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 e