The Mathematical Intelligencer volume 23 issue 2

) p i n i o n

Reply to Martin Gardner Reuben Hersh

The Opinion column offers

mathematicians the opportunity to

write about any issue of interest to

the international mathematical

community. Disagreement and

controversy are welcome. The views

and opinions expressed here, however,

are exclusively those of the author,

and neither the publisher nor the

editor-in-chief endorses or accepts

responsibility for them. An Opinion

should be submitted to the editor-in-

chief, Chandler Davis.

Dear Martin Gardner ,

Thanks for your in teres t in my writ- ings. As everyone knows, you ' re the most highly r e spec t ed science journal-

ist in the world. I jus t counted six of

your books on my shelf. Yet for interesting, mys te r ious reasons, you seem unable or unwill ing to unders tand my

writ ing about mathemat ica l exis tence. Your unhapp iness with me is not

new. You d issed The Mathematical Experience [1] by me and Phil Davis, in the New York Review of Books. In

the mos t recen t issue of The InteUi- gencer [4], you re turn to the task. You quote "myths 2, 3 and 4." from my Eureka art ic le ( repr in ted in What is Mathematics, Really? [5], pp. 37-39). Myth 3 is s o m e w h a t off the point; I will concent ra te on 2 and 4.

Myth 4 is objectivity. "Mathematical truth or knowledge is the same for

everyone. It does not depend on who in part icular discovers it; in fact, it is true whether or not anyone discovers it."

Your reaction: "What a s t range con- t en t i on" - - t o call it a myth.

Myth 2 is certainty. "Mathematics possesses a me thod cal led ' p r o o f ' . . . by which one a t ta ins absolu te cer ta in ty

of the conclusions , given the t ruth of the premises."

Your react ion: "Can Hersh be serious when he calls this a myth?"

In a way, I under s t and your difficulty. In c o m m o n speech, when some-

one says, "That 's jus t a myth!" he means something is false, untrue. But

in scholar ly writing, "myth" commonly has o ther meanings. I wrote, on the

very next page, "Being a myth doesn ' t entail its t ruth or falsity. Myths val idate and suppor t insti tutions; their t ru th may not be determinable ."

About certainty, I wrote: "We're certain 2 + 2 = 4, though we don' t all mean the same thing by that equation. It 's another mat ter to claim certainty for the

theorems of con tempora ry mathematics. Many of these theorems have proofs that fill dozens of pages. They're usually built on top of o ther theorems, whose proofs weren ' t checked in detail by the

mathemat ic ian who quotes them. The proofs of these theorems replace boring details with 'it is easily seen' and 'a

calculation gives.' Many papers have several coauthors, no one of whom

thoroughly checked the whole paper. They may use machine calculations that

none of the authors complete ly under- stands. A mathemat ic ian ' s confidence in some theorem need not mean she

knows every step from the axioms of set theory up to the theorem she's interes ted in. It may include confidence

in fellow researchers , journals, and ref- erees. Certainty, l ike unity, can be

claimed in p r inc ip l e - -no t in practice."

Now, you ' re ta lking abou t cer ta inty in principle. I do too. I recognize its impor tance as a posi t ive and valuable guiding myth. I a lso ta lk about cer-

ta inty in practice. The whole mathemat ica l communi ty recognizes its

value and is engaged in seeking it. Immedia te ly fol lowing in my book:

"Myth 4 is objectivity. This myth is more plausible than the first three. Yes! There ' s amazing consensus in mathe-

mat ics as to wha t ' s cor rec t or ac- cepted." On page 176 I elaborate: "Mathematical t ru ths are objective, in

the sense that they ' re accep ted by all qualified persons , regard less of race, age, gender, pol i t ical or rel igious belief. What 's cor rec t in Seoul is correct in Winnipeg. This ' invar iance ' of math-

emat ics is its very essence." On page 181: "Our convict ion when

we work with ma themat ics that we ' re working with someth ing real isn ' t a

mass delusion. To each of us, mathemat ics is an ex te rna l reality. Working with it demands we submit to its objec t ive character . I t 's wha t it is, not wha t we want it to be."

So we agree. Mathematical truth is objective! Then how can a sophist icated critic like you think I don't recognize the objectivity of mathemat ical truth?

The question, of course, is what we mean by "objective." To you, "objective" means "out there." To me, "objective" means "agreed upon by all qualified people who check it out." But I 'm

�9 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 3

unwilling to leave the mat te r at that.

Like many other people, I think objectivity is to be unders tood by reference to objects, things that really have the

proper t ies we discover. Mathematical objec ts are simply the things mathemat ical s tatements are about. Numbers, functions, operators, spaces, transfor-

mations, mappings, etc.

What sort of objects are they? They're not physical objects. Aristotle a l ready explained that the triangles and circles of Greek geometry are not physical entities. The first few natural num-

bers are abstracted f r o m physical sets. But the really big natural numbers are

not found in nature. The set of all natural numbers, N, is an infinite set, not found in nature. We made it up. The mos t important proper ty of N is mathe-

matical induc t ion- -an axiom said to be intuitively obvious. Intuition is "in there," not "out there." Certainly the use

and interest of the abst ract mathematical numbers come from their close connect ion with physical numbers. But

meaning and existence can ' t be untan- gled without acknowledging the dis-

t inction between physical numbers and mathematical numbers.

We also s tudy infinitely differen- t iable inf ini te-dimensional manifolds

of infmite connectivity. These are not found in the physical world. Not to ment ion the "big" sets of con t empora ry set- theorists .

Then, if not in phys ica l reality, could mathemat ica l objec ts exis t "in the mind"? Gott lob Frege famous ly de- r ided this idea. If I add up a row of fig-

u res and get a wrong answer , i t 's wrong even if I think it 's right. The the- o rems of Euclid remain af ter Eucl id 's mind is bur ied with Euclid.

So where are the objects about which mathemat ics is objective? The answer

was given by the French phi losopher/ sociologist Emile Durkheim, and expounded by the U.S. anthropologist

Leslie White [8]. But social sc ient is ts a ren ' t ci ted by phi losophers , nor by many mathemat ic ians . (Ray Wilder was the except ion.)

The universe conta ins things o ther than menta l objec ts and phys ica l objec ts . There are also inst i tut ions, laws, c o m m o n unders tandings, etc., etc., etc.--social-historical objects. (I say

jus t "social" for short .) We canno t

th ink of war o r money or the Supreme Court or the U.S. Const i tut ion or the doct r ine of the virgin bir th as e i ther

phys ica l or menta l objects. They have to be unde r s tood and deal t with on a different l e ve l - - t he social level.

Social ent i t ies are real. If you doub t it, s top paying your b i l l s - - s t op obeying

the speed limit. And social ent i t ies have real proper t ies . That 's how we manage to negot ia te daily life.

Social sc ient is ts don ' t say "object." They say "process" and "artifact" and

"institution." Social p rocesses and ar- t i facts and inst i tut ions are g rounded in

phys ica l and menta l o b j e c t s - - m a i n l y the bra ins and the thoughts of people . But they mus t be unders tood on a different level f rom the mental or physi-

cal. In o rde r to decide where mathemat ics belongs, I must cons ider all three-- - the physical , mental , and social. I need a word that can apply to all

t h ree - -phys i ca l , mental, and social worlds. "Object" seems suitable. The c o m m o n connota t ion of "object" as

only a phys ica l enti ty has to be set aside. Any definite en t i ty - - soc ia l , mental, o r p h y s i c a l - - w h o s e exis tence is

mani fes ted by real-life exper ience can be cal led an object. Mental ob jec t s ( thoughts, plans, intentions, emotions ,

etc.) are g rounded on a phys ica l ba- s i s - t h e nervous system, or the brain. But we cannot deal with our thoughts or the thoughts of each o ther as phys-

ical ob j ec t s - - e l e c t r i c currents in the brain. That is why there is a "mind- body problem." And social-his tor ical

objec ts a re on still a different level from ei ther the mental or the physical .

Now, Martin, if you recognize the ex is tence of social objects, you ought to ask, "Since mathemat ica l ob jec t s are ne i ther physical nor mental , a re they social?"

My answer is, "Yes, that is wha t they are."

That 's controversial. It 's "maverick."

That doesn't mean you can dispose o f it by distorting or denouncing it.

That ma themat ics is in the minds of people, including mathemat ic ians , is not a novelty. Everyone knows that.

It 's in minds connec ted by frequent communica t ion , in minds that fol low the her i tage of pas t mathemat ic ians .

My claim is this: to unders tand what mathemat ics is, we need not go beyond this recognized social existence. That's where it's at. Locating mathemat ics in the wor ld of social ent i t ies

DOESN'T make it unreal. Or imagi-

nary. Or fuzzy. Or subjective. Or rela- tivistic. Or pos tmodern .

Saying it 's real ly "out there" is a reach for a supe rhuman cer ta inty tha t is not a t ta ined by any human activity. A famous mathemat ic ian said to me, "I am willing to leave that question to the phi losophers ." Which phi losophers?

Profess ional ph i losophers who are not mathematic ians?! To obta in answers meaningful to us, I 'm afraid we' l l have

to get to work ourselves. Martin, 18 years ago you talked about

"dinosaurs in a clearing," in order to

prove that 2 + 3 = 5 is a mathematical truth independent of human conscious- ness. I answered that claim in my recent

book. In your review of it in the L.A. Times, you ignored my answer. In your letter to The Intelligencer, you ignore it again. You jus t repea t your dinosaur anecdote. I will explain again. Words like "2", "3", and "5" have two usages. Most basically, as adjectives--"two eyes," "three blind mice," "five fingers."

We call them "physical numbers," though they are also used for mental and

social entities. It 's a physical fact that two mama bears and three papa bears together make five great big bears. To put it in more academic terms, there are discrete s tructures in nature, and they

can occur in sets that have definite nu-

merosity. In mathemat ics , on the other hand,

we deal with "abs t rac t s tructures," not bears or fingers or dinosaurs. In mathematics, the words "2", "3", and "5" can be nouns, denot ing cer ta in abs t rac t

objects, e lements of N. As I explain above, N and its main p rope r ty are not

found in physical nature. Counting dinosaurs uses phys ica l numbers , adjectives, not the abs t rac t numbers we s tudy in mathemat ics . The physical numbers apply even if we don' t know about them. They are par t of physical

reality, not human culture. Mathemati- cal numbers , on the o ther hand, are a human creat ion, pa r t of our social-his- tor ical heritage. They were created, we presume, from the physical adject ive

4 THE MATHEMATICAL INTELLIGENCER

numbers , by abs t rac t ion and general- ization.

F rom t ime to t ime you call me a "cultural relativist." Cultural re lat ivis ts

say, "Western music (for ins tance) is not be t t e r or worse than New Guinea music. I t 's different, that ' s all." When I

say mathemat ics is par t of human culture, there ' s no relat ivism involved.

More mys te r ious is your conclusion: "To imagine that these awe-

somely compl ica ted and beautiful patterns are not 'out there ' i ndependen t of you and me, but somehow cobbled by our minds in the way we wri te poe t ry

and c o m p o s e music, is surely the ulti-

mate in hubris. 'Glory to Man in the highest , ' sang Swinburne, ' for Man is the mas te r of things. ' "

This song of Swinburne seems to be

"coming f rom left field." It suddenly de- nies your main contention. To under- s tand it I look at your books , Order and Surprise [2] and The Whys of a Philosophical Scrivener [3].

In Order and Surprise [2] you write, cri t iquing Ray Wilder, "One may, of course, adop t any way of talking one

likes, but the fact is that mathemat i - c ians do not ta lk like Wilder excep t for

a few who are mot iva ted by an in tense desire to make humani ty the measu re of all t h i n g s . . , to ta lk in a way so far r emoved from ordinary language, as

well as the language of great sc ient i s t s and mathemat ic ians and even mos t phi losophers , that in my layman 's opin-

ion adds nothing to mathemat ica l discourse excep t confusion." The confu- s ion here is your own. F rom the subs tant ive issue, the nature of math-

emat ica l reality, you swi tch to mere convenience of language, wi thout admiss ion or apology. More significant, you are a ler t to any poss ib le "desire to make humani ty the measure of all

things." You do not let that pass. You reac t by a gra tu i tous a t t r ibut ion of mo- tives.

Against Davis and me you raise the same non-issue of language, and make a s imilar gra tui tous a t t r ibut ion of mo- tives. "It is a language that also appea l s to those historians, psychologis ts , and ph i losophers who cannot bring them-

selves to ta lk about anything that t ran- scends human experience."

We can ta lk about the t r anscenden-

tal, Martin. We jus t don ' t think it ex- plains mathemat ics .

On page 72, you write, "The view that mathematics is grounded only in the cul-

tural process slides easily into the 'col- lective solipsism' that George Orwell satirizes in his novel Nineteen Eighty- Four�9 For if mathemat ics is in the folkways, and the folkways can be molded

by a political party, then it follows that the par ty can procla im mathematical laws." This easy sliding is the notorious

"slippery slope" pseudo-argument. Far- fetched political insinuation degrades and cheapens this controversy.

Later you write: " 'Mat ter ' has a way

of vanishing at the microlevel, leaving only patterns. To say that these pat terns

have no reality outside minds is to take a giant step toward solipsism; for, if you refuse to put the pa t te rns outside human

experience, why must you put them out-

side your experience?" Apart from your dubious vanishing of matter, you again

resort to "the sl ippery slope" toward solipsism as well as Stalinism! (This

t ime not jus t an easy slide, but a giant step!) (Some opponents of Social Secu- rity called it "the first s tep to socialism.")

You go on: "I am an unabashed realist (for emotional r e a s o n s . ) . . , if all men vanished, there would still be a sense

(exactly what sense is another and more difficult problem) in which spiral nebu- lae could be said to be spiral, and hexag-

onal ice crystals to be hexagonal, even though no human creatures were

around to give these forms a name." "Exactly what sense" is exactly the issue! Leaving it at that is on a pa r with your "out there, never mind where�9

I turn to The Ways of a Philosoph- ical Scrivener [3]�9 This book is a con- fession of faith. It is eloquent, touching, and immense ly learned. I was impressed by the chap te rs "Faith: Why I am not an atheist" and "Immortali ty: Why I am not resigned."

Starting on page 213, you write: "That the leap of faith springs from passionate hope and longing, or, to say the same thing, from pass ionate despair and fear, is readily admit ted by most fideists, cer-

tainly by me and by the fideists I admire. . . . Fai th is the express ion of feeling, of emotion, not of reason . . . . How can a fideist admit that faith is a kind of madness, a dream fed by passionate desire,

and yet maintain that one is not mad to make the l e a p ? . . . To believe what we do not know, what we hope for but can-

not see---this is the very essence of faith. � 9 To believe in spite of anything! This is the essence of quixotic fideism . . . .

With hope travels faith and with faith travels belief. But because it is belief of

the heart backed by no evidence, it is never free of doubt . . . . "

After reading this, I f inally appreci-

ate your bi t ter ly ironic quote from Swinburne. A se l f -named quixotic fideist has the hubris to tell me that say-

ing man is the creator of mathemat ics is the ult imate in hubris! I 'm sorry, Martin. I never wanted to disturb any-

one 's hope, faith, and belief. I 'm sorry. P.S. I in tended to answer your New

York Review of Books ar t ic le in my book, but my edi tor p e r s u a d e d me not to. Thanks for this chance to respond in The Intelligencer.

REFERENCES

[1] P. J. Davis and R. Hersh, The Mathemat-

icalExperience, Birkh&user, Boston, 1981.

[2] M. Gardner, Order and Surprise, Prome-

theus Books, Buffalo, 1983.

[3] M. Gardner, The Whys of a Philosophical

Scrivener, Quill, New York, 1983.

[4] M. Gardner, Letter to The Mathematical

Intelligencer, 23(2000), No. 4, 2000.

[5] R. Hersh, What is Mathematics, Really?,

Oxford University Press, New York, 1997.

[6] M. Kline, Mathematical Thought from

Ancient to Modern Times, Oxford Univer-

sity Press, New York, 1972.

[7] S. Korner, The Philosophy of Mathemat-

ics, Dover, New York, 1968.

[8] L. A. White, "The Locus of Mathematical

Reality," Philosophy of Science, 14, 289-

303; also, Chapter 10 in The Science of

Culture: A Study of Man and Civilization,

Farrar Straus, New York, 1949; also, in

The World of Mathematics, ed. J. R.

Newman, Simon and Schuster, New York,

1956, volume 4, 2348-2364.

[9] R. L. Wilder, Introduction to the Founda-

tions of Mathematics, Wiley, New York,

1968.

R. L. Wilder, Evolution of Mathematical

Concepts, Wiley, New York, 1968.

R. L. Wilder, Mathematics as a Cultural

System, Pergamon, New York, 1981.

[ l q

[11]

1000 Camino Rancheros

Santa Fe, NM 87501

VOLUME 23, NUMBER 2, 2001 5

ISTVAN HARGITTAI

John Conway Mathematician of Symmetry and Everything E sc

ohn Horton Conway (b. 1937 in Liverpool, England) is the John von Neumann Professor

of Applied and Computational Mathematics at Princeton University. He received his

B.A. and Ph.D. degrees from the University of Cambridge, England, in 1959 and 1962.

He was Lecturer in Pure Mathematics, then Reader, finally, Professor at the University

of Cambr idge before he j o ined Pr inceton Universi ty in 1987. He was e lec ted Fel low of the Royal Society (London) in

1981, and he received the P61ya Prize of the London Mathemat ica l Society in 1987 and the Freder ic Esse r N e mmers Prize in Mathemat ics in 1998. We reco rded our conversa t ion on August 5, 1999, at the Universi ty of Auckland, New Zealand, where both of us were Visiting

Professors for a br ief pe r iod (John in mathemat ics and I in

chemistry) . Is tv~in H a r g i t t a i ( I H ) : What does it mean to you to be

yon Neumann Professor at Pr inceton? J o h n C o n w a y ( J C ) : Von Neumann himself was a profes-

so r at Pr inceton at one t ime. He did a t r emendous number of different things in mathemat ics , many of them revolu- t ionary. The most famous one is the idea of the computer . He not only theor ized about it, he was also involved in the

bui lding and use of one. Ear l ier in his career, when he be- c ame establ ished, he des igned a sys tem of ax ioms of set

theory. He had this idea of cont inuous geomet ry in which the d imension funct ion took cont inuous values. With

Morgenstern, he wrote The Theory of Games and Economic Behavior. Many of the things von Neumann was

in teres ted in, I 'd been in teres ted i n - - s u c h as set theory, finite numbers , games, abs t rac t computa t ion , and this he lped me to accep t the job. It amused me tha t yon Neumann 's interes ts and mine were so closely related, with the excep- tion of making bombs. Also, I have never moved into his

sys tem of con t inuous geometry. IH: What is your main interest?

JC: I 've had so many. As you know, I 've been in teres ted in symmet ry for a long time, and that comes out as group theory. ! spent a good twenty years of my mathemat ica l life working intensively with groups, but I 'm not real ly a group- theorist . All the t ime I was at tending group- theoret ica l con-

ferences I felt myse l f a little bit of a f raud because all the par t ic ipants were concerned with the real ly big p rob lem of

6 THE MATHEMATICAL INTELLIGENCER �9 2001 SPRINGER-VERLAG NEW YORK

unders tanding all the s imple groups, the building b locks of group theory. They also had a lot of technica l knowledge

that I d idn ' t have. My in teres t is only in s tudying and ap- prec ia t ing all the beautiful pat terns , whenever you have a

group, and I was in teres ted in s tudying the assoc ia ted symmetr ica l objects .

I had a long odyssey. When I was a graduate s tudent I

was in te res ted in number theory, and my adviser was a famous number- theoris t , Harold Davenport . Then, while I was still officially his s tudent I became in teres ted in set

theory, and tha t ' s wha t I wrote my thes is on. After that, suddenly, these large groups began to be discovered, and

I j u m p e d into that field and made my profess ional name in it. That in teres t las ted for many years.

When I moved from Cambridge to Princeton, I d idn ' t

have anybody group- theoret ica l to ta lk to, and I became much more of a geometer , and that ' s wha t I cons ider my-

self now. In this, of course, I in teract with others s tudying

symmetry, but it doesn ' t have to be symmetry. The net effect of this long journey has been that I 've been in a good pos i t ion to not ice certain things. Fo r instance, I 've a lways

been in te res ted in games and regarded it as a mathemat i - cal hobby, but then the theory of games led to my discov-

ery of surreal numbers . (I wish I 'd invented the name but I didn' t .) There ' s a bizarre aspec t to the surreal numbers : You take a definit ion a priori and it looks as though it 's sor t of tame, giving you ord inary real numbers , one and a

half, roo t two, pi, and so on. But the same definit ion gives you infinite numbers and infinite dec imal numbers. I s tum- b led on these things as a consequence of s tudying game theory. The fact that I had a l ready s tud ied infinite numbers ,

as par t of my mathemat ica l development , meant that I was able to recognize that wha t I had come upon was a far- reaching general izat ion of var ious not ions of numbers :

Cantor ' s infinite numbers , the c lass ical real numbers , and everything else. So because I 've done so many subjects , I was able to grasp this, and I wro te a book cal led On Numbers and Games. It founded the theory of surreal numbers, and that includes the ord inary real numbers; this me thod of thinking of them as games turned out to give a

simpler, more logical theory than anybody had found before, even for the real numbers . That sor t of thing has happened to me a number of t imes. Fo r instance, one of the big d iscover ies in group theory was recogni t ion that the mons te r group, which is an absolu te ly enormous beautiful group, was connec ted with var ious things coming from

classical n ine teenth-century number theory. As s o m e b o d y who had done both, I was able to see those connect ions. Il l : Martin Gardner once told me that, while he was edit-

ing the mathemat ica l co lumn of Scientific American, wheneve r he s tumbled on a new p rob lem and asked you about it, it tu rned out that you had a l ready dealt with the problem, mos t ly had solved it, and ye t hadn ' t bo thered to

publ ish the solution. JC: I t 's a big j ob writ ing something for publicat ion, and I 'm lazy. I 'm not ambi t ious anymore. When I was a young man I was ambi t ious to be recognized as a great mathemat ic ian. I haven ' t l ived up to that ambi t ion because the kind of math-

John Conway in Auckland at the time of the 1999 conversation.

(Photos pp. 7-9 by Istvan Hargittai,)

emat ics I 'm doing is not the kind that had my ambit ion. In some sense I 've l owered my sights; I 've pul led in my horus. But I 'm enjoying myself. I 've got a good job, a l though I don ' t fit in with the Pr ince ton set. I 'm not the typical Pr inceton

mathemat ic ian , ye t I 'm recognized as such. I 'm at the top of the mathemat ica l tree, not the top pe r son but near the top. I don ' t feel any compuls ion to jus t i fy myse l f anymore.

What I think is this: "Princeton bought me, and whe ther it was a good buy or not is no longer my concern."

In my late twent ies I was quite worr ied that I d idn ' t seem

to have jus t i f ied myself. I had a job at Cambridge, and I got that j ob very easily. Then a few years la te r there came a sor t of c runch and nobody could get a job. There were very good people who were my near contemporar ies , who came jus t a year or so la ter than I and who had done be t t e r work

than I had, and they would not be gett ing anything. That made me feel guilty, and the guilt was e x a c e r b a t e d by the fact that I didn ' t s eem to have done any ma themat i c s wor th noticing after I got the job. That made me feel depressed.

Then something very nice happened. I w o r k e d out this s imple new group cal led the Conway group, which at the t ime was a real ly exci t ing contr ibut ion to knowledge. As soon as it was done I s ta r ted traveling all over the world.

I c rossed the Atlantic, gave a twenty-minute talk, and flew back. That was a round 1970. The upshot was that suddenly I s tar ted producing things. The next year I p roduced the surreal numbers , and then something else. Not only did I

become successful , but I also deserved the success . I r emember thinking one day, asking myself, "What's


happened? Why is it that I suddenly p roduced three or four

real ly good things and nothing in the previous ten years?" I suddenly realized that the l ack of guilt feelings was a good thing about it. Once I had just i f ied mysel f and was con- v inced that I deserved the job, I found the f reedom to think

abou t whatever I was in te res ted in and not wor ry about

how the rest of the wor ld evaluated this. IH: What lifted you out of your depress ion in the first p lace?

JC : Jus t the t r emendous ego tr ip of d iscover ing this new thing, which put me into the forefront. F rom then on it t ook

a lit t le while to convince myse l f that I was no t going to wor ry and that I was going to s tudy what s e e m e d interesting to me wi thout worry ing what the res t of the wor ld

th inks about it. I t 's been ra ther hard to live up to it at t imes. Fo r in-

s tance, when I moved f rom Cambridge to Pr inceton, I

s ta r ted giving some gradua te lectures about wha t I 'd been doing the last few years. There, in the audience, were very famous mathemat ic ians at Pr inceton who were all coming a long to hear me. My style of lecturing in Cambridge was

a lways elementary. Also, Cambridge is an informal p lace wi th a t radi t ion of to lera t ing eccentr ics . You're a lmos t ex- p e c t e d to be a little bi t odd. In Princeton, however , I felt

inhibi ted by the p resence of these big people . I s t a r t ed to lec ture more as a formal mathemat ic ian , as everyone else does, and then I real ized that it was a d isas ter because it

wasn ' t me. It t ook some effort to get back to my own style. By the way, those famous mathemat ic ians are no longer in my audience, the aud ience consis ts of graduate s tuden ts or

Professor Conway prides himself on interjecting humor and spon-

taneity into his lecture style.

undergradua te s tudents , depending on who I am lecturing to. When I am outs ide that climate, giving a lecture to a big in ternat ional meeting, then the aud ience is a lways mixed,

and tha t ' s a wonderfu l thing because then I can lecture at wha tever level I want to. IH: I mean t to ask you about your lecturing style. I re-

m e m b e r when we were both giving lec tures on symmet ry at the Smi thsonian Inst i tut ion and you were jumping on top

of a table and then hiding benea th it. JC: There ' s a cer ta in amount of a lmos t cynicism in this. Every now and then a joke appea r s to me spontaneous ly

while I 'm lecturing, and I incorpora te it. If it 's good then it s tays in tha t lec ture forever. If I give a lecture 20 or 30 times, the j o k e s jus t accumulate . The ne t effect is that the

lecture gets bet ter . I r emember a te r r ib le t ime when I was lecturing in Montreal and they asked me to let them video- tape it. Af ter the lecture it tu rned out that the man with the

video camera didn ' t arrive, but he ar r ived after the lec ture was over and they asked me to give the lecture again. I asked them to drag up an audience tha t was disjoint f rom

the prev ious one. So I gave the lec ture again. However, the audience was no t disjoint, because some of the same people still a t tended. This inhibi ted me t remendously , because

a j oke tha t looks as though it occurs to you on the spur of the moment , you can ' t tell a s econd time. IH: Did you come across Paul Erd6s? JC : He was a bi t strange. I met him when I was an under-

graduate. He used to pose problems, and I got involved in some of them. He did a lot of traveling, and I did a lot of traveling myself, though nowhere nea r Erd6s, but I t ended

to mee t him somet imes . I would mee t him in Montreal and a few days la te r in Vancouver or in Seattle. I wa lked into the cafe ter ia at Bell Telephone one day and sat next to Erd6s. My Erd6s number is 1.

IH: Donald Coxeter? JC : Coxe te r has been one of my heroes . When I was still at high school in England, g rammar school, I wrote to

Coxeter . He was the Edi tor of Rouse Ball 's Mathematical Recreations. I was absolute ly del ighted by that book. That was 1953-ish, and I have known him ever since. IH: Buckmins te r Fuller s ta ted that Coxe te r is the geome-

ter of the twent ie th century. JC: This mus t be one of the very few things I would agree with Bucky about . Coxeter is my hero. I r emember a s tory at one of the conferences in Coxeter ' s honor and peop le were tel l ing how this wonderful man had turned them into mathemat ic ians . I thought I mus t say something different. So when I got up, I said, "Lots of peop le have come here to thank Coxeter , I 've come here to forgive him." I told them

that Coxe te r once very near ly s u c c e e d e d in murder ing me. His m u r d e r weapon was someth ing that even Agatha Christie would never have thought of, a mathemat ica l problem. Then I to ld the story, which is ac tual ly true.

Coxete r came to Cambridge and he gave a lecture, then he had this p rob l em for which he gave proofs for se lec ted

examples , and he asked for a unif ied proof. I left the lecture room thinking. As I was walking through Cambridge, suddenly the idea hit me, but it hit me while I was in the


The Professor would like to have a 20-minute chat with Archimedes

or Kepler; he's not so sure about Newton and Gauss,

middle of the road. When the idea hit me I stopped and a large truck ran into me and bruised me considerably, and

the man considerably swore at me. So I pretended that Coxeter had calculated the difficulty of this problem so pre- cisely that he knew that I would get the solution just in the

middle of the road. In fact. I limped back after the accident to the meeting. Coxeter was still there, and I said, "You nearly killed me." Then I told him the solution. It eventu-

ally became a joint paper. Ever since, I've called that theorem "the murder weapon." One consequence of it is that in a group if a 2 = b 3 = c 5 = (abc) -1, then c 61~ = 1.

III: Other heroes? JC: Archimedes. Two thousand years ago, he had very clear ideas about difficult, subtle problems, the nature of the real

numbers. In my office I have painted on the wall all my friends. There was a young man who painted a caf~ in Princeton, and I got him to come and paint pictures on my wall. Archimedes is there and Leonhard Euler is there.

Johannes Kepler is also one of my heroes. He was the great- est mathematician of his age and a very interesting guy, too. There are some people about whom I have ambivalent feelings, Isaac Newton and Karl Friedrich Gauss, for instance.

They were really great mathematicians and great physicists too, but they don't seem to have been such nice people, and that rather distances me from them. I would like to have the opportunity to have a 20-minute chat with

Archimedes or Kepler, but I'm not sure about Gauss, though

he might be able to tell me more. I wouldn't enjoy the in-

terview with him so much.

Of the living heroes, I don't think there's anybody to

match Coxeter as an intellectual hero for me. The work he does is elegant and he writes beautifully. There is a paper by Coxeter, Miller, and Longuet-Higgins, and I just know

Coxeter wrote it, and I admire how beautifully it was written. If you look at any of Coxeter's papers you will fred this

beautiful craftsmanship in the design of his papers. That means his papers can be read smoothly. The really impor-

tant thing about Coxeter is that he kept the flame of geometry alive. There was a terrible reaction against geometry

in the universities 30 or 40 years ago, which has had tremen- dously bad effects. So geometry was not a popular subject, and Coxeter all the time did his beautiful geometry. And he

is a lovely man. I remember him at meetings; there's often this embarrassing time at the end of a lecture when the chairman asks for questions and comments, and there may

be none. Coxeter always had something to say, compli- menting the speaker. He's a true gentleman. IIt: What's your principal problem with Buckminster

Fuller? JC: His way of saying things is so obscure. To me, geom-

etry is nothing if you don't have precise proofs and clear enunciation and logical thoughts. There isn't any logical thought in Fuller, only a sort of simulacrum of logical

thought. You don't know what the rules are in manipulat- ing the words the way Bucky does. IH: Don't you think he deserves credit for having enhanced

interest in geometry and what is called today "design sci-

ence"? JC: He's certainly had a positive effect in that sense. On

the other hand, he says somewhere that you can't pack spheres with higher density than you get in face-centered cubic packing. I don't think he thought he had a proof, but

he has some words that almost constitute an argument, some plausible reason why this is true. But countless people say that Buckminster Fuller had proved this, years ago.

I experienced this when I was involved in a dispute over densest packing. And you look back at his words and find that he just sort of asserts it. Then they say, "Bucky wouldn't assert something unless he could prove it." They say this

because, to them, Bucky is a god who could do no wrong. IH: What is the situation today with the packing problem? JC: The situation is that in 1990 someone produced what

he called the proof, which never was a proof and which was heavily attacked. He had some good ideas. He has now actually withdrawn his claim to have a proof but he still

thinks he can patch it up.l One year ago now, Tom Hales announced that he'd fin-

ished his work on this. He has a 200-page paper supple- mented by computer logs of hours of interrogation between him and the machine. His student, Samuel Ferguson, is also involved. My view is, yes, this is probably a proof. On the

other hand, since it involves so much interaction with the machine, it will be very difficult to referee it.

1See Mathematical Intelligencer 16 (1994), no. 2, 5; 16, no. 3, 47-58; 17 (1995), no, 1, 35-42.


A gallery of John Conway's heroes watch over his work from the mural high on his office wall. (Photo by Magdolna Hargittai, 2000.)

I t I : So how can you assess it?

J C : I can bes t do that in a ra ther invidious way, by compar ing it with the prev ious claim. There the cr i t ic ism was

tha t he sor t of t ended to wave his hands, he had some in- equal i ty he had to prove, and in one notor ious case he eval- ua ted the inequali ty at one po in t and then main ta ined that it was t rue everywhere. Hales ' s way of proving inequali t ies is so much tighter, i t 's amazing. He cuts the integral into lots of little pieces, and in each piece he rep laces the func-

t ion that he is deal ing with by a l inear approx ima t ion based on the derivatives; he then reduces the p rob lem to a l inear- p rogramming p rob lem and uses the compute r to show the inequality. In the ar i thmet ic of the compute r he uses wha t

is cal led "interval ar i thmetic," which means that you at any t ime say, "This real n u m b e r is definitely grea ter than this and less than this." You don ' t jus t round it to the nea res t ten p laces of decimals. You have expl ici t upper and lower

bounds , and so on. Every inequali ty that Hales wan t s to p r o v e - - a n d the thing boi ls down to proving a large num-

be r of inequalit ies---is get t ing "interrogated" by the machine and examined by Hales. He shows that eveuvthing is one of the 2000 cases. The inequali t ies are p roved not jus t by gett ing some rough idea of how the funct ions are ar-

rayed but by gett ing prec ise ideas that the function is being be tween this number and this number , and so on. The whole thing is a lot t ighter and Hales has taken consider- able pa ins to provide an audi t trail. Anybody who disbe-

l ieves any asser t ion can fol low it th rough the t ree and find that this was actual ly shown on this day by the fol lowing computa t ion , which you can do again. Obviously, it wou ld take a t r emendous amount of work to check this. On the

o ther hand, the feeling of rel iabil i ty it gives to you is enormous. IH: Couldn ' t there be a s impler way of proving this? JC: My at t i tude to this is this: "I don ' t want to get involved, even reading it." My other feeling is that this isn ' t pa r t of the pe rmanen t furniture of mathemat ics , this type of proof.

My feeling is that eventual ly some s impler p roof will be produced . I have this idealist ic viewpoint : I am p repa red to wait. The wai t ing may mean that I die before I see the simpler proof, but still I 'm not in te res ted in anything that isn ' t going to be permanent . This is an a r i s tocra t ic viewpoint . I I t : May we move now to fivefold symmetry, Roger

Penrose, quasicrysta ls? You des igned the cover i l lustrat ion for Scientific American when Martin Gardner wrote about the Penrose tiling, which then b e c a m e an influential paper .

1 0 THE MATHEMATICAL INTELLIGENCER

JC : It 's funny that you have quoted Martin Gardner ' s say- his "pieces" from Gardner, and I re-proved some of the things ing that I had a lways been there before. I a t t end the Art that he had proved, but I didn' t know about the Penrose pat-

and Mathemat ics Conferences in Albany, organized by Nat tern. Gardner 's Scientific American article was largely Fr iedman, and the par t ic ipant l ist has informat ion abou t based on what I 'd done in Cambridge. I didn' t meet Roger the par t ic ipants ' f ields of interest . Somebody once said to again until a few years after the quasicrystals had been dis-

me that he admired what I 'd wr i t ten somewhere , and I covered. I still think the situation is ra ther funny: we still don' t a sked him what was it about , and he said, "Everything." know that the actual physical stuff is really behaving like the

Then it turned out that Nat hadn ' t ant ic ipated my re- Penrose pieces. To m y m i n d t h i s is annoying. It enables some sponding to his quest ion about my field of in teres t and he people to deny this possibility. Linus Pauling was a big hold- fi l led it out for me, including everything. This idea of being out, but in his case he just didn't unders tand what the new

in te res ted in everything is someth ing I a lmost consc ious ly configurations were. Certainly, it 's r idiculous to deny the pos-

t ry to be. sibility, because these things exist geometrically, why should- But wha t you are asking me abou t is a very s imple geo- n' t they exist physically. Those were interesting t imes for me.

met r ica l problem: Can you have some tiles tile the p lane II-I: Concerning the broadening in teres t in symmetry, you,

only aper iodical ly? I had a l ready been in teres ted in that as a mathemat ic ian , don ' t you feel some t imes that i t 's an p r ob l em and when Penrose came up with his solution, I be- infr ingement on your terr i tory that physicis ts , let a lone came t remendous ly exc i ted and s ta r t ed making the d a m n e d chemists and biologists , speak about symmet ry?

things and drawing them. I was s taying with Martin Gardner JC : No. I don ' t have any terri tory. If I 'm claiming for my one time, and I d rew out ra ther careful ly a small page full te r r i tory the ent i re world, I can ' t very well compla in if peo- of the tiles. Gardner had his own old-fashioned copying ma- ple t read on some of it. What I do feel in this r e spec t is this:

chine and we ran off a number of copies of this drawing The physic is ts and chemis ts have a t r emendous inves tment and p ieced them toge ther to p roduce a larger mosaic . Later, in all sor t s of things. Take, for example the names for these

when I was back in Cambridge, we pho tocop ied these groups. The crys ta l lographic point groups were enumer- smal le r and then made still larger ones, and so on. Martin a ted ages ago, the space groups were enumera ted in the took the initial vers ion I had made at his house in to the 1890s, and they 've got into the International Tables so peo-

Scientific American ple all over the wor ld

office, where the If I'm claiming for my territory the use the exist ing nota- graphics peop le re- t ions. There is no

did i t proper ly , a n d i t entire world, I can't very well corn- p r o s p e c t of changing became the cover, it to a rat ional sys-

I 've a lways felt plain if people tread on some of it. tem. If I p ropose a ra ther sad about our new sys tem of nam- dining room table. We had a ra the r nice dining room table ing, this means tha t I have to jus t t h row away that com- and we couldn ' t use it for about s ix months, and my wife munity because I can ' t get to them. I per fec t ly well under-

was fur ious with me because it was covered with thousands s tand the r easons and wouldn ' t even wan t to argue abou t of Penrose pieces, making a real ly beautiful pat tern, and I them, they ' re ju s t too invested in the sys tem as it is. never wan ted to dis turb it. I r e m e m b e r having d iscuss ions II-I: And it works.

about the poss ibi l i ty that chemica ls might crystal l ize in tha t JC: And it works, yes. But the poin t is, as a mathemat ic ian , sor t of manner , and I wish I had come out with that spec- my aims are different. I want to unde r s t and the thing. Let ulat ion in pr int because seven years la ter people found such me give you an example . There are these litt le shells in the

crystals , e lectronic s t ruc ture of the atom, the s, p, d, f shells, where I t I : Alan MacKay did come out with such a sugges t ion in s, p, d, f a re the initial le t ters of var ious words, which in-

pr in t pr ior to the exper imenta l d iscovery in 1982. dicate var ious p rope r t i e s of the spectra . But if you 'd s tar t JC : Martin Gardener ' s Scientific American art ic le ap- it ra t ional ly you 'd never use this sequence of letters. I would pea red in 1974, and we con jec tu red at that t ime about the s tar t calling them 1, 2, 3, or a, b, c. I don ' t want to be con- poss ib i l i ty of crystall ization, and I wish we had come out s t ra ined by having to agree to some preex is t ing usage, even

with it in print. I r e m e m b e r that I wonde red to mysel f how if I unders tand his tor ical ly how this usage came about. many different subs tances have been s tudied with r e spec t Let 's take the par t icu la r case of symmetry. The most re- to crystal l izat ion, and my guess was less than ten to the cent thing I 've done is a jo int work with severa l colleagues. seventh power . Then I thought, wha t is the probabi l i ty that We have comple te ly re -enmnera ted the 219 space groups

someth ing will crystal l ize in this manner , and one in ten to ab initio, and it t akes only ten pages. We were held up in the seventh p o w e r seemed a reasonab le guess; therefore, doing this by the feeling that we had to provide a dictio- such crystal l izat ion should happen, nary to the in ternat ional notation. Unders tanding the in- I t t : Did you ever d iscuss this with Roger Penrose? ternat ional no ta t ion for us was much more difficult than JC : No, I didn't . When I was a s tudent in Cambridge we unders tanding the groups. It actual ly held up the comple- got together; he and I were bo th in te res ted in puzzles. Then t ion of our p a p e r for ten years.

he wen t off to Oxford, and in the ear ly sevent ies I d idn ' t My aim is to under s t and something for me. I 'm less in- see him often. Soon, I didn ' t s eem him at all. I knew about t e res ted in publ icat ion. We're going to publ i sh this paper ,


of course , but I want to unde r s t and it myself. In doing that,

I can th row away the in ternat ional convention. I t 's a pity. Here I see this chemis t or physicist , and I can see he is talking abou t the same things, bu t I see him as l imited by having to accep t the baggage; he doesn ' t annoy me, rather, I

p i ty him. IH: Physics and chemis t ry are full of his tor ical notat ions. J C : And so is mathemat ics , but we ' re less re luc tant to give

up old notat ions in mathemat ics , s ince the whole aim of ma themat i c s is to get some kind of unders tanding of what ' s

going on. IH: You have in t roduced the t e rm gyration when speaking

abou t rotation. J C : That ' s a good example because gyration isn ' t j u s t a rotat ion. It 's really ra ther important . What is made c lear by the new way of thinking abou t things is that you should dis-

t inguish be tween rotat ions. Rotat ion means rotat ion, any rotat ion, but gyrat ion is a ro ta t ion when the axis of rota- t ion doesn ' t go through a mi r ro r line. We're ta lking abou t

a p lane pa t te rn or a pa t t e rn on a surface. A number of c rys ta l lographers have learned abou t the

new nota t ion but i t 's an uphil l struggle. My feeling is, in two

hund red years they' l l be thinking the correc t way. I 'm not saying that my nota t ion will be exact ly wha t it is, bu t even-

tual ly the baggage will be th rown away. This way of thinking about the groups is really Bill

Thurston 's idea. What actually happened was rather funny. We were discussing the 17 groups and I said, "Let me show you my way of thinking about it," and he said, "No, let me

show you my way of thinking about it." We agreed upon giving him ten minutes, when he explained his idea to me in ten minutes, I didn't bother to show him mine, and I have got quite a big ego. As soon as I saw his way of thinking about

things, I realized it was the correct way. Then I said, "We need a notat ion that conveys this way of thinking about things." I

set off for about two weeks to think what the notat ion should be, because to my mind notat ional matters are t remendously important . I finally designed the new system, which is very

s imple and which conveys Bill Thurston's philosophy. I haven ' t wri t ten it up very well, I 've only wr i t t en one

br ie f p a p e r about it, bu t that s i tuat ion is going to be changed soon. It 's a l ready on its way to becoming the stan-

da rd nota t ion for mathemat ic ians . There 's also a good chance that I can reach the so-cal led ar ty communi ty , that pa r t of the art communi ty tha t ' s in teres ted in mathemat ics . It ' l l t ake a long t ime to get th rough to the crys ta l lographers ,

the genuine chemis ts and phys ic is t s who have to use a lit- t le bi t of this stuff, and I don ' t see much poin t in trying, but we ' l l publ ish some papers . I have a young col league at pr ince ton , Daniel Huson, who is the pe r son who mos t

he lped to comple te the re -enumera t ion of space groups. He is a young man and he needs publ i shed pape r s to advance his career , to say what he has been doing for the last yea r

or so, so he 's very keen to get these things publ ished. The three-d imensional thing depends on the two-d imens iona l thing. We wrote the three-d imensional pape r knowing that we 'd have to wri te---paying a hos tage to for tune--- the two- d imens iona l paper . In the last few weeks before I left

Princeton, we wro te the two-dimens ional paper . We have a plan to wr i te a much longer paper , lavishly i l lustrated

with ar ty pic tures , and address it to a much wider community. I a lso have a plan to wri te a b o o k on these things.

That 's a real t roub le for me. So many of the things I do are e lementary that publishing papers is no t the right way to

do it. I wan t to r each a wider audience, I want to re-found some subjects . That demands wri t ing a book, but wri t ing a book is such a big hassle. There are about five books I ought

to wri te some time. IH: Does it bo the r you that physic is ts ta lk more about bro-

ken symmet r i es than symmetr ies? JC: It does wor ry me about the Universe. If we depend on

the breaking of symmetry, i t 's not as nice as it would be if symmetr ies were there. It does seem to be wha t the Universe does. I don ' t fault the phys ic is t s for talking about what ' s true. I under s t and that, and it h a ppe ns also on a very

e lementary level. In Aris totel ian phys ics there was a concept of "down," tha t was invariant. The direct ion down is

different f rom the direct ion up. If you ' re p repa red to jump, in o ther words , if you go to high enough energies, up becomes ra ther more similar to sideways. This is a very simple instance of symmetry breaking. If you really want to

travel as easily upwards, ra ther than horizontally, you need a t remendous amount of energy and to build yourself a rocket. This is a paradigm; this is something all over the

place. If you want to see the symmet ry be tween space and time, you have to travel at speeds close to the speed of light.

So I recognize that the symmetry breaking has actually happened. I del iberate ly chose some examples that are easier and pr ior to the examples worked out by the physicists.

IH: Would you ca re to tell us someth ing about your back-

ground? JC: I was bo rn in Liverpool in not a ter r ib ly well-off dis-

trict. My fa ther was a labora tory ass i s tan t who also did some minor teaching at the school whe re two of the Beat les went too. I was in teres ted in ma themat i c s from a very young age. My mo the r a lways used to say that she found

me reci t ing the power s of two when I was four. I t ended to be top or near ly top in mos t sub jec t s until I became an adolescent , when I went down and got in teres ted in o ther things. But s o m e h o w mathemat ics was a lways there. The

interes t in o the r sub jec t s was also a lways there, but I don ' t call mysel f a "somethingelsist ." When they couldn ' t t each me anything new at school, I dec ided to become a lightning calculator . That ' s a little hobby that I 'm gett ing back to now.

Tell me the da te when you were born.

IH: August 11, 1941. JC: OK. That was a Monday. Now, give me a three-digit

number.

IH: 999. JC: That ' s th ree t imes three t imes three t imes thirty-seven. IH: How did you develop this abil i ty? JC: I p rac t i ced it during the six months when I was still in

Liverpool af ter I 'd been accep ted to go to Cambridge as a s tudent on a scholarship. Then I wen t to Cambridge. I found it very hard because most of the s tuden ts were from ra ther posh homes, well off, had been to publ ic [i.e., private]


schools and I was a poor boy. However , I sort of gradual ly adap ted to the life. One thing that did happen was that there

were o ther people who were in te res ted in mathemat ics there, wha teve r their backgrounds . Then I got marr ied at quite an ear ly age and had four daughters by my first wife.

My pe r sona l life has been dec ided ly unhappy. I had two boys by my second wife. Over the break-up of my second marr iage I wen t suicidal and I a t t emp ted suicide and I was

in hospi ta l for a week af ter it. This was about five years ago. I t 's t aken me a long t ime to r ecover from that. Now

I 'm gett ing be t t e r and hoping to mar ry again when every-

thing can be sor ted out. IH: Did you become an ins ider in Cambridge socie ty? JC : My old college, Caius, in Cambr idge made me an hon- orary fel low las t year. That was very nice. But still I know

that I won ' t use this fact very much, because I still feel faintly uneasy; I don ' t feel that I be long in this par t icu la r

social grouping. IH: Is there an intel lectual social life in Pr inceton? JC : I 've a t t ended a few dinner par t ies in Pr inceton and a few pa r ty par t ies where things happen, and there is p lenty

of in te l lectual d iscuss ion going on there. I 've a lways lived in some intel lectual cen te r l ike this s ince I grew up. It 's n ice

when the newspaper s are saying something about some new discovery in as t ronomy, to be able to ask my neigh- bor who is a famous as t ronomer about it. Something I didn' t

know until very recent ly is that I 'm ra ther well known in Princeton.

I was trying to get Pr ince ton to buy these famous manuscr ip t s of Arch imedes that were up for sale

at Sotheby's . This involved going a round var ious depar tmen t s solicit- ing opinions. I saw a number of peo-

ple in the Classics Depar tment , Hellenic Studies, and others, and I found tha t a large number of these

peop le knew me or knew of me. I would 've expec t ed this of the mathemat ic ians or the physicists, maybe a few chemists , but to find that the c lass ic is ts know who I am was a surprise. So I am par t of the soc ie ty there a l though I still don ' t feel l ike a Princetonian.

When I became professor in Cambridge, and it means a lot more than it does in the States, I was ra ther hoping that s o m e b o d y would approach me and say something like, "Excuse me, Professor," and peop le would look a round to

see who this god-like figure was. It never happened. The s tudents wen t on jus t calling me "you" or "John," and that was that. But when I went to Princeton, people s ta r ted calling me Professor and the s tudents did, and then I found it

ra ther annoying because it d i s tanced them. One of the secre tar ies got it absolute ly beaut i ful ly right. If I came in by mysel f in the morning, she said, "Hi, John." If I came in with someone else, she said, "Good Morning, Professor

Conway." I 've also changed my appea rance a bit. My hair used to

be longer and my bea rd used to be longer. After I got my hai rcut I wen t into the local ice c ream shop next door and

I don't see why there should be this consis- tency in an abstract world that I don't really believe exists.

the girl said, "Oh, you look a lot younger." The secre ta r ies

in the depa r tmen t sa id the same thing. IH: So you care wha t o ther people say.

JC : I a lways thought that I didn ' t care abou t appearance , and I didn ' t care until recently, but I am getting a bit wor- r ied about gett ing old.

IH: You have said tha t you no longer had ambit ion. Aren ' t you looking fo rward to something?

JC : I don ' t th ink I am. What 's there at the end, death, and I don ' t l ike that very much. I am thinking about how much

t ime is there to go. I don ' t want to grow old. I don ' t feel old in my mind. On the o ther hand, I see myse l f behaving in var ious ways I wou ldn ' t have behaved when I was twenty.

Growing old is a bi t upsett ing. This is one of the reasons I 'm taking up this l ightning calculat ion again. I envisage myself in twenty years t ime hobbling in wi th a stick, sitting

down painfully. In an academic environment you ' re always sur rounded by young, very bright people, and I envisage one

of them looking over at this old fool, saying, "Oh, yes, he did some interest ing stuff once." But now he ment ions the date he was born and I instantly say it was a Fr iday and I

do this even though my physical frame is so fragile, and he thinks, "There mus t be something in there still working."

IH: Are you vain? JC: Very. I would like to think that I don ' t care wha t o ther people think, but i t 's not true, as the hai rcut t ing episode

showed. I do care wha t people think. But I don ' t care very much. The convent ions about the ways you act or d ress don ' t impress me at all. I jus t p re fe r to be comfort-

able. I 'm p r e p a r e d to go to some length to defend this. I somet imes del ibera te ly think, "What would

Conway do here?" and then do it. By behaving in some unexpec ted way you give yourse l f the right to

behave in an une xpe c t ed way. That ' s very, very nice. Here ' s a little thing I remember . I at- t empted suicide and, in fact, I usual ly th ink to myse l f I com-

mi t ted suicide but d idn ' t quite succeed; I woke up in a hospi ta l and then I was very glad. But then came the p rob lem of coming back to life. I was ra ther worr ied, I d idn ' t want people whisper ing behind my back. I thought, "What would Conway do here?" Conway would make it per fec t ly obvi-

ous that he knew. So I bo r rowed from Neff Sloane a T-shirt he had, which said "SUICIDE" in very large le t ters and then "Rock" undernea th it. It indica ted that he had c l imbed this

rock "Suicide," he is a v e r y keen rock cl imber, and "Suicide" is the second mos t difficult rock to cl imb in the United States. So I re -emerged in socie ty af ter m y suic ide a t tempt and went a round for three days in this shirt . I jus t met the p rob lem head-on, splash. That was a case when I actual ly thought about it because it would 've been painful for me to ignore this problem. I also r e m e m b e r when star t ing lec-

ture courses at var ious t imes, I 've felt, "What can we do to jus t shake these s tudents?" So I 'd jus t bur s t into the room with a big sc ream or a jump. It 's the same sor t of thing.

VOLUME 23, NUMBER 2, 2001 1 3

IH: You obviously do care. J C : I real ly do love my subject , and that includes the teach-

ing of it. I t 's not jus t developing the subject, hut the teaching is as important . I of ten say that I cons ider myse l f a t e ache r more than a mathemat ic ian . I spend a lot of t ime

thinking how to teach, I real ly do. Are we done?

IH: Is there a message? J C : There is something. I t 's how I feel about ma themat ica l

discovery. You're wander ing up and down, it 's l ike wander ing in a s t range town with beautiful things. You turn a round this c o m e r and you don ' t know whe the r to go left

o r right. You do something or o ther and then, suddenly, you

h a p p e n to go the right way, and now you are on the Palace steps. You see a beautiful bui lding ahead of you, and you d idn ' t know that the Palace was even there. There ' s a cer- ta in wonderful p leasure you get on discover ing a mathe-

mat ica l structure. It h a p p e n e d to me t remendous ly when I d i scovered the surreal numbers . I had no idea tha t I was going to go in there at all. I had no idea of wha t I was do-

ing. I thought I was s tudying games, and suddenly I found this t r emendous infinite wor ld of numbers . It had a beautiful s imple structure, and I was jus t lost in admira t ion of

it, and in a kind of s econda ry admira t ion of mysel f for having found it. For about s ix weeks I jus t wande red abou t in a pe rmanen t daydream. What happens af ter tha t is tha t I 'm

vainly trying to re-create that in the people I 'm trying to ta lk about it, trying to show what this wonderfu l thing is l ike and how amazing it is--- that you can reach it by study-

ing something else. I 'm perennia l ly fasc inated by mathematics , by how we can app rehend this amazing wor ld that appea r s to be there, this mathemat ica l world. How it comes

abou t is not really phys ica l anyway, it 's not l ike these con- c re te buildings or the trees. No mathemat ic ian be l ieves that the mathemat ica l wor ld is invented. We all be l ieve i t 's dis-

covered. That implies a cer ta in Platonism, implies a feeling tha t there is an ideal world. I don ' t real ly bel ieve that. I don ' t unders tand anything. I t 's a perennia l p rob l em to unde r s t and what it can be, this mathemat ica l wor ld we ' re

studying. We're s tudying it for years and years and years, and I have no idea. But i t 's an amazing fact that I can sit he re wi thout any expens ive equipment and find a world.

I t 's rich, it 's got unexpec t ed proper t ies , you don ' t know wha t you ' re going to find.

I can ' t comprehend how this can be. I don ' t know what it means . I don ' t know whe the r there is such an abs t rac t

world, and I tend not to bel ieve there is and to bel ieve that we are fooling ourselves.

We used to think that the ear th is flat and it was incon- ce ivable that it could be round. It was only some very

painful facts that eventual ly forced us to bel ieve that the ea r th is roughly spherical . What 's happened cont inual ly in the phys ica l sc iences is tha t the t ruth was not one of the poss ibi l i t ies that was cons ide red and then rejected, not even that. It was one of the possibi l i t ies that couldn ' t even

be cons ide red because it was so obviously impossible .

In ma themat i c s our deve lopment has come a little bi t

later, but the same sort of thing h a p p e n e d with G6del 's the- o rem and so on. What we thought was the truth was jus t a kind of approx ima t ion of the truth. Newtonian dynamics is

an approx ima t ion of relativist ic dynamics , and it 's not literally t rue if you go to high speeds and high energies; if you

go to smal l d i s tances it doesn ' t quite w o r k either, according to the quantum theory. In ma themat i c s we have these

beliefs that there are infinitely many integers and so on. Any bel ief l ike tha t about the na ture of things arbi t rar i ly far away has tu rned out to be false in physics. I think it is

false in ma themat i c s too. I think that eventual ly we'l l f ind

something wrong with the integers and then the class ical integers will be ju s t an approximat ion . That ' s a big puzzle for me. I don ' t quite bel ieve in this art if icial mathemat ica l

world. There appea r s to be a wonderfu l consis tency about it, which means that I can think of someth ing in some way and someone else can think about it in a different way, and we both come to the same conclusion. If we don't, there

must be a m i s t a k e - - a t least it has been so, so far. But I don ' t see why there should be this cons i s tency in a wor ld

that I don ' t real ly believe exists. So to me it 's a sort of fairy tale, and fairy ta les don ' t have to be cons i s ten t because they are human crea t ions ultimately. But this mathemat ica l

wor ld is cons i s t en t and I wonder , "What the hell is it?" without implying anything supernatural . I 'm non-religious.


llLVAl~"1|l[=-]i+nl+=-It|[,-]p:lll::ITi(,~li'r':+llili,[~liln'l,-1 A l e x a n d e r S h e n , E d i t o r ]

This column is devoted to mathematics

for fun. 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

from readers--either new notes, or

replies to past columns.

Constant-Sum Figures Li C. Tien

C ons ider two squares (shown in

gray) inside an isosceles tr iangle (Fig. 1):

(Fig. 3) the sum of squares of their s ides is constant .

Figure 3. 0 2 -~- b 2 = const,

To see why it is the case, let us d raw two equal right t r iangles of s ides a and b (Fig. 4). This is poss ib le s ince the bot-

tom sides of the squares form a seg- men t of length a + b that can be di-

v ided into segments of length b and a.

Please send all submissions to the

Mathematical Entertainments Editor,

A l e x a n d e r Shen, Institute for Problems of

Information Transmission, Ermolovoi 19,

K-51 Moscow GSP-4, 101447 Russia

e-mail:[email protected]

Figure 1. o + b = const.

When one of the squares becomes big- ger, the o ther becomes smaller. It is easy to verify that the sum of their

s ides (a + b) is a constant . The same is t rue for two tangent

semici rc les inside an isosceles trian-

gle:

Figure 2. dl + o+2 = const.

Here the sum of d iamete rs (dl + d2) remains the same. (Note that the centers

of the circles lie on the tr iangle 's base.) This is also an easy exercise.

A bit more difficult is to show that for two squares inside a semicirc le

Figure 4. Applying Pythagoras theorem:

0 2 -(- b 2 = r 2.

The hypoteneuses of the two tr iangles are equal to ~ + b 2, and they are or-

thogonal. Therefore the c o m m o n ver- t ex of these t r iangles is the cen te r of the circle and its rad ius is V ~ a 2 + b 2.

Now we come to our last and prob-

ably most in teres t ing example . (It is apparent ly new in the l i terature.) Here two small semic i rc les a re of equal size and their centers lie on the d iameter of

the big semicircle. The gray circles fit tightly. It turns out that dl + d2 = const.

Figure 5. dl + d2 = const.

�9 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 1 5

Why? There is a nice proof of this fact.

Let r be the radius of the small semi-

circles and R be the radius of the big

semicircle. These radii are fLxed, while

radii r l = dl/2 and r2 = d2/2 change (Fig. 5). I claim that r + r l + r2 = R in

general. To see why, let us draw this picture

"backwards." Take a fresh drawing pad

and construct three equal triangles

whose sides are a = r + rl, b = r l § r2,

and c = r + r2 (Fig. 6).

Then draw circles of radii r, rb r2

and again r, centered at the vertices of

these triangles, except for the common

central vertex (Fig. 7).

It is easy to check that the radii and

sides are chosen in such a way that

Figure 7. Three equal triangles and four circles.

since the "hole" determines uniquely

the size of the circle that fits into it.

The R = r + rt + r2 relat ion is apparent ly new. In the Mathematical Gazette, from 1937 to 1967 there were

several notes about the "curious rec-

tangle," involving the special case r :

r l : r2 = 3 : 2 : 1 of the relation. (The

centers of these three circles and the

center of the enclosing circle of radius

6 form a rectangle.) In 1986, Mathemat- ics in School discussed similarly formed

rectangles. I haven't found the general

case in the literature. Details and gen-

eralizations of constant-sum figures will

probably be published elsewhere.

Acknowledgment: I would like to

thank Mr. Martin Gardner for his en-

couragement.

Li C. Tien

4412 Huron Drive

Midland, MI 48642 USA

e-mail: [email protected]

Figure 6. Three equal triangles.

these four circles touch each other (ex-

cept for the two of radius r). Moreover,

they touch the circle of radius R = r +

r l + r2 centered in the central vertex

(the distances between centers are ex-

actly as they should be), and we come

to the picture we started with (Fig. 5).

The careful reader will complain

that I haven' t proved my claim. I have

proved that if r + r l + r2 = R, then the

configuration with touching circles is

possible, but not vice versa. The fol-

lowing argument completes the proof.

Imagine that for touching circles we

have r + r l + r2 r R. Then we change

(say,) r l and find some r~ such that r +

r~ + r2 = R. We have proved that there

is a configuration formed by circles of

radii r, r~', r2 and r inside a circle of ra-

dius R. Now we have two configura-

t ions where all the circles are the same

except for one, and this is impossible


DAVID W. HENDERSON AND DAINA TAIMINA

Crocheting thc I lyperbolic Planc

For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, m a y take up all your t ime and deprive you of your health, peace of m ind and happiness in life.

- -Wol fgang Bolyai urging his son Jfinos Bolyai to give up work on hyperbol ic geomet ry

In June of 1997, Daina was in a workshop watching the leader

of the workshop, David, helping the part ic ipants s tudy ideas of hyperbol ic geometry using a paper-and-tape surface in much the same way that one can s tudy ideas of spherical

geometry by using the surface of a physical ball. David's hyperbol ic p lane was then so ta t tered and fragile that he was afraid to handle it much. Daina immediate ly began to think:

"There mus t be some way to make a durable model." David made his first pape r hyperbol ic plane in the sum-

mer of 1978, while on canoe tr ip on the lakes of Maine, us-

ing the sc issors on his Swiss Army knife. He had jus t l ea rned how to do the cons t ruc t ion from William Thurs ton at a workshop at Bates College. This crude pape r surface

was used in David's geomet ry c lasses and workshops (becoming more and more ta t te red) until 1986, when some high school t eachers in a s u m m e r p rogram that David was leading co l labora ted on a new, larger paper-and- tape hy-

perbol ic surface. This second paper -and- tape hyperbol ic surface (used in c lasses and w o r k s h o p s for the next 11 years ) was the one that Daina wi tnessed in use.

Daina exper imented with knit t ing (but the resul t was no t rigid enough) and then se t t led on crocheting. She pe r fec ted

her technique during the workshop and c roche ted her first small hyperbol ic plane; then, while camping in the fores ts of Pennsylvania, she c roche ted more, and we s ta r ted ex- plor ing its uses. In this p a p e r we share how to c roche t a

hyperbol ic p lane (and make re la ted pape r versions). We also share how we have used it to increase our own unders tanding of hyperbol ic geometry. (What are horocycles? Where does the area formula qrr 2 fit in hyperbol ic geome-

try?) We will also prove that the intr insic geomet ry of these

surfaces is, in fact, (an approximat ion of) hyperbol ic geometry.

But, Wait! you say. Do not many b o o k s s ta te that it is

impossible to e m b e d the hyperbol ic p lane isometr ica l ly (an isometry is a funct ion that p rese rves all d is tances) as a comple te subse t of the Eucl idean 3-space? Yes, they do: For popular ly wr i t ten examples , see Rober t Osserman 's

Poetry of the Universe [9], page 158, and David Hilbert and

S. Cohn-Vossen's Geometry and the Imaginat ion [6], page 243. Fo r a de ta i led discuss ion and proof, see Spivak's A Comprehensive Introduction to Differential Geometry [10], Vol. III, pages 373 and 381.

All of the re fe rences are implici t ly assuming surfaces

embedded with some condi t ions of differentiabil i ty, and refer ( implici t ly or explici t ly) to a 1901 theo rem by David Hilbert. Hilbert p roved [5] that there is no real analytic isometr ic embedding of the hyperbol ic p lane onto a comple te

subse t of 3-space, and his a rguments also work to show that there is no i somet r ic embedding whose derivatives up to o rder four are continuous. Moreover, in 1964, N. V. Efimov ([2] Russian; d iscussed in English in Tilla Milnor 's [8]) ex tended Hilber t ' s resul t by proving that there is no

i sometr ic embedding defined by funct ions whose first and second derivat ives are continuous. However , in 1955, N. Kuiper p roved [7] that there is an i somet r ic embedding with

cont inuous first der ivat ives of the hyperbol ic p lane onto a c losed subse t of 3-space. For a more de ta i led discuss ion of these ideas, see Thurs ton [11], pages 51-52. The finite surfaces descr ibed here can apparent ly be ex tended indefinitely, but they appea r always not to be different iably em-

bedded (see Figure 12).

�9 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 1 7

C o n s t r u c t i o n s of Hyperbo l i c P l a n e s

We descr ibe several different i sometr ic cons t ruc t ions of the hyperbol ic p lane (or approx imat ions of the hyperbol ic

p lane) as surfaces in 3-space.

The hyperbolic plane from paper annuli

This is the paper-and-tape surface that David learned from William Thurston. It may be constructed as follows: Cut out many identical annular strips of radius p, as in Figure 1. (An

annulus is the region be tween two concentric circles, and

we call an annular strip a por t ion of an annulus cut off by an angle from the center of the circles.) Attach the str ips together by attaching the inner circle of one to the outer circle

of the other or the straight ends together. (When the straight ends of annular strips are a t tached together you get annular str ips with increasing angles, and eventually the angle will be more than 2~r.) The resulting surface is of course only an ap-

proximat ion of the desired surface. The actual annular hyperbol ic plane is obtained by letting 8--~ 0 while holding p fixed. (We show below, in several ways, that this limit ex-

ists.) Note that because the surface is constructed the same everywhere (as 8--~ 0), it is homogeneous (that is, intrinsically and geometrically, every point has a neighborhood that

is isometr ic to a neighborhood of any other point). We will call the results of this construct ion the annular hyperbolic p/ane. We urge the reader to try this by cutting out a few iden- t ical annular strips and taping them together as in Figure 1.

How to crochet the annular hyperbolic plane

If you t r ied to make your annular hyperbol ic p lane f rom p a p e r annuli you cer ta inly real ized that it t akes a lot of time. Also, la ter you will have to p lay with it carefully because

it is fragile and tears and c reases ea s i l y - -you may wan t ju s t to leave it sitt ing on your desk. But there is a way to get a s tu rdy model of the hyperbol ic p lane which you can work and p lay with as much as you wish. This is the c roche ted

hyperbol ic plane. To make the c roche ted hyperbol ic plane, you need jus t

a few very basic c rochet ing skills. All you need to know is

Figure 1. Annular strips for making an annular hyperbolic plane.

Figure 2. Crochet stitches for the hyperbolic plane.

how to make a chain (to s tar t) and how to single crochet . See Figure 2 for a pic ture of these s t i tches, which will be

descr ibed fur ther in the next paragraph. Choose a yarn that will not s t re tch a lot. Every yarn will

s t re tch a little, bu t you need one that will keep its shape. That 's it! Now you are ready to s tar t the sti tches:

1. Make your b e g i n n i n g c h a i n s t i t c h e s (Figure 2a). (Topologis ts may recognize that as the s t i tches in the Fox-Art in wild arc!) About 20 chain s t i tches for the be-

ginning will be enough. 2. For the f i r s t s t i t c h in e a c h row, inser t the hook into

the 2nd chain from the hook. Take yarn over and pull through chain, leaving 2 loops on hook. Take yarn over

and pull th rough both loops. One single c rochet s t i tch has been comple ted . (Figure 2b.)

3. Fo r t h e n e x t N stitches, p roceed exac t ly like the first stitch, excep t inser t the hook into the next chain (in-

s tead of the 2nd). 4. F o r t h e ( N + 1 ) s t s t i t ch , p roceed as before, except in-

ser t the h o o k into the same loop as the Nth stitch. 5. R e p e a t S t e p s 3 a n d 4 until you reach the end of the

r o w .

6. A t t h e e n d o f t h e row, before going to the next row, do one ex t r a chain stitch.

7. W h e n y o u h a v e t h e m o d e l a s b i g a s y o u wan t , you

can stop, j u s t by pulling the yarn th rough the last loop.

Be sure to c roche t fairly tight and even. That ' s all you need

from c roche t basics. Now you can make your hyperbol ic plane. You have to increase (by the above procedure) the number of s t i tches from one row to the nex t in a cons tan t ratio, N to N + 1 - - t h e rat io de te rmines the radius of the hyperbol ic p lane (cor responding to p in the former construction). You can exper iment wi th different ratios, but not in the same model . You will get a hyperbol ic p lane only if you increase the number of s t i tches in the same rat io all

the time. Crochet ing will take some time, bu t la ter you can work

with this mode l wi thout worrying abou t des t roying it. The

comple ted p roduc t is depic ted in Figure 3.

A polyhedral annular hyperbolic plane

A polyhedra l vers ion of the annular hyperbol ic p lane can be cons t ruc ted out of equilateral t r iangles by put t ing 6 tri-

angles toge ther at half the ver t ices and 7 t r iangles toge ther


Figure 3. A crocheted annular hyperbolic plane.

at the others. (If we were to put 6 t r iangles together at every vertex, then we would get the Eucl idean plane.) The precise cons t ruc t ion can be desc r ibed in three different (but,

in the end, equivalent) ways:

1. Cons t ruc t po lyhedra l annuli as in Figure 4, and then tape

them toge ther as with the annular hyperbol ic plane. 2. You can cons t ruc t two annuli at a t ime by using the

shape in Figure 5 and taping one to the next by joining:

a----> A, b----> B, c--> C.

3. The quickest way is to start with many strips, as p ic tured in Figure 6a. These strips can be as long as you wish. Then join four of the strips together as in Figure 6b using 5 ad-

dit ional triangles. Next, add another strip every place there is a ver tex with 5 triangles and a gap (as at the marked vert ices in Figure 6b). Every t ime a strip is added,

an addit ional vertex with 7 tr iangles is formed.

The cen te r of each strip runs pe rpend icu la r to

each annulus, and you can show that each of these curves (the cen te r lines of the s tr ip) is geodesic because they all have global ref lec t ion symmetry. This mode l has the advantage of being con-

s t ruct ib le more easily than the two mode l s above; however , one cannot make be t t e r and be t te r ap- p rox ima t ions by decreas ing the size of the triangles. This is t rue because at each sevenfold ver tex the cone angle is (7 x 60 ~ = 420 ~ no mat te r wha t

the size of the triangles, and the radius of the polyhedra l annulus will decrease because it is about 1-1

2 t imes the s ide length of the t r iangles (see Figure 4), whe reas the hyperbol ic p lane local ly looks like

the Eucl idean plane (360~

Figure 4. Polyhedral annulus.

Figure 5. Shape to make two annuli.

Figure 6a. Strips.

Figure 6b. Forming the polyhedral annular hyperbolic plane.


The hyperbol ic soccer ball

Polyhedral models of the hyperbolic plane can also be con- s tructed from equilateral triangles by putting 7 triangles at every vertex (the {3,7} model) or, dually, by putting 3 regular heptagons (7-gons) together at every vertex (the {7,3} model). These are difficult to use in practice because they are "pointy" with cone angles at the vertices of 420 ~ or 385.7. . .~ In addition, their radii are small (about the length of a side), and it is not convenient to describe the annuli and related coordinates.

Since the fwst version of this paper was written, Keith Henderson, David's son, showed us a better polyhedral model, which he named the hyperbolic soccer ball. The hyperbolic soccer ball construction is related to the {3,7} model in the sense that if a neighborhood of each vertex in the {3,7} model is replaced by a heptagon (7-sided form), then the remaining portion of each triangle is a hexagon. If you use regular heptagons and regular hexagons, then each heptagon is surrounded by seven hexagons; and two hexagons and one heptagon come together around each vertex (see Figure 7). This is the hyperbolic soccer ball. An ordinary soccer ball (outside the USA, called a "football") is constructed by using pentagons surrounded by five hexagons; and (especially if made from leather that stretches a little) is a good polyhedral approximation of the sphere. The plane can be tiled by hexagons, each surrounded by six other hexagons.

Because a heptagon has interior angles with 5~-/7 radi- ans (= 128.57. . .~ the vertices of this construct ion have cone angles of 3 6 8 . 5 7 . . . o and thus are much smoother than the {3,7} and {7,3} polyhedral constructions. The fin- ished product has a nice appearance if you make the heptagons a different color f rom the hexagons. As with any polyhedral construction, it is not possible to get closer and closer approximations to the hyperbolic plane by changing the size of the hexagons and heptagons, and again there is no convenient way to see the annuli.

The hyperbolic soccer ball also has a radius p that is large enough to be used conveniently. To calculate the radius, we first tile the hyperbolic soccer ball by congruent triangles

(see the triangle marked in Figure 7), which each contain a vertex of the hyperbolic soccer ball, where the curvature of the hyperbolic soccer ball is concentrated. We can then use the fact (which we prove at the end of this paper) that in the hyperbolic plane the area of a triangle is given by

where the ~i are the interior angles of the triangle. The triangles in the tiling have angles (Ir/3, ~-/3, 2~r/7), and their areas can be easily calculated (using ordinary geometry) to be (1 .3851. . . ) s 2, where s is the length of the sides of hexagons and heptagons. From this we calculate that the radius of the hyperbolic soccer ball is p = (3 .042 . . . ) s . For comparison, the radius of a spherical soccer ball is (2.404. . . )s , which can be calculated in a similar way.

Hyperbolic planes of different radii (curvature) Note that the construct ion of an annular hyperbolic plane is dependent on p (the radius of the annuli), which can be called the radius of the hyperbolic plane. As in the case of spheres, we get different hyperbolic planes depending on the value of p. In Figure 8 a, b, and c there are crocheted hyperbolic planes with radii approximately 4 cm, 8 cm, and 16 cm. These photos were all taken from approximately the same perspective, and in each picture there is a centime- ter rule to indicate the scale.

Note that as p increases the hyperbolic plane becomes flatter and flatter (has less and less curvature). For both the sphere and the hyperbolic plane, as p goes to infinity they become indistinguishable from the ordinary flat (Euclidean) plane. We will show below that the Gaussian curvature of the hyperbolic plane is - 1 / p 2. So it makes sense to call this p the radius of the hyperbolic plane, in agreement with spheres, where a sphere of radius p has Gaussian curvature 1/p 2.

How Do We Know that We Obtain the Hyperbolic Plane? Why is it that the intrinsic geometry of an annular hyperbolic plane is a hyperbolic plane? The answer, of course, depends on what is meant by "hyperbolic plane." There are

Figure 7. The hyperbolic soccer ball. Figure 8a. Hyperbolic plane with p == 4 cm.


Figure 8b. Hyperbolic plane with p ~= 8 cm.

four main ways of descr ibing the hyperbol ic plane; we hope

one of these is your favorite:

1. A hyperbol ic p lane sat isf ies all the pos tu la tes of

Eucl idean geomet ry except for Euclid's Fifth (o r

Parallel) Postulate. 2. A hyperbol ic p lane has the same local (intrinsic) geom-

etry as the pseudosphere. 3. A hyperbol ic p lane is a s imply connec ted comple te

Riemannian manifold with constant negative Gaussian curvature.

4. A hyperbol ic p lane is desc r ibed by the upper halj:plane model.

The i tal icized te rms will be exp la ined as we deal with each descr ip t ion in the sec t ions tha t follow. But first we cons ider na tura l coord ina tes that we will find useful.

In t r ins ic g e o d e s i c c o o r d i n a t e s

Let p be the f ixed inner radius of the annuli, and let H~ be the approx imat ion of the annular hyperbol ic p lane con-

s tructed, as above, from annuli of rad ius p and th ickness 8. On H~ p ick the inner curve of any annulus, calling it the base curve, p ick a posi t ive di rect ion on this curve, and p ick any po in t on this curve and call it the origin O. We can now cons t ruc t an (intrinsic) coord ina te sys tem x~ : R 2 ---) H~ by

defining x~(0, 0) = O, x~(w, 0) to be the point on the base curve at a d is tance w from O, and x~(w, s) to be the po in t at a d i s tance s from x~(w, 0) a long the radial (along the

radii of each annulus) curve through xs(w, 0), where the posi t ive d i rec t ion is chosen to be in the direct ion f rom oute r to inner curve of each annulus (see Figure 9). The r eade r can easi ly check that this coord ina te map is one-to-one and onto (if you were to c roche t indefinitely). Let x = lim x8 : R 2 ~ H 2, the annular hyperbol ic plane.

~ ~ that each coord ina te map xs induces a metric, ds, on R 2 by defining d~(p, q) to be the (intrinsic) d is tance be-

tween x~(p) and xs(q) in H~. Those readers who desire a more formal descr ip t ion of the l imit can check that, in the

limit as 8 ---> 0, the metr ics ds converge to a metr ic d on R 2, and this def ines the annular hyperbol ic plane a s R 2 with a special metric. In fact, this p roces s also defines a

Figure 8c. Hyperbolic plane with p ~= 16 cm.

Riemannian metric, but this will be eas ie r to see after we show the connec t ions with the upper half-plane model.

W h a t can w e e x p e r i e n c e a b o u t h y p e r b o l i c

g e o d e s i c s and i s o m e t r i e s ?

The following facts were observed by our s tudents during one class per iod in which, working in small groups, they explored for the first t ime the c roche ted hyperbol ic plane.

The radia l c u r v e s are g e o d e s i c s w i t h r e f l e c t i o n symme t ry . The radial curves (curves that run radial ly across each annulus) have intrinsic reflect ion symmet ry in each H~

because of the symmet ry in each annulus and the fact that the radial curves in tersec t the bounding curves at right angles. These ref lect ion symmetr ies car ry over in the limit to the annular hyperbol ic plane. Such bi la teral symmet ry is the

basis of our intuitive not ion of s t ra ightness (see Chapters 1 of references [3] and [4] for more details), and thus we can conclude that these radial curves are geodes ics (intrinsically straight curves) on the annular hyperbol ic p lane and that

reflect ion through these curves is an isometry.

The radia l g e o d e s i c s are a s y m p t o t i c . Looking at our hyperbol ic surfaces, we see the radial geodes ics gett ing c loser and c loser in one d i rec t ion and diverging in the other direction. In fact, let A and fr be two of the radial geodes ics

in H~. The dis tance be tween these radial geodes ics changes by p/(p + 8) every t ime they cross one annulus. (Remember, the annuli all have the same radii.) If we c ross n strips, then

the d is tance in Hs be tween A and tz at a d i s tance c = n6 from the base curve is:

( p )n I p \ c / ~ ~ _ ~ _ _

Now take the l imit as 8---) 0 to show that the d is tance be-

tween A and tz on the annular hyperbol ic p lane is:

d exp(-c/p). (1)

Asymptot ic geodes ics never happen on a Eucl idean plane

or on a sphere.


Figure 9. Geodesic coordinates on an annular hyperbolic plane.

T h e r e is an i s o m e t r y t h a t p r e s e r v e s the annul i . Because reflections through radial geodesics are isometr ies that pre- serve each annulus, the composi t ion of two such reflect ions

mus t also be an isometry that preserves each annulus. A brief considerat ion of what happens on a given annulus should convince us that this i sometry shifts the annulus along itself.

In the plane we would call such an isometry a rota t ion (about the center of the annulus). But, on the annular hyperbol ic plane, an annulus has no center and the isometry has no fLxed point because the radial geodesics (which are perpen-

dicular to the annulus) do not intersect. Also, we do not want to call this isometry a translation because there is no geo-

des ic that is preserved by the isometry. So, this is a type of i sometry that we have not met before on the plane. Such isometr ies are tradit ionally called horolations, and annular curves are tradit ionally cal led horocycles. Horolat ions can be thought of as rotat ions about a point at infmity (since the

radial geodesics are asymptot ic) , and the horocycles can be thought of as circles with infinite radius.

O t h e r g e o d e s i c s can b e f o u n d in a p p r o x i m a t e in tu-

i t i v e w a y s .

�9 Hold two points of the hyperbol ic surface be tween the

index finger and thumb on your two hands. Now pull gen- t l y - - a geodesic segment (with its ref lect ion symmetry) should appear be tween the two points. This is using the p rope r ty that a geodes ic is locally the shor tes t path.

�9 Fold the surface to a c rease with bi la teral symmetry . �9 You can lay a (s traight) r ibbon on the surface and it will

fol low a geodesic. This Ribbon Test for geodes ics on surfaces is d iscussed fur ther (with proofs) in re fe rence [3],

P rob lems 3.4 and 7.6.

The fol lowing proper t ies of geodes ics can be easily ex-

pe r i enced by playing with the annular hyperbol ic plane. These p rope r t i e s can be r igorously conf i rmed later by us-

ing the upper half-plane model.

G1. Every pair of points is joined by a unique geodesic.

G2. Two geodesics intersect no more than once.

G3. Every geodesic segment has a geodesic perpendicular bisector.

G4. Every angle (between two geodesics) has a geodesic angle bisector.

G5. Each non-radial geodesic is tangent to one annulus, and then, as you travel in both directions f rom that point, the geodesic approaches being perpendicular to the annuli that it crosses on the way to infinity.

Connect ions to Euclid's postulates

Euclid 's five pos tu la tes in modern wording are:

P1. A (unique) straight line may be drawn f rom any point to any other point.

P2. Every limited straight line can be extended indefinitely to a (unique) straight line.

P3. A circle may be drawn with any center and any radius.

P4. All right angles are equal.

PS. I f a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines ( i f extended indefinitely) will meet on that side on which the angles are less than two right angles.


face in perspective

cross-section of surface P P

Figure 10. Hyperbolic surface of revolution.

F

Figure 11. Relating R(z), p, 5z, and AR.

We will have to wai t for the analy t ic p o w e r of the up- p e r hal f -plane mode l to conf i rm r igorous ly these p roper -

t ies for the annular hyperbo l i c p lane , but we can give intui t ive a rgumen t s now. It is ea sy to convince yourse l f tha t the first t h ree pos tu l a t e s a re t rue by playing wi th the annu la r hyperbo l i c plane, bu t the o the r two take some m o r e

thought . All r ight angles are equal. What does this pos tu la te

mean? How is it poss ib le to imagine right angles that a re not equal? To see this we mus t look at Euclid 's defini t ion

of "right angle":

When a s t ra ight l ine in tersects another s t ra ight l ine

such that the adjacent angles are equal to one another,

then the equal angles are called r ight angles.

By this defini t ion, the r ight angles at a ve r t ex of a poly- h e d r o n are less than 90 ~ and thus any po lyhedron can no t

sa t i s fy Eucl id ' s Four th Postu la te . To show tha t the annular hyperbo l i c p lane sat is f ies this pos tu la te , cons ide r a r ight angle a at the po in t P def ined by the l ines 1p and mR

and ano the r r ight angle/3 at Q def ined by IQ and mv. Then, by ref lec t ing R in the p e r p e n d i c u l a r b i sec to r (see G3) of the l ine s egmen t PQ (see G1), the po in t P is t aken to the po in t Q; one or two more re f lec t ions through the b isec- to r s (see G4) of the angles def ined by the s ides of R ( a )

and fl will eventua l ly br ing the re f lec ted image of a into

co inc idence with ft. Eucl id 's F i f th Postulate. Consider two radial geodes ics

in te rsec ted by the geodesic 1 de te rmined by in tersec t ions

of these radial geodes ics with a given annulus curve. The radial geodes ics do not intersect , even though it is c lear that they make angles on the same side of I that are each less than a right angle. Thus Eucl id 's Fifth Postu la te does

not hold on the annular hyperbol ic plane. In many t rea tments of ax iomat ic geometry, Eucl id ' s

Fifth Pos tu la te is rep laced by

(Playfair's) Parallel Postulate: Given a l ine I and a

po in t P not on l, there is a un ique l ine through P that

is parallel to I.

Since any two geodes ics (great c i rc les) on a sphere intersect , it is c lear that Euclid 's Fifth Pos tu la te is true on a sphere while Playfair ' s Pos tu la te is not true, cont rary to the

s ta tements in many b o o k s that the two are equivalent. The correct s t a tement is that they are equivalent i n the pres-

ence o f all the other postulates.

Connect ion to the pseudosphere

Take the annulus whose inner edge is the base curve and

embed it isometr ical ly in the x - y plane as a complete annulus with center at the origin. Now at tach to this annulus por- t ions of the other annuli as indicated in Figure 10. Note that

the second and subsequent annuli form t runcated cones. Let the ver t ical axis be the z-axis; then at each z we have

the pic ture in Figure 11.

Thus

AR _ - R ( z )

Az X/(p + ~)2 _ R(z)2 .

In the limit as 6 (and AR and hz) go to zero, we get

dR - R ( z ) dz ~flp2 _ R(z)2" (2)

We can get the same differential equat ion by using (1)

above, which impl ies that the circle at height z has circumference 27rpe -s/p, where s is the arc length along the

surface from (0, r ) to (z, R(z)) . We can solve this differen-

tial equation expl ic i t ly for z:

z = h / p2 _ R2 _ p ln P + ~ R ~ ~ - R2 "

Here z is a cont inuous ly different iable funct ion of R and the derivative (for z r 0) is never zero, hence R is also a


Figure 12. Crocheted pseudosphere.

cont inuous ly different iable funct ion of z. Because R is

never zero, we can conc lude that this hyperbol ic sur face of revolut ion is a smoo th sur face ( t radi t ional ly ca l led the

pseudosphere) . Thus,

T H E O R E M : The pseudosphere is locally i some t r i c to the

a n n u l a r hyperbolic plane.

We can also c roche t a p seudosphe re by s tar t ing with 5

or 6 chain s t i tches and cont inuing in a spiral fashion, in- c reas ing as when crochet ing the hyperbol ic p lane (see Figure 12). Note that, when you crochet beyond the annular s t r ip that lies flat and forms a comple te annulus, the sur face begins to form ruffles and is no longer a sur face of

revolution. In fact, it appea r s that it is not even differen- t iable where the ruffles start, for the "top ridge" of the ruffles (see Figure 12) appea r s to be straight and thus not tan-

gent to the plane of the comple te annulus.

Connect ions to Riemannian manifolds with constant

negative Gaussian curvature

If a su r face is d i f ferent iably e m b e d d e d into 3-space by an i some t ry whose first and s e c o n d der ivat ives a re cont inuous (C2), then the su r face is sa id to be a R iemannian manifold. At a given po in t P on the surface, call the n o r m a l

d i rec t ion one of the two d i rec t ions that are pe rpend i cu - la r to the sur face at tha t point . The norma l c u r v a t u r e at a po in t of a curve on the su r face is def ined to be the com- ponen t of the curva ture of the curve that is in the no rma l d i rect ion. The co l lec t ion of all normal cu rva tu res of all the ( smooth) curves th rough P has a m a x i m u m and a min- imum value. These ex t r ema l va lues of the no rma l curva-

ture a re the pr inc ipa l cu rva tu res (and can be s h o w n to be the no rma l curva tures of two curves tha t a re pe rpend ic - u lar a t P). The G a u s s i a n curva tu re of the sur face at P is de f ined to be the p r o d u c t of these two pr inc ipa l curva-

tures . The pseudosphe re is a Riemannian surface, and at each

po in t [z, R(z) , 0] the pr inc ipa l curvatures are the normal

curvatures of generat ing curves, z ~ R ( z ) and the circle 0 ~-* [R(z), 0]. The curvature of the first curve is

- R " ( z ) [1 + (R'(z))2] 3/2

and is pe rpend icu la r to the surface, and thus is also (+_)

normal curvature. The curvature of the circle is 1/R(z), which mus t be p ro jec ted onto the d i rec t ion perpendicu la r

to the surface, giving the normal curva ture as

1

R(z)~/1 + (R ' (z) ) 2 "

We do not have a formula for R, but we do have a formula (2) for R' ( z ) . The Gaussian curvature is then the p roduc t

of these two norma l curvatures, which you can check [using (2)] is -1/p2; the minus sign occurs because the two

normal curva tures are in oppos i te direct ions. Thus, the

p seudosphe re has constant negative Gauss ian curvature. Gauss ' s famous Theorema E g r e g i u m s ta tes that the

Gaussian curva ture is independent of the (C 2) embedding,

hence is an intr insic p roper ty of the surface. Thus, s ince the annular hyperbol ic plane is local ly isometr ic to the pseudosphere , we can say it also has cons tan t negative

Gaussian curvature . Most differential geomet ry texts give intrinsic me thods for determining the Gauss ian curvature, which can be appl ied direct ly to the annular hyperbol ic

plane (see [3], P rob lem 7.7, for two such methods) . Note that in the c roche ted pseudosphe re (Figure 12) there are points that apparen t ly have no tangent p lanes and thus no normal direct ion, and therefore it is not poss ib le to define

(at these poin ts ) the principal curvatures . In addition, the resul t of N. V. Efimov [2] a l ready d i scussed shows that, no mat te r how the annular hyperbol ic p lane is p laced in 3- space, if it is ex t ended enough it canno t be C 2 embedded

(and thus cannot have pr incipal curva tures at all points).

Connection to the upper half-plane model

As shown above, the coordina te map x p rese rves (does not distort) d i s tances along the (vert ical) 2nd coordina te curves, but at x(a , b) the d is tances along the 1st coordina te curve are d i s to r ted by the factor of e x p ( - b / p ) when com- pa red to the d i s tances in R 2. To be more precise:

D E F I N I T I O N : Let y : A - - ) B be a m a p f r o m one metr ic space to another, and let t ~ A(t) be a curve i n A. Then, the

distortion o f y along A at the po in t p = A(0) is defined as:

arc length along y(A) from y[A(x)] to y[A(0)] lim x-~0 arc length along • f rom A(x) to A(0)

We seek a change of coord ina tes tha t will d is tor t distances equally in bo th directions. The reason for seeking

this change is tha t if d is tances are d i s to r ted to the same degree in bo th coord ina te direct ions, then the map will pre- serve angles. (We call such a map conformal . )

We cannot hope to have zero d is tor t ion in both coordi-

nate d i rec t ions (if there were no d is tor t ion then the char t would be an isometry) , so we try to make the dis tor t ion in the 2nd coord ina te direct ion the same as the dis tor t ion in


the 1st coord ina te direction. After a lit t le exper imenta t ion ,

we find that the des i red change is

z(x, y) = x[x, p in(y/p)]

with the domain of z being the upper half-plane

R 2+ ~- {(x, y) E R21y > 0}

where x is the geodesic coord ina te map defined above. This

is the usual upper half-plane model of the hyperbol ic plane, thought of as a map of the hyperbol ic plane in the same way that we use p lanar maps of the spher ical surface of

the earth.

LEMMA: The distortion o f z along both coordinate curves

x ~ z(x, b) and y ---* z(a, y)

at the po in t z(a, b) is p/b.

P R O O F . We now focus on the point z(a, b ) = x(a , pin(b/p)). Along the first coord ina te curve, x ---> z(x, b) =

x(x, pin(b/p)), the arc length f rom x(a , c) to x(x, c) is

Ix - a I e x p ( - c / p ) by (1) above. Thus, we can calcula te the distort ion:

lira Ix - a] e x p [ - [ p ln(b/p)]/p] x~a ix _ al = p /b .

Now, look at the second coord ina te curve, y ---> z(a, y) = x(a , p ln(y/p)). Along this coord ina te curve (a radial geo-

desic) the speed is not constant; but, s ince the second coord ina te of x measures arc length, the arc length f rom

z(a, y) = x(a , p ln(y/p)) to z(a, b) = x(a , p in(b/p)) is

IP in(Y/P) - P In(b/P)l

and the d is tor t ion is

lim IP in(Y/P) - P ln(b/P)l in(y/p) - in(b/p) ry bl = p y b

= P ~Y in(Y/P) y=b = p/b.

In the above situation, we call these dis tor t ions the distortion o f the m a p z at the po in t p and denote it d i s t ( z ) (p ) . Thus,

dis t (z ) ( (a , b)) = p/b

Hyperbolic Isometries and Geodesics We have seen that there are reflections in the annular hy-

perbolic plane about the radial geodesics , but we saw the ex is tence of o ther ref lect ions and geodes ics only approxi -

mately. However , we were able to see that non-radial geodes ics appea r to be tangent to one annulus and then in bo th d i rec t ions from that poin t to app roach being perpendicular to the almuli. To assis t us in looldng at t ransformat ions

of the a lmular hyperbol ic space, we use the upper half- p lane model . As the almuli co r r e spond to horizontal l ines in the uppe r half-plane model, geodes ics should then be curves that s tar t and end pe rpend icu la r to the bounda ry x- axis. Semici rc les with centers on the x-axis are such curves,

and we can show direct ly that they are geodes ics with bi- lateral symmetry. In part icular , we can show direct ly that

inversion in a semic i rc le cor responds to a ref lect ion isometry in the annular hyperbol ic plane.

D E F I N I T I O N : An invers ion wi th respect to a circle F is

a t ransformat ion f rom the ex tended p lane (the plane with 0% the poin t at infinity, added) to i tself that t akes C, the center of the circle, to oo and vice ve r sa and tha t takes a

point at a d is tance s f rom the cen te r to the po in t on the same ray (from the center) that is at a d i s tance of r2/s from

the center, where r is the radius of the circle (see Figure 13). We call (P, P ' ) an inversive p a i r because (as the reader

can check) they are t aken to each o the r by the inversion. The circle F is cal led the circle o f inversion.

Inversions have the following well-known propert ies (see

reference [1], Chapter 5, and reference [4], Chapter 14):

�9 Inversions are conformal .

�9 Inversions t ake c i rc les not passing th rough the center of inversion to circles.

�9 Inversions t ake c i rc les pass ing through the cen te r of in-

vers ion to s t ra ight l ines not through the cen te r of inversion.

If f is a t ransformat ion taking the u p p e r half-plane R 2+

to itself, then cons ide r the d iagram

H 2 g.~ H 2

z- iS t z

R2 + ~ R 2+ f

We call g = z �9 f �9 z - t the t ransformat ion o f H 2 that cor-

r e sponds to f. We will call f an i sometry o f the upper half- p lane model if the cor responding g is an i somet ry of the annular hyperbol ic plane.

THEOREM: Let f be the inversion in a circle whose center is on the x-axis. Then the corresponding g = z 0 f o z -1

has distortion 1 at every point and is thus an isometry.

PROOF. (Refer to Figure 14.)

1. Note that each of the maps z, z - I , f is conformal and

has at each po in t a (non-zero) d is tor t ion that is the same

p t

Figure 13. Inversion with respect to a circle.


Figure 14. Hyperbolic reflections correspond to inversions.

for all curves at that point. Using the defini t ion of distort ion, the reader can easi ly check that

d i s t ( g ) (p ) = d i s t ( z ) ( ( f 0 z -1) (p) ) x d i s t ( f ) ( z - l ( p ) ) x d i s t ( z - 1 ) ( p ) .

2. If z(a, b) = p, then, using (1),

d i s t ( z 1)(I9 ) = 1 / [ d i s t ( z ) ( ( z - l ( p ) ) ] = b/p.

3. Let r be the radius of the circle C which def ines f, and let s be the d is tance f rom the center of C to (a, b) = z - l ( p ) . The invers ion being conformal, the d is tor t ion is

the same in all direct ions. Thus, we need only check the d is tor t ion along the ray f rom C, the cen te r of circle, th rough p. The r eade r can check that this d is tor t ion

d i s t ( f ) ( ( a , b)) = r2/s 2.

One way to do this is to no te that, in this case, the dis- tor t ion is the speed (at s) of the curve t ~-) r2/ t .

4. By (1), d i s t ( z ) ( f ( z - l ( p ) ) = p/c, where c is the y-coordinate of f ( z - l (p ) ) = f(a, b). To determine c, look at Figure 14. By similar triangles, s/b = (r2/s)/c. T h u s c = b(r2/s 2) and

ps 2 d i s t ( z ) ( ( f 0 z -1) (p) ) - br 2 .

5. Putt ing everything together , we now have

ps 2 r 2 b _ 1. d i s t ( g ) ( p ) - br 2 s2 P

Since this is true at all poin ts p, the map g mus t be an isom- e t ry of the annular hyperbol ic plane.

We call these invers ions t h rough semic i r c l e s wi th cen- t e r on the x-axis (or the c o r r e s p o n d i n g t r a n s f o r m a t i o n s

in the annu la r hyperbo l i c p lane) h y p e r b o l i c r e f l e c t i o n s .

Thus the images of the semic i rc l e s in the u p p e r hal f -p lane

have b i l a t e ra l s y m m e t r y and so a re in t r ins ica l ly s t ra igh t (geodes ics ) .

We have es tabl ished that the annular hyperbolic plane is the same as the usual upper half-plane model of the hyper-

bolic plane. The usual analysis of the hyperbol ic plane can

Figure 15. Triangle with an ideal triangle and three 2/3-ideal triangles.


Figure 16. Ideal triangles in the upper half-plane model.

(-1,o) (1,o) o./

(-1,o) (1,0) Figure 17. 2/3-ideal triangles in the upper half-plane model.

now be considered as analysis of the intrinsic geometry of the annular hyperbolic plane. We give only one example here because it results in the interesting formula ~rr 2.

A r e a of Hyperbo l ic T r i a n g l e s

Given a geodes ic tr iangle with in ter ior angles/3i and exte-

r ior angles ai, we extend the s ides of the tr iangle as indica ted in Figure 15.

The three ex t ra l ines are geodes ics that are asympto t ic

at both ends to an ex tended side of the triangle. It is tra- di t ional to call the region enc losed by these three ex t ra geodes ics an ideal triangle. In the annular hyperbol ic p lane these are not actual ly t r iangles because their ver t ices a re

at infinity. In Figure 15 we see tha t the ideal tr iangle is di- v ided into the original tr iangle and three "triangles" that

have two of their ver t ices at infinity. We call a "triangle" with two ver t ices at infinity a 2~3-ideal triangle. You can use this decompos i t ion to de te rmine the area of the hyperbol ic triangle. Firs t we mus t de te rmine the areas of ideal and 2/3-ideal triangles.

It is imposs ib le to p ic ture the whole of an ideal t r iangle in an annular hyperbol ic plane, but it is easy to p ic ture ideal

t r iangles in the upper half-plane model . In the upper half- p lane mode l an ideal triangle is a tr iangle with all th ree ver t ices e i ther on the x-axis or at infinity (see Figure 16).

At first glance it appears that there must be many different ideal triangles; however:

T H E O R E M : All ideal triangles on the same hyperbolic p lane are congruent.

PROOF OUTLINE: Perform an invers ion (hyperbol ic re- f lect ion) tha t t akes one of the ver t ices (on the x-axis) to infinity and thus takes the two s ides from that ver tex to ver t ical lines. Then apply a s imilar i ty to the upper half- plane, taking this to the s t andard ideal tr iangle with ver- t ices ( - 1 . 0 ) , 0, 1), and ~ (see Figure 16).

T H E O R E M : The area o f an ideal triangle is 7rp 2. (Remember , this p is the radius of the annuli, and equal to ~L---1/K, where K is the Gauss ian curvature.)

PROOF: By (3), the d is tor t ion d i s t ( z ) ( a , b) is p/b, and thus the des i red a rea is

S:, :,. We now pic ture in Figure 17 2~3-ideal triangles in the

upper half-plane model .

T H E O R E M : All 2~3-ideal triangles w i t h angle 0 are congruent and have area (~ - O)p 2.

Show, using inversions, that all 2/3-ideal t r iangles with

angle 0 are congruent to the s tandard one at the right of Figure 17 and thus that the a rea is the double integral:

Combining these th ree theorems and Figure 15 we get:

T H E O R E M : The area o f a hyperbolic triangle is

( z4. REFERENCES

[1] Coxeter, H. S. M., and S. L. Greitzer. Geometry Revisited. New

Mathematics Library 19, New York City: L.W. Singer Company, 1967.

[2] N. V. Efimov. "Generation of singularities on surfaces of negative

curvature" [Russian], Mat. Sb. (N.S.) 64 (106) (1964), 286-320.

[3] Henderson, D. W. Differential Geometry: A Geometric Introduction.

Upper Saddle River, N J: Prentice Hall, 1998.

[4] Henderson, D. W. Experiencing Geometry in Euclidean, Spherical,

and Hyperbolic Spaces. Upper Saddle River, N J: Prentice Hall, 2001.

[5] Hilbert, D. "Uber FIAchen von konstanter gaussscher KrOmmung,"

Transactions of the A.M.S. 2 (1901), 87-99.

[6] Hilbert, D. and S. Cohn-Vossen. Geometry and the Imagination.

New York: Chelsea Publishing Co., 1983.

[7] Kuiper, N. "On Cl-isometric embeddings ii," Nederl. Akad.

Wetensch. Proc. Ser. A (1955), 683-689.

[8] Milnor, T. "Efimov's theorem about complete immersed surfaces

of negative curvature," Adv. Math. 8 (1972), 474-543.

[9] Osserman, R. Poetry of the Universe: A Mathematical Exploration

of the Cosmos. New York: Anchor Books, 1995.

[10] Spivak, M. A Comprehensive Introduction to Differential Geometry.

Vol. Ill. Wilmington, DE: Publish or Perish Press, 1979.

[11] Thurston, W. Three-Dimensional Geometry and Topology, Vol. 1.

Princeton, N J: Princeton University Press, 1997.


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I C o l i n A d a m s , E d i t o r J

The proof is in the pudding.

Opening a copy of The M a t h e m a t i c a l

I n t e l l i gence r you may ask yourself

uneasily, "What is this anyway---a

mathematical journal, or what?" Or

you may ask, "Where am I?" Or even

"Who am I?" 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.

Column editor's address: Colin Adams,

Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267 USA e-mail: [email protected]

U-Substitution

T his is comple te and ut ter humilia-

t ion in i ts nas t ies t form, thought the coach, as he glanced up at the scoreboard . His t eam was still in the

single digits while their opponen t s were about to b reak fifty. He looked down the bench at his demora l ized

team. It was not a br ight momen t for the Valparaiso Variables. What had the owner James Stewar t been thinking

when he moved the t eam up to the big leagues? They weren ' t in any way com-

parable to the Indianapol is Integrals, the t eam that was current ly scor ing at will as he wa tched from the sideline. f e i ~ d x stole the ball f rom hapless z

and effort lessly sco red again. His play- ers jus t couldn ' t keep up.

And not that they needed it, bu t the

Integrals had recent ly s igned f l / (1 + x 2) dx. Here was one of the mos t fa-

mous integrals on the planet, with en- dorsement dea ls galore. You couldn ' t turn on your te levis ion without seeing f l / (1 + x 2) dx bi t ing into a hot dog, or

hawking graphing calculators . He was scoring at will. The Variables were

scared of coming within 10 feet of him. As the coach looked down his

bench, all the p layers s ta red down at

their feet, afraid to mee t his eye. They didn ' t want to be pu t into the game jus t to be humiliated. All excep t that skinny kid tan u, sit t ing at the end. That k id ' s

got guts, thought the coach. He'd been hustl ing all semester . He had two left feet, but he wan ted to play so bad. And now he was looking at the coach with despera te hope in his eye. What the hell, thought the coach, this game is lost anyway.

"Okay tan u, you ' re going in for x. You cover the big integral." Tan u leaped off the bench. The coach sig- naled the referee.

"I 'm doing a u-subst i tut ion," he said. "I'm replacing x wi th tan u."

The res t of the bench looked up,

s tar t led to hear the call. x raced over to the sideline.

"What are you doing, coach? You're subbing that skinny kid in for me? Hey,

I 'm your top scorer . You can ' t do this." "Sit down, x," sa id the coach, as he

th rew him a towel. f l / ( 1 + x 2) dx laughed when he saw

the sc rawny p layer that was covering him.

"Hey look at this," he said. "They did a u-substi tut ion." His t eammates guf- fawed.

As play resumed, f l / ( 1 + x 2 ) d x

came down the cour t fast, with lit t le tan u trailing behind. But as f l / (1 + x 2)

dx made a move to the right, tan u s l ipped to the inside. As everyone wa tched in amazement , the 1 § x 2 became 1 § tan2u. A s tunned second later, 1 + tan2u b e c a m e sec2u. The dx

became sec2u du. The c rowd leapt to its feet. The sec2u's cancel led, and all

that was left of the mighty f l / (1 + x 2)

dx was f du. The o the r integrals s tood dumbfounded. The coach was waving

a towel over his head. The entire Variable bench was up screaming. Do it, do it! The f du b e c a m e jus t u + C. Pandemonium erup ted all over the sta- dium.

"Okay, x, finish it off," said the coach, grinning f rom ear to ear. A sheepish x went b a c k in for tan u, and

the u + C became arc tan x + C. The building reverbera ted with cheers. The Variables l if ted sc rawny tan u on their shoulders and pa r aded a round the cour t as the c rowd chanted, "Tan u, tan

u." Fans m o b b e d them from all sides. The Variables had won the game.

Later in the locker room, after all the repor te rs had come and gone, and all the champagne had been swal lowed

or dumped on heads, the coach gath- e red the team together .

"Well, I didn't think we could do it,

but thanks to tan u, we beat the Indianapolis Integrals. And I want you to savor this victory. You deserve it. But don ' t get carr ied away with it, either.

Next week we play the Pittsburgh PDE's, and if you think the Integrals are tough, wait until you try solving a PDE."

�9 2001 SPRINGER VERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 2 9

SPRINGER FOR MATHEMATICS

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iLvAl~'t||[=]t=l '=i|[,-~-Ii[="J[,]==l|=lml=n||[~--l i M a r j o r i e S e n e c h a l , E d i t o r I

A Personal Account of Mathematics in Bosnia Harry Miller, with assistance

from Naza Tanovi(~-Miller

This column is a foram for discussion

of mathematical communities

throughout the world, and through all

time. Oar definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater than one. What we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063 USA e-mail: [email protected]

I grew up in Chicago, in Hyde Park, near the University of Chicago.

Nothing in the first 25 years of my life connected me with Yugoslavia in any way. Then, when I was a graduate student at the Illinois Institute of Technology, I met an attractive student from Sarajevo, Naza Tanovid, in a course on Linear Operators in Hilbert Space, and my life took an eastward turn. We were married in 1965; I got my PhD in 1966, the same year our older daughter, Lejla, was born, and taught at DePaul University until Naza got h e r

degree in 1969. Then we moved to Sarajevo, arriving just days before Neff Armstrong took his frost step on the moon. Mica, our younger daughter, was born in Sarajevo in 1970. (Both of our daughters have become mathematicians: Lejla got her Ph.D. in 1994 under Fred Gehring at the University of Michigan and teaches at the Berkes campus of Penn State, and Alica is com- pleting work on her Ph.D. thesis under Cliff Weil at Michigan State University.)

When we arrived, we were--I believe--the eighth and ninth Ph.D.'s in mathematics in Bill (the Bosnian ab- breviation for Bosnia and Herzegovina), which had a population of about 41/4 million then. Naza's desire to help h e r

people and the notion that two 30-year- olds could make a difference seemed totally legitimate to me: our marginal utility to mathematics was vastly higher in Sarajevo than in Chicago. Besides that, living in Sarajevo in the latter part of the Tito era (he died in May 1980) was very peaceful even if the economic standard was low, and Yugoslavia--in particular Bosnia--is most beautiful. (Do you remember the scenes on ABC-TV of the '84 Winter Olympics?) During this period most people were happy to be Yugoslavs-- it wasn't perfect, but the majority w e r e

enjoying the best life-style they had e v e r known. Relations between people of different ethnic backgrounds were exemplary--certainly far better than in the United States at that time.

Our university was founded in 1949 in a city that was purposely kept in a distant third place, behind Belgrade and Zagreb, by the powers that be during the entire twentieth century. The language was Serbo-Croatian; the fffst (between the world wars) Yugoslavia was the Kingdom of Serbs, Croats, and Slovenians; and there was no Boshian (Bo~njak) nationality. The Bosnians, by edict, had to declare themselves Serbs or Croats. Later, Muslims were allowed to be called Muslim by nationality (as well as by religion), but not Bosnians. In this respect the situation was similar to being a Jew--by religion and na- t ionali ty-in the old U.S.S.R. Lejla's first-grade teacher at the Moritz Moco Salem Elementary School told us that she once asked her classmates the na- tionalities of their parents. Lejla heard them answer: father Muslim, mother Muslim; father Serb, mother Serb, etc., and became increasingly concerned as the teacher approached her row. Finally, when asked about us, she answered in a serious voice, "My mother is Muslim, I think my father isn't. Put down that he is Serbian." Oh, for those long-ago innocent days!

The chairman of the Mathematics Department for a long period was Mahmut Bajraktarevid, a man dear to me, who was born in 1909, the same year as my father. He encouraged, by example (is there any other way?), his younger colleagues to do research. In the 1960s and 1970s a great deal of progress was made in the world of Bosnian mathematics. Many of us published in Radovi-Akademija Nauka i Umjetnosti Bill, a journal published by the Bosnian Academy of Arts and Sciences. This journal was mainly of local character but was a first step for Sarajevo. In this period (and afterward) many mathematicians from America, Europe, and the U.S.S.R., who were "in the neighborhood," visited and lec- tured in Sarajevo. Many of these guests stayed in our home. These visits, besides helping us to keep in touch with


the great outside world of mathematics, also formed a base of contacts for later activities.

The Mathematical Gymnasium in Sarajevo, where Lejla and Alica at- tended high school, provided the Sarajevo Mathematics Department with a steady stream of talented students. This high school was very successful in producing winners of state (Bill), federal (Yugoslav), and even international mathematics competitions. The educational system in Yugoslavia, Austro-Hungarian in approach, was far superior in general to its American counterpart. (Young Bosnian war refugees have done very well in American schools, often performing at the top of their classes.) School in Sarajevo was serious business--the children did a lot of homework and were expected to place school at the top of their list of priorities.

Until the late 1980s our talented younger colleagues (I count nine) got their Ph.D.'s through our department (some of them had advisers outside of Sarajevo). All of them were associated with the Department of Mathematics of the Natural Science and Mathematics Faculty of the University of Sarajevo. Other faculties had smaller mathematics departments. In addition, universities were operating in other centers, including Mostar, Tuzla, and Banja Luka.

By the early 1980s our activities became much more international. Several people started publishing in leading journals abroad. Outstanding students elected to do their graduate work in America and they did us proud: every student who went to the States with a teaching assistantship or fellowship completed his or her Ph.D. degree! The schools they selected included Michi- gan State University, the University of Michigan, Illinois, Wisconsin, and UC Berkeley.

In 1985 with Professor Bajraktarevi6 and Naza leading the way, we founded an international journal, Radovi Mate- mati6ki. (We wanted to call it the Sarajevo Mathematics Journal, but the political forces mentioned earlier would not hear of Sarajevo or Bosnia being in the title.) Before the war in Bosnia, which started in 1992, this journal was being exchanged with over 400

journals. It appeared in two numbers each year and featured beautiful cover pages presenting interesting details from old manuscripts, of historical and artistic value, that are housed in libraries and museums all over the former Yugoslavia.

The Editorial Board consisted of leading Sarajevo mathematicians; a group of outstanding mathematicians from America, Europe, and Russia served as editorial advisers, adding stature and international standards to the journal. In the second edition of Radovi, each year, the Editorial Board presented information about recent scientific and academic activities of mathematicians and institutions in Bosnia and Herzegovina. In vol. 7, no. 2, which appeared at the end of 1991, thirteen are devoted to the Chronicle, reporting the following activities: twenty-eight research articles were published by our mathematicians, of which twenty-one appeared in foreign journals; during 1991 the mathematicians from Bill presented eleven talks on their research outside of Bosnia, and the Colloquium of the Department of Mathematics of the Natural Science and Mathematics Faculty of the Uni- versity of Sarajevo presented twenty- four research talks. In addition there were seminars in Algebra, Harmonic Analysis, Real Analysis, Differential Equations and Dynamical Systems, Dif- ferential Algebra and Algebraic Differ- ential Equations, and Differential geometry. The University of Banja Luka had three Colloquium talks, and the Uni- versity of Mostar had two. Six of our mathematicians made research visits abroad during the year, and one Chinese student visited our department and worked on his Ph.D. thesis. During the year six new textbooks authored by Bosnian mathematicians were published. In addition, three new editions of old texts appeared. Under the title "Other Publications and Communica- tions" (problem-solving and educational items) a total of thirteen items are presented. During the year, four people were elected to the rank of docent (assistant professor), two in Sarajevo and one each in Mostar and Tuzl& One of our students got his Ph.D. at the Univer- sity of California Berkeley, and one got

her M.S. in Sarajevo. The following graduate mathematics courses were offered (in Sarajevo): Non-Commutative Algebra, Functional Analysis, Semi- Groups of Linear Operators, and Analysis on Groups. Finally, the Society of Mathematicians, Physicists, and Astronomers of Bill reported on the year's state (Bill) elementary school and high school math contests.

Then the war began and everything changed. Of course, our relations with our Croatian and Serbian colleagues were affected by those turbulent times. Naza and I were always closer to the Zagreb Mathematics Department, since Naza got her undergraduate en- gineering education in Zagreb and per- sonally knew many of the people who had become professors in the Mathematics Department. But we are not anti-Serb. Naza's father risked his life in World War II by protesting the mistreatment of Serbs (and others) by the Nazis and Croatian fascists. Balance: in WWII there were also Serbian ultra-nationalists who commit- ted atrocious crimes. Fast forward. Slobodan Milo~evic is an evil man, the single person most guilty for the events leading to the bloodshed in the former Yugoslavia from 1991 to 1999. But the actions in the summer of 1995 by the Croatian Army against the Serbian pop- ulace in the Krajina are also war crimes. War crimes, perpetrated by any group, are war crimes and should be severely punished. With that caveat, let's get on to some more specific remarks.

Unfortunately, a fair percentage of colleagues of Croatian or Serbian ex- traction forgot or never learned that the politics of pitting one ethnic group against another is not a zero-sum game. By acquiescing to the extremists in their respective groups, they have actually injured the position of their co-religionists in Bill. (There were, of course, many notable exceptions. The Serbian General Divjak was a tower of strength in the Bosnia Army, and the Croatian Ivan Misi6 has been a model of integrity in his work as a spokesper- son for Bill since 1992.) Among colleagues, Veselin Peri6 was the most prominent mathematician who sprouted nationalist wings. Peri6 was born in Montenegro and received his graduate

3 2 THE MATHEMATICAL INTELUGENCER

educa t ion in Zagreb. He moved to Sarajevo in the 1960s and b e c a m e the leading algebrais t in Bill. Al though he had a very rewarding l i f e - - a n d was

well r e w a r d e d - - i n Bosnia, he suppor ted , wi thout hesi tat ion, the doc-

tr ine of greater Serbia at the expense of Croat ia and Bosnia. Foolishly, l ike

many Serbs and Montenegr ins he thought that with the Yugoslav National Army act ing on thei r beha l f

they would achieve thei r ter r i tor ia l ambi t ions quickly and with lit t le effort.

Let 's do a little more name-drop- ping. Bil jana Plavsi5 was the Dean of

the Natural Sciences and Mathemat ics Facul ty in Sarajevo during the per iod jus t before the war in Bosnia. Bil jana

is an at tractive, u rbane biologist , whose Ph.D. father worked at the

Museum of Natural His tory in Sarajevo. We were close to her, thinking that she sha red our pass ion to put Bi l l "on the

map" scientif ical ly (and she d i d - - a t the t ime). In the 1990s, Bil jana's life took a sharp c h a n g e - - f r o m sc ience to poli-

tics. She was e lec ted to the Pres idency of Bil l (the 7-member ruling body that

cons i s ted of two Serbs, two Croats, two Muslims, and one independen t ) as a m e m b e r of the Serbian Democra t ic Party, along with Nikola Koljevid. Bil jana underwent an amazing trans-

format ion from a modern civil ized lib-

eral ci t izen of the wor ld into a sup- po r t e r of the Grea ter Serbian Policy, which was based on ethnic c leansing and used rape, murder , and t e r ro r as

ins t ruments for execut ing policies. She

openly e m b r a c e d ( recorded for pos- ter i ty on v ideo) the most vicious eth-

nic c leanser Zeljko Re2njatovi5 (be t te r known as Arkan) jus t af ter his band of

barbar ians commi t t ed a ser ies of a t roc- i t ies in Eas te rn Bosnia in 1992. How cultured, civilized, wel l -educated "nor-

mal" peop le turn into psychopa ths under the in toxica t ion of rab id national-

ism is one of the myster ies of this war, and another example of a spec ies of human behav ior that is all too well doc-

umented in the twent ie th century.

Jus t as se ismic poli t ical upheavals can p roduce hor rendous changes in people , war and tragic events are of ten

the b a c k d r o p for acts of great courage and compass ion . It was a great plea- sure for us to become acquainted with

the then l i t t le-known CNN reporter , Christ iane Amanpour , early in the

Bosnian war. I was walking toward my home one day when her driver po in ted me out to her, saying, "He's the

American." I t ook Ms. Amanpour to our home, and there at the door a scene was p layed out that will a lways remain

vivid in my mind. Naza, s tanding 10 s ta i rs above Christiane, in a very serious voice, declared , "If you want to do

a ser ious j o b and explain what ' s rea l ly happening here, then you are we lcome in our home with all that we can offer

you, but if you want to do a three- minute p iece to amuse Amer ican housewives abou t how a bunch of sav-

ages are murder ing each o ther in the Balkans, then p lease leave and let us fight and die in dignity." My pr ide in my

Party of the Faculty of Natural Sciences and Mathematics, 1990. Standing at left, Prof. Biljana

Pav.~id; standing first at right, the author.

wife at that m o m e n t was off the

Richter scale. Chris t iane slowly, also with dignity, repl ied, "I exac t ly want to

do the former. I wan t to know and explain to o thers wha t is happening."

Fa te had brought toge ther two coura- geous, very strong, women. The CNN crew spent two full days following us

a round in our dai ly activities, collect- ing mater ia ls for the s tory they aired in

the summer of 1992. Christiane, a ha ted figure in Serb ex t remis t circles, r i sked her life repor t ing from all over Bosnia.

We love her dearly. Tragic events can p roduce humor-

ous moments . We were also inter- v iewed by Bob Simon of CBS news, who was based in Tel Aviv at the time.

Bob's c rew v ideo taped us as we ran from our home to the Bosnia Academy

of Science, a d i s tance of about 4 blocks. A couple of rounds of sn iper fire went off nea r us as we approached

the Academy. We were happy to get inside. After we caught our brea th Bob announced that the sound had been off

as he r ecorded the last 100 yards of our dash. He asked us to go out and repea t the fmal hundred or so yards of our s p r i n t - - s o they could re tape it with the

sound on! Jus t as we f inished our second sprint, sho ts were heard, and I said, "I feel l ike Paul Newman, only

these bul lets a re real." The f i l m - - a n d that q u o t e - - w e r e a i red on the CBS evening news. Needless to say, my

mother, seeing it in Chicago, was less

than enthusiast ic! Naza's book abou t our war experi-

ences, Testimony of a Bosnian, will be publ i shed by the Texas A&M Press early in 2001. Readers in teres ted in the his tor ical backg round of Bil l should

also consul t Noel Malcolm's A Short History of Bosnia.

Because of the ca tas t rophic effect of the war and the d ispers ion of many

of our mathemat ic ians abroad, it took some t ime to get Radovi MatematiSki back on its feet af ter the Dayton agreement. The first pos t -war Chronicle of mathemat ica l act ivi t ies in Bil l ap-

pea red in vol. 8, no. 2. Due to the ag- gress ion against our count ry and its consequences, this Chronicle covers the per iod 1992-1998. The repor t in-

c ludes scientif ic and academic activi- t ies of ma themat ic i ans from Bosnia


Harry and Naza Miller, in January 1998, on a bridge near their home in Sarajevo,

and Herzegovina, at home and abroad, wi th whom coopera t ion was main- t a lned during and af ter the war; a cer-

ta in number of ma themat ic i ans have chosen to s top all con tac t with their fo rmer inst i tut ions in our country, and so we were unable to r epor t on their

activities. We are aware that this re- View is therefore incomplete , and we will t ry to include missed da ta in our nex t Chronicle. During the per iod

1992-1998 we have r eco rded 56 re- s ea rch papers publ i shed f rom mathemat ic ians from Bill; a lmos t all of them a p p e a r e d in foreign journals . During

the same period, the ma themat ic i ans of Bi l l p re sen ted 91 lec tures on thei r re- s ea rch activities. Because of the con- s tan t shelling of Sarajevo during the war the regular Colloquium of the

Depar tmen t of Mathemat ics of the Univers i ty of Sarajevo was interrupted. It is now being react ivated: 14 talks were presented during 1996-1997. The act ivi t ies of the seminars ment ioned

prev ious ly (i.e., in the 1991 Chronicle) ceased with the beginning of the w a r - -

Apri l 5, 1992. The fifth pos t -war num- be r of the journa l has jus t been issued.

Af ter the war, seminars , in the form

of a ser ies of lectures, were p r e s e n t e d in Algebra, Phi losophy of Mathemat-

ics, Complex Analysis, Mathemat ica l Physics, Numerical Mathematics , and Appl ied Mathematics. In 1992 two

s tudents , one Chinese, one Bosnian, received their Ph.D. degrees in Mathemat- ics from the University of Sarajevo. In 1993 three Bosnians got their Ph.D.s at

Michigan State and the Univers i t ies of Wisconsin and Michigan, respect ively. In 1994 one Bosnian s tudent got he r

Ph.D. at the Univers i ty of Michigan. In

each of the years 1995 and 1996 one Bosnian s tudent got his Ph.D. from the

University of Sarajevo. Between 1993 and 1997 five s tuden ts got their M.S.

degrees, three in Sarajevo, one in Tuzla, and one at the University of Washington.

During the 1996-97 school year our graduate school r e sumed its work. Fif teen s tudents , mainly teaching as-

s is tants at the Universi t ies of Bihad,

Mostar, Sarajevo, and Tuzla, registered. They were offered courses in four areas: Funct ional Analysis, Harmonic Analy-

sis, Algebra, and Appl ied Mathematics. During the pe r iod f rom 1992 to 1997 six members of the Bosnian mathemat ica l

communi ty were kil led or pa s sed away.

The la tes t Chronicle presen ts the

extensive list of academic activit ies of the Bosnian mathemat ica l communi ty since the beginning of the war. It in-

cludes a list of the la tes t t ex tbooks and lecture notes wr i t ten by Bosnian au-

thors, as well as a list of educat ional art icles and act ivi t ies of the Bosnian Mathematical Society. We ment ion

with pr ide that f rom the summer of 1993, when Bosnia and Herzegovina was recognized as a sovereign state, until today, t eams of young Bosnian

mathemat ic ians have compe ted in the Internat ional High School Mathemat ics Olympics. In the s u m m e r of 2000, in

South Korea, our team, for the first time, scored higher than Croatia. Under the ausp ices of the Bosnian

Spring of 1998: a view from the home of the Millers in the Old Town Sarajevo, showing the

building of the National Library, whose interior was destroyed by Serbian nationalists' artillery

in 1992 with loss of 1,5 million books,


Mathematical Society, the magazineTriangle, for elementary and highschool students and teachers, has ap-peared in print quarterly since 1997.

The Math Department at the NaturalSciences and Mathematics Faculty ofthe University of Sarajevo is spic-and-span, in better shape than before thewar, including a new fully equippedcomputer lab. The reconstruction andimprovement efforts were made possi-ble with the help of Western monies.Naza and her colleagues report thepost-war students are eager beavers,anxious to work hard and get on witha normal life.

Many of Bosnia’s most productive in-tellectuals have settled and are pursuingcareers in Europe and North America.About 100 Bosnian medical doctors,aged 30 to 45, are currently in the UnitedStates. This represents a tremendousloss to BiH. We ase asking the govern-ment of the Bosnian Federation to takea constructive approach in dealing withtalented Bosnians who are living out-side of the country; we are suggestingthat, as a first step, the Prime Ministercompile a database of these people andwrite to them to express the govern-ment’s pride in their successes abroad.By taking a positive approach, it ishoped that talented scientists fromBosnia will contribute in their ownways to the development of the newcountry. We hope that academic andscientific personnel permanently set-tled abroad will spend sabbaticals orscientific visits in their homeland andin general be connected and aid scien-tific development in BiH. Paul Erdiiswas not physically present in Hungaryfor most of his life after the age oftwenty-two, but what a contribution hemade to Hungarian mathematics!

In my optimistic moments, I envi-sion the future of the territory of theformer Yugoslavia developing over thenext decade or two in concentric cir-cles. Bosnia, and particularly Sarajevo,are at the geographic center of the for-mer Yugoslavia. It is the region wherethe tradition of multicultural living was

strongest and lasted longest. If theWestern powers show more (muchmore!) resolve in resettling refugees intheir homes and remain steadfast intheir support of a single unified BiH(they get a D, in my opinion, so far, inthis area) over all of its territory, a vi-able democratic civil state will emerge.With the death of Tudjman and theelection of Mesii: as President ofCroatia, we are witnessing a 180-de-gree change in Croatian policy towardBiH. MesiC has firmly and unilaterallyended the Tudjman-MiloSeviC formulaof dividing BiH into Croatian andSerbian parts. It remains to disassem-ble the Serbian entity, RepublikaSrpska-the monstrous constructionof Richard Holbrooke at Dayton-andrestore BiH into a single unified civilstate. In Serbia we have seen the endof the MiloSeviC regime. Eventually,one hopes, the Serbian people will re-nounce, once and for all, as morallywrong, the concept of “Greater Serbia,”which was used to justify the land grabat the expense of its neighbors in thelast decade. We hope that, 20 yearsdown the road, these three states (andothers) will live in a loose (economic)confederation to everyone’s mutual ad-vantage. Oh, that this could have hap-pened in the late 198Os, and avoidedfour wars, hundreds of thousands ofdeaths, and billions of dollars of de-struction! Most of the people of BiHyearned for such a solution, but in 1992the negative forces of history pushedpast reason and ignited a holocaustagainst the non-Serbian citizens of BiH.

My personal plans are much easierto state. I started as a docent, at age 30,in 1969, at the University of Sarajevo.In for a dime; in for a dollar. Thirtyyears later, it’s back to Sarajevo. Naza’splan to make an academic impact inher homeland still seems like the rightthing to do. And we have our work cutout for us: with the assistance of manypeople from Bosnia and around theworld, Naza and I are trying to foundan international, private, English-lan-guage university-AUS, the American

University of Sarajevo. Despite its ex-cellent record in producing futurePh.D.s, the existing university organi-zation is outdated and inefficient. TheAUS, the first international academiccenter for higher learning and researchin BiH, will be a model for educationalreform. We also hope that the AUS willcontribute to a faster recovery of theacademic, economic, and social life inBiH and will help reduce the alarmingbrain drain. Being situated in a cityunique for the heritage of different civ-ilizations, the AUS will attract foreignstudents and scholars and help rebuildpeace and stability in this region.

VOLUME 23. NUMBER 2, 2001 3 5

WOJBOR A. WOYCZYI~ISKI

Seeking Birnbaum, or Ninc Lives of a Mathematician*

_ • riday. Northwest Airlines's DC-I O approaches the runway at the Seattle-Tacoma

l airport f rom the north. Lauren and Greg's admonitions, "Dad, don't forget

• J t : brsrin g us pres en ts fro m yo ur trip tothe Wild Wes t, " are s till ring i n g in m y

S e a t t l e The sky is intensely blue, unusual ly so for the Pacif ic Nor thwes t ' s rainy season; jus t a few cumulus c louds scat- t e r ed here and there. Two Japanese teenage girls, contin-

uing to Osaka, are craning the i r necks, now to the right, where, beyond the gl immering Puget Sound, one can see a sprawl ing Olympic Range, now to the left, where the lonely 14,000-foot volcanic cone of Mount Rainier domina tes the

landscape . The snow b lanke t s the mountains, but Seat t le b e l o w us is gray with the late winter grime. Only the Douglas firs just i fy the p r o u d n ickname the Evergreen

State. After renting a car and checking in at the Universi ty

Hotel, I call the number which Professor Bi rnbaum e-

mai led to me. It is about 1 p.m. A female voice wi th a slight German accent calls, "Bill, tha t ' s for you." I suggest that I

s top by to p i ck him up if he 'd give me direct ions, but he will have none of that. "I'll mee t you at your hotel in half an hour. I know exact ly where it is be c a us e we lived in that a rea for many years after we so ld our home in the Hilltop Community. Our apar tment was on the 24th floor of the

highrise you should be able to see f rom your room's window." His voice is quiet but de te rmined and energetic, his ins t ruct ions pol i te but firm. "Yes, I 'll d r i v e . . . No, p lease

do not wai t by the curb. I'll walk up the s t eps to your room. See you soon." We speak English, and there is not a shade of doubt tha t this is a natural choice of the language for our conversa t ion. This has its consequences : Polish affairs d iscussed in a different language acquire a new detach-

ment. Medium is the message. I 'm a lit t le embarrassed , because Birnbaum will cele-

bra te his 94th b i r thday in October and the three-s tory ho-

*The Polish version of this article was commissioned for the 100th anniversary issue of the Annals of the Polish Mathematical Society which appeared in 1997. The au-

thor appreciates comments from Monroe Sirken and Ron Pyke on the first draft of the English version.

36 THE MATHEMATICAL INTELLIGENCER �9 2001 SPRINGER-VERLAG NEW YORK

W. Birnbaum in his University of Washington office, February, 1997.

tel has no elevator. Energet ic knocking on the door re-

moves my misgivings. Birnbaum is slight, and his small- boned figure, seen against the light while I guide him in,

reminds me of Steinhaus f rom 1970. His pos ture is erect, and his high forehead is c rowned with a wrea th of hair, not a l together grey. Thick lenses in horn f rames add to the air

of intensity. " . . . Probably, very soon I will have to go through an eye s u r g e r y , . . , you know, ca tarac ts . . . . Hilde went through it about a month ago and now feels l ike new."

He is wear ing a long and heavy overcoa t covering a b rown j acke t and dark-colored plaid flannel shirt. The suggest ion is that we do not sit down but jus t repa i r to his univers i ty

office. "I keep all my pape r s and document s there. This way I can consul t them when my m e m o r y falls. It 's jus t a couple of b locks from h e r e . . . Yes, we ' l l drive my car because you do not have the park ing permit ."

The wea the r re turns to normal. It drizzles when we board a two-door Ford Escor t wi th au tomat ic t ransmission. He dr ives carefully, and c lose to the curb; the traffic is

heavy and the s t ree ts busy, filled wi th the usual univers i ty crowd. We pa rk on the t e r raced park ing lot behind the building hous ing the Mathemat ics Department . Lake Washington is sp read in front of us, bu t the mist prevents us from seeing the far shore or the downtown. " . . . This is

a gorgeous par t of the country. When we were younger we used to hike in the mounta ins and camp out quite a bit."

Bi rnbaum arr ived here in 1939, when the city was still small and provincia l and the Universi ty consis ted of a few buildings on the wes te rn shore of Lake Washington. The

Lake embraces Seatt le from the eas t and p re s se s it against the island- and inlet-fi l led Puget Sound, which spreads it-

self on the wes t s ide of the city. It 's a sa i lor ' s pa rad i se and Navy submarines ' haven. His choice of des t ina t ion was al- mos t accidental , the only al ternative was p re sen ted by an

Austral ian visa for which he had appl ied while still in Poland, but which r eached him only af ter his arrival in

America. The title of Ass is tan t Professor and the $2,000 annual salary were not impressive, but infini tely be t t e r than

the two-year uncer ta in exis tence in the Manhat tan caul-

d r o n - n o w as a spec ia l co r responden t of the Polish Illustrated Daily Courier, now as Fel ix Berns te in ' s assistant at New York University, s tudying corre la t ions be tween the progress of fa rs ightedness and longevity, now as an in-

dependen t consul tan t equipped with a wel l -used Monroe

electr ic ca lcula tor and a rented desk in an old office building in the ne ighborhood of the New York Publ ic Library. "I adver t i sed my services in the journa l Science, and all kinds

of people would show up on my office doors teps . A cer- ta in gent leman brought me some data, the origin thereof was not known to me. He would not vo lunteer thei r prove-

nance either, and asked me to do s imple regress ion on them. A week la ter he returned, checked my graphs, compar ing them against the light with curves that he had

brought along, gave me some more data, and p romised to come back soon. The scene repea ted i tself a few more

t imes. He was paying my fees wi thout delay, so I did not ask any probing questions. However, af ter a few weeks and another inspect ion of my graphs, he s h o o k his head and

resignedly al lowed, 'After all, this is unl ikely to work. ' It tu rned out that he p layed the s tock marke t relying on my predict ions. After that exper ience I p romised myse l f never to analyze da ta of an unknown origin."

T h e L w 6 w S t u d e n t

Birnbaum's grandfa ther was born in the Russ ian zone of

par t i t ioned Poland, but the p rospec t of being a draf tee in the tsar is t a rmy drove him to find shel ter in Galicia, which was then in the Aust r ian zone. There the young boy was

adop ted by the Bi rnbaum family. He quickly became independent . Initially he t ook to logging in Eas te rn Galicia 's Carpathian Mountains; the logs were then f loated on the

San and Vistula r ivers all the way to Gdatisk on the Baltic Sea, following an ancient route. It was back-break ing physical labor. In due t ime he had made good m o n e y on rafts- manship and bought a few thousand acres of logging for-

est in the Pysznica townsh ip of Nisko county. The business was good and the family prospered , assuming the life-style of the local Polish gentry. Of his ten children, four set t led in the capital ci ty of Lw6w. One of them was Birnbaum's

fa ther Izak. According to the records of the local Israel i te Registry Office, Zygmunt Wilhelm Bi rnbaum was born October 18, 1903, in the family of Izak and Lina, n6e Nebenzahl. The family was not par t icu lar ly rel igious and spoke Polish at home. " . . . Now we are m e m b e r s of a syn-

agogue," comment s Bi rnbaum about his cur ren t Seatt le life, "but in Amer ica this has a total ly different, more social s ign i f i cance . . . " . Izak ran the family sawmil l and o ther


businesses . They lived in a comfor tab le f ive-room apart- men t at 1 St. Ann's Street. In the school year 1913/14 Wilek, as the family cal led young Wilhelm, was enrol led in the first

grade of the private Gymnas ium run by Dr. Niemiec. But Wilhelm's educa t ion and the smooth family life were

in te r rup ted by the war of 1914. The family sought she l te r

in Vienna, where Wilhelm soon found himself annoyed and bo red by the low level of the school he was sent to, and

re fused to cont inue in it. Fo r the remainder of World War I he p roceeded by independen t study, with a dai ly one-hour

sess ion with a h i red tutor. In the consecut ive years 1915, 1916, 1917, and 1918, he pa s sed special exams admin i s t e red for home-schooled s tuden ts by the Viennese State

Examina t ion Commission. With the war over, it was t ime to re turn to the Lw6w homes tead , his repa t r ia t ion to the r eborn Poland made eas ier by the formal documen t testi-

fying that the family were Pysznica landowners . Independen t educat ion was, however, poss ib le only up

to a point. It was well unde r s tood that, af ter a cer ta in stage,

it was advisable to get acquain ted with the habi ts and standards of teachers who were going to conduc t the feared

comprehens ive Matura examinat ion, the manda to ry con- c lus ion of Gymnasium (High School) education. Univers i ty enro l lment was not poss ib le wi thout a successful pass ing of the Matura. So, for g rades VII and VI i i - - t he last two

years of the Gymnas iun l - -Wi lhe lm was enrol led in the H. Sienkiewicz State Gymnas ium No. X in Lw6w. By shee r co-

incidence, mathemat ics was taught there by a young doc- tora l s tudent , whose en thus iasm for set theory and topology exceeded his fear of the pr incipal ' s r ep r imand for

violat ing the official syllabus. He did not pay too much at- t en t ion to the logar i thmic tables and t r igonometry, and t a lked in class about ma themat ics in a way that left a per-

manen t impress ion on Wilhelm. Come vacat ion time, usually spen t in the region of the family sawmill in the east- ern Carpathian mountains , the ser ious t eenager was

lugging along col lege math tex tbooks . In 1921 he was a w a r ded the Matura cer t i f icate "with dist inction."

Majoring in mathemat ics was not an option. The family, whose bus inesses fa l tered as a resul t of the Great War, was

dec ided ly in favor of a more prac t ica l d i rect ion of study. The initial effort to enroll in the facult ies of Medicine and Engineer ing failed in view of the numerus clausus, then c o m m o n at many European and American universi t ies,

which res t r ic ted the pe rcen tage of Jewish s tudents in some depar tments . In the Law and Poli t ical Science Facul ty of the Jan Kazimierz University, luck was on Wilhelm's s ide and, for the full course of twelve t ~ n e s t e r s , he success-

fully s tudied law, suppor t ing himself with the Pol ish gov- e rnmen t scholarship. The high poin t of these legal s tudies was his seminar p resen ta t ion of George P61ya's mathemat ica l pape r on e lect ion systems. Birnbaum gradua ted

with the degree of Magister Utriusque Juris (i.e., mas te r of bo th lay and canon laws) in 1925, and for a yea r he ap- p ren t i ced as a "koncypient," a legal clerk, in the Lw6w law firm of his pa terna l uncle, Dr. Henryk Birnbaum.

Although legal educa t ion served him well on many a

la te r occasion, it was not someth ing that Wilhelm wan ted

to do as an occupat ion. He repea ted ly re turned to mathemat ics during his law studies, and for a decade was an ac-

tive par t ic ipant in the intel lectual ly electr ifying and un- o r thodox events that led to the c rea t ion of the Lw6w

mathemat ica l school. His first univers i ty mathemat ics in- s t ruc tor was Stanis law Ruziewicz. The wel l - read Wilhelm was conf ident that he knew everything that needed to be

known about real numbers, and suffered a shock when Ruziewicz p re sen ted an expos i t ion of Dedekind cuts. Then

came Zylinski 's a lgebra lectures, which were entered as the first record, da ted March 10, 1922, in his Indeks, the offi-

cial grade b o o k tha t all s tudents were required to car ry with them and submit for a p rofessor ' s en t ry whenever they

took an examinat ion. The grade, l ike all the others in the mathemat ica l subjects , was "excellent." An invitation to par t ic ipa te in the group theory seminar run by the same

professor fol lowed. Formally, for an addit ional twelve t r imesters , f rom 1925

to 1929, he was a s tudent in the facul ty of Mathemat ics and Natural Sciences, suppor t ing himself f rom another Polish government scholarship. It was the s tuden t generat ion of

Juliusz Schauder , Mark Kac, Stanis law Ulam, Wladystaw Orlicz, Marceli Stark, Henryk Auerbach, Ludwik Sternbach, Stanislaw Mazur and Julian Schreier. " . . . I was a par t of

that generat ion. Mathemat ics in that group of infatuated young peop le was kind of a fever. We would get together

at all t imes of day and night, ta lking incessant ly mathematics. In a r oom which served as a combina t ion of the seminar room and a small l ibrary there was a large ti le

stove, with one side a t tached to the wall. I r emember long hours of freezing winter nights when we were s tanding glued to the wa rm ti les of the three avai lable stove sides,

talking a round corners about mathemat ics ." Initially, he was in teres ted in complex function theory

and inf luenced by Steinhaus and Banach; he took classes

with both of them. The Indeks entry da ted October 24, 1924, records the probabi l i ty theory course in which Steinhaus used Markov 's tex tbook. Bieberbach ' s monograph on func-

t ions of complex var iables had jus t appeared , and the second volume con ta ined a chap te r on univalent functions. They were the subjec t of Birnbaum's doc tora l d isser ta t ion defended in 1929. "Around 1926 I wro te my first pape r in

complex funct ion theory. It was a s imple remark on the Cauchy formula. I initially showed it to Banach who prompt ly k icked me out of his office wi th a succinct com-

ment 'This is nonsense . ' But the nex t day he cal led me back to his office, apologized, and asked me to p repare the pape r for publ ica t ion . . . . During the Rigorosum, a Ph.D. qualifying examinat ion, my nonmathemat i ca l subject was

as t ronomy. Dean Ernst, who was s u p p o s e d to be my ex- aminer in that subject , fell ill, and two o ther examinat ion commiss ion members , Banach and Steinhaus, had been in- s t ruc ted to conduc t the examinat ion in his absence. The

only quest ion a sked was: 'Could you name at least one as- t ronomical r e sea rch ins t rument bes ides the t e l e s c o p e ? ' . . . Banach 's and Ste inhaus 's lecture s tyles were very different. Banach impressed one with the str iking and a lmost

b ru ta l p o w e r of his pure but cold intellect. Steinhaus was

,38 THE MATHEMATICAL INTELLIGENCER

warm, a good motivator , with broad, active in teres ts

s t re tching f rom poet ry to medicine."

Teacher The prac t ica l side of life, however , was not to be forgot- ten. " . . . My family could not suppor t me financially. I l ived

with my pa ren t s but had to cont r ibu te to daily expenses . �9 Birubaum shows me a four-page (wri t ten in tiny calli- graphic handwri t ing) DIPLOMA for high school t eachers is-

sued June 10, 1929. It br ings back memor ies of a s imilar documen t i ssued to my mother , who a few years la ter

pas sed s imilar examina t ions in Warsaw under Dickstein and SierpiAski 's tutelage. It s tar ts generical ly and in the

s t andard officialese: "Mr Zygmunt Wilhelm Birnbaum was admi t ted [ . . . ] on Februa ry 26, 1926, to the examina t ion

for a t eache r of mathematics as the main subject, and

civil educa t ion as an addi t ional subject , in high

schools wi th Polish as the language of instruction."

But the remainder of the d ip loma is an i l luminating document , wor th quoting at length, especia l ly in the contex t of our own current and

perennia l d iscuss ions on the quali ty of the school educa- t ion at the end of the twent ie th cen tury in general, and high schools in part icular . I p reserve the original 's punctuat ion.

The Lw6w State Examina t ion Commiss ion for candida tes to the profess ion of high school t eacher cons is ted of univers i ty professors Kazimierz Chylifiski, StanisIaw

Rdziewicz and Kazimierz Ajdukiewicz. "As home assignment the Commiss ion [ . . . ] a ccep t ed a seminar paper ,

submi t t ed by the candidate , ent i t led 'On weak convergence of sequences of funct ions of a real variable ' , which mem-

oir was judged to be 'very good' ." "The superv ised wri t ten examinat ion: a) in mathemat ics . May 25, 1926. Topics: '1) Prove that

the derivat ive of a cont inuous function, if it exis ts in each

poin t of a cer ta in interval, is cont inuous in the sense of D a r b o u x [ . . . ]. Is the converse true: is every function con- t inuous in the sense of Darboux the derivative of a cer ta in funct ion? [ . . . ] 2) Demons t ra te that the odd per fec t num-

ber (if it exis ts) does not conta in in its decompos i t ion into pr ime factors any factors of the form 4k + 3 in odd p o w e r [ . . . ] 3) Calculate the volume of a tr i-axial ell ipsoid. ' Score:

very good. May 25, 1926. Topics: '1) The re la t ion y + e xy = 0 de-

fines y = y(x); in wha t x interval does y = y(x) exist?

Calculate dy/dz for x = 0. 2) Prove that if Z ~ n = 0 an = A ,

n=0 bn = B, and Z ~ Z k=n n=0 Cn = C where Cn # = 0 an-kbk,

then AB = C.' Score: good. b) in civil education. May 26, 1926. Topics: '1) Posi t ion

of an individual in a monarchy and in a republic---differ-

ences in s tanding 2) Par l iament ' s pos i t ion in a monarchy and in a r epub l i c - -d i f f e rences in s tanding. ' Score: good."

But that was not the end of it. The oral mathemat ics exam took p lace June 24, 1926, and in civil educat ion June 19. The math exam commit tee was chaired by Professor

Mathematics in that group of infatuated young people was kind of a fever.

Chylifiski, and Ruziewicz and Steinhaus served as examin- ers. "The candida te was asked about the fol lowing matters:

Sets of the first and second Baire category, The problem of measure. Lebesgue measure. Nonmeasurab le sets. Euler 's

and Fermat ' s theorems. Proofs of the theorem about existence of infinitely many prime numbers . Proof of the Weierstrass theorem about the exis tence of a complex an-

alytic function with prescr ibed zeros. The fundamental theorem of algebra. Bernoull i-Laplace theorem, Bayes 's rule.

Cauchy's theorem in complex function theory. Proof of the

t ransformat ion of f f f dx dy dz (over the ball) into spherical coordinates . Geodesics. The surface c rea ted by lines tangent to a spat ia l curve. Curvature lines. Orthogonal

systems. Orthogonalization. ' The answers were judged very good. During the exam the candidate d isp layed a sufficient

knowledge of the German language." Comments Biru-

baum: "Probably I would flunk this exam today."

"In civil educat ion [ . . . ] the fol lowing issues were

asked about: 'The voyvod- ship selfrule. Voyvodships of the Eas te rn Malopolska. Ethnic minorit ies. The school- language bill. State budget. Legitimism. Agrar ian reform. Ar i s tocracy and oligarchy.

Propor t ional e lec t ions in general, and the i r organizat ion in Poland, in par t icular . The manner of se lec t ion of the

supreme commander . ' " The last question, a sked a few weeks af ter Marshal PiIsudski 's May coup, could have had

a special h idden significance. In the 1926-27 school year Birubaum taught mathemat-

ics in a pr ivate gymnas ium (grades 4-12) "with the rights to serve the public" run by Dr. Adela Karp-Fuchsowa, and,

during the next two years, in the pr ivate coeduca t iona l gymnasium of the Lw6w Evangelical community; in the spring t r imes te r of 1928 his income was supp lemen ted by

teaching in the Jan Kochanowski State Gymnas ium No. IX. "Initially I was ass igned to a coeduca t iona l c lass of twelve- year-olds. I was convinced that they would be enthusias t ic

about mathemat ics , and, following the example of my dis- t inguished univers i ty professors , I spoke in a low voice facing the b lackboard , concentra t ing on developing ideas, s topping to con templa te the next maneuve r when the p roof did not evolve smoothly. Already during my first lesson I

was a target of p a p e r a i rp lanes and spitballs; laughter and commot ion behind my back overwhe lmed my scholar ly discourse. I was facing the so-cal led disc ipl ine problem. Somehow, I survived it though. I moved to ano ther school

where an o lder and more exper ienced t eache r took me aside and expla ined a few basic pr inciples: 'Always pre- pare your class in advance so that you won ' t have to depend on thinking in front of your s tudents , ' 'Never turn your

back on your e n e m y ' . . . . It took a while, but I managed to shake off the initial shocldng exper ience and la ter was judged to be a compe ten t t eacher . . . . When, in 1974, I

reached the manda to ry re t i rement age I was mos t upse t by the fact that the event took away my regular contac t with undergraduates . For tunate ly , my Professor E m e r i t u s po-


sition permitted continuation of my research and work with graduate students."

To the final pedagogical examination "in the accelerated mode" Birnbaum was admitted by the special ordinance of the Ministry of Religious Denominations and Public Education. On June 10, 1929, Professors Chylifiski, Ajdukiewicz and Lempicki conducted an oral examination and " [ . . . ] the candidate was asked about the following matters: a) in philosophy. Deductive reasoning. Proofs by reductio ad absurdum. Axiomatics. Kant's philosophy of mathematics. Kant's and Brouwer's intuitionism, b) in ped- agogy and didactics. Presentation of axiomatic systems in high schools. Axiomatic and heuristic methods in mathematics. Synthetic and analytic methods. Mathematics teaching in schools (historical sketch). Piarist friars and Konarski. The school system in the Duchy of Warsaw and the Congress Kingdom. Comenianism and its founder. Secondary education in XVI century Poland. The answers were judged very good. The commission concluded that the candidate speaks fluent Polish as the language of instruction."

In combination with the just-acquired doctorate the life of a respected pedagogue awaited. But an opportunity to go to G6ttingen presented itself, and Birnbaum, for the sec-

Wladysfaw Orlicz, who collaborated with Birnbaum in G6ttingen and

Lw6w. This photograph is from the days when Orlicz was leader of

a well-known school of real and functional analysis in Poznar~,

ond time in his young life, abandoned a promising, stable career. He had been saving money for this trip during the previous three years. The day after his official Ph.D. award ceremonies, presided over by Steinhaus, Birnbaum boarded the train and left Lw6w.

G 6 t t i n g e n a n d t h e " B i r n b a u m - O r l i c z s p a c e s "

In October 1929, G6ttingen sti l l preserved a lot of its former splendor. Birnbaum arrived with a doctorate, and a few published papers to boot, and found immediately a mentor in the person of Edmund Landau. But in addition to continuing a vigorous program of original research, he decided to enroll in regular classes. The official records of G6ttingen University provide a complete listing of lectures for which he was registered: Courant--differential equations with recitation sessions (tuition, 17.50 marks); Courant--calculus of variations (only 5 marks); Landau-- power series (10 marks); Herglotz--higher geometry; Courant and Herglotz--mathematical seminar; Bernays-- probability calculus; Wegner--analysis of infinitely many variables. Also, during this academic year he met Felix Bernstein, with whom he took a course in insurance mathematics and a mathematical statistics seminar. The latter choice proved to be momentous.

Fast-forward to 1966 when I became Kazimierz Urbanik's assistant at Wroclaw University, and where my first assignment was to study the paper "Uber die Verallgemeinerung des Begriffes der zueinander kon- jugierten Funktionen" by Birnbaum and Orlicz, published in the 3rd volume of Studia Mathematica. The almost 70- page memoir, easily the longest article published in the Studia before World War II, contained the most important ingredients of the theory of L r spaces, which extended the concept of Lebesgue L p spaces to the case where the integrability of a p-th power is replaced by the integrability condition of the composition of a function with a given convex function (I). The paper utilized the notion of conjugate functions in the sense of Young to develop the duality theory of a new class of functional spaces, very much in the spirit of the Lw6w of those days. The paper was ac- cepted for publication on October 19, 1930 and the volume appeared in 1931. It was preceded by a short note "Uber Approximation im Mittel, I" by Birnbaum and Orlicz, which appeared in the 2nd volume of Studia, and which was ac- cepted for publication on July 31, 1930, where for the first time the notion of approximation with respect to arbitrary means was introduced, and by a solo Birnbaum paper "Uber Approximation im Mittel, II" on a similar topic but presented by Landau in G6ttinger Nachriehten. But for me, and in hundreds of papers of the mathematical literature of the subject, to which I myself contributed, the new spaces were just known as "Orlicz spaces."

Birnbaum's name disappeared without trace. I do not remember my Wroclaw mentors with Lw6w pedigree, Marczewski-Szpilrajn, Steinhaus, or Hartman, ever men- tioning Birnbaum, although because of my own family connections with pre-World-War-II mathematics, all kinds of

4 0 THE MATHEMATICAL INTELUGENCER

his tor ical deta i ls were d i scussed f rom time to time, in-

cluding the role of Mr. Henryk Majko, the famed cus todian of the Warsaw Mathematical Seminar (in Wroclaw after the war) who "befriended" and cor responded with several

p rominent mathemat ic ians from Henri Lebesgue to John von Neumann. But Birnbaum was never present in those remi-

niscences. I myself did not p ress the issue. Perhaps, l ike probably many others, I subconscious ly assumed that, like many Polish mathematicians, he per i shed in the Holocaust.

The question "What happened to the first half of Birnbaum- Orlicz?" was put away on the back shelf of my subcon-

sc iousness and disappeared from my intellectual radar. Since thei r appea rance in 1930-31 the L r spaces found

b road appl ica t ions in harmonic analysis, where f rom they in a sense arose, in funct ional analysis, in par t ia l differen- t ial and integral equations, and in probabi l i ty theory.

Besides the Lw6w "functional thinking," two inf luences on the Birnbaum-Orlicz papers are evident: W.H. Young's 1912 ar t ic le "On c lasses of summable funct ions and thei r Four ie r

series" pub l i shed in volume 87 of the Proceedings of the Royal Society, which conta ins a p r o o f of the inequali ty for pa i rs of conjugate functions, and an ear l ier 1907 Edmund

Landau p a p e r f rom the G6ttinger Nachrichten. HSlder 's in-

equali ty immedia te ly implies that if r = lug, ~ ( u ) =

lul q, 1/p + 1/q = 1, and both ser ies Zi ~P(ai), Zi T(bi) < ~,

then the ser ies Zi aibi converges. Landau proved a sor t of

converse to the effect that if the ser ies Zi aibi converges for each sequence bi for which the ser ies Zi xi~(bi) < ~, then the ser ies Zi dP(ai) < oo. A general izat ion of this the- o rem to the case of arb i t rary conjugate-in-the-sense-of- Young convex funct ions was one of the main resul ts of the

Birnbaum and Orlicz papers . The influential Antoni Zygmnnd monograph on trigono-

metr ic series, publ i shed in 1935, conta ins the first system-

atic expos i t ion of the theory of L(P-spaces, al though it never calls t hem Orlicz spaces . But the two-volume 1959 Cambridge edi t ion a l ready honors tha t name. As the bas ic

source Zygmund quotes Orlicz 's p a p e r "Ober eine gewisse Klasse von R i u m e n vom Typus B" publ i shed in 1932 in the Cracow Bulletin International de l'Acaddmie Polonaise, and the ear l ie r Birnbaum-Orlicz memoi r quoted above is

only men t ioned in the contex t of the above general izat ion of Landau 's theorem, which is given as a s imple corol lary to the Banach-Ste inhaus theorem.

Back to 1997. We are sit t ing in a corner office in the nons tanda rd shape of a nonconvex hexagon, which Birnbaum shares with the re t i red topologis t Ernes t Michael. A shelf is filled with more than sixty volumes of

Probability and Statistics, an Academic Press ser ies of monographs and tex tbooks , which Birnbaum has edi ted (jointly wi th Eugene Lukacs, f rom the beginning until the la t ter ' s dea th) for the last few decades . The desk top compute r ' s sc reen is frozen on the e-mail window; " . . . we are

trying, be tween Hilde, the computer , and myself, to run a fr iendly mdnage ~ t ro i s . . . ," he comments .

This is the present , but we re turn to the old days. "I deve loped the idea to work on those L r papers during my

GOttingen visit, and my conversa t ions wi th Landau had

something to do with it. I shared the thought wi th Orlicz, who, during the first yea r of my stay there, also res ided in

G6ttingen. F rom tha t moment onwards we w o r k e d on that pro jec t together, and in pr inciple the work was f inished before he left G6t t ingen in the Spring of 1930. Our motiva-

t ion was a pure ly inte l lectual curiosity, we had no concre te appl ica t ions in mind. Myself, having wr i t ten the second pa-

pe r on approx imat ion in the mean, I comple te ly abandoned the area. Orlicz, wi th whom I had s ince los t contact , obvi-

ously cont inued to work on the subject , and founded the whole school in Poznafi concentra t ing on re la ted topics. More recently, my Seatt le col league Edwin Hewit t em-

ba rked on a campaign to add my name to the L%spaces n o m e n c l a t u r e ; . . , in 1978, I.M. Bund publ i shed in Silo Paulo a monograph Birnbaum-Orlicz Spaces."

"At the beginning of my second yea r in G6ttingen, Landau offered me an assis tantship, bu t s imul taneous ly sober ly advised me no t to take it. He had cor rec t premo-

nitions. A few years la ter he h imself was unceremonious ly , in the middle of a semester , f ired from his cha i r by the Nazis. Simply, in l ieu of Landau, his fo rmer assistant ,

Werner Weber, showed up at one of his lectures. He brought with him Landau 's lecture notes, ra i sed his hand

in Heil Hitler and announced that from now on he would take over the lec ture and cont inue at a new, higher level guaranteed by NSDAP philosophy. I vividly r e m e m b e r him

from a jo in t d inner of the G6tt ingen Mathemat ische Fachschaf t at der Gasthof zum Kehr loca ted on top of a wooded hill r ising behind G6ttingen, as he enter ta ined a pre t ty female s tuden t under the p re tex t of a d iscuss ion on algebraische Fldchen.

"I took advantage of Landau 's advice and spen t my second year in G6tt ingen working on an ac tuar ia l d ip loma within the new p rog ram es tabl ished by Fel ix Bernstein, who was also a founder of the Inst i tut ftir Mathemat ische

Stochas t ik on LStze St rasse which still exis ts today."

Actuary The new career. The pos i t ion of an ac tuary in the Viennese insurance company, Phoenix, was too a t t rac t ive to turn down. The sa lary was excellent. "I was t aken aback when Professor Berger, who bes ides the Vienna Polytechnic

chair also occup ied the posi t ion of the chief ac tuary at Phoenix, w e l c o m e d me with a s t a t ement that he was famil iar with my jo in t pape r s with Orlicz and had a proba- bilistic in terpre ta t ion for our theore t ica l results . My immedia te superv isor was Edward Helly, who was a

wonderful boss, warm, compass ionate , and with sunny dis- posit ion. I l ea rned f rom him a lot about how mathemat ics can be of service in evaluat ion of actuar ia l t ransact ions.

He, of course, es tab l i shed his mathemat ica l reputa t ion by proving fundamenta l theorems in real and funct ional analy-

sis, but had no luck in the academic career . Throughout the res t of his life he remained jus t a Privat Dozent and was never p r o m o t e d to a well dese rved Professor Extraordinarius--in those days Jews had a hard t ime in

Vienna. My di rec t c o w o r k e r - - o u r desks abu t ted and we


faced each o t h e r - - w a s Eugene Lukacs . . . . " A f t e r the war

Lukacs had a dis t inguished academic career in the Uni ted Sta tes and authored a wel l -known monograph on charac-

ter is t ic functions; his name surfaced in this s tory before, he was Birnbaum's co-edi tor on the Academic Press b o o k

series. "The everyday ac tuar ia l w o r k did not have any signifi-

cant mathemat ica l or s ta t is t ica l component . Mainly it con- s i s ted of legal and computa t iona l chores. My Vienna math-

emat ica l contac ts were minimal. I did mee t Richard yon Mises, who ac ted surpr i sed when I told him tha t I wen t in

detai l through his t rea t ise on the foundat ions of probabi l - i ty theory; he used to boas t tongue-in-cheek that he was the only person who had r ead the book with any under-

s tanding. Later, in 1933, I wro te with Schreier a small pa- p e r for Studia Mathematica on the law of large numbers , showing how von Mises 's RegeUosigkeits Axiom could be

r igorous ly es tabl i shed within the Kolmogorov axiomat ics ." At that t ime the Polish commiss ione r of the State Office of Insurance Control had i s sued an execut ive o rde r that the

Pol ish subs id iary of the Phoen ix Corpora t ion be spun off into an independent entity. The chief ac tuary pos i t ion in

the newly formed firm with

ered that, miraculously, our run-down family proper t ies in the Pysznica townsh ip ended up r e c o r d e d in the f irm's financial b o o k s as Feniks, Inc. 's asse ts va lued at one mill ion

gold U.S. dollars. To this day I have no idea how this happened. By that t ime Pysznica had a lmos t no value, all the

t imber having been logged a long t ime before. "That pe r iod also wi tnessed my bap t i sm by fire as a la-

bor activist. After Feniks, Inc. filed for bankruptcy , the gov-

e rnmenta l admin i s t ra to r of lef tover asse t s f ired all the employees. In response , we organized a cell of the Labor

Union of Whi tecol lar Workers and I was e lec ted the shop s teward. Fac ing no progress in negot ia t ions with the man-

agement, we dec la red an occupa t iona l s t r ike of our third- floor offices. There were about thir ty str ikers, and the food suppl ies were del ivered by our families on ropes lowered

through the windows. After a few days the government threw in the towel and s u m m o n e d me for negot ia t ions in Warsaw. That was my first a i rp lane flight. We kept our

jobs but not for long. It was c lear that bo th my local posi- t ion and the genera l a tmosphere in Europe were not

promis ing . . . . "I left Lw6w on May 3, 1937, and was not to see my par-

ents nor my sister, Franciszka,

the Polonized name Fen iks was offered to Birnbaum, who

immedia te ly accep ted it wi th joy, and re turned for the second t ime in his life to his nat ive LwSw, where the Fen iks headquar te r s was to be located. The year was 1932, and for

the nex t five years Bi rnbaum led the life of an es tab l i shed and affluent businessman. " . . . during those years I t ook par t in the Scot t ish Caf6 and seminar life only sporadical ly ,

f rom t ime to t i m e , . . , the in teres t remained, but there was s imply no t ime for r e sea rch work."

But "business as usual" in the management of the Fen iks

did no t last long. Something s t range s tar ted happening in the Viennese center; as ff s o m e b o d y t r ied to hide the f i rm's funds f rom the Nazis. There was pressure on the Lw6w

subs id ia ry to cook the b o o k s and the annual f inancial report . " . . . I had a c o w o r k e r then by the name of Ludwik Sternbach, and the Vienna centra l office demanded tha t he and I turn our financial b o o k s over to them. Before ship-

ping the documenta t ion the two of us pho tog raphed the whole thing on glass negatives. Sure enough, af ter a few months the papers were r e tu rned to Lw6w but obviously the numbers had been doctored . Soon thereaf te r the f i rm

fi led for bankrup tcy and the Polish insurance commis- s ioner appoin ted a l iquidat ion commission. I was one of the commiss ion ' s members . As the external examine r the government named Zbigniew {~omnicki, an ac tuary at the

PKO Bank in Warsaw and the nephew of Antoni {,omnicki, p ro fe s so r of mathemat ics at the Lw6w Polytechnic (the lat- te r is bes t r emembered by mathemat ic ians as a pe r son who offered the first academic pos i t ion to a diploma-less Stefan Banach). The affair of doc to red books immedia te ly sur-

faced, but our glass negat ives he lped the commiss ion to re- cover the truth. The pr incipal legal counsel and the main b o o k k e e p e r ended up in jail. In the process , I had discov-

Orlicz spaces are seven years my junior, again. They were seen for the last t ime

Birnbaum-Orlicz spaces, in the Bergen-Belsen concen- t ra t ion camp . . . . An American

visa was not easy to obtain. People would wai t for them for ten years af ter filing an applicat ion. However , my cousin

Ludwik Rubel, who was the edi tor- in-chief of the Cracow paper Ilustrowany Kurjer Codzienny, famous in Poland under the n i ckname Ikac, took me to the Amer ican con-

sulate and in t roduced me as his n e w s p a p e r ' s reporter . My journey took me to Vienna where I bade farewell to my relat ives and acquaintances , and to Paris, where with my re-

por te r ' s c redent ia l s I managed to see the World Exhibi t ion a few days before its official opening. The then French pre- mier, L6on Blum, su r rounded by a f lock of officialdom, had

jus t been given a tour of the exhibi t ion grounds. I t ook some pic tures and exci tedly hurr ied to Le Monde to have them developed, sensing the occas ion as a breakthrough

in my journa l i s t ic career. However, n o b o d y showed any inte res t in using my services . . . . I left for Amer ica from Le Havre on the l iner M.S. Georgia of the Cunard White Line. Two years la te r she had the doubtful dis t inct ion of being

the first al l ied passenge r vessel sunk by the German U-

Boats."

Statistician The New York career of the Ilustrowany Kurjer Codzienny r epor t e r did not last long. "Short ly after my arrival in New York, I was stroll ing in Columbus Circle where a cer tain individual, pe rched on the proverb ia l soapbox, kept abusing Frankl in Delano Roosevel t with the most dis-

gusting a d j e c t i v e s - - b y the way, that pe r fo rmance significant ly enr iched my English vocabulary. A fat pol iceman, looking per fec t ly the role of a New York Irish cop from the Amer ican suspense movies, s topped brief ly in front of the


speaker , tu rned around, spa t a d i s tance that I still bel ieve

to be a wor ld record, and then jus t wa lked away. That was my first, shocking lesson in the Amer ican concept of free- dom of speech . . . . This did not mean that everything was

hunky-dory. F o r a while I r oomed at Columbia Universi ty 's In ternat ional House and bef r iended an at t ract ive b lack fe-

male s tuden t from one of the Car ibbean Islands. Soon I was warned that our walks toge ther were not rece ived

kindly by others. That was also a shocking exper ience, but of a different kind."

Bi rnbaum renewed his G6t t ingen acquaintances.

Courant and Fel ix Berustein were a l ready in New York, and the la t ter offered him an ass is tantship in the Biometr ics

Depar tmen t at New York University. The soft money did not last, however , and he had to look for new employment : the ca ree r of a s tat is t ical consul tan t in pr ivate prac t ice was

not very p romis ing either. "A suggest ion that I submit a j o b appl ica t ion in Seatt le came from Harold Hotell ing whose s ta t is t ical s emina r at Columbia Univers i ty I had s ta r ted to

attend: 'Do send an applicat ion, they need somebody to teach stat is t ics . But do not ment ion that it was my idea,

I 'm on the b lackl is t there . . . . ' My r ecommenda t ion le t ters were f rom Richard Courant, Edmund Landau, and Alber t

Einstein. I was in t roduced to the la t te r during a shor t formal visit. I could not go for an in terview to Seattle, so it was conduc ted in New York by two gentlemen. One was the Pres ident of the New School for Social Research; the

second, the chief execut ive of the Sun Oil Company. Apparen t ly I made a good impress ion, because soon there-

af ter I r ece ived a wri t ten offer f rom Professor Carpenter , cha i rman of the Mathemat ics Depar tmen t in Seattle. He also a t t ached an admoni t ion that I pol i sh up my English before my arr ival at the Universi ty of Washington to the

poin t that even f reshmen could under s t and it. I therefore spen t the res t of the Summer of 1939 at the Universi ty of

Vermont tak ing intensive language courses and swimming in Lake Champlain."

Bi rnbaum reached Seatt le in Oc tober 1939. The car he was t ravel ing in from San Franc i sco ran into a di tch on a

snowy mounta in pass in the Siskiyous, and the tr ip had to be comple t ed by train. "In the Depar tmen t of Mathemat ics of the Univers i ty of Washington the s ta t is t ic ians and appl ied mathemat ic ians were v iewed as if they had grease un-

der thei r fingernails. But we s tubborn ly deve loped the mathemat ica l s tat is t ics program, and in 1948 1 founded the Stat is t ical Research Labora tory which, for the next 25

years, was funded by the Office of Naval Research. At a cer ta in po in t we had twelve full-time professors of statistics. In 1979, three years af ter my ret i rement , the university finally agreed to emanc ipa te s ta t i s t ic ians and crea te a separa te Stat is t ics Department .

"My re sea rch was then concen t r a t ed in the area of prob- abil ist ic inequalit ies, in par t icu la r the two-dimensional Chebyshev inequality. When I p re sen ted it to George P61ya at Berkeley, the previous exper ience with Banach was re-

peated. P61ya's initial reac t ion was 'but this is trivial, ' bu t after a day 's reflection, he invited me to publ ish it. I was also in te res ted in the effects of p rese lec t ion on multivari-

ate distr ibutions. One of the first quest ions was how con-

s t raints on one c o m p o n e n t affect the jo in t dis t r ibut ion of the random vector. Here, the p rocess of admiss ion of students to a university, or of catching fish in a ne t with a

fLxed mesh size, a re good examples . This was a continua- t ion of Fel ix Berns te in ' s ideas.

"I met Johnny von Neumann at an IBM symposium. I

ment ioned to him tha t Kolmogorov had found an asymptot ic dis t r ibut ion of the so called Kolmogorov-Smirnov sta-

tistics. It was the yea r 1950 and von Neumann was ab- sorbed by cons t ruc t ion of the first digital computers . I

sugges ted that they be used to calculate the dis t r ibut ions of these s ta t is t ics for samples of finite size. He he lped me

to obtain funds for these calculations, which were done in the Western Compute r Center of the Nat ional Bureau of S tandards at the Univers i ty of California at Los Angeles.

My interes t in distribution-free statistics las ted much longer and during m y s tay at Stanford, jo in t ly with Herman

Rubin, we formula ted the conceptua l founda t ions for such s tat is t ics and p roved a few charac ter iza t ion theorems."

A col labora t ion with the Boeing Airplane Co., with its

headquar te rs and r e sea rch labs loca ted in Seattle, origi- na ted in 1958 and las ted for a decade, mos t ly in the area

of test ing mater ia l fat igue and rel iabil i ty of sys tems with many components . "Boeing's in teres t was spa rked by a se- r ies of c rashes of the new British Comet je ts . Boeing was

jus t gearing up for the p roduc t ion of its f irst 707 model. The British formed a Royal Commiss ion to s tudy the causes of the accidents , and af ter careful s tudy ar r ived at the con- c lusion that the reason was mater ia l fatigue: the sharp an-

gles of the rec tangular w indow openings, combined with the al ternat ing pressur iza t ion and depressur iza t ion cycles, were a deadly combinat ion. Of course, Boeing did not want

to commit the same errors. It is in teres t ing that fifty years ear l ier another Brit ish Royal Commiss ion found a similar cause for initially unexpla inable rail accidents . Boeing ini-

t ially re ta ined me as a consul tant for the i r r e sea rch orga- nization, the Boeing Scientific Research Laborator ies , to d i rec t bas ic r e s e a r c h - - ' f u t u r e work ' , as they cal led i t - - o n

mater ia l fatigue and on the rel iabil i ty of complex structures. With a t eam of mathemat ic ians , some of them my former students, we formula ted bas ic concep t s of the mathemat ica l theory of rel iabil i ty of mul t i -component sys-

tems, and ob ta ined many fundamenta l results . "I have also served as a consul tant to the U.S. National

Center for Health Statist ics, developing new mult iple-sam- pling designs and s tudying biases and var iances in infant mor ta l i ty rates. This w o r k led to publ ica t ions with Monroe

Sirken, another fo rmer student. Fur thermore , NCHS sup- po r t ed my work on the mathemat ica l theory of compet ing risks, which resul ted in a monograph. The jo in t work with Monroe Sirken ini t ia ted a sequence of fur ther pape r s that have appea red over many years.

"I became an act ivis t in the Inst i tute of Mathemat ical Stat ist ics (IMS) by accident . At one of the annual meet ings

of the society one of the topics deba ted was the racial seg- regat ion encoun te red at meet ings organized in the

Southern states. One of the shocking racial inc idents in-


volved David Blackwell , Professor of Stat is t ics at the Univers i ty of California, Berkeley. He was not pe rmi t t ed to lodge at the host dormi tor ies used by par t ic ipants or dine

at c o m m o n facilities. Harold Hotelling, who by then had al- r eady moved from Columbia to the Universi ty of North Carol ina at Chapel Hill, appea led for calm and not rocking

the boat; people here a re civilized, the racial r appor t is im- proving, and we should work harmonious ly within the ex-

ist ing structures. I s t rongly disagreed. As a resul t of this in- c ident I was appoin ted cha i rman of the euphemis t ica l ly

n a m e d Committee on Physical Facilities for Meetings, which, for the next few years, was supposed to verify whe the r the universi t ies offering to host IMS meet ings sat- isf ied certain min imum ant isegregat ion condi t ions. There

were cases when our commi t t ee re jec ted conference pro- posa l s because we felt tha t the guarantees were not suffi- c i e n t . . , my subsequent e lec t ion to the IMS p res idency was

par t ia l ly re la ted to my act ivi t ies on that commit tee . "I was the Editor-in-Chief of the Annals of Mathemat-

ical Statistics for three years, f rom 1967 to 1970. Towards the end of my term, which I accep ted with grea t relief, I made a r ecommenda t ion tha t the Annals be spli t into two

series: Annals of Statistics and Annals of Probability. This p roposa l has been implemen ted by my successors , Ingram

Olkin and Ron Pyke. "Life is a s tochas t ic process , and in many ins tants the

t r a jec to ry of my life could have deve loped in a to ta l ly dif-

ferent direct ion . . . . These days I 'm occupied with the problem of charac ter iza t ion of b imoda l dis t r ibut ions which are mix tures of two un imodal populat ions . Among topics I w o r k e d on I cannot r e m e m b e r even one which could be

ca l led ' real appl ied s ta t is t ics . ' My interes ts a lways lay in the mathematics of stat is t ics . Certainly it had someth ing

to do with my Lw6w background. I a lways get mos t satisfac t ion from doing nontr iviai mathemat ics . Perhaps , my computa t ions of corre la t ions , done while I was trying my hand at consult ing in New York City, were c loses t to the

defini t ion of true appl icat ions ."

Hi lde Saturday. We are eat ing lunch in the posh dining room of

the Ida Culver House on Greenwood Avenue, where the Bi rnbaums moved a couple of years ago. Their smal l apartment, filled with works of ar t and antique furni ture inher-

i ted from Hilde's parents , is par t of the large and upsca le c omp lex for affluent ret irees. Beyond windows that cover the whole wes tern wall of the dining room, one can see below, pas t the dense w o o d s of the coas ta l State Carkeek Park, pa le in the drizzle, the waters of the Puget Sound. Bi rnbaum and I o rder Reuben sandwiches , he will finish

only half of his, and Hilde eats a light seafood sa lad served by a wai te r in a fancy uniform. I have my own af te rnoon ta lk at the Universi ty of Washington to look fo rward to,

which leaves about three hours for our conversat ion. Hilde is d ressed with great a t tent ion to detail , and radi-

a tes energy and inte l lectual vitality. In her p r e sence the conversa t ion natural ly turns to her and Bill 's c o m m o n interes ts , of which there are many, some quite nontrivial . Her

father, Dr. Richard Merzbach, was a wel l -known lawyer in Frankfurt-on-Main, and she grew up there and s tudied law. " . . . as a s tudent , I spent the Summer of '31 in London, as

an intern at a law firm. The n e w s p a p e r headl ines an- nouncing the first e lectoral successes of the Nazis, when

their Reichs tag represen ta t ion j u m p e d from 5 to 112 dele- gates, caught me during a midnight stroll through the Piccadi l ly Circus . . . . I burs t into tears right there on the

street. The nex t day I wrote to my pa ren t s that as a w o m a n and a J ew I cou ld never prac t ice law in Germany, and that

I 'd l ike to se t t le in England. There was a good chance that I could remain employed at the law f irm at which I was doing my internship. My father t r ied to convince me to re-

turn. The family let me know that they bel ieved that my out look was too pessimist ic. They could not see Hitler com-

ing to power . So, I re turned to Germany and graduated with a law degree in 1932. According to the rules governing the legal p rofess ion in Germany, I had to prac t ice for a while

as a cour t clerk. "But a yea r la te r Hitler did come to power, and April 1,

1933, was dec la red the day of the ant i -Jewish boycot t . It

was no Fool ' s Day joke. On March 31 I was informed tha t

there was no need for me to show up at work the next day. At midnight we left with my s is ter for England. In the middle of the Depress ion it was not easy to obtain a work permit. But the lawyer I worked for two years ear l ier had a

cl ient who n e e d e d a London representa t ive . The pos i t ion was to be funded by a Swiss conglomera te . In this fashion

I became thei r representa t ive and managed their three product ion faci l i t ies and shops. I was not very fond of that j ob

but I d id all right. "Almost f rom the beginning of my s tay in England I was

trying to convince my family to emigrate. But in 1933 the major i ty of peop le in Germany and England did not th ink

it r easonab le to abandon the secur i ty of the family dwell ing for an uncer ta in exis tence of the exile. I often served as a cour ier moving money and o ther mater ia l s from Germany to England. I usual ly t raveled during hol idays as the bor-

der c ross ings were more c rowded and detai led cus tom checks less l ikely . . . . My pa ren t s had their roots in Frankfur t and owned cons iderab le p rope r ty there. The

thought of abandoning the home town was for them un- bearab le . . . . In 1936, I even t rave led to the Middle East in- vest igat ing a poss ibi l i ty of obta ining for them a visa to

Palestine, which in those days was a Brit ish mandate . The Nazis pe rmi t t ed Jews emigrat ing to Pales t ine to take 80~ of their proper t ies , whereas the Jews leaving for o ther des- t inat ions cou ld t ake only a small f rac t ion of their belong- ings. But my paren t s would not agree to that either. They

left jus t a few weeks before the KrystaUnacht . . . . In 1938 they jo ined my sister, who had ear l ier marr ied and se t t led in Seatt le . . . . The government decree , publ i shed in the newspape r s by the Nazis, dec la red me an enemy of the

Reich, s t r ipped me of cit izenship, and conf isca ted my whole proper ty . I became stateless. After that I s topped visiting Germany but did not give up on England, where I felt comfortable . However, in Sep tember of 1939 the atmo-

sphere in London became very tense, the civil defense o r -

THE MATHEMATICAL INTELLIGENCER

Hilde Birnbaum in the Birnbaums' Seattle apartment, 1997.

ganized exerc i ses in gas masks, t r enches were being dug in the City . . . . I was offered a pos i t ion of d i rec tor of our company ' s Brazilian subsidiary, but af ter many emot iona l

t ransa t lan t ic te lephone conversa t ions with my p a r e n t s - - quite unusua l in those d a y s - - I dec ided to emigrate to the

United States�9 "I met Bill through my brother- in- law . . . . For six months

he t r ied to make a mathemat ic ian out of me, but to no avail. Compet ing with the husband in the a rea in which he was

so much s t ronger did not make any sense . . . . F rom the very beginning we had many c o m m o n interests: law, social i ssues such as the terr ible s ta te of heal th insurance in America, and, in part icular , the dis t ress ing pos i t ion of

blacks, c o n s u m e r protect ion, l iberal polit ics. I enrol led in the economics p rogram at the Universi ty of Washington; the German law d ip loma is not easi ly exported. After grad- uat ion I b e c a m e a teaching ass is tan t with a p rospec t for a pe rmanen t pos i t ion in the Depar tmen t of Economics�9

Short ly thereaf ter , however, I rece ived a le t ter from the Universi ty 's p res iden t informing me tha t the fact that both Bill and I hold pos i t ions at the Universi ty viola tes

Universi ty 's nepo t i sm rules. If Bill he ld only a par t - t ime posi t ion then pe rhaps something could be ar ranged for me. Our choice was obvious. Bill 's ca ree r had to have the pri- ority. We had a small child and the s econd was on its way. � 9 After the chi ldren grew up I taught economics at one

of the local col leges and was the head of the Economics

depa r tmen t there for 10 years." The lunch is a lmost over; we o rde r decaffe inated cof-

fees and desserts . Hilde continues, "In 1946 1 was one of the founders of the Washington State b ranch of Amer icans for

Democrat ic Action, and la ter on worked as a Washington

State lobbyist for the Amer ican Federa t ion of Teachers . . . . Doctors had in those days an a lmost unl imi ted f reedom of

action, including the right to refuse t r ea tmen t to uninsured pat ients . . . . Bill and I he lped to organize the Group Health Cooperat ive of Puget Sound, taking over the prac t ice of a

small group of phys ic ians who during the war had a con- t rac t with the, then very active, Seatt le shipyards . Bill 's ex-

per ience in insurance affairs was very helpful. Initially he served on the Board of Trustees but res igned when we left for his sabbat ica l at Stanford. After our return, I was

e lec ted to the Board where I served for 23 years including four years as its chairperson. We fought p i t ched bat t les with the Amer ican Medical Associat ion, which was trying

to boyco t t us. Many physic ians would agree to work for us only on the condi t ion of anonymity. We bui l t severa l hos-

pi tals and clinics and today, with 700,000 subscr ibers , we are the largest heal th insurer in the Pacific Northwest . Bill is still a m e m b e r of the Grievance Commit tee . . . . Of course

by now this kind of heal th main tenance organizat ion has

become commonp lace all over the country. "In 1946 Bill was appo in ted cha i rman of the commit tee

to analyze the Univers i ty 's re t i rement and heal th insurance system, and p repa red a draft of s ta te legis la t ion which is current to this day. In 1955 he p lanned and supervised the

re fe rendum on merging the Universi ty sys tem into the na-

t ional Social Secur i ty System. "In 1947 we, toge the r with a group of friends, bought 70

acres of land on the hills be tween the towns of Bellevue and Renton, sou theas t of Lake Washington. In those days it was jus t a 20-year-old forest s la ted for logging�9 We buil t some forty homes and founded a township which we called

the Hilltop Community. Until this day it is an independent corpora t ion in spi te of the annexat ion efforts of the neigh- boring towns. We lived there for 27 years and our two children grew up t h e r e - - b o t h set t led in S e a t t l e - - r a r e luck for

paren ts in Amer ica . . . . But the name of Bi rnbaum is quite popular in the Uni ted States outs ide Seatt le as well. Perhaps it has someth ing to do with the fact tha t Bill 's old-

est pa terna l uncle had eighteen children." We talk about t ravels and Birnbaum reminisces: "In 1960

I spent my sabbat ica l in Paris. We lived in a two- room pent- house apar tment in a small hotel at 29 Rue Casse t te on the Left Bank. Initially we p lanned to s tay at the hote l only for

a few days, but Hilde d iscovered by acc iden t a newspape r ad informing the publ ic that the hote l wan ted to let the pen thouse apa r tmen t on a long-term basis. We had no

kitchen, but Hilde t ook advantage of our ear l ie r camping exper iences in the Seat t le a rea and organized a makeshi f t k i tchen in the b a t h r o o m using a small e lec t r ic stove. She even managed to p repa re roas t bee f there�9 I w o r k e d a lit-

t le bi t on finishing my book and met wi th Paul L~vy. But Paris i tself was very absorbing�9 We spent a lot of t ime at the museums and visit ing with European fr iends . . . . The Polish Academy of Science invited us in 1963, and we vis- i ted Warsaw, Wroclaw, and Cracow�9 We also t rave led to the mounta in resor t of Zakopane to see Steinhaus. It was

a very emot ional reunion . . . .


"I still r e m e m b e r an evening at the apa r tmen t of one of

the Warsaw mathemat ic ians . That was the t ime of a political t haw in Poland, and Stan Mazur, who held a high po- s i t ion in the Communis t Party, conf ided to me that tel l ing

pol i t ica l j okes was now O.K. People heard that I was su- ing the United States Government and worr ied whe the r it was safe for us to re turn home and if our j obs would still be wait ing for us on our return. I en te red the l i t igation jo in t ly with the Amer ican Civil Liberties Union, in which

I 'm act ive until this day, t rying to get the Loyalty Oath required of all Universi ty of Washington employees dec la red unconst i tut ional . The case ended up before the U.S.

Supreme Court, and my depos i t ion was the only one quoted in the highest cour t ' s opinion; the decis ion was in our favor.

"During my next sabbat ica l we spent four months in

Rome. At the beginning we lived far from the ci ty 's center . One day, stroll ing on a s ide s t ree t in the ne ighborhood of the Spanish Steps, Hilde no t iced a discreet , sma l lpens ione . The lady owner was su rpr i sed when we en te red asking for a room, and inquired who r e c o m m e n d e d us. It tu rned out

that the pensione was rese rved for a careful ly se lec ted

clientele, main ly internat ional tycoons and Hol lywood

types. But Hilde made a great impress ion on the owner who turned out to be a German-born I tal ian baroness . The con- tessa would br ing us a daily b reakfas t to our room and af- t e rnoons we par t i c ipa ted with o ther gues ts in lively wine part ies."

The lunch is over and we bid each o ther good-byes. "Please, s top in Seatt le and visit us again when you t ravel to Tokyo in June. It 's on the way."

Cleveland Sunday. My flight home leaves a round noont ime. Light rain and foggy. The day is merci less ly shor tened by the t ime

zones be tween the West and Eas t Coasts. During the pa s t 48 hours Bi rnbaum had become a na tura l and obviously missing par t of my own tradit ion. My pr ivate "rediscovery" of Bi rnbaum was, in a sense, serendipi tous . Two years ago

at the spec ia l conference in honor of Kazimierz Urbanik, my Ph.D. advisor, I ran into Roman Duda, a topologis t and pres iden t of the Wroclaw University, who asked me if I would be willing to write someth ing abou t Birnbaum for

the p lanned Centennial issue of the Annals of the Polish Mathematical Society. I re luctant ly agreed but p leaded my

ignorance of the subject. Besides, as far as "after-dinner" l i terary p ro jec t s were concerned, I a l ready had on my desk the English vers ion of Banach ' s biography. At the same

t ime I felt under pressure to oblige, as I had a debt to re- pay. A few decades ago, when bo th of us served his Excel lency Rec tor Marczewski as secre ta r ies of the

Colloquium Mathematicum, Roman did the overwhelming major i ty of the work. The writ ing was pu t on a fas ter t r ack when Chris Burdzy and Richard Bass invited me to give a

talk at the Pacif ic Nor thwest Probabi l i ty Seminar organized by the Oregon, British Columbia, and Seatt le probabi l is ts .

I cannot res is t looking at the Bi rnbaum story through the g lasses of my own exper iences . Toute proportion gardde--I did no t found any cities, and did not sue the Amer ican Government in the Court of Law. There is also

this trifling 40-year difference in our ages. But the para l le ls and poin ts of t angency are unmis takab ly there: the LwSw- Wroclaw t radi t ion and fasc inat ion with Banach and Steinhaus, a ma themat ics Ph.D. but only af ter a d ip loma in

another discipl ine, the Birnbaum-Orlicz spaces , GSttingen, Paris, and a s tubborn struggle for mathemat ics that is not an i so la ted enterprise, but ma themat ics in the fore- ground always, teaching as one of the great joys in life,

c rea t ion of new stat is t ics programs, social and edi tor ia l a c t i v i t i e s , . . .

In Cleveland the sky is ink-black, freezing, the visibil i ty

excellent. We land from the nor th over the par t ly frozen Lake Erie. On the left I seek the l ights of Shaker Hights. Lauren and Greg are impat ient ly wai t ing with the usual "Dad, i t 's good to have you back home." I have for them shr ink-wrapped samples of the minera l s from the Wild

West bought at the last minute at the Seatt le airport: ob- sidian, quartz, sulphur, t e r r a c o t t a , . . .


l l d , [= -aa~ ' j IR ' i~ l , z : - ] i i~ l i l z r D i r k H u y l e b r o u c k , E d i t o r J

Kepler in Eferding Karl Sigmund

I f you follow the summer crowd of tourists biking along the Danube, you

may discover, close to Linz, a short de- tour leading through shady woods to Eferding. This is a quiet little Upper Austrian town, offering the usual sight- seems' fare: castle, church, and market- place. The first house on the main

square used to be an inn. High up on the front wall, a plaque: on October 30, 1613, the astronomer Johannes Kepler celebrated here his marriage to Susanne Reuttinger, the daughter of a burgher from Eferding.

By then, Kepler was a widower of 42. Born and raised in Wiirttemberg, he

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cafd

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

I f so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 �9 Belgium e-mail: dirk.huylebrouck@ping,be

The Keplerhof Inn (formerly The Lion) on the main square of Eferding and the plaque com-

memorating Kepler's wedding. The house, which is well over five hundred years old, has lately

become derelict and will probably be taken over by a bank. Encased in one of its walls is the

tombstone of a Jewish refugee from Regensburg who had found shelter in Eferding in the

year 1410.

�9 2001 SPRINGERWERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 4 7

Kepler in 1620, age forty-nine. According to his friend, the poet

Lansius, this is how Kepler did not look. Lansius jocularly put the

blame on the motion of the Earth (a heresy at that time, as Galileo

came to learn): if the Earth had stood still, the artist's hand would

have been steadier.

Measuring barrel contents. The left barrel is measured in the Austrian

way, by help of a gauging rod. The other barrel's volume is deter-

mined by pouring its content into vessels of specific volume. This is

from the title page of a book on analysis, which appeared in 1980.

The drawing is from the title page of a treatise by Johann Frey, which

was published in 1531 in Nuremberg. The formula (in white) is

Kepler's barrel rule. (From the cover of Analysis 1, 126, I.)

had broken off his theological studies in Ttibingen to become professor of mathematics at a college in Graz. Later, he joined the as t ronomer Tycho Brahe in Prague, as one of the many scientists, astrologers, and alchemists at- t racted there by Rudolf II, the oddest of all Habsburg emperors, a dreamer suffering from fits of madness, who ended as a prisoner in his own castle. Kepler, who furnished his fair share of horoscopes, eventually held the job of Imperial Mathematician under three Habsburg rulers. Each was more nfili- tantly Catholic than his predecessor, unfortunately, and this made court life difficult for Kepler, a staunch Protestant. After Rudolfs death, he took a second job as "district mathematician" in Linz. This implied cartographical work, among other things. Kepler, by then somewhere between his second and

third law on planetary motion, was already a celebrity in European science, but this cut no ice with the suspicious farmers, who often chased him ignomi- nously away from their land.

Having gone through an unhappy first marriage, Kepler took great pains to avoid all mistakes on his second matri- monial adventure. We know from a long and almost comically candid letter (dated from Eferding one week before his wedding and addressed to a scholarly nobleman) that he had wavered for two years between no fewer than eleven candidates, among them a widow and her daughter. Some were too young; some, too ugly. Some seemed inconstant, and others lost their patience with his tem- porising, which became the talk of the town. Eventually, Kepler decided for number five, against the advice of all his friends, who deemed her too lowly.

Susanne was seventeen years younger than Kepler and seemed mod- est, thrifty, and devoted. He had met her in the household of a friend with a ring- ing name--Erasmus von Starhemberg-- whose palace dominated Eferding and whose family popped up in every century of Austria's history. Erasmus had studied in Strasburg, Padua, and Tfibingen, where he may well have met young Kepler for the first time. He was in sym- pathy with Kepler's religious plight--a few years later, at the outbreak of the Thirty Years War, he would himself be branded as a "main rebel" by the Catholic establishment--and had arranged for Kepler's transfer to Linz. (Later, when Starhemberg was imprisoned, Kepler wrote to the Jesuit priest Guldin, a professor at the University of Vienna and no mean mathematician himself, to ask him to intervene at the imperial court.)


As for Susanne, she was an orphan: she had no money, but on the o ther hand,

no in-laws either. She was a ward of Baron Erasmus ' s wife Elisabeth, who

pa t ronized an inst i tut ion for the up- bringing of impover i shed young ladies. After having taken the plunge, Kepler

never ment ioned his spouse again in all his copious cor respondence , excep t on

the seven occas ions when she gave birth. Based on this, all b iographers agree that the marr iage indeed was a

happy one. The Eferding wedding p lays a curi-

ous role in the preh is tory of calculus.

In Kepler ' s words:

After my remarr iage in November of

last year, at a t ime when bar re ls of wine f rom Lower Austr ia were

s to red high on the shores of the Danube near Linz after a copious vintage, on offer for a r easonab le

price, it was the duty of the new husband and devoted family-head to pu rchase the dr ink needed for his

household . Four days af ter severa l bar re ls had been brought to the cel- lar, the wine-sel ler came with a rod

which he used to measure the content of all barrels , i r respect ive of thei r form and wi thout any fur ther

reckoning or computat ion. The metal l ic end of the gauge-rod was in t roduced through the bung-hole

till it r eached the oppos i te poin t on the b o r d e r of the bar re ls ' s bot tom. � 9 I was amazed that the diagonal

through the half-barrel could yield a measure for the volume, and I doub ted that the me thod could work, s ince a much lower bar re l with a somewha t b roader bo t tom

and hence much less content could have the same rod-length. To me as a newlywed, it did not seem inop- por tune to investigate the mathematical principle behind the preci-

s ion of this pract ical and widespread measurement , and to bring to light

the underlying geometrical laws.

Pos ter i ty did not r ecord wha t Susanne

made out of this. Kepler, whose father had been an

innkeepe r when not abroad as a so ld ier of fortune, mus t have been on famil iar te rms with wine-casks of all shapes.

The fact that thei r content was mea-

sured by more compl ica ted means in o ther countries, for ins tance on the

Rhine, r endered him suspicious. But a few days suff iced to convince him of the validi ty of wha t he t e rmed the Austr ian method. He wrote a shor t

note, and ded ica ted it, as a New Year 's gift, to Maximil ian von Liechtenstein

and Helmhard Jhrger, two of his gen- erous suppor ters . He nex t t r ied to pub-

lish the l ea f l e t - - a t tha t t ime, an onerous enterpr ise that required, for

s tar ters , buying the neces sa ry reams of paper . Actually, Kepler had even to convince a printer, first, to set up shop

in Linz. The inevi table delays, which t ook a lmost two years, offered him the

oppor tun i ty to ex tend his resul ts con- s iderably. His Nova stereometria do- l iorum vinar iorum grew to a full-

f ledged book. The first par t dea ls with

Title page of the Nova Stereometria. When well-meaning experts told Kepler that a mathe-

matical text, and in Latin at that, would never find buyers, he produced a German version

(The Art of Measurement of Archimedes), which appeared in 1615 and must be one of the

first examples of popular science writing: it was considerably shorter than the Stereometria,

written in down-to-earth language, and divested of most proofs. Kepler also wrote the first

science fiction ever, an account of a voyage to the moon. He decided not to publish the in-

tegral text of his "Dream" during his lifetime, but it raised rumors of black magic which sur-

faced during the nearly fatal witchcraft trial that his mother had to undergo in her last years�9

Kepler seems to have been the first to see science as the cumulative effort of successive

generations.


cuba tu res in general, and in par t icu lar wi th the volumes of sol ids of revolu-

tion. The second par t dea ls wi th barrels. F o r Kepler, these were somet imes cylinders, somet imes they cons i s ted of

two t runks of a cone, and somet imes they were what he t e rmed "lemons" (ob ta ined by rotat ing a semic i rc le ' s arc

a round its chord) whose top and bottom had been sl iced off. The third par t

of his book dealt with prac t ica l problems in measur ing the con ten t of tota l ly or par t ly filled casks.

Kepler tried to avoid all algebra, and wrote in the style of Greek geometers.

But the content of his book was not at all classical. In a remarkable display of

intuition, he anticipated parts of calculus, arguing about infmities with a non- chalance quite foreign to the rigor of the

exhaust ion method of Archimedes (who

is invoked a great deal). For instance, Kepler cons iders the a rea of the circle as being made up of infinitely many triangles having one ver tex in the center ,

and the oppos i te base, r educed to a

point, on the circumference. If the circle rol ls a long a line for one full turn, the base l ines of the tr iangles cover an interval. The triangle with this base,

Kepler's proof that the area of the circle is half the product of the radius times the length of

the circumference (from the Art of Measurement of Archimedes). Early in the book, the num-

ber pi is given as 22/7, but Kepler adds that this is not to be understood too narrowly: "even

if one divides the diameter in twenty thousand thousand thousand times thousand parts of

equal length, something of the circumference will remain that is smaller than such a small

part," i.e., pi is irrational.

and the circle 's cen te r for a vertex, has

the same a rea as the circle. The same works for the full sphere: it is made up

of infinitely many pyramids whose ver- t ices meet in the center; their bases reduce to poin ts on the surface of the

sphere. In ano the r vein, s ince a torus is obta ined by rota t ing a circle a round a line that l ies in the circle 's plane (but

does not touch the circle), its volume is the p roduc t of the a rea of the circle

t imes the c i rcumference descr ibed by rotat ing its cen te r a round that axis.

Indeed, the t o m s is made up of infinitely many thin discs, whose volumes have to be added up. Kepler admits that

since such a disc is more like a wedge, he makes an er ror in assuming that it has uniform thickness; two errors, actually, s ince the ou te r par t of the wedge

is thicker, and the inner par t thinner. But these er rors cancel each other. The arguments run fast and loose, and a few

of the results are wrong. But they came decades before Bonaventura Cavalieri, Rend Descartes, and Pierre de Fermat ,

and they display in their blind groping toward calculus an uncanny sense of di-

rection. Kepler may well be the fore- most example of wha t Arthur Koest ler t e rmed a scientif ic "sleep-walker."

The Nova Stereometria's main re-

sult cons is ted in finding, among all cyl inders inscr ibed in a sphere, those with the maximal volume ( today an

easy exerc ise for f i rs t-year s tudents) . This implied that among all cyl inders having the same "measure" given by the rod-length, those have maximal vol-

ume whose height is equal to the diameter of the bo t t om mult ipl ied by square root of two. Kepler added judi- c iously that his resul t was still ap-

p rox imate ly valid for barre ls of c lose to cylindrical shape: indeed, "whenever there is a transit ion from smaller to larger and back to smaller again, the dif-

ferences are always insensible, to a de- g e e . " This anticipates an argtnnent explicit ly made only decades la ter by Fermat: close to a maximum, changes are small; i.e., opt ima are critical points.

So the rod-measurement works, as long as the barrels have approximately the right proportion: height to bottom, like diagonal to side of the square.

As it turns out, Aust r ian barrels had

(and still have) a height that is equal to


The best figure of all. Wine barrels in the run-

down entrance of the Keplerhof Inn. In

German textbooks, Kepler's name is associ-

ated with the so-called barrel rule of numer-

ical integration (Simpson's 1/3 rule). In spite

of Kepler's praise of Austrian barrels, the dis-

trict deputies of Upper Austria decided in

1616, after a formal scrutiny of all his publi-

cations, to dispense with his services. How-

ever, influential friends made sure that this

decision was never put into effect.

drinkable stuff m a y be around in copi-

ous quantities,

Et cum pocu la mil le mensi erimus,

Conturbabimus illa, ne sciamus."

Some ded ica ted t eache r s tr ied for five

years to teach me some Latin, but I cannot help you wi th the translat ion. It 's about wine and science, though.

EDITOR'S NOTE: About wine and ignorance, rather! A scholar informs us that

the two lines, a learned allusion to Catullus's poem "Vivamus, mea Lesbia,"

mean, "and if we have measured each other a thousand vessels, we will con-

fuse them, in o rder not to know."

REFERENCES

Mechtild Lemcke, Johannes Kepler, Rowohlts

Monographien, 1995.

Max Caspar, Johannes Kepler, Dover, New

York, 1993.

Arthur Koestler, The Sleepwalkers, Hutchinson,

London, 1959 (many reprintings).

For Kepler's relation to algebra, see P. Pesic,

Kepler's Critique of Algebra, Mathematical

Intelligencer 22 (2000), no. 4, 54-59.

There are several good Kepler s i tes on

the net, for s tar ters see www. kepler, arc. nasa. gov/j ohannes, hmtl www.es . r ice .edu/ES/humsoc/Gal i l ieo/

Fi les /kepler .html www.roups .dcs .s t -and.ac .uk/

h is tory/Mathemat ics /Kepler .h tml

Institut for Mathematik

Universit~t Wien

Strudlhofgasse 4

1090 Vienna

Austria


three t imes the radius of thei r bot tom. The fact that 1.41422 is c lose to 1.50000 suff iced to pe r suade Kepler tha t Austr ian bar re l s "had the bes t figure of

all" (figurae omnium optissimae): in fact, he inc luded this p roud claim in the full title, which covers half of the fron-

t ispice of his book. Kepler goes on to ask, "Who will

deny tha t na ture can teach geomet ry to humans th rough a vague feeling for form, wi thout any recourse to ra t ional arguments?" He toys with the possibi l -

ity that once upon a time, some pre- eminent geomete r could have taught the rule to Aust r ian barrel -makers ; bu t

then he d i scards it, with the a rgument that in this case, o ther wine-growing countr ies would also have adop ted the same rule. Kepler ends his t rea t i se wi th a hear ty p raye r that "our spir i tual and

mater ia l goods may be preserved, and


l l l l i [ : - I I J~ lE i i i [ ~ i n l ' ~ - I i [ , - l ' : l l i l , l ! l d l L - t l i l D i r k H u y l e b r o u c k , E d i t o r I

The Bolzano House in Prague P. Maritz

B ernardus Placidus Johann Nepomuk Bolzano was born in Prague, Bo-

hemia (now par t of the Czech Republic),

on 5 Oc tobe r 1781. His house of b i r th does not exis t any more, but it was on the si te of the presen t Maria Square

(Marihnsk~ nfim~st0 in the Old Town.

His mother , Caecil ia Maurer, was a daughter of a hardware t r adesman in Prague; a t the age of twenty- two she marr ied the e lder Bernard Bolzano, an

Ital ian immigrant who earned a mod- est living as an art dealer. Both pa ren t s

were p ious Christians. Bernardus was the fourth of twelve

children, ten of whom died before

reaching adul thood. He grew up with a high mora l code and a bel ief in holding to his principles . It was this back-

ground that a t t rac ted him to the church and the pr ies t ly life. F rom 1791 to 1796 he was a pupi l in the Piar is t Gymna-

sium, and in 1796 he entered the philo-

sophica l facul ty at Charles Univers i ty (es tabl i shed by Charles IV [ 1315-1378], Holy Roman Emperor and Bohemian King in 1348), where he fo l lowed

courses in phi losophy, physics , and mathemat ics . Bolzano's in teres t in ma themat ics was s t imula ted by read-

ing A. G. Kfistner's Anfangsgri~nde der Mathematik, mainly because Kfistner took care to prove s ta tements that were commonly unders tood as ev ident

in o rde r to make clear the a s sumpt ions on which they depended [12, p. 273]. After having f inished his s tudies in philosophy in 1800, Bolzano en te red the

theologica l facul ty and was o rda ined a Catholic p r ies t in 1804 [4].

In 1805 Emperor Franz I of the Austro-Hungarian Empire, of which

Bohemia was then a part, dec ided that a chair in the phi losophy of religion would be es tabl ished at each university. The reasons for this were mainly polit-

ical. The empire was comprised of many different ethnic groups that were prone to nationalist ic movements for inde-

pendence. The emperor feared the fruits of Enlightenment embodied in the French Revolution. The authori t ies con-

s idered the Catholic Church to be con- servative and hoped it would control the

liberal thinldng of the time in Bohemia. Bolzano was cal led to the new chair at

Charles University in 1805 [12, p. 273]. Bolzano, though a priest , spir i tual ly

belonged to the Enlightenment. He was a "free thinker"; his appoin tment was received in Vienna with suspic ion and

was not app roved until 1807. For the next 14 years Bolzano taught at the uni-

versity, lecturing mainly on ethics, social questions, and the links be tween mathemat ics and ph i losophy [4]. In

1815 he became a m e m b e r of the K6niglichen B6hmischen Gesel lschaf t der Wissenschaf ten and, in 1818, Dean

of the phi losophica l faculty. However , the Austro-Hungar ian authori t ies be-

came d isp leased with his l iberal views. On 24 December 1819, he was sus- pended from his professorship , forbid-

den to publish, and pu t under pol ice supervision. Bolzano refused to back down, and in 1825 the ac t ion came to

an end through the intervent ion of the influential nat ional is t l eader Jose f Dobrovsk~ (1753-1829). The la t ter had

been educa ted for the Roman Catholic p r ies thood and devo ted himself to scholarship af ter the 1773 dissolut ion of the Jesuit Order. He was an impor-

tant Enlightenment figure, and his tex- tual cri t icism of the Bible led him to study Old Church Slavonic and subse- quently the Slavic languages as a group.

F rom 1823 on, Bolzano spent sum- mers on the es ta te of his fr iend J. Hoff- mann, near the village of T~chobuz in

Southern Bohemia. He lived there permanent ly from 1831 until the death of Mrs. Hoffmann in 1842. He then returned to Prague where he cont inued

his mathemat ica l and phi losophical s tudies until his dea th on 18 December , 1848. A small pension, and the gen- erosi ty of Count Leo Thun-Hohenstein

(1811-1888, the Bohemian ar i s tocra t and la ter Austr ian s ta tesman) rel ieved him of all mone ta ry care [7].

In Prague, Bolzano and his b ro ther

s tayed in the house that had be longed

5 ~) THE MATHEMATICAL INTELUGENCER �9 200t SPRINGER-VERLAG NEW YORK

Figure 1. The house at 25 Celetnd Street, showing the Bolzano plaque centered over the key-

stone arch.

to thei r pa ren t s at 25 Celetmi Street (Celetn~ ulice 25) near Old Town

Square (StaromSstsk~ n~n~s t0 . The pho tographs (Figs. 1, 2, t aken by the author) are of the house and its Bolzano plaque. This house is owned

by the City of Prague, and all the space in the house is filled with apar tments .

Around the turn of the n ine teenth

century, European mathemat ic ians were mainly concerned with the s ta tus of Eucl id 's paral le l pos tu la te and with

the p rob lem of providing a sol id foundat ion for mathemat ica l analysis. Bolzano t r ied his hand at both prob-

lems. In 1804 Bolzano publ i shed a theory

of paral le l lines, which ant ic ipa ted Adrien Legendre ' s wel l -known theory. It is commonly accep ted that Bolzano

in his manusc r ip t Anti-Euklid, was the first to s ta te the theorem (now known as the "Jordan Curve Theorem") that a s imple c losed curve divides the p lane

into two par t s [12, p. 274]. The introduct ion of infmitesimals by

Isaac Newton and Gottfried Leibniz in

the seventeenth century had met with violent res is tance from phi losophers and mathematicians. To overcome the difficulties presented by infmitesimals, Joseph-Louis Lagrange p roposed to

base analysis on the exis tence of Brook Taylor 's expans ion for functions, while Jean d 'Alember t p roposed to found differential ca lculus on the not ion of limit.

Among the first to doub t the r igor of Lagrange's expos i t ion of the calculus were Abel Btirja (1752-1816) of Berlin, the Poles J. M. Ho~n6-Wron3(si (1776-

1853) and J. B. Sniadecki (1756-1830), and Bolzano [3, p. 258], who devoted his manuscr ip t Rein analytischer Bewe/s (1817) to a p roof of the "Bolzano

In termedia te Value Theorem." Bolzano argues that a sound p roo f of this theorem requires a sound definit ion of continuity. His defini t ion is the first that

does not involve infinitesimals. The definit ion as it was formula ted in

Volume I of his manusc r ip t Function- enlehre reads: If F(x + hx) - F(x) in

absolu te value becomes less than an ar- b i t ra ry given fract ion 1/N, if one takes

Ax small enough, and remains so the smal le r one takes Ax, the funct ion F(x) is sa id to be cont inuous in x [12, p. 275].

Bolzano 's definit ion of cont inui ty was rep laced in 1821 by Augustin-Louis

Cauchy 's e legant and general ly accep ted definition. Also in his p roof of the "Intermediate Value Theorem," Bolzano uses a lemma that later proved

to be a cornerstone of the theory of real n u m b e r s - - h e introduced the concept of the infimum of a nonempty set. His 1817

manuscr ipt also contains the theorem that is known as "Cauchy's Criterion

for Convergence of Sequences." The proofs given by Bolzano were incomplete, but he was aware of the difficul-

t ies involved. A fairly comple te theory of real

funct ions is conta ined in Bolzano's

Functionenlehre, including many of the fundamenta l resul ts that were re- discovered in the second half of the nineteenth century through the work of

Karl Weierstrass (1815-1897) and others. Bolzano proved in this manuscr ipt that a function that is unbounded on a c losed interval [a, b] cannot be continuous on [a, b] [12, p. 275]. In proving this,

Bolzano used the so-called Bolzano- Weierstrass Theorem, that a bounded in-

finite set has a cluster point. This theo-

Figure 2. Close-up of the Bolzano plaque on the facade of the house at 25 Celetna Street,

Prague.


rem rests on the "Bolzano-Weierstrass Selection Principle," a method frequently employed in mathematical analysis, which consists of successive subdivision of a segment into halves, one of which is selected as the new initial segment [6, pp. 419-420]. The most remarkable result of the Functionenlehre is the construction of the "Bolzano function." Bolzano con- structs a function which is continuous, but nowhere differentiable, on the interval [0, 1]. This example preceded by some forty years that of Weierstrass.

In August 1830, Charles X of France was forced to emigrate by the Revolution of 1830; in the autumn of 1832, he left his refuge in Scotland and took his family to Prague, where Emperor Franz I placed part of the Hradschin Palace (HradSany) at his disposal. In September 1830, Cauchy left France and went into voluntary exile, losing all his public positions in the process [1, p. 147]. Cauchy first went to Fribourg, Switzerland, then was ap- pointed as professor in sublime physics (that is, mathematical physics) in January 1832 at the University of Turin, Italy. In the summer of 1833 Cauchy was invited by Charles X to help with the education of his grand- son, the Duke of Bordeaux, in Prague.

While in Prague, Cauchy seems to have had only tenuous relations with the Prague scientific community. It is a matter of importance to inquire whether he knew Bolzano during his years (1833-1836) in Bohemia. The question remained unresolved for years. To the Struiks [11] it seemed rather improbable that there existed any interaction between Bolzano and Cauchy, even though both were members of the Royal Bohemian Society.

Cauchy was a famous French acaddmicien whose new method was already studied and followed in all parts of Europe, but he was also associated with a banished court, which maintained the severest reserve in a hos- pitable but foreign country. Bolzano, for his part, had been removed from his professorship since 1819, and he lived secluded from society and public notice in T~chobuz. Cauchy would have risked offending the imperial authori- ties of Austria if he communicated with the compromised Bolzano. An autobi-

ography of Bolzano published by his pupil J. M. Fesl in 1836 (see [11, p. 365]) mentions nothing about any influence of Cauchy on Bolzano.

Cauchy had already completed long before, as had Bolzano, his works on the foundations of the theory of real functions. He had published his investigations in 1821 and 1823. By 1834 he was chiefly occupied with theoretical physics, such as his investigations on the dispersion of light. His works do not contain any reference to Bolzano, not even to the latter's earlier definition of continuity. Bolzano, on the other hand, did not publish any pure mathematics after 1817, and was, about 1835, probably occupied by philosophical questions concerning theology, or problems in mechanics. His Versuch einer objektiven Begri~ndung der Lehre von der Zusammensetzung der Krdfte was not published until 1842, and neither that paper nor another paper on the aberration of light suggests the possibility of a connection between him and Cauchy.

The Struiks concluded in [11] that only a few letters of either Cauchy, or Bolzano had been published to that date, and that the contents of unpub- lished letters might clarify the relation between Cauchy and Bolzano. That is exactly what happened.

In a letter to F. P~ihonsk~, dated 24 April 1833, at T~chobuz, Bolzano writes about his esteem for Cauchy, stating that he would like to meet Cauchy in September of that year, hopefully ac- companied by P~ihonsl~. This letter was published in 1936 by E. Winters; see [8, p. 164] for a passage from it. It was found in 1962 by I. Seidlerov~ that there is a letter, dated 18 December 1843, from Bolzano to his student Fesl, in which he mentions several meetings with Cauchy in Prague, see [8, p. 164]. This was also confirmed by Winter in 1965, see [9, p. 99].

It is also mentioned in [1, p. 172] that around 1834 a meeting between Cauchy and Bolzano took place. That meeting appears to have been sought by Bolzano, who had sent Cauchy a tract on the problem of the rectification of curves (ideas developed by Bolzano in 1817 in Die drey Probleme der Rektifikation, Komplanation und Ku-

b ierung, . . . ) , which he had written in French for Cauchy's benefit [9, p. 99]. I. Grattan-Guinness assumed that Cauchy had plagiarized Bolzano's definition of continuity, but H. Freuden- thal and H. Sinaceur clarified the dif- ferences in approach of the two mathematicians [1, p. 255], [9, p. 99]. Boyer [2, p. 564] is of the opinion that the similarity in Bolzano's and Cauchy's arithmetization of the calculus, of their definitions of limit, derivative, continuity, and convergence were only co- incidental.

L. E. J. Brouwer in a 1923 paper giving examples of theorems whose proofs require the law of the excluded middle for infinite sets, mentioned in particular the Bolzano-Weierstrass theorem and the result on the existence of a maximum of a continuous function on a closed interval; see [5, p. 238].

Charles X and his court left the Hradschin Palace and the city of Prague in May 1836 for Toeplitz, to make way for the new Emperor Ferdinand to come to Prague to receive his investi- ture as King of Bohemia. Charles X died in GSritz on 6 November 1836. The Duke of Bordeaux reached his eighteenth birthday in September 1838, and that ended Cauchy's duties with the ex- iled court. In October 1838, Cauchy and his family returned to Paris, where a new period in his life began.

Like most of Bolzano's mathematical work, Functionenlehre remained in manuscript form and was published for the first time only in 1962. As a result, this bold enterprise failed to exercise any influence on the development of mathematics; Bolzano was "a voice crying in the wilderness" [2, p. 565], and many of his results had to be re- discovered in the second half of the nineteenth century. H. A. Schwarz in 1872 [3, pp. 367, 368] looked upon Bolzano as the inventor of a line of reasoning developed by K. Weierstrass.

Bolzano is buried in Olsany Ceme- tery (Olsansk~ h~bitovy), Cemetery III, Part 9, Grave 107, in Prague. There is also a Bolzano Street in Prague, some 100 meters north of the Main Railway Station.

The Bolzano stamp displayed in Fig- ure 3 was issued by Czechoslovakia in


Figure 3. The 1981 Czechoslovakian stamp

commemorating the mathematician’s birth.

1981 to commemorate the 200th an-niversary of his birth [lo, p. 621, SG25681.

ACKNOWLEDGMENTS

The author gratefully acknowledgesthe information supplied by Professor

Karel Segeth, Director of the Math-ematical Institute of the Academy ofScience in Prague, and the assistancereceived from Professor Marcel Wild,Mathematics Department, Universityof Stellenbosch.

REFERENCES

[I] Belhoste, B. Augustin-Louis Cauchy. A

Biography. Springer-Verlag, New York,

1 9 9 1 .

[2] B o y e r , C . B . A History o f M a t h e m a t i c s . J o h n

W i l e y a n d S o n s , I n c . , N e w Y o r k , 1 9 6 8 .

[3] Cajori, F. A Histo~ of Mathematics. Third

Edition. Chelsea Publishing Company,

New York, 1980.

[4] Golba, P. Bolzano, Bernard (1781-1848).

R e t r i e v e d J u n e 3,200O f r o m t h e W o r l d W i d e

Web: http://www.shu.edu/html/teaching/

math/reals/history/bolzano.html

[5] Kline, M. Mathematics. The Loss of

Certainty. Oxford University Press, New

York, 1980.

[6] Kudryavtsev, L. D. Bolzano-Weierstrass

Selection Principle: Bolzano-Weierstrass

Theorem. In: Encyclopeadia of Mafhe-

mafics. Volume 1, pp. 419-420. M.

Hazewinkel (Ed.). Reidel, Kluwer Aca-

d e m i c P u b l i s h e r s , D o r d r e c h t , 1 9 8 8 .

[7] Leimkuhler, M. Bernard Bolzano, The

C a t h o l i c E n c y c l o p e d i a . V o l u m e I I . R o b e r t

A p p l e t o n C o m p a n y , 1 9 0 7 . T r a n s c r i b e d b y

T h o m a s J . B r e s s . R e t r i e v e d J u n e 3 , 2 0 0 0

from the World Wide Web: http://www.

newadvent,org/cathen/02643c,htm

[8] Rychlik, K. Sur les contacts personnels de

Cauchy et de Bolzano, Revue d’Hi.sfoire

des Sciences 15 (1962) 163-l 64.

[9] Sinaceur, H. Cauchy et Bolzano, Revue

d’&forie d e s S c i e n c e s 2 6 (1973), 9 7 - l 1 2 .

[I 0] Stanley Gibbons Simplified Cafalogue,

Stamps of the World. Volume 1, 2000

Edition. Foreign Countries, A-J. Stanley

G i b b o n s L t d , L o n d o n , 1 9 9 9 .

[ll] Struik, D.J. and R. Struik. Cauchy and

Bolzano in Prague, lsis 11 (1928) 364-

3 6 6 .

[12] Van Rootselaar, B. Bolzano, Bernard. In:

D i c t i o n a r y o f S c i e n t i f i c B i o g r a p h y . V o l u m e

II, pp. 273-279. C. C. Gillispie (Ed.).

Charles Scribners Sons, New York, 1970.

D e p a r t m e n t o f M a t h e m a t i c s ,

U n i v e r s i t y o f S t e l l e n b o s c h

P r i v a t e B a g X 1

M a t i e l a n d

7 6 0 2 S o u t h A f r i c a

e - m a i l : [email protected]

I ld , , [~V~l~- ' l i l | [~ l= , l r~- l l | [ * "~- l l l l [ * ] ! |d~ l l D i r k H u y l e b r o u c k , E d i t o r I

The Kentucky Vietnam Veterans Memorial John W. Dawson, Jr.

F rankfort , Kentucky, l ies j u s t nor th of Interstate highway 64, about mid-

way be tween Lexington and Louisville. Often bypassed by motoris ts hastening

be tween those metropolises, it should be on the i t inerary of visiting math-

ematicians, for it is the site of a remarkable public monument whose in-

novative design, based on horological principles known from antiquity, evokes a powerful emotional response.

The Kentucky Vietnam Veterans Memoria l is a massive hor izonta l sundial I cons t ruc ted on a hi l l top over-

looking the s ta te capi tol (Figure 1). I ts s ta inless-s teel gnomon, over 5.3 meters in length, r ises a little more than

4.3 m above a rec tangular plaza, some 21.6 m wide and 27.1 m long, that is

c o m p o s e d of 327 b locks of grani te weighing abou t 195 metr ic tons (Fig- ure 2).

As in all such dials, the gnomon is

a l igned with the celest ial pole, so tha t the sun 's rays are perpendicu la r to it

at the equinoxes. On those days, the t ip of the gnomon ' s shadow t races a s t raight line across the plaza. On all

o ther days, i ts pa th is a hyperbol ic arc. But this shadow is not employed to in-

dicate the hour of the day. Its pu rpose is ra ther to c o m m e m o r a t e the 1065 Kentuckians who were lost during the Vietnam conflict.

Inscr ibed along radial l ines extend-

ing ou tward f rom in front of the gnomon are the names of all those kil led in action. They are pos i t ioned in such

a way that the shadow cast by the t ip

o f the g n o m o n fa l l s on each n a m e on the a n n i v e r s a r y o f that ind iv idua l ' s death. The names of those missing in act ion or held as p r i soners of war are also inscr ibed on the plaza, in the re-

gion south of the gnomon where the shadow never falls. 2

Figure 3 shows the layout of one of the granite slabs, which are a r ranged

in sec tors co r respond ing to the years of the conflict, f rom 1962 through 1975.

The sec tors a re in te rsec ted by curves marking the equinoct ia l shadow line and the solst i t ial shadow trajectories ,

and are s epa ra t ed by walkways analogous to the hour l ines on a t radi t ional sundial (Figure 4). The concent ra t ion

1In principle; to ensure proper drainage, the dial is actually inclined 2% away from the horizontal.

2In the event that one of those listed as missing is later declared dead (as has happened eight times to date),

the name of that individual is inscribed in the position corresponding to his or her presumed date of death,

and the date of the official declaration of death is inscribed after the name on the MIA list.

.,,j

TO LOUISVILLE

U.S. 60 I

d l State 676 EAST-WEST ~ Capitol

CONNECTOR

Library& " ~ ' ~ V I'~ETN Archives �9 AM

V E T E R A N S MEMORIAL

i-64

O~ O

TO LEXINGTON

Figure 1. Location of the memorial.

56 THE MATHEMATICAL INTELLIGENCER �9 2001 SPRINGER VERLAG NEW YORK

Figure 2. Gnomon and plaza: view from the

southeast.

of names in cer ta in of the sec tors

vividly i l lus t ra tes the major engage- ments of the war, especia l ly the Tet of-

fensive of 1968. The pos i t ion r eached by the shadow

at 11:11 a.m. on Veterans ' Day, Novem- ber 11, is ind ica ted by a special ma rke r

that the shadow t raverses during the minute of s i lence observed at that time, and a round the base of the gno-

mon the words of Eccles ias tes are engraved: "For everything there is a sea- s o n . . , a t ime to be born, and a t ime

to d i e . . , a t ime to kill, and a t ime to h e a l . . , a t ime for war, and a t ime for

peace." The p lan for the memor ia l was con-

ceived by Lexington archi tect Helm Roberts, who submi t t ed his idea to a design compet i t ion sponsored by the

Kentucky Vietnam Veterans Memorial Fund. The judges unanimously selected his design as tha t which bes t ex-

emplif ied the cr i ter ia es tabl ished for

the monument , among which were that it " n o t . . . imitate o ther monuments" and that it "evoke an emot iona l re-

membrance whils t being aes thet ica l ly authentic." They exp re s sed some

doubts , however, regarding its feasi- bility.

To determine the precise azimuth and

elevation of the sun for various dates and t imes the project employed the astro- nomical computer program ACEcalc

(Astrosoft Computer ized Ephemeris) . Drawings of the individual s labs were

p r o d u c e d using AutoCAD, and the ac- curacy of the calcula t ions was so good that the names could be engraved in the

s tone before the slabs were set in place. Only the equinoctial and solstitial lines were incised in s i t u . 3

Comple ted in 1988 at a cos t of over one mill ion dollars, the memor ia l was ded ica ted on November 12 of that year

and has been open to vis i tors a round the c lock every day since.

3For further details concerning the design and construction of the monument, see [1].

Figure 3. Slab detail (AutoCAD drawing. Courtesy of Helm Roberts)


Figure 4. Overhead plan of the memorial, with superimposed shadow trajectories. Radial bars indi-

cate placement of names. (Courtesy of Helm Roberts.)

Armchai r t ravelers can view pictures of the memorial , as well as ar-

chi tec tura l drawings of it, onl ine at ht tp: / /www.helmr.com/ky.htm; o ther pho tos may be v iewed at http://grunt. space . swr i . edu /kymem.h tm. Informa- t ion and a downloadab le brochure

are ava i lab le at h t tp : / /www.s ta te .ky . u s / a g e n c i e s / k h s / m u s e u m s / m i l i t a r y / vietnam.

A c k n o w l e d g m e n t s

I am indebted to Mr. James Halvatgis, super in tenden t of the memorial , and to

Mr. Helm Roberts , its designer, for informat ion and il lustrations.

REFERENCES

[1] Aked, Charles K., "Vietnam Veterans

Memorial." Bulletin of the British Sundial

Society 93.1 (February 1993), 15-16.

[2] Waugh, Albert E., Sundials, Their Theory

and Construction (Dover Publications, New

York, 1973).

John W. Dawson, Jr.

Pennsylvania State University

York, PA 17403

USA



m'.ir: l~-lw'-.,[.~- Je remy Gray, Editor I

Symbols and Suggestions: Communication of Mathematics in Print* Jeremy Gray

Column Editor's address:

Faculty of Mathematics, The Open University,

Milton Keynes, MK7 6AA, England

M a themat ic ians are creators (or

discoverers , if you wish). They are also communica tors , and receivers. In all these roles they are people with

sophis t ica ted ways of assembl ing and re-assembling ideas. Fo r several cen-

turies, the mos t effective communica- t ion medium has been print. His tor ians of mathemat ics somet imes used to

claim that wi thout this or that p iece of nota t ion some idea was unthinkable. This does not s eem ent i rely satisfac-

tory, especia l ly to mathemat ic ians who know very well tha t new symbols are easy to devise, bu t the claim has some

merit. It is more prof i tab le to argue that mathemat ica l notat ion, l ike any lan-

guage, is torn be t we e n syntax and semant ics and of ten p roceeds by relying on taci t unders tand ings about mean-

ings. Two examples will be cons idered here, one drawing on recent work

about the impl ica t ions of ratio and equality, and one on examples of logical notation.

Equality and Proportion in Geometry and Algebra Franqois Vi~te, who was the leading

French mathemat ic ian of the turn of the seventeenth century, wrote his al-

gebra with a la tent appea l to geometry, which survives in the balancing of the

terms, so that each monomia l in an equation has the same weight or dimension. This is to prevent the nonsense of adding a s ide to an area. There

is also a convent ion about what le t ters s tand for. It 's not the m o d e m one: here vowels s tand for the unknowns. This could catch on; it bare ly needs to be said and seems so natural that it doesn ' t

t ravel with an explanat ion. Such taci t unders tanding is an impor tan t par t of pr in ted communica t ion , and it implies the exis tence of a communi ty in on the

secret . The convent ion tha t did catch on is,

of course, due to Descartes . He intro-

duced lowercase letters, ra ther than

capitals , with x and y for unknowns. He argued his way a round the dimen- s ional i ty convent ion by showing how a

length t imes a length can be thought of,

not as an area, but as ano ther length. Because all his magni tudes have the s ame dimension, the d imens ion concep t does no work and can be forgot-

ten, which is wha t happened . For Descar tes , p rob lems s ta r ted in geometry, were t rans la ted into algebra, where

they were solved, and then re tu rned to geometry. 1 So a curve for him was a t ru ly geometr ic object. This gave him a

problem: What express ions truly define curves? Descar tes ' s cr i ter ion for

a curve to be t ruly geomet r ic was (roughly, because Descar tes could not be more precise) that it be an algebraic

curve defined by a po lynomia l equation in two variables. Curves of double motion, as he cal led them, including

roule t tes l ike the cycloid, were not de- f ined exact ly enough.

In the 1660s, Descar tes ' s greates t fo l lower was the young Isaac Newton,

but by the 1680s the middle-aged Newton had turned agains t the great F r enchman on mat te r s of physics, theology, and mathemat ics . He did not

share the implied p re fe rence for alge- b ra over geometry, or his cr i ter ion for s implici ty (roughly: degree of the defm-

ing equation). What, Newton protes ted, was more s imple than the cycloid? And

as he turned back to the o ld Greek me thods of geometry, he found that geomet ry was the na tura l language for

his physics. Here is one of the key passages in

Newton ' s Principia, where he ad- d resses Johannes Kepler ' s law that p lane t s sweep out equal a reas in equal

t imes. Crucially, the p rogress in the diagram is to be enac ted by the reader, as the accompanying tex t makes clear. At each moment the a rea actual ly

swep t out is specified, shown to be

*Based on a paper presented to the Math ML Conference, Urbana-Champaign, October 19-21, 2000.

1As Bos carefully describes (Bos 1981).


Soit pa r exemple A B l'vnit6, & qu'il faille

mul t ip l ier BD par BC, ie n 'ay qu 'a ioindre les po ins A&C, puis t i rer

DE para l le le a CA, & BE eft le produi t de cete Multiplication.

Oubien s'il faut d iufer BE par BD, ayant ioint les poins

E&D, ie t ire AC para l le le a DE, & BC eft le p rodu i t de cete

dinifion. Figure 1. Descartes's proof that multiplying a line by a line gives a line, from his La Geometrie.

equal to another , and so they are all the same. Finally, the r eade r is to imagine

tha t the impulsive ac t ion of gravity as it is dep ic ted here (happening in dis- c re te intervals of t ime) b e c o m e s con- t inuous, by shrinking the t ime inter-

vals. This is seductive. Newton typically wrote his conclu-

sions in the language of ratio and proportion, but with a significant modifica- tion. His immediate p redecessor and to

some extent model, Christian Huygens, did not appreciate this change, and when he read the Principia he tried

whenever possible to put the hill appa- ratus of ratios back. 2 Huygens 's geo-

metr ical language cannot t reat Galileo's

law of falling bodies, that the distance a body has travelled is propor t ional to the square of the elapsed time, s ~ t 2, except as a comparison of two ratios: to be

wri t ten as s l : s2 :: t 2 : t2 2. It seems that Newton made the move to thinking of

s ta tements of proport ion like statements of equality only during the writ-

ing of the Principia. It has the signal benefit of involving only two terms, not

four, and so allowing terms to become

more obviously variables. Al though Newton ' s language is de-

te rminedly geometr ic , he did not use the full pa rapherna l i a of p ropor t ion

theory, with i ts bu rdensome symbolism and its res t r ic t ion on what can be said, or rather, written. His guide here

was Descar tes , al though Newton did not go the final s tage and in t roduce co-

ordinates. F o r Newton, the fundamental objec ts were quantities, chiefly

physical ones (posit ion, velocity, force), which evolve in interre la ted

ways as t ime goes b y - - s a y , as a p lanet moves round the Sun, now close in, now further away. They were natural ly

expressed mathemat ica l ly as geometric quantities, a l though Newton was

bri l l iant in his b lending of geometry and dynamics. But thinking and writing rat ios be tween variables, as

Newton did, is not the same as thinking and writ ing about functions, and

this is a central po in t in any s tudy of the relat ion be tween notat ion, ideas,

and understanding.

Newton, Calculus, and Mechanics We have arrived at the calculus. For

Newton in the 1680s this was a set of techniques applied to curves. He did not, as a discredited rumour has it, write his

Principia using the calculus and then rewrite it for popular consumption. He

did occasionally resor t to the method of infinite series when he had to. But the Principia is a geometry book, marvel-

lously adapted to mechanics. What happened in the years after Newton was that more and more mathematicians, espe-

ciaUy Continental ones who had rela-

Figure 2. Newton, by Kneller, 1689. Figure 3. Newton's proof of Kepler's equi-area law for any central force, from his Principia.

2This point is eloquently made in Guicciardini [1999].

60 THE MATHEMATICAL INTELUGENCER

tively easy access to the Leibnizian cal-

culus, found that they wanted to use it to understand, then to replace, and then to extend Newton's geometric argu-

ments. But what does the calculus apply to? Most efficiently, to formal expres- sions, fimctions (however they are de-

fmed). It says things like this:

if y = f (x ) then dy = ~ dx.

So when Leonhard Euler set about rewri t ing all of pure and appl ied mathemat ics in the mid-eighteenth century, he cas t it all in the language of func-

t ions ra ther than of geomet ry and pro-

port ions. In short, the idea that "this ratio of

variables is equal to that" is not the same as "these variables are functions of those." Equality has its origins in simple

problems that can be writ ten a s - - note the word- -equa t ions . It pushed its way into geometry, and into mechanics;

finally, because of its efficacy in the calculus, it replaced the old ratio idea almost entirely. As it did so the quantities

being equated were a l lowed to become variables, and gradually, as that happened, the equations were taken to ex-

press functional dependence.

Logic, Set Theory, and Language The delightfully vexed his tory of logic offers a different domain of mathe-

matics, one with a rich, ongoing history, that might suggest the sor ts of p rob lems future wr i te rs of mathematics may encounter , indicat ive of those

genera ted by o ther profess iona l groups that use mathemat ics but do not con- s ider themselves mathematicians. There are two paral le l famil ies of ideas in early modern logic. There is the idea

that among a class of ob jec t s there are e lements with this o r that property , and we can form the class of all elements that are e i ther this or that, or are

bo th this and that. Paral lel to c lasses are s ta tements : we can asser t this and

asser t that, and asser t e i ther this or that, o r both this and that.

George Boole was an ob jec t man: I p ick at r andom this express ion,

x(1 - y) + y(1 - x),

which he glossed as things that are x ' s but no t y ' s or y ' s but no t x 's . 4 The ex-

p re s s ion 1 - y s tands for not y 's , because Boole had a universal set, which

he deno t ed 1, and complemen t s he deno ted by analogy with subtract ion. Similarly for 1 - x. The p roduc t or jux-

taposi t ion he interpreted as we do intersect ion, so x(1 - y) is all objects (in

the universal set) that are both x ' s and not y's. The problem comes with the "or," as we shall see shortly, because

Boole 's was the disjunctive "or"; for him a or b meant either a or b but not both.

Boole was insis tent that his Laws of Thought obey, so far as possible , the rules of ari thmetic. In his b o o k The

Laws of Thought (1853) he po in ted out that the symbol ic rules he found to apply were very near ly those that also ap- p l ied to ari thmetic. The only po in t at

which the laws of logic and of arith- met ic differed, he felt, was the germ or seminal pr inciple of logic: the asser t ion that x 2 -- x. He set great s tore by the

a lgebra ic nature of his equat ions and his abi l i ty at solving them. Fo r exam-

ple (p. 105), he exp res sed the statement "No men are perfect" as

y = v(1 - x),

where y represen ts men, v is an indef- inite class, and x per fec t beings. By el iminat ing v he turned this into

0 1 - x = y + - ~ ( 1 - y ) ,

"Imperfect beings are all men with an indefini te remainder of beings, which

are not men." Note the new symbol

0

0

which he in t roduced to a l low for the

symbol ic express ion of the idea of "some." In Boole 's day, and for some years before, many logicians felt that if

3See Gray [1994].

4Boole, p.105,


the re was any way in which the s tudy of logic could surpass the state to

which Aristot le had brought it, it would be in "quantifying the predicate ," a res- onant phrase that mean t being able to

ta lk about "all," "some," and "none." Boole 's exclusive "or" caused prob-

lems with the algebra. Not many years

later, William Stanley Jevons replaced it with the inclusive "or," thus abolishing

the problem of unders tanding expres- s ions of the form x + y when x and y have elements in common (part icularly

acute in the case when x + x is to be understood). An eloquent example of how much he simplified the process of

el imination was his demonst ra t ion that it could be mechanised. After some ten years of work, he exhibi ted a logical ma-

chine at the Royal Society of London in 1870 (a descript ion was also published

in his The Principles of Science, 1874). It became known as Jevons 's logical

piano.

Peirce and the Logic of Relations A little later, in the late 1870s and early 1880s, leadership in logic passed to

C. S. Peirce and the s tuden ts he briefly ga thered around him at Johns Hopkins.

Pe i rce began by ex tending Boole ' s opera t ions for c lasses to b inary relations.

He in t roduced the relat ive p roduc t of two relations, which conta ins existen-

t ial s ta tements implicit ly. If I is a rela- t ion (Peirce sugges ted "lover of") and w is a class ("woman"), then lw s tands for "lover of woman." This mus t be un-

packed. With Peirce, let l(i, 3) s tand for "i is a lover of j " and w(?) for ':?' is a woman," then lw(i) means "i is a lover of a woman," or, more formally, "There

is a woman j such that i is a lover of j . " Exponent ia t ion works l ike this: lW(i) means "i is a lover of every woman," or more formally, "For e v e r y j such that

j is a woman, i is a lover o f j . " Pei rce ' s group, in line wi th o ther au-

thors, used + to denote the inclusive

"or." Signs were in t roduced to asser t exis tence, of which Pe i rce ' s w a s - - < . He wrote,

G r i f f m - - < brea th ing fire

"to mean that every griffm (if there be such a crea ture) b rea ths fire; that is, no griffin not breathing fire exists," and

Animal - - < Aquatic

"to mean that some animals are not aquatic, o r that a non-aquatic animal does exist."

In 1885, Peirce made the breakthrough to the product ive in t roduc t ion of quantif iers into logic. He used the

symbols E and H, somet imes with sub- scripts , as he put it, "in o rde r to make

the no ta t ion as iconical as poss ible , we may use E for some, suggest ing a sum,

and II for all, suggesting a p roduc t . . . . If x is a s imple relation, I]iHjxij m e a n s

that every i is in this re la t ion to a j , ZiIIjxij tha t to e v e r y j some i or o the r is in this relation, ~,iEyxij that some i is

in this re la t ion to some j " (in Brady, p. 187). His reasoning was that if the xij were e i ther 0 or 1, and one wro te

Z~ZF~J = 1 to indicate that the corresponding s ta tement was true, then the

s t a tement "ZiZjxij = 1 was t rue pre- c isely when the sum was not zero. Analogous arguments deal t wi th the

use of the p roduc t symbol. He re fe r red to his symbols as quantifiers, and the indices as pronouns, thus bringing out

a l inguist ic analogy. Note that the quantifiers, l ike funct ions in mathemat ics ,

mus t here be read from right to left. Pei rce then showed how to ex tend this analysis to deal with severa l re la t ions at once and d ropped the r edundan t ex-

p ress ion "= 1."

Schr6der's Logical Algebra Just as Peirce's career began its long and painful decline, a fortunate reading of Peirce's Studies in Logic [1883] by the German mathematician Ernst Schr6der persuaded him that the algebra of relatives held the key to the domain of formal algebra. He therefore embarked, as he wrote to Christine Ladd-Franklin (Peckhaus, p. 273), on Part III of the book he was writing before finishing

Part II. In the event, the three publ ished volumes of the Algebra der Logik (some

of which were published posthumously) cover the calculus of classes ("the connect ion of ideas"), the calculus of propo- sit ions ("the connection and relation be-

tween judgements"), and the calculus of relatives. The fundamental relat ion in the first is that of inclusion, which SchrSder did not denote C , but

and which has these propert ies :

1. a C a (reflexivity)

2. If a C b, and b C c, then a C c (tran- sivity)

He defined equal i ty this way: a = b if and only if a C b and b C a.

A feature of his presenta t ion is its emphas is on duality, so that dual con-

s t ruct ions are p resen ted in para l le l

columns (as was often the case in con- t emporaneous project ive geomet ry books) . The ident ical ly null or nothing,

denoted 0, and the identical ly one, or all, is in t roduced as satisfying

0 C_ a and a C 1 for all domains a,

making them initial and terminal ob-

j ec t s in la te r terminology. What he cal led identical mult ipl ica-

t ion and ident ica l addi t ion he def ined

this way: for all domains a, b, c:

i f c C a and c C b then c C a b

and

if a C_ c and b C c then a + b C c.

SchrSder then es tab l i shed the distr ibutive laws, much as Peirce had done,

wi thout use of the concept of negation, and in the form ab + ac C a(b + c) and a(b + c) C a b + ac. He introduced mod-

els of sys tems sat isfying these ax ioms to show that the two distr ibutive laws

were i ndependen t of each other. Finally, he def ined the negation of a domain a to be a domain a l such that

aal C_ 0 and 1 C a + al.

Frege In the work of Boole, Peirce, and SchrSder, ma themat ica l nota t ion was

adap ted to the needs of the logician, in par t icular of the inventive or creat ive logician. Nota t ion was adop ted with an eye to its meaning in the old domain; what was wan ted was a calculus, and

the new ideas tha t were expres sed were par t ly mathemat ica l , par t ly philosophical, par t ly linguistic, par t ly logical. More radical ideas about language

were held by a man SchrSder could not abide (and who subjec ted some of his ideas to his cus tomary des t ruct ive crit- icism): Got t lob Frege. Frege wrote

B

6 2 THE MATHEMATICAL rNTELLIGENCER

for wha t he cumbersomely expla ined was the asser t ion (marked by the th ick

vert ical l ine) that the judgemen t holds

that it is not the case that A is den ied and B holds. A little work al lows us to

read this as "A or not B," or, if you pre- fer, "B impl ies A." Denying a j udgemen t he wro te this way:

[ , A

Since he was t ra ined as a mathemat i -

cian and worked at Halle, where Georg Cantor was, Prege had no p rob l ems with funct ion notation. So express ions

l ike

' ~ F(A) !

mean t that for all a it is the case tha t

F(a). It isn ' t too difficult to see that

' va F ( )

means that F(a) is not the case for all

a, or that there is some a that does not have p rope r ty F.

[ F (z)

I 8 F (a) I(

a f (&a) ~ ] - - - - - F (o)

f (x,a) 'y f (x~, yz)

f (Y,Z) Figure 4. A Frogean diagram, from his Begriffsschrift nr. 87, p. 65.

Matters rap id ly got complicated, because Frege ' s who le aim was to pro-

duce proofs based on logic wi thout ap- peal to facts, and in par t icu lar to found

our concep t of number on such proofs. He therefore set h imself the t ask of prevent ing "anything intuitive f rom

penet ra t ing here unnoticed," and "in at- tempt ing to comply with this require- ment in the s t r ic tes t poss ib le way," he wrote, "I found the inadequacy of lan-

guage to be an obstacle." The search

for prec is ion led him to this ideography (whence the tit le of his book,

Begriffsschrift, literally, idea script), which he also re fe r red to as a formula language for pure thought. He com-

pa red it to the mic roscope , an instru- men t general ly infer ior to the human

eye but for se lec t p u r p o s e s a vast (and indispensable) improvement . Frege

was opt imist ic tha t his ideography could be used to s tudy the foundat ions of the calculus and, wi th less work, of

geometry. It was, in any case, he reck- oned, an advance in logic.

As a language, it did not catch on.

One has to rewrite it before one can unders tand it. But this example (Figure 5) from Noam Chomsky 's Aspects should

remind us of the complexi t ies of natural (and indeed formal) languages.

P e a n o a n d I n t e r n a t i o n a l

L a n g u a g e s

But there is ano ther a spec t of Frege 's work that I should l ike to mention. The

pe r iod around 1900 was the heyday

(3S) #-S-#

NP Predicate-Phrase I

N Aux VP

[ + A ~ Present Copula Predicate

Compar Adjective

A.

Predicate-Phrase NP

I N VP

Copula Predicate I

Adjective 7X clever

i u x I

Present

John

clever

Bill

Figure 5. A Chomskyian diagram, from his Aspects, p. 178.


of internat ional languages such as Esperanto. 5 Frege was not averse to

them, at least for scient i f ic purposes , and Schr6der was keen. He hoped to

show that Peirce had a l ready defined enough symbols for all of mathemat ics . Ano the r enthusias t was Giuseppe

Peano, to whom we owe further ad- vances in mathemat ica l symbolism. He

in t roduced N for "and" and U for "or," �9 for "is" (as in a �9 K for a is a class).

But his aim, rea l ised in his journa l Rivista and his encyc lopaed ic Formu- lario, was to e l iminate natural lan-

guage in mathemat ica l pape r s ent irely in favour of a heavily symbol i sed pre-

senta t ion (often a c c o m p a n i e d in prac- t ise by t ransla t ions into Italian, or perhaps Peano 's favoured internat ional

language, Latino sine Flexione or Interlingua). These a t t empts failed. 6

They remind us that mathemat ica l symbol i sm cannot a lways be held apar t

f rom other symbol isms, including those of languages themselves .

The reasons for this failure are interesting. Prior to Esperanto, there had

been an international language called Volaptik. It had not las ted very long before it col lapsed into schism, split be-

tween users who wished to al low the language to evolve and simplify and others, grouped around its inventor, who wished it to stay as it was. The failure

weighed heavily on the minds of the Esperantists , who dec ided instead that

there should be no head office or central commit tee capable of defining an orthodoxy. Nonetheless a conference

held in 1906 to debate the rival merits of Esperanto and other international languages, such as Peano's , led only to

a vicious feud, with allegations of be- t rayal and deceit. There can, of course, be only one international l a n g u a g e - -

once there are two you have the translation p rob lem all over a g a i n - - a n d un- seemly behaviour among prosely t i sers

for internat ional ism and its benefi ts un- dermined their cause corrosively.

It would seem that the efforts to

c rea te an internat ional language (at leas t for mathemat ics and pe rhaps for

genera l use) serve to indicate the need for co-operat ion, s tandardisa t ion, and inter-translatabil i ty. P resen ta t ionssuch

as Frege ' s Begriffsschrift are too narrow to ca tch on. But an-embracing pro- posa l s for a truly scientif ic interna-

t ional artif icial language have also never succeeded . Language, including

ma themat i ca l language, may be too human a ma t t e r for consensus ever to emerge.

BIBLIOGRAPHY

Boole, G. 1853 An Investigation of the Laws of

Thought, Walton and Maberly, Cambridge

and London: Dover reprint, New York (1958).

Bos, H. J. M. On the representation of curves

in Descartes's Geometrie, Archive for History

of Exact Sciences, 24 (1981 ), 256-324.

Brady, G. From the Algebra of Relations to the

Logic of Quantifiers. In Studies in the Logic

of Charles Sanders Peirce, N. Houser, D.

Roberts and J. Van Evra, eds. Indiana

University Press, Bloomington, Indiana (1997),

173-192.

Chomsky, N. Aspects of the Theory of Syntax,

MIT Press, Cambridge, Mass. (1965).

Descartes, R. 1637 La Geom6trie, Appendix in

Discours de la M~thode, etc Leiden 1637, tr.

D. E. Smith, M. L. Latham, The Geometry of

Rene Descartes, Dover, New York (1954).

Frege, G. 1879 Begriffsschrift, eine der arith-

metischen nachgebildete Formelsprache des

reinen Denkens, English translation. In From

Frege to Gddel, ed. J. van Heijenoort,

Harvard University Press, Cambridge, Mass.

(1967), 1-82.

Gray, J. J. Complex curves--origins and in-

trinsic geometry. In The Intersection of

History and Mathematics: Proceedings of the

Tokyo Symposium in the History of Mathe-

matics, 1990, ed. C. Sasaki, Birkhauser,

Basel (1994), 39-50.

Gray, J. J. Languages for mathematics and the

language of mathematics in a world of na-

tions. In Jevons, S. 1874 The Principles of

Science, London (200I).

Guicciardini, NiccolS. Reading the Principia.

The Debate on Newton's Mathematical

Methods for Natural Philosophy from 1687

to 1736, Cambridge University Press, Cam-

bridge (1999).

Mathematics Unbound: The Evolution of an

International Mathematical Community, 1800-

1945, ed. K. H. Parshall, to be published by

the American and London Mathematical

Societies, Providence, RI.

Newton, I. 1687 Mathematical Principles of

Natural Philosophy and His System of the

World, tr. A. Motte (1729) rev. F. Cajori,

University of California Press (1962), 40-42.

Peckhaus, V. Logik, Mathesis universalis und

allgemeine Wissenschaft, Akademie Verlag,

Berlin (1997).

Peirce, C. S. I885 On the Algebra of Logic,

American Journal of Mathematics, 7, 180-

202.

Peirce, C. S. 1883 Studies in Logic. By Mem-

bers of Johns Hopkins University, Little,

Brown and Company, Boston.

SchrOder, E. 1890, 1891, 1895, 1905 Algebra

der Logik, 3 vol. reprint, Chelsea, New York

(1966).

5See Gray [2001] forthcoming.

6Although new proposals circulated a while back for something not too different.


I I - - , ~ . . . . . �9 [ ~ , ~--~ J e t W i m p , E d i t o r I

Feel like writing a review for The

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Column Editor's address: Department

of Mathematics, Drexel University,

Philadelphia, PA 19104 USA

Geometry from Africa: Mathematical and Educational Explorations MATHEMATICAL ASSOCIATION OF AMERICA,

WASHINGTON D.C., 1999, ix + 210 pp. ISBN: 0-88385-

715-4. 2.

Le Cercle et le Carr L'HARMA'FI-AN, PARIS, 2000, 301 pp. ISBN: 2-7384 9235-5

by Paulus Gerdes

REVIEWED BY DONALD W. CROWE

A l though Paulus Gerdes has lived for decades in Mozambique, a

country known to mos t of us mainly for its ear l ier long-standing civil war wi th its result ing land-mine hazard, and in the year 2000 for devasta t ing floods,

but until now not for its intellectual achievements, he has produced a body of work that is remarkable for its depth and inventiveness. Unfortunately, until

recently most of it was not readily accessible to those who only read English, in part because much of it was published

in Portuguese, German, or French, and in part because many of his English publications, while attractively presented, were published (several with financial

support from Sweden) in Mozambique. Some of these are l isted at the end of the present review. Now, however, the pub-

lication by the Mathematical Associa-

t ion of Amer ica of Geometry f rom Africa has given him a chance to present in readily accessible form material from many of his earl ier publications,

extended in par t by new problems and

exercises. The general t e rm for the act ivi ty in

which Gerdes has been engaged is "eth- nomathemat ics" (a te rm popular ized by Ubira tan D'Ambrosio) , and he might

legi t imately be cons ide red to be the leading act ive r e sea rche r in this field. But, in the mind of this reviewer, in its original formula t ion this te rm had polemical conno ta t ions whose ab-

sence is an asse t in the p resen t books . These two b o o k s emphas ize the dis-

covery of what Gerdes has aptly cal led the "hidden" or "frozen" mathemat ics

(geometry, in this case) in cultures, or cul tural activities, that mos t of us imagine have no mathemat ica l s ignificance

or component . This idea that there is unrevealed

mathematical insight in areas outside of

formal mathematics, though it has become more widespread in recent t imes beginning with Claudia Zaslavsky's

Africa Counts (1973) and continuing with Marcia Ascher 's Ethnomathe- matics (1991) and now with Paulus

Gerdes 's Geometry from Africa, turns out not to be complete ly new. Several

years ago R. L. Wilder called my atten- t ion to a quotation that s imultaneously expresses this idea quite eloquently

(with apologies for the word "savage"!) and reminds us that the recognit ion of it dates back at least a hundred years:

A careful s tudy of all woman ' s work in basketry, as well as in weaving

and embroidery, revea ls the fact that both in the woven and in the

sewed or coil ware each st i tch takes up the very same a rea of surface. When women invented basketry,

therefore, they made ar t possible. Along with this fact, tha t each st i tch on the same baske t made of uniform mater ia l occupies the same number

of square mil l imetres, goes another f a c t - - t h a t mos t savage women can count to ten at least. The produc t ion of geometr ic f igures on the surface

of a baske t or b lanket , therefore, is a mat te r of counting. If the enumer- a t ion is cor rec t each t ime the figures will be uniform.

Now, many of the figures on savage baske t ry conta ined intr icate se- r ies of numbers , to r e m e m b e r which

cos t much menta l effort and use of numerals . This constant , every day and hour use of numera ls developed a facility in them, and, coupled with

form in ornament , made geometry possible.

- - F r o m Otis Tufton Mason, Women's Share in Culture, D. Appleton and

Co., N.Y. (1894), 52.

�9 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 ~5

In Le Cercle et le Carrd Gerdes presents the beginning of a ca ta log of shal- low woven baske t s and t rays from var-

ious par t s of the world, made from a var ie ty of mater ia ls by bo th women and men (convenient ly in F rench "van-

n i t r e s et vanniers") depending on the locality. Most of these are f rom Africa

and South America, the two par ts of the wor ld Gerdes knows best , but there

a re severa l examples f rom Vietnam, Indonesia , China, the Phil ippines, and Tonga. Almost all of these are initially

woven in the shape of a square, and then sewn and t r immed to a c ircular rim, so that the final shape is circular.

Par t icular ly a t t ract ive and famil iar examples are made by the Yekuana in the

uppe r reaches of the Orinoco in south- e rn Venezuela, as shown in Figure 1 (Planche 9.8). This example , l ike sev-

eral o thers from the same source, appea r s at first glance to have 90 ~ rota- t ional symmetry, but in fact there is

only half-turn symmetry. (In the b l ack -wh i t e schemat ic of Figure 1,

there is also ref lect ion symmet ry in two perpendicu la r axes. But in the

original weave there is direct ionali ty, which des t roys the ref lec t ion symmetry. Gerdes provides de ta i led drawings in which the d i rec t ional i ty is shown in

detail .) It is aes thet ica l ly des i rab le that the cen te r of the square be at the cen- te r of the circle, and he gives a deta i led

descr ip t ion of ways weaver s throughout the world vary the weave or co lor in o rde r to make the cen te r of the

square immedia te ly obvious, often by c lear ly indicat ing the two diagonals of

the square as lines of symmetry .

Figure 1. Yekuana.

Figure 2. Making a knot.

Geometry f rom Africa goes much fur ther than mere ly cataloging the va- r iety of African designs and pa t t e rns

with a descr ip t ion of the mathemat ica l pr inc ip les inherent in them, though its Chapter 1 does consis t of an overview

of suggest ive African images. Many con t empora ry examples , such as the geometr ic house paint ings of sou thern

Afr ica or the mult i -colored kente cloth from Ghana, a re relat ively familiar, bu t the in t r ica te combina tor ics of old (e leventh cen tury to fifteenth century)

Tel lem woven mater ia l f rom Mali will be new to most . The remaining th ree

chap te rs develop the underlying mathemat ica l pr inc ip les into pedagogica l tools in the form of new geomet r ica l and combina tor ia l problems. These are

the chap te r s where we find the unique cont r ibu t ions of the author.

Chapter 2 shows how var ious designs can lead to finding the Pytha- gorean Theorem. Among these is a simple knot used in Mozambique baske t ry

woven of two s t rands as the beginning of a bu t ton for closing a basket , as in Figure 2a. [Gerdes Figure 2.2]. When

pul led tight, as in Figure 2b [Gerdes 2.4a] a d i ssec t ion of a square appears ,

which can easi ly be t ransformed to yield a famil iar d i ssec t ion p roof of the Pythagorean Theorem, as in Figure 2c

[Gerdes 2.6]. Many o ther examples follow. The last sec t ion of Chapter 2, "From mat weaving pa t te rns to Pyth-

agoras, and Latin and magic squares," shows how an "over 4, under 1" mat

weaving pa t t e rn in a Chokwe mat leads in a natural way to a 5 x 5 Latin square, from which a set of five pai rwise or-

thogonal Latin squares is derived, f rom which a 5 x 5 magic square is con- strncted. It will require a pers is tent and

pat ient t eache r to implement this in the classroom, but the s teps and p ic tures are clear ly presented .

Chapter 3 dea ls mainly with basketry. This includes the p lanar weave

with hexagonal holes in common use throughout the world, for example in chair-caning. When baske t -makers want to conver t such a p lanar weave

into an ac tual container , they realize that some holes smal le r than hexagons, often pentagons , mus t be introduced. If a c losed ball is desired, the intuit ive discovery is made that 12 pentagonal holes will be needed. Such a ball, wi th

the fewest poss ib le number of hexago-


Figure 3. Woven ball (left), soccer ball (right).

nal holes (namely 0), the centuries old "sepak raga" ball from southeast Asia shown schematically in Figure 3a [Gerdes Fig. 3.92a], is closely related to the modern soccer ball of Figure 3b [Gerdes 3.92b], which is combinatori- ally the truncated icosahedron, now known to the chemists as buckmin- sterfullerene. Gerdes has already elab- orated on this theme in his paper "Molecular modeling of fullerenes with hexastrips" in the Mathematical Intel- ligencer (vol 21 (1), 1999, 6-12, 27), where he shows that under natural weaving rules only certain fullerenes are obtainable, and that these are among those that have special chemi- cal properties.

Other sections of Chapter 3 use common African plaiting designs to suggest interesting combinatorial and geometrical exercises. For example, "toothed squares" suggest the question, "Find all the different toothed squares that have fourfold symmetry and diagonals of length n." Four of the 16 examples for n = 7 are shown in Figure 4 [excerpts from 3.113]; all 64 examples for n = 9 are shown in the book. Some intricate questions about placing large numbers of toothed "near-squares"

(with diagonals of lengths n and n + 1) have been solved in imaginative ways by weavers in the lower Congo. Other weavers in East Africa use a diagonal plaiting to make strips that have a variety of symmetries. Examples of all seven symmetry types of strip patterns are produced by ',sipatsi" weavers of Mozambique. These are shown in Figure 5 [Gerdes Fig. 3.149]

Chapter 4 treats "sona" sand drawings of the Chokwe people. These draw-

ings are an especially productive source of exercises and problems. They have led Gerdes, Jablan, Chavey, and Straffm to the invention and study of "mirror curves" obtained by placing suitable in- ternal reflecting barriem on an imagined billiard table before starting the billiard ball at a 45 ~ angle to the sides and trac- ing its path. Figure 6 a and b [4.87 a,b] show the placement of three mirrors and the mirror curve that results. The designs the Chokwe call "chased chicken" and "lion's stomach" are obtained as mirror curves. But surprisingly, as pointed out by the above-mentioned authors elsewhere, many Celtic knots, a variety of mathematical knot projec- tions, and Tamil threshold designs are also examples of mirror curves. Gerdes derives from the mirror curves a class of 0,1 matrices, which in theft black-white visualization he calls "Lunda designs," and finally gives a complete combinatorial characterization of those 0,1 matrices which correspond to Lunda designs.

Figure 4. The 7 x 7 design. Figure 5. Sipatsi strip patterns belonging to each of the seven symmetry classes.


a --

I Q �9 D LD �9 9

�9 �9 B ~ D LP B

B �9 t t ; �9

! �9 tl Q �9 �9 �9

�9 t

b

Figure 6. Mirror design (top), corresponding

mirror curve (bottom).

As ment ioned at the beginning,

much of this mater ia l has appea red in Gerdes ' s less access ib le publicat ions,

somet imes in greater detail . In the un- l ikely event that the r eade r does not f ind enough here to keep occupied, further rewards await , bo th there and in

future work, which is sure to appear .

ERRATA: An unfor tunate edi tor ia l slip- up led to the mislabel ing of mos t of the f igures (Planches) in Chapter 2 of Le Cercle et le Carrd. The r eade r can cor-

rec t these labels as follows:

2.1 to 2.10 should be 2.6 to 2.15;

2.11 to 2.25 should be 2.18 to 2.32; 2.26 to 2.30 should be 2.5, 2.4, 2.1, 2.2,

2.3. 2.31 to 2.34 should be 2.33 to 2.36; 2.35 to 2.36 should be 2.16 to 2.17.

FURTHER REFERENCES=

1. Gerdes, Paulus, "On Ethnomathematical

Research and Symmetry," Symmetry:

Culture and Science, Vol. 1, No. 2(1990),

154-170.

2. , Lunda Geometry." Designs,

Polyominees, Patterns, Symmetries, Uni-

versidade Pedag6gica, Maputo (1996).

3. ., On Lunda-Designs and Some

of their Symmetries, Visual Mathematics (elec-

tronic journal: http://members.tripod.com/

vismathh, Vol. 1, No. 1 (1999).

4. Gerdes, Paulus, and Bulafo, Gildo, Sipatsi:

Technology, Art and Geometry in Inham-

bane, Universidade Pedag6gica, Maputo

(1994).

5. Jablan, Slavik, Mirror generated curves,

Symmetry." Culture and Science, Vol. 6,

No.2,275-278 (1995). (Followed by an elec-

tronic presentation at http://members.tri-

pod.corn/- modularity/mir.htm.)

Donald W. Crowe

Department of Mathematics

University of Wisconsin-Madison

Madison, WI 53706 USA

e-mai: [email protected]

Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences by Jody Azzouni

CAMBRIDGE: THE UNIVERSITY PRESS, 1994. x+249 pp.

US $54.95, ISBN: 0-521-44223-0

R E V I E W E D B Y D I E G O B E N A R D E T E

B y the 1930s, there were th ree main

app roa c he s to the ph i losophy of mathemat ics : logicism, intui t ionism, and formalism. In Jody Azzouni 's book,

a lgor i thmic p rocedure is t aken to be the ha l lmark of mathemat ics , and it

can be cons idered as a succes so r to formalism. By careful a t tent ion to how mathemat i c s is done, the b o o k at-

t empts to mee t the usual objec t ion that rec ipes for mere ly making marks on a shee t of p a p e r cannot account for the

beauty, truth, and usefulness of mathematics.

It somet imes seems that mathemat i - cians have acquired a heredi ta ry wari-

ness of phi losophy, arising f rom en- counters such as those repor ted by Plato be tween the dis t inguished ancient mathemat ic ians Theodorus and

Theaete tus and that respectful but t r icky dialect ic ian Socrates. Theodorus p leads his old age as an excuse to avoid answer ing the quest ion "What is

knowledge"; the young Theae te tus is finally forced to confess that all the answers to that question which he has

come up with in response to Socra tes ' s p rodd ing are "mere wind-eggs."

Mathemat ic ians may fear that the i r vital intui t ions about the na ture of thei r

subject, whe the r t rue or false, will be

stifled or r id iculed by the misappl ied logical r igor of the phi losophers . An example is the fol lowing remark of Jean

Dieudonnd made in response to a ques- t ion at the end of a lecture [D]:

On foundat ions we believe in the reality of mathemat ics , but of course

when ph i losophers a t tack us wi th their p a r a d o x e s we rush to hide be-

hind formal ism and say, "Mathe- matics is jus t a combinat ion of mean-

ingless symbols," and then we bring out Chapters 1-3 of Bourbaki 's Set Theory and so are left in peace to

go back to our mathemat ics and do it as we have a lways done, with the feeling tha t each mathemat ic ian has

that he is work ing with something real. This sensa t ion is p robably an

illusion, but is very convenient. That is Bourbaki ' s a t t i tude toward foun-

dations.

The current genera t ion of mathemat ic ians is even more removed from

the ph i losophy of mathemat ics of our day than the founders of Bourbaki were from theirs. Some think that the

erect ion of formal ba r r i cades has been accompl i shed with Zermelo-Frankel set theory, while o thers share in the

general waning of in teres t in founda- t ions for science. Nonetheless , some familiari ty wi th the subjec t might not

only help us ref lect on our own activity, but also enable us to speak more

effectively in cur ren t debates about the educat ional and research roles of mathemat ics . After all, ma themat ics held its p lace in the curr iculum in par t

because of i ts d i rec t applicabili ty, bu t also because of its p re sumed meri t as a training in abst ract , r igorous think-

ing. During the last half-century the most

impor tant and influential Amer ican

phi losopher of sc ience and mathemat - ics has been Willard Quine. This influence ar ises not so much from his technical cont r ibut ions to symbol ic logic. The great cha rm of Quine 's work ra ther

is how he manages to combine a no- nonsense be l ie f tha t con tempora ry science offers our bes t account of the universe toge ther wi th a wil l ingness to revise prevai l ing phi losophic views on


how to art iculate, justify, and e labora te that belief. We can cer ta inly note these

Quinean charac ter i s t ics in his s tudies of the being of mathemat ica l entit ies.

Roughly speaking, universal terms,

such as c o m m o n nouns like "dog" o r "chair," have been cons idered in th ree

ways. For a realist , these te rms refer to extra-physical , ext ra-mental entit ies, somet imes pic turesquely p laced in what is called with dubious historical ac- curacy "Platonic heaven." For the mentalist, the terms refer to ideas in the mind; for the nominalist, they are merely

names which denote many objects as opposed to proper names which denote only one. A similar range of views exists

concerning mathematical entities such as the integers or the real numbers, even

though for many thinkers a term such as "three" is not considered to be a univer-

sal term but rather a proper noun de- noting the number three.

Quine argues that we mus t ascr ibe real i ty to those enti t ies whose exis- t ence is aff i rmed in natural science,

our bes t theory of the universe, unless such aff i rmat ions can be successful ly p a r a p h r a s e d away [Q1, Q2]. Natural

sc ience rel ies on several b ranches of ma themat ics such as number theory and real analysis. These discipl ines in their usual formulat ions are reple te

with exis tent ia l commi tments to various numbers , sets, functions, and spaces of functions. In a 1947 article,

Quine and Nelson Goodman a t t empted the first s t eps towards a nominal is t re-

formula t ion of these subjec ts tha t would free them of commi tments to

such abs t rac t entities. Quine did not see a way to comple te this project , and with some re luc tance set t led into the

real is t camp in the phi losophy of mathematics. One thinks of the old j o k e about the t own rat ional is t who when asked about the rabbi t ' s foot in his wal- let repl ied that it worked even if one didn ' t be l ieve in it. A poker - faced Quine would not a l low such shenani-

gans, such t r icky bookkeeping. Quine 's has been cal led a "prag-

matic" rea l i sm because it regards our exis tent ia l commi tments as devices

used to s implify our unders tanding of on o therwise confusing hubbub of sen- sory input. It was o therwise with the

d is t inguished logicim-I Kurt GOdel [Wl,

W2]. While one could not decide using

Zermelo-Frankel se t theory whe ther

there was a set wi th a cardinal i ty between that of ~0 and that of the p o w e r

set oflr G0del thought that there was a cor rec t answer to this quest ion because the sets involved exis ted independent ly of any of our theor ies regarding them. This correct answer

would ul t imately be found using not only cons idera t ions of s implici ty and

scientif ic appl icabi l i ty but also a kind of intuition. In fact, G0del 's pos i t ion is shared by many mathemat ic ians ,

though few of them pursue the issue with G0del ' s tenaci ty. For example , over wine and cheese at a recept ion fol-

lowing a lecture at Hunter College in

New York, Andrew Wiles told the story about being pes te red by a correspon- dent who wanted to know whether he

cons idered ma themat i c s to be discovered (real ism) or invented (formalism or pe rhaps mental ism). When asked for his reply, Wiles said, "Discovered, of

course. Doesn ' t every mathemat ic ian

believe that?" These beliefs of many ph i losophers

and mathemat ic ians were chal lenged in two very influential ar t icles by Paul Benacer raf [B]. In the first article,

"What numbers cou ld not be," which appea red in 1965, Benacer ra f r easoned that for bo th prac t ica l and mathemat i -

cal pu rposes there was nothing to choose be tween the von Neumann cons t ruc t ion of the na tura l numbers as

the set of sets [}, [{]}, [{{}}},... , where {} is the empty set, and the Zermelo cons t ruc t ion of these numbers as the

set of sets [}, {{]}, {{}, [{}}}, . . . . Because any such iteratively const ructed se-

quence could serve the purpose, he concluded that the natural numbers could not be ident i f ied with any set of sets. The ma themat i ca l real is t would

have to look e l sewhere for the natura l numbers or pe rhaps concede that they

did not exist. In the subsequent paper "Math-

emat ical truth" [B], which appea red in 1973, Benacer ra f accep t ed a posi t ion of

Quine that our theo ry of knowledge could go so far but no further than assuming all the conclus ions of natura l science. Such a natura l ized epis temol- ogy was thought to require that a causal connec t ion ex is ted be tween the

knower and the thing known, as for example was p rov ided by the ref lect ion

of light in the s impler case of sight. However , even if one granted to the re- a l is t the exis tence of var ious extra-

natura l mathemat ica l beings, it would be imposs ib le to expla in in a natural way how these beings could causal ly inf luence and thus b e c o m e known to

animals such as ourselves. In the following decades, na tura l ized theor ies of

knowledge no longer ins is ted on the need for causal connect ions , but the

p rob lem of how natura l beings could know abs t rac t ob jec t s remained. A rela ted problem, which is pe rhaps more

basic, is how our mathemat ica l language can successful ly refer to mathemat ica l objects. The s t a tement that the Eiffel Tower is on the right bank of the

Seine is false, but we have still suc- c eeded in referr ing to this famous s tructure. Some sor t of connec t ion be-

tween the user of language and the re- ferred-to objec t s eems to be necessary. Such connect ions appea r to be absent

in the case of ma themat ica l objects , at leas t as they are conce ived by the

realist . Various responses to Benace r r a f s

two art icles have shaped the ensuing

decades of ph i losophy of mathemat ics IF, K, M1, M2, S]. Jody Azzouni 's thinking is in some way similar to that of the

th inkers gathered in Tymoczko ' s collect ion New Directions in the Philoso- phy of Mathematics [T] and vigorously

expounded in the mathemat ic ian Reuben Hersh 's new b o o k What is Mathematics, Really? [He]. Like them, Azzouni bel ieves that one should

c losely observe the p rac t ices of mathematicians. But while they wish to s idel ine the concern with foundat ional questions, he bel ieves that he can provide the correc t an swer to the founda-

t ional quest ions and, be t t e r yet, without requiring revis ion of s tandard

accounts of t ruth or reference. An approach like his (which, while

drawing on e lements of all three ap- p roaches to the foundat ions of mathematics , mos t obviously resembles formal ism) can easi ly a rouse the in- dignat ion of many mathemat ic ians . They bel ieve that it could only be held by someone who does not have the ex-

per iences of a real mathemat ic ian .


However , Azzouni bel ieves that his account al lows for these exper iences . He

might differ on how to in te rpre t them. The book has three parts . In Part 1,

Azzouni presents the puzzles that in-

volve knowledge of, and reference to, mathemat ica l objects. In Par t 2, he gives his solut ion to those puzzles. In Par t 3, he d iscusses o the r topics in-

cluding the appl icabi l i ty of mathematics. The Appendix p resen t s a formal

theo ry that includes a t ru th predicate . My review will not fol low this order, but ra ther go direct ly to a recons t ruc-

t ion of the book ' s posi t ive doctr ine. Fo r Azzouni, the key ingredient of

ma themat ics is a lgor i thmic pract ice.

"One sets down an arbi t rar i ly given set of axioms, an arbi t rary set of inference rules, and then der ives t heo rems from

them" (page 80). He is certainly aware that at the core

of cur ren t mathemat ics there are sets

of ax ioms that are impor tan t because of thei r appl icabi l i ty to na tura l sc ience

and to other a reas of mathemat ics . However , if I in terpre t him correct ly, he bel ieves that the neophy te who tr ies all sor t s of odd var ia t ions on the ax-

ioms of a group or of a r ing is still doing mathemat ics , though pe rhaps of a very insignificant kind. He does not re-

quire that this prac t ice be car r ied out in a formal language. The ax ioms and inference rules need only to be suffi-

c ient ly explici t that one can de te rmine wha t counts as a deduc t ion in the system. This gives enough la t i tude so that

Euler ' s manipula t ions wi th infinite se- r ies a re still ma themat ics even though they ignore quest ions of convergence. Needless to say, he inc ludes within mathemat ics various nons tandard prac-

t ices, such as intuit ionism, construc- tivism, and mult i-valued logics. How- ever formal is t this all might seem, it is

impor tan t to realize that mathemat ics is no t cons idered to be a p rocess of making marks on a shee t of paper . Rather it is cons idered to be a ser ies of

sen tences expressed e i ther in a natural language such as English or in a formal language such as f i rs t -order p red ica te calculus. In fact there is no essent ia l

need for the mathemat ica l d i scourse to be wr i t ten down at all.

A pos i t ion like Azzouni 's will be as-

soc ia ted in many minds with the formal i sm of David Hilbert. In fact, Par t II of this book bears as an epigraph the

fol lowing quote from a le t ter tha t Hilbert wro te to Gott lob Frege: "If arbi t rar i ly given axioms do not contra-

dict one ano ther with all thei r consequences, then they are true and the things de te rmined by the ax ioms ex-

ist." Many mathemat ic ians bel ieve that Hilber t ' s app roach was fatal ly weak-

ened by the famous incomple teness results of Kurt G6del. This conclus ion is drawn, for example , in a wonder fu l

book by Howard Eves and Carrol l Newsom [E, page 305]. The p rob lem is

twofold. Hi lber t bel ieved that his formal a p p r o a c h to the t ransfini te par t s of

"If axioms do not contradict one an- o t h e r , . . , then they are true and the things deter- mined by the axioms exist." mathemat ics required there to be a proof, not jus t a belief, that the sys tem was consis tent , i.e., that the "arbi trar-

ily given ax ioms do not con t rad ic t one another." However , G6del showed tha t any formal sys tem strong enough to

conta in e lementa ry ar i thmet ic is not s t rong enough to prove its own con- sistency. Fur thermore , it s eems that

such a formal sys tem conta ins sen- t ences which are unprovable but true. This seems to indicate that ma themat - ical t ru th cannot be r educed to a matter of drawing consequences in a for-

mal system. Azzouni scarce ly repl ies expl ic i t ly to the first objection, i.e., to the lack of a consis tency proof. It is

l ikely that his implici t a rgument is that all of us are in the same boat, wha teve r our ph i losophy of mathemat ics . If to- m o r r o w a cont rad ic t ion should be de- r ived within Zermelo-Frankel se t the-

ory, bo th p la tonis t and formal is t would be equally obl iged to t inker with the ax-

ioms in o rde r to escape the di lemma.

As for the supposed ly true but unprovable sentences , Azzouni points out that v iewed in a s t r ic t ly syntact ic way

all G6del shows is tha t if we assume the cons is tency of ar i thmetic then there is a sen tence of ar i thmetic such

that nei ther it nor i ts negat ion is provable (page 134). This sentence is "true" only if we s tep outs ide our formal sys-

tem and cons ider its terms to refer to

the natural numbers unders tood in the usual way. As we shall see below, Azzouni is all in favor of our tran- scending any given formal sys tem and

expanding it o r l inking it to others. However, such l inkages do not require

any bel ief in mathemat ica l objects existing apar t f rom formal systems.

What then is the mode of ex is tence enjoyed by mathemat ica l objects? According to Azzouni they are "posits"

(Section II.6). He is here accept ing a not ion of Quine, who also cons iders mathemat ica l ob jec t s to be posi ts [Q3].

For Quine, a pos i t is an enti ty that we int roduce to s impl i fy our unders tanding of the complex i t i es of exper ience.

These include the se ts of mathemat ics , but they also include theoret ica l enti- t ies from phys ics such as e lect rons and quarks. In fact, even familiar everyday

objects such as tab les and chairs can be v iewed as pos i t s in t roduced to sim-

plify the wel te r of sense-da ta in which we are engulfed. F o r Quine all these pos i ted ent i t ies have equal claim to ex-

is tence s ince they are inel iminable par ts of science, which is our bes t account of reality.

There is, however , a crucial difference be tween Quine and Azzouni. For the latter, pos i t s come in three "widths": thick, thin, and ultrathin.

Enti t ies such as e lec t rons and quarks are th ick posi ts , for we believe that we can have causal in terac t ions with them by means of var ious exper imenta l pro-

cedures. He rese rves the te rm "thin posit" for pos i t s which have the sim- plifying util i ty which Quine insists on,

but which are not involved in these causal relationships. He is hard pressed

to name a specific example, but con- s iders cer ta in "propert ies" to be candidates for thinness. Perhaps he has

in mind a p rope r ty such as electr ic


charge. Finally, ul trathin pos i t s have no o ther being than that of te rms in-

t roduced in some kinds of s t ruc tu red

discourse. A good example is the king in chess. One descr ibes a game in

which there a re several k inds of p ieces

that can move and capture in specif ic ways on a cer ta in grid. The king is one

such piece. Mathemat ical ent i t ies a re included among the ul trathin posits .

But wha t is a posit? What more can we say abou t i ts mode of being? After all, we usual ly cons ider physical ob- j ec t s to exis t whe ther or not there a re any minds perceiving them. Are pos i t s also mind- independent? In an impor-

tant footnote at the end of his b o o k (page 213), Azzouni s ta tes that there can be no pos i t s wi thout posi tors . I am

not sure whe the r this is mean t to apply to phys ica l posi ts such as e lec t rons jus t as much as to the posi ts of math-

ematics. In any event, it s eems to leave these la t ter pos i t s as being mind-dependen t in an impor tan t way. In t e rms

of my ear l ier classif icat ion of th inkers as realists, mental is ts , or nominal is ts ,

one would have to say that Azzouni is at leas t in par t a mentalist .

Let me pause here to note that with

their not ion of posit, both Quine and Azzouni have answered the Benacer ra f chal lenge as to how the abs t rac t enti-

t ies of ma themat ics can be known. F o r both these thinkers, the mathemat ica l objec ts have no need of causal ly im-

pinging on human mind/bodies in order to be known. They have simply been posited. For Quine they are needed aux-

iliaries to the smooth running of natural science. Fo r Azzouni, they have been brought into existence by the mathematician engaged in axiomatic algorith-

mic procedures . We must now see how these seemingly evanescent ultrathin posi ts are robus t enough to account for all of pure and applied mathemat ics

both pas t and present. One major difficulty that ar ises is

that ma themat ica l te rms are ident i f ied

across many different a lgori thmic systems, both pas t and present . Azzouni gives many examples , and it is easy to supply many more (Sect ions 1.4,1.5, 1.6, II.6, II.7). Egypt ian papyr i are t aken to

refer to the number 12 even though the Egypt ians did not employ the Peano

axiomat iza t ion of the natural numbers .

Mathemat ic ians such as Euler are

taken to have got ten correc t the sum of the infinite geometr ic ser ies 1 + i/2 + 1/4 + . . . even though they did not employ our cur ren t definit ions of convergence. Similar ly Newton was able to integrate the function x 2, even

though he did not have the benefi t of the theory of Riemann or Lebesgue integration. Also, of ten we do not distin-

guish be tween the number 1 under- s tood as a natura l number , an integer,

a rat ional number , or a real number, even though each of these cases involves a different sys tem of axioms.

Finally, the ax ioms of a group are t aken to apply to a whole range of formal sys-

Egyptian papyri

are taken to refer to the number 12 even though the Egyptians did not employ the Peano axiomatization. tems from the in tegers modulo n to the set of cont inuous homeomorph i sms of a topological space. All these examples

suggest that ma themat ica l entities, if they are posits , a re not local posi ts . That is they are not res t r ic ted to a par- t icular ax iomat ic system.

It would seem that a mathemat ica l

real is t would have litt le difficulty with these examples . The real is t would say that the ancient Egypt ian and the mod- e m American are ab le to refer to the

same numbers (e.g., 12) in much the same way as they are able to refer to the same celes t ia l bod ies (e.g., the

sun). In both cases the enti t ies have an object ive ex is tence which, not surprisingly, can be d i sce rned with greater or lesser clari ty by different thinkers. A formalis t such as Azzouni needs to come up with an explanat ion of his

own. He argues tha t there is, in fact, no absolute ly compel l ing need for us to identify enti t ies ac ross the boundar ies

of mathemat ica l systems. However, he e labora tes and accep t s the many rea-

sons which induce mathemat ic ians and his tor ians of ma thema t i c s to do

jus t that. He takes these ident i f icat ions to be s imply a ma t t e r of "stipulation" (page 121). We s imply identify one t e rm with another. Fo r example , when

const ruct ing the real numbers , we identify the natura l number 1 with a

cer ta in Dedekind cut. Taken more generally, it seems to me that this business of "identification" is a famil iar par t of

mathemat ica l pract ice . It is informally t a lked about whenever some quotient

ob jec t is being cons t ruc ted in, say, topology or group theory. Of course, in

con tempora ry mathemat ics , which is usual ly mode led in set theory, these cons t ruc t ions are e l abora t ed as equiv- a lence classes de te rmined by equiva-

lence relations. Azzouni 's identifica- t ions have a s imilar flavor, but they are accompl i shed by identifying bits of

syn tax across systems. The mat te r of group theory, which I a l luded to above, is somewha t more compl ica ted . Such

theor ies have many non- i somorphic mode l s none of which is cons idered s tandard. He takes these to be collec-

t ions of sentence s c h e m a ra ther than col lect ions of sen tences (Sect ion II.7).

Another usual ob jec t ion to a convent ional is t account of mathemat ics

such as Azzouni 's is that it cannot explain the uncanny effect iveness of ap- p l ied mathematics . The s t ruc tures of

ma themat ics do not s eem to be mat ters of agreement or s t ipulat ion, but ra ther par t and parce l of the formal bed rock of the universe. In his account of these

mat ters , Azzouni begins f rom the ob- serva t ion that there are many formal geometr ica l systems, and that it is a ma t t e r of empir ical sc ience to decide

which one bes t fits the cosmos. So, for example, if a model of the universe is

taken t o be a Lorentz manifold with underlying topological space S 3 x R, then the identifications be tween the universe

of physical theory and the mathematical entity would presumably be handled by mechanisms similar to those which explain the cross references within pure

mathematics: that is, by stipulation, and/or by viewing some mathematical theories as sentence schemata which


can be turned into sentences by fixing some abstract or empirical interpreta- tion. Azzouni has a similar understand-

ing of all other applied mathemat ics including applied arithmetic. He takes the

supposed "unreasonable effectiveness of mathematics" to be not so unreasonable. It is simply a mat ter of our evolu-

t ionary adaptat ion to the world in which we live (Sections 1].3, III.6).

A final difficulty is that mathemat i - c ians for the mos t pa r t do not think of themse lves as opera t ing in the way that

Azzouni suggests. This he acknowl- edges. Fo r example: "this is not how Euler saw what he was doing; and gen-

eral ly no mathemat ic ian sees things in this way. Mathemat ic ians invariably

have a semant ic in te rpre ta t ion in mind. AS a resul t they have no compunc t ion abou t augmenting sys tems at any t ime

on the basis of this in terpreta t ion" (page 84). He does not v iew this ob- j ec t ion as fatal, for, as his t i t le pro-

claims, he is trying to be faithful to the p rac t ice of mathemat ic ians and not necessar i ly to their own unders tanding of this pract ice. Cons ider the following

s t rong statement . "In general , real ism

is never needed to expla in why a group of profess ional sc ient is ts have the p rac t i ces they have: The collect ive de lus ion that wha t they are s tudying

exis ts can do jus t as well" (page 138). He does not dismiss the non-formal motivat ions, insights, and intuit ions

tha t guide mathemat ic ians . They jus t b e c o m e par t of the psycho logy and so- c iology of the subject. "In pract ice , one p r o d u c e s proofs in somewhat the same

way that humans play chess: we see patterns, proof patterns, that we then sketch out informally. Seeing these things, this way, is not a mechanical matter: we do not work within algorithmic

systems algorithmicaUy" (page 137). Lest we be alarmed by the comparison with chess, Azzouni soon concludes that

"mathematics is a more creative and in- triguing subject than anything afforded

by a game" (page 139). Broadly spealdng, Azzouni gives a

conventionalist account of mathemat-

ics. Today, when the so-called "science wars" are raging and threatening to spill over to mathematics, and when conventionalist accounts of science are damned

as pernicious for their polit ical consequences, it may be appropriate for me to discuss his posi t ion in the light of such attacks. First, it must be emphasized

that Azzouni is in no way a conventionalist in regards to science. The world is

real. So also are the objects of science, our knowledge of which continues to

grow. To proceed to mathematics, I think it is interesting to try to apply Azzouni's account to a famous l i terary

confrontat ion of politics and mathematics. Many of us recall reading in Orwelrs

novel 1984 about the terrifying experi- ences in Room 101 by which the Inner

Party member O'Brien induced the

I remain puzzled. However, this is not necessarily a bad thing. Outer Party member Winston to abandon his view that 2 § 2 = 4. The objec-

tive truth of mathematics is here presented as one of the last bulwarks of f reedom against totalitarianism. How

might Azzouni deal with O'Brien's arguments? (This wicked man combined discourse with torture as means of per-

suasion.) Well, if it is a mat ter of pure mathematics, and if O'Brien accepts the axioms and rules of inference of Peano

arithmetic, then the denial that 2 + 2 = 4 is s imply a mistake. If it is a mat ter of appl ied mathematics, Azzouni accepts

the argument of Quine that there are no absolutely incorrigible "co-empirical" statements. However, the denial that 2

volts plus 2 volts equals 4 volts would be bad science, which if generally followed would soon lead to the destruct ion of Big Brother 's tyranny. We see, therefore, that O'Brien's t r iumph remains the tri-

umph of a bully, with no laurels con- ferred by Azzouni's philosophy.

Some mathemat ics texts claim little

in the way of formal prerequisi tes but do s tate a need for that al l - important mathemat ica l sophistication. However, this b o o k does p re suppose a cer ta in sophis t ica t ion in the phi losophy of math-

ematics. Even without such prepara - tion, many sec t ions are access ib le and

interesting. These include those which

presen t the core of his teaching (Sect ions II.2, II.3, II.6, II.7, III.6) and others which are dense with mathe-

mat ical examples (I.4, 1.5, 1.6). How- ever, the neophy te who wishes to get

the most out of it wou ld do well to r ead first the two Benacer ra f ar t icles [B], and then fol low with Maddy's Realism in Mathematics [M1, Hi].

This is an impor tan t book and

should be in every college library. As far as I can tell the re are no er rors

in mathemat ics , logic, or phi losophy. Fur thermore , it shows some familiarity with the ways of mathemat ic ians .

Azzouni 's b o o k not only has many well- chosen his tor ical examples , but also contains some of those informal re-

marks which are made as we chat over coffee and scr ibble on napkins. For ex-

ample, "a p ro fe s so r of mathemat ics once told me that his opinion of some-

one 's ma themat ica l abil i ty would d rop sharply if he lea rned that he or she had

read a mathemat ica l tex t from cover to cover" (page 168, note 27).

However, the key reason for my rec- ommenda t ion is tha t this is the bes t con tempora ry defense of a minimal is t pos i t ion in the ph i losophy of mathe-

mat ics that I am famil iar with. Why is such minimal ism impor tant? Let us lis-

ten to the poe t Wallace Stevens, who in Credences of Summer [St] asks us to "Exile des i re /For wha t is not" and replace it with the "visible announced"

because

This is the successor of the invisible. This is its substitute in stratagems

Of the spirit. This, in sight and memory Must take its place, as what is possible Replaces what is not.

The gist is that we mus t j e t t i son the impossible, that which can no longer be believed, and t ry to get by on the

possible, and even t ry to find a new charm in such low and humble ground. Most mathemat ic ians bel ieve that they are uncover ing objec t ive t ruths about some preexis t ing reality. But, as Reu-

ben Hersh po in ts out [He, page 11], when asked to expla in how this k ind

of being and this k ind of knowledge is poss ib le they quickly change the sub-

'72 THE MATHEMATICAL INTELLIGENCER

ject. Now everyone would grant the little that Azzouni presupposes for our cognitive powers. While one may quib- ble on the details, it seems plausible that mathematics could be recon- structed on this narrow foundation. Therefore, even mathematicians of a realist persuasion should welcome this account. For them, it provides a solid base which could serve as a makeshift until replaced by something more glo- rious fashioned out of our intuitions

and longings. I still do not know the nature of

mathematical objects. I see arguments which support each of the standard positions. I remain puzzled. However, this is not necessarily a bad thing. For as we are told by Aristotle, philosophy begins with wonder.

REFERENCES

[B] Paul Benacerraf, "What numbers could not

be" and "Mathematical truth," in Philosophy

of Mathematics: selected readings (second

edition), Paul Benacerraf and Hilary Putnam

(editors), Cambridge Univ. Press, 1983.

[D] Jean Dieudonne, The work of Nicholas

Bourbaki, Amer. Math. Monthly, 77(1970),

134-145.

[E] Howard Eves and Carroll Newsom, An

Introduction to the Foundations and Funda-

mental Concepts of Mathematics (revised

edition), Holt, Rinehart, and Winston, 1965.

[FJ Hartry Field, Science Without Numbers: a

defence of nominalism, Princeton Univ.

Press, 1980.

[He] Reuben Hersh, What is Mathematics,

Really?, Oxford Univ. Press, 1997.

[Hi] Morris Hirsch, Review of Realism in

Mathematics by Penelope Maddy, Bull.

AMS, 32(1995), 137-148.

[K] Jerrold Katz, Realistic Rationalism, MIT,

1998.

[M1] Penelope Maddy, Realism in Mathemat-

ics, Oxford Univ. Press, 1990.

[M2] Penelope Maddy, Naturalism in Mathe-

matics, Oxford Univ. Press, 1997.

[Q1] W. V. Quine, "On what there is," in From

a Logical Point of View, (second edition, re-

vised), Harvard Univ. Press, 1961. Also in

Philosophy of Mathematics: selected read-

ings (first edition), Paul Benacerraf and Hilary

Putnam (editors), Prentice Hall, 1964.

[Q2] W. V. Quine, "Truth by convention," in

Philosophy of Mathematics: selected read-

ings (second edition), Paul Benacerraf and

Hilary Putnam (editors), Cambridge Univ.

Press, 1983. [Q3] W. V. Quine, "Posits and reality," in The

Ways of Paradox, and Other Essays (revised

and enlarged edition), Harvard Univ. Press,

1976. [S] Stewart Shapiro, Philosophy of Mathema-

tics: structure and ontology, Oxford, 1997.

[St] Wallace Stevens, Collected Poetry and

Prose, Library of America, 1997.

[T] Thomas Tymoczko (editor), New Directions

in the Philosophy of Mathematics (revised

edition), Princeton Univ. Press, 1998.

[W1] Hao Wang, Reflections on Kurt Gddel, MIT

Press, 1987.

[VV2] Hao Wang, A Logical Journey: from Gddel

to philosophy, MiT Press, 1996.

Diego Benardete

Department of Mathematics

University of Hartford

West Hartford, CT 06117 USA


Proof A Play by David Auburn

with Mary-Louise Parker,

Larry Bryggman, Johanna Day,

and Ben Shenkman

Walter Kerr Theater, New York City

REVIEWED BY JET WlMP

T he dramatic crux of Proof depends on a mathematical proof of a result

in number theory, so people should be staying away in droves, right? They're not staying away. The matinee perfor- mance I saw on a Wednesday was filled to overflowing with a very receptive audience, and the play, by 31-year-old playwright David Auburn, is absolutely thrilling. At heart the play is a mystery, but a witty, intellectually engaging, and viscerally compelling one.

Proof has little in common with the British playwright Michael Frayn's Copenhagen--another very popular Broadway play that chronicles in a highly fictionalized manner a meeting between physicists Niels Bohr and Weruer Heisenberg--even though both plays discuss men of science and their

sometimes tenuous engagement with the world. Like Copenhagen, Proofdeals with personalities who have achieved near grandeur through their abilities to resolve scientific enigmas, but Proof depends less on technical jargon and ab- struse moral dicta than on rich and easily accessible human concerns. Proof was such a critical and popular success at its Manhattan Theater Club off- Broadway run that it was soon moved to Broadway.

Catherine, the young daughter of a brilliantly successful but tragically un- balanced University of Chicago mathematician, is, at the play's beginning, about to bury her father. She has spent six onerous years caring for him, nurs- ing him through bouts of madness. The battle, which has caused her own college career to be aborted, has left her pugnacious and disparaging of everyone and everything, clearly symptoms of a treacherous nihilism that only anger can keep at bay, and then only for a little while. She suspects that a younger former graduate student of her father's, to whom she has allowed temporary access to her father's study, is trying to smuggle out and publish mathematical results contained in car- tons of old notebooks, thereby bolster- ing his own professional stature. The student insists that the notebooks hold only her father's ravings, and although Catherine believes him sufficiently to give in to a rare moment of trust that puts them in bed together, her trust is soon challenged. Catherine's character, the complete confidence with which it is limned, is impressive dramaturgy.

What is compelling about the play is the structure, imaginative and also completely coherent. The play darts between past and present and dream and reality in a way that is both skilled and agreeably trusting of the audience. It comes rather as a shock to discover the disposition of a leading character in the first scene. However, that plot device is just the touch needed to set the play properly in motion, to augur the mystery that inhabits the core of the play, the mystery of the origins and compass of the creative act. The play has many surprises, but the surprises are more than just pinchbeck pulled


out of a p laywright ' s bag of tricks. The p lo t twists grow logical ly out of the

events and the cha rac te r s the playwright has worked very hard to put before us. Nevertheless , the surpr ises

make it difficult to d i scuss the p lay wi thout indulging in, to use Netnews

film discussion par lance , "spoilers." It is not giving away too much to re-

veal that at the apex of the d rama a note- book is discovered that contains a rev-

olutionary result in number theory in a p roof occupying some 40 pages. This re-

suit has been sought by mathemat ic ians for centuries, and it p romises to have substantial implicat ions for the future

of mathematics. The questions posed for the audience are two: First, is

the proof, which utilizes the most modern mathemat ica l techniques- -a l - gebraic geometry and representa t ion

theory are men t ioned - - r ea l ly a proof, or is it jus t one of the pseudo-proofs that surface so often in mathematics?

Second, who is the author of the proof?. In the course of the play, both of these

questions are answered, subtly and sat- isfyingly.

Several scenes s t ick gratifyingly in my mind. One, a f lashback, opens with

the father, apparen t ly emerging from his long struggle wi th insanity, scrib- bl ing in a no t ebook with childlike en- thusiasm. What he is writing, he vol-

ubly assures his daughter as he wrings his hands in self-satisfaction, is good

stuff; it signals the beginning of his return to profess ional mathemat ics . He shows the n o t e b o o k to Catherine, but

she somehow is conce rned only that the porch is too chilly for him, and begs him to come inside. He insis ts she read two lines of wha t he has done, and she again implores him to come inside.

Finally, red-faced and raging, he demands she read f rom the proof. As she begins to read s lowly and tonelessly, we are shockingly conf ron ted with the

d imens ions of the fa ther ' s t ragic ill-

ness. All the ac tors deserve the grea tes t

plaudits . Mary-Louise Parker , who has

had an abundant career on s tage and in film, gives a wonderful ly nuanced and highly pra ised pe r fo rmance as

Catherine: at first abrasive, bu t vulner- able and fearful of the degree to which

her fa ther ' s fate may have infected her, she spi ts her lines with a laughing snarl

l aced with au courant col lege-age pa- tois as she quaffs cheap champagne r ight f rom the bottle. Yet during the

course of the d rama she is revea led to

be a pe r son of imposing spir i tual and

in te l lec tual resources. Larry Bryggman as the fa ther is

equally impressive. Soap fans will recognize him as the charac te r Dr. John Dixon, which he has p layed in "As The

World Turns" since 1956. With his res- onant voice and urgent physical i ty, he conveys with comple te convic t ion

bo th the father 's immense love for his daughter and also his despa i r when he realizes, as in the above-ment ioned

scene, the i r reparable devas ta t ion of his mind. Johanna Day and Ben Shenkman provide great suppor t , re-

spectively, as Cather ine 's very conce rned but unabashedly achievement-

o r ien ted s is ter and as the gradua te s tuden t whose ambiguous c h a r a c t e r - - is this a knight in shining a r m o r or the

gradua te s tudent from he l l ? - - i s ye t an- o ther t r ibute to the p laywright ' s skill.

The mat inee audience was not a typ-

ical one. About half were high school s tudents , but, mouse quiet, they at- t ended to the play raptly, somet imes respond ing with a unified gasp o r sigh

at narra t ive high points. My disap- po in tmen t was with the behav ior of some of the oldsters who had fai led to ha rness thei r e lectronic a c c e s s o r i e s - -

l is tening devices, beepers , cell tele- phones. Twice the thea ter manage r had

to remind us tha t these implements must be control led. The d iscomfi ture

of the aud ience at these incurs ions was extreme. The end of this p rob lem is nowhere is sight, and soon I expec t we will read in our morning papers of the

first inc ident of Broadway thea te r rage. The reviews the drama has gathered

provide an interesting insight into how the rest of the world views mathematicians. A review in The New York Times

suggests that we mathematicians are

both blessed and bedeviled by our pen- chant for abstraction, that our gift both

bonds us and separates us from the rest of manldnd, and the critic expresses sur-

prise that the playwright has managed to reach into the forbidding terrain of intellectual pursuit and discover "good peo-

ple." No one seems to want to take the less dramatic viewpoint that mathematicians are no different from anyone else

who is lucky enough to possess a talent for doing what he or she loves to do.

There is not much mathemat ics as

such in the play, only that required to es tabl ish context . I think a little more would have enhanced my resonance

with the p lay ' s central mystery. So many of the grea t unsolved p rob l ems in number theory can be communi-

ca ted easi ly to a lay audience. Was the p roof that of the the twin pr ime con- jec ture? Goldbach ' s conjecture? But

the use of mathemat ics as a me taphor for the creat ive exper ience is unerr ing

and tota l ly dramatic . The play has much to say abou t the corros ive effect

of academic compet i t iveness , and its rueful obse rva t ion that t ruth of ten chooses as i ts human vessels those too

fragile to conta in it is forcefully made. I can ex tend to this work no grea ter t r ibute than to say that at the p lay 's end, when a cha rac te r delivers the line, "This is a s t andard number- theore t ic

result," the hair on the back of my neck s tood up. And I'll be t yours will too.

'74 THE MATHEMATICAL INTELLIGENCER

SPRINGER FOR MATHEMATICSROBIN WILSON, The Open University, Oxford, UK; andJEREMY GRAY, The Open University, Milton Keynes, UK

MATHEMATICAL CONVERSATIONSSelections from the Mathematical Intelligencer

Since its first issue, The Mathematicallntelligellcer has beenthe main forum for expositionand debate between some of the !world's most renowned math- Iematicians, covering not onlyhistory and history-making mathematics, but also includingmany controversies that surround all facets of the subject.This volume contains forty arti

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MARTIN AIGNER, Freie Universitiit and GUNTER M. ZIEGLER,Technische Universitiit, both, Berlin, Germany

PROOFS FROM THE BOOKSecond Edition

The (mathematical) heroes of this book are "perfectproofs": brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems ,from Number Theory, Geometry, Analysis, Comb-Iinatorics, and Graph Theory. Thirty beautiful exam- IpIes are presented here. They are candidates for TheBook in which God records the perfect proofs - :according to the late Paul Erdos, who himself suggested many of the topics in this collection. Theresult is a book which will be fun for everybody withan interest in mathematics, requiring only a verymodest (undergraduate) mathematical background. For .this second edition several chapters have been revisedand expanded, and three new chapters have been added.2000/224 PP./HAROCOVER/S29.95/ISBN 3-540678654

l. BADESCU, Romanian Academy, Bucharest, Romania

ALGEBRAIC SURFACES

K. JANICH, Universitiit Regensburg, Regensburg, Germany

VECTOR ANALYSISClassical vector analysis deals with vector fields; thegradient, divergence, and curl operators; line, surface,and volume integrals; and the integral theorems ofGauss, Stokes, and Green. Modem vector analysis distills these into the Carlan calculus and a general formofStokes's theorem. This essentially modern text carefully develops vector analysis on manifolds and reinterprets it from the classical viewpoint (and with theclassical notation) for three-dimensional Euclideanspace, then goes on to introduce de Rham cohomology and Hodge theory. The material is accessible toan undergraduate student with calculus, linear algebra, and some topology as prerequisites. The manyfigures, exercises with detailed hints, and tests withanswers make this book particularly suitable for anyone studying the subject independently.2000/288 PP., 114 ILLUS./HAROCOVER/$34.95ISBN 0387-98649-9UNDERGRADUATE TEXTS IN MATHEMATICS

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UNIVERSITEXT

B.---l~.uJlu,]ao-~.]i=[:~i R o b i n W i l s o n I

World Mathematical Year 2000

Argentina

O n 6 May 1992, in Rio de Janeiro, the International Mathematical

Union designated the year 2000 as World Mathematical Year. This special year was commemorated philatelically by a number of countries that issued stamps with a mathematical theme.

The first country was Belgium, with a WMY2000 stamp featuring S tokes ' s theorem and Fermat's last theorem inside a circle, together with the normal curve and a representation of the notches on the Ishango bone, the earliest known mathematical artefact. Next came Luxembourg, also with Forma t ' s las t theorem and S tokes ' s theorem, and with the value o f ~ and the Riemann zeta-function, all featured next to the attractive logo of WMY2000. A variety of mathematical pictures also appears on the stamp from Monaco.

The WMY2000 logo also appears on

the Spanish stamp, next to Jul io Rey Pastor, a dominant figure in geometry and analysis in Spain and later in Argentina. The Argentina stamp features the number 2000 next to the infinity symbol, first introduced by John Wallis in 1665.

Format's last theorem is the central theme of the stamp from the Czech Republic, with Fermat's equation can- celled by ANDREW WILES 1995 across the equals sign. The Slovakian stamp features the Slovak mathematicians Juraj Hronec and Stefan Schwarz with a lattice diagram. The Croatian stamp features graph theory, with a representation of the "Blanu~a snark," presented by Danilo Blanu~a in 1946; this is the smallest 3-regular graph (other than the well-known Petersen graph) that cannot be 3-edge- coloured in such a way that adjacent edges receive different colours.

Croatia

Czech Republic Luxembourg

Belgium

Slovakia Spain

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes, MK7 6AA, England


Monaco

7 6 THE MATHEMATICAL INTELLIGENCER �9 2001 SPRINGER VERLAG NEW YORK

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