No title

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

Of course, the change in title had ab­

Compared to What? When my article "On the Unreasonable Effectiveness of Mathematics in Mo­

solutely no effect on my remarks. Now Prof. Gelfand's remark has

lecular Biology" appeared

(The Math­ ematical InteUigencer 22, no. 2), I

forced me to rethink

observed with wry surprise the box

fective in biology, for rationalizing ob­

containing the Wigner-Gelfand princi­

servations. However, biology lacks the

inef­

magnificent compression of the physi­

ple, asserting "the unreasonable

Mathematics is unquestionably ef­

fectiveness of mathematics in the bio­

cal sciences, where a small number of

logical sciences."

basic principles allow quantitative

Perhaps the genesis of the title of

diction of many observations to high

my article would interest Prof. Gelfand

precision. A biologist confronted with a

and other readers.

large body of inexplicable observations

In 1998, The Isaac Newton Institute

does not have faith that discovering the mathematical

structure

will

for Mathematical Sciences in Cam­

correct

bridge, U.K., held a programme, "Bio­

make sense of everything by exposing

molecular Function and Evolution in

the hidden underlying regularities.

the Context of the Genome Project." I

The problem is that historical acci­

proposed to one of the organizers (a

dent plays much too important a role.

biologist) that I might speak at the gala

A famous physicist once dismissed my

final symposium. I made this sugges­

work, saying: "You're not doing sci­

tion with some diffidence-the sym­

ence, you're just doing archaeology!"

posium would be attended by numer­ ous mathematical dignitaries, it would

I'd like to think this was unfair to me

take place in the same room where

it emphasizes a genuine and severe ob­

(it certainly was to archaeologists), but

Andrew Wiles initially announced the

stacle to applications of mathematics

proof of Fermat's Last Theorem . . .

in biology.

The following exchange ensued:

Prof. Gelfand may consider it not

"What would you speak on?"

only wrong but ingenuous to believe

"I propose an echo of E.P. Wigner

that mathematics will overcome these

as my title: 'On the unreasonable inef­

and perhaps other limitations. In my

fectiveness of mathematics in molecu­

case, on the other hand, it was the con­

lar biology.' "

viction

A prolonged and uneasy silence. Then: "But, you see, this is not quite the message that we want to send

that

mathematics will

ulti­

mately succeed, that motivated my original title. What is the conclusion?

Is mathe­

these people." More silence. Then:

matics effective in biology? I must fall

"Would you consider changing 'inef­

back on Henny Youngman's famous re­

fective' to 'effective'?" Had I the stature of a Gelfand I

sponse when asked "How's your wife?" He retorted, "Compared to what?"

should no doubt have refused, but with my eye on the opportunity I acqui­

Arthur M. Lesk

esced. After all (it was easy to per­

Department of Haematology

suade myself), isn't it merely a ques­

University of Cambridge Clinical School

tion of whether the cup is half full or

Wellcome Trust Institute for Molecular

half empty? I thereby missed what will

4

pre­

Mechanisms in Disease

surely be my only chance to have my

Wellcome/MR C Building

name linked with Wigner and Gelfand.

Hills Road, Cambridge, CB2 2'XY, U.K.

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

Look Again at Vilnius

It was saddening to read "Vilnius Between the Wars" (Mathematical Tourist, The Mathematical InteUigen­ cer 22, no. 4)-frrst because the au­ thors seem oblivious to the terrible or­ deal of Wilno during World War II, and second because they give such inade­ quate treatment of Antoni Zygmund, who was surely one of the great glo­ ries of Polish mathematics. The article starts with a casual men­ tion of the expansion of the Polish mathematical school "in the beginning of the 20th century." Poland gained in­ dependence from Russia in 1919 and lost it to the Third Reich in 1939; it was not by chance that Polish mathematics flourished precisely during that 20year period. The authors describe a historical stroll around the city of Vilnius-Wilno up to 1940, and its learned institutions, without even a single mention that Wilno was not only a Polish university town but had a long tradition as a thriv­ ing center of Jewish culture. In 1939 Wilno (or Vilna) was home to about 60,000 Jews, of which less than 3,000 remained after WWII. I mention these facts as significant in the assessment of Zygmund's singularity at the Uni­ versity of Wilno. And the years of Zygmund at Wi1no are the city's main claim to mathematical fame. Zygmund became professor and head of the Mathematics Department at Wi1no in 1930, as mentioned in the article, and the years he and his family spent there were probably the very best of his long and productive life. The impact of Zygmund's personality was strongly felt at the University. He not only put Wilno on the international map mathematically, but he, a Catholic Pole by birth, took a leading role in the academic fight for the rights of mi­ norities (Jewish, Lithuanians, Ukrain­ ians, Russians-all systematically dis­ criminated against in Poland), and of those striving for social justice. At that time students' associations were re­ quired to have "curators," that is, fac­ ulty sponsors, responsible for them be­ fore the University Senate. Zygmund

became curator of various minorities' associations and of the Association of Independent Socialist Youth (the great Polish poet Czeslaw Milosz was a member). Members of this Association were frequently jailed, and it was up to their curator to get them out. During one of the periods of repression of Polish universities by the nationalist­ militarist regime in power, Zygmund was dismissed from his post. Letters of protest from the world mathematical community, especially from his fa­ mous colleagues in Great Britain and France, succeeded in getting him rein­ stated. [1] During his tenure at Wilno, Zygmund was one of only two professors at the University who refused to accept the ban of Jewish students from his classes. He only mentioned this brave stance many years later, when accused of antisemitism by colleagues who were much concerned for the rights of Jewish mathematicians in the Soviet Union but not for Jewish mathemati­ cians persecuted elsewhere. Zygmund, international in science and cosmopolitan in culture as he was, was ardently attached to Poland, and so was his wife. They lived exile as a permanent loss, and their apartment in Chicago was kept as if they had still been in Warsaw. After the end of WWII the Zygmunds considered returning home; personal circumstances delayed the project beyond realization. Pro­ fessor Zygmund visited his three sis­ ters in Poland frequently, and kept in touch with the mathematical life of his native country, especially through his involvement with Studia Mathemat­ ica, which became a main journal of publication for his large (and interna­ tional) school. Zygmund's lasting fame is due to his deep impact in developing harmonic analysis both in the classical setting and going far beyond, to that of n-di­ mensional function theory and the the­ ory of operators, with great impact in tum on probability, partial-differential equations, and several complex vari­ ables. He should be remembered equally as teacher, and for his unique

eye for highly talented students. He spotted them quickly, encouraged them intensely, introduced them early to his level of research, and took them as collaborators on equal footing. When Zygmund was in his early thir­ ties, his frrst extraordinary student was J6zef Marcinkiewicz, a first-year stu­ dent at the University of Wilno. [1] The results of their nine-year-long fruitful collaboration are still central to all of harmonic analysis and its many appli­ cations, even more than 60 years after Marcinkiewicz's death. He was killed at age 30 during the war, probably by the Soviet Army at Katyn Forest. 1 At the out­ break of World War II, both Zygmund and Marcinkiewicz were mobilized into the Polish Army; while Marcinkiewicz was captured by the Soviet Army and disappeared, Zygmund, after the Polish Army's defeat, succeeded in returning to Wilno-already renamed Vilnius and in Lithuania. From there he tried to save his family and his life from the immi­ nent danger of Nazi invasion. The terrible months from November 1939 to March 1940, when he was try­ ing to secure a way of escape--a job, any job, a visa, some route to safety not yet cut by the Nazis-are summarized in the article by "Thanks to friends such as J. Tamarkin, Zygmund would move to the USA in 1940, where his mathe­ matical career flourished." Those in­ volved in saving Zygmund's life were not only his friend J. Tamarkin, but also Norbert Wiener, with whom he had already collaborated, and Jerzy Neyman, who admired him greatly. It was not easy in those days, not even for people of their standing, to help others immigrate. And when after many efforts they succeeded, it was certainly not with a flourish: Zygmund went from his sig­ nificant and world-recognized position at Wilno to teach for five years at Mount Holyoke College, a girls' college isolated from the mathematical cen­ ters, with heavy teaching of trigonom­ etry and analytic geometry. Not only did he never complain, but he was al­ ways grateful and appreciative of Mount Holyoke College. Thanks to his

'This is what Zygmund always believed most probable. The authors of 'Vilnius between the wars" state it as definite, perhaps on the basis of recently opened Soviet archives.

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001

5

extraordinary scientific output, he was

Stefan Banach" (or should it be "assis­

even though they refer to Zygmund's in­

able to move, first to the University of

tant to Professor Banach"?), he never

troduction to Marcinkiewicz's collected

Pennsylvania in

works, of which he was the editor?

1945, and two years

worked with Banach, and he did con­

later to the University of Chicago,

tinually work with Zygmund from his

Zygmund's list of honors appears in

where his impact on world mathemat­

thesis on. To say that he "MIGHT

the biography by W. Zelazko contained

ics became legendary.

HAVE BECOME A FIRST-RANK MATH­

in Zygmund's selected papers, which

Only those who never suffered dan­

EMATICIAN [caps mine] if destiny had

the authors cite. It is too bad that

ger of persecution, or even thought

been as favorable to him as it was to

they didn't bother to cite those from

about its effects on others, can see

Zygmund" is utterly wrong-HE DID

"minor" countries like Argentina and

forced exile and dislocation of life as

BECOME

MATHE­

Spain, which made Zygmund an acad­ emician and honorary doctor at differ­

A

FIRST-RANK

an opportunistic career move. Appre­

MATICIAN EVEN IF HE DIED AT 30-

ciative though Zygmund was for being

and it insults both his memory and

ent universities. From Argentina came

able to survive the war, settle down in

Zygmund's.

Marcinkiewicz is recog­

A.P. Calderon, the second extraordinary student of Zygmund, and his collabora­

the USA, surmount the difficulties of

nized today largely because Zygmund

exile, and continue doing mathematics

survived

more and more fruitfully, he never con­

champion.

the

war

and

became

his

tor during the long second half of his ca­ reer. I find it ironic that the authors

the

There can be no doubt about the in­

chose to omit those countries in which

University of Wilno as a career im­

trinsic value of Marcinkiewicz's con­

the influence of Zygmund as a teacher

provement.

tributions. But there can be no doubt

and spotter of talent was largest outside

sidered

In

being

the

expelled

from

correspondence

either that Zygmund's devotion to his

of the USA-the ones the master him­

1939,

memory was unique. It may be of in­

self always compared with his native Poland: small countries where mathe­

between

Zygmund and Tamarkin in late

there are repeated mentions of the dis­

terest to your readers to learn that in

appearance of Marcinkiewicz, as well as

1939, and while already in captivity,

matics could flourish through local tal­

concern for Saks-his best friend, who

Marcinkiewicz was able to write a

ent, but only when favorable political

was ultimately killed by the Gestapo­

mathematical letter to Zygmund with

conditions permitted.

Kuratowski, Sierpiff;ki, Mazurkiewicz,

some ideas on how to expand the re­

and many others with whom he had lost

sult of a short note to the

contact. Many of his colleagues were

Rendus

killed during the war. When years later

Sciences (which was to be his last pub­

to The Mathematical Tourist. I reproach

Cambridge University Press published

lication). In the note Marcinkiewicz

the article for displaying loss of histor­

Zygmund's monumental two-volume

had sketched one of his claims to pos­

ical memory prevalent in countries with

treatise

it ap­

terity, the now famous Marcinkiewicz

a not-too-distant past of repression and

peared "dedicated to the memories of

Interpolation Theorem, for the so-called

exclusion. In those communities there

Trigonometric Series,

A. Rajchman and

J. Marcinkiewicz, my

teacher and my pupil."

I have been reacting to the article, re­

Comptes

ally, in terms appropriate to the section

of the Paris Academie des

Mathematical Communities more than

"diagonal case." Zygmund explained

is a tendency to blame the victims, and

that result to his Chicago students and

to reproach as unpatriotic the survival

presented in his seminar the general

in exile of those forced to leave their

astonishing.

case-which is by no means an evident

homelands, while ignoring the circum­

They praise his talent, and quote a re­

extension-while insisting that he was

stances that often make their return dif­

sult in his thesis, but then they sum up

only developing Marcinkiewicz's own

ficult or impossible.

by saying that Marcinkiewicz "spent

ideas. In

About Marcinkiewicz, the authors' portrayal

is

positively

1956, when the whole result

some time at the famous Lvov mathe­

was worked out, Zygmund published a

References

matical school and THIS LED TO SEV­

paper with the theorem giving his for­

1 . Wirszup, I . , Antoni Zygmund, 1 900-1 992, in

ERAL PAPERS IN THE LVOV STUD/A

mer student full credit. The fame of the

The University of Chicago Record,

MATHEMATICA

result is proportional to its constant

2 1 , 1 995, pp. 1 2-1 3.

[my caps]." It is a bit

biased to say that only Lvov had a "fa­

use by analysts everywhere. It is hard

mous" school of mathematics, but it is

to

bad history to conceal that the papers

body paying such homage to a long-dead

Department of Mathematics

in Studia were the product of the con­

colleague. How could the authors fail to

Howard University

tinuing

Washington, DC 20059

work

realize the relation between the two greatest mathematicians associated with

USA

ever "qualified as assistant professor of

Wilno during the period of their article-

e-mail: [email protected]

indeed

author

Cora Sadosky

with

6

the

of another example of some­

Marcinkiewicz

Zygmund. If

of

think

THE MATHEMATICAL INTELLIGENCER

January

Opinion

Is Mathematics W

ith very few exceptions mathe­ maticians have always believed,

lowed to say that n is either prime or

composite before anyone makes such

and still believe, that mathematical

a test?

truths have a strange kind of abstract

give his game away?

If Hersh agrees, does

this not

llOut There''t

reality that is discovered, not created.

Primality is a timeless property of

In recent years a tiny minority of mav­

certain integers, as independent of hu­

erick mathematicians have joined the

manity as pebbles and stars. Humans

Martin Gardner

postmodem ranks of the social con­

are not needed to test a pile of pebbles

structivists who see both math and sci­

for primality. It can be done by mon­

ence as cultural artifacts, unrelated to

keys or even mindless machines. To re­

any sort of timeless truth domain.

ply to this by saying that humans are

Reuben Hersh, a distinguished mathe­

necessary to have the

matician, has long defended this anti­

prime and to call numbers prime is to

realist view, notably in his 1997 book

say something utterly trivial.

the international mathematical

What is Mathematics, Really? If all Hersh means is that mathe­

women?) from Quasar .X9 sent us their

community. Disagreement and

matics is part of human culture, then

math textbooks, we would fmd again

of course he is right, but the statement

A = 1rr2." If the aliens haven't advanced to

The ()pinion column offers mathematicians the opportunity to write about any issue of interest to

controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-chief endorses or accepts

is vacuous.Everything humans say and

Myth

concept of a

3: "If the little green men (and

do is part of culture. Hersh obviously

plane geometry, their textbooks of

means something less trivial. He is con­

course would not contain this theo­

vinced that mathematicians do not dis­

rem, but if they knew about circles and

cover timeless

areas how could they not discover that

theorems-theorems

true in all possible worlds. Rather they

a circle's area is

1r

r 2? Given the axioms

responsibility for them. An ()pinion

create ever-changing, uncertain con­

of plane geometry, the theorem holds

should be submitted to the editor-in­

jectures in much the same way that

in all possible worlds.

chief, Chandler Davis.

others create art, music, religion, wars,

Myth 4: "Mathematics possesses a

and traffic regulations. In an article in

Eureka (March 1988), he discusses

attains absolute certainty of conclu­

several "myths" which he claims have

sions, given the truth of the premises."

been mistakenly defended by great mathematicians.

Can Hersh be serious when he calls this a myth?

No one can quarrel with Hersh's first myth, that Euclid put plane geom­ etry on

method called 'proof' ...by which one

In mathematics, unlike in

science, proof is the essence. Given the symbols, and the formation and trans­

a firm formal foundation. All

theorems are tautologies. They are, as

2: "Mathematical truth or

synthetic. Even in the center of the

mathematicians today agree that he did not. Myth

formation rules of a formal system,

all

Kant was the first to say, analytic, not

knowledge is the same for everyone. It

sun, Bertrand Russell once wrote, two

does not depend on who in particular

plus two equals four.

discovers it; in fact, it is true whether or not anyone discovers it."

I once put it this way. If two di­ nosaurs joined two other dinosaurs in

What a strange contention! In no

even though no humans were around

the Pythagorean theorem not certain

to observe it, and beasts were too stu­

culture on earth, or anywhere else, is

within the formal system of Euclidian geometry.

a clearing, there would be four there

pid to know it. Mathematical structure

was deeply embedded in the universe

Consider a heap of n pebbles. The

number is prime only if, when you take

long

before

sentient

life

evolved.

Indeed, the structure was there a mi­

k each

crosecond after the big bang, and even

(k not 1 or n) , there always will be one

before the bang because there had to

away pebbles in increments of

or more pebbles left over. Are we al-

be quantum fields to fluctuate and ex-

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1, 2001

7

plode. Is Hersh willing to say that galaxies had a spiral structure before creatures were around to use the word "spiral"? No mathematician, Roger Penrose has observed, probing deeper into the intricate structure of the Mandelbrot set, can imagine he is not exploring a pattern as much "out there," indepen­ dent of his little mind and his culture, as an astronaut exploring the surface of Mars.

To imagine that these awesomely complicated and beautiful patterns are not "out there," independent of you and me, but somehow cobbled by our minds in the way we write poetry and compose music, is surely the ultimate in hubris. "Glory to Man in the high­ est," sang Swinburne, "for Man is the master of things."

In the light of today's physics the en­ tire universe has dissolved into pure mathematics. The cosmos is made of molecules, in tum made of atoms, in tum made of particles which in tum may be made of superstrings. On the pre-atomic level the basic particles and fields are not made of anything. They can be described only as pure mathe­ matical structures. If a photon or quark or superstring isn't made of mathe­ matics, pray tell me what it is made of?

3001 Chestnut Road Hendersonville, NC 28792, USA

Mathematical Olympiad Challenges Titu Andreescu, American Mathematics Competitions, University of Nebraska, Lincoln, NE Riizvan Gelca, University ofMichigan, Ann Arbor, MI This is a comprehensive collection of problems written by two experienced and well-known mathematics educators and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from decades of national and international competitions are presented, encouraging readers to move away from routine exer­ cises and memorized algorithms toward creative solutions and non-standard problem-solving techniques. The work is divided into problems clustered in self-contained sections with solutions provid­ ed separately. Along with background material, each section includes representative examples, beautiful diagrams, and lists of unconventional problems. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on stimulating readers to find ingenious and elegant solutions to problems with multiple approaches. Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teach­ ers and students training for mathematical competitions and for teacher professional develop­ ment, seminars, and workshops. From the foreword by Mark Saul: "The book weaves together Olympiad problems with a com­ mon theme, so that insights become techniques, tricks become methods, and methods build to

I 280 PP., 85 ILLUS.

mastery... Much is demanded ofthe reader by way of effort and patience, but the investment is

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Textbook evaluation copies available upon request, call ext. 669. Prices are valid in North America only and subject to change without notice.

Birkhiiuser Boston -International Publisher for the Mathematical Sciences

8

THE MATHEMATICAL INTELLIGENCER

7/00 Promotion #YI097

I $29.95 $59.95

XIAORONG HOU, HONGBO Ll, DONGMING WANG, AND LU YANG

"Russian Killer" No. 2: A Challenging Geometric Theorem

with Human and Machine Proofs

� •

n February 1998 Sergey Markelov [7] from the Moscow Center for Continuous Mathematics Education sent a set of five geometric theorems to Dongming Wang for test­ ing the capability of his GEOTHER package [8], with the aim of presenting a chal­ lenge to computer provers to prove really hard theorems. These theorems have been

used to prepare the Moscow team for the all-Russia school mathematics Olympiad, and are called kiUers to analytic ways ofgeometric problem-s olv ing. They can be proved in geometric ways, but no analytic proof could be found even by expert geometers. Let us call these five theorems the Russ ian kiUers for short. After a quick look at the five killers, Wang was convinced that some of them can be proved by GEOTHER in princi­ ple. For experimental purposes, he took the second of the killers which is stated below. This killer is very easy to ex­ plain and to understand, and it provides a beautiful repre­ sentation of the area of an arbitrary quadrilateral in terms of its four sides and four internal angles. Theorem. Let ABCD be an arbitrary quadrilateral with s ides !ABI = k, !Bel = l, ICDI = m, IDAI = n, and intern al angles 2a, 2b, 2c, 2d at v ertices A, B, C , D respectiv ely; and let S be th e area of th e quadrilateral. Then (k + l +m + n)2 48 = cot a + cot b + cot c + cot d (l +n- k - m)2 tan a + tan b + tan c +tan d' The beauty of the expression lies partially in the sepa­ ration of the sides and internal angles. The theorem gen­ eralizes the well-known Brah maguptaformula for the case where the quadrilateral is inscribed in a circle. After a few trials, Wang announced a machine proof of the theorem using Wu's method [11] in GEOTHER in Aprill998; this proof requires heavy polynomial computations. Meanwhile, he posted the theorem to several colleagues, so­ liciting other machine or human proofs. Soon after that,

Hongbo Li announced another machine proof using Clifford algebra formalism, followed by the third machine proof given by Lu Yang using complex numbers, both in May 1998. The proofs of Li and Yang are short and took only a few seconds of computing time. Finally in later May 1998, Xiaorong Hou discovered an elegant and short geometric proof of the the­ orem. This proof reflects the common features of traditional geometric proofs, in which one may just have to be inspired. This article collects the four proofs, together with an al­ ternative approach using rules of trigonometric functions provided by an anonymous referee. Our purpose is twofold: on the one hand, different kinds of possibly new proofs of a difficult geometric theorem are presented that have clear interest for geometers. On the other hand, the proofs demonstrate the power, capability and features of auto­ mated deduction methods and tools-which reduce quali­ tative difficulty to quantitative complexity instead of rely­ ing on individual ingenious ideas-for proving hard theorems: the Chinese provers against the Russian killers! D

Fig. 1

B

© 2001 SPRINGER-VERLAG NEW YORK. VOLUME 23. NUMBER 1, 2001

9

We look for the advantages and disadvantages of machine proofs versus human proofs. Proving mathematical theorems automatically has been an active area of research with appealing prospects for ed­ ucation. It has been advocated by many mathematicians and computer scientists. For geometry in particular, there are several very successful algebraic approaches including the well-known method of Wu [11]. We may also mention a recent interesting article about RENE by Ekhad [ 1] who has predicted the future of plane geometry (around 2050! ). The present case study not only provides another example for one to see the machine power and intelligence against human ingenuity, but also contributes to our understand­ ing about effective algorithms and software tools for au­ tomated theorem proving. A Traditional Geometric Proof In this section we give a geometric proof of the theorem by the first author. The non-trivial ideas and special tech­ niques used in the proof, besides their own value, may be illuminating as a contrast to machine proving, which is au­ tomatic, straightforward, and fast. Let 8 = {a, b, c, d) and

T = k + l +2 m + n ' t = l + n -2 k - m .' f3 I cot e. a= I tan(}, =

liE8

B Fig. 2

Corollary 2.

+ lcnl

V!FI+ IB'£1 -Icnl =!FBI- ViEI + IB'£1-Icnl T- CVtRI + lcnl) = t, IB'C'I + IDA'I = ICEI+ IDAI - IFB'I = !Bel - IBEI + InAI - IFB' 1 = ciBC1 + InAI) - r = t, Vt'B'I

=

=

so A'B'C'D has an inscribed circle. Moreover,

liA'B'C'D Corollary 3.

Proof Let

=

liES

=

T2 t2 73- -;; ·

STEP 1. We first note the following lemma, which is known

and can be easily proved.

8

= I!_ ' {3

I

=

(cot a

II AB;

So FBEB' has an inscribed circle. The details of construc­ tion are given in step 5.

STEP 3. In what follows, let liABcD or liABc denote the area of quadrilateral ABCD or triangle ABC. We have the fol­ lowing corollaries, of which the first two follow immedi­ ately from Lemma 1.

10

1,FBI -

THE MATHEMATICAL INTELLIGENCER

_

(cot

a + cot b) · T {3

2

.

b·'

a·f3 sin(a+ b) · sin(a+ d) · t

·

a·f3·y

sin(c

f3

which yields five points B', E, F, A', C', such that

T2 fiFBEB' = 73'

..!.sin d·cos

(tan a + tan b) · (cot a+ cot d)

c

STEP 2. Let ABCD be an arbitrary quadrilateral. Without loss of generality, we assume that iBCI + IDAI > W11 + ICDI and IBCI 2:: foAl. Then one can construct a diagram as in Fig. 2,

Corollary 1.

a

b) . t

·

t · T · sin

·sin 2c ·(cot c +cot d) · t · T ·sin

·

2c

a·f3

+ b) · sin(c + d) t · T ·

a·f3·y sin( 7T - a - d) · sin( 7T- a

= fiFAA'B'·

a·f3·y

STEP 4. Let I be the intersection Fig. 2. By Corollary 3, we have fiABCD = fiABEI + fi!ECD =

2a

T

(tan + tan b)

+ cot b) · T

FB' II AD, EB' II CD, B'C' II BC, A'B' IFBI + �B'I = iBEI + IFB'I = T.

(tan a+tan

=

= Vt'B'I · IFB'I · sin 2a

liEcC'B' = IB'C'I · IEB'I

Lemma 1. If ABCD has an inscribed circle, then pv 1 AD

liFAA'B'

Vt'B'I

'Y =

we have

The proof of the theorem consists of the following five steps.

,

liFAA'B' = liEcC'B'·

The formula to be proved becomes

s

f._ a

- b) · t ·T

point of B'E and A'D in

(fiFBEB' - fiFAIB) + fi!ECD

= fiFBEB'- (fiFAA'B' + fiB'A'I) + fi!ECD = fiFBEB'- (fiECC'B'- fi!ECD) - fiB'A'I = fiFBEB'- fiA'B'C'D·

The following theorem is therefore established. Theorem 1.

liABcD = liFBEB'- liA'B'C'D·

STEP 5. Construct the diagram in Fig. 3 according to the steps detailed below. It is a simple exercise to verify that the con­ structed diagram satisfies the requirements given in step 2.

•

•

•

• • • • •

Draw the inscribed circle of the three sides AB, BC , D A . Draw the tangent line GH to the circle, parallel to line CD . This produces a point G on the segment BC and a point H on the segment DA. Mark off a segment BP of length T and a segment BQ of length CW1J +!BGj + jGHj + �Aj)/2 on line BA from point B. Draw the parallel to line QH through point P, intersect­ ing line BH at point B'. Draw the parallel to line DC through point B', intersect­ ing line BC at E. Draw the parallel to line DA through point B', intersect­ ing line BA at F. Draw the parallel to line AB through point B' , intersect­ ing line DA at A' . Draw the parallel to line BC through B' , intersecting line CD at C' .

This completes the proof of the theorem. The geometric constructions used to reduce the prob­ lem in the above proof are clearly crucial and well thought out. It is not trivial to figure out such constructions and proofs even for experts. The reader is urged to work out his own geometric proofs. A Machine Proof Using Wu's Method

In contrast with the geometric proof presented in the pre­ ceding section, the machine proof explained below is straightforward. Instead of ingenious ideas, we simply write a short natural specification of the theorem, apply a geometry theorem prover-Wprover in GEOTHER [8]­ developed on the basis of Wu's method [ 11), and let the machine do the computation and proving. According to the geometric hypotheses, we have the following relations:

h1 = 28- kn sin 2a- lm sin 2c = 0, h2 = 28 - kl sin 2b- mn sin 2d = 0, h3 = k2 + n2- 2kn cos 2a- l2 - m2 + 2lm cos 2c = h4 = k2 + l2- 2kl cos 2b - m2 - n2 + 2mn cos 2d = h5 = sin (a + b + c + d) = 0.

Let e

= (a, b, c, d) as before, and Xo

=

sin 0,

Yo

=

cos 0,

0, 0,

0 E e.

By expanding the sine and cosine of double angles and sum of angles with simple substitution, the above h1, . . . , h5

will become expressions in Xa, Clearly,

l, m, n.

ho = � Thus

h1 =

..., xd, Ya, ...Yd and 8, k,

+ Jlo - 1 = o,

0, . . . , h5

=

e

E e.

0, h a = 0, . . . , hd = 0

(H)

constitute the hypothesis of the theorem, and the conclu­ sion to be proved is

Yo +tdL, Xo= . O IeEe Xo IeEe Yo

g =8-T2 1L

The statement of the theorem implies that the denomina­ tors do not vanish (e.g., x0 * 0 and y0 * 0 for every e E e). So we only need to prove that (H) implies that the nu­ merator g* of g is 0. For this purpose, let us simply apply Wu's method of automated geometry theorem proving [11). Without loss of generality, take n = 1. Then g* becomes a polynomial consisting of 91 terms. With respect to the variable ordering

Yb < xb < Yc < Xc < Yd < xd < Ya < Xa < l < k < m < 8, the set of hypothesis-polynomials h1, . . . , h5, ha .. . , hd may be easily triangularized by variable elimination into an "almost equivalent" set (called a quasi-characteristic set) of 8 polynomials:

C2 = he, C3 = h ,d � + b X d b b Y � X d b- YY Y 2( 2YEYcYdXc + 2YY 2 + ) X d - YcYdXc 4y� E Ya YbYcXbXc 4YY b Y c X a X b c + 2yElfc + 2yEy� + 2�y� YE- ic -y�, C5 = l5Xa + YY a bXcXd + YaYY a Y c X d b- YX b cX,d b X d c + YaYY + C5 = I�+ (YY b Y � X d c YY b X c d YbY�d Y�XbXcXd + Y�YdXb- YcYdX)b l - yy b X � c +YcYdXb + YbYdXc + - YcYhb YbYcY�Xd Y�XbXcX,d c7 = hm + ( 2YlE - l- 2!/a + 1 )k- l2 + 1, c8 = 8- Y�blk- yx d �, C1 C4

= =

h b,

where

/5 h

= YbYcYd- YbXcXd YdXbXc- YcXbX,d

3YEYcYdXb- YX E bXX c d + Y�YdXc + 3yY � X c d - YbYdXc- 2YY b X c d - YY c X d b + 4yy E x � x b x c d - 4yy E y � x d b- 4yy � y � x d c- 4y�y�xd + Y�YdXb- Y�XbXcXd + 3yblfcYdXc + 3yby�x,d h 2lic - l - 2y� + 1. =

=

During the computation, a polynomial factor l2 - 1 is re­ moved. The "almost equivalence" means that (H) is equiv­ alent to c1 = 0, . . . , c8 = 0 under the subsidiary condition that (D)

F Fig. 3

A

B

With this condition assumed, proving the theorem is re­ duced to verifying whether the pseudo-remainder R of g* with respect to [c1 , .. . , c8] is identically equal to 0. The remainder R is indeed 0: the verification is not easy and takes about 3 20 seconds of CPU time in Maple V.3 on an Alpha station. Some of the polynomials occurring are very

VOLUME 23, NUMBER 1, 2001

11

large. R is obtained successively as follows: Compute first

tations are performed with polynomials. It is known that

the pseudo-remainder

geometric problems may also be formalized in other al­

R8 of g* with respect to c8 in S, then the pseudo-remainder R7 of R8 with respect to c7 in m, and so forth. R is the last pseudo-remainder R1 of Rz with re­ spect to c1 in xb. Here, S, m, . . . , xb are the leading vari­ ables of c8, c7, . . . , c1, respectively. Let us use an index triple [t v 8] to characterize an ar­ bitrary polynomial P, where t is the number of terms in P, v the leading variable of P, and 8 the degree of P in v. The reduction of g* to 0 using c8 , . . . , c1 may be sketched as

computations have to be carried out according to the

follows:

proving.

g*

�

�

�

=

!

(93 S 1] � [106 m 2]� [680 k 2]

gebras. In the next proof, Clifford algebra is used to rep­ resent geometric relations, where each algebraic expres­ sion has a clearer geometric meaning. In this case, rules in Clifford algebra. The reader is referred to Chapter 1 of

[3] for a geometric introduction to Clifford algebra, [2, 5, 6, 9) for some recent developments on Clifford

and to

algebra approaches for automated geometric theorem­

A, B, C , D be considered as vectors from k = B - A, I = C - B, m = D - C , and n = A - D are also v�c!ors. Their correspond­ Let the points

the origin to the points. Then,

1

[4529 Xa 2]� [6541 Ya 6]� [19013 Xd 9], [3432 Xa 2] � [5221 Ya 6] � [17586 Xd 9], [2540 Xa 2] � [3690 Ya 4]� [11066 Xd 8], [1015 Xa 2] � [1543 Ya 2] � [6276 Xd 7], (4034 Xa 2] � [6067 Ya 6] � [17813 Xd 9] (687 Xc 9], [210 Xc 8], (549 Xc 9], (656 Xc 8], [327 Xc 9], [803 Xc 9], (697 Xc 9], [647 Xc 9], (524 Xc 9], [688 Xc 9], [667 Xc 9], [420 Xc 9], [697 Xc 9], (688 Xc 9], [283 Xc 9], [684 Xc 9], [432 Xc 9], (622 Xc 8], [523 Xc 8], [554 Xc 9], (549 Xc 9], [549 Xc 9], (544 Xc 9], [376 Xc 9], (732 Xc 8], (699 Xc 8], [696 Xc 8], [711 Xc 8], [602 Xc 7], [799 Xc 9], [810 Xc 9], [790 Xc 9], (558 Xc 9], (556 Xc 9], (549 Xc 9], (494 Xc 9], [437 Xc 8], [313 Xc 8], [165 Xc 6], [425 Xc 7], [641 Xc 7], (649 Xc 7], [621 Xc 7], [157 Xc 8], (308 Xc 9], [665 Xc 9], [545 Xc 9], [796 Xc 9], (780 Xc 9], (804 Xc 9]

ing unit vectors are denoted by Let

ll

geometrically the oriented parallelogram formed by two

[505 Xc 7], (582 Xc 9], [693 Xc 9], (519 Xc 9], [538 Xc 9], [730 Xc 8], [646 Xc 7], [218 Xc 8], [170 Xc 7], [293 Xc 7], [444 Xc 9], [718 Xc 9],

Then the hypotheses of the theorem may be expressed as follows. •

•

•

k +I +- m +n = 0. � = kk , I� li, m = �m, n = nii. Angle constraints: iik = -Za, ki = -zb, I m = - Zc, m ii

=

Unit magnitude constraints:

1,

Quadrilateral constraint:

Length constraints:

braic condition can be interpreted (automatically) in most cases, and we do not enter into the details of interpreta­

• •

•

nn = 1.

Inequality constraints:

Area constraint: As

Let •

=

s

1 -2 (m

Let

R

x;

a)

r=

Direct computation of the pseudo-remainder

on our machines without this technique is still not possi­ ble. With some thought and reasoning, the machine proof may be considerably simplified by using different formu­ lations. This can be seen from the formulation and proofs using complex numbers in the last section below.

A Machine Proof Using Clifford Algebra In the previous machine proof, geometric relations are ex­ pressed as polynomial equations with lengths of segments and sines and cosines of angles as variables, and compu-

12

THE MATHEMATICAL INTELLIGENCER

= 1,

mm

=

k, 1, m , ii * 0, Za, Zb, Zc, Zd * 0. LlADc + LlAcB, we have

k 1\ I)

1

4cn m - mn + Ik mn + Ik - kl.

=

Ze - 1 , Ze

and r =

/ eEe tan 8ll

-

kl).

= ll tan x for

8 E e.

+1

1

4 r z/I

Rll.

Then

+ 4t21I

/eEe

tan 81l.

So the conclusion to be proved has the form

However, this is already within the reach of a PC Pentium nowadays. The splitting technique in the reduction is due

ll

then

= 4(T2!{3 - t2/

In the above proof of the theorem, we made no attempt to use special techniques to simplify the algebraic formu­

= 1,

=

tan 81l =

by using the same method.

lation, so the algebraic computations are very heavy.

1\ n +

kk

Trigonometric transformations: Let tan xll any scalar

some degenerate cases in which the geometric theorem may be false or meaningless. Moreover, whether the theo­

S

= 4SI1; then s = nm -

tion. In general, the subsidiary condition corresponds to

rem is true in a special or degenerate case can be checked

-

-zd.

Therefore, the theorem is proved to be true under the sub­ sidiary condition (D). The geometric meaning of the alge­

8 E e.

e28rr,

Ze =

SI1

[10].

and n.

unit vectors. Let

1 0 }.

to Wu

k, I, m,

be a unit bivector of the plane, which represents

[j

=

s-

r =

0.

The proof of the theorem now proceeds in a way simi­ lar to that in Wu's method. The hypothesis-expressions are first triangularized and then used to reduce the conclusion­ expression to

0.

The triangulation and reduction process,

which is described in

[5, 6]

tion-solving method,

is however different.

and is called a

vectorial equa­

To make triangulation simple, we choose the following basic variables together with an ordering:

k


g:=s-r:

st:=time():g:=numer(g):time()-st; 782

t (u+ o �0).

sin 20

. u0+I , uo I

cot 0=�

tan

---

=

-t (u-o �0).

(J I .Uu=-�--+ Uo I

ll

for 0 E e. Note that U,a ub, U,c ud are not all independent; in fact, uu a u b cu= d I because 2a+2b+ 2c+ 2d= 2'lT. Hence,

u= d

I . a UUbUc

---

For brevity and without loss of generality, we may let

k= I, S = iF. Substituting all the above expressions into the polyno­ mials j3, !2, fi and removing all the extra/trivial factors, re­ spectively, we obtain

� I) h1= m2(uu � u+ � �- uu � u � �u+ (uu � u� u�+ I)u� � +4F(uu � �- I)u,� h2= -lu(a u� I) � I)uc+mn(u�uu � - 4FUU a bUc, a c. h3= lmua(U� I)+ n(u�- I)uc+4FuU Doing the same substitution and simplification for the conclusion-polynomial

T2 t2 s-+ /3

a'

VOLUME 23, NUMBER 1, 2001

13

we have g

= 1 - 2(1 + m)(l + n)p 1 + P2 + P3

P1 = Uc + Ub + Ua, UbUc + UaUc + UaUb, P2 P3 = U�U�U� + UaUbU� + UaU�Uc

Ua, Ub, Uc,

S=

l, m, n,

F.

(n sin 81 + k sin 84)/2 = -i(2u1u�u�ui - ufu�u� - uiu�ui- uiu�ui - u�u�u� + ui + u� + u� + ui - 2)/[4(u1u�- 1)(u�ui - 1)].

The identity is then easily proved by substitution of the above expressions with simplification. All the computa­

tions for the proof can be done in less than one CPU sec­

Now, what we need is only to verify whether the fol­

V. It is not surprising that we have arrived at such a short

ond in Maple

lowing holds

proof with little computation. For the proof process has in­

h1 = 0, h2 = 0, h3 = 0 :::::) g = 0, where

8E8 ,

8'

and

�

When expanded, g is a polynomial of 84 terms in

u�u§ + 1 , 2 2 U2U3- 1

cot

=

+ U UbUc.

.

=�

1 tan0= --

- 2(1 + m) (l + n) (2 + P2)UaUbUc 2 2 + (t + m2 + n + 2ln + 2m)(1 + P2 + P3) - 4F(1 - P2 + P3 - 2u�u�u�),

where

c

cot

Ua, ub, Um

volved the referee's ingenious use of trigonometric-func­

F are considered as independent pa­

l, m, n as dependent variables. This may be done by computing frrst the pseudo-remainder ii2 of h2 with respect to h3 in l, and then the pseudo-remainders ofg with respect to h 3 in l, ii2 in n and h1 in m successively. The fi­ rameters and

tion relations; the referee's identity-proving power .is brought into full play. A method for proving geometric theorems involving trigonometric identities similar to those used in the above proofs may be found in

nal pseudo-remainder was found to be 0, so the theorem is proved. With the pseudo-division function in Maple V . 3 ,

the computation of the pseudo-remainders took about 3 seconds on a Pentium 200MHz, while the whole program, including substitution, simplification and pseudo-division, may be

run in less than 6 seconds of CPU time.

[4].

Remark. Finding machine proofs for the other four Russian killers is also an attractive challenge. In fact, D. Wang has found a simple proof (in less than one second of comput­ ing time) for the fifth killer, and L. Yang has given a ma­ chine proof for the third killer. The details will be reported

Finally, we present an alternative proof using rules of

elsewhere.

trigonometric functions, which is due to an anonymous ref­ eree. Let

jACI = 1

and

Acknowledgments

lh = LCAD, fh = LACD, fh = LACE, 84 = LCAB. Denote

ei1V2

by

-t (

uk for k = 1, ..., 4.Then

cating the killers and for interesting discussions on these beautiful geometric theorems, and the anonymous referee

) ) -t (u� - :�}

ei0k 1 :S k :S 4; = 8k = ok sin 2b =sin( fh + 84) . . . u�ui - 1 , =Sill fh COS 84 + COS 83 Sill 84 = -� 2 U32 U42 sin 2d = sin( 81 + fh) . ufu�- 1 . . =Sill 81 COS 82 + COS lh Sill 82 = -� 2u2u 22

sin

The authors wish to thank Sergey Markelov for communi­

1

for suggesting the alternative approach explained in the last section. This work has been supported partly by CAS, CNRS, and the Chinese National

REFERENCES

[ 1 ] Ekhad, S.B.: Plane geometry: An elementary school textbook (ca. 2050 AD). The Mathematical lntelligencer 21/3: 64-70 (1 999).

·

[2] Fevre, S., Wang, D.: Proving geometric theorems using Clifford al­ gebra and rewrite rules. In: Proc. CADE-1 5 (Lindau , Germany, July

It follows that ur(u� - 1) sin 82 81 = Cui- 1)u� n = --m= = 4 4 U 1U 2 - 1 ' sin 2d uiu� - 1 ' Sin 2d u§(u�- 1) . (u� - 1)u� sin 83 sin 84 k= = 4 4 sin 2b U3U4 - 1 ' sin 2b u�u� - 1 ' {= --i eib = ei(7T-03-o4)12 = __ U3U4 ' i ; eid = eiC o2)/2 = _ sin

5-1 0, 1 998), LNAI1421, pp. 1 7-32 (1 998). [3]

7r- fi1-

14

THE MATHEMATICAL INTELLIGENCER

_

. u§u�cot b =t

2

2 U3U4 +

U1U2

1

1

,

Hestenes, D., Sobczyk, G . : Clifford algebra to geometric calculus. D. Reidel, Dordrecht Boston (1 984).

[4] Gao, X.-S.: Transcendental functions and mechanical theorem proving in elementary geometries. J. Automated Reasoning 6:

=

. uru� + 1 , cot a = � 2 2 U1U4 - 1

973 Project "Mathematics

Mechanization and Platform for Automated Reasoning."

403-41 7 (1 990).

[5] Li, H.: Vectorial equations solving for mechanical geometry theo­ rem proving. J. Automated Reasoning 25: 83- 1 2 1 (2000).

[6] Li, H . , Cheng, M . : Clifford algebraic reduction method for auto­

mated theorem proving in differential geometry. J. Automated

Reasoning 21: [7]

1 -21 (1 998).

Markelov , S.: Geometry solver. E-mail communication of February 1 1 , 1 998 from ([email protected]) .

[8] Wang, D . : GEOTHER : A geometry theorem prover. In: Proc.

CADE-1 3 (New Brunswick, USA, July 3D-August 3, 1 996), LNAI 1 1 04,

pp. 1 66-1 70 (1 996).

[9) Wang, D.: Clifford algebraic calculus for geometric reasoning with application to computer vision. In: Automated deduction in geom­ etry (Wang, D., ed. ) , LNAI1 360, pp. 1 1 5- 1 40 (1 997).

[1 0] Wu, W.-t . : Some recent advances in mechanical theorem-proving of geometries. In: Automated theorem proving: After 25 years (Bledsoe, W.W., Loveland, D.W., eds.), Contemp. Math. 29, pp.

235-241 (1 984).

[1 1 ) Wu, W.-t.: Mechanical theorem proving in geometries: Basic prin ­ ciples.

Springer, Wien, New York (1 994).

A U T H O R S

XIAORONG HOU

Chengdu

HONGBO Ll

Institute of Computer Applications Academia Sinica

Institute

of Systems Science

Academia Sinica

Chengd u 610041

Beijing 1 00080

China

China

Institute of Computer Applications, and has been at the

Hongbo U got his doctorate at Peking University in 1 994. He

Laboratory for Automated Reasoning and Programming at that

theorem-proving, hyperbolic geometry, Clifford algebra , and

Institute since 1 994; he has been professor there since 1 997.

computer vision. In 1 998, he won the Qiushi Excellent Youth

He has been working on symbolic computation, automated

Scholar Award of Hongkong.

Xiaorong Hou received his MSc in mathematics at the Chengdu

is now a professor of mathematics, specializing in automated

theorem-proving, and constructive real algebraic geometry.

DONGMING WANG

Laboratoire d'lnformatique de

LU YANG

Paris 6

Universite Pierre et Marie Curie -CNRS

4 place Jussieu

75252 Paris Cedex 05

Chengdu Institute of Computer Applications Academia Sinica

Chengdu 61 0041 China

France e-mail: [email protected]

Lu Yang received his diploma from Peking University in 1 959. His research interests include automated theorem - proving,

Dongming Wang has been a senior researcher at CNRS since

symbolic computation,

1 992. Before that, he was an assistant professor at Johannes

received the Natural Science Prize of Academia Sinica in 1 995.

and intelligent software technology . He

Kepler University in Austria after getting his PhD from

He is now a professor at Academia Sinica and concurrently

Academis Sinica. Antedating his academic interest in symbol ic

at Guangzhou University, and

computation and automated reasoning, he has an amateur in­

Informatics at Peking University.

is

Chair

of

the Department of

terest in Chinese seal cutting and writing poetry.

VOLUME 23, NUMBER 1 , 2001

15

i,i,@jj:l§rr6'h¥11+J·'I,'I,p!,hii¥J

Between Discovery and Justification

Marjorie Senec h al , Editor

j

With some notable exceptions, mathe­

and second, to generate debate on the

maticians-and mathematics-stayed

issues posed by his title. The responses

on the fringes of the so-called "Science

by Mary Beth Ruskai and Michael

Wars" that raged in academic circles in

Harris, which comprise this issue's col­

Europe and the United States in the

umn, were solicited in hopes of start­

late twentieth century. For those who

ing a discussion that may continue in

missed them, "Science Wars" was a

these pages from time to time.

catchy phrase for a wide-ranging con­

We can agree at the outset that in

troversy, among social scientists, his­

different times and places scientific re­

torians of science, scientists, philoso­

search has been (and continues to be)

phers, literary critics, and others, on

directed,

the nature of science (as opposed to

sticks, toward the solution of problems

the science of nature). To oversimplify,

by

society's

carrots

and

that society deems important. The

the central question was whether, or to

Science Wars debated the deeper and

what extent, science and mathematics

more difficult question of the ways in

are "constructed" by the societies in

which "science itself"-whatever that

which they are practiced. Most sci­

may mean-is influenced by society

entists and mathematicians became

and culture.

aware of the fray only through Alan

To help prepare this introduction to

Sakal's famous hoax, a parody of ex­

the responses to Graham, I turned to

treme

an article written over 20 years ago by

social

constructivists

which

This column is a forumfor discussion

a journal innocently

of mathematical communities

straight scholarship, or by reading Paul

throughout the world, and through all

Gross and Norman Levitt's uninten­

in what today would be considered at­

tional parody of the "extreme science

tenuated form. Hubbard pointed out

time. Our definition of "mathematical community" is the broadest. We include

published

as

side," Higher Superstition. Now, in the early twenty-first cen­

the

biochemist

Ruth Hubbard,

in

which she raised some of these issues

that our consciousness is shaped by the place and time in which we live,

"schools" of mathematics, circles of

tury, the debates are cooling down:

and that science is the picture of na­

correspondence, mathematical societies,

most warriors have come to realize

ture that scientists construct as they

student organizations, and informal communities of cardinality greater

that all sides have much to contribute

"transform the seamless unity of na­

and no side has a monopoly on arro­

ture into a carefully patterned patch­

gance. This may be a good time to look

work quilt." She points out that "it is

than one. Uthat we say about the

again at the lasting questions. Is math­

important to be aware of the elements

communities is just as unrestricted.

ematics a "locality" inhabited by eter­

of both patchwork and patterning: for

nal truths, or do culture and society

both involve choices that are far from

mathematicians of all kinds and in

subtly influence our subject as well as

arbitrary . . . the way in which the

how and why we practice it? Such

patches are selected-the unconscious

all places, and also from scientists,

questions have no easy answers-per­

decisions

historians, anthropologists, and others.

haps there are no definitive answers at

ground and what gets pulled into the

all-but it is important, for our self-un­

foreground-and the way in which

We welcome contributionsfrom

on

what

remains

back­

derstanding and for the picture of the

they are stitched together, are deter­

world of mathematics that we transmit

mined only in part by the explicit pos­

to our students, to discuss them.

tulates and rules of the 'scientific

I invited Loren Graham to con­

method.' For they, as well as our total

tribute his paper "Do Mathematical

selective mind-set are social products

Equations Display Social Attributes?"

that depend on who we are and where

Please send all submissions to the

to the Mathematical Communities col­

and when. Our scientific reality, then,

Mathematical Communities Editor,

umn

like all reality, is a social construct­ by which I don't mean to say that I am

Marjorie Senechal,

Department

(Vol. 22 , no. 3, 31-36) for two rea­

sons: first, to bring his thoughtful and

of Mathematics, Smith College,

lucid study of the philosophical impli­

an 18th-century idealist and believe

Northampton, MA 01 063, USA;

cations of the equations of general rel­

that there is nothing out there. I am

e-mail: [email protected]

ativity to the readers of this journal,

quite sure that there is something out

16

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

tions---may illuminate different aspects

there, but I believe that what we see

thus even more threatening-question

out there, and our interpretation of it,

of what it is that mathematicians actu­

of the object we are studying. Insisting

depend on the larger social context

ally do. The old, inward-looking, arro­

on their logical "equivalence" and ig­

that determines what we actively no­

gant

noring their differences masks their dif­

tice and accept as real."

"Mathematics is what mathematicians

ferent mathematical and even philo­

But is this what mathematicians do?

do," is not an acceptable reply. (Is it

sophical implications. Conversely, our

At first sight the answer would seem

merely coincidental that, in the "West,"

awareness of context may lead us

to be no, Graham's article and Ruskai's

extreme forms of abstract art, abstract

to those different proofs or equations.

response notwithstanding. The equa­

music, and abstract mathematics all

tions for Einstein's theory of general

were touted early in the twentieth cen­

Hence, Graham argues, it is fruitful to

relativity and the theory of quantum

tury and became problematic by the

of scientific and mathematical theories,

mechanics play the dual roles of math­

century's end?)

and "to deny the existence and signifi­

twentieth-century

definition,

think of social influences as "attributes"

ematical theory and description of an

We all were taught, and most of us

cance of these social attributes in the

external reality. It is this second role

continue to teach, the crucial distinc­

body of science and mathematics would

that Graham and Ruskai emphasize

tion between mathematical discovery

be misleading." Harris broadens the

here. Graham notes that Fock's frame­

and mathematical justification.

Dis­

spectrum from discovery to justification

work for general relativity "had to be

coveries, we tell our students, can be

still further, noting that "mathemati­

compatible with that of Einstein, but

inspired in an endless variety of ways,

cians, and scientists for that matter,

carry different philosophical implica­

from a careful reading of a proof to a

judge our peers not by the truth of their

tions"; that is, Fock's equations had to

dream. But once discovered, a mathe­

work but by how interesting it is."

be logically equivalent to Einstein's,

matical statement can only be consid­

Of course, any definition of "inter­

ered to be true if it has been justified

esting" is subjective. To me, a piece of

but the universe that they describe is

mathematics is interesting if it explains

not the same. And it is surely the de­

by rigorous proof. Moreover, we be­

scriptive role that Ruskai is referring

lieve, and the editors of mathematics

diverse phenomena, or gives me a

to when she remarks that "to my mind,

journals vigilantly reinforce this belief,

fresh perspective on something I had

the most convincing verification of

that we know what rigor is: deductions

thought I understood, or opens up new

quantum theory is not the individual

from explicit assumptions by estab­

vistas; "interesting" may mean some­

microscopic experiments but the fact

lished rules of inference. Yet-and

thing very different to you. But how­

that no other theory can even come

here

close to explaining so many diverse

agree-a view of mathematics as dis­

cide to give it any weight at all we find

Graham,

Harris,

and

Ruskai

ever we defme "interesting", if we de­

covery and justification alone is nar­

that the boundaries between mathe­

some sense, mathematicians too

row and sterile. It obscures the fact

matics and the other sciences, and

try to fit pieces of jigsaw puzzles to­

that much of our creative energy lies

other forms of knowledge, has sud­

gether, to use Ruskai's apt metaphor.

in a broad middle ground excluded by

denly blurred.

As Martin Gardner notes in an article

the discovery/justification dichotomy.

phenomena " Or is it?

In

this issue

Graham, Ruskai, and Harris have given us a lot to think about. The

to

Graham), "With very few exceptions

ferent ways of understanding a theo­

Mathematical InteUigencer welcomes

mathematicians have always believed,

rem--different proofs, different equa-

your response.

in

(not

in

response

For example, we all know that dif­

and still believe, that mathematical truths have a strange kind of abstract reality that is discovered, not created." But whether we believe that mathe­ matical truths (and mathematical ob­ jects) are "out there" or are products of the human mind, mathematicians agree that truths exist in some sense (2

+2

=

4 everywhere), and they agree with the spirit,

if not the

letter, of the fust half

of Kronecker's famous remark that

Although only 3, Adam knows all about cancer.

"God made the natural numbers; all

He's got it. Luckily, Adam

else is the work of man. " As Harris

has

points out, "this situation is not an ob­

doctors and scientists are

stacle to mathematics, much less ra­ tionality. The real absurdity is to claim otherwise." The discussion in these pages, then, is not the ancient and un­ ending conundrum of Platonism, but the

somewhat more

tractable-and

St. Jude Children 's

Research Hospital,

where

making progress on his disease. To learn how you can help, cal l :

1-800-877-5833.

j,itfflj.i§,Fhl£119.'1.1rrlll,hhfj

I

Marjorie Senec h a l , Ed itor

)

t is impossible to address the ques­

hard-pressed to figure out who is the

tion posed by Loren Graham's title,

mainstream and who is the margin in

"Do Mathematical Equations Display

this debate, and (2) the primary moti­

Social Attributes?", without bearing in

vation for hidden variables theories

Contexts of Justification

mind the current state of the debate on

seems to be the desire to preserve de­

the relation between science and soci­

terminism (and "objective reality") at

ety. As Graham's first paragraph makes

all costs-probably the strongest ar­

Michael Harris

clear, his article was written on the

gument I've yet seen in favor of the so­

(fortunately peaceful) fringes of the

cial constructivist view of science.

Science Wars, in which arguments for

I erijoyed Sakal's hoax when it came

and against the notion of social influ­

to light in 1996, but it has become de­

ence on science tend to be framed in

pressingly clear that its main legacy h�

the

been reinforcement of the low level of

crudest

possible

terms.

Even

Graham feels compelled to include a

debate characteristic of the Science

ritual affirmation of his belief in "the

Wars, especially since the publication

existence of objective reality. " A de­

of his book with Bricmont [4]. For in­

bate in which one has to talk that way

stance, researchers from the GERSULP

to be taken seriously obviously took a

(Groupe d'Etude et de Recherche sur la

wrong turn a long time ago [ 1 ] .

Science de l'Universite Louis Pasteur, in

Graham's article defmes what he

Strasbourg) could witness a marked de­

This column is a forum for discussion

means by "social attributes" implicitly,

terioration in their longstanding col­

of mathematical communities

through examples. He does hint, how­

laboration with scientists from this

throughout the world, and through all time. Our definition of "mathematical

utes" are pervasive, or even universal,

University after the publication of Im­ postures intellectueUes. A case in point:

in science. I don't think it detracts in

A scientist, who for years had graciously

ever, that, in his view, "social attrib­

community" is the broadest. We include

the least from Graham's contribution

allowed some of the GERSULP's gradu­

"schools" of mathematics, circles of

to ask whether the Marxist influence

ate students to carry out fieldwork in

correspondence, mathematical societies, student organizations, and informal

on Fock's relativity theory is of a kind

his laboratory, wrote a very aggressive

with the philosophical motivations

letter commenting on the paper writ­

behind the recent revival of hidden

communities of cardinality greater

variables theories

in quantum me­

ten by the last student intern. In this

than one. What we say about the

chanics. Hidden variables theories as

to the Sokal-Bricmont book, implying

communities is just as unrestricted.

an alternative to "quantum orthodoxy"

that papers

are a recurrent Science Wars theme.

showed how right these authors were

N. Higher Superstition, by S. Goldstein in an article in The Flight from Science and Reason, and by Alan

to defend science against unqualified

tack on the exact sciences whatsoever.

Sakal's coauthor Jean Bricmont in a

A lot of energy on the part of the mem­

We welcome contributionsfrom

mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

They are defended by P. Gross and

Levitt in

Mathematical Communities Editor, Marjorie Senechal,

Department

like the one

at hand

criticism. But a conscientious scrutiny of the student's paper showed no at­

separate publication [2]. As a mathe­

bers of the GERSULP is being spent

matician, I can only be cheered by the

these days to try to cope as best they

et al. [3]

can with these new prejudices. Scien­

may have constructed a consistent de­

tists in Strasbourg now tend to be a pri­

terministic account of quantum me­

ori suspicious of whatever they receive

news that Diirr, Goldstein,

Please send all submissions to the

letter, the scientist explicitly referred

chanics, and the fact that it comes at

from philosophically or sociologically

the cost of adding variables that can

oriented researchers, whereas in the

never be measured is the least of my

past many of them were quite open­

concerns. As a mathematician, fur­

minded [5].

of Mathematics, Smith College,

thermore, the question is strictly none

Northampton, MA 01 063, USA;

of my business, but it does seem to me

in the context of a more general ques­

e-mail: [email protected]

that (1) sociologists of science will be

tion, namely: Can interaction with so-

18

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

I prefer to frame Graham's question

ciologists (and sociologically-minded historians and philosophers) help sci­

More interestingly, one can ask what Tr

ics; they seem to think their job is to

was before the formal

explain mathematical truth. Edinburgh

entists understand, not only incidental

definition of real numbers. To assume the

sociologist David Bloor and philosopher

features of their subject (the "context

real numbers were there all along, wait­

purposes as an irresponsible relativist

of Platonism [9). Dedekind wouldn't

and a moderate realist, respectively [ 17],

of discovery" as Sokal would put it), but even what is seen as the core of the subject ("context of justification")?

kind of object

ing to be defined, is to adhere to a form

have agreed [10]. In a debate marked

Philip Kitcher, cast for Science Wars

have both attempted to develop empiri­

Replace "scientists" by "mathemati­

by the accusation that "postmodern"

cians" in the above question (the rela­

writers deny the reality of the external

edge [18]. (Knowledge and truth are not

tion between the two is by no means

world, it is a peculiar move, to say the

synonyms but they are on the same

clear), and it seems to me that the an­

least, to make mathematical Platonism

wavelength [ 19].) They have their own

swer is an unequivocal "yes." I would

a litmus test for rationality [ 1 1] . Not

(very different) reasons, but in so do­

even go further (down this particular

that it makes any more sense simply to

ing I'm convinced they have missed the

cist accounts of mathematical knowl­

philosophical gangplank) and argue

declare Platonism out of bounds, like

that such interaction is unavoidable if

point of mathematics. As is typical in

Jean-Marc Levy-Leblond, another con­

such discussions, their examples are

mathematicians ever hope to under­

tributor to

Impostures Scientifiques,

drawn either from mathematical logic

stand how we actually do see our dis­

who calls Steven Weinberg's gloss on

or from mathematics no more recent

cipline, not to mention how we "ought"

Sokal's comment "une absurdite, tant

than the 19th century. If the sociolo­

to see it. Indeed, a particularly regret­

il est clair que la signification d'un con­

gist, at least, had done some field work,

table feature of the Science Wars is

cept quelconque est evidemment af­

he couldn't have helped observing that

that quite a lot of non-mathematicians,

fectee par sa mise en oeuvre dans

un

what mathematicians seem to value

but hardly any mathematicians, are at­

contexte nouveau" [l2] ! Now I fmd it

most are "ideas" (not necessarily of the

tempting to instruct the general public

hard

a

Platonic variety); the most respected

on how mathematics is to be under­

straight face, and my personal prefer­

mathematicians are those with strong

ence is to regard the formula Tr =

pher assures us, is philosophically in­

mathematics, for instance, is summa­

� as a creation rather than a dis­

covery.

corre­

defensible; Sokal and Bricmont add

rized in the sentence "A mathematical

spond to the familiar experience that

doesn't change, even if

that "intuition cannot play an explicit

there is something about mathematics,

role in the reasoning leading to the ver­

the idea one has about it may change"

and not just about other mathemati­

(p. 263). This claim, referring to a

ification (or falsification) of these the­

"crescendo of absurdity" in Sokal's orig­

away with saying "Evidemment" [ 13] !

independent of the subjectivity of in­ dividual scientists" [20].

stood. The Sokal-Bricmont philosophy of

constant like

Tr

inal hoax in Social

to

defend

Platonism

But Platonism does

with

cians, that precisely doesn't let us get

Text, was in turn crit­

This experience is clearly captured by

icized by anthropologist Joan Fqjimura,

Alain Connes, a self-avowed Platonist,

Impostures

in his dialogue with neurobiologist

in an article translated for

"intuition." Now intuition, the philoso­

ories, since this process must remain In barring intuition from the context

of justification, Sokal and Bricmont are

scientijiques [6], the most extended

J.-P. Changeux, who (to oversimplify)

using the language of truth (and falsity).

response

Sokal­

expects to find mathematical struc­

Bricmont book Most of Fqjimura's ar­

tures in the brain [ 14]. I don't think

In fact, truth is a secondary issue in

ticle consists of an astonishingly bland

Connes (or Roger Penrose, another

prove true theorems, but this is hardly an adequate or even useful description

in

France

to

the

mathematics. Of course we want to

account of the history of non-euclidean

prominent Platonist) is confused about

geometry, in which she points out that

reality, and I have a hard time imagin­

of our objective. Mathematicians, and

the ratio of the circumference to the

ing a neuronal representation that does

scientists for that matter, judge our

diameter depends on the metric. Sokal

justice to the concept of Tr. But the on­

peers not by the truth of their work but

and Bricmont know this, and Fqji­

tological issues are far from settled,

by how interesting it is [21]. The differ­

mura's remarks are about as helpful

and while there is no reason to assume

ence between the true and the interest­

as FNs referral of Quine's readers

they will ever be settled, the important

ing even has a market value. Professor

to Hume (p. 70). Anyway, Sokal ex­

point is that this situation is not an ob­

Ed Fredkin of Carnegie-Mellon once of­

plicitly referred to "Euclid's pi", pre­

stacle to mathematics, much less to ra­

fered to give you $100,000---almost

sumably to avoid trivial objections

tionality [ 15]. The real absurdity is to

enough to buy a bachelor's degree at a

like Fqjimura's-wasted effort on both

claim otherwise.

prestigious private university-if you

sides [7]. If one insists on making triv­

This leaves the "context of discov­

ial objections, one might recall that the

ery vs. context ofjustification" in a cu­

interesting

theorem that "' is transcendental can

rious

mathematicians

[22]. Whereas any beginner can program

be stated as follows: the homomor­

have largely given up worrying [ 16]

the computer to prove a true theorem

Q [X] � R taking X to "' is in­ jective. In other words, "' can be iden­ tified algebraically with X, the variable par exceUence [8].

about the nature of what it is they need

after a single lesson [23].

phism

limbo,

since

were first to make a computer prove an theorem

in

mathematics

to justify. This doesn't bother philoso­

On the other hand, it is surprising to

phers and philosophically-minded so­

see just how little we seem to be con­

ciologists concerned with mathemat-

cerned with "truth" these days. Math-

VOLUME 23, NUMBER 1 , 2001

19

ematicians rarely discuss foundational issues any more, so it was significant that an article by Arthur Jaffe and Frank Quinn, reaffirming the impor­ tance of rigorous proof in the current context of strong interaction between physics and mathematics, provoked no fewer than 16 responses by eminent mathematicians, physicists, and histo­ rians. No two of the positions ex­ pressed were identical, which already should suggest caution in laying down the law on rationality, as Sokal and Bricmont (and Levy-Leblond) seem in­ clined to do. Remarkably, almost none of the responses had much to say about "truth."[24] "Truth" was central, predictably, only to the responses of Chaitin and Glimm. Chaitin's branch of mathematics treats "truth" as a techni­ cal term, without metaphysical conno­ tations, and Chaitin's claim to have "found mathematical truths that are true for no reason at all" suggests that it may be harder than Fredkin suspects to know just when to award his prize. Glimm's brand of truth is quite the op­ posite: it "lies not in the eye of the be­ holder, but in objective reality. . . . It is thus reproducible across barriers of distance, political boundaries and time" [25]. Turning to the introduction to the book Quantum Physics, by Glimm and Jaffe , one fmds the unusual assertion that "mathematical analysis must be included in the list of appro­ priate methods in the search for truth in theoretical physics." I can't help thinking it's not a coincidence that both Bricmont and Sakal are amply represented in the Glimm-Jaffe bibli­ ography. Ironically, given the starting point of Graham's exchange with Sakal, the substance of the Jaffe-Quinn article, and the subsequent debate in the Bulletin of the AMS, is precisely the context of justification, specifically the extent to which physical intuition should be admitted as an alternative to rigorous proof [26]. (Admitted by whom? That is a question for the soci­ ologists!) Fredkin's theorem-proving machine may share the Sokal-Bricmont commitment to truth as that-which-is­ to-be-justified, but what are we to make of Thurston's emphasis on the "continuing desire for human under-

20

THE MATHEMATICAL INTELLIGENCER

standing of a proof, in addition to knowledge that the theorem is true" [27]? We know what he means, as we know what Robert Coleman means, when, having discovered a gap in Manin's proof of Mordell's col\iecture over function fields, he nevertheless writes, "I believe that all this is testi­ mony to the power and depth of Manin's intuition" [28]. Is Coleman try­ ing to slip a counterfeit coin between the context of discovery and the con­ text of justification? Do these offhand comments touch on something gen­ uine and profound about mathemat­ ics? Or is it just my indoctrination that makes me think so? Let's try a thought experiment. How do we know Wiles's proof of Fermat's Last Theorem, completed by Taylor and Wiles, is correct? Although this particular theorem, better publicized than any in history, has been treated with unusual care by the mathematical community, whose "verdict" is devel­ oped at length in a graduate textbook of exceptionally high quality, I'd guess that no more than 5% of mathemati­ cians have made a real effort to work through the proof [29]. Some scientists (and some mathematicians as well) apparently view Wiles and his proof as an "anachronism" [30]. The general public is not entirely convinced. Why are we? Can a sociologist study this question without knowing the proof? Can mathematicians pose the question in terms sociologists would find mean­ ingful? Knowing the truth of the mat­ ter is obviously of no help, and rela­ tivism is not the issue: it's not clear what kind of "reality" would be rele­ vant to settling the question, but the fact that no one has found a coun­ terexample is certainly not a good candidate. Few of us would choose to treat our belief that Wiles proved Fermat's last theorem as "a mythical and false ide­ ology" [31] , but is it possible that our attempts to justify this belief always in­ volve an element of self-delusion? And how are we to convince a skeptical outsider that this is not the case? The only reasonable answers that come to mind are empirical in nature, and specifically historical and sociological, rather than philosophical [32]. We

would have to pay attention to the ql,les­ tion of how knowledge is transmitted among mathematicians. Fermat's Last Theorem provides a particularly good test case. Wiles's proof generated an un­ precedented [33] number of reports, sur­ vey articles, colloquium talks, working seminars, graduate courses, and mini­ conferences, as well as books, newspa­ per and magazine articles, television reports, and other forms of communi­ cation with non-mathematicians. Not to mention the spate of announcements, designed to impress public policy-mak­ ers and the public at large, citing Wiles's work as proof that mathemat­ ics "has never been healthier." Has anyone been keeping track of all these incitements to belief formation, chec�­ ing them for contamination by myth and false ideology? Studying questions like these pro­ vides a second answer to the thought experiment proposed above, comple­ mentary to the answer we would nat­ urally provide based on our experience as mathematicians, and potentiallyjust as interesting. These two answers are not in competition, much less on op­ posite sides of a battlefield. Arriving at the second answer would be the work of sociologists. For this, full coopera­ tion with mathematicians would be necessary. I hope such cooperation is still possible. Acknowledgments I am indebted to many people for help in formulating and clarifying the ideas developed in this note, and in the arti­ cle from which it was extracted (see note [4]); I am especially grateful to Marie-Jose Durand-Richard, Catherine Goldstein, Catherine Jami, Patrick Petit­ jean, and Jim Ritter, co-organizers of the 1997 IHP workshop on the Sokal affair. REFERENCES

[1 ]

The mathematics department may be the only spot on campus where belief in the reality of the external world is not only op­ tional but frequently an annoying distrac­ tion.

[2] Higher Superstition,

p. 261 , note 9; Gold­

stein, "Quantum Philosophy: the Flight from Reason in Science" in Gross, Levitt, Lewis, The Flight from Science and Reason,

pp.

[4] The avoidance of serious debate in

marks are contained in his article "Sakal's

is one of the main themes of Yves

Hoax," in the N e w York Review o f Books,

Jeanneret's L 'affaire Sakal ou Ia querel/e

August 8, 1 996.

des impostures

(Presses Universitaires de

[1 3) Metaphors from virtual reality may help

France, 1 998), the best book I've seen on the Sokal affair and its ramifications in France. This theme is also addressed in

MICHAEL HARRIS

VII

Cedex

example, Connes writes, "Je pense que

le mathematician developpe un 'sens', ir­

which can be viewed at

http://www.math .jussieu. fr/-harris. Most

roouctible a Ia vue, a l'ou'ie et au toucher,

of the present article was extracted from

aussi contraignante mais beaucoup plus

this review.

stable que Ia realite physique, car non lo­

qui lui permet de percevoir une realite tout

calises dans l'espace-temps" (p. 49). In

and Strasbourg mathematician Norbert

fact, the debate is even more complex,

05

France

Schappacher for providing this informa­

because Changeux often comes across

tion.

as a social constructivist, though one who

[6] Impostures scientifiques

e-mail: [email protected]

[1 4) Matiere a penser. Odile Jacob, 1 989. For

[5] I thank Josiane Olff-Nathan of GERSULP

2 place Jussieu

75251 Paris

here.

my unpublished review of Fashionable N onsense,

Universite de Paris

Michael Harris studied at Harvard

(henceforth IS},

sees society as materialized in the human

sous Ia direction de Baudouin Jurdant,

brain; thus he sees mathematical objects

Paris: Editions La Decouverte, 1 998. Also

as

published as a special issue of the journal

(PhD 1 977 with Barry Mazur), and taught at Brandeis University before moving to Paris. Among his research

"

'representations culturelles' suscepti­

bles de se propager, se fructifier et de pro­ liferer et d'etre transmises de cerveau a

Alliage. [7]

contributions, the most recent is "On the geometry and cohomology of

p. 38, footnote 26. Weinberg's re­

[ 1 2] IS,

Science Wars literature-on both sides­

There are many circles in Euclid, but no

cerveau. "

pi, so I can't think of any other reason for

[1 5] See Barry Mazur's astonishing "Imagining

Sokal to have written "Euclid's pi," unless

Numbers (particularly Y-15)," particu­

some simple Shimura varieties" (with Richard Taylor). He has been a visi­

this anachronism was an intentional part

larly for (among other things) an attempt

of the hoax. Sakal's full quotation was "the

to get beyond sterile ontological debates.

tor at the Institute for Advanced Study, Bethlehem University in the

thought to be constant and universal, are

West Bank, the Tata Institute, the Steklov Institute, and other institu­

now perceived in their ineluctable his­

quirement in any university with which I am

toricity. " But there is no need to invoke

familiar. One would think this fact would

tions. In 1 986-1 989 he helped orga­

non-Euclidean geometry to perceive the

be of interest to sociologists of science,

nize a program of scientific coopera­

historicity of the circle, or of pi: see

but I have not seen it addressed in the lit­

tion between US and Nicaraguan

Catherine Goldstein's "L'un est !'autre:

erature. As a graduate student at Harvard,

universities. He organized with five historians of science a workshop at

pour une histoire du cercle," in M. Serres,

I saw foundations actively discussed only

Bordas,

in the graffiti in the men's room on the sec­

[8] This is not mere sophistry: the construc­

[ 1 7] What they really think hardly matters. The

title "La guerre des sciences n'aura

tion of models over number fields actually

"strong program" of Bloor and Barry

pas lieu."

uses arguments of this kind. A careless

Barnes is criticized at length in the philo­

Photo credit: Emiliano Harris.

construction of the equations defining

sophical "intermezzo" of FN ; but Sokal

7T

of Euclid and the G of Newton, formerly

Elements d'histoire des sciences,

the lnstitut Henri Poincare in 1 997 on

1 989, pp. 1 29-149.

the subject of this article, under the

Fashionable Nonsense

(henceforth FN)

In public, at least. A course on founda­ tions of mathematics is not a core re­

ond floor of the science library.

modular curves may make it appear that

"agrees with nearly everything" in Kitcher's

1r

attempt to "occupy middle ground" in

is included in their field of scalars.

[9) Unless you claim, like the present French 1 HH 25. These two books, together with

[1 6)

Minister of Education, that real numbers

Koertge, op. cit. [2] [1 8] D. Bloor, Knowledge and Social Imagery

exist in nature, while imaginary numbers

(U. of Chicago press, 1 991 ), chapters 5-8;

1r

P. Kitcher, The Nature of Mathematical

by Sokal and Bricmont, first published in

were invented by mathematicians. Thus

French as Impostures intellectuelles, and A

would be a physical constant, like the

Knowledge

(N. Koertge, ed.) are

mass of the electron, that can be deter­

1 984).

House Built on Sand

(Oxford

University

Press,

the canonical texts of what passes for the

mined experimentally with increasing ac­

pro-science camp of the Science Wars.

curacy, say by measuring physical circles

discourse. Thurston's extended response

[3] DOrr, D., S. Goldstein and N. Zanghi,

with ever more sensitive rulers. This sort

to the Jaffe-Quinn article, in BAMS, April

"Quantum equilibrium and the origin of ab­

of position has not been welcomed by

1 994, discussed below does refer to truth,

solute uncertainty,"

most French mathematicians.

but he seems more interested in knowl­

(1 992):

843-907;

Goldstein and V.

J.

Statistical Phys. 67

also

DOrr

Zanghi,

D.,

S.

"Quantum

Mechanics, randomness, and determinis­ tic reality" Phys. Letters; 1 72 (1 992): 6-1 2.

[1 0) Cf. M. Kline, Mathematics: The Loss of Certainty,

p. 324.

[1 1 ) Compare Morris Hirsch's remarks in BAMS,

April 1 994.

[ 1 9] One might also say they share a common

edge and especially in understanding. [20] Chapter 3 of Kitcher's op. cit. [1 8] is de­ voted to a refutation of Kantian or Platonist intuition as a means to mathematical knowl-

VOLUME 23. NUMBER 1, 2001

21

edge, and what we mean when we use the

Corry, "The Origins of Eternal Truth in

temative (or complementary) standa�d of

word informally is presumably even less de­

Modern Mathematics," Science in Context

justification to rigorous proof. This point is

fensible. The quotation is from Fashionable

1 0 (1 997), 297-342. Corry is so intent on

a cliche of the popular literature on chaos,

developing his theme (that "the idea of eter­

and it is repeated in the article by Amy

[21 ] As Barry Mazur reminded me, "interest" in

nal mathematical truth . . . has not itself

Dahan-Dalmedico and Dominique Pestre

this context is generally used as shorthand

been eternal") that he completely misses

for an intellectual criterion, such as

the near absence of the word "truth" from

Nonsense,

pp. 1 43-44, note 1 83.

in Impostures Scientifiques (pp. 95-96). [27] Thurston, op. cit. [25]. Thurston's com­

"enhancement of understanding" (see

the debate, claiming with no attempt at jus­

ment referred to the computer-assisted

Thurston's comments, quoted below).

tification that "the eternal character of

proof of the Four Color Theorem, and

Such a criterion is by nature not well-de­

mathematical truth" was "implicit at the very

echoes Deligne's remarks on the same

fined, yet we have the sense that we know

least" in the Jaffe-Quinn proposal.

topic, quoted by Ruelle in Chance and

[25] Glimm goes on to say that mathematical

what it means.

pp. 3-4.

Chaos,

[22] Actually a "major mathematical discovery."

truth is to be compared with the stronger

(The author obtained this information

standard of truth in science, "the agree­

conjecture

November 24, 1 997 from Fredkin's web

ment between theory and data." The

L 'Enseignement Math.

page on Radnet. This web page has ap­

whole discussion can be found in Bulletin

parently been discontinued and the infor­

of the AMS,

mation regarding the prize has not been

July 1 993 and April 1 994.

[26] There is a similar irony in the discussion of chaos in FN (pp. 1 34-1 46). As in sim­

reconfirmed since 1 997 .)

[28] Coleman, "Manin's proof of the Mordell over

function

fields,"

36 (1 990), p. 393.

[29] The text book is G. Cornell, J. H. Silverman, G. Stevens, eds. : Modular Forms

and

Fermat's

Last

Theorem,

Springer-Verlag (1 997). If I believe, or u � ­

ilar discussions in The lntelligencer, the

derstand, or have some meaningful rela­

says something similar in IS (p. 39), and

non-post-modern

position emphasizes

tion to the proof, it's mainly because I have

[23] This point is hardly novel; Levy-Leblond Dieudonne distinguished further between

the continued role of proofs in the theory

been collaborating with Richard Taylor to

"math8matiques vides" and "mathema­

of non-linear dynamical systems, insists

generalize parts of the argument to auto­

tiques significatives" (quoted in Dominique

on the determinism inherent in dynamical

morphic forms of higher dimension.

Lambert,

systems, and points to quotations from

[30] John Horgan, "The Death of Proof, "

"L'incroyable

efficacite

des

mathematiques," La Recherche, janvier

Maxwell and Poincare to argue that no

Scientific American,

1 999, p. 50). Truth is also not what inter­

conceptual revolution has taken place.

92-1 03.

October 1 993, pp.

(See also Higher Superstition , p. 92 ft.,

[31 ] As in David Bloor: "What if scientists need

verite qui inspire Ia philosophie, mais des

and Bricmont's contribution to The Flight

to believe a mythical and false ideology,

categories comme celles d'lnteressant, de

from Science and Reason.)

This is all rea­

because they would lose motivation with­

Remarquable ou d'lmportant qui decident

sonable (though I would need to be a his­

out it?" Social Studies of Science 28/4

de Ia reussite ou de l'echec." This quota­

torian to explain why a few late-1 9th-cen­

(August 1 998), p. 658.

tion is taken from Qu'est-ce que Ia philoso­

tury quotations are hardly decisive in

[32] As Thurston wrote, Wiles's proof-still in­

phie? (1 991 ), p. 80, one of the books sub­

discussions of conceptual revolutions),

complete at the time-"helps illustrate how

jected to extensive ridicule by Sokal and

but once again the interesting point has

mathematics evolves by rather organic psy­

Bricmont.

been missed. Namely, in the wake of

chological and social processes." Thurston,

[24] The debate provoked by the Jaffe-Quinn

chaos, computer modeling ("experimental

op. cit.

article is taken up in a recent article by Leo

mathematics") is being proposed as an al-

ests Deleuze and Guattari: "ce n'est pas Ia

[33] Or so I would assume.

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22

THE MATHEMATICAL INTELLIGENCER

[25]

l,�@ii • f§rr@h(¥119.111J,II!Ihh¥J

G Response to Graham: the Quantum View

M a rj o rie Senechal , Ed itor

raham's article discusses the pos­ sibility

of social influence

or below a certain standard amount.

on

But this already assumes something

I will not

about the markings on the stick. Even

comment directly on his example: rel­

with accurate calibration, a low read­

mathematical

Mary Beth Ruskai*

I

equations.

ativity is far from my specialty. The his­

ing could result from not inserting the

tory of quantum theory provides ample

stick properly. Before adding oil, one

evidence that different people with dif­

often inserts the stick a second time or

ferent views may formulate a theory in

(perhaps not trusting the gas station at­

quite

tendant) checks oneself. 1 In science

different mathematical terms.

The issue is to what extent this implies

this is referred to as reproducibility:

that science is shaped by social and

experimental result is not accepted un­

cultural factors. The development of

less it can be repeated by an indepen­

quantum theory, from its inception to

dent observer.

an

the most recent novel experiments ,

However, reproducibility of single

provides fertile territory for discussing

experiments is not enough. For exam­ ple,

this question. Before doing so, I would like to comment on the distinction between

G.P. Thompson's observation of dif­

fraction rings due to the transmission of electrons through a

thin metal foil ap­

I prefer the term "verifica­

of mathematical communities

pears to give unequivocal evidence that

tion" for the latter. Such a term sup­

only when one remembers the many

throughout the world, and through all

poses the existence of an external

other experiments which seem to offer

objective reality. Without such an as­

convincing evidence that electrons are

This column is a forum for discussion

time. Our definition of "mathematical community" is the broadest. We include

the context of "discovery" and of "jus­ tification".

electrons are waves. 2 Problems arise

sumption it makes no sense even to

particles. Verification of a theory re­

discuss

quires what one might term "consis­

the

concept

of

science.

"schools" of mathematics, circles of

Histories of science often focus on the

tency"

correspondence, mathematical societies,

role of

Reproducibility guards against the sub­

student organizations, and informal communities of cardinality greater

a small number of critical ex­

periments. However, it is important to

as

well

as

reproducibility.

jectivity or bias of a single observer.

remember these are only part of the

Consistency is more complex. At the

legacy and that virtually all experi­

simplest level, it requires verification by

than one. What we say about the

ments have multiple interpretations

different types of experiments as well as

communities is just as unrestricted.

and/or hidden assumptions.

independent observation. Modem sci­

We welcome contributions from

ex­

ence often requires complex experi­

tremely simple process of using a dip

Consider,

for

example,

the

stick to check the oil level in a car en­

ments with sophisticated instruments

all places, and also from scientists,

gine (one of the few measurements not

and rely for their interpretation on the

historians, anthropologists, and others.

yet replaced by an electronic device

assumption of the validity of a large

with digital readout). The results may

body of scientific evidence and theories.

seem unambiguous: the oil level on the

Thus, even experimental observations

stick tells how much the total is above

can be viewed as human constructs.

mathematicians of all kinds and in

which permit only indirect observation

*Partially supported by National Science Foundation Grant DMS-97-06981 and Army Research Oftice Grant

DAAG55-98- 1 -037 4 1 1f one continues scrutinizing this experiment, questions arise that may seem similar to those of quantum the­ ory. Before starting. oil is wiped off the dipstick, so that the stale of the oil reservoir is necessarily changed by

Please send all submissions to the

the measurement. However, unlike quantum theory, in which all information about the initial state may be lost,

Mathematical Communities Editor,

with careful procedures a good estimate of the initial state can still be made. Nevertheless, even in such a rudi­

Marjorie Senechal,

Department

mentary situation, the observer can not be completely separated from the experiment.

of Mathematics, Smith College,

2The more familiar Davidson-Germer experiment in which electrons incident on a nickel crystal show diffrac­ tion patterns was the first to confirm the wave-like properties of electrons. However that evidence is not un­

Northampton, MA 01 063, USA;

equivocally in favor of waves because the full set of experimental data contains evidence consistent with par­

e-mail: [email protected]

ticle properties in addition to the peak of electron emission associated with diffraction.

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001

23

My own view of physics is that it is like a giant mosaic or jigsaw puzzle. When key pieces are found, sections fits together so well that the result seems evident. But many different pieces have similar shapes. Thus, it is possible to find edges that seem to match and build up promising aggre­ gates which eventually break down. The "old quantum theory," developed in the first two decades of the previous century, fits this metaphor. One had bits and pieces of the puzzle­ Einstein's description of the photo­ electric effect, de Broglie waves, the Bohr atom--each piece explained physical phenomena in strikingly con­ vincing ways, yet gaps and inconsis­ tencies remained in the larger puzzle. In 1925 and 1926, two theories were proposed-the matrix mechanics of Heisenberg, and the wave mechanics of Schrodinger-whose mathematical for­ mulation was so different that one could hardly expect both to be valid. Yet, within a few months of Schrodinger's second paper, a link was found by at least three people independently, Eckart, Pauli, and Schrodinger.3 Subsequently, Dirac [4) and von Neumann [31) formulated quan­ tum mechanics in terms of abstract Hilbert spaces, in which the Heisenberg and Schrodinger pictures are but two of many possible equivalent representa­ tions. It is worth noting that both Heisenberg and Schrodinger's original formulations required some intuitive leaps over logical obstacles-Heisenberg required that "matrices" satisfy commu­ tation relations which can not hold in finite dimensions, while Schrodinger replaced the second derivative of time in the wave equation by a first deriva­ tive with a pure imaginary coefficient. As far as I am aware, there is no ev­ idence that their different approaches were motivated by political or social

factors.4 However, they had very differ­ ent attitudes toward the then-emerging Nazi philosophy. (See, e.g., [3,21,27].) Heisenberg, if not an active proponent, was at least sufficiently partisan to be comfortable as head of German sci­ ence under Hitler; Schrodinger, if not a courageous opponent, was unwilling to make the accommodations neces­ sary to remain in Austria. Suppose, for the sake of argument, that Heisenberg and Schrodinger's5 different formula­ tions of quantum theory were moti­ vated by opposing political views. What difference could it have made? The two theories were still equivalent. Indeed, if they had proposed inequiva­ lent theories, one would have been verified and the other discredited by subsequent experiments-and the out­ come would not have depended on the political views of the proposer. It is often said that mathematics is the language of science. But it would be more accurate to say that mathe­ matics is a family of related languages. The languages of algebra, analysis, geometry, etc. often give very different insights into a problem. Barry Simon's views on p. xv of the Introduction to [28) describe one such situation. I was partly motivated by a not atyp­ ical functional analyst's suspicion of probability as nothing more than a subset of functional analysis with strange names . . . any statement in probability theory has a translation into functional analysis. But this somehow misses the point. Like any other foreign language, probability theory is structured around its own natural thought patterns and so is critical to a mode of thinking; but more prosaically, certain exceed­ ingly natural constructions of prob­ ability theory look ad hoc and un-

natural when viewed as functidnal analytic constructions. . . . I speak probability with a marked func­ tional analysis accent: lecturing in Zurich, I couldn't help feeling that I was speaking not hoch probability theory but only a kind of Schweitzer probability. Simon was referring to the spectacular progress in constructive quantum field theory which followed Nelson's syn­ thesis of several earlier ideas into what became known as "Euclidean quantum field theory." Many such examples can be found. In 1982 Donaldson used ideas from classical field theory to solve problems in 4-dimensional topol­ ogy. The recognition that what alge­ braic topologists called a "connection" on fibre bundles was but another name for a gauge field,6 and that this seem­ ingly abstract language was important in physics, led to advances in both fields. Inspired by electric-magnetic duality, Seiberg and Witten's insightful work in 1994 led to another impressive series of rapid advances, in which long proofs of Donaldson were dramatically shortened and long-standing questions resolved [ 1 7). Hilbert space is the language of quantum mechanics. Although other languages can give useful insights, they can also mislead. In particular, experi­ ence with classical waves is helpful in understanding some aspects of super­ positions; however, attempts to give too literal an interpretation to the no­ tion of "wave function" have also been a major source of confusion. There was at least one other math­ ematical approach which appeared in the development of quantum theory. In 1925, C. Lanczos [18] tried to formulate quantum theory in terms of integral, 7 rather than differential, equations, but

3Aithough this is generally attributed to Schri:idinger, there is clear evidence of independent discovery by Eckart [7,29], and [33] contains the text and interpretation of a letter from Pauli to Jordan. 4There are, however, historians who have argued that that both Heisenberg and Schri:idinger were infiuenced by the Weimar culture to find an indeterministic compo­ nent in physics. See [1 1 , 1 4]. 5Schr6dinger's biographer raises another social element. Moore [2 1 ] devotes considerable attention to Schri:idinger's sexual philandering, and even raises the possi­ bility that the psychological impact of a liaison during the Christmas holidays at the end of 1 925 may have stimulated the creative period of 1 926 in which wave me­ chanics was developed. 6Aithough this relationship was known to a few experts, prior to Donaldson's work algebraic topology was generally regarded as an abstract area of mathematics with little relevance to physics. In fact, I recall an anecdote from the early 1 970's in which an advisor told an enthusiastic graduate student that the outstanding problem in algebraic topology was "finding an application." 7See [32] for a summary of this work and a response by Lanczos 35 years later.

24

THE MATHEMATICAL INTELLIGENCER

was unable to specify the kernels needed

to

complete

his

theory.

However, his approach did lead quite

A Modern View of EPR for pedestrians*

naturally to operators which satisfied

Suppose that every morning when

Heisenberg's commutation relations.

you log onto your computer the

In view of that, one wonders why

his

work had so little impact.8 Was it be­

lar internet provider seem to be paired up in a similar way. When two members of a pair choose the same box, one, and only one, wins.

It appears that the internet entre­

screen shows three boxes

preneur is sending out paired mes­

cause physicists were less comfortable

sages

with the dialect of operator theory

boxes are complementary--e.g., if

programmed

so

that

the

differential

flashing on and off with the words

your boxes are coded W W L, your

equations? Because Lanczos's equa­

CLICK ME. You can not proceed to

partner's will be L L W.

based

on

integral

than

tions required singular kernels which

check your e-mail (or do anything

However, one astute pair notices

seemed inconsistent with Heisenberg's

else) until you choose one of the

something curious. On the days

diffe-rent boxes,

discrete matrices? Or because Dirac's

boxes. As soon as you click, the

when they choose

delta function was not yet the familiar

other two boxes disappear and the

both win 3/8 of the time and both

tool needed to complete the picture?

remaining box changes to either

Despite the overwhelming experi­

I Win I

or

ILosel

indicating that

lose

3/8 of the time; only 114 of the

time does one win and the other

mental evidence in support of quan­

your "frequent web buyer" account

lose with different boxes. Yet, an el­

tum mechanics, some of its implica­

has won or lost 500 points.

ementary calculation shows that

tions made many reluctant to accept

You choose at random but, in

it. Difficult questions about its inter­

the hope of finding a better strat­

probabilities should be both win 114

pretation led to controversies, some

egy, keep careful notes of your

and both lose 114 of the time.

of which are still debated. The best re­

choice and the result. The game ap­

seems to eliminate the complemen­ tary box hypothesis.

with

complementary

pairs,

the

This

"God

pears fair, in the sense that you win

does not play dice with the universe"

50% of the time; however, no strat­

What other explanations are pos­

is "Indeed, She does not." For quan­

egy appears. In January, you attend

sible? The internet provider (located in Kansas) may be sending entan­

sponse to

Einstein's

famous

tum theory, although it has a proba­

the annual AMS meeting where

bilistic aspect, is not described by the

you meet a colleague from the op­

gled pairs of polarized photons.

classical random probabilities associ­

posite coast who uses the same in­

Clicking on box A, B, C chooses one

ated with dice games. Quantum states

ternet provider and has kept simi­

among polarization filters set at 120°

are based upon superpositions which

lar records. You compare notes

angles. If the photon passes through

give rise to non-classical correlations

and discover an amazing coinci­

the filter and hits the detector you

which permit experimental tests of

dence. On those days when you

win. The observed results are con­

some of quantum theory's most per­ verse features.9 When "gedanken ex­

both choose the same box one

sistent with the predictions of quan­

wins and the other loses. Further

periments,"

investigation

such

Podolsky-Rosen and

as

the

(EPR)

"Schrodinger's

Einstein­

experiment

cat"

assumed

to

be

that

other

mathematicians using this particu-

tum theory [18, 1 9,30]. Finding any

other explanation poses a challenge, as discussed in [30].

paradox,

were first suggested, the outcomes were

reveals

obvious.

"inspired by Nelson's 1 980's version (24) using "game cards" from a fast food chain.

However, modern technology has per­ mitted similar experiments to be per­

cepted, particularly by philosophers,

in the puzzle, than by experimental ev­

formed, and, in every case, the out­

despite the fact that the experimental

idence, which left room for alternative

the

evidence was much less compelling.

theories of gravitation. One attempt at

predictions of quantum theory and

come

Indeed, the initial verifications, such as

a modified theory is the search for a

seem

is to

consistent preclude

with

"hidden

vari­

the red shift and deflection of light

"fifth force" in the 1980's [30]. Several

1 ables." 0 Any alternative to quantum

around the sun, were based on mea­

theory must be consistent with these

surements of distinctions so fine that

periments seemed to support gravita­

experiments as well as replacing an

they were the same order of magnitude

tional corrections. But the need for

impressively successful framework.

extremely careful and reproducible ex­

as the error bounds. Scientists may

consistency, as well as reproducibility,

By contrast, Einstein's theory of

have been influenced more by the ele­

proved crucial-some experiments in­

general relativity was more readily ac-

gance with which the theory filled gaps

dicated repulsive deviations from the

81t should be noted that Lanczos's work preceded that of Schrbdinger, who cited it, but seems to have misinterpreted a critical feature [32). See also [6) .

9See Faris [8,9] for brief accounts of the important differences between quantum theory and classical probability, especially pp. 205-206 of [8]. See Mermin (1 9,20) for a very readable discussion of the EPR experiment, Wick [33) for a overview of the entire history of paradox and controversy, and Bell's original papers in (1 ].

10'fo be precise, these experiments only preclude local hidden variables. They are most important as evidence that quantum theory, with or without hidden variables, requires non locality or "spooky action at a distance."

VOLUME 23. NUMBER 1 . 2001

25

conventional theory, while others sug­ gested attractive forces. One of the most extensive attempts at an alternative to quantum theory is E. Nelson's "stochastic mechanics." In 1966, Nelson, building on ideas of Bohm and Feynes, proposed an alter­ native theory based on classical me­ chanics and a type of Brownian motion arising from interaction with a back­ ground field. For the next 20 years, Nelson [22,23] continued to develop this remarkable theory. However, in 1984, at a conference in Boulder, he reported that "Markovian stochastic mechanics is untenable as a realistic physical theory" [24]. Compatibility with the results of the increasingly convincing EPR experiments required a non-Markovian diffusion process. 1 1 In the jigsaw metaphor, Nelson's the­ ory is a good example of an alternative whose pieces fit together and whose edges can even be attached to part of the puzzle, but still does not quite fit. 12 Adding a non-Markov process may ex­ tend this arm of the puzzle, but gaps are likely to remain unfilled, and the fi­ nal resolution must come from exper­ iments. This assumes, of course, that some experiment can always be found to distinguish between mathematically inequivalent theories. One might also ask, instead, what it might mean if this assumption were false. This question is discussed further within the partic­ ular context of Bohmian mechanics in Appendix B. One can identify several periods in the development of quantum theory. In the first, physicists tried to come to grips with a new theory with unfamil­ iar properties. A mathematical frame­ work which satisfied most physicists was found more readily than answers

to many of the perplexing questions which arose. This led to a period of about 50 years during which the Copenhagen Interpretation based on "complementarity" pushed fundamen­ tal questions under the rug. While some 13 have derided this as a period of "cognitive repression" [ 14], I think it was natural that most physicists ig­ nored foundational questions while they learned the language needed to explore their strange new land. In the past 20 years sophisticated experi­ mental tools, such as ion traps and scanning tunneling microscopes, have given physicists new ways to observe quantum phenomena. For many this was the icing on the cake of a mature theory whose status was secure; for others, it opened the door to new in­ sights and more ways to examine quan­ tum paradoxes. We are now entering a new era, in which quantum phenom­ ena are no longer viewed as perverse, but as the potential source of exciting new practical tools for computation and communication. (See, e.g., [12] for an enthusiastic summary of this view­ point, and [25,26] for comprehensive treatments of quantum computation and communications.) These stages seem a necessary part of our development in understanding the physics of a world far from our or­ dinary experience. What is not so clear is whether different cultural and social influences could have length­ ened or shortened some of them sig­ nificantly. It should be noted that even though studying the foundations of quantum mechanics has long been far from the mainstream, it has never been suppressed. The papers of Bohm, Bell, et al were published in reputable journals, and heads of lab-

oratories which had the neces,sary equipment agreed to perform the ex­ periments proposed by people like Clauser and Shimony. 14 Delicate experiments, such as those done by the laboratory groups as­ sociated with Aspect, Leggett, and Zeilinger are useful in clarifying some critical aspects of quantum theory. However, to my mind, the most con­ vincing verification of quantum theory is not the individual microscopic ex­ periments, but the fact that no other theory can even come close to ex­ plaining so many diverse phenomena (including macroscopic as well as mi­ croscopic phenomena); literally, "from atoms to stars" [16]. Few jigsaw puz­ zles fit together so neatly. We a,re forced to overcome the biases arising from our experience with the familiar macroscopic world of classical me­ chanics despite the challenge of re­ solving all questions about the founda­ tions of quantum theory. In the end, quantum theory remains a human con­ struct subject, in principle, to social forces. But it is a theory so remarkable, so different from ordinary experience, that it transcends social and cultural forces. Afterword: It may be worth men­ tioning the personal bias I bring to these matters. As a student I was suf­ ficiently fascinated with the mathe­ matics and foundations of quantum mechanics to seek out a postdoctoral position in Geneva, then a major cen­ ter in research on the foundations, based on Jauch and Piron's ap­ proaches. However, I found the for­ malism of Yes-No experiments too far from the physics to which quantum theory was applied. I realized that my interest lay more with the mathemati-

1 1To be precise, it is not the EPR probability distribution that require the process to be non-Markovian but compatibility of that process with the assumption of "active locality." Mhough the EPR experiments seem to imply some form of passive non-locality" in the sense of correlations between distant events, this does not necessarily imply that action at one point can instantaneously affect the probabilities of events at an arbrtrarily distant location. Nelson regarded active locality (which precludes such effects) as es­ sential to a realistic physical theory. Others disagree; in particular, some advocates of Bohmian mechanics accept active non-locality and even regard rt as an asset. As far as I

am aware, a non-Markovian form of stochastic mechanics has not been developed. For more information on the subtle issues of active and passive localrty see [1 0].

12Wick [30] reports (p. 79 and pp. 1 72-1 73) unpublished work by Wallstrom which raises another experimental objection to Nelson's theory and a possible modifica­

tion which would reconcile the difficulties. 1 3Aithough Keller [1 5] never uses the term "hidden variables," her assertion that "knowability" must be relinquished would appear to imply a belief in some form of hid­ den variables. In view of this, it is curious that this essay, originally published in 1 979, was reprinted in 1 985 without change, desprte widely publicized and impressive experiments of Aspect, et al in 1 982 agreeing with the predictions of traditional quantum theory [33]. To retain belief in hidden variables one now had to give up an­ other traditional belief, locality. The notion of physical science implies a subject whose "cognitive development" requires the development of an understanding consis­ tent with the results of experiments. It would be interesting to know why Keller did not even comment on the issue of locality or the EPR experiments. 14See the chapter on "Testing Bell" in [33].

26

THE MATHEMATICAL INTELLIGENCER

cal problems which arose from exam­

the insights of gender can play a role

of theory connecting phenomena with

ining physical models within the con­

analogous to that of a mathematical

the output from modern devices.

text of Hilbert space and operator al­

language such as geometry.

gebras that are naturally associated

It is harder to see a role for gender

Appendix B. Some thoughts on

with Dirac and von Neumann's ap­

in a field like quantum theory. Does

proach. I do not believe that a focus on

this mean that quantum theory is "gen­

Nelson's

the mathematical equations implies a

der-free"? I think this question misses

"Bohmian mechanics" are theories

Bohmian mechanics

stochastic mechanics and

lack of appreciation for the founda­

the point. Those who see a role for gen­

which assume that particles have tra­

tional puzzles. Rather, I believe that

der do so in the expectation that

jectories; they differ from classical me­

some of the confusion arises from try­

women's experiences will provide dif­

chanics by introducing a probabilistic

ing to force quantum phenomena into

ferent and valuable insights. However,

a cognitive framework developed from

quantum phenomena are far from

a different set of experiences, and that

every physicist's personal experiences.

component. In Nelson's approach, the

trajectories are random. In Bohm's the trajectories are deterministic; all the

mathematical concepts are needed to

All physicists, regardless of their gen­

randomness is in the initial conditions.

supplement that framework My views

der or ethnicity, must enter a strange

Both theories are designed to agree

may not satisfy others; but they are not

land and learn its customs and lan­

with quantum mechanics for single­

those of one who has ignored founda­

guage in order to understand quantum

time distributions.

tional issues.

theory and develop an intuition for its

The advocates of Bohmian me­

phenomena. Scientific inquiry and the

chanics claim [2,5] that even joint time

Acknowledgments

language of mathematics provide an

distributions or multi-particle experi­

It is a pleasure to thank Professors Eric

opportunity for people to go beyond

ments can not distinguish it from stan­

Carlen, William Faris, Chris Fuchs,

their ordinary experience. To insist

dard quantum theory. Their argument

Stephen Fulling, Chris King, and Alan

relies on the fact that both Nelson's

C. King for reference [ 13) and to S.

that gender play a special role in such inquiries confines women instead of aUowing them the freedom to learn new languages and explore the uni­ verse.

Fulling for [30]. I owe a particularly

The cultural milieu in which boys

Sokal for comments on earlier drafts. I am grateful to C. Fuchs for bringing references [ 1 1, 14] to my attention, to

theory and Bohmian mechanics imply that the Schrodinger equation holds. However, there is more to quantum theory than the Schrodinger equation,

which yields only wave mechanics. 1 5

large debt to W. Faris for clarifying a

often begin the study of science with

In the Nelson and Bohm theories, the

number of issues related to both

more exposure to the phenomena of

position operator plays a special role and the equivalence between position

stochastic mechanics and

classical physics may be changing-ir­

Bohmian mechanics. Any errors or

respective of whether or not girls are

and momentum representations is lost.

misinterpretations remain my own re­

encouraged to engage in comparable

More to the point, there are other ob­

Nelson's

sponsibility. Moreover, I should em­

activities. The days when children

servables, such as spin and angular

phasize that the opinions and biases

could take apart a radio or tinker with

momentum. I am not convinced that no

expressed here are entirely my own

a car engine to see how it works are

experimental distinction is possible us­

and do not necessarily reflect the

rapidly disappearing. Circuits and vac­

ing these other observables. (See [30]

uum tubes have long been replaced by

for a lively discussion of this issue and

views of Professor Faris. Appendix A. Gender Issues

much smaller electronic devices, and

[ 13] for a recent proposal of such an

modern car engines use sophisticated

experiment.)

My own involvement in the "science

fuel injection devices. Neither girls nor

Precisely because of the role played

wars" began in 1986 with a letter to the

boys will have-as I did-the opportu­

by trajectories and position, the unitary

editor in the A WM Newsletter, and my

nity to learn to read a scale and observe

equivalence of different representations

interest has remained focused on gen­

thermal expansion when my mother

is lost. However, the concepts of unitary

der issues. Some social constructivists

checked my temperature when I was

gates and outcome measurements are

ill. All they see is a digital readout from

essential to quantum computation. Su­

ity might also provide special insights

a mysterious device. The playing field

perposition gives quantum computers

affecting the development of science.

may be leveled because modern tech­

their massively parallel power; and uni­ tary transformation provides a natural

have suggested that gender or ethnic­

In those cases where gender has been

nology has removed many phenomena

shown to have an impact, the most

from our direct experience and obser­

tool for (reversible) gates. But these

convincing examples have been in sub­

vation. This raises new pedagogical

tools only push the computational com­

specialties of biomedical research,

challenges: it may be necessary to

plexity problems from the computa­

psychology, or anthropology, where

teach children to explore and observe

tional phase to the output phase.

gender is directly relevant to the sub­

at the most basic level before they can

quite remarkable that algorithms can be

ject. In these situations, I believe that

recognize the necessity of a long chain

developed that leave the computer in a

It is

1 5See [32] for an explicit discussion of the impossibility of deriving Heisenberg's hypothesis about transition probabilities from Schrbdinger's theory.

VOLUME 23, NUMBER 1, 2001

27

state upon which a suitable measure­ ment yields useful information. Even if quantum computation is consistent with Bohmian mechanics (which im­ plies reversible trajectories), the key concepts rely upon the standard Hilbert space formalism. It is difficult to imag­ ine how the recent advances in quan­ tum computation and communication could have occurred without the in­ sights obtained from the usual Hilbert space view. Some advocates of quantum com­ puting believe (e.g., [ 12]) that it will yield new information about the foun­ dations of quantum mechanics. Thus far, the early experiments have been too close to those of EPR type to pro­ vide a basis for distinguishing the usual Dirac/von Neumann theory from Bohmian or stochastic mechanics. However, the development of larger multi-bit devices may eventually pro­ vide such tools. As I wrote this final section, I real­ ized that I was describing a situation re­ markably similar to Graham's example. The equivalence of position, momen­ tum, and other representations plays an important role in the Hilbert space ap­ proach to quantum theory developed by Dirac and von Neumann. Bohmian me­ chanics singles out a particular repre­ sentation with special properties. What is missing is an explicit acknowledg­ ment of political or cultural motivation for that view. Although there are certainly situa­ tions in which a particular representa­ tion or coordinate system may give use­ ful insights, that is quite different from an insistence on using that representa­ tion exclusively. Quantum mechanics not only explained the puzzles of the early decades of this century, it also predicted the results of experiments not performed until many years later and explained phenomena not anticipated. I find this far more convincing than a sub­ sequent theory that is merely consistent with results known at its inception. It is curious that its proponents assert the impossibility of experimental verifica­ tion as a virtue rather than seeking new phenomena to explain or test their the­ ory. Whether or not the Bohmian view is useful, this seems to place it outside the realm of physics.

28

THE MATHEMATICAL INTELLIGENCER

REFERENCES

[1 ] J.S. Bell, Speakable and unspeakable in quantum mechanics

A U T H O R

(Cambridge Univer­

sity Press, 1 987). [2] K.

Berndl, M. Daumer, D. Durr, S.

Goldstein, and N. Zangh, "A SuNey of Bohmian Mechanics" II Nuovo Cimento

1 1 08,

737-750(1 995).

[3] J. Bernstein and D. Cassidy, "Bomb Apologetics: Farm Hall, August, 1 945" 32-36 (August, 1 995).

Phys. Today 48(8),

[4] P.A.M . Dirac, The Principles of Quantum (Oxford, 1 930).

Mechanics

MARY

"Quantum equilibrium and the origin of ab­ solute uncertainty" J. Stat. Phys. 61, 843-

BETH RUSKAI

Department of Mathematics

[5] D. Durr, S. Goldstein, and N. Zangh,

University of Massachusetts Lowell Lowell, MA 01 854

907 (1 992).

USA

[6] C. Eckart, "The solution of the Problem of

e-mail: [email protected]

the Simple Oscillator by a Combination of the Schrodinger and Lanczos Theories" Proc. NAS

12, 473-476 (1 926).

M . Beth Ruskai got her Ph.D. at Wisconsin in 1 969. She has been at

[7] C. Eckart, "Operator Calculus and the So­

her present University most of the

lution of the Equations of Quantum Dy­

years since, but with numerous tem­

namics" Phys. Rev. 28, 71 1 -726 (1 926).

porary positions, including Geneva,

[8] W. Faris, "Shadows of the Mind: A Search

the University of Oregon, MIT, the

for the Missing Science of Consciousness"

University of Michigan, Case Western

1 996).

Reserve, Mittag-Leffler, and Paris IX

[9] W. Faris, "Review of Roland Omnes, The

(Dauphine). She is known for her work

AMS Notices 43(2), 203-208 (Feb. ,

Interpretation

of Quantum

AMS Notices

43(1 1 ) , 1 328-1 339 (Nov.,

Mechanics"

on operator-theoretic problems aris­ ing

from

quantum

mechanics ­

among others, the "Ruskai-Sigal the­

1 996). [1 0] W. Faris "Probability in Quantum Mechanics"

orem" that an ion with sufficiently large excess of electrons can not

Appendix to [33]. [1 1 ] P. Forman, "Weimar Culture, Causality,

have a bound state. In recent years

and Quantum Theory, 1 91 8-1 927: Adap­

she works especially on quantum in­

tation by German Physicists and Mathe­

formation. She is also known for her

maticians to a Hostile Intellectual Environ­

off-duty writing on gender and sci­

ment," in Historical Studies in the Physical

ence, and other social issues of the

Sciences,

3,

R. McCorrnmach (U.

ed.

profession. Her outdoor enthusiasms are gardening, hiking, and most es­

Pennsylvania Press, 1 97 1 ) pp. 1 -1 1 5. [1 2] C.A. Fuchs, "The Structure of Quantum

pecially SNOW.

Information," available at http://www.its. caltech.edu/-cfuchs/lwl.html or directly from the author at [email protected]. [1 3] P. Ghose, "The Incompatibility of the de Broglie-Bohm theory with Quantum Mechanics" posted at xxx.lanl.gov/abs/ quant-ph/0001 024. [1 4] P.

A.

Hanle,

"Indeterminacy

Before

on Gender and Science

(Yale University

Press, 1 985). [1 6] E. Lieb, "The Stability of Matter: From Atoms to Stars" Bull. AMS 22, 1 -49 (1 990).

[1 7] D. Kotschick "Gauge Theory is Dead!­

Heisenberg: The Case of Franz Exner and

Long Live Gauge Theory!" AMS Notices

Erwin Schrodinger," in Historical Studies

42(3), 335-338 (Mar., 1 996).

in

the

Physical Sciences

10, ed. R.

McCormmach, L. Pyenson, and R. S. Turner

(Johns

Hopkins

U.

Press,

Baltimore, 1 979), pp. 225-269. [1 5] E.F.

Keller, "Cognitive Repression

[1 8] C. Lanczos, Z. FOr Physik 35

[ 1 9] N. D. Mermin, "Is the Moon Really There When Nobody Looks? Reality and the Quantum Theory" Physics Today 38(6),

in

Contemporary Physics" Am. J. Phys 48(8) 7 1 8-721 (1 979); reprinted in Reflections

38-47 (April, 1 985). [20] N. D. Mermin, Boojums All the Way Through (Cambridge University Press, 1 990).

The M IT Pres [2 1 ] W. Moore. Schr6dinger: Life and Thought (Cambridge University Press, 1 989).

[22] E. Nelson, Dynamical Theories of Brownian Motion (Princeton University Press, 1 967).

[23] E. Nelson, Quantum Fluctuations (Prince­ ton University Press. 1 985). [24] E.

Nelson. "Quantum Fluctuations-an

Introduction" in Mathematical Physics VII,

W.E. Brittin, K.E. Gustafson, W. Wyss,

eds, pp. 509-51 9 (North Holland, Amster­ dam. 1 984).

I.L. Chuang,

[25] M.A. Nielsen and

Quantum

Computation and Quantum Information

(Cambridge University Press, in press) . [26] J. Preskill, lecture notes on Theory of Quantum

Information

and

Quantum

Computation available at http://theory.cal­

tech.edu/preskill/ph229.

Women Becoming Mathematicians C reating a Professional Identity in Post-World War I I America Margaret A. M. Murray "A sophisticated, schol ar l y,

and readable study - this is

without a doubt the best book

yet written on American

M AT H [ M A T I C l A N S

women mathematician s . ·

[27] R. Rhodes, The Making of the Atomic Bomb (Simon and Schuster, 1 988).

[28] B. Simon, The P(h Euclidean Quantum Field Theory (Princeton University Press,

1 974).

- Ann Hibner Kobl itz, Women's Stud ies Program, Arizona State U n i versity 304 pp . ,

23 illus. $29.95

[29] K. R. Sopka, Quantum Physics in America: The years through

[30] M.O.

Scully,

1 935 (AlP, 1 988).

"Do

Bohm

Trajectories

Always Provide a Trustworthy Physical Picture

of

Particle

Motion?"

Physica

To order call SOo-356-0343 (US & Canada) or 617· 625-8569. Prioes subject to change without notioe.

Scripta T76 4 1 -46 (1 998).

[3 1 ] 8. Schwarzschild "From Mine Shafts to Cliffs-the 'Fifth Force' Remains Elusive"

Phys. Today 41(7), 2 1 -24 (July, 1 988).

[32]

J . von Neumann tion

of

Mathematical Founda­

Quantum

Mechanics

(English

translation, Princeton University Press, 1 955). [33] B.

L. van der Waerden, "From Matrix

Mechanics and Wave Mechanics to Unified Quantum Mechanics" in The Physicists Conception of Nature,

J. Mehra, ed., pp.

276--293 (D. Reidel Publishing, Dordrecht,

Holland, 1 973); reprinted in AMS Notices 44(3), 323-328 (1 997).

[34] D. Wick, The Infamous Boundary (Birk­ hauser, 1 995).

VOLUME 23, NUMBER 1, 2001

29

N.G. KHIMCHENKO

From the " Last I nterview" with A. N . Kol mogorov I, for one, have followed all my life the precept that truth is sacred, that it is our duty to seek it out and to defend it, regardless of whether it is pleasant or not. A.N.K., 1984

While I pursued a fairly wide range of practical mathematical applications, and at times obtained useful results, I remain predominantly a pure mathematician. I admire those mathematicians who became significant in technology; I fully recognize the importance of computers and cybernetics for the future of mankind; nonetheless I feel that pure mathematics in its traditional form has not yet ceded its deserved place of honor among the sciences. The only thing that could kill it would be too sharp a division of mathematicians into two tendencies: those who cultivate the newest abstractfacets without account for their ties to the real world which bred them, and those who busy themselves with "applications, " neglecting the need for in-depth analysis. A.N.K., 1963

I consider my scientific career, in the sense of getting new results, to be completed. This saddens me, but I yield to the inevitable. In recent years I have been directing my energies elsewhere, on textbooks for secondary schools and books for the mathematically talented. Ifeel the desire to participate in this project with the vigor of youth. A.N.K., 25 April 1986 (His next-to-last birthday.) The great Russian mathematician Andrei Nikolaevich Kolmogorov was open and outgoing with friends, but rarely granted interviews; few direct records of conversations with him survive. Fortunately, the film-maker Aleksandr Nikolaevich Marutyan, in planning his successful 1983 film "Stories on Kolmogorov," tape-recorded long, wide-ranging conversations in which he explored areas of potential use for the film. After Kolmogorov's death in 1987, the unique interest of these tapes was recognized by V.M. Tikhomirov, with whom they had been left. The enormous task of transcription of the fragmentary (sometimes incomprehensible) materials was undertaken by Natal'ya Grigor'evna Khimchenko. The full printed text she prepared circulated privately, and recently became available in the book Yavlenie Chrezvychainoe (Extraor­ dinary phenomenon) devoted to Kolmogorov. l For presentation to the general reader, it seemed ap­ propriate to sift the materials and put them in some kind of order. Fortunately for the mathematical public, N.G. Khimchenko returned to her labors, editing and organizing the text and translating it into English. Thanks also to 1 Edited by V.M. Tikhomirov. FAZIS, MIROS, Moscow, 1 999. See pp. 1 83-21 4.

30

THE MATHEMATICAL INTELLIGENCER ©

200 1

SPRINGER-VERLAG NEW YORK

V.M. Tikhomirov and Ya. G. Sinai for advice, to Smilka Zdravkovska for great help in the fmal editing-and of course to A.N. Marutyan for conducting the interview in the first place. In the original interview, Marutyan often doubled back on a topic several times. The editing process, aimed at uni­ fying and at reducing repetition, sometimes juxtaposed passages from different points in the tapes.-Editor's Note A.N.

Marutyan: Andrei Nikolaevich, I wanted to ask how you

began your journey into mathematics, what influenced your choice of direction, your choice of specialty in mathematics. A.N. Kolmogorov: It seems to me that for young mathe­ maticians the most common scenario does not involve afree choice of direction, but rather an attempt to solve concrete problems presented by the older generation. This is the norm. My first works in trigonometric series were all of this nature. Ideas of undertaking, let us say, a reconstruction of an entire branch of science arise at a later time. In my case, it was not that much later; namely, when I was investigat­ ing the basic tenets of probability theory, I aspired from

View(s) of Andrei N. Kolmogorov by the famous portraitist Dmitrii I. Gordeev. (Used by permission of A.N. Shiryaev.)

the start to build a more logical system of the concepts of this whole science. This was in the early 1930s, when I was around 30 years old. Generally mathematicians start from a certain catalogue of existing problems of interest to a given mathematical cir­ cle-the Luzin school in Moscow, for instance. And the young people struggle over the solutions. When they can­ not solve one problem, they take up another. There are al­ ways plenty of those closest unsolved problems to choose from. At times an excessive insistence on solving precisely one stated problem is quite detrimental to a mathemati­ cian's career. The majority of mathematicians begin their work under someone's tutelage; the supervisor has many such unsolved problems, and his task becomes one of matching the young students with the most suitable prob­ lems. If success is long in coming, then he may suggest a switch to some other similar problem.

M.: How do you face the fact that you have worked all your life in a field which most people do not understand? K.: Calmly . . . I think that the achievements of mathemat­ ics prove useful to mankind, while to us, mathematicians, they bring such inner satisfaction! It is a perfect solution, to have such a peaceful coexistence. A friend of mine, a pure humanist, used to say that to him mathematicians were like useful domesticated animals [Marutyan laughs], that they had to be treated in a utilitar­ ian way. To him, all true cultural values were humanistic; but, technology being necessary, mathematics is necessary, so we have to give mathematicians what they need to stay alive and keep going. M.: Have you ever felt jealous of someone accomplishing the same task you were working on, or doing it more elegantly? K.: No.

VOLUME 23, NUMBER 1, 2001

31

M.: Why? Because you didn't see them as rivals, or are you simply free of such a complex?

K.: Probably the latter. If a problem is solved-this is good and I am simply glad to know it. I understand that what I am saying sounds like copy-book morality. But looking back, I really do not recall such a situation of rivalry. Upon discovering that something I had been working on had al­ ready been done. I would feel a relief of sorts: Thank God, it's done! I am not boasting. M.: You have probably felt that some of your abilities sur­ pass those of your colleagues?

K.: Well, in some cases yes, in others no. Of course, when it does happen, the feeling is a pleasant one, I guess. M.: Have you ever encountered ill-will because you were generally quicker, deeper, more talented?

K.: No, I must say, I don't think I have. M.: Would you say this is because of the fairness and ob­ jectivity of mathematicians?

K.: Mathematics is generally a reasonably objective science. The potential of a new idea to go far and to solve existing problems becomes evident fairly quickly. Mathematics is a tremendously pleasant field of work precisely because real progress is not lost, and is in the vast majority of cases ac­ knowledged in time-to a greater degree than in any other sphere of human activity. M.: But was this on your mind when you were choosing mathematics?

A.N. Kolmogorov at the blackboard.

K.: That it would be a calmer life? [Laughs.] No, I was not thinking that. My final decision to opt for mathematics as my main field came fairly late, at a time when I had already

Especially, I was always extremely interested in work­ ing in general education.

produced works of my own. At that point it had become

I studied in an altogether extraordinary school, founded

the clearly logical path. [Pause. ] Until then I had been turn­

by two dedicated women, Repman and Fedorova. And one

ing over several possibilities for myself. My first serious

wish that I have always felt is to concentrate on realizing

adult idea of a future career was forestry. I was also seri­

a somehow

ously interested in history.

when my greatest ambition was to be director of a school-

A.N. Kolmogorov with a hiking party. V.N. Tikhomirov is the man on the right.

32

THE MATHEMATICAL INTELLIGENCER

ideal school. There was a rather long period

I don't mean specially a mathematical school. This was the influence of my own experience as a student. The Repman high school was set up by a group of Moscow intellectuals specially for their own children. [Pause. ] I finally decided to become a mathematician only when I became convinced that that would

suit me, whereas I had

no idea whether anything else would work or not. M.: You didn't want to risk trying out as a school director, eh?

K.: Right. But I was secretary of a school soviet for two and a half years. At Potylikh. Did you know about that episode in my life, the Potylikhin school? M.: But Andrei Nikolaevich, that Potylikhin school was so

experimental. . . . Were you also involved in the experiments?

K.: I wasn't involved, I was carrying out the experiments:

the so-called Dalton plan. To this day I believe there was a great deal of good in them. The scheme was, the teacher of each school subject, say mathematics, if there were 5 hours for it [per week] , would

tell about the subject, in an entertaining way, with demon­

strations-but only for one of those hours. The remaining time the students would follow a monthly schedule of tasks: look at such-and-such a book, read such-and-such, solve such-and-such problems, find such-and-such a rela­ tionship and represent it graphically. M.: What are your interests outside of mathematics?

K.: If I were to rank them, then after mathematics comes my interest in educating the young, in all fields.

As to prosody [the study of verse forms] , which even among humanistic fields is a very special nook-that I re­

Kolmogorov in his study.

gard as another branch of my scientific work. I am taken seriously there. I even served as

opponent

[external ex­

aminer] of the doctoral dissertation of the philologian Gasparov. My works are published;

Zhirmunski'i valued

Once I was returning from Rome to Moscow, and found myself in the same car with Cardinal Wyszyriski, who was

them, and among foreign experts, Jacobson. Among other interests, let me call them

K.: No, I was attracted by art itself, but I rarely got in step, so to say, with the latest trends in our intelligentsia.

consumer inter­

returning from Rome to Cz�stochowa. And in the same

ests (where I do not produce anything myself), I would name

two-person

music and also early pictorial arts, especially Russian.

Poznanski.

An indelible impression was made on me by my adven­

Christianity

compartment We and

talked about

with

for

me

hours

was in

Archbishop

German

non-religious

about

humanism-of

tures in Russia's North. I would set off on such voyages,

Thomas Mann, say. . . . But the Archbishop, naturally, took

having found the location of old wooden churches in

the position that true humanism not based on religion is

Grabar's

History of Russian Art. From one church to the

impossible; I, naturally, set about trying to prove to him

next I would travel alone, sometimes on foot, sometimes

that on the contrary, faith in the eternal is not necessary

by rowboat, sometimes on board ship. The project would

for a positive human philosophy. . . .

lead me to priests, who would often put me up for the night.

much all the way from Rome to Cz�tochowa, where

This went on pretty

Traveling with a companion was also a favorite pursuit.

though he hadn't reached Warsaw he had to get off together

In these activities people trust each other and fully open up. I traveled with Dima Amol'd2 for 20 days, with Igor'

with Wyszynski. Right at the end of the conversation, my Archbishop took a tape recorder out from under the seat

Zhurbenko. Earlier, it would be with my own age group, of­

[Marutyan laughs] and said, "Herr Professor, I hope you

ten with Gleb Seliverstov. He was one of my closest friends.

have no objection to my having recorded our instructive

One of my first works on trigonometric series was done

exchange." [Both laugh. )

jointly with him; he was something of a mathematician.

M.: So, a tricky agent, I'd say!

A few people

were

really close

to

me:

Volodya

Tikhomirov, Igor' Zhurbenko; and Il'dar lbragimov, my

K.: I told him I had no objection; I don't think I had been

saying anything terrible . . . . And we parted friends.

friend in Leningrad.

M.: Whereas I am openly taking a recording, Andrei

M.: You were probably also attracted by the art world?

Nikolaevich.

21J.I. Arnol'd.

VOLUME

23,

NUMBER 1 , 2001

33

What attracted you in peopl�onunon interests, or were

-�

you intrigued by people from spheres distant from yours?

K.: No, in most cases I was drawn to people with similar interests to my own. M.: How important in your life was your friendship with Pavel Sergeevich Aleksandrov?

K.: Very important. Although the difference in age between us was only seven years, in 1929 when we grew close this

\

was still noticeable. I was the junior in our relationship, a sort of protege or ward. And this tinge of patronage per­ sisted all our life, completely accepted by me. Have you seen the little note Pavel Sergeevich wrote for my jubilee? M.: Yes, certainly, and also the memoir he wrote.

K.: That little note about me he wrote less than a year before his death. You remember how he speaks there of this friend­

ship which in the whole 53 years was always unclouded.

M.: People say of him, and he has said himself, that at the

beginning of his mathematical career there was a time when things didn't go well, he got no results, and he was

)

of a mind to give up mathematics.

K.: Yes, that's true. Nikolai Nikolaevich Luzin, the teacher of both of us, took great pleasure in constructing hy­

I,

potheses, which sometimes worked out and sometimes didn't, about who in his inunediate circle of students should work on which problem and would get somewhere. In the case of Pavel Sergeevich, Nikolai Nikolaevich got the no­ tion-how, I don't know-that a famous problem for which there was then no known avenue of approach, the Problem of the Continuum, was going to be solved by Pavel

A portrait that hangs in Komarovka. (Artist unknown.)

Sergeevich Aleksandrov. With great insistence, and with

dents would take turns presenting papers, but the instruc­

the great persuasiveness he possessed, he planted this no­

tor would speak more than the other participants. The stu­

tion in Pavel Sergeevich's soul, where it led to such a cri­

dents with whom one eventually undertakes individual

sis as to make him decide to leave mathematics.

work tend to come from these seminars.

M.: So you and Pavel Sergeevich had difficult relations with

M.: Do you take naturally to collective work, or does it take

Luzin. . . .

away the joy of your own creativity?

K.: Yes. M.: But just when you were beginning your independent

K.: It need not. Maybe, with the critical problem of over­

specialization, we must find new forms of collective work,

work, you left the Luzin entourage and the two of you

finding ways somehow to divide up a problem and solve it

founded a new circle.

piecemeal. Among my

To what extent was your work with graduate and un­

students, by the way, are

masters

of organizing collective work, with the ability to divide an

dergraduate students a creative process?

area of research into pieces, overcoming the difficulties

K.: Well, even when a seminar covers elementary mater­

that this entails by constant contact, distributing the work

ial, and the teacher is already familiar with the solutions

among close colleagues: Izrail' Moiseevich Gel'fand, and

to all the problems, one has the challenge of putting one­

now Vladimir Igorevich Arnol'd.

self on the same level as the student.

M.: Have you been successful in such collective work, or

M.: What role have your students played in your life?

do you have more of a "loner style"?

K.: A very significant one, certainly. Emotionally, of course

K.: No, compared with them I did not find it easy to orga­

some were more important than others. Some are very

nize

close to me personally.

out. . . . I must say that when I wasn't working entirely on

All my years of active work at the university would in­

large groups. When I try to think of my work I carried

my own I did best in a collaboration of

two. This worked

clude two hours a week of some required course. I con­

with Pavel Sergeevich Aleksandrov, with Boris Vladimiro­

ducted many of these core classes: Theory of functions of

vich Gnedenko, with various collaborators.

a real variable, Functional analysis, Differential equations,

Pavel Sergeevich and I collaborated intensively only once,

Theory of probability. This was a conunon distribution of

on a fairly narrow question of topology; after that, only oc­

his works and he in mine, but

work for all our professors: one such required course, and

casionally. I was interested in

a two-hour special course in lecture format addressing re­

a really shared project occurred only that one time.

cent work, including one's own. Then normally there would

The first people with whom I worked closely and suc­ cessfully were Dmitri! Evgen'evich Menshov-on trigono-

be one or two seminars a week, in which about ten stu-

34

THE MATHEMATICAL INTELLIGENCER

Here is the list of Kolmogorov's students,* as found in V.M. Tikhomirov's article in

Mathematics

(ed.

S.

Golden Years of Moscow

Zdravkovska and P.L. Duren),

American Mathematical Society, 1993, pp.

125-127.

A.M. Abramov (education)

P. Martin-Lef (complexity) A.V. Martynov (probability) R.F. Matveev (stochastic processes) Yu.T. Medvedev (mathematical logic) L.D. Meshalkin (probability, ergodic theory)

V.M. Alekseev (classical mechanics)

V.S. Mikhalevich (probability)

A.M. Arato (probability)

M.D. Millionshchikov (turbulence)

V.I. Arnol'd (superpositions, classical mechanics)

AS. Monin (oceanology, turbulence)

E.A. Asarin (complexity)

S.M.

G.M. Bavli (probability)

A.M. Obukhov (atmospheric physics, turbulence)

G.l. Barenblatt (hydrodynamics)

Yu.S. Ochan (set theory)

L.A. Bassalygo (information theory)

Yu.P. Ofman (complexity)

ikol'skll (approximation theory)

Yu. K. Belyaev (stochastic processes)

B. Penkov (probability)

V.I. Bityuskov (probability)

A.A. Petrov (probability)

E.S. Bozhich (mathematical logic)

M.S. Pinsker (information theory)

L.

. Bol'shev (mathematical statistics)

A.V. Prokhorov (study of prosody)

A.A. Borovkov (probability)

Yu.V. Prokhorov (probability)

A.V. Bulinskii (stochastic processes)

Yu.A. Rozanov (stochastic processes)

N.A. Dmitriev (stochastic processes)

M. Rozenblat-Rot (stochastic processes)

R.L. Dobrushin (probability)

B.A. Sevast'yanov (stochastic processes)

AN. Dvoichenkov (theory of functions)

A.N. Shiryaev (stochastic processes)

E.B. Dynkin (stochastic processes)

F.l. Shmidov (theory of functions)

V.D. Erokhin (approximation theory)

Ya.G. Sinai (ergodic theory)

M.K. Fage (functional analysis)

S.Kh. Sirazhdinov (probability)

S.V. Fomin (ergodic theory)

V.M. Tikhomirov (approximation theory)

G.A. Gal'perin (dynamical systems)

A.

I.M. Gel'fand (functional analysis)

V.A.

B.V. Gnedenko (probability)

I.Ya. Verchenko (theory of functions)

O.S. Ivashev-Musatov (theory of functions)

V.G. Vinokurov (probability)

. Tulal"kov (theory of functions) spenskll (mathematical logic)

AT. Kondurar' (theory of functions)

V.G. Vovk (complexity)

M.V. Kozlov (stochastic processes)

A.M. Yaglom (turbulence)

V.V. Kozlov (probability)

B.M. Yunovich (theory of functions)

V.P. Leonov (probability)

V.

L.A. Levin (complexity)

I.G. Zhurbenko (probability)

AI. Mal'tsev (mathematical logic)

V.M. Zolotarev (probability)

. Zasukhin (stochastic processes)

·Someone whose name does not appear on this list of formal advisees, but who in effect was Kolmogorov's last student and is often listed

as

such, is the fre­

quent Mathematical lntelligencer contributor, Alexander Shen.

metric and orthogonal series-and Aleksandr Yakovlevich

often in science: a problem is solved in a roundabout way,

Khinchin-on application of function-theoretic methods,

and only later is a more direct approach discovered.

especially trigonometric series and orthogonal functions,

M.:

to important aspects of probability theory. This led to joint

would you name as the greatest contributors to the

publications with both of them. The collaboration with

progress of science?

Among

twentieth-century mathematicians,

whom

Khinchin was the most productive of all, because we got

reaUy important results: we found the criteria for conver­ gence of random series, and conditions for applicability of the law of large numbers, and more. Then I should mention the relatively brief period of

In the country house where P.S. Alexandrov and

working with Arnol'd. The so-called Hilbert 13th problem

A.

should really be counted as a joint result of the two of us.

many years, there was a seminar room with a black­

The decisive step was taken by Arnol'd, although all the

board. Several years after Kolmogorov's death it had

foundation for this and related problems was laid by me. The particular problem stated by Hilbert was settled by Arnol'd; soon after, I was able to find a much simpler al­

. Kolmogorov held their famous gatherings for

not been erased, and still had in his hand, in English,

the motto

MEN ARE CRUEL, BUT MAN IS KIND.

ternative solution, but it was published later. This happens

VOLUME 23, NUMBER 1, 2001

35

flights of fancy not always directly related to anything in the real world. Can you explain your point of view on this?

K.: [Hesitates.] The essential thing is, what is an applied mathematician? There is really no separate science of ap­ plied mathematics. An applied mathematician is a mathe­ matician who knows how to apply ordinary mathematics to real problems. Thus a real applied mathematician is in­ terested in the real problems of some related field. [Pause. ] H e effectively ceases t o be a pure mathematician. M.: So it all depends on the nature of the problem he is solving?

K.: Yes, and an applied mathematician working on some­ thing like ocean hydrodynamics is treating the study of

ocean with mathematical tools. M.:

the

This leads into another question: theoretical physics. I

know that often mathematicians are very skeptical about theoretical physicists, because they apply mathematics in a "dirty" way. You personally, when you worked on turbu­ lence-how far were you from the real problems of physics?

K.: I would first like to reply to your reference to "dirty" mathematics.

You see, mathematicians always want

mathematics to be as pure as possible, in the sense of A.N. Kolmogorov at the seashore.

being rigorous, proof-oriented. But generally the most in­ teresting problems brought before us are not tractable in this manner. Then it is very important for the mathe­

K.: [Pause.] Hilbert, of course . . . . Hadamard. . . . After that

matician himself to be able to find, not a rigorous, but an

it gets more difficult. . . .

effective treatment of the problem. Anyway for me this

M.: Did you pursue administrative jobs, or did they over­

was always the way: if I am studying turbulence, then I

take you?

am studying

K.: I never pursued them. In some cases there was a feel­

do not work, then I look into experimental materials,

ing of duty, a belief that if I took on a task it would be bet­ ter done.

In the case of my deanship, for example.

turbulence. If purely mathematical methods

seek to discover in them a thread of coherence, and then proceed to make rigorous mathematical deductions, but

speculative assumptions.

M.: Do you think you were successful in this?

starting from such entirely

K.: To a degree. For our Department, at least, this was one

for one value most highly this type of applied mathe-

of the better periods. I never approached administrative duties with revulsion. When Otto Yulievich Schmidt asked me to join the Presidium of the Academy of Sciences, as the academic secretary of the physics and mathematics section, in 1939, I was straightaway interested. The underlying reason for Schmidt's bringing me in was this. The physics and math­ ematics section didn't give much administrative role to mathematics (there are many more physics institutions, and their material resources are much greater). At that time there were two physicists with every claim to leadership of all of physics: Kapitsa and Ioffe. So Schmidt took it into his head to set between them a very young mathematician! And it really didn't work out too badly. [Marutyan laughs. ] When the astronomers had t o select a location for a large geophysical observatory somewhere in the South, I visited the proposed sites-in private capacity, just as a tourist­ to get a better perspective on them. My Academy work was only three years, 1939-42. M.: Andrei Nikolaevich, you have said several times, if I understand correctly, that the distinction between pure and

applied mathematics is not at all sharp. But in the popular image, applied mathematics is especially computers and computation, whereas pure mathematics is more abstract

36

THE MATHEMATICAL INTELLIGENCER

The bust at Kolmogorov's grave in Novodevichi Cemetary.

I

matician, who essentially ceases to be a mathematician

to some scientific lecture or reading something new, you in­

and simply solves problems of physics-if possible by

evitably try to figure: perhaps I could do something here?

rigorous "pure" deduction, but if that doesn't work, then

M.: Andrei Nikolaevich, in the days of Archimedes or even

by introducing assumptions.

Newton, the study of the surrounding world was accessi­

M.: So you favor flexibility of thought?

ble to any educated person.

K.: And where possible, participating in

experiment with

M.:

K.: That is true. M.: But now it is not sufficient simply to be educated. One

the physicists.

In the course of solving a problem, have you ever

thought about whether it was important? Did you have

may have understanding, and not that full, in a single field

of theory. Do you feel that this condition is a natural result

global goals?

of progress, or is it a stage through which we will pass and

K.: No. To be sure, in the overall planning of one's work, in

perhaps once more return to some kind of common un­

choosing which new books to read, in combing the scientific

derstanding?

journals to decide which works need closer study-there this

K.: Are you familiar with Shklovskii's The

rational element is present, must be present. But on the other

and the Mind?

Universe, Life,

hand, spontaneous sparking of interest in a hypothesis which

M.: Yes.

just leaps into one's head can often be crucial. M.: Has intuition played a role in your thought process, and

K.: He maintains that the development of every culture, if it

to what degree?

what might befall humankind now-culminates in a stage of

is not aborted by some catastrophic events-and we all know

K.: Of course, a very important role. This is very common

loss of interest in technology. Perhaps he really is right.

for a mathematician.

M.: What does "loss of interest in technology" mean? You

M.: How do you work, Andrei Nikolaevich?

mean that people occupy themselves more with humanis­

K.: Real scientific work? Usually it goes as follows: You

tic problems?

read books, you prepare your own lectures with some new

K.: Not really humanistic problems. But it must be possi­

variations, and suddenly, an unexpected idea springs up

ble to return to a more basic and child-like

if the problem

Do you know the German writer Hesse?

out of the soil of this everyday work: what

joy in living.

at hand can be solved entirely differently from known ap­ proaches? Another way becomes vaguely visible . . . . Then for a mathematician, and probably for any other scientist,

A U T H O R

it is overwhelmingly important to set aside everything else and simply think, think unrestrictedly on this new way which has just appeared. Fortunately I usually had the op­ portunity to do this. There's a mathematician named Boris Nikolaevich Delone. You must have heard a lot about him. M.: Yes, sure.

K.: When he would speak to students and they would ask him what is the essence of a scientist's creative work, he would answer like this: "Suppose you are in a mathe­ matical Olympiad. They give you four hours to solve five problems (that's how it goes in the Olympiad). So you

N.G. KHIMCHENKO

devote around an hour to a problem. Now imagine a prob­

Mathematics Department

lem which would take you not one hour to solve, but

Moscow State University

some idea of what a real scientist does." Now no doubt

Moscow 1 1 9899

Boris Nikolaevich was greatly exaggerating. For me, any­

e-mail: [email protected]

(say)

5000 hours of constant thought! Then you'll get

Russia

way, in the making of all of my scientific discoveries, such utter concentration, excluding all else, would last

Natal'ya Grigor'evna Rychkova (married name Khimchenko)

sometimes a week, maybe sometimes two weeks, but no

has been an associate professor in the Chair of Probability

more.

Theory

M.: What considerations led you sometimes to divert your

ment since 1 964. She worked under the supervision of A.N.

at

the Moscow State University Mathematics Depart­

attention sharply into new areas?

Kologorov on mathematical problems of linguistics and poetry.

K.: I do not believe you put the question correctly, because the various fields of mathematics in which I have worked

Currently

works left by Kolmogorov. Both interests are exemplified in the

have usually led directly into one another, so that passing

article "Analysis of the rhythm of Russian verse and probabil­

from one to another was

natural.

is much occupied with recovering and editing

ity theory" prepared from an unpublished manuscript of his,

In principle, you see-not with a conviction that I will un­ doubtedly achieve something, but out of general curiosity­

aU mathematics more or less interests me.

she

published in Teoriya Veroyatnostei i ee Primeneniya 44 (1 999), 4 1 9-431 .

When listening

VOLUME 23, NUMBER 1. 2001

37

M.: Yes. K.: In Das Glasperlenspiel, Hesse depicts such a society, and quite brilliantly, I would say. A society which has lost interest in technological progress. M.: What role has chance played in your life? [Both laugh.] After all, you worked on stochastic processes. K.: I would be hard pressed to say. On the whole I believe that in a slightly different time, with a different form, still es­ sentially what I was able to contribute to science would have been done if the distribution of roles had been different. M.: In other words, if you had been surrounded by other people, worked in different circumstances . . . ? K.: It is likely that the objective outcome would have been more or less the same.i

Andrei Nikolaevich, you know there will have to be music in this film I'm making. . . . K.: Yes, certainly. M.: And I'd like the music heard in the film to be some­ thing close to you. Do you have some favorite pieces? K.: I hope there will be a place in the film where you tell about the musical evenings at Komarovka for our friends. Pavel Sergeevich and I would regularly have a good many guests for those occasions. At that point in the film I would like you to play Bach's Concerto for Two Violins. M.: That was the favorite piece of you and Pavel Sergeevich? K.: I think we had that in common. We would listen often to Mozart's G minor Symphony. M.:

S P R I N G E R F O R M AT H E M AT I C S ROBIN WILSON, The Open University, Oxford, England

PETER HILTON, State University of New York, Binghamton, New York; DEREK HOLTON, University of Otago, Dunedin,

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book, Mathematical Reflections, and can be read

independently. 2001/APPROX.

JAMES J, CALLAHAN, Smith College, Northampton, MA

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THE MATHEMATICAL INTELLIGENCER

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A Story of a Pai nting

Wojbor A. Woyczynski The lUldistinguished low-sltiDg building of the A.M. Obukhov Institute of Atmospheric Physics of the Russian Academy at 3 Pyzhevski'i per. in Moscow, which I visited in January 2000, is dwarfed by the massive, ornate, and marble-clad edifice of the Ministry of Medium Machine Construction across the street. The Ministry's name is a euphemistic cover for the bureaucracy that long supervised the Russian nuclear weapons program. My friend and host, Valeri'i Isakevich Klyatskin, a mathematical physicist, tells me that in 1937 the Institute's mul­ tilevel basement contained the laboratory of Academician Igor Vasilevich Kurchatov, the father of the Soviet atomic bomb, and it is here that the first chain reaction

in Russia was achieved. Curiously, there is no plaque

commemorating that event.

A second-story office, obviously little used, contains a second surprise, an oil painting of A.

. Kolmogorov.

Valeri! explains to me that the portrait was painted in 1966 by Dmitrii Gordeev, who was then Kolmogorov's student. Gordeev was a teacher at the time in the well-lawwn school for mathematically talented children l'tlll

in Moscow by Kolmogorov, and the painting was intended to hang in the school building. However, Kolmogorov disliked the painting, which shows him

in an idiosyncratic knuckle-cracking gesture well remembered by all

those who knew him. He vetoed the proposal. Gordeev's wife, who worked at the Institute, mentioned to several yotiDg turbulence researchers there that the painting was looking for a home. They put together 40 rubles to purchase it. (A professor's monthly salary was 105 rubles.) Since then Gordeev's career as a painter blossomed, and his portraits of many m::Yor figures are widely known. Yost Hall

220

Case Western Reserve University Cleveland, OH

441 06-7058

USA e-mail: waw @ po.cwru . ed u

VOLUME

23,

NUMBER 1 , 2001

39

I,�Mj,i§,Fh1¥11@i§#fii,i,i§,id

This column is devoted to mathematics forjun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed

only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions

Alexander S h e n , E d itor

The Importance of Being Formal K.S. Makarychev and Yu . S. Makarychev

from readers-either new notes, or replies to past columns.

T

I

(say) A's cards does not prevent him having any specific card, because each of the cards

0, . . . , 5

occurs on every

row: Sum

Possible combinations

0 1 2 3 4 5 6

{0,2,5}, {0,3,5 }, {0,4,5}, {0, 1,2}, {0, 1,3}, {0,1,4}, {0, 1,5},

{0,3,4}, { 1,2,5}, { 1 ,3,5}, { 1 ,4,5}, {2,4,5} {0,2,3}, {0,2,4},

{ 1,2,4} { 1,3,4} {2,3,4} {2,3,5} {3,4,5} { 1,2,3}

C cannot name any of A's cards. C cannot name any of B's cards. If C has any other card instead of 6 , the situation is similar: cir­ Thus

he following problem (suggested by A. Shapovalov) was given to the

participants

in

the

Olympiad in spring

Moscow

Math

2000.

For the same reason

cular shift of the cards does not change anything.

The deck of cards contains seven However, the problem is subtler cards labeled 0,1,2,3,4,5,6. The cards than the organizers realized. To ex­ are shuffled and distributed among plain why, let us consider the follow­ three people A, B, C. A and B receive ing "solution." IfA holds {p, q,r}, he says three cards each; the remaining card to B, "If you don't have card p, then I have cards {p , q ,r )." Likewise B, hold­ is given to C. Show that A and B can exchange information about their ing {u,v,w }, says to A, "If you don't have cards (ensuring that B knows A's u then I have {u,v,w } ." Each of them cards and vice versa), speaking in the knows that the other's hypothesis is presence of C, in such a way that C true and so has been informed of the stiU cannot name any card (other other's cards. But if their hands were than his own) and say whether A or exchanged (so that A held {u,v,w } and B held {p , q ,r}) , both hypotheses would B has it. be false and therefore both statements

The organizers of the Olympiad con­

would be true. So

C, if he is

allowed to

sidered this problem as well posed,

draw conclusions only from the state­

having in mind the following solution:

ments and not from the knowledge

Each of the players A and B declares the sum modulo 7 of the three cards he has.

not exclude either scenario,

The sum of the declared cards will be

you agree that something is wrong

(0 + 1 + 2 + . . . + 5 + 6) minus C's card, so A and B will

would not have enough information to

needed by

A and B to assert them, can hence

does not know the location of any card. Is this solution "good"? Probably

the sum of all cards

know C's card and thus each other's cards. Please send all submissions to the

with it In fact

if

A held

{u,v,w},

he

say truthfully to B, "If you don't have p then I have {p, q,r} " (for he would not

To show that the solution is valid, it

know which three of the four cards

remains to show that this does not give

it fail to satisfy the conditions stated?

Information Transmission , Ermolovoi 1 9,

his own). Assume for example that

What is a "good" solution anyway?

K-51 Moscow GSP-4, 1 01 447 Russia;

has the card with

e-mail:[email protected]

ing table shows that fixing any sum of

Mathematical Entertainments Editor, Alexander Shen, Institute for Problems of

C the

location of any card (except for

6

C

on it. The follow­

hidden from him B held). But how does We see that to make the problem

clear we need a formal definition.

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001

It 41

turns out that there are several natural definitions which are not equivalent. Let us describe two of them briefly. (1) A solution is a pair of algorithms A and B that prescribes the behavior of A and B during the game: what to say given the cards and any messages received from the other player. We re­ quire that for any configuration of cards, the protocol of exchange to­ gether with A's cards determines B's cards uniquely and vice versa. On the other hand, for any configuration and for each card x other than C's card, there should exist another configura­ tion that produces the same protocol but gives x to a different player. (2) A solution is a rule that says what statements A and B may make on see-

A Bet With Leonid Levin T

he axiom of choice is a well­ known source of paradoxes. Here is one more suggested by Leonid Levin from Boston University (e-mail: [email protected]). Let us begin with a standard con­ struction. We call two reals x,y E [0,1) equivalent if (x - y) is a fmite decimal fraction, i.e., if (x - y) 10" is an inte­ ger for some positive k E N. Select one element from each equivalence class. Let A be the set of all selected ele­ ments. Then the sets A+f (mod 1) for all (finite decimal) fractions j E [0, 1) are disjoint and cover [0,1). If we con·

42

THE MATHEMATICAL I NTELUGENCER

ing their cards. A statement determines a subset of the set of all configurations. We require that A's statement together with B's cards determine A's cards and vice versa, but that A's and B's state­ ments together with C's card do not de­ termine the position of any other card. These definitions can be applied not only to this game but also to other sim­ ilar games, and one can prove that de­ finition (1) is always stronger. The in­ tended solution of the Olympiad problem satisfies both defmitions (1) and (2), but the second one, the "bad" one, satisfies only (2). For the same problem with only five cards instead of seven, there is no solution according to definition ( 1), but the "bad" solution still works for definition (2).

We know of another natural defmi­ tion intermediate in strength between (1) and (2). What is the moral of the story? The importance of being formal is well un­ derstood in logic-and in computa­ tional cryptography, which, by the way, is essentially the way our prob­ lem is solved. But this problem illus­ trates that even in very simple cases the absence of a formal defmition can lead to ambiguity-which is better avoided, especially in a math competi­ tion problem!

vert [0,1) into a circle, all A+f are im­ ages of A under rotation. Letfi andf2 be two different (finite decimal) fractions. Consider the fol­ lowing bet: we pick at random some x E [0,1) (by rolling a die for each digit). If x E A+fi (mod 1) then I pay you $1; if x E A + !2 (mod 1) then you pay me $1; otherwise (if x is outside both sets) nothing happens. This is a fair bet since A+fi and A + !2 differ only by a rotation. To make the bet at­ tractive to you I will even pay you $2 against your $1. If you agree to play this game, I pro­ pose we make many bets simultane­ ously. For any finite decimal fraction! E (0, 1), letf' denote the fraction!' E [0,1) that is the fractional part of 10!, i.e., f' = lOf mod 1. (For example, if f = 0.502 1 thenf' = 0.021, and if j = 0.6 thenf' = 0.) Let us make (J, f')-bets for all frac­ tionsjat the same time. (Note that each

individual bet is good for you, so the whole game should be also good.) If you agree with me, we start play­ ing immediately before you realize the consequences. Imagine what happens if (say) x E A + 0.057 (mod 1). Then you win the (0.057,0.57)-bet and get $2, but at the same time you lose the (0.0057, 0.057)-bet, the (0. 1057,0.057)-bet, the (0.2057,0.057)-bet, etc. (altogether 10 bets). So in fact you pay $10 and get only $2. Ifx E A + 0, you win no bets and lose 9 bets, so the situation is even worse. Note that only two people participate in the game, and for each x only finitely many bets (at most 1 1) are implemented, so this scheme does not resemble pyra­ mid schemes where infinitely many par­ ticipants pay one another in such a way that each has positive balance. Doesn't charity make donors richer? [and the recipients more miserable?­ Ed.]

PO Box 65 Moscow 1 27322 Russia e-mail: [email protected]

LAN WEN

Sem anti c Parad oxes as Eq u ati o n s1 By a paradox we mean generally an argument that leads to contradiction for no clear reason. The most ancient and most influential paradox in history is perhaps the paradox of the Liar. Here is a well-known version of it:

The Liar Paradox.

"This sentence is false."

If it is true then it is false, and if it is false then it is true. This popular argument is quite short. So we repeat the argument with some commonly understood explanation in­ serted: If it is true then what it says should be the case and hence it is false. If it is false then what it says should be negated and hence it is true. The argument leads to contradiction. It is unclear at first glance what goes wrong. A number of theories have been proposed in the literature to resolve the Liar paradox, no­ tably the hierarchy theory of language of Tarski [ 1 1], which separates sentences into different levels, and the truth­ value gap theory of Kripke [5], which adopts three-valued logic. Nice accounts can be found in [1], [2], [4], [5], [6], [7], [9], [10], [11]. In this paper I present a different solu­ tion to the Liar paradox. It is not hierarchic, and adopts the classical two-valued logic. The main observation is this: Main Observation. There is an assumption implicitly used in the Liar argument. With this assumption un­ covered, and announced explicitly in front, the Liar ar­ gument wiU be found to be a normal ''proof by contra­ diction", but not at aU paradoxical.

This is supported by a "Three Cards paradox" I found recently, which uncovers the connection between Liar-like paradoxes and inconsistent Boolean systems. To make my

point precise I need to introduce some basic notions into our ordinary language, such as "sentence given", "sentence unknown", "sentence equation", and "sentence solution", for sentences. Note that analogous notions are standard in elementary algebra, for numbers. When these notions be­ come available for sentences, we can solve not only para­ doxes of the Liar type, but also some others. LOb's para­ dox will be an example. After giving a thorough exposition of the ideas, I will proceed in the last section of this paper to a more formal treatment. The Liar Paradox as an Equation

First note that what the Liar paradox displays is not a sin­ gle object, but a relation between two. I hinted at this by using quotation marks in stating it. From the argument we can see that the paradox needs to use the term "This sen­ tence" to refer to the statement "This sentence is false". Indeed, the word "it" printed above in italic appears six times and serves as a link: In first, second, fourth, and fifth occurrences it stands for "This sentence is false" (four words), but the third and sixth times it stands for "This sentence" (two words) (take a few seconds to verify this). Since the relation is "refer to", we may call it a "referential relation." In symbols, if we use A to abbreviate "This sen­ tence", F to abbreviate "is false", and ": =" to abbreviate "refer to", then what the Liar paradox displays is not a sin­ gle object AF, which reads "This sentence is false", but a referential relation

A : =AF, which reads " "This sentence" refers to "This sentence is false" ". Some people take the referential relation to be straight equality A =AF. All conclusions of the present pa­ per hold automatically under such a stronger identification, but it is clearer to keep the referential relation less special. A further observation is that this relation should better be considered a "presumed" one but not a "verified" one.

1Work supported partially by NSF of China and Qiu Shi Science & Technologies Foundation.

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1, 2001

43

The term "This sentence", denoted by A, is like a pronoun that can be, but is perhaps not yet, specified to be some specific sentence. This is like an unknown x in algebra, which can be, but is perhaps not yet, specified to be some specific number, that is, some given. Thus the Liar refer­ ential relation should better be regarded as a "referential equation," and written as

sentence says is the case, hence the first sentence shotlld be true, which contradicts that the first sentence is false. This way we have run out of possibilities with contradic­ tions everywhere. This is a new paradox I discovered just recently. Both the statement and the argument are of the same type as the Liar. In symbols, it is written as

{

X : = XF. Of course, to regard a relation as an equation does not exclude the possibility that the relation might be verified later, hence it is not a loss but just a precaution. Also note that, though the notions of "given", "unknown", and "equa­ tion" are standard for numbers in elementary algebra, they are not customarily applied to sentences in our ordinary language. The Three Cards Paradox: The Secret of the Liar

To present my solution to the Liar paradox, perhaps the best way is first to present a new paradox, the Three Cards paradox (the device of using cards is inspired by the fa­ mous Jourdain's Cards paradox, which involves two cards [2]). Uncovering the secret of it leads directly to the solu­ tion to the Liar paradox.

The Three Cards Paradox. Consider three cards with the following three sentences: The sentence on the second card is true, and the sen­ tence on the third card is false. Either the sentence on the first card is false, or the sentence on the third card is true. The sentences on the first and second cards are both true. This paradox looks fancier than the Liar (and Jourdain), what with the logical connectives "and" and "or". The ar­ gument must be more complicated, but may yield insight. Assume the first sentence is true. Then what it says should be the case, which means the second sentence is true and the third sentence is false. Hence what the third sentence says should be negated, which means that either the first sentence is false, or the second sentence is false. Putting these together, we have that either the second sen­ tence is true and the first sentence is false, or the second sentence is both true and false. But this "or" is impossible, because we adopt the classical two-valued logic. Thus this "either" must hold. That is, the second sentence is true and the first sentence is false. This contradicts the assumption that the first sentence is true at the beginning. Thus the first sentence must be false. Then what the second sentence says is the case, hence the second sentence is true. Moreover, what the third sen­ tence says is not the case, hence the third sentence is false. In summary, the first and third sentence are each false, but the second sentence is true. In particular, what the first

44

THE MATHEMATICAL INTELLIGENCER

X : = IT !\ ZF, Y : = XF V zr,

Z : = XT !\ IT,

where !\ stands for "and", and V stands for "or". But where does this paradox come from? Let me reveal the secret: In writing such a complicated "paradoxical" ar­ gument, I had in mind the following Boolean system.

The Boolean Model for the "Three Cards. " The Boolean

{

system

x

=

yz,

x + z, z =xy

y=

has no solution. Proof Assume there exists a solution. That is, assume there are three givens x, y, and z that satisfy the system. This will lead to a contradiction. Assume x = 1. Then, by equation 1, y = 1 and z = 0. By equation 3, z 0 yields either x = 0, or y = 0. Putting these together, we have either y = 1 and x = 0, or y = 1 and y = 0. But this "or" is impossible. Thus this "either" must hold. That is, y = 1 and x = 0. This contradicts the assumption x = 1 at the beginning. Thus x = 0. Then z =0 from equation 3, and y = 1 from equation 2. In summary, x = z = 0, but y = 1. However putting x = z = 0 and y = 1 into equation 1 yields a contradiction. This proves there are no three givens x, y and z that sat­ isfy the system. That is, the system has no solution. The reader may have noticed a clear resemblance be­ tween the Boolean system and the Three Cards paradox. The difference is clear too: In their statements, one has a phrase "has no solution", the other does not; in their argu­ ments, one has a standard frame of proof by contradiction, that is, the "head" "Assume there is a solution, we derive the following contradictions" and the "tail" "This contra­ diction proves there is no solution", while the other does not. In fact my analysis of Three Cards was just translation of the Boolean problem into ordinary language, only I cut off the phrase "has no solution" from the statement, and cut off the head and the tail from the argument. As ex­ pected, the normal Boolean proof became a mysterious ar­ gument that leads to contradiction apparently with no rea­ son, that is, a "paradox." However, removing the head does not affect the argu­ ment. Because the head "Assume there exists a solution" is merely an announcement for the assumption. The actual use of this assumption takes place not in the head, but in =

the body of the Boolean proof. Removing the tail does not affect the argument either, because the argument has fm­ ished already. Thus the solution to the Three Cards para­ dox must be (informally) this: It is (the translation of) the assumption of existence of a solution that causes contradiction in the Three Cards paradox. The assumption is tacit and goes unnoticed. It is believed traditionally that Liar-like paradoxes are logically different from Boolean problems. It is believed that in Boolean "proofs by contradiction" one assumes ex­ istence of solution and hence derives contradiction, but in Liar-like paradoxes one does not assume anything, except some basic rules of language and logic, hence contradic­ tions must have some deep, yet unknown cause in our lan­ guage or logic. The Three Cards paradox shows this is not the case.

An Informal Solution to the Liar Paradox

The Liar paradox involves the same secret. Indeed, the Liar paradox corresponds in the same way to the following Boolean model.

The Boolean Model for the Liar. x = x has no solution.

The Boolean equation

Proof Assume there exists a solution, that is, assume there is a given x that satisfies the equation. We derive the following contradiction. If x = 1 ("If it is true"), then x = 0 ("then it is false"). And if x = 0 ("And if it is false"), then x = 1 ("then it is true"). This contradiction proves that there is no given x that satisfies the equation. That is, the equation has no solution. We have enclosed the Liar argument in the parentheses for comparison. It is indeed the translation of the Boolean proof, with the head and tail removed. (Here by translation I mean logically, but not historically. Historically, the Liar paradox is perhaps 2500 years older than Mr. Boole.) Thus the solution to the Liar paradox is informally this:

one that corresponds to the algebraic term "equation" could be called "referential equation" (I have already in­ formally called the Liar relation X : = XF a "referential equa­ tion"), and this reduces to establishing the notion " : = " or "refer to", for sentences. The one that corresponds to the algebraic term "given" could be called "sentence given", which should be established in a way that is compatible with the notions of truth and "refer to". We need also some related notions such as "sentence unknown" and "sentence solution", to a "referential equation". While the ideas are very natural, the formal work involved is deferred to the last section of this paper. Assuming this formal work done, I can state the formal solution to the Liar paradox as follows:

Solution to the Liar paradox (Formal version). It is the assumption of existence of a sentence given that satis­ fies the Liar equation X : = XF that causes contradiction in the Liar paradox. In other words, there can be no sen­ tence given that says, of itself, that it is false. Thus the solution to the ancient Liar paradox is simply the negation of the original Liar relation, with only one word "given" put in! This sounds like cheating. But actu­ ally this is the right conclusion, expressed in the new ter­ minology. The Huge Class of Liar-like Paradoxes

The reader can create a huge class of "Liar-like paradoxes," corresponding in this way to Boolean systems that have no solutions. The Liar paradox, the Three Cards paradox, and Jourdain's paradox are just the three simplest examples in the class. The number of sentences or cards involved can be arbitrarily large. Without the help of Boolean theory, we would not suspect there is such a huge class of "paradoxes" in ordinary language. All Liar-like paradoxes involve the same secret, and can be solved the same way. Criteria in Boolean algebra that determine which Boolean systems have no solution become automatically criteria in ordinary language that determine paradoxical referential systems of sentences. The Truth-teller

Solution to the Liar paradox (Informal version).

It is (the translation of) the assumption of existence of a so­ lution that causes contradiction in the Liar paradox.

Thus a referential system of sentences is paradoxical if the corresponding Boolean system has no solution. But what if the Boolean system does have solution? Here is such a problem, known as the Truth-teUer.

The Formal Solution to the Liar Paradox

The above solution to the Liar paradox is informal. What it needs is how to translate the term "existence of solution" from Boolean algebra into our ordinary language. The term "existence" needs no translation. It is a universal term used in many disciplines. The term "solution" reduces to two other terms, "given" and "equation" (in algebra a solution is just a given that satisfies an equation), which need some preparation. There are not yet corresponding notions for sentences in our ordinary language. We need first to es­ tablish these notions before we can do the translation. The

The

Truth-teUer."This sentence is true."

Again, this is understood as not merely a single object, but a relation between two. More precisely, the Truth­ teller gives a referential relation X : = XT. The corre­ sponding Boolean equation is hence x = x, which cer­ tainly has solutions. Thus Boolean diagnosis reveals nothing wrong. But in some sense something is wrong with the Truth­ teller. A diagnosis for Truth-teller is given below in the last

VOLUME

23,

NUMBER 1 , 2001

45

section about the formal work According to the diagno­ sis, the Truth-teller equation has solutions respecting some interpretations of "refer to", but no solution re­ specting some other interpretations. This fact is not per­ ceivable by Boolean diagnosis. Boolean diagnosis is coarse. If it says fine", things may not be really fine, as the Truth-teller shows. (But if it says "ill", things must be seri­ ous. Contradictions that appear in Liar-like arguments are of serious Boolean nature.) An Application to Lob's Paradox

Establishing the notions of "sentence given" and so on does more than just solve Liar-like paradoxes. The well known Lob's paradox, which is not Liar-like, is an example.

Lob's paradox (1955). Let A be any sentence. Let B be the sentence: "If this sentence is true, then A." Then a contradiction arises. In fact, B asserts: "If B is true, then A." Now consider the following argument: Assume B is true. Then, by B, since B is true, A is true. This argument shows that, if B is true, then A. But this is exactly what B asserts. Hence, B is true. Therefore, by B, since B is true, A is true. Thus, every sen­ tence is true, which is impossible. This is the argument of Lob's paradox, quoted from [8]. Observe that the argument is based on a referential rela­ tion. Indeed, in the statement, the letter B denotes the sen­ tence "If this sentence is true, then A", and so does the let­ ter B at the beginning of the argument. But right next to that, inside the quotation marks, the same letter B denotes merely the term "this sentence". Thus the paradox uses "this sentence" to refer to "If this sentence is true, then A". In symbols, this is the referential equation

x := cx� A) . Here is the solution of Lob's paradox. With the notions of "sentence given" and so on at hand, it reduces, as do the Liar-like paradoxes, to a proof by contradiction. The se­ mantic statement is that LOb's equation has no solution. The proof is exactly LOb's argument. Formal Work

This section contains the formal work The main issue is to establish formally the notions of "refer to" and "sentence given". Then the notions of "sentence unknown", "sentence equation", and "sentence solution" will follow. Another is­ sue is to make precise the correspondence between Liar­ like paradoxes and inconsistent Boolean systems. I also in­ clude some different interpretations for the notion of "refer to", which are not essential to the Liar paradoxes, but es­ sential to explaining the Truth-teller. I start with the standard sentential logic, denoted by L. The notions of sentence, truth value, sentence connectives --, ("not"), 1\ ("and"), V ("or"), � ("if . . . then"), � ("if and only if"), etc., are standard, for instance from [3] or [8]. I use capital letters to denote sentences. With these con­ nectives several sentences could combine into a longer sen-

46

THE MATHEMATICAL INTELLIGENCER

tence. As convention I will not write long Boolean exp:r;:es­ sions below, but shorter ones such as (A /\ B) V (C /\ D) instead. Subscripts could be resorted to if needed. I use T(A) to denote the truth value of a sentence A. Thus T(A) is an element of the two-valued Boolean algebra; it is either 1, meaning A is true, or 0, meaning A is false. The truth value of a long sentence is determined in the natural way by the truth values of its component sentences. For instance, the truth value of the sentence (A 1\ B) V (C 1\ D) is T(A)T(B) + T(C)T(D).

To study our paradoxes I follow Kripke [5] by extend­ ing L to a language :£ by adding to L a truth predicate "is true", denoted by T, together with its negation "is false", denoted by F. T and F are supposed to possess the fol­ lowing natural properties: Tl. AT � --..AF. T2. (A 1\ B)T � (AT 1\ BT). T3. (A V B)T � (AT V BT), where A, B are sentences in :£. All this is standard. Now I introduce a relation REfer to, denoted by " : = ", between sentences of :£. It is to obey the following axiom.

Axiom of truth-REference.If a sentence A REfers to a sen­ tence S, then A is true if and only if S. In symbols, if A : = S, then AT � s.

This axiom restates formally the two rules used in the argument of the Liar paradox: If a sentence A is true, what it "says", or what it "REfers to", which is S, should "be the case". And conversely, if a sentence A is false, what it "says", or what it "REfers to", which is S, should "be negated". We will see below that it is this axiom that trans­ fers Liar-like arguments into Boolean arguments. Like many other axiomatically defmed objects, the no­ tion of "REfer to" is to be without properties other than the axiom which defmes it. To distinguish it from the usual verb "refer to" in the dictionary, I have written its first two letters in capital. It hence allows many interpretations, not just different words such as "says that" or "refers to". I name two, one very fine, and one very coarse.

Example A. Here ": =" is interpreted as the "quotation mark

name." That is, a sentence enclosed within two quotation marks, and only this sentence, is considered to REfer to the sentence with the quotation marks removed. A classi­ cal example of Tarski is that "It is snowing" is a true sen­ tence if and only if it is snowing [1 1, P.156]. This interpre­ tation is very fme. For instance, the interpreted relation is not reflexive, nor symmetric, nor transitive. To admit such an interpretation one has to extend :£ further because, for a sentence A in :£, "A" is not in :£. This is compatible with Tarski's hierarchy theory of language.

Example B. �.

Here ": =" is interpreted as the biconditional

sentence solution of a

set of sentence givens is called a

�

REferential system if, for every equation of the system, re­

A for

placing uniformly the unknowns by these sentence givens,

any A. Thus, under this interpretation of "REfer to", the

the obtained sentence given on the left-hand side of the

relation

This interpretation is very coarse. Since A

A for any A, the axiom gives immediately that AT

�

truth predicate T has no effect. This is often done in math­

equation REfers to the obtained sentence given on the

ematics.

right.

Now

I proceed to the central issue, "sentence given".

Definition. Sentence givens of 5£ are defmed recursively: G 1. Any sentence of L is a sentence given of 5£. G2. AT is a sentence given of 5£ if and only if A is a sen­ tence given of 5£. Likewise AF.

G3. If B : = A, then B is a sentence given of 5£ if and only

A special feature of REferential equations that is not

shared by the usual algebraic equations is that whether or

not a REference system has a solution may depend on in­ terpretations of "REfer to". For instance, the Truth-teller equation

tion is so strict that only "AT", but not A, can REfer to AT.

However, the same equation

if A is a sentence given of 5£.

G4. If A is part of a sentence B, then replacing A with a sentence with the same 5£-givenness does not change the

from Example B above, any sentence given A in this case Now I make precise the correspondence between

G5. Only sentences that reduce to sentences of the orig­ procedures

X : = XT does have solutions In fact, as seen

under the interpretation of biconditional.

satisfies A � AT and hence is a solution of Truth-teller.

5£-givenness of B. inal sentential logic

X : = XT has no solution under the interpretation

of Tarski's "quotation mark name", because this interpreta­

L

in finitely many steps of reduction

REferential systems and Boolean systems. A REferential equation gives rise to a Boolean equation as follows.

G1 through G4 are sentence givens of 5£.

Replace ": = " by Then we defme a

sentence unknown X of 5£

to be a

place-holder that can be occupied by sentence givens

"=",

replace respectively 1\ and

V by

Boolean multiplication and addition, and replace capital letters by corresponding small letters in the following way:

Replace the capital letters on the left-hand side of the equa­

of 5£. Note that G3 indicates a relation between givenness and

tion by the corresponding small letters, replace the capital

REference. This is like the situation in algebra, where not

letters together with predicate T on the right-hand side of

3 and so on are givens, but

the equation also by the corresponding small letters, but

also symbols 7T and sin 7T/2 are, as long as these symbols

replace the capital letters together with F on the right-hand

only the actual numbers 1, 2,

side of the equation by the opposite of the corresponding

refer to some givens. On the other hand, "This sentence", denoted by A in the

small letters. Let us call the Boolean equation so obtained

associated

Liar paradox, is not a sentence given. Indeed, it does not

the

satisfy the above defmition for sentence givens. This is be­

tion. For instance, the associated Boolean equation of the

cause, among the four reduction procedures G 1 through G4, only G3 applies. But by G3, its givenness reduces to

REferential equation

G2, back to the givenness of "this sentence" itself, or reduces, by G3 and G4, to the givenness of " "This sentence is false" is

is

X:=

the givenness of "This sentence is false". The givenness of "This sentence is false" then either reduces, by

false", and so on so forth. This will not give a sentence given of

L in fmitely many steps. Thus this A does not sat­

Boolean equation of the REferential equa­

V (VF 1\ WT)

(YT 1\ ZF)

x = yz + mv. The following theorem shows that this correspondence preserves solutions and hence

is

a "homomorphism." This

isfy the defmition. Likewise, the Liar sentence "This sen­

justifies the claim that semantic diagnosis

tence is false" is not a sentence given either. This reminds

Boolean diagnosis.

us of the celebrated notion of

grounded sentences

of

Kripke [5]. However, they are different. For instance, for sentences in the original sentential logic

L, while the no­

tion of groundedness depends on the choice of the exten­ sion 81 and the antiextension

S2 of the truth predicate

[5],

Having made precise the notions of "REfer to", as well as "sentence given" and "sentence unknown", I define ref­ erential equations and their solutions. By a

X : = (YT 1\ ZF) where

REferential

x = yz + 'i5w has a solution. In fact, the truth values of any solution ated Boolean system.

X through W are sentence unknowns of 5£.

(Remember that by convention this could be a long ex­

REferential system

has a solution respecting one, not necessarily all, inter­

to the REferential system form a solution to the associ­

V (VF 1\ WT),

pression with subscripts.) By a

Transfer Theorem. If a REferential system X : = (YT 1\ ZF) V (VF 1\ WT) pretation of ": = ", then the associated Boolean system

the notion of givenness defined above does not.

equation I mean an expression of the form

is fmer than

we

mean a system of finitely many REferential equations.

A

Proof Let A through E be a solution of the REferential sys­ tem, respecting some interpretation of ": = ". Then A : = (BT 1\ CF)

V (DF 1\ ET).

VOLUME 23, NUMBER 1, 2001

47

Hence by the Axiom of truth-REference,

A U T H O R

AT � (B T A CF) V (DF A ET).

This implies by properties

Tl

through T3 that

AT � ((B A -.c) V (-,D A E))T.

This means that

T(A) = T((B A -.C) v c-.n A E)),

which is the same as

T(A) = T(B)T(C) + T(D)T(E).

Thus the truth values satisfy the associated Boolean system

x = yz which proves the theorem.

LAN WEN

School of Mathematics

+ vw,

Peking University 1 00871

Beijing

China

Acknowledgment

This paper has been written with interaction from many of my colleagues. I thank Manuel Blum, Steve Smale, Xinghua Wang, Jian Wen and Jingzhong Zhang for many good ref­ erences, comments, and encouragement. I thank Shiming Guo and Elliott Mendelson for many critical comments and stimulating discussions that sharpened the ideas signifi­ cantly. Finally, I thank the Department of Mathematics and the Department of Computer Science of City University of Hong Kong for kind hospitality.

e-mail: [email protected]

Lan Wen was educated, and now teaches, at Peking University a campus with a beautiful lake named Not Yet Named. He did his doctoral work at Northwestern University on the shore of the beautiful Lake Michigan. His specialty is ,

dynamical systems, a beautiful field noted for dramatic fea­ tures like evolution and chaos.

AEFiiRENCI!S

[1] J. Barwise & J. Etchemendy, The Liar, Oxford University Press,

[7] Benson Mates, Skeptical Essays, The University of Chicago Press, 1 981 .

1 987. [2] N. Falletta, The Paradoxicon, John Wiley & Sons, Inc., 1 990.

[8] E. Mendelson, Introduction to Mathematical Logic, Third edition,

[3] H. Kahane & P. Tidman, Logic & Philosophy: A Modern

[9] R. Sainsbury, Paradoxes, Second edition, Cambridge University

Introduction.

Wadsworth Publishing Company, 1 995.

[4] R. Kirkham, Theories of Truth, MIT Press, 1 995. [5] S. Kripke, Outline of a theory of truth, The Journal of Philosophy, 72 (1 975), 690-7 1 6.

[6] R. Martin, Recent Essays on Truth and the Liar Paradox, Oxford University Press, 1 984.

48

THE MATHEMATlCAL INTELUGENCER

Wadsworth & Brooks/Cole Advanced Books & Software, 1 987. Press, 1 995. [1 0] K. Simmons, Universality and the Liar, Cambridge University Press, 1 993. [1 1 ] A Tarski, Logic, Semantics, Metamathematics, Hackett Publishing Company, Second edition, 1 983.

M athe m a t i c a l l y B e n t

C o l i n A d a m s , Ed itor

Overcoming Math Anxiety The proof is in the pudding.

Opening a copy of The Mathematical In.telligencer you may ask yourself uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Colin Adams

T

here is a crippling disease that has a vice grip on the nation. It is low­ ering the Gross National Product, causing whole communities to break out in hives, and convincing many peo­ ple to stay home with the covers over their heads. Of course, I could only be talking about Math Anxiety. This poison ivy of the soul has a long and mangy history. Isaac Newton himself had such a bad case that while he wrote the Principia Mathematica he was shaking from head to foot. Joseph Louis Lagrange was bedridden for a week before he could bring him­ self to write down his famous multi­ plier. And Evruiste Galois preferred risking his life in a duel to grappling with the mathematics that made him so nauseous. Of course, math anxiety is not the only ailment associated with mathe­ matics. There is math incontinence, math male pattern baldness, math itch, and math runny nose. However, today, we will focus our attention on math anxiety, leaving those other maladies to a series of articles that I am writ­ ing for the Notices of the American

Mathematical Society.

How do you know if you suffer from math anxiety? Here is a quick test. Check off each of the symptoms that you experience when confronted with mathematics. Symptoms of Math Anxiety.

Column editor's address: Colin Adams, Department of Mathematics, Williams College, Williamstown, MA 0 1 267 USA e-mail: [email protected]

A. Hyperventilation. B. Holding your breath. C. Sweating profusely, while holding your breath.

D. Sweating profusely, while holding Spanier's Algebraic Topology. E. Eating other people's bag lunches. F. Uncontrollable shaking, hopping, or doing the rhumba. G. Weruing a heavy winter coat in the Math Resource Room. H. Putting pencils in your nostrils or ear holes. I. Sucking your thumb. J. Sucking your T.A.'s thumb. K. Rapid heart beat. L. Rapid pulse. M. Rapid heart beat but no pulse. N. Rapid pulse, but no heart beat. 0. No pulse or heart beat. P. K, L, but not 0. Q. P, M, N but not L. R. Not R. S. Extreme nausea, accompanied by hallucinations of large mammals lec­ turing you on Euclid's parallel postu­ late. T. The feeling that you and calculus are in a custody battle over your math­ ematical future and the judge has or­ dered you to make child support pay­ ments. U. The sensation that someone has poured soda water up your nose, and now expects you to thank him for do­ ing so. V. Dizziness, accompanied by an in­ ability to stand straight on an inclined plane. W. The feeling that the alphabet is endless. X. A thousand red ants are crawling over your body, biting and stinging you until you want to scream. Y. The impression that a thousand red ants are crawling over your body, bit­ ing and stinging you until you want to scream. Z. The feeling that you are running out of ideas, but you must complete a list. For each of the symptoms that you checked off, write down the number 6.9986. Add these numbers together. Divide by 27T. Take the natural log of

© 2001 SPRINGER·VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001

49

the result. Add 1 . 145, and subtract 1.946. Exponentiate the result.

If you

are now sweating profusely and feel as if you

rings a bell. The student is immediately

there are still many important ques­

forced to nm a maze, at the end of which

tions to pursue.

he or she is force-fed a pellet of rat food.

Will there ever be a vaccine for

fessor begins the lecture, and as soon

math anxiety? And if so, will it be one

you may have eaten bad

as he sees a student looking uncom­

pink cube of sugar? Should triskadeko­

tuna If so, you may be suffering from

fortable, he stops the lecture, comes

phobia be considered a type of math

gastronomic masochism. I should have

over and gives the student a warm hug.

anxiety? Are math anxiety and math

an article out on that in about a week.

He says, "Don't worry, you can do it.

phobia the same or slightly different?

Try to hang on until then.)

You're special. We're all special. Love

These are just a few of the issues addressed in my upcoming anthology,

Math Anxiety, including all of the pres­

is all around, if you just let it in." The other students in the auditorium come

appearing in a special issue of the

idents from 1872 to 1891, and Teddy

over, take hands, form a giant circle

Journal of the Mathematical Psychoses Institute.

had

eaten bad tuna, you have

math anxiety. (Note:

Many

Or

presidents

suffered

from

2. The Nurturing Approach: The pro­

Roosevelt, who had to wear diapers as

around the student, and sway back and

he charged up San Juan Hill, lrnowing

forth, singing songs about how great

he would need to count the enemy

Coke tastes.

once he got to the top. Sharon Stone

3. Confronting Your Fear Approach:

of those ones where you swallow a

Until this scourge can be cured, we will need dedicated facilities: ambu­ lances to rush those with sudden-onset

asked to give a proof of the central

The student is tied to the chair. A drill

limit theorem, as does Woody Har­

want to lrnow about real anxiety? You

quarantine wings in hospitals to pre­

relson. Ed Begley Jr. refuses to appear

have

vent Ebola-like epidemics. And most

in any movie involving a covariant

you real anxiety. I'm going to make you

importantly, we will need substantial

fimctor.

wish you could hide your head in a big

federal grants to support those of us

breaks into a torrential sweat when

Psychologists have settled on the following four treatments for math

math anxiety to emergency rooms staffed by ready Ph.D. math educators, .

sergeant screams in his face, "You no

idea!

I'm

going to

show

fat textbook and never come out."

who are at the cutting edge of research

4. Nature's Own Approach: The stu­

in this seminal field.

anxiety:

dent is tied to a large rock and thrown

L B.F. Skinner Approach: Here, the

in a pond.

student is hooked up to an anxiety de­

surface, the rock is replaced by a larger

sign. Lunches will not be lost to loga­

tector-usually a rabbit taped to the

rock and the process is repeated. lfthe

rithms. Then, researchers like me will

Some day, perhaps, no one will

If the student floats to the

tremble at the sight of a percentage

student's leg. A trigonometry lecture

student does not float to the surface,

need to find other sources of support.

begins. As soon as the rabbit senses

he or she is declared cured.

But in the meantime, continue to read

anxiety on the part of the student, it

my papers on the subject.

Although much has been learned,

Errata We recently have had some trouble correctly attributing articles. In vol. 22, no.

3, misspellings occurred of the names of

Bernard Geneves and Nikolai V. Ivanov. And now in vol. 22, no. 4, we inadvertently interchanged the photographs of the authors, Oleksiy Andriychenko and Marc Chamberland (see below for the correction). Our apologies to these authors, and to the readers.

OLEKSIY ANDRIYCIIENKO

50

THE MATHEMATICAL INTELLIGENCER

MARC CHAMBERLAND

Magellan's and Elcano's Proof . . .

.

.

•

and the gap pointed out by Michael Little-Endings in the last issue of the Transactions of the SubAntarctic Mathematical Society

" . . . Leaving on March 9, 1521, Magellan steered west-southwestward. . . . Less than two months later, however, Magellan was killed in a fight with natives on Mactan Island. . . . It had been left for Elcano [originally master on the ship "Concepcion"], returning by the Cape route, to give practical proof that the Earth was round." Encyclopedia Britannica, 1999

Sasho Kalajdzievski Department of Mathematics University of Manitoba Winnipeg, R3T 2N2 Canada e-mail: [email protected]

VOLUME 23, NUMBER 1, 2001

51

Curves in Traditional Architecture in East Asia Hiroshi Yanai

O

ne of the pleasures for people

In

travelling to East Asia may be to

by an envelope of stretched strings.

see various curves decorating tradi­

Some of the procedures and equations

tional

are illustrated in the figures.

architecture.

Curves

are

ob­

other cases, they are approximated

served everywhere: in gables, in eaves,

The author knows little about the

over the entrances, and sometimes in

mathematical aspects of the curves in

jacket walls of castles. In particular,

China, Korea, and in other countries

roofs play the most important role in

like Indonesia, Kazakhstan, and Thai­

their appearance, as the fac;ade does in

land where beautiful and characteris­

European architectures.

tic curves are also observed in archi­

However, if you watch the curves

tecture. Mathematical forms are basic

very carefully, you will notice the dif­

not only to architectural engineering,

ference among the curves in different

but also to the cultural connotation

regions in East Asia. The curves in

hidden in architecture.

Chinese architecture have, in general,

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where thefamous conjecture was made, the desk where thefamous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either

stronger curvature than those in Japan.

Figures are reprinted with permission

Korean curves are somewhere in be­

from

tween. One might say that they reflect

Yanai,

H.,

"Curves

of

there must be also some philosophical

the Operations Research Society of Japan

or religious connotations.

33 (1 988), No. 6 (in Japanese)

Some old procedures are known, in

Yanai, H . , "Curves in Traditional Japanese

Japan, to draw such curves, which can

Architecture,"

be translated into western mathemat­

Opera-tions Research Society of Japan

ics, although many traditional archi­

(1 991 ), No. 3 (in Japanese)

tects copy and modify older curves nowadays.

Communications

English)

or textbooks, Japanese curves are most

In

some cases,

Prof. Hiroshi Yanai

may follow in your tracks.

result from the process of numerical so­

KEIO University

lution of the differential equation y" =

3-1 4-1 , Hiyoshi, Kohokuku

0 with boundary condition at both ends.

Please send all submissions to Mathematical Tourist Editor, Aartshertogstraat 42,

8400 Oostende, Belgium e-mail: [email protected]

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

the

36

Civil Engineering," Forma Vol. 14, No. 4 (in

According to many old manuscripts frequently parabolas.

of

Yanai, H . , "Curves in Traditional Architecture and

they are formed by line segments which

52

Walls­

Mathematics in Style," Communications of

a map or directions so that others

Dirk Huylebrouck,

Stone

the aesthetic senses of the nations;

Faculty of Science and Technology

Yokohama 223-8522, Japan

Xian, China

Wak:ayama, Japan Pavilions y

y(x) = --=--cosO (sinO - (LcosONL-x)) ' N L

X

where is the length of the beeline, is the number of the nodes and 0 is the angle of the inclination.

Drawing Procedure of Gable

y

y = d (1 = IJlJ, d=AD. bu. + h

where

-

h = AB,

3u)2 2h

b

B

C

II+· -b-�·1 -

B

Drawing Procedure of Stone Wall

Kanazawa, Japan

VOLUME 23. NUMBER 1 , 2001

53

The Shape of Divinity Kim Williams

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck,

Aartshertogstraat 42,

8400 Oostende, Belgium e-mail: [email protected]

54

T

he mathematical tourist visiting medieval cathedrals in Europe certainly has a wealth of material to sort out. We have already seen tracery [A] and hammerbeam ceilings [H3] dis­ cussed in these pages. I would like to add another item to the list of note­ worthy architectural features: the sym­ bolism of geometric shapes, and in par­ ticular, the vesica piscis. The vesica, which translated literally means "fish bladder", is also known as a mandorla, Italian for almond. The shape is created by the inter­ section of two circles of equal diame­ ter, the perimeter of each passing through the center point of the other (Fig. 1). Its frequent appearance in me­ dieval ornamentation, particularly in the sculpted reliefs that fill the arches over entrance portals (an element properly known as a tympanum) sig­ nals the importance of this shape. It is associated with the themes of ASCEN­ SION or ASSUMPTION, and most usu­ ally forms the frame around the figure of Christ, but is used as a frame for the Virgin Mary as well. It is said that orig­ inally the almond-like shape repre­ sented the cloud which carried the saintly figures into heaven, but it grad­ ually assumed the role of an aura or kind of "glory" [H1: 197]. The first use of the vesica in art appears in the Byzantine art of the 5th or 6th century, when it was used to represent the in­ carnation of Christ in the womb, in a figure known as a Platytera [H1: 337]. From there the symbol passed into western Europe in the middle ages. The Gothic may be the single most geometric style of architecture. John Harvey made a strong case that the rise of Gothic architecture was directly re­ lated to the new availability of Euclid's Elements, as well as Arabic astronom­ ical texts, in western Europe in about 1 120. "It can be no mere accident," Harvey has noted, "that this placing of the world of thought within a strictly scientific framework parallels the sud­ den rise of the new Gothic art and ar­ chitecture. [ . . . ] The study of practi­ cal geometry was indeed the essence of architectural design" [H2]. When we encounter a geometrical form in Gothic architecture, we are likely en­ countering a symbol. The iconographic

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

content of the vesica, created from two intersecting circles, could have been the reconciliation of opposing duals, such as terrestrial-celestial and hu­ man-divine into a harmonious unity [CG: v.2, 59]. The geometrical proper­ ties follow from the simplicity of the construction [K: 54]. The vesica is also important because it could form the basis for a cohesive proportional sys­ tem (Fig. 1). In Romanesque architecture, Christ is found in a vesica in an altar apse fresco of the Chapel at Berze-la-Ville, France (ca. 1 100 AD); in the tympanum over the entrance portal of Ste. Madeleine, vezelay, France (ca. 1 120 AD); and in the tympanum of the en­ trance portal of Sant'Trophine, Aries (ca. 1 170). In Gothic architecture, Christ appears in a vesica on the west facade of Notre-Dame-la-Grande, Poitiers (ca. 1 162), and in the tympa­ num of the central portal of the west fat;ade of Chartres Cathedral (ca. 1 145) (Fig. 2). The Virgin Mary appears in the vesica of the tympanum of the north entrance to the Cathedral of Santa Maria del Fiore in Florence, the so­ called "Porta della Mandorla," in a re-

root-S

root-4

root-3 root-2

--!-'f-++-�-

Figure 1 . The vesica piscis, and root-n con­ structions (drawing by S. Hisano).

Figure 2. Christ in the vesica. Central portal, west facade, Chartres Cathedral, France. Photograph by the author.

lief executed by the artist Nanni di Banco in 1418 (Fig. 3). The Virgin also appears in a vesica in the glittering glass mosaics that decorate the me­ dieval fru;ade of the Cathedral of Orvieto (Fig. 4). One needn't travel further than a good research library to find other me­ dieval examples of vesicas. The arm­ chair tourist will find them in many a medieval illuminated manuscript. As the Gothic age gave way to the Renaissance, ideals and concepts changed, and the vesica appears to have fallen out of favor. One late ex­ ample appears in the tomb slab com­ memorating Cosimo de' Medici, de­ signed by Verrocchio in 1467, and laid in the pavement under the crossing of the Basilica of San Lorenzo in Florence [W] (Fig. 5). The tomb slab is noted by historians for its abstract design, because it features a veritable vocabulary of shape, but no figurative imagery. Two vesicas of green por­ phyry flank a central 3:4:5 rectangle. Cosimo had wished to be eternally present at the celebration of the Eucharist, hence the location of the tomb slab at the foot of the altar (though the actual tomb is in the crypt

Figure 3 {Top). Mary in the vesica. Tympanum,

north entrance, Cathedral of Santa Maria del Fiore, Florence. Photograph by the author. Figure 4 (Bottom). Mary in the vesica. Mosaics, main facade, Cathedral of Orvieto, Italy. Photograph by the author.

Figure 5. Verrocchio's tombslab for Cosimo de' Medici, Basilica of San Lorenzo, Florence. Drawing by the author from Italian Pavements.

Patterns in Space (Houston: Anchorage Press, 1998). Reproduced by permission.

Figure 6. Christ in the vesica, intrados of the entrance portal, Certosa of Pavia, Italy. Photograph by the author.

below), and the appearance of the

I hope that other tourists will let me

vesica, the fish-shaped symbol of

hear of more examples.

Bridge Between Art and Science.

Christ. However, note that the pro­ REFERENCES

the "classic" vesica. Where the tradi­

[A]

root-3 rectangle, Verrocchio's vesica is circumscribed by a root-2 rectangle. I do not know the reason for this. A last late example of Christ in the

vesica is found in the reliefs decorating the intrados (the inner surface) of the arch over the main entrance portal of the Certosa of Pavia (ca. 1497) (Fig. 6).

New York:

McGraw Hill, 1 991 .

portions of these vesicas differ from tional vesica is circumscribed by a

[K] Kappraff, Jay, Connections: The Geometric

Artmann,

[W] Williams, Kim, "Verrocchio's Tombslab for Benno,

"The

Cloisters

of

Cosima

de'

Medici:

Designing with

a

Hauterive," Mathematical lntelligencer, val.

Mathematical Vocabulary," in Kim Williams,

1 3 , no. 2, pp. 44-49.

ed. Nexus: Architecture and Mathematics.

[CG] Chevalier, Jean and Alain Gheerbrant, Dizionario

dei

Simboli,

2

vols.

Milan:

Fucecchio,

Florence:

Edizioni

deii'Erba,

1 996, pp. 1 91 -205.

Biblioteca Universals Rizzoli, 1 986. [H 1 ] Hall, James, Dictionary of Subjects and Symbols in Art.

London: John Murray, 1 97 4.

[H2] Harvey, John, The Medieval Architect. London: Wayland Publishers, 1 972.

Kim Williams Via Mazzini 7

Though I have personally tracked

[H3] Horowitz, David, "The English Hammer­

down vesicas in Italy and France, I

beam Roof, " Mathematical lntelligencer, val.

Florence, Italy

have not yet done so in other countries.

1 8 , no. 4, pp. 61 -64.

e-mail: [email protected]

50054 Fucecchio

The Better the Workers the Fewer It Takes Q: How many pure mathematicians does it take to unscrew a light bulb? A: None are needed! With gravity pulling steadily down, and ambient vi­ bration (trucks passing in the street, Brownian motion, distant earthquakes

and meteorite impacts, etc.), it will eventually come out on its own. Robert Haas 1 081 Carver Road Cleveland Heights, OH 441 1 2 USA

VOLUME 23, NUMBER 1, 2001

57

PAUL L. ROSIN

On Se r i o ' s Co n stru ct i o n s of Ova s

fter a dearth of exposition in the Middle Ages, when architectural principles were kept secret by the guilds, the Renaissance was witness to an explosion of architectural treatises and handbooks [ 11]. These were mostly written by architects, in manuscript form until the end of the fifteenth century, and provided a combination of theories, rules and patterns con­

The first treatise of the Renaissance (written around

cerning all aspects of architecture. Contents range over the

1450 and published in 1485, 1 3 years after his death) was

suitability and preparation of building materials, the design

Alberti's

of plans, fa =

v5/ 1 "" 1.618

is

the

Golden

Ratio

which

often appears in sacred geometry and theories of propor­ tions and aesthetics. The radius of the arc is thus ¢>, and the intersection of the arc with the

X axis is at x = W.

Drawing another circle centred at the origin and passing through the intersection (i.e., with radius circle whose circumference

\,I;{J) produces a

is 2 7TW "" 7.99. The perime­

ter of the square circumscribing the initial circle equals eight, and thus provides a very close estimate to the final circle's circumference. Figure l la also provides a means of constructing a circular approximation to an ellipse, us­ ing the inner circles (radius ing at

..

a k = a·, -2 h,

.

. .

.

t) and the arcs of radius ¢>join­

(xt, yt). The centres are at

the fiXed aspect ratio of the oval is ¢>, and the ratio of the radii of the arcs is 2¢>. Fidelity to the Ellipse Having defined a substantial range of oval constructions, the obvious question is: how well do they approximate the desired ellipse? The analytic solutions are complicated by the involvement of the elliptic curve and so a numerical approximation to the approximation errors is calculated

(a)

instead. The circular arcs are sampled at approximately

-- Scrlio l

I I ' I •. I .

• • •' ·.

' ' '

' '

e • 0 • e

' ' ' .. • · .

r

I I

'�

·� I �

•*

.·

·

,

·

I

1

.

a

+

't'

••••

Serlio JV

3 squares

4 circle

half-square triangle Vignola Kitao

-- Moll - - • Bianchi - - · Simp on

A

golden ratio

- optimai C J

- -......... - . ....... - - - � ... ,

.. ..

-

... .... _

_

-

(b) Figure 1 1 . Squaring of the circle.

66

Serlio l l Serlio lll

Tl-IE MATl-IEMAnCAL INTELUGENCER

--

a/b

Figure 1 2. Approximation errors of the ovals with respect to the de­ sired ellipse.

(a) Serlio

(b) Stirling

(c) Lockwood

(d) Walker

Figure 13. Various oval constructions showing their discrepancies against the ellipse.

equally spaced points, and at each point the distance along the normal to the ellipse is approximated. See [26, 27] for more details. Fixing b = 100 and using 1000 sample points in total, the graph in Figure 12 was generated. Although only the maximum error is displayed, the average error ex­ hibits a similar pattern. The error incurred by the optimal tangent-continuous (Cl) oval (which was numerically estimated by a 1D search) [27] is included for reference. We can see that Serlio's con­ structions do reasonably well, but are certainly not the clos-

(a)

est to the ellipse (although of course this may not reflect their aesthetic qualities). For instance, Simpson's construc­ tion does uniformly well and is generally superior (it was previously found to outperform most of the more modern methods too [27]). In addition, Vignola's construction does especially well. Nevertheless, the extensions of all of Serlio's constructions (i.e., half-square triangle, 3 squares, 4 circles, and l(itao's generalisation) mostly perform poorly (the ex­ ception is the half-square triangle construction at low ec­ centricities). This shows how apparently plausible con-

( b)

Figure 14. The best-fit ellipse and oval overlaid on Serlio's church plan.

VOLUME 23, NUMBER 1 , 2001

67

A U T H O R

overlaid in Figure 14a and the discrepancies at the diago­ nals are evident. The best-fit oval was determined by per­ forming an optimisation over all parameters using Powell's method, and as can be seen in Figure 14b, not only does it provide a better fit to the church's perimeter, but it is clearly Serlio's Construction IV. Acknowledgments

I would like to thank Ian Tweddle for providing a copy of his translation of Simpson's construction from the original Latin. PAUL L. ROSIN

Department of Computer Science

REFERENCES

Cardiff University

[O]

Cardiff CF24 3XF

[1]

UK

Sebastiana Serlio on Architecture, translated with commentary by Vaughan Hart and Peter Hicks. Yale University Press, 1 996.

e-mail: Paui. [email protected]

R. Arnheim. The Dynamics of Architectural Form . University of California Press, 1 977.

[2]

W. Blackwell. Geometry in Architecture. John Wiley and Sons.

[3]

C. Bouleau. The Painter's Secret Geometry. Thames and Hudson,

search scientist at the Institute for Remote Sensing Applica­

[4]

J.A. Brown. Technical Drawing. Pitman, 1 962.

tions, Joint Research Centre, lspra, Italy, and lecturer at Curtin

[5)

Paul Rosin is senior lecturer at Cardiff University. Previous posts include lecturer at the Department of lnfonmation

1 984.

Systems and Computing, Brunei University London, UK, re­

1 963 .

University of Technology, Perth, Australia. His research interests include the representation, seg­

W.B. Dinsmoor. The literary remains of Sebastiana Ser1io: I . The Art Bulletin, XXIV( 1 ) :55-91 , 1 942 .

[6) V.

mentation, and grouping of curves, knowledge-based vision

Fasolo. Sistemi ellitica nell'architettura. Architettura e Arti

Decorative, 7:309-324, 1 93 1 .

systems, early image representations, machine vision ap­

[7) J .C .

proaches to remote sensing, and the analysis of shape in art

[8)

and architecture. A common factor of much of his research is a near obsession with the theme of the ubiquitous ellipse.

Golvin. L 'Amphitheatre Romain. Boccard, 1 988.

N.T. Gridgeman. Quadrarcs, St. Peters, and the Colosseum. The Mathematics Teacher, 63:209-2 1 5, 1 970.

[9) [1 0)

J.H. Harvey. The Mediaeval Architect. Wayland, 1 972. D.E. Hewitt. Engineering Drawing and Design for Mechanical Technicians.

structions do badly, and suggests that some care was taken in developing Serlio's original constructions. This argument is supported by the fact that Serlio's Construction I is always better than Bianchi's, and mostly better than Mott's con­ struction. It is interesting to note that for ovals with aspect ratios just less than two, Serlio's construction I approaches the optimal approximation. The oval approximations of ellipses with low eccen­ tricity are mostly good, and the errors are barely notice­ able. For more elongated ellipses we can see the discrep­ ancies more clearly, as shown in Figure 13. The first two show four-arc ovals discussed in this paper, and the supe­ riority of Stirling's approximation over Serlio's construc­ tion is obvious. The following two ovals are constructed using eight arcs-details are given in [28]. Naturally, using more arcs it is possible to improve the accuracy of the ap­ proximation to the ellipse, as is achieved by Walker's method. Perhaps surprisingly, several eight-arc ovals were found to be inferior to four-arc ovals, as demonstrated by Lockwood's oval; see also Rosin [27].

[1 1 )

Renaissance Architectural Treatise. [1 2)

THE MATHEMATICAL INTELLIGENCER

T.K. Kitao. Circle and Oval in the Square of Saint Peter's. New York University Press, 1 974.

[1 4)

G. Kubler and M . Soria. Art and Architecture in Spain and Portugal 1500..1 800. Penguin Books, 1 959.

[1 5) 0.

Kurz. Durer, Leonardo and the invention of the ellipsograph.

Raccolta Vinciana; Archivio Storico del Comune di Milano, 1 8: 1 5-24, 1 960.

[1 6)

B. Langley. The London Prices of Bricklayers Materials and Works. Richard Adams and John Wren, 1 750.

[1 7]

R. Lawlor. Sacred Geometry. Thames and Hudson, 1 982 .

[ 1 8)

W. Lotz. Die ovalen Kirchenraume des Cinquecento. R6misches Jahrbuch fOr Kunstgeschichte, 7 : 7-99, 1 955.

[1 9)

G.B. Milani and V. Fasolo. Le Forme Architettoniche, volume 2 . Casa Editrice Vallardi, 1 934.

[20)

C.F. Mitchell and G.A. Mitchell. Building Construction. B.T. Batsford, 1 925.

[2 1 )

68

New Haven, 1 998.

M. lliescu. Bernini's "idea del tempio". http://www.arthistory.su. se/bernini.htm, 1 992.

[ 1 3)

An Application

Serlio himself made good use of his methods of oval con­ struction. Figure 14 shows his plan for an oval church from the Tutte le Opere d'Architettura. The best-fit ellipse is

Macmillan, 1 975.

P. Hicks and V. Hart, editors. Paper Palaces: the Rise of the

L.C. Mott. Engineering Drawing and Construction. Oxford Univer­ sity Press, 2nd edition, 1 976.

[22]

E. Panofsky. Ga/i/eo as a Critic of the Arts. Martinus Nijhoff, 1 954.

[23] R.T. Peterson, G. Mountfort, and P.A.D. Hollom. Birds of Britain and Europe.

[24] I. Preussner. Jahrhunderts.

A Field Guide to

[28] P.L. Rosin and M.L.V. Pitteway. The ellipse and the five-centred

Houghton Mifflin Co, 1 993.

arch.

Ellipsen und Ovate in der Malerei des 15. und 16.

Weinheim, 1 987. Physis,

[30] W.K. West. Problems in the cultural history of the ellipse.

1 2:37 1 -404, 1 970.

for the History of Technology,

[26] P.L. Rosin. Ellipse-fitting using orthogonal hyperbolae and Stirling's oval.

CVG/P: Graphical Models and Image Processing,

[31] R. Wittkower.

60(3):209-

2 1 3, 1 998.

[32] R. Wittkower.

cular approximations to the ellipse.

Society

pages 709-71 2, 1 978.

Art and Architecture in Italy 1600-1 750.

Yale

University Press, 1 992.

[27] P.L. Rosin. A survey and comparison of traditional piecewise cir­ Design,

85(502), 2001 .

James Stirling: 'This about series and such things'.

Scottish Academic Press, Cambridge, UK, 1 988.

[25] P.L. Rose. Renaissance Italian methods of drawing the ellipse and related curves.

The Mathematical Gazette,

[29] I. Tweddle.

Architectural Principles in the Age of Humanism.

John Wiley and Son, 1 998.

Computer Aided Geometric

[33] F. Zozorra.

1 6(4):269-286, 1 999.

Engineering Drawing.

McGraw-Hill, 2nd edition,

1 958.

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Expansion of the IMU (1 955-1 958)

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VOLUME 23, NUMBER 1, 2001

69

I a§IH§'.ifj

.J et Wi m p , Editor

I

Inverse Problems: Activities for Undergraduates by Charles W. Groetsch

temperature gradient at the Earth's surface, and the medical inverse prob­ lem of interpreting tomographic scans. Groetsch argues persuasively for the value of "inverse thinking" as a part of an undergraduate mathematics cur­

WASHINGTON, DC: THE MATHEMATICAL ASSOCIATION OF AMERICA, 1 999, xii+222 pp.

riculum. While forward problems dom­ inate the training of mathematics ma­

US $26.00; ISBN 0-88385-716-2

jors, sole reliance on this direction is too limiting; viewing problems from

REVIEWED BY STEPHEN HUESTIS

the other (inverse) direction leads to a

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited

H

richer understanding of the whole pic.­ ow does one defme an inverse

ture.

problem? Charles Groetsch intro­

This book is not intended to be a

duces this volume with a discussion of

student's text on inverse theory. Nor is

review of a book of your choice; or, if

the difficulties of arriving at a formal

it a monograph on important concepts

definition and of recognizing a given

of inverse theory which an instructor,

you would welcome being assigned

problem

who might be ill-prepared to teach it,

a book to review, please write us,

(Groetsch uses the term "direct" for

could use to develop personal back­

the latter, but "forward" seems more

ground. For that, one might refer to

telling us your expertise and your

predilections.

as

inverse

or

forward.

suggestive of the duality between the

Groetsch ( 1 ] , to which the present

two). Though there may be a certain

book can be considered a companion

arbitrariness in choosing which prob­

volume. Rather, it is meant to be a

lem is "inverse" to which, in practice

source of ideas for the teacher inter­

this seems not to be so problematical.

ested in incorporating inverse prob­

It comprises

A collection of data is measured to

lems into the curriculum.

make quantitative inference about a

a miscellany of problems from various

physical cause. Suppose that a mathe­

disciplines, some also discussed in [ 1 ] ,

matical model is available so that, were

which illustrate aspects o f inverse

this cause known, the values of the

thinking.

measurements could be calculated un­

Groetsch takes a very broad view of

ambiguously. Then it is logical to call

what comprises an inverse problem,

this calculation the forward problem,

with the advantage that he can then ad­

and to call the data interpretation the

dress some of the concerns of inverse

problem inverse to it.

theory from precalculus onward. To

Groetsch's interesting first chapter

Groetsch, the unknown of an inverse

provides a series of historical vignettes

problem might be a finite collection of

of familiar important problems that, in

discrete parameters, a continuous ftmc­

the broadest sense, are inverse prob­

tion of one or more independent vari­

lems. His goal, effectively achieved, is

ables, or even the nature of a physical

to demonstrate the importance that in­

law. There is not, however, universal

verse problem solutions have had in

agreement on this very inclusive defi­

the development of the sciences, even

nition.

if such problems were not explicitly

As an Earth scientist, I came to in­

stamped with the label "inverse the­

verse theory as it is applied to geo­

ory." Examples include Newton's de­

physical inference, receiving my tute­

duction of the inverse-square law for

lage from Robert Parker, one of its

gravity from observations of orbital

foremost practitioners. Parker devel­

Column Editor's address: Department

forms, Kelvin's (incorrect) inference of

oped his ideas from his unique view that

of Mathematics, Drexel University,

the age of the Earth from a conductive

the unknown of an inverse problem is

Philadelphia, PA 1 91 04 USA.

cooling model applied to the current

some ftmction representing a parame-

70

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

ter distribution, often corresponding to the spatial variation of a physical prop­ erty in the inaccessible interior of the Earth (density, temperature, seismic velocity, etc.). As such, it is a member of an appropriately defmed infmite­ dimensional space of functions, and data are values of linear or nonlinear functionals defmed on this space. Parker's notion requires at least ele­ mentary functional analysis. Normed spaces, Hilbert spaces, and optimiza­ tion theory play large roles. Such top­ ics might be beyond the lower-division preparation of the students targeted by Groetsch. Instructors drawing ideas from Groetsch, however, would be well served by also reading Parker's [2] very accessible text. Inverse problems often do not share certain nice properties enjoyed by for­ ward problems. A forward problem is cast as a well-posed transformation (linear or nonlinear), mapping every member of a well-defined domain to a unique member of its range. The for­ ward problem is generally stable, with output depending continuously on in­ put. The inverse problem on the other hand, can be expected to be ill-posed. Recovery of a solution from the data is likely unstable: arbitrarily small per­ turbations to the data can lead to large variations in the solution. Such ex­ treme sensitivity becomes a practical problem when data are contaminated by random noise: the noise translates to wildly improbable oscillations in the constructed solution. Another new concern is the possible non-existence of any solution at all, if the candidate data function falls outside the range of the forward transformation. In such a case, one can conclude that the for­ ward problem is incorrectly posed. Perhaps a priori constraints on the do­ main are too stringent. Here I mention examples from my geophysical experi­ ence: (1) Measured gravity anomalies are to be interpreted in terms of subsur­ face density contrast. A data collection containing values of both signs would be incompatible with an a priori as­ sumption of solution positivity. Such data would force a redefmition of the domain to allow two-signed density contrasts.

(2) Potential field data are some­ times downward-continued toward the source, through the source-free region. This is an unstable, roughening process, inverse to the stable forward problem of upward continuation. If we plan to continue downward by a specified dis­ tance, through what we believe to be the source-free region, then certain functions yield no solutions, lacking sufficient smoothness to be in the range of the forward upward continuation op­ erator. Discovering conditions which must be met by data, in order to guar­ antee solution existence, is an impor­ tant part of the complete formal analy­ sis of any inverse problem. Arguably the most severe difficulty which can be encountered in inverse theory, not shared by the forward problem, is the possibility of solution nonuniqueness. In the theoretical (but unattainable in practice) situation of perfect data (complete and accurate), certain inverse problems can be shown to have a unique solution; others are fundamentally nonunique. A goal of formal inverse theory is a characteri­ zation of the complete solution set. In practice, data are incomplete, comprising only a fmite collection of numbers from which we cannot hope to learn which member of an infinite­ dimensional domain is the true solu­ tion. Nonuniqueness is inevitable; data are consistent with an infinite number of different solutions, possibly with tremendous variation in form. We must abandon the hopeless goal of recover­ ing the single true solution: the data do not specify it. Yet, they certainly contain some information about the solution set, and the inverse problem becomes one of inferring such information. One approach is the construction of some extremal solution, which is the simplest possible by some chosen mea­ sure of complexity. Such a solution might be as smooth as possible, thereby avoiding any artificial features of rapid variation not dictated by the data. The conclusion to be drawn is not that an ex­ tremal solution is correct, but instead that the true solution can be no simpler. Alternatively, we might seek prop­ erties common to all solutions without constructing any particular solution. In linear theory, the famous Backus-

Gilbert method is an example of such an approach, wherein certain weighted averages are constructed that are shared by all solutions. Another exam­ ple which has seen much geophysical application is the bounding of various solution properties-e.g., placing a greatest lower bound on the uniform norm of the solution. Notice that both extremal solution construction and bounding of solution properties are op­ timization problems that might rely on methods, such as linear programming, which are more advanced than the background of the students targeted by Groetsch, in their first two years of un­ dergraduate training. It is Groetsch's position that the teaching of mathematics, even at the lower division, can be significantly en­ riched by viewing problems with in­ verse, as well as forward thinking. By taking a broader view of the purview of inverse theory than does Parker, he is able to propose activities which stimulate students to appreciate and investigate some of these concerns even before their exposure to calculus. Following the introductory chapter, Groetsch divides his book into four subsequent chapters that follow the student's progress through the first years of an undergraduate curriculum. Each chapter comprises a set of six in­ dependent modules, each of which be­ gins with a brief description of a par­ ticular forward problem, followed by a collection of activities designed to prompt investigation into the rela­ tion between the forward and inverse problem. Activities are of six types: Questions, to be answered in discus­ sion form; mathematical Exercises which are illustrative, but routine; more challenging Problems; Calcula­ tions, which require numerical explo­ ration using a graphical calculator; computer-based Computations of a more intensive numerical nature (Groetsch supplies ail Appendix of MATLAB scripts which are useful for some of these); and occasional open-ended Projects, inviting the most flexible de­ gree of student exploration. For example, in the precalculus chapter we see simple projectile mo­ tion, with the associated inverse range problem, and an elementary geophysi-

VOLUME 23, NUMBER 1 , 2001

71

cal problem of recovering the position

tinuously on the data. Or, it might be

of a buried point mass, using one or

explored numerically by developing

erage difficult to interpret. (For this rea­

more surface gravity measurements.

an approximate solution construction

out of favor with many geophysicists.)

son, Backus-Gilbert theory has fallen

The calculus chapter contains prob­

method, then applying it to a data set

In an example, Groetsch observes a

lems as diverse as the forward/inverse

without and with the addition of ran­

significant difference between a partic­ ular solution value and the associated,

problem pair for the density and cen­

dom noise. In each module, activities

troid functions of a nonhomogeneous

exploring such issues are nicely tied

universally valid solution average. He at­

bar, and the continuously compounded

together in a sensible development.

tributes this to the sparseness of the

data set, but fails specifically to look

interest model with variable interest

Construction of the true solution

rate, to be recovered from the value

would seem to be the ultimate goal in

deeper into the nature of the weighting

history. Examples from the chapter on

any practical inverse problem. When the

function he has constructed, and in gen­

inverse problems in differential equa­

data function is known completely and

eral at what Backus-Gilbert analysis is

tions include the problem of recover­

accurately, analysis might then show

really giving (and not giving).

ing resistance laws from the shapes of

that the forward problem is invertible,

Because this book is neither a text

equitemporal curves for descent down

with a formal solution construction

on inverse theory, nor a tutorial on

inclined planes, and parameter estima­

method. So, for example, Groetsch de­

practical inversion, these shortcom­

tion (mass, spring constant, damping

votes one module to the application of

ings in no way detract from its value.

coefficient) for a one-dimensional dy­

the Laplace transform in a problem re­

namic system. The fmal chapter, tapping linear al­

lating weir notch shape to flow rate us­

ing Torricelli's law. The convolution the­

tors involved in the first years of t�e

gebra, begins with a module casting

orem for Laplace transforms allows a

highly recommended even to teachers

It is a rich source of ideas for instruc­

mathematics curriculum, and can be

the general problem of solving systems

formal inversion of flow rate function

who didn't realize they were looking

of linear equations in the language of

for notch shape. Application to the prac­

for such a book!

inverse theory. Solution nonunique­

tical case is problematical, however, be­

ness or nonexistence are, of course,

cause the inversion is a deconvolution,

REFERENCES

the problems of under- or overdeter­

whose instability Groetsch has us ex­

(1] Groetsch, C.W.,

mined systems. Instability for invert­

plore in several activities.

ible matrices is illustrated using the

Inverse Problems in the

Mathematical Sciences,

Where Groetsch provides the least

familiar concept of the condition num­

guidance is in inference issues for real

ber. The chapter ends with a module

problems with incomplete, inaccurate

Vieweg, Braun­

schweig, 1 993. [2] Parker, R.L.,

Geophysical Inverse Theory,

Princeton University Press, Princeton, 1 994.

underdeter­

data. As discussed above, when data

mined problem of inferring informa­

don't specify a single true solution, we

Department of Earth and Planetary Sciences

tion about the unlmown density struc­

can still ask what they do tell us.

University of New Mexico Albuquerque, NM 871 31

on

the

fundamentally

disc,

Construction of some particular solu­

knowing only its total mass and mo­

tion is of little value. In one module,

USA

ment of inertia. Here, for the first time,

Groetsch develops an algorithm, the

e-mail: [email protected]

ture

of a radially symmetric

Lagrange multipliers are introduced to

algebraic

impose auxiliary constraints; the no­

which is a geometrically based itera­

reconstruction

technique,

tion of Lagrange multipliers is a tool

tion method for tomographic recon­

crucial to the optimization-oriented ap­

struction. He admits that such prob­

proach to inverse theory championed

unique object satisfying the data. The

lems

are

undetermined,

without a

by Parker. In addition, this problem is of special interest as a relative of the

algorithm puts out a solution, but its

geophysical inverse problem of recov­

value is not justified. How different

ering quantitative information about a

might the true solution be?

planet's radially stratified density struc­

Groetsch does deal with the problem

The Development of Prime Number Theory: From Euclid to Hardy and Littlewood by Wladystaw Narkiewicz

ture from the astronomically measur­

of finding properties common to all so­

able mass and moment of inertia.

lutions compatible with the data when

BERLIN: SPRINGER, 2000.

Groetsch also gives plenty of activi­

there is nonuniqueness. For the mass

xii + 448 pp.

ties in which an assumed data set is

/moment-of-inertia problem for a radi­ ally

nonuniqueness. He might then propose

the Backus-Gilbert formalism which,

an activity of fully characterizing the so­

for linear problems, constructs certain

The title of this volume could mislead

lution set, or one of investigating how a

weighted averages shared by all solu­

the potential reader. It misled me. I ex­

more complete data set can eliminate

tions.

An average is designed to attempt

pected a largely historical narrative

the nonuniqueness. Instability might be

to estimate the value of the solution at

outlining the principal achievements in

shown by demonstrating that the size

a chosen point, but the shape of the

explorations of the primes. The author

of the solution does not depend con-

weighting function often makes the av-

quickly disabuses us of this notion,

THE MATHEMATICAL INTELLIGENCER

disc,

he

introduces

US $ 94.00, ISBN 3-540-66289-8

limited enough to illustrate solution

72

symmetric

SPRINGER MONOGRAPHS IN MATHEMATICS

REVIEWED BY GERALD L. ALEXANDERSON

however, when he says in the preface

(The answer is not known.) The cast

most of us probably don't know many

(do only reviewers read prefaces?):

of characters even for the variants of

proofs. Here we can contrast the proof

"This is not a historical book since we

Euclid's proof is impressive: Hermite

refrain from giving biographical details

and Stieltjes; for variants of Euler's

in this development and we do not dis­

proof: Sylvester, Kummer, and Thue.

cuss the questions concerning why

about history and biography, he does

each particular person became inter­

generously supply footnotes

of the people who have played a role

In spite of the author's disclaimer

giving

of Hadamard with that of de la Vallee­

Poussin,

for

example.

Hadamard's

proof was simpler, something admitted

by de la Vallee-Poussin: "I proved for

s) + f3i.

the first time . . . that the function ?C does not have roots of the form

1

Mr. Hadamard, before knowing about

ested in primes, because, usually, ex­

minimal

act answers to them are impossible to

about those

obtain. Our idea is to present the de­

dates, as well as lists of principal ap­

orem in a simpler way. " There is ex­

velopment of the theory of the distrib­

pointments. Occasionally he cannot re­

ution of prime numbers in the period

tensive discussion of Landau's work in

sist the temptation to add something to

simplifying the proof, and descriptions

starting in antiquity and concluding at

a biographical footnote, such as his

the end of the first decade of the 20th

biographical

information

cited: birth and death

my research, also found the same the­

of elementary proofs (not using ana­

comment on Paul Erdos, "Authored

lytic means) beginning with the fa­ mous Selberg-Erdos proof.

century." This too is slightly mislead­

more than 1300 papers in number the­

ing, though the author goes on to ex­

ory, combinatorics and analysis, which

plain that he does indeed move beyond

seems to be a world record," or on

While these variations on the PNT

theme will be of considerable interest

1910, though with less detail. This

Alfred Pringsheim, "Father-in-law of the

book is no general historical narrative,

writer Thomas Mann." The footnotes

easily read by the casual reader inter­

are more than usually rewarding. Philip

sections more rewarding. There is no

ested in the primes. It's chock full of

J. Davis wrote a whole book motivated

shortage of amazing results.

mathematics. And throughout the au­

by the question of the origin of Pafnuti

thor exhibits a dazzling display of

Chebyshev's first name

scholarship.

here Narkiewicz worries in a footnote

The bulk of the text, as one would expect, given the topic and the time

frame, is devoted to the Prime Number

Theorem (PNT), both the work leading

(The Threaff);

about the various transliterations of Chebyshev from the Cyrillic ! He comes

to readers, I suspect that many readers

like myself will find some of the other

In Chapter 1, for example, we learn that x12 + 488669 assumes composite values for all integers x satisfying lxl < 616980. (K McCurley, 1984) Is that widely known? Does everyone remember the polyno­

up with fifteen different spellings and

mial in

tells where they appear. Those cited are

positive values are prime at nonnega­

26 variables, degree 25, whose

up to it over centuries and the work fol­

again stellar: Poincare, Dickson, Hardy

lowing, principally efforts to provide

and Wright, Sierpiriski, Markov, Niven,

easier or different proofs. The Riemann

Cesaro, Ribenboim, and Landau, among

prime can be obtained in this way?

?-function plays a central role.

others. The author prefers Cebysev.

number of variables is

The book starts off, naturally, with

proposition 20 of the ninth book of

tive values of the variables and every (Jones, Sato, Wada, Wiens,

1976) The

convenient,

The first chapter deals with early

given our alphabet! The latter result de­

questions about the primes: Are there in­

rives from the work of J. Robinson, M.

Euclid: there are infinitely many prime

numbers. But not content with that

finitely many? What is the sum of the

proof, the author proceeds to give

mulas giving prime numbers? Chapter 2

reciprocals of the primes? Are there for­ theorem

on

the attempts to get closer and closer to

eleven more proofs of the theorem,

deals

primes in arithmetic progressions. From

the proof in Euclid; proofs suggested

there on the concentration is on 11(x),

by Euler's early explorations of his

with Chapter 3 on Chebyshev's theorem,

large even integers can be expressed as

Chapter

the sum of a prime and a product of at

product formula; those derived from

Dirichlet's

on Hilbert's Tenth Problem. In Chapter 6 there is a fairly extensive account of

gathered together by type: variants of

certain sequences of pairwise coprime

with

Davis, H. Putnam, and Yu. Matijasevic

4 on the Riemann ?-function

the Goldbach Col\iecture, right down to and beyond Chen's result

(1966), that

and Dirichlet series, and Chapter 5 on

most two primes.

Chapter 6 gives information on what fol­

it from cover to cover, but many will

Few will pick this book up and read

one involving Fermat numbers, attrib­

the Prime Number Theorem (PNT).

lowed the proofs of Hadamard and de

fmd it a useful source of information

one derived from topology. Not con­

la Vallee-Poussin, with Landau's ap­

to use when teaching a number theory

the col\iectures of Hardy and Little­

browsing. There's no end to the star­

positive integers (the most famous the uted by Hurwitz to P6lya); and even thor describes at some length work

proach to the PNT, Tauberian methods,

prompted by this proof, questions such

wood, among other things.

tent with Euclid's simple proof, the au­

as the following: if

While most sections of the book con­

class. And it's a great text for casual tling results proved about primes-or

{an} is a sequence of primes, a1 = 2, a1, a , . . . , an- 1 al­ 2 ready defined, then let an be the largest prime divisor of Pn = 1 + a 1 a2 . . . an-1. Does this sequence contain all

casional proof, the sections on the PNT

not anticipate using it as a textbook

are considerably more detailed and

"Exercises" may be a misnomer; many

proofs are actually given at length.

of them look like hard problems.

primes? (The answer is no.) Does it

Anyone interested in number theory has

contain all sufficiently large primes?

probably seen a proof of the PNT, but

sist of accounts of results with the oc­

in

some

cases

only

col\iectured.

Though there are exercises, I would

The author's command of English is

excellent. The text reads very well and

VOLUME 23, NUMBER 1 , 2001

73

the number of misprints and errors is

vehicle to teach or re-teach a variety of

requires an entire chapter because the

relatively small, given the extent of the

mathematical concepts: ingenious log­

limit cannot be computed separately

coverage. Most errors are trivial and

arithms,

beautiful

on each side of the parentheses. Here

most unimportant. (Einar Hille's death

geometry, fascinating number theory,

Maor provides another excellent re­

date is clearly wrong, for example, and

and mind-bending complex function

fresher course, this time on the bino­

there are other little problems of that

theory. Maor is one of a small handful

mial formula, which in turn leads to a

sort.) The cutoff point (the Hardy­

of mathematical authors who create a

Martin Gardner-like subject: Pascal's

Littlewood conjectures) can be frus­

satisfying blend of historical anecdote

fascinating symmetric triangle of coef­

trating. While the author has tried to

and mathematical information that ap­

ficients.

bring up to date certain problems, oc­

peals to amateur and expert alike.

casionally this falls short of bringing us

useful

calculus,

Following this is a delicious list of

The first three chapters give us a

curious numbers related to e. For ex­

completely up to the present. For ex­

fresh look at logarithms, that primor­

ample, what x do you think yields the

ample, the conjecture (which would

dial ooze out of which e arises.

maximum possible value of

have implied the Riemann Hypothesis)

= In s xA(n) :::; 0, where the Liouville function A(n) ( - 1)0Cn) and ncn) is the number of prime factors of n counting multiplicities, was correctly disproved by C.B. Haselgrove (1958), but R.S. Lehman (1960) did more than confirm

that L(x)

=

Haselgrove's

computations.

Lehman actually found the first coun­ terexample. (Haselgrove only proved

1

Chapter

covers Napier's life and

Chapter

5

y = xllx?

paints the pre-calculus

200 B.C., comprised of

his fetal logarithmic table. Maor ex­

landscape, circa

plains the value of finding numbers

well-understood straight-lined geomet­

with a common base, which can be ex­

ric figures. On the horizon are curved

pressed as a function of their expo­

figures, waiting to be measured. But,

nents. It was a little tough getting

alas, calculus was not to be. The jump

through the description of Napier's

from a geometric way of thinking to an

table without an illustration.

algebraic one was too much of a Grand

2, Maor introduces other

Canyon even for Archimedes, who

men who invented, improved, or pro­

held the crown of mathematics longer

In chapter

moted the use of logarithms. Unfortu­

than anyone before or since. It's amaz­

found the least coun­

nately, history often attaches only one

ing to think that calculus was that

terexample. Though I am sure the au­

name to a great invention. Maor con­

close to our grasp, yet had to wait al­

thor was aware of these points, his

tinues

most

chronological time frame limited his

Napier while throwing some crumbs to

exposition on this point, and, I'm sure,

Burgi, Briggs, and others.

that counterexamples exist.) Further, Tanaka

(1980)

this

tradition

by

crediting

2,000 years to be discovered (in­

vented?).

In chapter 6, Viete and Wallis relate

on others as well. But as he said in

Maor's book has peripheral sections

a finite quantity to an infinite series of

his preface when he pointed out the

scattered throughout, when there is

multiplied quantities, thus uncovering

time limitation, "The following years

more to be said about a subject that is

one of those mathematical gems that

brought a great leap forward in our

related to e, but doesn't warrant a

ject. Maor uses this as a lead-in to the

knowledge of prime numbers deter­

whole chapter. The first is a refresher

debunk the myth of math as a dry sub­

mined by the birth of new powerful

course on using logarithm and antilog­

acceptance of infinite processes, nec­

methods, but this should be, possibly,

arithm tables. This scary section will

essary for the calculus.

a subject of another book" We can

turn any Luddite into a computer geek.

only hope that the author is hard at

Maor gets these math class flashbacks

ideas in chapter

work on the sequel.

out of the way early-stick with him!

much about the hyperbola. It doesn't

Chapter

Department of Mathematics & Computer Science Santa Clara University Santa Clara, CA 95053-0290

3 connects money and e, a

e: The Story of a Number by Eli Maar

are

some

7.

pretty

amazing

I'd never thought

have the intricacy of a lemniscate. It's

surprise for those of us anxious to read

not a nice neat closed figure. It's a de­

about

and

creasing function and so we wouldn't

seashells. Kind of like a meeting be­

e's

role

in

sunflowers

want it to describe our retirement sav­

tween yuppies and hippies, e is an un­

ally. It has no beginning and no end.

likely thread between nature's poetic

USA

There

ings over time. Its hard to plot manu­

subtlety and humans' harsh acquisi­

Yet the desire to "square" (measure the

tiveness.

area of) the hyperbola led to interest­

Chapters

4 through

10 are a Trojan

Horse, used to sneak a history of cal­

PRINCETON, NJ: PRINCETON UNIVERSITY PRESS

culus into a book on e. These seven

chapters describe the fascinating per­

ing

spin-off

discoveries,

course, have to do with e. Calculus

then

that,

evolved

of

from

Descartes's and Fermat's determina­

1 998, 232 pp.

sonalities, controversies, and versatil­

tions of particular cases to Newton's

US $1 4.96, paperback, ISBN: 0691 058547

ity associated with calculus, a tool that

and Leibniz's general algorithms, de­

REVIEWED BY MELISSA HOUCK

is to e what lasers are to an eye with

signed to handle any case that might

I

n this delightfully thought-provoking yet digestible book, Maor uses e as a

74

THE MATHEMATICAL INTELLIGENCER

cataracts.

Chapter

4 gets to the crux of e by

computing the limit of

(1

+

1/n)n. This

come along. The case

llx,

however,

steadfastly

to

conform.

Chapter

8

refused introduces

the

missing

link-Mercator's

t - t2/2 + &/3

-

series:

t414

+ . . .

log(l +t) =

Chapter 9 brings to life the battle over kudos for inventing calculus. The average calculus course would be more memorable if students were introduced to the subject via a lively account of the priority dispute between Newton and Leibniz. Maor describes the different perspectives each had on the subject­ Newton: physics, Leibniz: philosophy­ and how this drove their approaches. In historical retrospect, having these two divergent-seeming paths both converge on the same method was almost neces­ sary for the universal appreciation ofthe subject. "The Evolution of a Notation" is pre­ sented next as an aside. Maor stresses how essential an effective notation is. Perhaps the most taken-for-granted aspect of mathematics (and, quite frankly, the most boring), notation can stymie mathematical progress (e.g., ro­ man numerals) or grease the tracks. As Maor demonstrates, Leibniz's nota­ tion was butter compared to Newton's "pricked letters." Chapter 10 fmally describes the connection between calculus and e. The beauty of Maor's discourse on e is that he presents e's meaning in several different ways: geometrically-as a key player in the hyperbola's area; ap­ plied-as in the solution to a para­ chutist's problem; naturally-as it de­ scribes the relationship between pitch and frequency. And, in this chapter, al­ gebraically-as a function that equals its own derivative. Geometry is meant to be generously illustrated, and Maor doesn't let us down. He uses several advanced mathe­ matical concepts in chapter 1 1-polar coordinates, mapping, invariance, cur­ vature-to describe the beautiful loga­ rithmic spiral and its narcissism: It is its own evolute, just as ec is its own deriv­ ative. If this geometric chapter is indica­ tive of what Maar's book Trigorwmetric Delights is like, then I'm sold. Next is an interlude about a ficti­ tious meeting between Bach and Johann Bernoulli, where they discuss music scales. It is fascinating that these two overlapped in their time on earth, studied complementary sub-

jects, and probably did affect one an­ other, albeit indirectly. Those who pos­ sess a strong correlation between their mathematical and musical neurons will enjoy this passage. Another interlude-this time it's the sunflowers and seashells we've been waiting for. And some Escher litho­ graphs as the cherry on top. Again, Maor isn't stingy with the pictures. This chapter should suffice as the an­ swer to the question: "Why do I need to study math?" Most students will not grow up to engineer great structures or invent the next computing device. But everyone should have a chance to mar­ vel over nature and its unexpected connection with mathematics. Ah, the stuff of mental blocks: hy­ perbolic functions. In chapter 12, Maor transforms these from dryly memo­ rized, funny-sounding (sinh, cosh) cal­ culus 101 mysteries to a river of geo­ metric and analytic meaning. Maar uses the hanging chain (catenary), (f!C + e-x)/2, to take us to these tran­ scendental, underrated analogies of trigonometric functions. Another interlude, this time on the analogies between x2 + y2 = 1 and x2 y2 = 1. The circle and hyperbola look so unalike but the identities within them are close (identical, in some cases). A topic that lends itself to both number theory and geometry is always enlightening. The properties of most numbers pale in comparison to e. Maor could have elaborated on e's role in three key integrals that he discusses all too briefly in this next interlude. Probability Theory, Exponential Integrals and Laplace Transforms are all sophisti­ cated concepts that should not have daunted the author, who has a gift for bringing sophisticated concepts down to earth. I think a deeper handling of these would have shed light on e's util­ ity as well as its beauty. In chapter 13, Maar does elaborate on one of the formulas he discusses in the previous interlude: eia:. The author gives a very satisfying account of Euler bring­ ing e into the complex (as in i) world. Kudos to an author who not only dares to address complex number theory, but who does so in a way that preserves the -

nature of the beast. Maor points out that

ix describes 4 cycles, not numbers in the usual sense. Maar derives Euler's for­ mulas connecting trigonometry with ex­ ponentials in a way that is surprisingly easy to understand. Chapter 14 continues Maar's won­ derfully accessible account of complex number theory. He boils i down so that it is seen as a compact way of intro­ ducing rotation into analysis, rather than as a hard-to-handle blockade be­ tween problem and solution. Maar ap­ plies complex function theory to e", and shows us how to map equidistant lines to exponentially increasing circles, thus circling back (literally) to Napier's loga­ rithms. Maor describes i as the least in­ teresting constant in the equation em + 1 = 0. After reading his chapters on imaginary numbers, I would have to dis­ agree. He points out some differences between complex and real numbers in math: Exponentials become periodic; logarithms become multivalued; circu­ lar (sin, cos) and hyperbolic (sinh, cosh) functions become directly related. Best of all: ii is multivalued and real! Perhaps i is uninteresting to Maor, but he writes an interesting account of it. Another of those unexpected con­ nections is the following interlude­ this time between integers and tran­ scendentals (namely e!). Maor gives a typically lucid explanation of primes and the prime number theory, high­ lighting e's prominent role. The last chapter summarizes e's place in mathematical history. Maor keeps the discourse lively by contrast­ ing e with 1r, that other transcendental wonder, and by including a brief synopsis of the different categories of numbers-rational, irrational, alge­ braic, transcendental, etc.-showing how e fits into a teeny tiny hole in the number line. Having read Maor's To Infinity and Beyond, I was sure that e, The Story of a Number would not disappoint me. I was right. Now its time to pick up a copy of Trigonometric Delights. 232 Yale Road Wayne, PA 1 9087 USA e-mail: [email protected]

VOLUME 23, NUMBER 1, 2001

75

Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory by Jifi Herman, Radan Kucera, and Jaromir Si'mSa

equations:

thors state (without proof) a valuable

q =/=

transformation theorem: if

X� + X� = 464 XI + X2 = 4.

1 is

real, then there exists a polynomial

Q(x) of degree m and a real number d

One can eliminate one of the variables,

such that

but then one is faced with the problem

S(q,n)

=

of solving a quartic equation. Rewrite

d + Q(n)qn.

the equations using symmetric func­

The authors show how the polynomial

tions:

u1 - 5uYu2 + 5uw�

determined coefficients.

MATHEMATICS

£TI = 4,

There is a valuable discussion of

vii + 344 pp.

polynomials and their properties. One

US $69.95 ISBN 0-387-989420

common practice in proving an iden­

REVIEWED BY JET WIMP

l

an example, consider the simultaneous

call the above a finite q-series. The au­

Q may be found by the method of un­

CANADIAN MATHEMATICAL SOCIETY BOOKS IN

NEW YORK: SPRINGER PUBLISHING,

m. A special function specialist would

tity is to show the quantity in question

=

464

This leads rapidly to the solutions x1

2 ± v2, x2

=

=

2 + v2. (It turns out that

MAPLE can solve the original equations,

satisfies a polynomial equation with

but it is not difficult to devise similar

have a long-standing addiction to

one or more known roots, and to rule

problem books, and have over a

out the presence of the other roots. For

equations solvable this way which will

dozen of them on my bookshelf. Some present a concatenation of assorted problems, organized roughly by type,

�3V2i + 8 - �3v2i - 8 =

with little explanatory material and no

denote

exposition of systematic techniques.

Elementary algebra gives

An

example

is

the

recent

book

Berkeley Problems in Mathematics, edited by de Souza and Silva, Springer,

1998.

Some-and the present book is

an example-are much more tech­

the

left

hand

side

A3 + 15A - 16 =

Thus A is a zero F(x) = x3 + 15x zero

cripple computer algebra systems.)

Section 6 of this Chapter deals with

instance, to prove

the solution of irrational algebraic

1,

by

A.

arcane equations considered here is

Vx+1 + � + Vx+3 = 1.

0.

of the polynomial 16.

equations, and introduces a variety of techniques. A typical example of the

x= 1

of this polynomial.

As might be expected, computer alge­

is also a

bra systems fail on these sorts of equa­

We

tions.

find

nique-oriented. They explain how large

F(x)l(x - 1) = x2 + x +

classes of problems may be attacked,

mial which has only complex roots.

and the problems themselves are much

Thus

16, a polyno­

A = 1, as required.

more like the exercises found in texts.

The discussion of symmetric poly­

Of the latter type, this book is about

nomials is unusual and fascinating. I

the most successful I have seen, and

wish I had been aware of some of this

the editors of the series, Jonathan and

material when attempting to solve

2, Algebraic Inequalities, is

Chapter

one of the most illuminating studies of the subject I have ever seen. Again, symmetry becomes a powerful tool. If one is attempting to show that

fi:x1>X2, . . . , Xn) > 0

f is symmetric in all its variables,

Peter Borwein, are to be congratulated

some equations arising from the trans­

and

for bringing it before a large public in

formation theory of hypergeometric

then it is no loss of generality to assume

a superbly smooth translation.

functions. The following result, which

XI 2: x2 2:

The book is divided into four parts: Algebraic Identities and Equations; Al­ gebraic Inequalities; Number Theory; and Hints and Answers. Each chapter presents an assortment of techniques for dealing with the problems in that area. Chapter

1 discusses fmite sums and

combinatorial identities. As might be

has generalizations to more than two

(The proof of this and its extensions is in van der Waerden's algebra book.)

rem is sufficient to prove the formula

The authors give a nice little table ex­

in question. One unusual aspect of this

pressing powers

chapter is the attention devoted to

and

76

THE MATHEMATICAL INTELLIGENCER

The same technique will work to

prove that

xf + x� in terms of £TI a(a - b)(a - c) + b(b - c)(b - a) + c(c - a)(c - b) 2: 0 . . . , 5, and a sim­

u2 for n = 1, 2,

ilar table for the case of three vari­

· · ·

This assumption

for an arbi­ ple, suppose we wish to show that trary polynomial F(x1,x2) symmetric 1oo < 3100 (a iOO + b ioo + l� c in the variables x1 and x2 there exists (a + b + c) a unique polynomial H(y1,y2) such for a, b, c, positive. Assume a 2: b 2: c. that We get F(x1,x2) = H(avr2), (a + b + c) Ioo ::; (3a) Ioo < 3 100 (a !OO + b iOO + c lOO . £TI = X1 + X2, £T2 = X1X2. )

imaginative use of the binomial theo­

S(q,n) = P(l)q + P(2)q2 + P(3)q2 + + P(n)qn, where P(x) is a polynomial of degree

Xn.

variables, is invaluable:

suspected, for many such sums the

sums of the form

. . . 2:

can have a magical efficacy. For exam­

ables. Symmetric functions are very useful in solving simultaneous nonlin­ ear algebraic equations when the equa­ tions are symmetric in the variables. As

for positive

a, b, c, since the left-hand

side above may be written

a(a - b) (a - c) + (c - b) [c(c - a) - b(b - a)],

a ::::; b ::::;

x = 1, y = 2, or x = 0, 0). The chapter closes with a dis­

c

multiple and greatest common divisor,

(SOLUTIONS:

guarantees that both the term in brack­

prime numbers,

function. The number of bizarre results

y =

As an exercise, the authors ask the

that may be proved using very ele­

nomials

reader to demonstrate an intriguing ho­

mentary

Eisenstein irreducibility criterion.

mogeneous nonnegative lower bound

boundless. I will whet the reader's ap­

for the difference between the arith­

petite by giving just a few of them:

and the assumption that

0
� ��� � --� - 8(a + b)(a2 + 6ab + b2 ) --

between

22n + 1

means

is

apparently

and

numbers. ii) Let

the results that struck my fancy was

� +V'bc +� - 1 2 ::::; -(a + b + c). 3 The third chapter, Number Theory, was, to me, a revelation. The chapter is self-contained, with a treatment of basic concepts such as divisibility, the Euclidean algorithm, least common

a, b, and Va + Vb be posi­

tive rational. Then tional.

generalizations of Cauchy's inequality receive a sound treatment also. Among

cannot be expressed as a dif­

ference of fifth powers of two natural

--

•

Inequalities

techniques

i) Any prime number of the form

--

--

primality, Euler's ­

iii) The number by

13 .

Ya and Vb are ra­

260 + 730 is divisible

cussion of the factorizability of poly­ and

a

statement

of

the

The fourth chapter has the solu­ tions for the exercises. Sometimes these are only sketched, which is ap­ propriate.

After

all, in a problem

book, the reader has the responsibil­ ity for doing at least some of the work. I have nothing but praise for this book, and I can't imagine a working mathematician who wouldn't want to own it. As a matter of fact, my unwill­ ingness to surrender the book to some­

Congruences receive a thorough

one else prompted this review.

treatment, including systems of con­ gruences and nonlinear congruences. The discussion of diophantine equa­ tions is extensive and systematic, and the equations treated include many strange nonlinear diophantine equa­ tions, for instance

1 + x + x 2 + x3 = 2Y.

Department of Mathematics and Computer Science Drexel University Philadelphia, PA 1 9904 USA e-mail: [email protected]

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VOLUME 23, NUMBER 1 , 2001

77

4jfi i.MQ.ip.i§i ..

Robin Wilson

Romanian Mathematics

IMatematicii

n 1895 the monthly periodical Gazeta made its first appear­ ance. Founded in order "to improve the knowledge of mathematics of high­ school students," but also featuring orig­ inal papers in mathematics, it has since appeared without a break, and at one time had over 120,000 subscribers. Romanian stamps were issued to commemorate its 50th and 100th an­ niversaries. One of the 1945 stamps featured "the four pillars of Gazeta Matematicii," Ion Ionescu, Andrei loachimescu, Gheorghe 'fi leica and Vasile Cristescu, while the centenary stamp depicted Ion Ionescu alone, the "spiritus rector" of the Gazeta who ran it for over fifty years. Gheorghe Titeica, the only pure mathematician among the "four pillars," made distin-

guished contributions to differential geometry. The Romanian mathematical educa­ tion system was established in 1898 by the Minister of Education, the mathe­ matician Spiro Haret. With his support and encouragement, Gazeta Matematicii initiated a series of mathematical text­ books that became of great importance in secondary schools for many years. Romania has always been at the forefront of mathematical competj­ tions at high-school level. Since 1897 it has regularly organized national com­ petitions, and in 1959 launched the flrst International Mathematical Olympiad. Forty-one such Olympiads have now taken place in over twenty countries, and Romania has played host on no fewer than four occassions.

Gazeta Matematka

Please send all submissions to

Jon lonescu

the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,

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

80

THE MATHEMATICAL INTELUGENCER 0 2001 SPRINGER·VERLAG NEW YORK

The "four pillars of the Gazeta"

Gheorghe Tilelca

Spiru Haret