No title

(25)

z(O) = x E H, z ' (t) = - ('V
(-->

and consequently u is a critical point of ¢. Call w, y E H equivalent provided there is u E H such that if x is either of w and y, and z satisfies (25), then (26) holds. This notion of equivalence gives a foliation of H, two members of H being in the same leaf of the foliation if and only if they are equivalent. This leads to a setting in which each leaf of the foliation contains precisely one critical point of ¢. An analytical, topological, or algebraic (prefer­ ably all three) understanding of this foliation tags in a po­ tentially useful way each critical point of

,v·


v E H.

Suppose x E H and B(x) = 0, and assume that both PB and F'(·)*F(·) are C1 . Then there is z : [O,oo) � H so that if

t 2: 0,

then

B(z)' (t) = B'(z(t))z' (t)

=

-B' (z(t))('ilBcfJ)(z(t)) = 0, t 2: 0,

since

('ilB
2:

=

x E H.

II Tx llk/2,

Then

For each v E H, define PB(v) to be the orthogonal projec­ tion of H onto HB Then

z(O) = x E H, = z ' (t) ('ilB cfJ)(z(t)),

0.

u = lim00 z(t) exists,

(26)


('VB
2':

Assume also that for each such x E H the corresponding solution z to (25) satisfies

and for each v E H, define

HB, v = {h E H : B'(v)h = O J.

t

0.

Thus continuous steepest descent with this Sobolev gradi­ ant, starting at x, preserves the supplementary condition satisified by the initial condition, i.e.,

B(z(t)) = B(x) = 0, t 2: 0, and so B(u) = 0 if u = limt---.ocZ(t). Here is a second rather general suggestion about the "supplementary condition" problem. I make no attempt for maximum generality; more on this issue can be found in [ 1 1 ] and references therein. Assume H is a Hilbert space,


x E H.

('il
Pick x E H and consider z satisfying (25) . Note that not only does u = lim z(t) exist, but u is the nearest zero of T to the initial v�u� x. An understanding of the resulting fo­ liation would give a hold on the set of all solutions to (2 1). Numerical Approximations

Quite a number of numerical implementations have been worked out for Sobolev gradients. The Ginzburg-Landau de­ velopment for superconductivity is chosen as a represen­ tative application. Work described is that of the present au­ thor and his long-time collaborator R. J. Renka (see [ 14] and references therein). The Ginzburg-Landau functional

(27)
Jn ( 11Vu - iuAII2

II'V X A - Hol l2 +

+

�2 clul2 - 1)2}

(u,A) E H,

where H0 is a given C1 function on D. The complex-valued function u is an "order parameter," meaning here that lul2 indicates density of superconducting electrons. The func­ tion 'il X A is an induced magnetic field, and Ho an imposed magnetic field.

© 2005 Springer Science +Business Media, Inc., Volume 27, Number 3, 2005

53

Fig. 1. Magnetic field on a "squid" with smaller ''flippers."

A Sobolev gradient for ¢ is constructed along the lines of the simple example. The figures give some representa­ tive results. In each of these figures, the magnitude of the magnetic field V X A corresponding to a critical point of (27) is depicted. In Figure 1 , the "flippers" of the squid-like hole are shorter than corresponding flippers in Figure 2. In

Fig. 2. Magnetic field on a "squid" with longer "flippers."

54

THE MATHEMATICAL INTELLIGENCER

Figure 1, there is a "free" vortex, one not pinned by a given structure, whereas in Figure 2 there is no such free vortex. These illustrations show how small changes in a super­ conducting device may lead to substantially different crit­ ical points. The models, from a code by Robert Renka, were obtained by Barbara Neuberger as part of a program seek-

ing to aid design of superconductors by means of simula­ tions. For purposes of superconducting electronics, one wants to understand how to place holes or moats that will attract vortices in order to leave substantial parts of the de­ vice free of vortices, places in which it is attractive to put superconducting circuits. A recent reference for this pro­ gram is [ 14]. Other applications are [17], [ 1 1 ] and references therein. These other applications include transonic flow problems and elasticity problems as well as a variety of Ginzburg-Landau type problems. Yang-Mills [6] functionals are very close in structure to those of the Ginzburg-Landau functional of superconductivity. It is an interesting research problem to code the Yang-Mills functionals using Sobolev gradients. A principal interest of this writer in these applications of Sobolev gradients is that they look like developments moving in the direction of a central theory of partial dif­ ferential equations. Sobolev gradients have worked in all known instances in which they have been seriously tried.

A U T H O R

JOHN W. NEUBERGER

Department of Mathematics University of North Texas Denton, TX 76203-1 430 USA

e-mail: [email protected] John W. Neuberger did his doctoral work at the University of

S. Wall . In addition to differ­

CONCLUSIONS

Texas under the direction of H.

The sketch I have given above seeks to place a broad part of PDE into a variational setting, a setting that includes both numerical and theoretical considerations. A key ele­ ment is the systematic use of self-duality of the Sobolev spaces involved. It is briefly indicated how a wide variety of boundary conditions (more accurately, supplementary conditions) may be dealt with in a systematic way.

ential equations and numerical analysis, his interests include nonlinear semigroups, quasi-analyticity, superconductivity, and algebraic geometry. He has supervised twenty-seven PhD students. Ever since graduate school he has been active in

consulting. He is now Regents Professor at the University of North Texas. He rides his various bicycles about 8000 kilo­

meters a year.

REFERENCES

[ 1 ] R. A Adams, Sobolev Spaces, Academic Press, 1 978.

[2] R . Chill and M . A Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Non­ linear Analysis 53 (2003), 1 01 7-1 039. [3] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Dif­ ferential Equations, CBMS Reg. Cont. Ser., Amer. Math. Soc. 1 990. [4] Farago , 1 . , Karatson, J . , Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications. Advances in Computation, Volume 1 1 , NOVA Science Publishers, New York, 2002. [5] S.-Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (200 1 ) , 675-698. [6] A Jaffe and C. Taubes, Vortices and Monopoles. Structure of Sta­ tic Gauge Theories, Progress in Physics 2 , 1 980.

[7] S. t.ojasiewicz, Une propriete topologique des sous-ensembles an­ alytiques res, Colloques internationaux du C.N.R.S.: Les equations aux derivees partielles, Editions du C.N.R.S., Paris (1 963), 87-89. [8] D. Lupo and K. R. Payne, On the maxim um principle for general­ ized solutions to the Tricomi problem, Commun. Contemp. Math. 2 (2000), 535-557. [9] J . Moser, A Rapidly Convergent Iteration Method and Non-Linear Differential Equations, Ann. Scuola Normal Sup. Pisa, 20 (1 966), 265-31 5. [ 1 0] John M . Neuberger, A numerical method for finding sign chang-

ing solutions of superlinear Dirichlet problems, Nonlinear World 4 (1 997), 73-83. [1 1 ] J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer Lecture Notes in Mathematics 1 670, 1 997. [ 1 2] J . W. Neuberger, Continuous Newton's Method for Polynomials, Mathematical lntelligencer, 21 (1 999), 1 8-23. [1 3] J. W. Neuberger, A near minimal hypothesis Nash-Moser Theo­ rem, Int. J. Pure. Appl. Math. 4 (2003), 269-280. [1 4] J. W. Neuberger and R. J. Renka, Numerical determination of vor­ tices in superconductors: simulation of cooling, Supercond. Sci. Technol. 1 6 (2003), 1 -4. [1 5] B. Sz-Nagy and F. Riesz, Functional Analysis, Ungar, 1 955. [1 6] L. Simon, Asymptotics for a class of non-linear evolution equations with applications to geometric problems, Ann. Math. 1 1 8 (1 983), 525-571 . [1 7] S. Sial, J. Neuberger, T. Lookman, A Saxena, Energy minimiza­ tion using Sobolev gradients: applications to phase separation and ordering. J. Comp. Physics 1 89 (2003), 88-97. [1 8] M. Struwe, Variational Methods, Ergebnisse Mathematik u. Grenzgebiete, Springer, 1 996. [1 9] J . von Neumann, Functional Operators II, Annals. Math. Stud. 22, (1 940). [20] E. Zeidler, Nonlinear Functional Analysis and its Applications Ill, Springer-Verlag, 1 985.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 3, 2005

55

1$ffij.t§j.fh1¥11=tfii§4£11.1..t§!'d 1

Weirdoes by Michael Kleber

Four answers in this puzzle are weirdoes;

3

4

5 14

16

17

19

20 24

23

7

6

13

22

their clues appear below

2

Michael Kleber and Ravi Vakil, Editors

8

9

11

10

12

15 18 21

25

26

27

28

29

30

in italics. Eight answers

32

31

are unclued; each of

33

34

36

35

these somehow colludes

37

with another to form

39

38

one of the weirdoes. Solvers will need to

43

42

40

41

44

45

discover what 's odd (and what 's not) about

46

47

the weirdoes and method of collusion to

48

49

52

51

50

53

finish the puzzle.

Triangle man

( SEE

INSTRUCTIONS )

Used to hold feathers Tear

15 Pen 16 Leave to reader 17 It comes between a lady and her protest

18 Singularity Hawking bet against

19 ( SEE INSTRUCTIONS ) 20 Herded fish? 22 Craze 24 Most coquettish 29 Emblem (or - oo , if approached right)

31 33 35 37

56

Unaided Cash recipient Man or Wright Proboscis

64

Down 38 39 40 41

Change phase on a plane

58 61

63

Across 1 7 10 13 14

57

60

59 62

56

55

54

[The solution will appear in the next issue, volume 27, no. 4.]

That woman

1 Mathematician's output

Parts Cut out, roughly Bond - an attractive one

42 It's less lovely than a tree

44 Eject 45 Mathematician's input 47 Opposite of 58D

Mathematician's output 53 ( SEE INSTRUCTIONS )

48

54 U.S. chip, in U.K. 55 Thompson or Bovary 57 What you're lousy with 59 Heads in a rows 60 Acknowledge 61 Help! 62 So far 63 ( SEE INSTRUCTIONS ) 64 Rue

THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+ Bus1ness Media, Inc.

2 3 4 5 6 7 8 9 10 11 12 15 21 23

Goal Observed

( SEE

INSTRUCTIONS )

Integrates, discretely Beast Rex Decimate religiously Sound bounce Janet or Nevada Movie shot Like wine and cheese

28 30 32 34 36 37 43 46 47 48 49

Between prime and sext Nobel city Minima Urge Lecherous look

( SEE

INSTRUCTIONS )

Humble

Nikita, for one Composure

P V --,p What some can't take

650 nanometers

50 Australia's national

Dynkin diagram of

51 Try again 52 Over-gratify 53 ( SEE INSTRUCTIONS )

Muse ( SEE INSTRUCTIONS )

SLtwo

24 Female combinatorist? 25 Some nerve 26 Gulliver's-bane.com? 27 Type of type

gem

54 Tear 56 Russian high-altitude outpost

58 Opposite of 47A

The Honors Class: Hilbert's Problems and Their Solvers by Benjamin H. Yandell NATICK, MA. A K PETERS 2002 x + 486 PAGES. US $39.00. ISBN 1 -5688 1 - 1 4 1 - 1 ; PAPERBACK U S $19.95. ISBN 1 -46881 -2 1 6-7

REVIEWED BY GERALD L. ALEXANDERSON

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: [email protected]

A report on the current state of a Mllilbert problem is always of in­

terest to mathematicians. Everyone who cares about mathematics knows of that famous list of 23 problems pre­ sented by David Hilbert at the second International Congress of Mathemati­ cians (ICM) held in Paris in 1900. In a way that list laid out an agenda for the mathematical community for the twen­ tieth century. Hilbert presented only ten of the problems at the Congress; the others appeared in reports published later. It is a tribute to Hilbert's standing in the field-he was only 38 years old at the time-and his prescience, that the list became such a motivating force in mathematics over the next 100 years. It was Hermann Weyl who wrote that, by solving one of Hilbert's problems, a mathematician "passed on to the hon­ ors class of the mathematical commu­ nity." This provided the title for the present book, which surveys the state of the problems at the time of publica­ tion, 2002. The Honors Class is a brilliant piece of work, surely destined to become a classic. There have been other surveys that attempted to inform the reader with technical knowledge of mathe­ matics about the current state of each problem. P. S. Aleksandrov edited a collection of essays on the problems that was translated from Russian into German in 1971 [ 1 ] , and in May 1974 a conference devoted to the Hilbert problems was held at Northern Illinois University, with the resulting proceed-

ings appearing in 1976 [2]. Later a par­ tial manuscript of a survey by Irving Kaplansky circulated, but it seems it was never published. A brief history of the problems by lvor Grattan-Guinness was published in 2000 [4], and in the same year Jeremy Gray's The Hilbert Challenge (5] appeared. All of these are most easily read by those with some prior knowledge of the mathematics involved. Prior to Yandell's book there has not been a detailed survey appro­ priate for students or a wide general audience. Although it would have been rewarding to see a completed work by Kaplansky, with his insights into the value of the problems and their solu­ tions, we can take comfort in the fact that Yandell's survey is truly breath­ taking in its scope and readability. Writing a book of any sort on twen­ tieth-century developments in mathe­ matics is a daunting task. That it can be done at all is amazing. To see it done so well gives this reviewer great plea­ sure. Yandell was well aware of the dif­ ficulties, and he has presented a book that is er\ioyable at several levels. In it he provides a setting for each of the problems by explaining with extraor­ dinary clarity and elegance what the problem is about. Then he gives some details on the place the problem has in mathematics and the solution (where there is one); finally, he gives bio­ graphical information about the prin­ cipal contributors. Of course, the gen­ eral reader will fmd the material in the middle part of that list the most diffi­ cult to read, even though for the most part an undergraduate majoring in mathematics should have little diffi­ culty getting the gist of it. As in so many other parts of the book, though, the au­ thor gives some charming advice: "My mathematical readers will know how to read this book. they will read it the way they read mathematics books. If they don't understand something (a state they are used to), they will keep reading in the hope that they will un­ derstand the next thing. They will skip

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 3, 2005

57

sections that don't interest them. So this is to my nonmathematical readers. If you are reading for the story, keep reading if you don't understand some­ thing. Skip a bit if you want-the bio­ graphical narrative will pick up again. Pretend you are reading Moby Dick and you've come to another section on whaling." The author spent almost ten years writing this book. I am not at all sur­ prised; it has been meticulously re­ searched. The author, with only a bachelor's degree in mathematics from Stanford University, has an impressive grasp of the mathematical background and the ability to state the problems in terms that persuade the reader that it is possible to have, at least at some level, an idea of what they are about. Because the problems cover such a wide range of mathematics, this would not be an easy task even for a research mathematician of considerable breadth of experience. There is the occasional slip that a knowledgeable reader can fix up and a student or general reader probably won't worry about. Yandell succeeds by having read just about everything on the problems that he could lay his hands on and then aug­ menting his reading with many per­ sonal interviews. The conversations he had with the major players or members of their families provide fresh new ma­ terial that probably does not appear anywhere else. Hilbert ordered his 23 problems roughly by sub-discipline, and Yandell generally follows the same plan. How­ ever, because some of the problems seemed to lie within one field in Hilbert's time but then had solutions deriving from other fields, the order in the book does not agree completely with Hilbert's. For example, the first two on Hilbert's list were on the cardi­ nal number of the continuum and on the compatibility of the arithmetical axioms, both clearly falling within the area of logic and foundations. But Yan­ dell discusses these two with the tenth problem (the determination of the solv­ ability of a Diophantine equation). The tenth is, at first glance, a number the­ ory problem, but the solution followed work of Martin Davis, Hilary Putnam, Julia Robinson, and Yuri Matiyasevich,

58

THE MATHEMATICAL INTELLIGENCER

all of whom applied methods from foundations, with the exception of some insights from number theory provided by Robinson. Yandell there­ fore places it in the section titled "The Foundation Problems." Other cate­ gories are "The Foundations of Spe­ cific Areas," "Number Theory," "Alge­ bra and Geometry: A Miscellany," and "The Analysis Problems." The book opens with a quick outline of the significance of Gottingen in mathematics in 1900 and the impor­ tance of Hilbert in establishing Gottin­ gen's preeminent position. Or, as the author puts it, "To understand some­ thing of Hilbert's problems and their solvers we must start with the 'origin of the coordinates,' as Gottingen math­ ematicians jokingly called a point on the town square from which four churches could be seen. That origin is David Hilbert and Gottingen." This 16page survey is fascinating by itself, re­ lying as it does on Constance Reid's tireless research for her biography of Hilbert. The cast of characters repre­ sents much of late nineteenth- and early twentieth-century mathematics. This was Gottingen in its heyday. We are told that the motto on the Rathaus in Gottingen reads: "Away from Got­ tingen there is no life." In this introductory material we are treated to the first of many little stories about mathematicians. Here's one. "Once when Klein had filled an entire blackboard with numbers about Ger­ man middle schools, Minkowski said, " 'Doesn't it seem to you, Herr Geheim­ rat, that there is an unusually high pro­ portion of primes among those fig­ ures?' " Later there are many stories about Carl Ludwig Siegel, who appar­ ently had an unusual sense of humor. He "would make dinner plans with someone new to Gottingen and say, 'Meet me at the restaurant near the palm trees.' " Siegel knew full well there were no palm trees in Gottingen. Siegel once "took an empty baby car­ riage up to the top of a steep hill and pushed it down, to see what people would do." But Yandell warns us that much of what we know about Siegel is from stories told about him or that he told himself. The stories "were not written down when they were told, and

those who heard them are now either old or dead, heard them third- or fourth-hand, or heard them from Siegel when he was fairly old." Are they all true? Who can tell? To extract only parts from the math­ ematical sections would produce gib­ berish. I shall concentrate here on the more historical and biographical sec­ tions, which are the strongest aspect of the book. The first section, on foundations, gives an extraordinary survey of the contributions of Georg Cantor, Kurt Godel, Emil Post, and Paul Cohen, among others. Unfortunately, with the first three of these one detects a pat­ tern of mental instability, perhaps re­ lated to their obsessive pursuit of very deep and difficult problems. About Cantor, we read of his various hospi­ talizations, and Yandell notes that "Cantor blamed the strain of his math­ ematical work and took up other in­ terests, becoming more passionate in his study of the Scriptures and inter­ ested in different spiritual paths: freemasonry, theosophy, and Rosicru­ cianism. He also became intent on proving that Bacon was the real author of Shakespeare's works . . . none of these changes helped in the long run." Of Godel: "His depressions grew . . . [he] became more afraid of food poi­ soning than ever and didn't eat. . . . The death certificate listed the cause of death as 'malnutrition and inanition' caused by 'personality disturbance.' Wang heard that Godel weighed sixty­ five pounds at the end." And of Post: "He adopted the strategy of always working on two problems at the same time to keep himself from becoming overexcited. He would work on one for two weeks and then no matter how promising the work was, would switch to the other. . . . Post suffered his final breakdown in 1954 . . . and died of a heart attack shortly after an adminis­ tration of electroshock therapy at the age of fifty-seven." John Nash did not suffer alone. Perhaps mathematicians often appear eccentric-some even cultivate this image-but Yandell is quick to point out that "if we looked at the rate of occurrence of severe mental illness in mathematicians (as in the case of Nash), I suspect that it wouldn't

appear much different from that of the general population." Most of the stories are lighter. Who would have thought that GOdel, who wore those round eyeglasses worn more recently only by the late architect Philip Johnson, and who looked, to put it bluntly, strange and rather scary, was fond of the films of Walt Disney? Snow White and the Seven Dwarfs was his favorite. And he liked American show tunes and operettas. (By contrast, his close friend Einstein, predictably, liked classical music.) There are often refer­ ences or themes that run through var­ ious sections of the book; "Snow White" is one of them. A few pages on, in the section on Paul Cohen, we read that the "question of the continuum hy­ pothesis lay like Snow White where Godel left it, in the middle years of the Second World War." And yet later the author says of Alan Turing, who reput­ edly killed himself by eating an apple poisoned with cyanide in 1954: "Like Godel, [Turing] had been fascinated by Disney's 'Snow White' and, sadly, as early as 1937 had talked of suicide in­ volving a poison apple." Ernst Straus called Godel "by far Einstein's best friend." But the last time Straus saw Einstein, in 1953, Ein­ stein said of Godel, "You know, GOdel has really gone completely crazy." So Straus said, "Well, what worse could he have done?" Einstein answered, "He voted for Eisenhower." Einstein had, of course, voted for Adlai Stevenson. Godel's wife, Adele, who had been a dancer in a Vienna nightclub before mar­ rying Godel, also liked Ike but was re­ ally an admirer of Douglas MacArthur. The story of another logician, Jean van Heijenoort, is told at some length in Anita Feferman's biography [3] , but it is also told here quite succinctly-his contributions to logic, his having been a bodyguard for Leon Trotsky, and his having been at one time Frida Kahlo's lover. Yandell also tells of van Hei­ jenoort's death: " [he] was murdered by his [former] wife . . . in Mexico City in 1986--three bullets in the head as he slept." The mathematical world is full of drama. One of the most touching sections of the book is that on Hilbert's third problem, whether the Bolyai-Gerwin

G l oba l Pra i se for AK

::Jeters,

-fhe Honors Class r 1rlbert's Froblems and Their Solvers

ltc

0 �

'

"'

"Felicitations a !'auteur pour avoir reuni dans un texte tres agreable a lire une telle

masse d'inforrnations interessantes, souvent peu connues et difficiles a trouver". Armand Borel

"Bokin er fyrirtak.

-

lAS, Prino:ton, NJ

St:l£r<'lfrreoinni r b6kinni er lyst mea nakvremni og refiagripin

eru mjt}g ftarleg og frlx'lleg".

Sigurdur Helgason - MIT, Cambridge, MA

... 1?1JM' ,;'I 1�':0 'l!l::l l1::l1:>;"1 m� ,.01,) 'lin �7 '1M"1 1!)0 K1:1 "The Honors Class" .C'l11V l'lll 'l.l1nnl.l C'!Vllt '"ll :'lltl.,::l M"1Pl 1:::11, 7:>

,Wit '1:::1,_:::1, ,!lC ::ln:> In;"!

Philip J. Davis Brown University, Providence, RI -

"I predict your book will be a great success; the historical comments are fascinating". Peter Lax - CIMS, New York, NY

·KHMra HanMCaHa 0\leHb lKHBO M �MTaeTes� c yBI1e'feHMeM. 0Ha noMolKeT MHOfMM nOHATb, "f'TO T8K08 38HATMA M8T8M8TMKOM, K8K M8T8M8TMKM lKMBYT, M KaK H8 HMX BnMAeT MX npocpeccMA." Yurl Manin - Max-Planck lnstitut fur Mathematik, Bonn, Germany

,[Yandell] liefert im ausfiihrlichen intellektuellen wie politischeu und

personlichen Portriit des grolkn so\\jetischen Mathematikers Andreij Nikolajewitsch Kolmogorov ... ein dichtes, gelungeues Zeit· und Denkbild, das seinem Buch einen angenehm unpomposen, iiberzeugenden SchluBakkord beschcrt". Dietmar Dath

-

FAZ, Germany

tJ1� t:1 �;Yii�?tr!�c�=+i!t*c:'fJJ��r&�rp�MlR����. tffiJ%���$�1iWW�IJ�. �M�Jt.����tcr:�zJfH�tcr:�to/1 Yum-Tong Siu - Harvard University, Cambridge, MA

In

To P l a ce O r d e rs

All other countries

Europe

Transatlantic Publishers Group www.transatlanticpublishcrs.com [email protected]

A K Peters, Ltd.

www.ak.peters.com

service@_,ak.peters .com Tel : + 1 -78 1 -4 16-2888

Tel : +44- 1 202-665432

Fax: + 1 -78 1 -4 1 6-2889

Fax: +44- 1 202-6662 1 9

·•saunders :Mac Lan� is an immense presence in modern mathematics.

One of the last Americans to be educated in G1ittingen prior to the Nazi era, Mac Lane was ideally and propitiously situated to become a leader in modem algebra and modem algebraic topology. His influ­

ence on our subject has been powerful and widespread. This chronicle

tells

of an influential and important life." Steven G. Krantz,

Washington University in St. Louts

In Memoriam

1909 . 2005

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 3, 2005

59

theorem on equidecomposability can be extended to three-dimensional fig­ ures. It was solved shortly after the Paris Congress by Max Dehn. The problem was, perhaps, one of the "eas­ iest" posed by Hilbert, but that does not detract from Dehn's overall standing as a mathematician, as Yandell points out. Andre Weil, who was not given to overly generous appraisals of his con­ temporaries, said of Dehn, "I have met two men in my life who make me think of Socrates: Max Dehn and Brice Parain [a French philosopher] . . . . [Dehn] left behind a body of work of very high quality." The story of Dehn is a fascinating one. When it became nec­ essary for him and his wife to leave Germany in 1940 they took roughly the same route that Godel and his wife had taken, across Russia and Manchuria via the Trans-Siberian Railway, thence to Japan, and on to San Francisco. But where Godel proceeded on to Princeton, Dehn was able to find a job only at the University of Idaho in Pocatello-now Idaho State University. (Yandell does not make it clear that this was not even the principal campus, which is in Moscow, Idaho, but the Southern Branch of the University, located in a fairly remote and small town in south­ eastern Idaho.) Dehn, a distinguished German professor who had solved one of Hilbert's famous problems, must have appeared rather an exotic crea­ ture in Pocatello, known then as now mainly as a section point on the Union Pacific Railroad. Dehn went on to teach at the Illinois Institute of Technology, St. John's Col­ lege in Annapolis (where he had to teach Homer's Odyssey in one week, in English), and eventually to Black Mountain College in North Carolina, where there were few students, almost none interested in mathematics. Still, he lived almost all of the rest of his life at Black Mountain (it eventually went out of business), and there he and his wife spent happy years in part because of the distinction of others on the fac­ ulty: poets Robert Greeley, Robert Dun­ can, Denise Levertov; painters Willem de Kooning, Franz Kline, Robert Moth­ erwell, Josef Albers, Robert Rauschen­ berg; composer John Cage; dancer Merce Cunningham; and filmmaker

60

THE MATHEMATICAL INTELLIGENCER

Arthur Penn. One of Dehn's friends was Buckminster Fuller, who built an early geodesic dome on the campus, but "it collapsed and was renamed the 'supine dome.' " Dehn is buried in the woods on what was once the campus.

H i l bert ' s p roblems .

.

came 1 n vanous forms . For this reason it is not clear j ust how many have been "solved . " So, although the mathematical life there was not that of Gottingen or Berlin, for Dehn and his wife, Black Mountain turned out to be a good in­ troduction to some of the best of Amer­ ican culture in the postwar years. Before moving on to the next set of problems I should explain that Hilbert's problems came in various forms. For this reason it is not clear just how many have been "solved." Yandell says it best: Hilbert's problems have the charac­ teristics of any good founding doc­ ument. Each one is a short essay on its subject, not overly specific, and yet Hilbert makes his intent re­ markably clear. He leaves room for change and adjustment. Hilbert's goal was to foster the pursuit of mathematics. He helps us fmd the vital center of each problem by us­ ing italics-often in more than one place. So the question of how many of the problems have been solved becomes part of a more ambiguous question: How many have been posed? Sometimes Hilbert suggests a plan for investigation of an area. Or he may be pursuing an intuitive feeling, and the problem can be paraphrased: Why don't you look in that direction? However, most of his problems have been identified-in something approaching a consen­ sus-with a single, clearly stated

mathematical question. That con­ sensus, when it exists, is what I have taken to be the problem. The fourth problem, on alternative geometries resulting from weakened axioms, was rather vaguely presented, and it wasn't entirely clear what con­ stituted a solution. But you can read of the contributions of Georg Hamel, Her­ bert Busemann, and A. V. Pogorelov. The fifth problem, on Lie groups, is technical and difficult to describe, and it was solved in two stages, the first by Andrew M. Gleason, the second by Deane Montgomery and Leo Zippin. The biographical material on all three is rich in detail and tells a fine story. Yandell cannot refrain from looking at patterns. He points out how much good mathematics seems to be done while mathematicians are out on long walks: Hilbert, Hurwitz, and Minkowski, Ein­ stein and GOdel, Ulam and Erdos, and Montgomery and Zippin. Also in the context of the fifth problem, he ob­ serves that Hilary Putnam made major contributions to the tenth problem without having a degree in mathemat­ ics, and Gleason at Harvard became the Hollis Professor of Mathematicks and Natural Philosophy (a chair en­ dowed in 1 727) yet "never received a single graduate credit toward a Ph.D., much less the degree." Of course, Glea­ son shared the distinction of not hav­ ing a doctorate with another eminent Harvard mathematics professor, Gar­ rett Birkhoff. Being on the faculty with­ out a Ph.D. was possible for Junior Fel­ lows at Harvard in those days. We also learn that when Gleason took the Put­ nam Competition at Yale, he was un­ happy because he had been able to do only thirteen of the fifteen problems. "It was probably the first time he could­ n't solve all the problems on a mathe­ matical test." Nevertheless, he placed in the top five contestants that year and was awarded the Putnam Fellowship to Harvard. In his sixth problem Hilbert pro­ posed the axiomatization of physics, which has not been done, at least to everyone's satisfaction. Yandell points out that if one of the recent theories in physics is right, it should be axiomati­ zable, in which case Hilbert's problem

could have a "clean" solution. But then he observes that "the relationship be­ tween mathematics and physics­ clearly a case of opposites attracting­ continues to be tempestuous." He suggests that "maybe string theory will go to live with mathematics-at the present time this 'physical' theory has no connection to experimental data." The section on number theory is one of the richest, with six problems (though one of them went to the sec­ tion on foundations, as we saw), this is perhaps not surprising, since Weyl claimed that Hilbert's best work was in number theory. In the introductory re­ marks Yandell describes what he calls "near misses," and gives as examples both L. G. Shnirelman's theorem that every even integer is expressible as the sum of no more than 300,000 primes (not quite Goldbach's conjecture that every such number is the sum of 2 primes) and Alan Baker's work that implies that if the difference between one perfect cube and twice another is 1, then neither can be larger than 1.5 X 10 1 3 17. Hilbert's seventh problem asked for a proof that 2Y2 and some similarly formed numbers are transcendental. This was proved independently in 1934 by A. 0. Gelfand and Theodor Schnei­ der, but with a contribution along the way by George P6lya and many con­ tributions by Siegel. Yandell points out that in 1920 Hilbert said in a lecture that "he thought no one [present] would live to see 2Y2 proved to be tran­ scendental . . . [but he] said he thought he himself might live to see Riemann's hypothesis proven . . . and that the youngest members of the audience might see Fermat's last theorem proven. He had the order of solution the wrong way around." The author is always alert to the pos­ sibility of making something clearer with a clever choice of words. He quotes Gleason as saying, " [The positive solu­ tion of the fifth problem] shows that you cannot have a little bit of orderli­ ness without a lot of orderliness. " In talking about Zermelo's work on sets, Yandell says, "it forbids infinite nests of brackets with nothing in them, like parallel mirrors in a deserted barber shop."

Yandell moves quickly through the history of the eighth problem, the Rie­ mann hypothesis. With several recent popular books devoted exclusively to this problem, anything here becomes somewhat redundant. Still, it gives Yandell a chance to talk about Hardy and Ramanujan, in particular Hardy's receiving Ramanujan's letter from In­ dia explaining some of the work he had done, essentially in a vacuum. Yandell quotes Thomas Gray's "Elegy Written in a Country Churchyard": "Some mute inglorious Milton here may rest." He then quotes H. L. Mencken: "There are no mute, inglorious Miltons, save in the hallucinations of poets. The one sound test of a Milton is that he function as a Milton." Yandell goes on: "Hardy had just received a letter from a marginally educated Indian, whose English was poor, who was in a mathematical sense functioning as a Milton." Further along we read that "Siegel used mainly alge­ bra in proving the Thue-Siegel theorem and for some time couldn't convince ei­ ther Schur or Landau he had done it. Schneider solved . . . the seventh prob­ lem using fairly elementary methods, apparently without being aware he was working on it." How many others have been out there working, but with their accomplishments unrecognized?

Raman ujan was no m ute i ng lorious M i lton . Throughout the first half of the twentieth century the contrast be­ tween classical methods and the more modem trends that took hold later is obvious. Yandell points out that, in a rather provocative letter to Weil, one of the founding members of Bourbaki, Siegel wrote in 1959: "It is entirely clear to me what circumstances have led to the inexorable decline of mathematics from a very high level, within about 100 years, to its present nadir. The evil be­ gan with the ideas of Riemann, Dedekind and Cantor, through which the well-grounded spirit of Euler, La­ grange and Gauss was slowly eroded. Next the textbooks in the style of Hasse, Schreier, and van der Waerden,

had further a detrimental effect upon the next generation of scholars. And fi­ nally the work of Bourbaki here pro­ vided the last fatal shove." Weil's re­ sponse is not recorded. The modernists were clearly not deterred by Siegel's conservative views. Yandell raises the question of how Siegel would have re­ sponded to Wiles's proof of the Fermat problem, where he used modem meth­ ods and structures extensively. There are so many great stories in this book-How much can one pass along in a review? In the discussions of the ninth, eleventh, and twelfth prob­ lems on class field theory and related problems, we read of a party given by Teiji Takagi at the 1932 ICM in ZUrich, with guests Chevalley, Iyanaga, Hasse, Nagano, Noether, Taussky, and van der Waerden, among others. Yandell points out that Noether so admired Takagi's work that she had started learning Japanese in order to converse with him. At the party "Noether enjoyed her­ self, talking the most, . . . and then, as the party ended, requesting that the Japanese teach her how to bow . . . 'She asked, bending forward . . . "Like this, or more?" Her funny look remained long in the memories of the young Japanese mathematicians.' " She was the senior mathematician at the party. We can only wish we could have been there. Building on Takagi's development of class field theory, Emil Artin solved the ninth problem, on a law of reci­ procity; Helmut Hasse, the eleventh problem, on quadratic forms. These sections are filled with a warm account of Artin and his work, along with a guarded appraisal of Hasse, who was later criticized for staying in Germany during World War II and cooperating with the Nazi regime as director of the Institute in Gottingen. Yandell points out that Hasse's father was descended, on his mother's side, from the Jewish composer, Felix Mendelssohn. How did that get past the Nazis? Sanford Se­ gal in his excellent Mathematicians under the Nazis [6] explains that Hasse had a great-great grandmother who was Jewish but baptized, so he was 1/16 Jewish. To be an academic in Germany at that time it was only nec­ essary to have no Jewish grandparents.

© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number 3, 2005

61

But that condition was not sufficient for him to be a member of the Nazi party, which he was. For that "honor" one could have no Jewish ancestors who were alive after 1800. This should have excluded Hasse from the party, but a series of bureaucratic blunders made it possible for him to remain a member until the war was over. Hasse came to the United States af­ ter World War II and taught at several universities. Yandell says, "Hasse was an excellent mathematician. But com­ pare him to Hilbert in the same time and place, who saw clearly even from senility's downward slope. A political order that destroys mathematics and makes mathematicians flee for their lives is almost certain to be evil. To our lrnowledge Hasse never saw this." Se­ gal put it this way: "[Hasse held to) some­ what rigid principles, patriotism, and na­ tionalism-Manfred Knebuschdescribes Hasse in the postwar years as having 'all the strengths and wealrnesses of a Prussian officer'-[that] caused him to view Hitler as a national hero and to ap­ ply for membership in the Nazi party in 1937." Segal goes on to say that Hasse thought that "slavery in America had been a good institution for blacks." Hasse may have been a member of "The Honors Class," but that did not make him an admirable human being. In the discussion of the fourteenth and seventeenth problems in algebraic geometry we have an essay on Masayoshi Nagata. The cast of charac­ ters in the chapter on the fifteenth problem-Schubert's Variety Show­ includes Solomon Lefschetz, van der Waerden, Edward Witten, all the way up to Maxim Kontsevich. We learn that Hermann Schubert "calculated 666,841,048 quadric surfaces tangent to nine given quadric surfaces and 5,819,539,783,680 'twisted cubic space curves tangent to 12 given quadric sur­ faces.' " And he did it without comput­ ers! But computers have since con­ firmed Schubert's calculations. The eighteenth problem also in­ volves counting. Hilbert asked whether the number of crystallographic groups in four dimensions is finite, which was only part of the problem and a question fairly quickly answered in 191 1-12 by Ludwig Bieberbach. Yandell treats the

62

THE MATHEMATICAL INTELLIGENCER

life and career of Bieberbach in some detail. A talented mathematician but a complex man, Bieberbach was a stu­ dent of Felix Klein and Paul Koebe at Gottingen. He was at Konigsberg and Basel before going to Berlin to replace Constantin Caratheodory in 192 1 (after the post had been turned down by Brouwer, Herglotz, Weyl, and Heeke!). Louis de Branges's proof of the famous Bieberbach conjecture on schlicht functions was one of the great achieve­ ments in mathematics in the 1980s. But able as he was as a mathematician, Bieberbach acted very badly during World War II. P6lya claimed that while some mathematicians in Germany used the system for personal advance­ ment-Wilhelm Blaschke, for exam­ ple-Bieberbach was a true believer in

Yandell g ives n i ne p roblems not on H i l bert ' s l ist that cou ld have been on a l i st of Poi n care ' s . the Nazi cause. Yandell is careful to give a balanced view of Bieberbach, telling of his many accomplishments as a mathematician, an author of text­ books, and a teacher, mainly in his early years. But Yandell does not de­ fend his actions after 1933. By that time Bieberbach had joined the S.A. (the storm troopers), and from that point on he became more combative, eventually establishing the infamous journal Deutsche Mathematik, "which mingled race politics with 'good German math­ ematics' . . . According to Bieberbach there were two types of mathemati­ cians, the S type, short for Strahltypus (Jewish, French, etc.-bad) and the J type (German-possessing Anschau­ ung-good). The J type had three sub­ divisions J1, J2, J3. So the man who had helped [to) clarify the classification of crystals into types [the eighteenth problem], making his way in the world, extended classification into human life, an obscene and sloppy tiling."

The last section is on the problems from analysis (with a more admirable cast of characters), not taken in the natural order but starting with a long section on the twenty-second problem, on automorphic functions, with a care­ ful exposition of the work of Paul Koebe and Henri Poincare, who is of­ ten cited, along with Hilbert, as one of the last mathematicians who had a grasp of the whole field of mathemat­ ics. Yandell gives us an additional list of nine problems not on Hilbert's list­ not surprisingly somewhat more topo­ logical-that could have been on a list of Poincare's, had he outlined an agenda for the century. The nineteenth, twentieth, twenty­ first, and twenty-third problems are discussed in two rather short sections. They are all concerned with differen­ tial equations and the calculus of vari­ ations. The twenty-third problem is not a problem at all but a program for fu­ ture research in the latter. Hilbert es­ sentially called on the mathematical community to look again at the calcu­ lus of variations-in Hilbert's words: "[it] does not receive the general ap­ preciation which in my opinion, it is due." Yandell describes briefly the ex­ tensive work that this program in­ spired. There is interesting biographi­ cal material on Josip Plemelj, Andrei Bolibruch, S. N. Bernstein, and I. G. Petrovsky. In this section we learn that it is not always a straight path to a so­ lution to one of these problems. The twenty-first problem, on the existence of linear differential equations having a prescribed monodromic group, con­ cerns those equations "that have points where the coefficients . . . have singu­ larities . . . [Solutions) to this kind of problem, as with many things in com­ plex analysis, are not single-valued. One circles a singularity (like a horse in a hippodrome), jumping from one lo­ cal value of the solution to another. The monodromy group captures this." Hilbert produced a partial solution in 1905, and Plemelj in 1908 gave a sim­ pler proof of a more general result. In 1989 Bolibruch found a counterexample to Plemelj's work, not just a gap in the proof. The result was false. Something similar happened in the solution of the sixteenth problem.

The most moving essay in this sec­ tion is about A. N. Kolmogorov, who solved, along with V. I. Arnol'd, the thir­ teenth problem on the general equation of seventh degree, and who also made contributions to the unfinished sixth problem on axiomatizing physics. This biographical essay is warm and touch­ ing, even lyrical. We learn that Kol­ mogorov's mother died in childbirth and that he was brought up by an aunt, first on an estate in the country and subsequently in Moscow. The descrip­ tion of his childhood calls to mind beau­ tiful and poignant scenes described by Tolstoy and Chekhov. V. M. Tikhomirov, editor of Kolmogorov's collected pa­ pers, wrote of Kolmogorov's childhood home . . . "the grand old house with a second floor, constructed, most prob­ ably, at the beginning of the 19th cen­ tury . . . with four massive columns at the main entrance. . . . Around the house . . . were courtyards-front and back, flower gardens, a kitchen garden, a garden (so big that it could be called a park), barns, a cattle yard, a stable, a bathhouse-in a word, all those things infinitely remote from our present-day lives, and of which we now know only from books." The Kolmogorov family followed Tolstoy's moral philosophy, so it is not surprising that Kolmogorov was named for Tolstoy's estimable character, Prince Andrei Nikolaevich Bolkonski. In 1929 Kolmogorov and P. S. Alek­ sandrov, later to become a world­ renowned topologist, along with an­ other friend, rented a boat for a trip down the Volga. They took with them one book, Homer's Odyssey. Kolmogorov's description of floating down this great river reads in part: "During the first days of our journey we often swam at night, in the white sum­ mer nights, gliding along past the over­ grown osier beds on the bank, the air filled with bird song, made a lasting im­ pression on us. We wished that this could go on forever." (Those interested in American literature will be reminded of Mark Twain's descriptions of Huck and Jim floating down the Mississippi.) Twenty-one days and 1300 kilometers later, the trip on the river ended. The three turned in their small boat and continued by steamer to Baku and

Yerevan before returning to Moscow. Much later Kolmogorov wrote, "Pavel Sergeevich Aleksandrov and I date our friendship . . . which lasted 53 years [from the 1929 trip] . . . . For me these fifty-three years of close and indissolu­ ble friendship were the basic reason why all my life turned out to be filled, on the whole, with happiness, and the basis of my good fortune was the un­ ceasing thoughtfulness of Aleksandrov." Kolmogorov, though his name is usu­ ally associated with probability, worked in other areas as well, producing impor­ tant results in geometry, algebraic topol­ ogy, approximation theory, ergodic the­ ory, and, during World War II, ballistics

Kol mogorov was named for Tolstoy ' s esti mable character, Pri nce And rei N i kolaevich Bol konski . and turbulence. It was in the 1950s that Kolmogorov did the work that resulted in the solution of the thirteenth problem, with Arnol'd, who also contributed to the solution of the sixteenth problem on the topology of algebraic curves and sur­ faces. Arnol'd, who always insisted on clarity in writing, shared some views with Siegel, mentioned earlier. He re­ ferred to "criminal bourbakizers." In the 1960s Kolmogorov wrote a se­ ries on the metrical structure of Rus­ sian poetry, not a frivolous enterprise. A. A. Markov earlier had claimed "that the string of vowels and consonants in [Pushkin's] Eugene Onegin was as pure an example of a Markov chain as he could find." At this point we should note that many of the pictures in this book are a treat for the reader, often fresh and un­ common, taken as they are from vari­ ous private collections. Arnol'd is shown, for example, standing on the Golden Gate Bridge! The story ex­ plaining it is worth checking out.

Yandell, in summary, quotes from Hermann Weyl in his mid-century re­ view of progress in mathematics [7]: "David Hilbert . . . formulated twenty­ three unsolved problems which he ex­ pected to play an important role in the development of mathematics during the next era. How much better he predicted the future of mathematics than any politician foresaw the gifts of war and terror that the new century was about to lavish upon mankind! We mathemati­ cians have often measured our progress by checking which of Hilbert's questions had been settled in the meantime." Of course, in the end, the natural question is how many of the twenty­ three problems have been solved. Yan­ dell summarizes: "Sixteen problems have been solved in a discrete way. The core of the question Hilbert asked re­ ceived an answer that is complete with respect to the original question (within reason), mathematically rigorous, and unlikely to be dramatically improved upon in the future. These problems are 1, 2, 3, 4, 5, 7, 9, 10, 1 1, 13, 14, 15, 17, 18, 21, and 22. Four problems-12, 19, 20, and 23-are what amount to pro­ grams for research and are somewhat vague in their statement. These have prompted a great deal of successful re­ search, and I think it is time to count them as 'solved,' as problems of Hilbert's. Their concerns can be embodied in new problems written in new language. Three problems-6, 8, and 1 6--have not been solved." Not bad. Here I have given only a quick overview of some of the fascinating, amusing, and evocative accounts in this book There are plenty of other sections that are equally interesting, mathematically stimulating, and his­ torically illuminating. This is a splen­ did piece of work and an essential book for anyone interested in the vast panorama of twentieth-century mathe­ matics. It should be in every mathe­ matician's library. Further, we should recommend it to our friends or our stu­ dents who will be inspired by this ele­ gant book about the accomplishments, not of the ancients, but of twentieth­ century people, many of whom lived during our lifetimes and contributed richly to our mathematical community and to society as a whole.

© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 27, Number 3, 2005

63

[2] Felix Browder, ed. , Mathematical Develop­

While Ben had interviewed many peo­

25, 2004,

ments Arising from Hilbert Problems. Pro­

ple, including the grandchildren of Max

Bel\iamin Yandell, only 53 years old,

ceedings of Symposia in Pure Mathematics,

Dehn, I was able to supply some more

died of a heart attack in Pasadena, Cal­

Vol. 28. Providence, Rl: American Mathe­

ifornia. In 1993 he had been diagnosed

matical Society, 1 976.

Epilogue

On the morning of August

contacts, including Peter Lax, who pro­

vided valuable information about recent developments. Finally, after a labor of

as having multiple sclerosis, but this

[3] Anita Burdman Feferrnan, Politics, Logic,

disease did not appear to be related to

and Love: The Life of Jean van Heijenoort.

love of nine years

his death. He is survived by his wife,

Wellesley, MA: A K Peters, 1 993.

formed) the final manuscript emerged.

Janet Nippell, and a daughter, Kate Louise, born in 1988. In the years since the appearance of

The Honors

(if I am correctly in­

[4] lvor Grattan-Guinness, A Sideways Look at

Ben's intellectual curiosity and his

Hilbert's Twenty-three Problems of 1 900.

ability to communicate his insights in

Notices of the American Mathematical So­

words that were of interest to a broad

ciety 47:7 (2000), 752-757.

audience including experts and ama­

Class he had been working on two writing projects: (1) Chasing a Wave, a story of the discovery and defi­

[5] Jeremy J. Gray, The Hilbert Challenge. Ox­

nition of solitons that was to be a his­

[6] Sanford L. Segal, Mathematicians under the

torical-biographical account, similar to

Nazis. Princeton, NJ: Princeton University

is hard to accept. The following text,

Press, 2003.

found in his papers, explains better

that in The Honors

Class, and (2) a short

biography of John von Neumann. Yandell's undergraduate years were spent at Occidental College and Stan­

ford: Oxford University Press, 2000.

courses. Instead of pursuing a doctoral

tainly his future readers with a loss that

[7] Hermann Weyl, A Half-Century of Mathe­

than I could Ben's motivation, born out

matics, American Mathematical Monthly 58

of an awareness of his talents and a

(1 95 1 ) , 523-553.

modest intention to use them in the best way he knew. We all are benefit­

ford University, to which he trans­ ferred so he could take some graduate

teurs is impressive. His unexpected death left many of his friends and cer­

ing from his decision.

Department of Mathematics & Computer Science

degree in mathematics, he spent the

Santa Clara University

One of the jokes about "pure" math­

years after graduating from Stanford

Santa Clara, CA 95053-0290 USA

ematicians is that they are always

writing poetry and working as a televi­

e-mail: [email protected]

concerned with proving a given

later in Los Angeles. During the latter

Reminiscences

be solved and to what extent the so­

part of his career in television repair he

The following are recollections of Yan­

lutions are unique. Perhaps they will

was sitting in on physics classes at Cal­

dell's life and his book from some of

supply a general outline of how such

tech. His first book was coauthored

the people who knew him:

a solution might be accomplished.

Mostly on Foot: A Year in L.A. (Floating Island, 1989).

From Klaus Peters, Publisher,

interested parties. I was beginning

AK Peters, Ltd.

to think that maybe I was such a

Hilbert (Springer, 1970), Yandell wrote

Several years ago I answered a call in

pure mathematician that I would be

in some autobiographical notes: "I real­

the office and the caller ordered a num­

satisfied with the determination that

ized how well-written it was. I didn't

ber of high-level research monographs

yes, it was possible that I could be

know, at the time, that Freeman Dyson

on various mathematical subjects. After

a mathematician. I began to lose in­

had written, of the book, 'beyond this it

I secured the order, I asked the caller:

terest in working out the details of

"Who are you? I cannot imagine anyone

the actual problem. The attitude of

class of equations can in principle

sion repairman, first in Palo Alto and

Any actual solution is left to other

with his wife:

Inspired by reading Constance Reid's

is a poem in praise of mathematics.' . . .

Hilbert

in this era of specialization, who would

those around me was that I had

I . . . was looking up into the sky. It was

be interested and able to read these

something like a duty to become a

clear and cold and I could see the stars.

books profitably."-"! am writing a book

mathematician. But the graduate

There was a light dry wind, and I was

on the Hilbert Problems and the people

students I knew also worked to

aware that one could aspire to writing

who solved them."-"Who

is your pub­

counteract that. I felt, whether I was

One night when I had just read

lisher?" I responded, bracing myself for

right or not, that I had a little more

I might bring something to trying to un­

some disappointing news.-"You are,"

raw ability than they did. I couldn't

derstand the mathematical picture of

he answered; and when I declared that

really know. The only way to deter­

something like . . .

Hilbert. . . . I thought

It was then that he

I knew nothing of the plan, he answered,

mine that was for all of us to spend

started working on this remarkable

in what I later found to be characteris­

the next

tic matter-of-factness, "That's because I

cians. But I didn't feel there was any

Hilbert's problems." book, The

Honors Class, an inspiring

legacy of this all-too-short career. REFERENCES

[ 1 ] P. S. Aleksandrov, ed. , Hilbertschen Prob­

years being mathemati­

have not sent it to you yet, but your

gross discrepancy. My adviser and

name is associated with most of the

others thought I could get into Har­

books that I have used in my research."

vard

or

Princeton

for

graduate

Ben then graciously agreed to send

school. I, too, felt I would have done

me his current draft. I read it enthusias­

well there. However, when I was

Hannelore Bernhardt. Leipzig: Akademische

tically, and we both knew that there

still at Occidental, Benedict Freed­

Verlag, 1 97 1 .

were many open areas to be treated.

man had arranged for me to meet

leme. Translated from Russian to German by

64

20

THE MATHEMATICAL INTELLIGENCER

his son, Michael. In the short term

very hard interviewing different experts

original result was conditional . . . they

this had helped excite me about

in order to be accurate. Despite his hand­

had to assume that there existed arbi­

mathematics, but in the long term it

icaps he was lively, hard-working, and

trarily long arithmetic sequences of

supplied me with the suspicion that

full of ideas for the future. He had iden­

prime numbers . . . (and] even as of this

I wouldn't be missed if I didn't be­

tified solitons as the topic of his next

writing [that] has not been proved." In

come a mathematician. Michael was

book This alone showed great discern­

1959 it required an additional contribu­

just a graduate student when I met

ment, since solitons were one of the

tion by Julia Robinson to obtain the re­

him,

about him

great discoveries of the second part of

sult without this assumption. However,

seemed clear: He had a powerful

the twentieth century with wide appli­

today, thanks to the very recent sur­

mind and a powerful desire to do

cations and deep mathematical theory. I

prising discovery by Ben Green and Ter­

mathematics. When allowed insight

strongly encouraged him and gave him

but two things

to the workings of a great mind, as

what advice I could. I am sure he would

ence Tao, we know that such arbitrarily long arithmetic progressions of primes

in the Cohen book, one comes away

have produced an excellent book which

do exist, so that the original proof is no

with a sense of ease, power, grace,

would have attracted a wide readership.

longer "conditional." Were Ben Yandell

is sad that fate prevented the realisa­

still alive, he really would have ef\ioyed

and intensity. Michael seemed to

It

have the mathematical energy to

tion of this project. Mathematics needs

achieve something like that. At the

people like Ben who can convey the in­

time, I had no way to tell that he was­

tellectual excitement of our subject to

From Peter Lax, Professor Emeritus,

n't just a very good graduate student.

the general public. I hope others will fol­

Courant Institute of Mathematical

low in his footsteps.

Sciences, New York University

From Martin D. Davis, Professor

tremely pleasant. We had a long tele­

It later became clear that I had, in fact, met someone in that first rank. But in

1971, when I met him, Michael

this new footnote to the story.

My contacts with Ben Yandell were ex­

merely supplied me subliminally, and

Emeritus, Courant Institute of

phone conversation when his book on

then after the fact, with a feeling of

Mathematical Sciences, New York

the solvers of the Hilbert problems was

assurance that I could do what I

University and Visiting Scholar,

in the planning stage. We discussed

University of California, Berkeley

bringing in Poincare, who had strong

wanted with a clean conscience. I began to focus on the romantic

Although I never met Ben Yandell in per­

views of the shape of things to come in

prospect of becoming a poet. Such a

son, we had an extensive interaction by

mathematics, and perhaps goaded by

choice would validate my indiscrimi­

e-mail and telephone over a five-year pe­

Hilbert's example, described it in a re­

nate curiosity. In the wake of the '60s

riod, and his untimely death leaves me

port to the

with a very real sense of loss. When he

of Mathematicians. We agreed that hav­

such a choice seemed to have special

integrity. I was sure I could make a

contacted me in April

living somehow. Poetry and writing in

proposal to write about the Hilbert prob­

1997 about his

1908 International Congress

ing a foil for Hilbert would enhance the book We met in person at Cal Tech; I was

general were what I thought I might

lems and their solvers, I was struck by

have an abiding interest in. This has

the sheer audacity of his project. With

surprised

proved to be the case. I am interested

only an undergraduate's knowledge of

bound, which in no way diminished his

in working out the details here. I had

mathematics, he intended to explain to

zest for life. I was shocked and sad­

little, if any, concrete evidence of tal­

a general audience the

full sweep of the

dened to learn that he is gone, in the

ent, but I felt I could learn. I felt I had

mathematical ideas needed to make

the energy necessary to concentrate

sense of the Hilbert problems. With my

on the essential problems, and re­

own limited expertise, I could only be of

gardless of the quality of the results,

assistance with the first four chapters.

I have proven to have that energy.

As the project developed I became more

Writing has kept my attention. I am

and more impressed by his pleasant, easy-going style and by his ability to con­

happy when I write.

verge to sufficient understanding of the mathematical ideas.

From Sir Michael Atiyah,

It was fun for me to read his account

University of Edinburgh

I met Ben Yandell on the occasion of the

of the lives and interactions of the four

Berkeley panel discussion of The

people involved with the negative so­

ors Class,

Hon­

his account of the Hilbert

lution of the tenth problem. I am par­

problems. I had very much ef\ioyed read­

ticularly amused by the realization that

ing the book, which was designed for a

in one small detail, his account would

general audience, and so I was keen to

be

meet the author. I was surprised to find

different today.

Concerning the

proof by Hilary Putnam and me, that

to

find

him

wheelchair

midst of his labors.

Science in the Looking Glass: What Do Scientists Really Know�

by E. Brian Davies

NEW YORK, OXFORD U NIVERSITY PRESS 2003. $US 29.95 (hardcover) Pp. x + 295

REVIEWED BY JAMES ROBERT BROWN

B

rian Davies takes a shotgun ap­ proach in

Glass.

Science in the Looking

There are rambling discussions

someone who was not a professional

every recursively enumerable set of

of a very wide range of topics: Psy­

mathematician, but who had a deep feel­

natural numbers has an exponential

chological work on perception is de­

ing for the subject and who had worked

Diophantine definition, he wrote: "The

scribed, an argument against Platon-

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 3, 2005

65

ism is presented, mind-body dualism is

be easily expressed as one about the

not descriptions of independently ex­

rejected, there are speculations on

existence of (timeless) triangles ori­

isting entities, as Platonism holds. We

thinking machines, intrinsically hard

ented at angle (} for every (} from

make them the way we make chairs

0° to

problems, observation in quantum me­

360°. There is an interesting question

and cities and constitutions and games.

chanics; evolutionary biology is upheld

here about the relation between the

But once made, we can make discov­

and reductionism is rejected. That's

idealized physical examples in mathe­

eries about their consequences; for

just a sample. Needless to say, nothing

maticians' thought experiments and

instance, we discovered that three­

is pursued in depth. I can't hope to dis­

the mathematical entities themselves.

legged chairs don't wobble and that a

cuss it all, but I will take up a few of

How does the one give us knowledge

given sequence of moves leads to

Davies's points.

of the other? But the fact that we can

checkmate. The view has all sorts of at­

and do reason this way does not un­

tractions for an anti-Platonist such as

dermine Platonism.

Davies. But it has drawbacks. Mathe­

The arguments against Platonism are quite unconvincing. Davies (Pro­ fessor of mathematics, King's College,

Davies very briefly mentions (but

matical practice is concerned with a

London) cites GOdel and Penrose as

neither endorses nor rejects) an argu­

great deal more than merely establish­

representative speak of

Platonists who

seeing mathematical

often

objects.

He then reminds us of what he reported

earlier in his book about sense percep­

ment that many philosophers accept. Readers of

The Intelligencer may like

ing logical truths of the form: If axioms

then theorem. We worry about the truth

to know what it is. It begins with what

of the axioms themselves. In some

is known as the causal theory of knowl­

fields this is more evident than in oth­

tion, in particular that what we see is

edge:

To know about X one must

ers. Set theorists, for instance, are con­

not the way things actually are. This is

causally interact with X. For instance,

stantly proposing new axioms con­

hardly damaging to Platonism. GOdel

I know about the coffee cup on my

cerning large cardinals and arguing

repeatedly stressed the fallibility of

table because

coming

their respective merits. They seem to

mathematics in general and of mathe­

from it to me. If I didn't interact with

be searching for axioms that are true.

photons

are

matical perceptions or intuitions in par­

the cup in some way, then I simply

This flies in the face of Davies's claim,

ticular. "I don't see any reason why we

wouldn't know about it at all. The sec­

since he would have it that axioms are

should have any less confidence in this

ond premise of the argument is a stan­

not true-they are merely interesting

kind of perception, i.e., in mathemati­

dard feature of Platonism: Mathemati­

or useful or fun. Davies's view also

cal intuition, than in sense perception,

cal entities are perfectly real, but they

clashes with Godel's theorem. The fa­

which induces us to build up physical

do not exist inside space and time.

mous incompleteness theorem says

theories and to expect that future sense

Third, is the assumption that there

that no set of axioms can capture all

perceptions will agree with them and,

can be no causal interaction between

the truths of arithmetic. It's hard to

moreover, to believe that a question not

things inside and outside of space and

avoid the conclusion that the unprov­

decidable now has meaning and may be

time. (Religious people might claim

able sentence is actually true, which

decided in the future. The set-theoreti­

this can happen, but, of course, they

means that truth and theoremhood

cal paradoxes are hardly more trouble­

also say it is a miracle when it hap­

(i.e., being derived from axioms) are

some for mathematics than deceptions

pens.)

It follows from this that we can­

distinct things. If this is so, then the

of the senses are for physics." 1 The fact

not know mathematical entities, if they

Peano axioms do not create the truths

that we don't directly see things as they

are as Platonism claims. Since we do

of arithmetic, but rather partially de­

are in the physical world is not a rea­

seem to have mathematical knowl­

scribe something that exists indepen­

son to say the physical world does not

edge, it follows that mathematics must

dently. Davies mentions Godel's theo­

exist or that it is unknowable. Exactly

be about some other kind of thing.

the same can be said in defence of the Platonic realm.

rem in connection with minds and

This is the best argument against

machines, but not in connection with

Platonism. Not all philosophers accept

Platonism, where it seems to refute the

In another argument he asks us to

it. I don't, but I won't take the time to

kind of postulationism Davies advo­

consider a small triangle inside a larger

say why, except to say that I think the

cates. I readily admit that the argument

one. The problem to consider is this:

first premise is false, in spite of seem­

for Platonism from Godel's theorem is

Can the smaller be freely rotated inside

ing quite plausible. Nevertheless, it's a

contentious,

the larger without hitting the edges? In

much better argument than the kinds

Davies doesn't mention it.

thinking about this, we imagine rotat­ ing the smaller to see

if

it will go

of considerations Davies brings against Platonism.

but I'm surprised that

There is another view he also holds, a kind of finitism. Not only does he re­

around all the way. Davies points out

Davies's own view of mathematics

ject actual infinities, but he takes large

that this rotation involves time, but the

is a kind of postulationism. Axioms

fmite numbers to be a kind of fiction,

Platonic world is timeless. Well, this is

such as Peano's are things we create.

as well. Constructivists and Intuition­

hardly a problem, since the puzzle can

They are not discovered by us; they are

ists are well known to draw a line be-

1 Kurt Godel. "Russell's Mathematical Logic," reprinted in Benacerraf and Putnam (eds.), sity Press,

66

1 97 4, 484.

THE MATHEMATICAL INTELLIGENCER

Readings in the Philosophy of Mathematics,

Cambridge, Cambridge Univer­

tween the finite and the infinite. The

interesting example that occurs in a

that this is an admission that conser­

common view is that the finite can be

Newtonian framework of five bodies

vation of energy is not an automatic

calculated "in principle," no matter

set up in such way that their gravita­

consequence of Newton's laws. In any

how large. Davies dismisses this view

tional interaction leads to one of the

isolated system, it is, but in an infinite

and takes the divide to be between

bodies flying off to infmity in a fmite

universe, it must be a separate as­

what can and what cannot be calcu­

time. Davies rightly finds the example

sumption. In his subtitle, Davies asks,

lated in practice. If one is going to

impressive. But instead of milking it for

"What do scientists really know?", sug­

make any distinction, then Davies, I

its considerable philosophical content,

gesting that they don't know a lot of

think, is right to draw the line at what's

he remarks that it is physically unreal­

things they think they do.

practical. However, the price to be

istic and that when bodies start mov­

pointing out that they don't know that

Perhaps

paid-which is much worse than the

ing fast, special relativity takes over

Newtonian dynamics is deterministic

price to be paid for constructivism-is

and prevents this sort of thing from

might have been a juicy example.

A very great deal of

happening. He is much too practical­

When it comes to quantum me­

mathematics would be under a cloud.

minded. I suspect readers would pre­

chanics, Davies's views are difficult to

But why, after all, should we think that

fer the musings of a more philosophi­

pin down. After a briefaccount of some

everything real must be directly acces­

cally

of the physics, he discusses EPR and

sible to human beings? We cannot see

sneers at reality and wants to explore

SchrOdinger's cat. He suggests that the

electrons and we cannot count all the

the nature of Newtonian dynamics, re­

cat doesn't go into a state of superpo­

members of a transfinite set, but these

gardless of whether Newton's theory is

sition. But he seems unaware of the

entities are central to the best beliefs

true. What can one say about this ex­

cost of saying this. It means that there

we have.

ample? There are two remarkable con­

are two physical realms, a micro-realm

sequences.

that obeys the quantum laws and a

simply too great.

Science and mathematics

would be impoverished without them. And that's a good reason to think them

minded

mathematician

who

First, Newtonian dynamics, our par­

macro-realm that does not. You don't

adigm example of a deterministic the­

have to be a mad dog reductionist to

Turning to physics, Davies takes up

ory, is not, after all, deterministic. The

chaos, determinism, and related themes

argument is simple: Start with a world

rebel at this. It's the desire for one phys­

perfectly real and objective.

ical world that leads people such as Eu­

in a chapter on mechanics and astro­

Science in the Look­

in which, among other things, the five

gene Wigner to say that the cat does in­

physics. Because

particles are arranged so that one will

deed go into a state of superposition and that human consciousness is what puts

ing Glass

has philosophical aspira­

be sent off to infinity. Let it run, say, a

tions, it might be useful to contrast

second longer. Since Newtonian dy­

Davies's view with how philosophers

namics is time-reversible, use the lo­

think about these issues. First, con­

cations of the remaining particles at

sider the definition of

this stage with all the motions reversed

mathematical

as the initial conditions for a world W 1 .

world, not in the world itself. But he fails

determinism.

The old idea was this: If we know exactly/approximately the laws and ini­

For a world W2 use exactly the same

it into an eigen-state. Davies doesn't like

this solution either and goes so far as to

claim that superpositions are just in the representation

of

the

to note that without actual superposi­

tial conditions, then we can know ex­

initial conditions. The initial conditions

tions in the physical realm, we are at a

actly/approximately the final conditions.

in these two worlds, though identical,

loss to explain interference effects such

This is the idea captured in Laplace's

do not require or forbid a particle com­

as the pattern that results in a split

famous demon and often called Lapla­

ing in from infinity one second after the

screen experiment.

cian determinism. Chaos has revealed

start. Their histories could be different,

When discussing EPR he claims that

a problem with this account, since

in spite of their being governed by

the entangled state shows that the re­

even

approximate

Newton's laws. That is, in one world a

mote particles cannot be considered as

knowledge of the initial conditions

highly

accurate

particle comes in from infinity after one

separate entities.

won't let us predict very far ahead.

second, but in the other world no such

what about non-local effects? There is

prima facie

Well,

maybe,

but

Among philosophers, the current

particle enters. These are both com­

a

favourite definition of determinism has

patible with the initial conditions and

relativity, but Davies, unfortunately,

Newton's laws. The moral should make

doesn't mention it.

abandoned any connection to knowl­ edge or predictability.

A theory is

de­

conflict with special

our heads spin: Newtonian dynamics is

Chapters at the end are on Darwin­

not deterministic, because there are

ian evolution, which he endorses, and

the theory with identical initial condi­

models with identical initial conditions

on reductionism, which he does not.

tions have identical final conditions. I

but different fmal conditions.

One of Davies's reasons for rejecting

terministic provided that all models of

mention this change in the concept of

If we impose a relativistic speed

reductionism is a poor one: our inabil­

determinism only in passing. The ex­

limit, then determinism can be recov­

ity to predict. Knowledge of physics

ample I now want to discuss is equally

ered. But this would not be Newtonian.

won't allow us to predict someone's

bizarre and instructive on either defi­

We could also recover determinism if

behaviour. Reductionism, however, is

nition.

we demanded conservation of energy

a doctrine about reality, not about our

in the initial specifications. But note

knowledge of reality. The fact that we

Davies describes the wonderfully

© 2005 Spnnger Sc1ence+Business Med1a, Inc , Volume 27, Number 3, 2005

67

least

cannot predict is irrelevant. (Note the

those which are the

mathemati­

2. The notion itself of mathematics has

relation of this issue to the rival defi­

cal-for example evolution, plate tec­

to be redefined. Joseph defines it as

nitions

tonics, and the existence of atoms." (v)

of

determinism

mentioned

above.) He does, however, mention more interesting examples that do put

This is not only true, it is profoundly true.

to numbers or spatial configurations

the doctrine of reduction under a great deal of pressure: money or subjective

any activity that arises out of, or di­ rectly generates, concepts relating together with some form of logic.

Department of Philosophy

Hence he includes in his study

consciousness, for instance. It seems

University of Toronto

proto-mathematics.

almost impossible to reduce a social in­

Toronto M5S 1 A 1

stitution such as money to the laws of

Canada

physics. I should point out that the

e-mail: [email protected]

Joseph justifies this decision by There is a close link between mathe­

philosophical literature on this is far

matics and

from any consensus. The

last

chapter,

"Some

Final

Thoughts," contains brief musings on chaos, the anthropic principle, Hume and Popper on induction, realism vs anti-realism, the sociology of science, and technology. This chapter, like the whole book, is a series of observations, often off-hand, that are sometimes in­ sightful and sometimes not. It's hard to know for whom such a book is written. It is rambling, undisciplined, and un­ focussed throughout.

This in itself

needn't be a bad thing. Littlewood's

A Mathematician's Miscellany is wildly unfocussed and rambling, but it is one

of the most interesting books I've ever

read. What's the difference? Science in

the Looking Glass seems to be written

for a general audience. That, too, is

fine. But then explanations are needed of the relevant mathematics, physics, biology, and philosophy. Almost none are given. Scientists who have never given a moment's thought to philo­

sophical implications of chaos or quan­

tum mechanics or Godel's theorem

might profit from this book, because

they won't need to be filled in on the

background that Davies is assuming. But those who have thought about

these issues will find many of the mus­

ings here obvious or misleading.

But there are also some notable ob­ servations that offset the book's disap­ pointment. One of these occurs at the outset. "My conclusion is surprising, particularly coming from a mathemati­ cian. In spite of the fact that highly mathematical theories often provide very accurate predictions, we should

not, on that account, think that such

theories are true or that Nature is gov­

erned by mathematics. In fact the sci­ entific

theories

most

likely

to

be

around in a thousand years' time are

68

three aspects of proto-mathematics:

THE MATHEMATICAL INTELLIGENCER

The Crest of the Peacock: Non-European Roots of Mathematics, new edition

by George Gheverghese Joseph

PRINCETON AND OXFORD, PRINCETON UNIVERSITY PRESS, 2000 ISBN: 0-691 -00659-8, $US 22.95, 4 1 6 pp.

REVIEWED BY E. KNOBLOCH

T

his extraordinary book, first pub­ lished in 1992, lives on in this sec­

ond edition, an enlarged and improved

version of the first. It is in search of our "hidden" mathematical heritage, which can be found all over the world and cer­ tainly not only in European countries. It is a plea against "the parochialism that lies behind the Eurocentric per­ ception of the development of mathe­ matical knowledge" (p. 348).

The title itself is taken from an In­

dian source of the fifth pre-Christian century

(Vendanga Jyotisa) : "Like the

crest of a peacock, like the gem on the head of a snake, so is mathematics at the head of all knowledge." Joseph re­

veals his driving passion behind the book: the global nature of mathematical pursuits and creations. Hence he con­ tinuously emphasizes that scientific cre­ ativity and technological achievements existed long before the incursion into these areas by Europe. His guiding prin­ ciple is to recognize that different cul­ tures in different periods of history have contributed to the world's stock of mathematical knowledge.

This principle implies some crucial

consequences: 1. The study of the history of mathe­ matics should not be confined to

written evidence.

astronomy.

Early man­

kind's capacity to reason and to con­ ceptualise was not different from that of today's modem peoples. Conjec­ tures about the mathematical pursuits of early humankind have to be exam­ ined in the light of their plausibility. Joseph would like to attribute math­ ematical thinking, practice, knowledge to as many peoples, and as early peo­ ples, as possible, pleading against a re­ strictive view of what is proof. Conse­ quently, he begins by speaking about numerical recording devices on bones of Central Equatorial Africa (35000 B.c.),

South American knots, counting sys­

tems of people in Nigeria, Mayan nu­ meration. The next eight chapters deal with Egyptian, Babylonian, Chinese, In­ dian, and Arab mathematics. The rich­ ness of this information cannot conceal the fact that most is well known to pro­ fessional historians of mathematics. No serious historian of science will contest either the importance of these non-Eu­ ropean cultures or Joseph's statement

that mathematics is a pancultural phe­

nomenon which manifests itself in a number of different ways as counting,

locating, measuring, designing, playing, explaining, classifying, sorting. Joseph's

too

general

criticism

("most standard histories," "many text­ books," etc.) is outmoded to a large ex­ tent. Several decades later, M. Kline's evaluation of 1962 does not represent the state of the art any longer (p. 125). Yet Joseph does not stop there. He be­ lieves in the existence of East-West links he hopes would come to light if research were to be channelled in the direction of a westward transmission of mathematics. Fortunately he avows that much research needs to be done before we can be more certain about the nature and mode of the interchange

of mathematical ideas that took place between China and the other cultural centres (p. 2 12). Yes, we should distin­ guish among claims, beliefs, and his­ torical facts. Sometimes Joseph does not notice that he disproves his own argumenta­ tion. He complains about Eurocen­ trism because it cannot bring itself to face the idea of independent develop­ ments in early Indian mathematics, even as a remote possibility. But he does not concede this possibility to the Greeks with regard to the earlier cul­ tures of the Near East. By all means, it is a too condescending attitude to con­ cede only "that the Greek approach to mathematics produced some [!] re­ markable results" (p. 346). Thus the reader is left with mixed feelings. While Joseph rightly rejects the hegemony of a Western version of math­ ematics, he is inclined to replace it by another one, although he explicitly states that "since the first edition we are no closer to gathering further definitive evidence of transmission of mathemati­ cal knowledge to Europe" (p. 354). lnstitut fUr Philosophie, Wissenschaftstheorie Wissenschafts- und Technikgeschichte Technische Universitat Berlin 1 0587 Berlin Germany e-mail: ehkn01 [email protected]

Mathematics and Music. A Diderot Mathematical Forum

edited by G. Assayag, H. -G. Feichtinger, and J. F. Rodrigues BERLIN, HEIDELBERG, SPRINGER-VERLAG, 2002, 288 PP., US $84.95. ISBN 3-540-43727-4

• • • •

• •

ity," also the source of the Music of Spheres that Pythagoreans referred to when required to swear (Fig. 1 ) . It was perhaps because he was im­ pressed by the mathematical consis­ tency of consonance that Pythagoras devised the idea that Number is the substance of the Universe. Be that as it may, on an instrument consisting of a single taut string vibrat­ ing on a sounding board and fitted with keys that make it possible to select suit­ able lengths of the string being vibrated, one obtains with the Tetraktys the in­ tervals known as octaves, fifths, and fourths. Figure 2 represents such a sin­ gle-stringed instrument (e.g., the Vosges spinet, still used today by certain folk groups in Eastern France) with the cor­ responding modem names of the notes. Musical instruments such as the tetrachord lyre may also be built with four strings having these same lengths (L, U2, U3, U4) that produce simulta­ neous sounds. The respective tensions are adjusted so that the sound pro­ duced by each string is that of the string having the same length on the monochord. The article describes the improve­ ments brought to this theory by Philo­ laus and others. The chief result of that period was obtained by Archytas, who demonstrated the need for unequal divisions in order to obtain all the con­ sonants comprised in an octave. He recognised the importance of arithmetic, geometric, and harmonic means. This

• • •

•

Fig. 1

and calculation in music," while in Paris the Forum dealt with "Mathe­ matical logic and musical logic in the twentieth century." These three topics are covered in a fairly balanced way in this book, five articles dealing with the first topic, seven with the second, and four with the last. All these articles are of signif­ icant interest, whether from a histori­ cal or theoretical point of view. Bringing them together in the same publication sheds magnificent light on the dialogue and mutual enrichment that Mathe­ matics and Music have developed over the centuries [ 1 ] [2] [3] [4] . The first article, by Manuel Pedro Ferreira, deals with the musical the­ ory constructed by Pythagoras. Two sounds from the same taut string are said to be consonant when they are pleasing to listen to simultaneously. In the Greek cultural arena of that period such sounds are produced by lengths of string that are inversely proportional to the numbers 1, 2, 3, and 4. These compose the famous Tetraktys (1 + 2+3+4 10), a diagram of figured numbers symbolising pure harmony, the "vertical hierarchy of relation be­ tween Unity and emerging multiplic=

... ..

................

Length L/4 : obtained sound Sol3,fourth ofRe3,and octave of Soh ......... ........

......

Length L/3 : obtained sound Re3,fifth of Sob .................................. ........................

......

REVIEWED BY SERGE PERRINE

Length L/2 : obtained sound Soh ,octave of Sol1 .................

.......

he book under review brings to­ gether sixteen contributions to the Diderot Mathematical Forum held un­ der the auspices of the European Math­ ematical Society, simultaneously in Lisbon, Paris, and Vienna, with tele­ conference exchanges, on 3 and 4 De­ cember 1999. The conference in Lisbon covered "Historical aspects," the topic in Vienna was "Mathematical methods

T

Total length L of the vibrating string: its vibration gives the sound designated by So11

�

1

I

I

I

.

1

Sounding box with keys

Fig. 2

© 2005 Springer Sc1ence+ Business Med1a, Inc , Volume 27, Number 3, 2005

69

G1

G2

D3

G3

B3

D4

L

U2

U3

U4

U5

U6

Octave

Perlect

Perlect

Major

Minor

112

fifth

fourth

third

third

2/3

3/4

4/5

5/6

Fig. 3

allowed him to enrich the range of sounds used and their associated in­

tervals (Fig.

3).

The reference

made

to Aristox­

as new ways of combining rhythms.

such a project. Yet, once rid of its ab­

However,

surd objective, this statement aptly

this

culmination

of the

pythagorean musical base that had de­

sums up the concept of music prevail­

veloped over many centuries eventu­

ing in Renaissance and Baroque times, founded on number and its symbolism,

enus-a pupil of Aristotle who totally

ally degenerated in the following cen­

rejected pythagorean harmony in fav­

tury because it proved to be inadequate

a source of beauty and harmony. It ac­

our of a musical theory based on the

for responding to the new aesthetic

tually sets it in an oriental tradition

continuous sounds perceived by the

trends that were appearing as well as the

considerably older than the Greeks,

ear, as well as on the tensions of the

practical needs of musicians. Those who

that considered number as the handi­

strings and their relaxation time­

were concerned with the tuning of their

work of God who ordered all things in

shows the rich diversity of musical

keyboard instrument were led to con­

measure, number, and weight (Wisdom

thinking in Ancient Greece.

sider the problem of temperament

However, the most interesting as­

[5].

Having had one's mind brilliantly

1 1 . 17) and on which all the

of Solomon

work of Man rests. Kronecker's well­

pect of this article, whose numerous

stimulated by such an article, one is led

known phrase "God created number,

references provide ample scope for

to wonder about the Byzantine evolu­

all the rest is the work of Man," draws its inspiration from the same source.

digging deeper, is certainly the de­

tion, geographically so close to the

scription of the rich musical evolution

Greek source; regrettably, however,

In fact, Kircher's book develops the

flowing from the Greek roots into the

this aspect is not touched upon in the

musical ideas of the minim monk Mar­

Latin world and right up to the four­

article. One wonders too what was

inus Mersennus (Marin Mersenne ) , in

teenth century of our era. In St. Au­

the contribution of the ancient manu­

particular the combinatory approach

gustine's De Musica, written at the end

scripts passed on by the "sons of the

of the fourth century, rhythms are also

Greeks," as the Arabs of the time called

Harmonia Univer­

contained in his

salis,

written in

1636. The article un­

classified according to their propor­

themselves. Fortunately the article refers

fortunately does not speak of Mersenne's

tions (the proportional notation used

to the influence of Arabian and Persian

activity as the science correspondent

today came much later). Then in the

music in the

of the whole of Europe, nor of his

ninth century, Carolingian policy in

Peninsula, and the contribution of the

creation

educational and ecclesiastical matters

reading of the ancients, thanks to the

Parisiensis, the ancestor of the future

It encouraged

translation in the twelfth century of the

Academie des Sciences; nor does it

musical treatise by AI Farabi.

mention the measurement of the speed

vides a novel answer by introducing

his discovery of the higher harmonics

defmed new practices.

the use of neumes that indicate the in­ flexions of the voice, but not the pitch of the sounds. The names Do, Re, Mi, Fa,

Sol, etc., appeared with Guido

d'Arezzo in the eleventh century, de­

Cantigas

of the Iberian

Eberhard Knobloch's article pro­

the concepts of Athanasius Kircher, who in

in

1635 of the Academia

of sound that he obtained in

1636, nor

of a string. No mention is made either

Musurgia Univer­

of his systematic use of the notion of

Kircher quotes Hermes Tris­

frequency, introduced at the time by

1650 wrote

riving from the syllables at the begin­

salis.

ning of the stanzas (voces) of a hymn

megistos, the mystical author who was

Galileo Galilei. Mersenne was a student

addressed to St John the Baptist, writ­

so loved by the Medicis and Pico della

of the latter's work and was familiar with the law that gives the frequency

770 A.D. The notes (claves)

Mirandola: "Music is nothing else than

are also designated by letters, a prac­

to know the order of all things." This

of the fundamental vibrationf of a vi­

tice that is still in use today in English­

very pythagorean concept postulates

brating string having a length L with a linear mass p and with tension F:

ten around

speaking countries (La = A, Ti = B,

that Music is a part of Mathematics

Do = C, . . . ) and in Germany (with

(and

some specificities). Finally, polyphony

Kircher, this is a relevant concept

created

harmonic

when seeking to help someone having

mastery, the response coming from

virtually no knowledge of the mastery

This formula is merely mentioned in

Philippe de Vitry in the fourteenth cen­

of sounds to acquire an in-depth knowl­

the

edge of musical composition. Pythago­

Galileo, the son of the musician Vin­

ras doubtless would have disowned

cenzo Galilei, and it was written in this

new

needs

for

tury with his Ars Nova: in this work he

defined new musical notations as well

70

THE MATHEMATICAL INTELLIGENCER

consequently

a science).

For

f=

Discorsi,

� If·

2

written

in

1638 by

modem form only in 1715, by Brook Taylor. The limited part of Mersen­ ne's work mentioned in the book is nonetheless of major interest and sets the record straight regarding a nwnber of misconceptions as to the history of science at that time. In his Harmonia Universalis, Mer­ senne sets out the table of all the values of the nwnber of permutations with n elements up to n 64. He discusses non-repetitive arrangements P(n, p) n(n - 1) . . . (n - p + 1) and combina­ tions C(n, p) = P(n, p) P(n, p)/P(p, p). He solves the problem of calculating the number of combinations presented by a given type of repetitions. This he does thirty years before Leibniz succeeds in obtaining, with a few errors, the same results in his "schoolboy's essay" De Arte Combinatoria, and well before the combinatorial work of Fermat and Pascal. If n is the maximum number possible of notes for a song composed with p different notes, of which r1 dis­ tinct notes appear once, r2 distinct notes appear twice, . . . , and using in fact r r1 + r2 + . . . + r, distinct notes in all, Mersenne gives the total number of possibilities for the corre­ sponding songs:

tury bell-ringers and the rules laid down by Fabian Stedman [7] is but a short step, but one which none of the articles in this book dares to take. On the other hand, the approach taken does shed light on musical analysis, as may be seen in the article by Laurent Fichet, and makes it possible to extend one's horizon, as in Marc Chemillier's article dedicated to ethnomusicology. The formula mentioned earlier re­ lating to the fundamental frequency of a string, in tum allows a better under­ standing of the problem of tempera­ ment. It consists of seeking to divide an octave into twelve equal intervals, and therefore to identify rational num­ bers that simultaneously come as close as possible to the irrational real num­ bers 2(1/1 2), 2(2/1 2) , 2 (3/12) , . . . , 2c11112) , being aware that a trained ear will per­ ceive any deviation that is too signifi­ cant. This leaves plenty of margin for numerous systems, and the remarkable article by Benedetto Scimeni pre­ sents the choice proposed by Gio­ seffo Zarlino in his work Le Istitutioni harmonicae [8], published in 1558:

n! r1 !r2 ! . . . r,!(n - r)!

Galileo's father quarrelled with Zarlino because he preferred 18/17 to 10/9. But of course, whatever choice one makes, the practical issue is the tuning of instruments, in particular harpsichords with several octaves and the largest possible number of tones. The article referred to here mentions the remarkable work undertaken on these questions by Giuseppe Tartini, Daniel Strahle, and Christoph Gottlieb Schroter. One of the most fascinating aspects is the connection with the so­ lution to Pell!Fermat's equation in Tar­ tini's Trattatto di Musica:

=

=

=

=

For 22 possible notes, of which 7 dis­ tinct ones are repeated according to the type 2, 2, 1, 1, 1, 1, 1, Mersenne shows that there are 3,581 ,424 possible songs. Is it therefore not understand­ able that it was the analogy with the combinatorics derived from gaming that led Mozart to devise a musical game allowing the players to produce waltzes by throwing dice [6]? This then poses the question of the link between musical creativity and chance. One may indeed wonder if certain of Haydn's compositions were not in­ spired by similar methods. This point is not mentioned, even though his 41st piano sonata is quoted in the article by Wilfrid Hodges and Robin J. Wilson dedicated to musical forms. Speaking of combinatorics raises the possibility of using the group of permutations of objects that one arranges and com­ bines. . . . From there to seeing Galois's theory in the practice of sixteenth cen-

10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5.

x2 - 2y2

=

1.

In fact, this becomes obvious when one realises that the above also leads to coming as close as possible to the irra­ tional 2C6112) = v2 with a rational nwn­ ber, a classic problem of diophantine analysis, which is much simpler than the previous problem of simultaneous approximation of the twelfth roots of 2. The book is incomplete if one con­ siders it from the point of view of the

history of acoustics, a word invented by Joseph Sauveur, who professed mathematics at the College de France from 1686. Dumb until the age of seven and deaf for the whole of his life, it is he who looked more closely at the ob­ servation made by Mersenne that there exist higher harmonics: a string may vibrate in several parts around nodes that remain fixed. The book makes scant mention of the work of Bernoulli or Euler. It remains almost completely silent concerning the discovery in 1 747 by d'Alembert of the partial differential equation of vibrating strings: a2y

-

at2

=

�

a2y

-

ax2

·

It is in fact the solution of this equation that makes Sauveur's discoveries un­ derstandable. However, research ac­ tivity on sound was so extensive at the time that to describe it would be an al­ most impossible task We would need to mention Wallis, Newton, La Hire, not forgetting Bach, Rousseau, and so many others; one would necessarily have to be selective. The selection made in the book is particularly rele­ vant, but makes one want a new Fo­ rum, to take the question deeper by re­ ferring to the activities of other authors who have been left out. The article by Jean Dhombres ex­ plores another major historical mile­ stone, by referring to the interest shown by Lagrange around 1760 in musical texts and the theory of instruments. In his Recherches sur la nature et la prop­ agation du son he gives a definition of the integral of a function as a limit. No more nwnber theory and geometry. He shows that the same differential equa­ tion appears in the vibrations of strings and those of air. He discovers the or­ thogonal relationship of sine and cosine. Yet Lagrange cannot be considered to be the inventor either of series or of the Fourier analysis. It was indeed Fourier who recognised the universality of the calculus discovered by Lagrange in his study of musical sounds. From a math­ ematical point of view, the next stages in this millennia! adventure, which are not covered in this book, are the march towards distributions [ 10] and the deeper understanding of spectral analy­ sis [11] and group representations [ 12].

© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 2 7 , Number 3, 2005

71

Today the Music of corpuscles and solitons is taking the place of the Mu­ sic of spheres and mermaids. Consid­ erations of the multiple infinitely small (chaos?) are replacing those on the sin­ gle infinitely great (the cosmos?). The bifurcation took place at the end of the eighteenth century, at the very moment when musicians were being pushed into the category of artists, whose role was to provide pleasure for the pre­ sent, and mathematicians into the cat­ egory of scientists, building the society of the future. The remainder of the book presents four articles by Giovanni De Poli and Davide Rocchesso, by Erich Neuwirth, by Xavier Serra, and by Jean-Claude Risset on the application of modem digital sound technology. There are new acoustic domains being explored, such as the impact of non-linearity, the hearer's perception, the use of computerised toolboxes to produce sounds, texture compositions, and acoustic illusions. The proliferation is huge and shows how that mathemati­ cal machine par excellence-the com­ puter-is invading music. Far from slackening, the interaction between the two fields is continuing to develop and is as strong as ever. The major change seems to have been that math­ ematics now has its instruments­ computers--whereas classical musical instruments are left standing. Experi­ mental practice seems to have provi­ sionally changed sides, but the process of mutual enrichment is continuing [ 13] . It is consequently natural to ask about the logic and meaning behind this evolution of the two fields [ 14] . Logic has always been essential to mathematics, but in the recent period it would appear to be less natural in music. Of course, one may consider the computerised machine learning music, as do Shlomo Dubnov and Gerard As­ sayag. But does this have anything to do with logic? Another article by Marie­ Jose Durand Richard, which retraces the history of logic, shows that the is­ sue isn't clear. It refers us to the arti­ cle by Franc;ois Nicolas dealing with just that question: What is the logic in music? The answer given by Nicolas is as anti-pythagorean as could be, be­ cause it results in the impossibility of

72

THE MATHEMATICAL INTELLIGENCER

defining this concept today, and hence leads to falling back on the study of the practices involved in musical produc­ tion, free from mathematical, acoustic (physical), and psycho-physiological tutelage. Such a loss of meaning is in total contradiction of the tradition of a relationship between music and logic, as illustrated in the double organisa­ tion of ancient knowledge of liberal arts in the form of trivium (grammar, rhetoric, logic) and mathematical quadrivium (arithmetic or the number in itself, geometry or the number in space, music or the number in time, as­ tronomy or the number in space and time). Yet it is thoroughly contempo­ rary. It also sets itself completely apart from the theories that Marin Mersenne proclaimed in his Traite de l'harmonie universelle published in October 1627 under the pseudonym Franc;ois de Ser­ mes [15]. Theorem 1: Music is a part of mathematics and consequently a sci­ ence, capable of showing the causes, effects, and properties of sounds, songs, concerts, and anything related thereto. Theorem 4: Music is both a speculative and practical science, and an art, and consequently is a virtue of understanding, which it leads to the knowledge of the truth. In this under­ standing, which one may consider to be outdated (wrongly, for the joint evo­ lution of the two fields is continuing, as the present book shows), the logic of music finds profound meaning, which the author of the article submits to the meditation of its readers. To this end, the book contains one last article that I have not yet mentioned. It is by Guerino Mazzola and is titled The Topos Geometry ofMusical Logic. (See the review by Shlomo Dubrov, this is­ sue.) On the mathematical side, he relies on the theory of categories and Grothen­ dieck constructions; on the musical side, on Riemann's harmony (not G. F. Bern­ hard but K. W. J. Hugo, i.e., not the math­ ematician but the author of Mathema­ tische Logik published in 1873!). He develops a Galois theory of musical con­ cepts which locks Beauty and Truth into the same kingdom. So might there after all still be some pythagoreans in our day and age, lost among our contempo­ raries, Guerino being one of them? At any rate, his article is fascinating from

an intellectual point of view. He con­ firms that the new alliance between mu­ sic and mathematics announced by Pierre Lamothe in 2000 on his Web site is forging ahead, though using paths other than those he had envisaged [16]. This new alliance between pleasure and science cannot but enrich both parties. It might even constitute a rem­ edy for the desertion from mathemati­ cal studies observed in our times, when knowledge and work are parcelled out piecemeal. The pleasure derived from reading this remarkable work is very great. The reviewer is convinced that other Mathematics and Music initia­ tives need to be taken, and that there is no lack of topics to be covered. Acknowledgement

I am indebted to the non-mathemati­ cian Andrew Wiles for the English translation. BIBLIOGRAPHY

(1 ] http://www. medieval.og/emfaq/harmony/ pyth.html [2] J. G. Roederer, Introduction to the Physics and Psychophysics of Music, Springer­ Verlag, New York and Berlin, 1 975. [3] N. H. Fletcher, T. D. Rossing, The Physics of Musical Instruments, Springer-Verlag, New York and Berlin, 1 991 . (4] P. Bailhache, Une histoire de J'acousti­ que musicale, CNRS editions, Paris, 2001 http://bail hache. human a. u niv-nantes. frI thmusiques [5] E.

Neuwirth,

Musical

Temperaments ,

Springer-Verlag, New York and Berlin, 1 997. [6] http://sunsite. u nivie .a c. at/Mozart/dice/ mozart.cgi [7] W. T. Cook, Fabian Stedman, 1 64Q-1 730, Ringing World, 29 October 1 982, pp. 900-901 . (8] http://virga.org/zarlino/ (9] J . Cannon, S. Dostrovsky, The Evolution of Dynamics: Vibration Theory from 1 687 to 1 742,

Springer-Verlag,

New York and

Berlin, [1 98 1 ] . [ 1 0] J . LUtzen, The Prehistory of the Theory of Distributions, Springer-Verlag, New York and Berlin, 1 982. [1 1 ] C. Gordon, D. Webb, S. Wolpert, You cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27, pp. 1 34-1 38, 1 992. [1 2] A. W. Knapp, Group representations and

harmonic analysis from Euler to Lang­

into notebooks known as "common­

truth-values by a "subobject classifer,"

lands, Notices Amer. Math. Soc. 43,

place books." These notebooks were

which is something more general than

1 996, pp. 4 1 0-41 5, pp. 537-549, http:�

commonly indexed and arranged for

the Boolean algebra of True and False.

www.ams.org/notices/1 99604/knapp.pdf

easier reference, and maybe it is not sur­

This,

[1 3] http://cnam .fr/bibliotheque/tables/table

prising that the book by Mazzola opens

Grothendieck on algebraic geometry,

by placing music in a new "encyclo­

allows Mazzola to define complex mu­

0302.html

together

with

the

work

of

[1 4] R . Steinberg (ed.), Music and the Mind

space," a space where human knowl­

sical structures of Global Music corn­

Machine, Springer-Verlag, New York and

edge production is assumed to be cou­

positions, with earlier categories being

Berlin, 1 995.

pled to navigation in a topologically

embedded as "patchworks" of local ob­

arranged concept space.

jects in a global theory, leading to cat­

[1 5] Marin Mersenne, Traits de l'harmonie uni­ verselle, Corpus des oeuvres de philoso­ phie en langue franr;:aise, Fayard, 2003. [1 6] http://www.aei.ca/�plamothe/

In mathematics, category theory is known as a study of abstract mathe­ matical structures and relationships.

egorization of music as constructions on geometric manifolds. The book opens with a very gen­

Groups are often used to describe sym­

eral,

metries of objects, and they were used

tion, which seems vague or somewhat

philosophical-historical motiva­

Conseil scientifique de France Telecom R&D

by music theoreticians for describing

ambiguous, to provide an intuitive basis

38-40 rue du General Leclerc

musical

scales,

for dealing with the forthcoming for­ malisms. In Part II the author goes from

properties

such

as

92794 lssy les Moulineaux Cedex 9

pitch classes, or rhythms. Every ele­

France

ment of the group creates a corre­

e-mail: serge. [email protected]

spondence to some other set of ob­

tions to very abstract concepts of forms

jects, and Cayley's theorem states that

and denotators, assuming prior knowl­

every group

The Topos of Music: Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola, with Stefan Goller and Stefan Muller BIRKHAUSER VERLAG, BASEL. BOSTON. BERLIN. 2002 1 368 PP. Hardcover. rncl. CD-ROM. € 1 28 ISBN 3-7643-573 1 -2

REVIEWED BY SHLOMO DUBNOV

I

n the context of classical Greek phi­ losophy, a

topos

(literally "a place")

G

is isomorphic to the

gories,

Yoneda lemma in category theory is a

reader to "recall" these concepts from

generalization of Cayley's theorem that

appendix G would probably require also

allows the embedding of any category

"recalling" earlier concepts from ap­

into a category of mappings (called

pendices C-F on set theory, rings, alge­

functors) defined on that category. Us­

bras, and algebraic geometry, and so on.

and logic.

Asking the

Part III of the book deals with the

jects, categories of musical composi­

next level of describing musical con­

tions are defined as elementary objects

structs, such as scales and chords,

of music. Then, describing the Yoneda

terming them

Perspective, Mazzola claims that in re­

This brings up a discussion of musical

lation to the arts, "understanding paint­

symmetries in the local composition

"local

compositions."

ing and music is a synthesis of per­

objects, such as Messiaen modi and se­

spective

rial techniques. But there appears to be

variations."

Therefore,

by

considering functors as the represen­

a deeper aspect of local composition

tations, one is led to defining art and

related to the use of functors and their

music as a set of operations (symme­

concatenations, needed in preparation

tries) that leave the object invariant.

for Part IV. This aspect (culminating in

Music composition becomes the "in­ variant" or the "essence" of a set of per­

the art of oration or per­

topoi,

ing denotators to describe musical ob­

referred to a method of constructing

rhetoric,

edge of advanced concepts of cate­

group of its symmetric operations. The

and presenting an argument, being part of

concrete examples of note representa­

formances,

the Yoneda Perspective) employs the

fact that in the denotator representa­

an idea related also to

tion one has the mathematical struc­

suasion. Plato greatly opposed rhetoric,

Adorno's esthetic principle in music.

ture of a topos, which offers properties

claiming that it values style or manner

The emphasis is on the rhetoric func­

such as unions, products,

of persuasion over the discussion of

tion "as a means to express under­

(somewhat as in set theory), and al­

or limits

substance. Then carne Aristotle's "rec­

standing, and in this respect perfor­

lows for enumeration or classification.

onciliation" of rhetoric and dialectic

mance is not only a perspective of

Musical or visual examples could help

saying that while dialectical methods

action but instantiation of understand­

in clarifying these developments, but

are necessary to find truth, rhetorical

ing, of interpretation given structures."

the author offers rather general claims about the utility of the mathematical

methods are required to construct an

Mazzola further assumes that math­

argument in order to communicate it.

ematical study in the context of art will

methods to analysis of an Escher draw­

The concept of topos was extended

lead to objects which are "meant to de­

ing or appreciation of the fractal Julia

later to literature by a German scholar,

scribe beauty and truth. " This brings

set shape, without much detail. Amer­

Curtius, as a study of ways to compile

topos theory to being a way for com­

ican Music Set Theory is called "thor­

knowledge by selecting and indexing

bining logic and geometry. In topos

oughly out of date from the point of

important phrases, lines, and/or pas­

theory one replaces the set by a cate­

view

sages from texts and writing them down

gory, function by a morphism, and

conceptualization."

of

20th-century

mathematical


73

Part IV, on global theory, is a required development for dealing with larger mu­ sical structures, such as rules of har­ mony based on local chord construc­ tions. For instance, a standard example of local versus global topology is a Mobius strip in which a local topology can be taken as a combination of a cir­ cle and a line segment. The "twisting" in the band is apparent only globally, while locally the ribbon structure defines the topology. In harmony, placing triad chords on a strip in triangles, so that de­ grees a fifth apart appear at the bases and the third at the vertex (such as I-III­ Y chords, so that I-V are at the base and III at the vertex), creates a sequence of triangles, with two similar triangles ap­ pearing at the two ends of the strip, one of them being "upside down." Twisting the strip to connect the two similar ends creates a Mobius strip. This construc­ tion of the harmonic relations can be used to resolve ambiguities in the Tonic­ Dominant-Subdominant functions of the chords or explain modulations, etc. More such "simple" examples would have helped to explain the musical as­ pects of the work. The following Parts V, VI, and VII use related topological methods to an­ alyze rhythms, motives, harmony, and counterpoint. Parts VIII and IX introduce what probably are the main highlights of this work-relations and transformations of mental activity into music realiza­ tions. Part VIII deals with performance theory, defining performance fields and the hierarchy of performance cells. These are related to semiotics in Part IX, preceded by an overview of works by Langner, Clynes, Gabrielson, and others, and performance grammars of the KTH school (Stockholm) and Todd. The goal here is to provide tools for rhetoric expression of musical text, considering interpretation as evidence of musical understanding. These meth­ ods are formally developed into a soft­ ware system RUBATO, which is de­ scribed in Part X. Other topics include statistics of analysis methods and per­ formance, principles of music criticism ("inverse performance theory"), and other contributions that are broadly re­ lated to uses of formal computer meth­ ods and object-oriented programming

74

THE MATHEMATICAL INTELLIGENCER

to music representation and computer­ aided composition. The accompanying CD-ROM contains Presto and Rubato software that are re­ alizations of the theory into composition and analysis-performance applications, respectively. Presto is written for an Atari computer, which makes it hardly accessible. Rubato has two versions, one for Nexstep and the other for Mac OSX. The OSX version comes with little doc­ umentation and the user is referred to documentation of the Nexstep version, which provides a good overview of the program, but lacks tutorials or usage ex­ amples, except for mp3 recording of one result. It should be noted that, in terms of computation, the various analysis modules of Rubato might be computa­ tionally demanding, and some simplified approaches to analysis are needed to find the "right orientation" in the great variety of these objects. The huge work presented in the book is highly mathematical and I doubt how much it will allow understanding of the utility of the topos theory for musicians, or of learning about music for the math­ ematician. It is a very particular view of music focusing on the representation! rhetoric aspects of describing musical objects, disregarding perceptual, cogni­ tive, or other dialectic/discursive aspects in music creation or listening. However, it should be clear that the mathematical theory of music is not obliged to explain or place human intelligence in the mod­ eling of music, although to me such a link would be welcome. Department of Music University of California in San Diego LaJolla, CA 92093-0326 USA

and a New York Times Notable Book. The New York Times Book Review is quoted: "The biography of choice . . . Newton the man emerges from the shadows." The Wall Street Journal: "Succinct, elegant . . . A sharp, beauti­ fully written introduction to the man." All true. There's no doubt that this is a bril­ liant piece of writing. "Elegant" seems on occasion too mild a word to describe it. Is "poetic" too strong? The language is evocative and beautiful but at the same time conveys in very general terms the scientific achievements of this genius-no easy task (The author has, of course, on other occasions ex­ hibited his own genius at casting scien­ tific results in terms that have meaning for the lay reader.) Consider, for exam­ ple, the opening paragraph of the book: Isaac Newton said he had seen farther by standing on the shoulders of giants, but he did not believe it. He was born into a world of darkness, obscurity, and magic; led a strangely pure and ob­ sessive life, lackingparents, lovers, and friends; quarreled bitterly with great men who crossed his path; veered at least once to the brink of madness; cloaked his work in secrecy; and yet discovered more of the essential core of human knowledge than anyone before or after. He was chief architect of the modern world. He answered the an­ cient philosophical riddles of light and motion, and he effectively discovered gravity. He showed how to predict the courses of heavenly bodies and so es­ tablished our place in the cosmos. He made knowledge a thing of substance: quantitative and exact. He established principles, and they are called his laws.

e-mail: [email protected]

Isaac Newton by James Gleick NEW YORK: PANTHEON BOOKS. 2003. xii + 272 pages. US $29.95. ISBN 0-375-42233-1

REVIEWED BY GERALD L. ALEXANDERSON

P

ublishers' advertisements point out that this is a Pulitzer Prize finalist

That sets the scene very well. At the same time one has to point out that there's very little mathematics in this short volume, and the mathematician or professional scientist is going to find the treatment good reading but rather light. Of course, I don't think the au­ thor had professionals in mind when he decided to write it. To reduce the life and work of a tow­ ering figure like Newton to 189 pages plus notes and references (octavo, not even folio) is to set oneself an impossi-

ble task. This is very different from the more extensive biographies by Richard S. Westfall at 908 pp. (Never at Rest, Cambridge, 1980); Gail Christianson at 624 pp. (In the Presence of the Cre­ ator/Isaac Newton and His Times, Free Press, 1984); A. Rupert Hall at 468 pp. (Isaac Newton/Adventurer in Thought, Cambridge, 1992); or even the psy­ chobiography by Frank A. Manuel at 478 pages (A Portrait ofIsaac Newton, Har­ vard University Press, 1968). Gleick gives a helpful overview of where things stood in science prior to Newton. There were some pretty strange ideas floating around-for ex­ ample, Athanasius Kircher's claim that "the ocean waters continually pour into the northern pole, run through the bowels of the earth, and regurgitate at the southern pole." By the 17th cen­ tury, views were changing. Newton the scientist was perhaps without equal. Newton the human being was deeply flawed. Because readers of The InteUigencer will be familiar with the major scientific achievements of Newton, I shall talk more about New­ ton the man. It is curious that when Newton said that he had been "like a boy playing on the sea-shore, and di­ verting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me"-yes, all the classic quotes are here-we are also told that "he proba­ bly never set eyes on the ocean." And he lived in an island nation, never that far from the sea. Unlike his contempo­ rary Leibniz, who was well-travelled and worldly, Newton never travelled, and his whole life was spent between Grantham (where he was born), Cam­ bridge, and London (where he was Master of the Mint)-a little over 100 miles. When we think of Newton, two im­ ages immediately come to mind: a car­ toon-like figure sitting under an apple tree, observing a falling or fallen apple, or the heroic and confident stance of the Newton of L.-F. Roubiliac's statue in the Trinity College Chapel. For those who look further, there are the rather benign and probably flattering images in the portraits by Sir Godfrey Kneller. Gleick tells us that late in life "New-

ton told at least four people that he had been inspired by an apple in his Wools­ thorpe garden-perhaps an apple falling from a tree, perhaps not." The benign Newton would probably not have been recognized by Flamsteed or Hooke or Leibniz, or others whom he attacked vociferously. We can all probably think of ge­ niuses-perhaps not as great as New­ ton-whom we regret never having had a chance to meet or at least ob­ serve from a distance-Rembrandt, Mozart (at least before Peter Shaffer wrote Amadeus), Euler, Einstein, . . . I can say without equivocation, how­ ever, that I am relieved not to have en­ countered Newton. Gleick sometimes generously lets Newton off rather lightly. Consider New­ ton's relations with the Astronomer Royal, John Flamsteed. Newton, who needed observations to test his theo­ ries, asked for data from Flamsteed, (admittedly a difficult man to deal with), promised to keep the data for his use only, and then published them. Gleick writes that when Newton needed the data he visited Flamsteed at Greenwich and "pried loose fifty lu­ nar observations and a promise of one hundred more. Flamsteed was reluc­ tant, and he demanded secrecy, be­ cause he considered these records his personal property. Soon Newton wanted more . . . [he] cajoled Flam­ steed and then pressured him." Manuel, by contrast, is blunt in his de­ scription: "[Newton wrote to Flam­ steed] 'I will neither publish them [the observations] nor communicate them to any body whilst you live, nor after your death without an honourable ac­ knowledgment of their Author.' This was an unequivocal promise which Newton had already made once before. . . . Both promises were later broken." Robert Hooke, seven years New­ ton's senior, held an important post, Curator of Experiments for the Royal Society. Gleick describes him as "New­ ton's goad, nemesis, tormentor and vic­ tim." Their great dispute derived from, among other things, their opposing views on light, whether it was "particle or wave," and in doubting Newton's views Hooke had an ally in Christiaan Huygens. Much of the correspondence

cloaks their enmity for each other in flattery and compliments. Manuel says of one exchange, "Hooke was fulsome in his praise . . . 'I doe justly value your excellent Disquisitions . . . I judge you have gone farther in that affair much than I did . . . I believe the subject can­ not meet with a fitter and more able person to inquire into it than yourself . . . ' . . . Newton . . . replied in kind 'You defer too much to my ability for searching into this subject . . . [I know] no man better able to furnish me wth them then your self. ' But interspersed with the baroque compliments are con­ descending jibes. Hooke, casually al­ luding to his recognized priority in the field, implied that Newton was merely putting finishing touches on what he had already initiated.'' This was calcu­ lated to drive Newton into a fury and it did. Much later, when Newton headed the Royal Society, he "purged the . . . Society of all remnants of Hooke.'' Gleick gives a straightforward and balanced account of what was perhaps the most important of Newton's battles with colleagues, the Newton-Leibniz controversy on priority in the discov­ ery of the calculus. Normally Newton operated by stealth, attacking either through his disciples or by writing anonymous reviews himself. Leibniz was also adept at getting his friends to write on his behalf. But when Leibniz wrote a strong letter to the Royal So­ ciety making his claims for the inven­ tion of the calculus, the Society issued a report condemning Leibniz, "accus­ ing him of a whole congeries of pla­ giarism. It judged Newton's method to be not only first-'by many years'-but also more elegant, more natural, more geometrical, more useful, and more certain.'' Newton felt vindicated by the report. Of course, he wrote it himself. By the time some of Newton's attacks were made, it no longer mattered to Leibniz. He had died in 1 716. As with others of Newton's adversaries-Barn­ abas Smith (Newton's stepfather), Flamsteed, and Hooke-Newton won by outliving them. Obsessed with secrecy, Newton not only published seldom; when he did communicate his ideas they were often in code. Gleick gives an example, in a

© 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc , Volume 27, Number 3, 2005

75

letter to Leibniz asserting possession of a "twofold" method for solving in­ verse problems of tangents: "At present I have thought fit to report both by transposed letters . . . 5accdm 1 Oefjh1 1 i4l3m9n6oqqr8s1 1 t9v3x:1 1 a b3cdd1 Oem g1 Oillrm 7n603p3q6r5s 1 1 t8vx,3acm 4egh5i4 14m5n8oq4r3s6t4vaaddm eeeeeiijmmnnooprrs­ ssssttuu. " It appears that when he did "communicate" with peers, he didn't convey much information. He was not only obsessively secre­ tive, choleric, and disputatious, he also exhibited unattractive traits of ambition and excessive concern for power and money. He wielded intellectual power as President of the Royal Society and fi­ nancial power as Master of the Mint. Gleick gingerly handles one aspect of Newton's life that others have ad­ dressed more candidly. He quotes Voltaire who said in his Letters on Eng­ land, "In the course of such a long life he had neither passion nor weakness; he never went near any woman. I have had that confirmed by the doctor and the surgeon who were with him when he died." Gleick goes on to treat very delicately the role that Fatio de Duil­ lier, the minor Swiss mathematician, mystic, and adventurer, played in New­ ton's life. He says that "Newton felt real affection for this brash and hero-wor­ shiping young man, who lodged with him increasingly in London and visited him in Cambridge." Manuel suggests more: "Isaac Newton died a virgin . . . Newton's inhibitions probably pre­ cluded both cohabitation with women and inversion, though not feelings of tenderness toward a succession of younger men, budding scientists and philosophers. In his relations with Fa­ tio de Duillier, the aristocratic Swiss genius who appeared in England when Newton was about forty-five, these feelings reached a high pitch, posing a threat and creating a demand for re­ pression that was an element in the breakdown of 1693." 1 Westfall tells us that when Fatio wrote to Newton in the fall of 1692 that "I have Sir allmost no hopes of seeing you again. With coming from Cambridge I got a grievous cold, which is fallen upon my lungs," Newton responded "I . . . last night received your

76

THE MATHEMATICAL INTELLIGENCER

letter wtll wch how much I was affected I cannot express. Pray procure y" advice & assistance of Physitians before it be too late & if you want any money I will supply you. . . . sr wtll my prayers for your recovery I rest, Your most affec­ tionate and faithfull friend to serve you Is. Newton." It is not clear just what games Fatio was playing here. He went on to live another 61 years! Newton worried a great deal about his own health, in particular about the plague and the pox. According to Gleick, "he treated himself preemptively by drinking a self-made elixir of turpen­ tine, rosewater, olive oil, beeswax, and sack" By Newton's thirties his hair was already gray. Yet, for his time, he lived a long life-84 years. With Newton's genius for science, modem readers have to wonder about the amount of effort he expended on alchemy. Gleick correctly cautions us, however, not to come down too hard on Newton for this. The distinction be­ tween chemistry and alchemy was not as clear then as it is today, just as as­ trology and astronomy were not as dis­ tinct. After all, "Kepler and Galileo had trafficked in horoscopes." Another aspect of Newton's life work that might surprise some today is his great concern for questions in the­ ology. Gleick writes: "the fatherless man, the fellow of the college named Trinity, turned to Christian theology with the same sleepless fervor he brought to alchemy." It is interesting to note that though Newton spent his whole academic life at a college named for the holy Trinity, he himself did not believe in the Trinity (or the divinity of Christ) and used "A.C." instead of "A.D." to emphasize the association of a year with Christ and not the divinity. It amounted to heresy, but who would dare to accuse Newton? Newton expended enormous amounts of time and energy on hermeneutics, though, and in the last days of his life he was still working on his Chronology of the Ancient Kingdoms Amended in which he measured the reigns of the kings and calculated the date of the sailing of the Argonauts to find that the ancient kingdoms were much younger than was generally assumed. His sci-

ence has survived, but, alas, his theo­ logical work has not. Newton died an agonizing death from kidney stones. He probably would have approved of his monument at the screen in Westminster Abbey, erected in 1731 and designed by John Michael Rysbrack Gleick gives a succinct de­ scription, but I'll describe it with more detail: behind the actual marker on the grave a marble catafalque under a large sarcophagus, with a frieze of putti per­ forming scientific experiments, on which Newton reclines against four books (the Principia, Opticks, Divin­ ity, and the Chronology), with two putti displaying a scroll with a diagram, and, well above, a large globe showing the constellations, signs of the zodiac, and the comet of December 24, 1680, whose path Newton predicted. And on the globe a large female figure of As­ tronomy, the Queen of Sciences, re­ clines-all under a neo-Gothic screen. Westfall calls it a monstrosity. But it's very grand, the tomb of a rich and pow­ erful man. By contrast, the location of Leibniz's grave is unknown. Gleick gives us a splendid account of the life of Newton. It's such a fasci­ nating tale, it makes one long for more. Department of Mathematics & Computer Science Santa Clara University Santa Clara, CA 95053-0290 USA e-mail: [email protected]

Across the Board: The Mathematics of Chessboard Problems b y John J. Watkins PRINCETON UN IVERSITY PRESS, 2004. 264 PP. US $24.95. ISBN 0-691 - 1 1 503-6

REVIEWED BY ARTHUR T. BENJAMIN

think I became a mathematician be­ cause I loved to play games as a child. I learned about probability and expec­ tation by playing games like backgam­ mon, bridge, and Risk But I experi­ enced the greater thrill of careful

I

deductive reasoning through games like Mastermind and chess. In fact, for many years I took the game of chess quite seriously and played in many tournaments. But I gave up the game when I started college and turned my attention to more serious pursuits, like learning real mathematics. So when I first picked up a copy of Across the Board: The Mathematics of Chessboard Problems, I was pleasantly surprised. I think I expected to find mathematical analysis of questions like "How should White play to mate as quickly as possible?" or "What sequence of moves produced this sequence of po­ sitions?", as were effectively described in the recent Intelligencer article by Elkies and Stanley [ 1 ] . Instead, what I found was a delightfully written book on "real mathematics," loaded with the­ orems with elegant proofs, directed at problems that arise on the chessboard. Aside from knowing how each chess piece moves, the reader does not need to know how to play chess nor any of its notation. The book begins with a discussion of knight's tours, whereby a chess knight visits every square on a chess­ board exactly once, beginning and end­ ing the tour at the same square. Al­ though knight's tours exist on an 8-by-8 chessboard, they do not exist for boards of all sizes. For instance, it is easy to see that a knight's tour is im­ possible when m and n are both odd, since the color of the square alternates at each move; the knight could not re­ turn to its original square in mn moves. Knight's tours are clearly impossible for 1-by-n and 2-by-n boards, and by colorful arguments can be proved im­ possible for boards of size 4-by-n, 3-by6, and 3-by-8. The book then outlines a proof of "Schwenk's Theorem," which says that all rectangular chessboards have a knight's tour, except for the aforementioned cases. Since knight moves alternate colors, then on an 8-by-8 chessboard, we could place 32 knights on all the black squares or all of the white squares, and no two knights will attack each other. No other arrangements of 32 or more knights is possible, for such an arrange­ ment would necessarily use two adja-

cent squares in a knight's tour. In gen­ eral, it can be shown that the maximum number of "independent" knights on an n-by-n chessboard is ln2/2 l (where lxl denotes the ceiling of x). How many non-attacking bishops can be placed on an n-by-n board? Since there are 2n - 1 positively slop­ ing diagonals, including the length-one diagonals of the lower right and upper left square, there can be at most 2n 2 such bishops. One way to achieve this is by placing bishops on every square of the first and last row, except for the rightmost squares. Watkins goes on to prove that there are exactly 211 ways to accomplish this, and all of them must have the bishops only oc­ cupy the outer ring of the chessboard. In a similarly thorough fashion, the book also addresses the related prob­ lems of "domination," such as how many kings can be placed on an n-by­ n board so that every square is either occupied or attacked. (Here the an­ swer is ln!W.) As with the indepen­ dence questions, the book provides satisfying answers to domination ques­ tions for all pieces, and on a variety of surfaces, including toroidal chess­ boards and "Klein bottle" boards. I found Watkins's style of writing very engaging, as one would expect for a book on recreational mathematics. Each chapter begins with some easy re­ sults and builds gradually in sophisti­ cation, much like a Martin Gardner ar­ ticle, culminating in numerous open problems suitable for exploration by undergraduates. Indeed, many of the theorems in the book arose from joint work of the author and several under­ graduates. I frequently found myself earmarking theorems and problems that could be used as interesting home­ work questions for the next time I teach discrete mathematics or graph theory. So at the risk of offending "rank and file" mathematicians, I would say that if you are looKING for a book that can capture your imagination or igKNIGHT a passion for discrete mathematics, then buying a copy of Across the Board would be a great move. I guarantee that these "board problems" will not lead to "bored readers."

REFERENCE

[ 1 ] N. D. Elkies and R. P. Stanley, The Mathe­ matical Knight, Math. l ntelligencer 25 (2003), no. 1 , 22-34. Harvey Mudd College Claremont, CA 91 71 1 USA e-mail: [email protected]

The (mis)Behaviour of Markets by Benoit Mandelbrot and Richard L. Hudson BASIC BOOKS, NEW YORK, 2004, ISBN 0-465-04355-0, $27.50

REVIEWED BY ERIC GRUNWALD

andelbrot, on whose behalf this book is written in the first person singular despite the presence of a sec­ ond author, believes that modem fi­ nancial theory is wrong. He thinks that the theory underestimates the proba­ bility of large price movements, and that markets are therefore riskier than current theory indicates. Written for the general reader, the book mixes anecdotes and theoretical discussion cleverly and entertainingly. Anybody who hasn't yet read about fractals will find them well exposed here. Mandelbrot devotes a lot of space to demonstrating his view that price variations (month-to-month, day-to­ day, minute-to-minute) are not distrib­ uted normally, but according to a power law. He would replace conven­ tional models with one based on two parameters: a, the exponent in the power law; and H, a measure of the ex­ tent to which the direction of price variations persists over time. How should one judge a book on a scientific theme written for the non­ expert? I think there are three key questions.

M

1 . Is It Well Written?

Page 13 is discouraging. Here, the pe­ riod from 1916 to 2003 is described as "Truly, a calamitous era that insists on

© 2005 Spnnger Sc1ence+ Bus�ness Med1a, Inc., Volume 27, Number 3, 2005

77

flaunting all predictions." Either the era displays an unusual form of exhi­ bitionism or the authors mean "flout­ ing." On page 148 we learn that "For more than a century, the New York Cotton Exchange had kept exacting, daily records of prices . . . . " Do the au­ thors really mean "exacting," i.e., "mak­ ing great demands, calling for much ef­ fort," or should it be "exact"? There is a more serious problem on page 1 1 , where Mandelbrot defines the Efficient Market Hypothesis (EMH): " . . . in an ideal market, all relevant in­ formation is already priced into a se­ curity today." The next sentence reads, "One illustrative possibility is that yes­ terday's change does not influence to­ day's, nor today's, tomorrow's; each price change is 'independent' from the last." But what is an "illustrative possi­ bility"? Is it a consequence?-if so, the chain of logic should be explained, be­ cause it isn't obvious to me that the second quoted sentence follows from EMH. Or is it something that doesn't contradict EMH? Or is it another tenet altogether? The difference between these interpretations is critical, be­ cause the book is largely a picking apart of the notion that price changes are independent of each other. 2. Do I Trust the Authors'

Judgment?

In a book like this, many of the tech­ nical details are necessarily omitted, and the reader needs to take the au­ thors' word on trust. So it is important for the authors to build up the readers' confidence. In this particular case, the field is one where great care needs to be taken over the interpretation of con­ fusing, noisy data. The evidence here is generally presented graphically, to which I have no objection, but the chart on page 205 caught my eye. It compares the Cisco Systems stock price with what Mandelbrot calls the stock's "real" value, i.e., earnings per share, and purports to show how the price rose well above the "real" value during the dot-com bubble before the 2001 collapse. Unfortunately, however, a cursory glance at the chart suggests that the "real" value was even more turbulent than the price. I give three

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THE MATHEMATICAL INTELLIGENCER

further examples that made me won­ der whether to accept the authors' judgment. Having started off by assassinating modem financial theory, which is "founded on a few shaky myths" (chap­ ter 1) and "riddled with false assump­ tions and wrong results" (chapter 5), Mandelbrot desecrates the corpse in chapter 12 with "ten heresies of fi­ nance." He is particularly harsh on EMH. Now, I had always thought of EMH as a useful simplifying assump­ tion that has a strong element of truth without being absolutely correct (how could every piece of information be known to every market player?), but nevertheless enables useful deductions to be made, the variations of which from reality can yield rich insights. Mandelbrot, however, regards EMH as something in which all "conventional" economists and financiers believe im­ plicitly and absolutely, and picks it apart with the glee of a child attacking a scab. Mandelbrot portrays himself as a doughty warrior against the academic establishment. An early paper was pub­ lished as an internal research report by IBM. "The noise from academia was loud. Who was this Mandelbrot fellow, a grimy industrial scientist with a de­ gree in applied mathematics, to chal­ lenge the elaborate models of the eco­ nomics elite?" (p165). When he tried to publish in economics journals, some people even had the cheek to ask "me to spell my name" (p166), but he was not above using a connection to "skip the usual, time-consuming process of having a paper read, critiqued, and dis­ sected by academic 'referees' " (p166). The whole business of peer review is tiresome; on page 191 we read how Mandelbrot went straight to a journal editor. "He agreed on the spot to pub­ lish, without the usual routine of call­ ing academic referees. Such forceful editors are the salt of the earth, but rare in scientific publishing. More com­ mon is the risk-avoiding bureau­ crat, nailed to an influential editorial chair." Even colleagues sometimes balk him (p190): "I tried collaborating with a Harvard hydrologist, but his computer program produced garbage.

(For which he blamed me: The course of scientific collaboration seldom runs smoothly)." I began to wonder what it must be like to be Mandelbrot's col­ league or editor. On page 255 we are told that "We place control of the world's largest economy in the hands of a few elderly men, the central bankers. We do not understand what they do or how, but we have blind faith that they can some­ how induce the economic spirits to bring us financial sunshine and rain. . . . " And so on. Well now, three points spring to mind. (a) Do "we" really have such blind faith? (b) Are these old men really in control of the US economy? (c) Considering the overall perfor­ mance of the US economy since the Second World War, it looks as if the el­ derly men, if indeed they are in control, haven't done a bad job. 3. After Reading the Book, Will I

Look at the World in a New Way?

Buried like an undescended testicle in the body of this book is a bizarrely one­ sided conversation under the heading, "In financial markets, the idea of 'value' has limited value" (p249). The imag­ ined reader's responses are so differ­ ent from my own that I give my reac­ tions in italics. The authors' voice tells us that econ­ omists and financial analysts like to calculate the "value" of a commodity or a company because, although prices fluctuate, "there is something in the hu­ man condition that abhors uncertainty, unevenness, unpredictability. People like an average to hold onto, a target to aim at-even if it is a moving target." The voice continues, "But how useful is this concept, really? What is the value of a company? Well, you say, is it the price the market in its collective wisdom hangs on it?" Well, perhaps I might say that. The authorial voice then points out that prices of companies fluctuated wildly during the technology boom and bust: "Ah, you say, it was not the com­ pany's business fundamentals, but the market's appetite for technology com­ panies that changed-and this is as much a part of the measure of intrinsic value as balance sheet or cash flow."

Er, I'm not sure that I would say that. The voice then points out that the

By whom ?

bulent," "inherently uncertain," and "deceptive. "

"So how, you ask, does one survive in such an existentialist world, a world

"real" value of a company would then

without absolutes?"

change all the time and asks, "What use

What can I say?

• They are strong determinists. "Why

even talk about chance in financial

markets? The very idea clashes with

is a valuation model with new param­

I dwell so long on this passage be­

every intuition we have about the

eters for every calculation?" The answer

cause it brought to a head the strange

way society, commerce, and finance

comes back, "Point taken, you say."

feeling of dislocation I had while read­

work" (p27). Really? • They are the purest of the pure. They

No, really, I don 't say that.

ing the book. It is not aimed at me, or

The voice, gaining confidence with

at anybody I know. Intended readers

each rhetorical triumph, now suggests

need, as we have seen, to believe in

demic purists, you would find lots of

that value is perhaps a function of cost,

something called the "intrinsic value"

things that look plain wrong on a typ­

and asks, "What is the cost of Microsoft

of a commodity and to have blind faith

ical, real-world trading floor" (p80).

Office software? Easy, you say."

in US central bankers. Such readers, let

I absolutely don't say that. How could anybody say that?

us call them mandelbrothers, will find

It might well have been possible to

their beliefs exposed to relentless ar­

fashion a paradigm-changing book out

Brushing the reader aside dismis­ sively with a mere glimpse of the diffi­

gument. What else can be deduced

of Mandelbrot's ideas. But while man­

about mandelbrothers?

delbrothers may look at the world in a

culties of calculating costs, the voice concludes once again, "Point taken, you say. "

need to be told that "in the eyes of aca­

different way after reading this par­ • They

believe

that

markets

are

smooth, predictable, and transpar­

ticular book, earthling mathematicians are likely to be less excited.

I'm getting .fed up with this. I say no such thing.

of finance," which appear to be "a

Perihelion Ltd.

Winding down now, the author con­

collection of-to m�bvious facts"

1 87 Sheen Lane

cludes "that value is a slippery concept,

(p226) that the author thinks are

London SW1 4 SLE

and one whose usefulness is vastly

heretical to received wisdom. Three

UK

over-rated."

of these state that markets are "tur-

e-mail: [email protected]

ent. This follows from the "heresies

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Volume 27, Number 3, 2005

79

..jfi 1.19·h•i§i ..

Robi n Wi lson

The Philamath's Alphabet- ! l

The main achievements of early Egyptian mathematicians in­ volved the practical skill of measure­ ment. The oldest of the Egyptian pyra­ mids is King Djoser's step pyramid in Saqqara, built in horizontal layers and dating from about 2700 BC. It was sup­ posedly designed by Imhotep, the cel­ ebrated court physician, Grand Vizier and architect. Impossible obj ect: Several artists, notably Maurits Escher, the 'master of optical illusion', have incorporated into their designs certain 'impossible features' that cannot exist in threemhotep:

I

dimensional space. The impossible ob­ ject shown here was one of several drawn in 1934 by the Swedish artist Os­ car Reutersvard. Integral sign: In the fall of 1675, Gott­ fried Wilhelm Leibniz introduced two symbols that would forever be used in calculus. One was the d (or dyldx) no­ tation used for derivatives. The other was the integral sign. Attempting to find areas under curves by summing lines, he defined omnia l (the sum of the ls ), which he then represented by an elongated S. Integral signs appear on two of the stamps shown here. Integrated circuit: The development of printed circuit boards in the 1960s led to the introduction of the all­ important integrated circuit, an assem­ bly of thousands of transistors, resis­ tors, capacitors, and other devices, all interconnected electronically and pack­ aged as a single functional item. This led, in tum, to the first personal com­ puters in the 1970s.

POSTA

International Congresses of Math­ ematicians: As part of the 400th an­ niversary celebrations of Columbus's voyage to America, a World Congress of Mathematicians took place in Chicago in 1893. Since the first meet­ ing, more than twenty ICMs have taken place, usually every four years. Those in Moscow (1966), Helsinki (1978), Warsaw (1983), Kyoto ( 1990), and Berlin (1998) were commemorated on stamps. International date line: The mea­ surement of time has long been a mat­ ter of concern. In 45 BC Julius Caesar introduced his 'Julian Calendar', but the Julian year proved to be 1 1 minutes too long. In 1582 Pope Gregory introduced the Gregorian calendar, eventually to be adopted by Britain and the American colonies in 1752. In 1884 the line from which time is measured (0° longitude) was located at the Royal Observatory in Greenwich; the Astronomer Royal at the time was Sir George Airy.

�55

§fy,ds •

Impossible object

BANI

Integral sign

Integrated circuit

lmhotep

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

80

International Congress of Mathematicians

THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Busrness Medra, Inc

International date line