quantum physics,

266n. 24; and relativity, 268n. 26; as

science of space, 129

Ghirardi, Giancarlo, 245n. 11

Glossematics (Louis Hjelmslef's), 142

Godel, Kurt, 25, 128, 158, 166, 175,

264nn. 14, 16, 280n. 20, 287n. 2,

289n. 29; theorem(s), 25, 118, 158,

166, 280n. 29; on Russell, 289n. 29;

on undecidability, 120, 158, 175,

280nn. 16, 20

Goldstein, Sheldon, 208, 210-11

Gottfried, Kurt, 259n. 51, 274n. 4

Gravity (gravitation), 47, 186, 215,

218,226, 283nn. 32, 38; classical

(Newtonian), 47, 215; Einsteinian or

relativistic (see also Relativity, gen-

eral theory of), 47, 186, 215, 226;

gravitational constant, 178,226,

283n. 32

Greenberger-Horn-Zeilinger experi-

ment, 98

Griffiths, Robert R., 212, 251n. 15,

288n. 21,289n. 25

Gromov, Misha, 133-34, 269n. 32,

281n.23

Gross, Paul R., and Norman Levitt

(also Higher Superstition), xvi, xxii,

109, 112-13, 154, 157, 161-62,

166-78, 181, 198, 203-19 passim,

261n. 1,262n. 5, 273n. 2, 274nn. 3,

4, 276n. 7, 281nn. 21, 23,282nn.

27, 29, 287nn. 9, 12; on Derrida,

166-77, 198; on quantum mechan-

ics, 203-19

Grothendieck, Alexandre, 128, 246n.

27, 248n. 3; topos theory, 128,248n.

Index * 309

Heisenberg, Werner: key discussion,

219-34; xiv, xxii-xxiii, 2, 19-22,

25-26, 32, 34-36, 40, 49, 55-56, 75,

79, 81, 102, 111, 120, 126, 159,

162, 165, 202-4, 244n. 10, 246nn.

20, 22, 246n. 1, 249nn. 3, 9, 250nn.

9, 14, 252n. 20, 253n. 28, 254n. 31,

255-56n. 34, 257n. 41, 260nn. 65,

69, 268n. 24, 270n. 42, 287n. 3,

289n. 30; on antiparticles, 252n. 20;

on Bohr, 102, 202, 225; on concepts

and language, 226; on continuity and

revolution in science, 19; critique of

particle and wave concepts, 225-27;

"deconstitution of nature," 230-33;

and deconstruction, xxii-xxiii,

219-34; on experiment, 17, 22; and

Heidegger, 26, 260n. 69; kinematics

(see Kinematics, Heisenberg's); and

mathematical character of quantum-

mechanical concepts, 257n. 41; and

matrices, 225, 232, 270n. 42; on

observable quantities in quantum

mechanics, 223-25; on theoretical

physics as experimental, 202; on

visualization (Anschaulichkeit), 126;

and uncertainty relations (see Uncer-

tainty relations)

Heraclitus, 13

Heterogeneity, 27; interactive, 27

Hidden variables, hidden-variables the-

ories. See Bohmian mechanics

Hilbert, David, 129, 267n. 24

Hilbert spaces, 81, 124, 127-28, 131,

205, 244n. 10, 251n. 15, 258n. 47,

264n. 18, 267n. 24; finite-dimen-

sional, 258n. 47, 270n. 42; infinite-

dimensional, 124, 128-29, 205,

251n. 15, 258n. 47, 264n. 18; opera-

tors in, 270n. 42

Hiley, Basil, 258n. 51

Hipassus, 119, 264n. 12

Hjelmslef, Louis, 142, 273n. 58

Hume, David, 15

Husserl, Edmund, 69, 134, 158, 192,

284-85n. 44; "The Origin of Geome-

try," 158; phenomenology of con-

sciousness, 69

Huth, John, 274n. 4

Hyppolite, Jean, 155, 160, 164, 172,

175, 177-99, 276n. 8, 280n. 18,

281n. 23, 283nn. 32, 38, 286n. 51

Hyppolite-Derrida exchange, 157, 160,

164, 172, 177-99, 280n. 18, 283n.

32

i (the square root of -1). See Complex

and imaginary numbers, i

Idealism (philosophical), 266-68n. 24

Idealization, 1, 45-50, 53, 55, 130; in

classical physics, 1, 40, 45-48, 53; in

quantum mechanics (nonclassical ide-

alization), 4-8, 40, 49-50, 54-55,

70, 130

Imaginary numbers. See Complex and

imaginary numbers

Incommensurability (of magnitudes in

mathematics), 118-19

Indistinguishability (of objects in quan-

tum physics). See Quantum

310 * Index

Interpretation (see also entries on inter-

pretation in "Classical physics" and

"Quantum Mechanics"); and models

(see Models, and interpretation);

nonclassical interpretation of a the-

ory, 10-11; role in physics, 47, 213,

244n. 10

Intuition, intuitive representation. See

Visualization

Irigaray, Luce, 22

Irrationality, 23, 119; conception of,

xxi; and rationality, 119-20, 236;

and rationality in quantum mechan-

ics, xxi, 23, 236

Irreversible amplification, 63

Jacob, Frangois, 276n. 8, 281n. 23

Jacob, Margaret C., 279n. 13

Jeanneret, Yves, 275n. 5

Jordan, Pascual, 32

Joyce, James, 25-26

Kafka, Franz, 25-26, 169-71, 186

Kant, Immanuel, xxiii, 4, 15, 34, 153,

191, 225, 245n. 17, 250n. 12,

267-68n. 24, 268n. 26; critical phi-

losophy, 225; and non-Euclidean

geometry, 268n. 26

Kant-Derrida axis, xix, 225

Keats, John, 261n. 71

Kepler, Johannes, 154, 266n. 24, 276n.

8

Kinematics, 34-35, 249n. 9; and

dynamics, 35; Heisenberg's ("new

kinematics"), 34-35, 49, 81, 223-24,

249n. 9

Kinetic theory of gases, 45, 53

Klein's bottle, 116, 154, 171

Kleist, Heinrich von, 267n. 24

Knowable, the, xiii-xiv, 2, 10, 21,

119-20, 238-41; and classical, 2;

and the unknowable, xiii-xiv, 7, 10,

21, 119-20, 238-41

Knowledge, xiii, 5, 7, 10, 14, 17-18,

20-21, 30, 64, 73-74, 99, 102,

133-34; and action, 17-18; and the

double-slit experiment, 64; scientific,

102; theoretical, 44; and the

unknowable, xiii, 5, 7, 21; and the

unknown, 21

Kochen-Specker theorem, 88, 247n. 2

Koertge, Noretta, 275n. 4

Kojeve, Alexandre, 116, 273n. 55

Koyre, Alexandre, 191

Krips, Henry, 262n. 5, 270-71n. 42

Kronecker, Leopold, 127

Kuhn, Thomas, 25, 132, 246n. 1, 249n.

7, 250n. 10, 260n. 66, 287n. 3; nor-

mal science, 132-33; scientific revolu-

tion, 132-33

Labinger, Jay, 274n. 4

Lacan, Jacques: key discussions,

109-16, 141-56; xiv, xx-xxii,

19-20, 22, 105, 158, 172, 175, 190,

202, 204, 261n. 2, 262nn. 4, 5, 7,

263n. 9, 265n. 22, 269n. 37,

270-71nn. 42, 44, 272nn. 50, 52,

53, 54, 55, 273nn. 55, 56, 60, 61,

281n. 23; "algebra," 113, 142,

148-50, 153, 175, 273n. 57; and

complex and imaginary numbers,

Index * 311

the image of the penis), 110, 150-52,

273n. 56; and philosophy, 115-16,

154-56, 263n. 9; and quantum

mechanics, 142-43, 153, 270-71n.

42; and the Real (order or register

of), 142-44, 153, 270-71n. 42,

272n. 54; and sacrifice, 152, 272n.

55; and science, 262nn. 5, 7;

signification of the phallus, 151; and

signifier and signified, 142, 147-52;

"square root of 1" or (L) 1 (of

Lacan's system) (see Lacan, Jacques,

"the erectile organ"); and subject,

subjectivity, 113, 141, 150-51; and

the Symbolic (order or register of),

150-52; and topology, 110, 153-54,

262n. 6; and tragedy, 118-19; and

visualization and de-visualization,

141, 154

Lacour, Claudia Brodsky, 264n. 13,

265n. 21

Lagrange, Joseph-Louis, 48, 276n. 8,

287n. 4

Lakatos, Imre, 25, 260n. 66, 289n. 29

Langlands, Robert, 129, 132, 246n. 27,

268n. 25

Laplace, 207; Laplace's demon,

207

Latour, Bruno, 22, 158, 260n. 66,

274-75n. 4, 279n. 13, 284nn. 42,

43, 286n. 48

Laugwitz, Detlef, 132, 134-35, 269nn.

31-35

Lautman, Albert, 191

Le Gaufey, Guy, 262n. 4

Leggett, Anthony J., 64-66, 256n. 35,

289n. 25

Leibniz, Gottfried E., 109, 134-36,

145, 172, 184, 186, 191, 196, 201,

263n. 8,266n. 24, 271n. 45; on

complex numbers, 109, 145; debate

with Clarke, 191

Leray, Jean, 281n. 23

Levinas, Emmanuel, 25, 105, 120, 134,

238, 240, 284-85n. 44; and alterity

or otherness (Autrui), 284-85n. 44;

and infinity, 285n. 44; and nonclassi-

cal thought, 284-85n. 44

Levi-Strauss, Claude, 187, 193, 286n.

49

Levitt, Norman (see also Gross, Paul

R., and Norman Levitt), 287n. 10

Light. See Radiation

Light, speed of. See c

Line, 138, 205; in philosophy, 205; real

(as representation of real numbers),

138-39

Linearity, 204-5; mathematical, 205;

philosophical, 205; in quantum

mechanics, 204-6

Literature, 25, 239-40

Locality and nonlocality, xix-xxi, 11,

30, 209-10, 244n. 11; and Bohmian

mechanics (see Bohmian mechanics,

nonlocality of); and quantum

mechanics (see Quantum mechanics,

locality and nonlocality of)

Logic, mathematical, 110; multivalued,

67; ordinary, 67

312 * Index

Mathematical logic, 14, 15, 128, 198,

287n. 2

Mathematics: key discussion, 109-40;

Arab, xv, 118, 120; Babylonian, 121,

193; Euclidean, 126, 128, 265n. 23;

foundations of, 25, 120, 264nn. 14,

16, 269n. 37; Greek, xv, 117-20,

193; and language, xv; and material-

ity, 130-31, 271n. 44; non-Euclid-

ean, xxi, 126-32, 135-36, 155,

171-72, 220, 223; philosophy of,

264n. 14, 269n. 37; proof in, 264n.

11

Matrices (in quantum mechanics). See

Heisenberg, matrices; Quantum

mechanics, matrix (Heisenberg's) ver-

sion of

Matter. See Nature

Maxwell, James Clerk, 44, 48, 196

Mazur, Barry, 264n. 10, 266n. 24

Measurable quantities (in physics), 1,

43, 45

Measurement (see also Measuring

instruments in quantum theory), 118,

121; classical, 38, 74, 77, 250n. 15;

and immeasurable, 118; and (real)

numbers, 121

Measuring instruments in quantum the-

ory (also measuring and experimental

apparatus, arrangements, and proce-

dures; or agencies of observation), 2,

7, 9, 14, 31, 38, 40, 42, 49-51, 56,

59-60, 63, 65, 67-68, 71-72, 74-90

passim, 92, 93, 99-101, 130, 143,

187, 223,231-32, 236, 248n. 2,

249n. 9, 250-51n. 15, 259n. 55,

270-71n. 42, 288n. 13; as interacting

instruments, 68; interaction with

quantum objects, 2, 7, 9, 14, 35-38,

40, 42, 49-51, 56, 61, 63, 65, 67-68,

71-72, 74-88 passim, 90, 92-93,

99-101,223,231-32, 236, 259n. 55,

288n. 13; and irreversible amplifi-

cation in Bohr, 63; vs. measuring

instruments in classical theory, 38,

40; quantum aspects of, 40, 86; in

relativity, 172, 187

Mermin, N. David, 96-97, 247-48n. 2,

249n. 8, 250n. 11, 256nn. 36, 38,

259nn. 51, 52, 57, 58, 260nn. 61,

62, 264n. 19, 274n. 4, 284n. 42

Metaphor, 127-28, 271-72n. 50, 280n.

16; spatial-geometrical, 128; systemic

vs. direct, 271-72n. 50

Milton, John, 106-7, 222, 26 1n. 72; on

chaos, 106; Paradise Lost, 106-7

Mind, 30, 130

Minkowski, Hermann, 182, 184

Mittelstaedt, Peter, 251n. 15, 253n. 25

Models, 2, 40, 45-49, 53-55, 86-87,

93, 244n. 10, 252n. 21; and classical

ideal in physics, 45-47; classical

mathematical, 126; classical physical,

2, 40, 45-48, 53-54, 87, 127, 244n.

10; and interpretation, 47-50, 54,

93, 244n. 10; mathematical, in

physics, 48; mathematizable, in

physics, 48; quantum-mechanical, 40,

48-49, 54, 87

Index * 313

limits of science and logic, 21; on

metaphor, 280n. 16; on physics, 17

Nihilism, 10

Nonclassical physics (also new physics),

44-45

Nonclassical theory (thinking, views)

(see also Epistemology, nonclassical),

key discussion, 1- 13; and classical

theory (see Classical theory, and non-

classical theory); cultural context and

scope, 13-27; of history, 195-96;

and the humanities (or social sci-

ence), xvii, 201; and mathematics

and science, xiii, xv, 111, 162, 164,

171, 201, 220-21; and postclassical

theory, 14-15

Non-Euclidean mathematics. See Math-

ematics, non-Euclidean

Nonlocality. See Locality and nonlocal-

ity

Noumena, 250n. 12

Nucleus (atomic), 48, 73

Numbers, 117-18, 139-40, 153; arith-

metical representation of, 118; com-

plex (see Complex and imaginary

numbers); geometrical representation

of complex (see Complex and imagi-

nary numbers, geometrical represen-

tation of); geometrical representation

of (other than complex), 118,

123-27; imaginary (see Complex and

imaginary numbers); irrational,

117-18, 235-38; irrational complex

(see Complex and imaginary num-

bers, as irrational numbers); irra-

tional real, 111, 117-20, 123, 146,

235-38; natural, 117; negative,

264n. 15; rational, 117, 124; real,

111, 120-21, 124-25, 133, 138, 140,

144, 266n. 23, 267n. 24; roots,

120-21; square roots, 117-20;

whole, 117, 120

Number theory, 25

Object(s), 56, 63, 100; of classical

physics, 1-3, 38-40, 47, 74-76, 87,

217; of classical theories, xiii, 1-3;

concept of, 56, 63, 100; infinite-

dimensional (mathematical), 57,

251n. 15; mathematical, 57, 128-30,

268n. 27; natural (material), 47; of

nonclassical theories, xiii, 2-5, 7,

130-31; non-Euclidean, 128-29;

quantum (see Quantum objects)

Objectivity, 23, 119; vs. subjectivity, 23

Observation (see also Measurement),

55-56; classical nature of, 56; as a

concept borrowed from everyday life,

55

Occam's Razor, 207, 212

Old quantum theory, 32-33

Omnes, Roland, 103, 251n. 15, 256n.

39, 288-89nn. 21, 25

Operators. See Hilbert spaces, opera-

tors in

Optics, 267n. 24

Pais, Abraham, 41, 249n. 6, 251n. 17,

257n. 43

Parmenides, 41, 152, 251n. 16, 264n.

11, 276n. 8, 287n. 4; the One, 152

314 * Index

Phenomenon (continued)

74; closed (in Bohr's sense), 71, 73;

collective, 39, 66, 69, 84-85; collec-

tive vs. individual, 69, 84-85; indi-

vidual, 34, 39, 61, 67, 69-74, 78,

82-87, 252n. 20, 285n. 44; indivisi-

ble (wholeness of phenomena),

71-73, 92, 252n. 20; in philosophical

vs. Bohr's sense, 69, 88; physical,

30-31, wavelike, 30, 53, 69

Philosophy, xv, 17, 22-23, 25-26,

115-16, 147, 154-56, 188-91, 198,

201, 225, 263n. 9, 276n. 8, 277n. 8;

and concepts (in Deleuze and Guat-

tari's sense), 22, 115, 154, 190; Con-

tinental vs. analytical, 201; in

Deleuze and Guattari's sense, 22,

115, 154, 190; and literature and art,

201-2; and mathematics, 116, 147,

154-56; of mathematics, 268n. 27,

289n. 29; and mathematics and sci-

ence, 189-90, 198, 201-3, 276n. 8,

277n. 8; and physics, 16-17; and

psychoanalysis, 155

Photon(s), 31

Physical law(s), 30, 44, 99-101, 104-5

Physics: as mathematical science, 43,

57, 102-5; as mathematical-experi-

mental science, 104-5; and philoso-

phy, 16-17; theoretical, as experi-

mental, 202

Pickering, Andrew, 267n. 24, 274n. 4

Pictorial representation. See Visualiza-

tion

Planck, Max, 30, 32, 35, 37, 40, 44,

49, 56-57, 60-61, 65, 71-73, 83, 85,

221, 246n. 1, 248n. 3, 252n. 20;

constant (h), "the quantum of

action," 30, 40, 56-57, 60-61; law

(also "Planck's discovery") 30, 37,

72, 83, 221, 246n. 1, 248n. 3, 252n.

20

Plane, real two-dimensional (vs. com-

plex plane), 138-39, 266n. 23; geo-

metrical representation of, 138-39,

266n. 3

Plato, xv, 16-17, 41, 117-18, 153-54,

171-72, 186, 191,196, 251n. 16,

275n. 5, 276n. 8; chora, 171-72,

186, 196

"Platonia," 251n. 16

Platonism, mathematical, 268n. 27

Play. See Derrida, Jacques, and

play/game

Poe, Edgar Allan, 272n. 50

Poetry, xv

Poincare, Henri, 26, 45, 135, 207, 215,

218, 265n. 22, 281n. 23

Position or coordinate (as a physical

variable), 1, 6, 37, 39, 56-57, 66-67,

76-77, 79, 81-84, 90, 206, 249n. 9,

250n. 13

Postclassical theory, 14-15

Postmodernism (postmodern, postmod-

ernist), 13, 16, 153-54, 158, 166,

174, 203, 211, 219, 222, 245n. 12,

279n. 13; and nonclassical mathe-

matics and science, 287n. 2; and sci-

ence, 222, 287n. 2

Index * 315

objects, behavior, and motion, 1, 3,

6, 34-36, 40, 43, 45, 53, 63, 74, 79,

250n. 13, 252n. 20; dynamic

(mechanical), 35; as "elements of

reality" (Einstein), 54; kinematic, 35;

of mathematical objects, 127, 129,

131, 140, 250n. 13; particlelike, 40,

63, 66, 228, 230, 249n. 4; and quan-

tum objects, behavior, and motion, 7,

34-36, 39-42, 50, 56, 60, 66-68, 71,

76-77, 82-83, 88-89, 100, 131, 228;

wavelike, 40, 63, 66, 71, 228, 230,

249n. 4

Proportions (in mathematics), 119

Proust, Marcel, 42, 251n. 18

Psychoanalysis, 19, 110-16, 141-56

passim, 272n. 50

Pythagoreans, 15, 117-20, 123

Quantum computing, 88

Quantum cryptography, 88

Quantum electrodynamics (QED), 32,

252n. 20, 260-61n. 70

Quantum field theory, 6, 179, 206,

246n. 2, 252n. 20, 261n. 70; Yang-

Mills, 206

Quantum gravity, 6, 213

Quantum information. See Information,

quantum

Quantum information theory, 41,

256n. 35

Quantum mechanics: key discussion,

29-107; 126, 129-32, 142-43, 146,

153, 159-60, 162, 172, 186-88, 198,

203-32, 243n. 6, 244nn. 8, 9, 10,

245nn. 11, 18, 246nn. 1, 2,

247-48n. 2, 250-51n. 15, 252n. 20,

253nn. 26, 30, 256nn. 34, 36, 39,

258n. 50, 261nn. 70, 72, 263n. 8,

264nn. 13, 14, 266-68n. 24, 269n.

28, 270-71n. 42, 276n. 8, 280n. 20,

283n. 38, 287n. 2, 288n. 13, 289n.

21; classical-like interpretation of,

11-13, 48-51, 81-82; and classical

physics (see Classical physics, and

quantum physics); completeness and

incompleteness in: key discussion,

86-101; xix, 5, 11-12, 36, 66, 258n.

50; and complex numbers (see Com-

plex and imaginary numbers, and

quantum mechanics); and conscious-

ness, 269n. 29; and consistency, 5,

12, 67, 101; different interpretations

of, 244n. 9, 247n. 1; Dirac formal-

ism, 247-48n. 2; experimental data

in, 4, 10, 11, 14, 43, 51, 67-69, 72,

94, 99-101, 223, 230; explanatory

and descriptive capacity of, 32-33;

Feynman's version of, 247n. 2, 251n.

15; Ghirardi-Rimini-Weber interpre-

tation of, 247n. 2; histories interpre-

tation of, 247n. 2, 251n. 15, 288n.

21; idealization in (see Idealization,

in quantum mechanics); interpreta-

tion of (the role of interpretation in)

xiv, 10-13, 48-51, 94-95, 213,

244n. 10, 246n. 1; Ithaca interpreta-

tion of, 247-48n. 2, 251n. 15; linear-

ity in, 204-6; locality and nonlocality

of: key discussions, 87-101, 210-11;

316 * Index

Quantum mechanics (continued)

33, 39, 51, 69, 78-88,231,256n.

35, 288n. 13, 289n. 33; relational

interpretation of, 247n. 2; relativistic

quantum theories in, 32; and relativ-

ity, 93-94, 187; statistical nature of,

24, 64, 83-86, 216-17; symbolic

character of, 57; and Von Neumann's

formalism, 244n. 10, 247-48n. 2,

270n. 42; Von Neumann's interpre-

tation of, 251n. 15, 270n. 42; wave

(Schrodinger's) version of, 32, 39,

223,228-29,235,247n. 2

Quantum objects (also quantum

processes, behavior, and states), xiii,

2-9, 11, 14, 29, 31, 33-34, 37-43,

49-51, 54, 56, 60-61, 63, 67-92, 96,

100, 130-31, 143, 244n. 8, 247n. 2,

248n. 3,252n. 20, 253n. 30, 257n.

41, 259n. 55, 265n. 23, 288n. 13;

existence, 41-42, 130-31; identity

and indistinguishability of, 252n. 20,

257n. 41; and measuring instruments

(see Measuring instruments in quan-

tum theory, interaction with quan-

tum objects); macro quantum objects,

248n. 3; "motion" of, 34, 41, 49;

properties of (see Properties)

"Quantum world." See Bohr, Niels, on

the "quantum world"

Quantum postulate, 35, 73; as a tech-

nological concept, 73

Quasi-mathematics, quasi-mathemati-

cal, xxi, 113-14, 176; in Lacan, xx,

113-14

Quaternions, 128, 267n. 24

Radiation (light), 30, 73, 83; discontin-

uous nature of, 30-31, 73; particle

character of, 31

Radical, radicality, 16, 20; and nonclas-

sical theory, 16, 20

Rationality, 119-20, 235-36; and irra-

tionality, 119-20, 235-36

Reading, xxii, 160, 162, 165; the ethics

of, 160, 162, 165

Reality, realism, 1, 7, 15, 24, 41-46,

50, 52-55, 57, 76-77, 86-87, 89-91,

99, 101-3, 121,206, 216-18,248n.

2, 253n. 30, 255n. 33, 258n. 47,

268n. 27, 271n. 44, 283n. 38; and

causality (see Causality, and reality);

"elements of reality" (Einstein), 54,

81, 83, 86, 91; and locality, 89;

mathematical, 268n. 27, 271n. 44;

objective, 41, 76; realist theories as

classical theories, 54

Reed, David, 260n. 68, 264n. 10, 265n.

21

Referent (of sign), 142

Reichenbach, Hans, 98, 260n. 63

Relativity (also relativity theory): key

discussion, 177-99; xx-xxi, 25, 27,

30, 44-45, 47-48, 66, 79, 93, 97,

101, 105, 129, 157-58, 160, 162-64,

167, 172, 206, 209, 213, 215, 220,

225-26, 239,263n. 8,268n. 26,

276n. 8, 280n. 18, 283nn. 32, 38,

39, 287n. 2; conceptual framework

of, 193-94, 197; and cosmology,

Index * 317

Riemannian metric, 207

Riemann's surface, 127

Rilke, Rainer Maria, 26

Rimini, Alberto, 245

Rokhlin, Vladimir A., 281n. 23

Romanticism, 266-67n. 24

Rosen, Nathan, xix, 29-30

Rosenfeld, Leon, 243n. 1

Ross, Andrew, 206

Rbtzer, Florian, 276n. 9

Rovelli, Carlo, 248n. 2, 289n. 24

Riidinger, Eric, 250n. 10

Ruelle, David, 133

Russell, Bertrand, 289n. 29

Rydberg-Ritz formulas (for atomic fre-

quencies), 224

Saussure, Ferdinand de, 142, 153,

273n. 58

Schaffer, Simon, 274n. 4, 279n. 13

Schilpp, Paul Arthur, 254n. 32, 258n.

48, 261n. 71

Schonberg, Arnold, 106

Schrbdinger, Erwin, 11, 16, 32, 39, 43,

45-48, 54-56, 87, 89, 165, 202, 207,

223, 230, 251n. 20, 258n. 50; "cat

paradox," 89, 207, 213-14; "cat

paradox" paper, 43; on classical

ideal and models in physics, 45-46;

on quantum mechanics, 45-46,

258n. 50; on wave mechanics, 32,

39, 223, 229

Schrbdinger's (wave) equation, 39,

47-49, 65, 205-7, 223,287-88n. 13;

and causality and determinism,

287-88n. 13; as linear, 205-6; as

partial differential, 205, 207; and

uncertainty relations, 288n. 13

Schweber, Sylvan S., 260n. 70

Science: basic principles of (see Basic

principles of science); normal (in

Kuhn's sense), 132-33

Science studies (also social studies of

science), 260n. 66

Science Wars, xvi, xxi-xxiii, 16-18,

20-21, 25, 30, 102, 112, 115,

157-64, 177, 185, 198, 202, 205,

211, 213, 219, 222, 260n. 66, 269n.

37, 274n. 3, 275n. 7, 280n. 18,

287n. 2; and quantum mechanics,

204-19; representation of mathemat-

ics and science in, xxii-xxiii, 162,

201-19

Searle, John, 275n. 7

Serre, Jean-Pierre, 281n. 23

Serres, Michel, 189, 193, 261n. 72,

281n. 23, 286n. 48, 287n. 8; on

structuralism, 193

Shakespeare, William, 42, 238; Romeo

and Juliet, 42; Hamlet, 238

Shapin, Steven, 279n. 13

Sheaves (in topology), 116, 263n. 8

Shelley, Percy Bysshe, 222; on poetry

and poets, 222

Shimony, Abner, 259n. 51, 260n. 61

Signifier and signified. See Lacan,

Jacques, and signifier and signified

Singularity, 73; and black holes, 73; as

individuality and uniqueness of phe-

nomena in Bohr, 70, 73

318 * Index

Space(s) (continued)

268n. 26; without points (in topos

theory), 128

Space-time (in relativity), 163, 181-88,

194, 197; decentered, 194, 197; in

general relativity (curved by gravita-

tion), 186; vs. space and time in clas-

sical physics, 182, 184

Space-time physical events and

processes (motion), 11-34, 37,

54-55, 59, 65, 71, 74, 78-80, 83,

257n. 41, 288n. 13

Spatial, spatiality, 80, 128, 136, 169,

187; intuition, visualization, 128-29

Spatialization, 123-27

Spectra, atomic, 249n. 9

Spin, 89, 257n. 47

Stapp, Henry, 256nn. 36, 38, 259nn.

53-55, 260n. 61

State-vector (in quantum mechanics),

253n. 30

Statistics. See Probability and statistics

Stevens, Wallace, 222

String theory, 6, 73, 220

Structuralism, 193, 228

Structure. See Derrida, Jacques, on

structure

Subject, psychoanalytic. See Lacan,

Jacques, and subject, subjectivity

Subjective, 23

Subjectivity, 285n. 44

Symbols, (in quantum mechanics), 57

Symmetry, 257

Taylor, Paul, 246n. 2, 248n. 2

Technology, 1, 7, 26, 72-73, 159,

187-88; of experiment and measure-

ment, 1, 7, 26, 220; in nonclassical

theory (vs. classical theory), 7,

72-73, 159; and writing in Derrida's

sense, 159, 187-88

Theaetetus, 118

Thermodynamics, 1, 17, 44

Things-in-themselves, 43

Thom, Rene, 281n. 23

Time, 34, 182-83, 187-88; absolute (in

Newton), 184, 186, 191; and mea-

suring instruments (in relativity),

187; retroaction in, 66

Tononi, Giulio, 269n. 29

Topography, 168-70; vs. topology,

168-69

Topology, 25, 116, 120, 124, 127-28,

135-36, 154, 167-77, 186, 188, 198,

220, 262n. 5, 263n. 8, 265n. 22,

266n. 24, 281n. 23; of complex num-

bers, 127; differential 167-74, 188,

281n. 23; philosophical, in Greek

philosophy (as philosophy of spatial-

ity), 136

Topos, topoi, 168-69

Topos theory. See Grothendieck,

Alexandre, topos theory

Tragedy, 118; and mathematics, 199

Trajectories, 47, 63, 244n. 8; continu-

ous in classical physics, 47; impossi-

bility of, in quantum mechanics, 63

Transformation theorems (of quantum

mechanics), 78

Truth, 119

Index * 319

Variables (in mathematics and physics),

40, 48, 56-57, 66, 75-77, 83, 89, 91,

98, 205, 258n. 47; conjugate (or

complementary), 56, 75, 89, 91, 98;

kinematical and dynamic, 258n. 47;

symbolic (in quantum mechanics),

57

Vectors, vector spaces, 124-25, 128,

264n. 18

Velocity, 39, 205

Visible and invisible, 238, 240-41

Visualization (Anschaulichkeit) (also

pictorial representation), 123-29,

135, 136, 139, 144-45, 231,250n.

9, 252n. 20, 265nn. 18, 22, 23,

266n. 23, 266-68n. 24, 285n. 44

Von Neumann, John, 35, 80, 225,

244n. 10, 247-48n. 2, 251n. 15,

270-72n. 42; "observables," 270n.

42; projection postulate, 80

Wave, 6, 14, 30-31, 36-40, 53, 62-63,

69, 74, 80, 226-30, 249n. 4, 265n.

18; and effects (wavelike effects),

36-39, 74; and particles (see Parti-

cles, and waves); quantum vs. classi-

cal, 226

Wave-function (v-function), 51, 65, 97,

205

Weber, Tulio, 245n. 11

Weil, Andre, 129, 193

Weil, Simone, 193

Weinberg, Steven, 179-81, 205-6,

273-74n. 3, 277n. 12, 282n. 31,

283nn. 32, 34, 35, 287nn. 6, 7, 8,

288n. 13; on Derrida and "the Ein-

steinian constant," 179-81

Weininger, Stephen, 274n. 4

Wessels, Linda, 259n. 60

Wheeler, John Archibald, 21, 211,

246n. 25

Wigner, Eugene, 247n. 2, 289n. 23;

"Wigner's friend," 289n. 23

Wilson, Kenneth, 274n. 4

Wittgenstein, Ludwig, 15, 25, 245n. 16

Woolf, Virginia, 26

Wordsworth, William, 222

Writing. See Derrida, Jacques, on writ-

ing

Zeno, 264n. 11

Zizek, Slavoj, 270-71n. 42; on quan-

tum mechanics and the Real, 271n.

42