No title

)pinion Reply to Martin Gardner Reuben Hersh The Opinion column offers mathematicians the opportunity to write about a...

Author: M.L. Senechal | C. Davis (Editors in Chief)

5 downloads 304 Views 22MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

)pinion

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-inchief, Chandler Davis.

Dear Martin Gardner, Thanks for y o u r interest in m y writings. As e v e r y o n e knows, y o u ' r e the m o s t highly r e s p e c t e d science journalist in the world. I j u s t c o u n t e d six of your b o o k s on m y shelf. Yet for interesting, m y s t e r i o u s reasons, you s e e m unable o r unwilling to u n d e r s t a n d m y writing a b o u t m a t h e m a t i c a l existence. Your u n h a p p i n e s s with me is n o t new. You d i s s e d The Mathematical Experience [1] b y m e and Phil Davis, in the New York Review of Books. In the m o s t r e c e n t issue of The InteUigencer [4], you r e t u r n to the task. You quote "myths 2, 3 and 4." from m y Eureka article (reprinted 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 c o n c e n t r a t e on 2 a n d 4. Myth 4 is objectivity. "Mathematical truth o r knowledge is the same for everyone. It does not depend on who in particular discovers it; in fact, it is true whether or not anyone discovers it." Your reaction: "What a strange cont e n t i o n " - - t o call it a myth. Myth 2 is certainty. "Mathematics p o s s e s s e s a m e t h o d called ' p r o o f ' . . . by which one attains a b s o l u t e certainty of the conclusions, given the truth of the premises." Your reaction: "Can Hersh be serious w h e n he calls this a myth?" In a way, I u n d e r s t a n d y o u r difficulty. In c o m m o n speech, w h e n s o m e one says, "That's j u s t a myth!" he m e a n s s o m e t h i n g is false, untrue. But in scholarly writing, "myth" c o m m o n l y has o t h e r meanings. I wrote, on the very n e x t page, "Being a myth d o e s n ' t entail its truth o r falsity. Myths validate and s u p p o r t institutions; their truth m a y not be determinable." About certainty, I wrote: "We're certain 2 + 2 = 4, though we don't all m e a n the s a m e thing b y that equation. It's another m a t t e r to claim certainty for the t h e o r e m s of c o n t e m p o r a r y mathematics. Many of these t h e o r e m s have proofs that fill dozens of pages. They're usually built on top of o t h e r theorems, w h o s e proofs w e r e n ' t c h e c k e d in detail by the

mathematician w h o quotes them. The proofs of these t h e o r e m s replace boring details with 'it is easily seen' and 'a calculation gives.' Many p a p e r s have several coauthors, no one of w h o m thoroughly c h e c k e d the whole paper. They m a y use machine calculations that none of the authors completely understands. A mathematician's confidence in some t h e o r e m n e e d not mean she knows every step from the axioms of set theory up to the t h e o r e m she's interested in. It m a y include confidence in fellow researchers, journals, and referees. Certainty, like unity, can be claimed in p r i n c i p l e - - n o t in practice." Now, y o u ' r e talking a b o u t certainty in principle. I do too. I recognize its i m p o r t a n c e as a positive and valuable guiding myth. I also talk a b o u t certainty in practice. The w h o l e mathematical c o m m u n i t y recognizes its value and is e n g a g e d in seeking it. I m m e d i a t e l y following in m y book: "Myth 4 is objectivity. This myth is m o r e plausible t h a n the first three. Yes! There's amazing c o n s e n s u s in mathematics as to w h a t ' s c o r r e c t or accepted." On p a g e 176 I elaborate: "Mathematical truths are objective, in the sense that t h e y ' r e a c c e p t e d by all qualified persons, r e g a r d l e s s of race, age, gender, political o r religious belief. What's c o r r e c t in Seoul is correct in Winnipeg. This 'invariance' of mathe m a t i c s is its very essence." On page 181: "Our conviction w h e n w e w o r k with m a t h e m a t i c s that w e ' r e w o r k i n g with s o m e t h i n g real isn't a m a s s delusion. To e a c h of us, mathem a t i c s is an e x t e r n a l reality. Working with it d e m a n d s w e s u b m i t to its obj e c t i v e character. It's w h a t it is, not w h a t w e w a n t it to be." So we agree. Mathematical truth is objective! Then h o w can a sophisticated critic like you think I don't recognize the objectivity of mathematical truth? The question, of course, is what we mean by "objective." To you, "objective" means "out there." To me, "objective" means "agreed u p o n by all qualified people who c h e c k it out." But I'm

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

3

unwilling to leave the m a t t e r at that. Like m a n y other people, I think objectivity is to be u n d e r s t o o d by reference to objects, things that really have the p r o p e r t i e s we discover. Mathematical o b j e c t s are simply the things mathematical statements are about. Numbers, functions, operators, spaces, transformations, mappings, etc. What sort of objects are they? They're not physical objects. Aristotle already explained that the triangles and circles of Greek geometry are not physical entities. The first few natural numb e r s 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 m a d e it up. The m o s t important p r o p e r t y of N is mathematical i n d u c t i o n - - a n axiom said to be intuitively obvious. Intuition is "in there," not "out there." Certainly the use and interest of the abstract mathematical numbers come from their close connection with physical numbers. But meaning and existence can't b e untangled without acknowledging the distinction between physical n u m b e r s and mathematical numbers. We also study infinitely differentiable infinite-dimensional manifolds o f infmite connectivity. These are not f o u n d in the physical world. Not to m e n t i o n the "big" sets o f c o n t e m p o r a r y set-theorists. Then, if not in p h y s i c a l reality, could m a t h e m a t i c a l o b j e c t s exist "in the mind"? Gottlob F r e g e f a m o u s l y der i d e d this idea. If I a d d up a r o w o f figu r e s a n d get a w r o n g answer, it's w r o n g even if I think it's right. The theo r e m s o f Euclid r e m a i n after Euclid's m i n d is buried with Euclid. So where are the objects about which mathematics is objective? The answer was given by the French philosopher/ sociologist Emile Durkheim, and exp o u n d e d by the U.S. anthropologist Leslie White [8]. But social scientists a r e n ' t cited by philosophers, n o r by m a n y mathematicians. (Ray Wilder w a s the exception.) The universe contains things o t h e r t h a n mental o b j e c t s a n d p h y s i c a l objects. There are also institutions, laws, c o m m o n understandings, etc., etc., etc.--social-historical objects. (I say

4

THE MATHEMATICALINTELLIGENCER

j u s t "social" for short.) We c a n n o t think o f w a r o r m o n e y o r the S u p r e m e Court o r the U.S. Constitution o r the d o c t r i n e o f t h e virgin birth as e i t h e r p h y s i c a l or m e n t a l objects. T h e y have to be u n d e r s t o o d and dealt with on a different l e v e l - - t h e social level. Social entities are real. If y o u d o u b t it, s t o p paying y o u r b i l l s - - s t o p o b e y i n g the s p e e d limit. And social entities have real properties. That's h o w w e m a n a g e to negotiate daily life. Social scientists don't say "object." They s a y "process" and "artifact" a n d "institution." Social p r o c e s s e s a n d artifacts a n d institutions are g r o u n d e d in p h y s i c a l a n d mental o b j e c t s - - m a i n l y the b r a i n s a n d the thoughts o f people. But t h e y m u s t be u n d e r s t o o d on a different level from the mental o r physical. In o r d e r to decide w h e r e m a t h e matics belongs, I m u s t c o n s i d e r all three---the physical, mental, a n d social. I n e e d a w o r d that can a p p l y to all t h r e e - - p h y s i c a l , mental, a n d social worlds. "Object" s e e m s suitable. The c o m m o n c o n n o t a t i o n of "object" as only a p h y s i c a l entity has to be set aside. Any definite e n t i t y - - s o c i a l , mental, o r p h y s i c a l - - w h o s e e x i s t e n c e is m a n i f e s t e d b y real-life e x p e r i e n c e c a n be called an object. Mental o b j e c t s (thoughts, plans, intentions, emotions, etc.) are g r o u n d e d on a p h y s i c a l bas i s - t h e n e r v o u s system, or t h e brain. But w e c a n n o t deal with our t h o u g h t s or the t h o u g h t s of each o t h e r as physical o b j e c t s - - e l e c t r i c currents in t h e brain. That is w h y there is a "mindb o d y problem." And social-historical o b j e c t s a r e on still a different level from either the mental o r the physical. Now, Martin, if you recognize the e x i s t e n c e of social objects, y o u ought to ask, "Since m a t h e m a t i c a l o b j e c t s are n e i t h e r physical n o r mental, a r e they social?" My a n s w e r is, "Yes, that is w h a t t h e y are." That's controversial. It's "maverick." That doesn't m e a n you can dispose o f it by distorting or denouncing it. That m a t h e m a t i c s is in the m i n d s of people, including mathematicians, is n o t a novelty. Everyone k n o w s that. It's in m i n d s c o n n e c t e d b y frequent c o m m u n i c a t i o n , in minds that follow the heritage o f p a s t mathematicians.

My claim is this: to understand w h a t m a t h e m a t i c s is, w e need not go beyond this recognized social existence. That's where it's at. Locating mathematics in the w o r l d o f social entities DOESN'T m a k e it unreal. Or imaginary. Or fuzzy. Or subjective. Or relativistic. Or p o s t m o d e r n . Saying it's really "out there" is a reach for a s u p e r h u m a n certainty t h a t is not attained b y a n y h u m a n activity. A famous m a t h e m a t i c i a n said to me, "I am willing to leave that question to t h e philosophers." Which p h i l o s o p h e r s ? Professional p h i l o s o p h e r s who are n o t mathematicians?! To obtain a n s w e r s meaningful to us, I'm afraid we'll have to get to w o r k 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 o f h u m a n consciousness. I a n s w e r e d that claim in m y recent book. In your review of it in the L.A. Times, you ignored m y answer. In your letter to The Intelligencer, you ignore it again. You j u s t r e p e a 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 t h e m "physical numbers," though they are also used for mental and social entities. It's a physical fact that two m a m a b e a r s and three p a p a bears together m a k e five great big bears. To put it in more a c a d e m i c terms, there are discrete structures in nature, and they can occur in sets that have definite numerosity. In mathematics, on the other hand, w e deal with "abstract structures," n o t b e a r s or fingers o r dinosaurs. In mathematics, the w o r d s "2", "3", and "5" c a n be nouns, denoting certain a b s t r a c t objects, e l e m e n t s of N. As I explain above, N and its m a i n p r o p e r t y are not found in physical nature. Counting dinosaurs uses p h y s i c a l numbers, adjectives, not the a b s t r a c t n u m b e r s w e study in m a t h e m a t i c s . The physical n u m b e r s apply even if we don't k n o w a b o u t them. They a r e p a r t of physical reality, not h u m a n culture. Mathematical numbers, on the o t h e r hand, are a h u m a n creation, p a r t o f our social-historical heritage. They w e r e created, w e presume, from the physical adjective

numbers, b y a b s t r a c t i o n a n d generalization. F r o m time to time you call m e a "cultural relativist." Cultural relativists say, "Western music (for instance) is not b e t t e r o r w o r s e than New G u i n e a music. It's different, that's all." When I say m a t h e m a t i c s is p a r t of h u m a n culture, t h e r e ' s no relativism involved. More m y s t e r i o u s is y o u r conclusion: "To imagine that t h e s e awes o m e l y c o m p l i c a t e d and beautiful patterns are n o t 'out there' i n d e p e n d e n t o f y o u and me, but s o m e h o w c o b b l e d b y o u r m i n d s in the w a y w e write p o e t r y a n d c o m p o s e music, is surely the ultim a t e in hubris. 'Glory to Man in t h e highest,' sang Swinburne, 'for Man is the m a s t e r o f things.' " This song o f Swinburne s e e m s to b e "coming f r o m left field." It s u d d e n l y denies y o u r m a i n contention. To unders t a n d it I l o o k at y o u r books, Order and Surprise [2] and The Whys of a Philosophical Scrivener [3]. In Order and Surprise [2] y o u write, critiquing Ray Wilder, "One may, of course, a d o p t any w a y of talking one likes, b u t the fact is that m a t h e m a t i cians do n o t talk like Wilder e x c e p t for a few w h o a r e m o t i v a t e d b y an intense desire to m a k e humanity the m e a s u r e o f all t h i n g s . . , to talk in a w a y so far r e m o v e d from ordinary language, as well as the language o f great s c i e n t i s t s a n d m a t h e m a t i c i a n s and even m o s t p h i l o s o p h e r s , that in m y l a y m a n ' s opinion a d d s nothing to m a t h e m a t i c a l disc o u r s e e x c e p t confusion." The confusion here is y o u r own. F r o m t h e substantive issue, the nature of mathe m a t i c a l reality, you switch to m e r e c o n v e n i e n c e of language, without admission o r apology. More significant, you are alert to any p o s s i b l e "desire to m a k e h u m a n i t y the m e a s u r e o f all things." You do not let that pass. You r e a c t b y a gratuitous attribution o f motives. Against Davis and m e you raise the s a m e non-issue of language, a n d m a k e a similar gratuitous attribution of motives. "It is a language that also a p p e a l s to t h o s e historians, psychologists, a n d p h i l o s o p h e r s w h o cannot bring themselves to talk a b o u t anything that trans c e n d s h u m a n experience." We can talk a b o u t the t r a n s c e n d e n -

tal, Martin. We j u s t don't think it explains m a t h e m a t i c s . On page 72, you write, "The view that mathematics is grounded only in the cultural process slides easily into the 'collective solipsism' that George Orwell satirizes in his novel Nineteen EightyFour9 F o r if mathematics is in the folkways, and the folkways can be molded by a political party, then it follows that the party can proclaim mathematical laws." This easy sliding is the notorious "slippery slope" pseudo-argument. Farfetched political insinuation degrades and cheapens this controversy. Later you write: "'Matter' has a way of vanishing at the microlevel, leaving only patterns. To say that these patterns have no reality outside minds is to take a giant step t o w a r d solipsism; for, if you refuse to put the p a t t e r n s outside h u m a n experience, why m u s t you put them outside your experience?" Apart from your dubious vanishing o f matter, you again resort to "the slippery slope" t o w a r d solipsism as well as Stalinism! (This time not just an e a s y slide, but a giant step!) (Some o p p o n e n t s of Social Security called it "the first step to socialism.") You go on: "I a m an unabashed realist (for emotional r e a s o n s . ) . . , if all m e n vanished, there w o u l d still be a sense (exactly w h a t sense is another and more difficult problem) in which spiral nebulae could be said to b e spiral, and hexagonal ice crystals to b e hexagonal, even though no h u m a n creatures were around to give these forms a name." "Exactly what sense" is exactly the issue! Leaving it at that is on a p a r with your "out there, never mind where 9 I turn to The Ways of a Philosophical Scrivener [3]9 This b o o k is a confession o f faith. It is eloquent, touching, and i m m e n s e l y learned. I w a s i m p r e s s e d by the c h a p t e r s "Faith: Why I a m n o t an atheist" a n d "Immortality: Why I a m n o t 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 passionate despair and fear, is readily admitted b y most fideists, certainly by me and b y the fideists I admire. . . . Faith is the expression of feeling, of emotion, not of r e a s o n . . . . How can a fideist admit that faith is a kind of madness, a d r e a m fed by passionate desire,

and yet maintain that one is not mad to m a k e the l e a p ? . . . To believe what we do not know, what w e hope for but cannot 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 b e c a u s e it is belief of the heart b a c k e d b y no evidence, it is never free of doubt . . . . " After reading this, I finally appreciate y o u r bitterly ironic quote from Swinburne. A self-named quixotic fideist has the hubris to tell m e that saying m a n is the creator of mathematics is the ultimate in hubris! I'm sorry, Martin. I never w a n t e d to disturb anyone's hope, faith, and belief. I'm sorry. P.S. I intended to a n s w e r y o u r New York Review of Books article in m y book, but m y editor p e r s u a d e d me not to. Thanks for this c h a n c e to r e s p o n d in The Intelligencer. REFERENCES

[1] P. J. Davis and R. Hersh, The MathematicalExperience, Birkh&user, Boston, 1981. [2] M. Gardner, Order and Surprise, Prometheus 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 University Press, New York, 1972. [7] S. Korner, The Philosophy of Mathematics, Dover, New York, 1968. [8] L. A. White, "The Locus of Mathematical Reality," Philosophy of Science, 14, 289303; 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 Foundations of Mathematics, Wiley, New York, 1968. [ l q R. L. Wilder, Evolution of Mathematical Concepts, Wiley, New York, 1968. [11] R. L. Wilder, Mathematics as a Cultural System, Pergamon, New York, 1981. 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 o f C a m b r i d g e before he j o i n e d Princeton University in 1987. He w a s elected Fellow of the Royal Society (London) in 1981, a n d he received t h e P61ya Prize o f the L o n d o n M a t h e m a t i c a l Society in 1987 and the F r e d e r i c E s s e r N e m m e r s Prize in M a t h e m a t i c s in 1998. We r e c o r d e d o u r c o n v e r s a t i o n on August 5, 1999, at the University of Auckland, New Zealand, w h e r e b o t h of us w e r e Visiting P r o f e s s o r s for a brief p e r i o d (John in m a t h e m a t i c s a n d I in chemistry). Istv~in H a r g i t t a i ( I H ) : What d o e s it m e a n to y o u to be yon N e u m a n n P r o f e s s o r at Princeton? J o h n C o n w a y ( J C ) : Von N e u m a n n himself w a s a profess o r at Princeton at one time. He did a t r e m e n d o u s n u m b e r o f different things in m a t h e m a t i c s , m a n y of t h e m revolutionary. The most f a m o u s one is the idea of the c o m p u t e r . He n o t only theorized a b o u t it, he w a s also involved in the building and use of one. Earlier in his career, w h e n he bec a m e established, he d e s i g n e d a system o f a x i o m s o f set

6

THE MATHEMATICAL INTELLIGENCER 9 2001 SPRINGER-VERLAG NEW YORK

theory. He h a d this i d e a of c o n t i n u o u s g e o m e t r y in which the d i m e n s i o n function t o o k c o n t i n u o u s values. With Morgenstern, he w r o t e The Theory of Games and Economic Behavior. Many of the things v o n N e u m a n n was i n t e r e s t e d in, I'd b e e n interested i n - - s u c h as set theory, finite numbers, games, a b s t r a c t c o m p u t a t i o n , and this h e l p e d m e to a c c e p t t h e job. It a m u s e d m e t h a t y o n N e u m a n n ' s interests and m i n e w e r e so closely related, with the exception of m a k i n g b o m b s . Also, I have n e v e r m o v e d into his system of c o n t i n u o u s geometry. IH: What is y o u r m a i n interest? J C : I've h a d so many. As you know, I've b e e n interested in s y m m e t r y for a long time, and that c o m e s out as group theory. ! s p e n t a g o o d t w e n t y y e a r s of m y m a t h e m a t i c a l life working intensively with groups, b u t I'm n o t really a grouptheorist. All t h e t i m e I was attending group-theoretical conferences I felt m y s e l f a little bit of a f r a u d b e c a u s e all the p a r t i c i p a n t s w e r e c o n c e r n e d with the really big p r o b l e m of

u n d e r s t a n d i n g all the simple groups, the building b l o c k s of group theory. They also had a lot of technical k n o w l e d g e that I didn't have. My interest is only in studying and appreciating all the beautiful patterns, w h e n e v e r you have a group, a n d I w a s interested in studying the a s s o c i a t e d symmetrical objects. I h a d a long odyssey. When I was a graduate s t u d e n t I was i n t e r e s t e d in n u m b e r theory, and m y adviser w a s a fam o u s number-theorist, Harold Davenport. Then, while I was still officially his s t u d e n t I b e c a m e interested in set theory, and t h a t ' s w h a t I w r o t e m y thesis on. After that, suddenly, t h e s e large groups b e g a n to be discovered, a n d I j u m p e d into that field and m a d e m y professional n a m e in it. That i n t e r e s t lasted for m a n y years. When I m o v e d from C a m b r i d g e to Princeton, I didn't have a n y b o d y group-theoretical to talk to, and I b e c a m e m u c h m o r e of a geometer, and that's w h a t I c o n s i d e r myself now. In this, of course, I interact with others studying symmetry, b u t it doesn't have to be symmetry. The net effect o f this long j o u r n e y has b e e n that I've been in a g o o d p o s i t i o n to n o t i c e certain things. F o r instance, I've always b e e n i n t e r e s t e d in g a m e s and r e g a r d e d it as a mathematical hobby, b u t then the t h e o r y o f g a m e s led to m y discovery o f surreal numbers. (I wish I'd invented the n a m e b u t I didn't.) There's a bizarre a s p e c t to the surreal numbers: You t a k e a definition a priori a n d it l o o k s as though it's sort o f tame, giving you o r d i n a r y real numbers, one and a half, r o o t two, pi, and so on. But the s a m e definition gives you infinite n u m b e r s and infinite d e c i m a l numbers. I stumb l e d on t h e s e things as a c o n s e q u e n c e of studying g a m e theory. The fact that I h a d a l r e a d y s t u d i e d infinite numbers, as p a r t o f m y m a t h e m a t i c a l development, m e a n t that I was able to recognize that w h a t I h a d c o m e u p o n was a farreaching generalization of v a r i o u s n o t i o n s of numbers: Cantor's infinite numbers, the classical real numbers, and everything else. So b e c a u s e I've d o n e so m a n y subjects, I was able to grasp this, and I w r o t e a b o o k called On Numbers and Games. It f o u n d e d the t h e o r y of surreal numbers, a n d that includes the o r d i n a r y real numbers; this m e t h o d of thinking of t h e m as g a m e s t u r n e d out to give a simpler, m o r e logical t h e o r y t h a n a n y b o d y had found before, even for the real numbers. That s o r t of thing has happ e n e d to m e a n u m b e r of times. F o r instance, one of the big d i s c o v e r i e s in group t h e o r y was recognition that the m o n s t e r group, which is an a b s o l u t e l y e n o r m o u s beautiful group, was c o n n e c t e d with various things coming from classical nineteenth-century n u m b e r theory. As s o m e b o d y w h o h a d d o n e both, I w a s able to see t h o s e connections. Ill: Martin G a r d n e r once told m e that, while he w a s editing the m a t h e m a t i c a l c o l u m n of Scientific American, w h e n e v e r he s t u m b l e d on a n e w p r o b l e m and a s k e d you a b o u t it, it t u r n e d out that you h a d a l r e a d y dealt with the problem, m o s t l y h a d solved it, and y e t h a d n ' t b o t h e r e d to publish the solution. J C : It's a big j o b writing s o m e t h i n g for publication, and I'm lazy. I'm not ambitious anymore. When I was a young m a n I was a m b i t i o u s to be recognized as a great mathematician. I haven't lived up to that a m b i t i o n b e c a u s e the kind of math-

John Conway in Auckland at the time of the 1999 conversation. (Photos pp. 7-9 by Istvan Hargittai,)

ematics I'm doing is not the kind that h a d m y ambition. In s o m e sense I've l o w e r e d m y sights; I've p u l l e d in m y horus. But I'm enjoying myself. I've got a g o o d job, although I don't fit in with the P r i n c e t o n set. I'm not the typical Princeton mathematician, y e t I'm recognized as such. I'm at the top of the m a t h e m a t i c a l tree, n o t the top p e r s o n b u t n e a r the top. I don't feel any c o m p u l s i o n to justify m y s e l f anymore. What I think is this: "Princeton b o u g h t me, a n d w h e t h e r it w a s a g o o d b u y o r n o t is no longer m y concern." In m y late t w e n t i e s I w a s quite w o r r i e d that I didn't s e e m to have justified myself. I h a d a j o b at Cambridge, a n d I got that j o b very easily. Then a few y e a r s l a t e r t h e r e c a m e a s o r t of c r u n c h a n d n o b o d y could get a job. There w e r e very good people who were my near contemporaries, who came j u s t a y e a r o r so later than I and w h o had d o n e b e t t e r w o r k t h a n I had, a n d t h e y w o u l d not be getting anything. That m a d e m e feel guilty, a n d the guilt w a s e x a c e r b a t e d by the fact that I didn't s e e m to have done any m a t h e m a t i c s w o r t h noticing after I got the job. That m a d e m e feel depressed. Then s o m e t h i n g v e r y nice happened. I w o r k e d out this simple n e w group called the C o n w a y group, w h i c h at the time w a s a really exciting contribution to knowledge. As s o o n as it w a s d o n e I s t a r t e d traveling all o v e r the world. I c r o s s e d the Atlantic, gave a twenty-minute talk, and flew back. That w a s a r o u n d 1970. The u p s h o t w a s that s u d d e n l y I started p r o d u c i n g things. The n e x t y e a r I p r o d u c e d the surreal numbers, a n d t h e n something else. Not only did I b e c o m e successful, b u t I also d e s e r v e d the success. I r e m e m b e r thinking one day, asking myself, "What's


7

h a p p e n e d ? Why is it that I s u d d e n l y p r o d u c e d t h r e e or four really good things a n d nothing in the previous t e n years?" I s u d d e n l y realized that the l a c k of guilt feelings w a s a g o o d thing a b o u t it. Once I h a d justified myself a n d w a s conv i n c e d that I d e s e r v e d the job, I found the f r e e d o m to think a b o u t w h a t e v e r I w a s i n t e r e s t e d in and n o t w o r r y a b o u t h o w t h e rest of the w o r l d evaluated this. I H : What lifted you out o f y o u r d e p r e s s i o n in the first p l a c e ? J C : J u s t the t r e m e n d o u s ego trip of discovering this n e w thing, which p u t m e into t h e forefront. F r o m t h e n on it t o o k a little while to convince m y s e l f that I w a s n o t going to w o r r y and that I w a s going to study w h a t s e e m e d interesting to m e without w o r r y i n g w h a t the rest o f the w o r l d t h i n k s about it. It's b e e n r a t h e r h a r d to live up to it at times. F o r instance, when I m o v e d f r o m Cambridge to Princeton, I s t a r t e d giving s o m e g r a d u a t e lectures a b o u t w h a t I'd b e e n doing the last few years. There, in the audience, w e r e very f a m o u s m a t h e m a t i c i a n s at Princeton w h o w e r e all c o m i n g along to hear me. My style of lecturing in C a m b r i d g e w a s a l w a y s elementary. Also, Cambridge is an informal p l a c e with a tradition of tolerating eccentrics. You're a l m o s t exp e c t e d to be a little bit odd. In Princeton, however, I felt inhibited by the p r e s e n c e of t h e s e big people. I s t a r t e d to lecture m o r e as a formal mathematician, as e v e r y o n e else does, and then I realized that it was a d i s a s t e r b e c a u s e it w a s n ' t me. It t o o k s o m e effort to get b a c k to m y o w n style. By the way, those f a m o u s m a t h e m a t i c i a n s are no longer in m y audience, the a u d i e n c e consists of g r a d u a t e s t u d e n t s or

Professor Conway prides himself on interjecting humor and spontaneity into his lecture style.

8

THE MATHEMATICAL INTELLIGENCER

u n d e r g r a d u a t e students, d e p e n d i n g on w h o I a m lecturing to. When I a m outside that climate, giving a lecture to a big international meeting, then the a u d i e n c e is always mixed, and t h a t ' s a w o n d e r f u l thing b e c a u s e t h e n I can lecture at w h a t e v e r level I w a n t to. IH: I m e a n t to a s k you a b o u t y o u r lecturing style. I rem e m b e r w h e n w e were both giving l e c t u r e s on s y m m e t r y at the S m i t h s o n i a n Institution a n d you w e r e j u m p i n g on t o p of a table a n d t h e n hiding b e n e a t h it. J C : T h e r e ' s a certain a m o u n t o f a l m o s t cynicism in this. Every n o w and then a j o k e a p p e a r s to m e s p o n t a n e o u s l y while I'm lecturing, and I i n c o r p o r a t e it. If it's g o o d then it stays in t h a t lecture forever. If I give a lecture 20 o r 30 times, the j o k e s j u s t accumulate. The n e t effect is that the lecture gets better. I r e m e m b e r a t e r r i b l e time w h e n I w a s lecturing in Montreal and they a s k e d m e to let t h e m videot a p e it. After the lecture it t u r n e d out that the m a n with the video c a m e r a didn't arrive, b u t he a r r i v e d after the lecture was over a n d t h e y a s k e d m e to give t h e lecture again. I a s k e d t h e m to drag up an a u d i e n c e t h a t was disjoint from the p r e v i o u s one. So I gave the lecture again. However, t h e a u d i e n c e w a s n o t disjoint, b e c a u s e s o m e of the s a m e people still attended. This inhibited m e tremendously, b e c a u s e a j o k e t h a t l o o k s as though it o c c u r s to you on the s p u r o f the m o m e n t , y o u can't tell a s e c o n d time. IH: Did y o u c o m e a c r o s s Paul Erd6s? J C : He w a s a bit strange. I m e t him w h e n I was an undergraduate. He u s e d to p o s e p r o b l e m s , a n d I got involved in s o m e o f them. He did a lot of traveling, and I did a lot o f traveling myself, though n o w h e r e n e a r Erd6s, but I t e n d e d to m e e t him sometimes. I w o u l d m e e t him in Montreal a n d a few d a y s l a t e r in Vancouver o r in Seattle. I w a l k e d into the c a f e t e r i a at Bell Telephone one d a y and sat n e x t to Erd6s. My E r d 6 s n u m b e r is 1. IH: Donald Coxeter? J C : C o x e t e r has b e e n one of m y heroes. When I w a s still at high s c h o o l in England, g r a m m a r school, I wrote to Coxeter. He w a s the Editor of Rouse Ball's Mathematical Recreations. I w a s absolutely delighted b y that book. That was 1953-ish, a n d I have k n o w n him ever since. IH: B u c k m i n s t e r Fuller stated that C o x e t e r is the geometer of the t w e n t i e t h century. J C : This m u s t b e one of the v e r y few things I w o u l d agree with B u c k y about. C o x e t e r is m y hero. I r e m e m b e r a s t o r y at one of the c o n f e r e n c e s in C o x e t e r ' s h o n o r and p e o p l e w e r e telling h o w this wonderful m a n h a d turned t h e m into m a t h e m a t i c i a n s . I thought I m u s t s a y something different. So w h e n I got up, I said, "Lots of p e o p l e have c o m e h e r e to t h a n k Coxeter, I've c o m e here to forgive him." I told t h e m that C o x e t e r o n c e very nearly s u c c e e d e d in murdering me. His m u r d e r w e a p o n w a s s o m e t h i n g that even A g a t h a Christie w o u l d n e v e r have thought of, a m a t h e m a t i c a l problem. Then I told the story, which is actually true. C o x e t e r c a m e to Cambridge a n d he gave a lecture, then he had this p r o b l e m for w h i c h he gave p r o o f s for s e l e c t e d examples, a n d he a s k e d for a unified proof. I left the lecture r o o m thinking. As I was walking through Cambridge, s u d d e n l y the i d e a hit me, but it hit m e while I was in the

he might be able to tell me more. I wouldn't enjoy the interview with him so much.

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 precisely 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 eventually 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? J C : 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 greatest 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

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 important 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 tremendously 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, complimenting 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, geometry 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 manipulating 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 science"? 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? J C : 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 finished his work on this. He has a 200-page paper supplemented 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.

VOLUME23, NUMBER2, 2001

9

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 h o w can you a s s e s s it? J C : I can b e s t do that in a r a t h e r invidious way, b y comp a r i n g it with the p r e v i o u s claim. There the criticism w a s t h a t he sort of t e n d e d to w a v e his hands, he h a d s o m e inequality he had to prove, a n d in one n o t o r i o u s c a s e he evalu a t e d the inequality at o n e p o i n t and then m a i n t a i n e d that it w a s true everywhere. H a l e s ' s w a y of proving inequalities is so m u c h tighter, it's amazing. He cuts the integral into lots o f little pieces, and in e a c h piece he r e p l a c e s t h e function that he is dealing with b y a linear a p p r o x i m a t i o n b a s e d on the derivatives; he t h e n r e d u c e s the p r o b l e m to a linearp r o g r a m m i n g p r o b l e m a n d u s e s the c o m p u t e r to s h o w the inequality. In the a r i t h m e t i c o f the c o m p u t e r he u s e s w h a t is called "interval arithmetic," which m e a n s that y o u at any t i m e say, "This real n u m b e r is definitely g r e a t e r t h a n this a n d less than this." You d o n ' t j u s t r o u n d it to t h e n e a r e s t t e n p l a c e s of decimals. You have explicit u p p e r a n d l o w e r b o u n d s , and so on. E v e r y inequality that Hales w a n t s to p r o v e - - a n d the thing boils d o w n to proving a large numb e r o f inequalities---is getting "interrogated" b y the machine and e x a m i n e d b y Hales. He s h o w s that eveuvthing is o n e of the 2000 cases. The inequalities are p r o v e d n o t j u s t b y getting s o m e rough i d e a of h o w the functions a r e ar-

10


r a y e d b u t b y getting p r e c i s e i d e a s that the function is being b e t w e e n this n u m b e r and this number, and so on. The w h o l e thing is a lot tighter and Hales h a s t a k e n considerable p a i n s to p r o v i d e an audit trail. A n y b o d y who disbelieves any a s s e r t i o n can follow it t h r o u g h the tree and find that this w a s actually s h o w n on this d a y b y the following c o m p u t a t i o n , w h i c h you can do again. Obviously, it w o u l d t a k e a t r e m e n d o u s a m o u n t of w o r k to c h e c k this. On the o t h e r hand, the feeling o f reliability it gives to you is enormous. IH: Couldn't t h e r e be a s i m p l e r w a y o f proving this? J C : My attitude to this is this: "I d o n ' t w a n t to get involved, even r e a d i n g it." My o t h e r feeling is that this isn't p a r t o f the p e r m a n e n t furniture o f m a t h e m a t i c s , this type of proof. My feeling is that eventually s o m e s i m p l e r p r o o f will b e p r o d u c e d . I have this idealistic viewpoint: I a m p r e p a r e d to wait. The waiting m a y m e a n that I die b e f o r e I see the simpler proof, b u t still I'm not i n t e r e s t e d in anything that isn't going to b e p e r m a n e n t . This is an a r i s t o c r a t i c viewpoint. IIt: May w e m o v e n o w to fivefold symmetry, Roger Penrose, quasicrystals? You d e s i g n e d the c o v e r illustration for Scientific American w h e n Martin G a r d n e r w r o t e a b o u t the P e n r o s e tiling, which then b e c a m e an influential paper.

J C : It's funny that you have quoted Martin G a r d n e r ' s saying that I h a d always b e e n t h e r e before. I a t t e n d the Art a n d M a t h e m a t i c s Conferences in Albany, organized by Nat F r i e d m a n , and the p a r t i c i p a n t list has information a b o u t the p a r t i c i p a n t s ' fields of interest. S o m e b o d y once said to m e that he a d m i r e d w h a t I'd w r i t t e n somewhere, a n d I a s k e d him w h a t was it about, a n d he said, "Everything." Then it t u r n e d out that Nat h a d n ' t anticipated m y res p o n d i n g to his question a b o u t m y field of i n t e r e s t and he filled it out for me, including everything. This i d e a of being i n t e r e s t e d in everything is s o m e t h i n g I almost c o n s c i o u s l y try to be. But w h a t you are asking m e a b o u t is a very simple geom e t r i c a l problem: Can you have s o m e tiles tile the p l a n e only aperiodically? I had a l r e a d y b e e n interested in that p r o b l e m a n d when P e n r o s e c a m e up with his solution, I bec a m e t r e m e n d o u s l y e x c i t e d and s t a r t e d making the d a m n e d things a n d drawing them. I w a s staying with Martin G a r d n e r one time, a n d I d r e w out r a t h e r carefully a small page full of the tiles. G a r d n e r had his o w n old-fashioned copying machine and w e ran off a n u m b e r o f copies of this d r a w i n g a n d p i e c e d t h e m t o g e t h e r to p r o d u c e a larger mosaic. Later, w h e n I w a s b a c k in Cambridge, w e p h o t o c o p i e d t h e s e s m a l l e r a n d then m a d e still larger ones, and so on. Martin t o o k the initial version I h a d m a d e at his h o u s e in to the

Scientific American office, w h e r e the graphics p e o p l e redid it properly, a n d i t b e c a m e the cover,

his "pieces" from Gardner, and I re-proved s o m e of the things that he had proved, b u t I didn't k n o w a b o u t the Penrose pattern. Gardner's Scientific American article was largely based on w h a t I'd done in Cambridge. I didn't meet Roger again until a few years after the quasicrystals had been discovered. I still think the situation is rather funny: we still don't know that the actual physical stuff is really behaving like the Penrose pieces. To m y m i n d t h i s is annoying. It enables some people to deny this possibility. Linus Pauling w a s a big holdout, but in his case he just didn't understand what the new configurations were. Certainly, it's ridiculous to deny the possibility, b e c a u s e these things exist geometrically, why shouldn't they exist physically. Those were interesting times for me. II-I: Concerning the b r o a d e n i n g i n t e r e s t in symmetry, you, as a m a t h e m a t i c i a n , don't you feel s o m e t i m e s that it's a n infringement on y o u r territory that physicists, let alone chemists and biologists, s p e a k a b o u t s y m m e t r y ? J C : No. I d o n ' t have any territory. If I'm claiming for m y territory the entire world, I can't v e r y well c o m p l a i n if people t r e a d on s o m e of it. What I do feel in this r e s p e c t is this: The physicists a n d chemists have a t r e m e n d o u s investment in all s o r t s of things. Take, for e x a m p l e t h e n a m e s for t h e s e groups. The crystallographic point g r o u p s w e r e enumerated ages ago, the s p a c e groups w e r e e n u m e r a t e d in the 1890s, and they've got into the International Tables so peop l e all o v e r the w o r l d

If I'm claiming for my territory the entire world, I can't very well cornplain if people tread on some of it.

I've always felt r a t h e r s a d a b o u t our dining r o o m table. We h a d a r a t h e r nice dining r o o m table a n d w e c o u l d n ' t use it for a b o u t six months, and m y wife w a s furious with m e b e c a u s e it w a s c o v e r e d with t h o u s a n d s o f P e n r o s e pieces, m a k i n g a really beautiful pattern, and I n e v e r w a n t e d to disturb it. I r e m e m b e r having d i s c u s s i o n s a b o u t the possibility that c h e m i c a l s might crystallize in t h a t s o r t of manner, and I wish I h a d c o m e out with that speculation in print b e c a u s e seven y e a r s later p e o p l e f o u n d s u c h crystals, I t I : Alan MacKay did c o m e o u t with such a suggestion in print p r i o r to the e x p e r i m e n t a l d i s c o v e r y in 1982. J C : Martin G a r d e n e r ' s Scientific American article app e a r e d in 1974, and w e c o n j e c t u r e d at that time a b o u t the p o s s i b i l i t y o f crystallization, and I wish we h a d c o m e o u t with it in print. I r e m e m b e r that I w o n d e r e d to m y s e l f h o w m a n y different s u b s t a n c e s have b e e n studied with r e s p e c t to crystallization, and m y guess w a s less than t e n to the seventh power. Then I thought, w h a t is the p r o b a b i l i t y that s o m e t h i n g will crystallize in this manner, and one in ten to the s e v e n t h p o w e r s e e m e d a r e a s o n a b l e guess; therefore, such crystallization should happen, I t t : Did you ever discuss this with Roger P e n r o s e ? J C : No, I didn't. When I w a s a s t u d e n t in C a m b r i d g e w e got together; he and I w e r e b o t h i n t e r e s t e d in puzzles. Then he w e n t off to Oxford, a n d in the early seventies I didn't see him often. Soon, I didn't s e e m him at all. I k n e w a b o u t

u s e the existing notations. There is no p r o s p e c t o f changing it to a rational system. If I p r o p o s e a

n e w s y s t e m of naming, this m e a n s t h a t I have to j u s t t h r o w a w a y that community b e c a u s e I c a n ' t get to them. I p e r f e c t l y well unders t a n d the r e a s o n s a n d w o u l d n ' t even w a n t to argue a b o u t them, they're j u s t t o o invested in the s y s t e m as it is. II-I: And it works. J C : And it works, yes. But the p o i n t is, a s a mathematician, m y aims are different. I w a n t to u n d e r s t a n d the thing. Let me give you an example. There are t h e s e little shells in the electronic s t r u c t u r e o f the atom, the s, p, d, f shells, w h e r e s, p, d, f a r e the initial letters of v a r i o u s words, which indicate various p r o p e r t i e s of the spectra. But if y o u ' d start it rationally y o u ' d n e v e r use this s e q u e n c e of letters. I w o u l d start calling t h e m 1, 2, 3, or a, b, c. I d o n ' t w a n t to be constrained by having to agree to s o m e p r e e x i s t i n g usage, even if I u n d e r s t a n d historically h o w this u s a g e c a m e about. Let's t a k e the p a r t i c u l a r case of symmetry. The m o s t rec e n t thing I've d o n e is a joint w o r k with s e v e r a l colleagues. We have c o m p l e t e l y r e - e n m n e r a t e d t h e 219 s p a c e groups ab initio, and it t a k e s only ten pages. We w e r e held up in doing this b y the feeling that w e h a d to p r o v i d e a dictionary to the international notation. U n d e r s t a n d i n g the international n o t a t i o n for us was m u c h m o r e difficult than u n d e r s t a n d i n g the groups. It actually held up the completion of o u r p a p e r for t e n years. My aim is to u n d e r s t a n d s o m e t h i n g for me. I'm less int e r e s t e d in publication. We're going to p u b l i s h this paper,


11

o f course, but I w a n t to u n d e r s t a n d it myself. In doing that, I can t h r o w a w a y the international convention. It's a pity. Here I see this chemist o r physicist, and I can see he is talking a b o u t the s a m e things, b u t I see him as limited b y having to a c c e p t the baggage; he d o e s n ' t a n n o y me, rather, I p i t y him. IH: Physics and c h e m i s t r y are full of historical notations. J C : A n d so is m a t h e m a t i c s , b u t w e ' r e less r e l u c t a n t to give up old notations in m a t h e m a t i c s , since the w h o l e aim of m a t h e m a t i c s is to get s o m e kind of u n d e r s t a n d i n g o f w h a t ' s going on. IH: You have i n t r o d u c e d the t e r m gyration w h e n s p e a k i n g a b o u t rotation. J C : That's a g o o d e x a m p l e b e c a u s e gyration isn't j u s t a rotation. It's really r a t h e r important. What is m a d e c l e a r b y t h e n e w w a y of thinking a b o u t things is that you s h o u l d distinguish b e t w e e n rotations. Rotation m e a n s rotation, any rotation, but gyration is a r o t a t i o n w h e n the axis o f rotation d o e s n ' t go t h r o u g h a m i r r o r line. We're talking a b o u t a p l a n e p a t t e r n or a p a t t e r n on a surface. A n u m b e r of c r y s t a l l o g r a p h e r s have l e a r n e d a b o u t the n e w n o t a t i o n but it's an uphill struggle. My feeling is, in t w o h u n d r e d years they'll b e thinking the c o r r e c t way. I ' m n o t saying that m y notation will b e exactly w h a t it is, b u t eventually the baggage will b e t h r o w n away. This way of thinking a b o u t the groups is really Bill Thurston's idea. What actually happened was rather funny. We w e r e discussing the 17 groups and I said, "Let m e s h o w you m y w a y of thinking a b o u t it," and he said, "No, let m e s h o w you my w a y of thinking about it." We agreed u p o n giving him ten minutes, w h e n he explained his idea to m e in ten minutes, I didn't bother to s h o w him mine, and I have got quite a big ego. As soon as I saw his w a y of thinking about things, I realized it was the correct way. Then I said, "We n e e d a notation that conveys this w a y of thinking about things." I set off for about two w e e k s to think what the notation should be, b e c a u s e to my mind notational matters are t r e m e n d o u s l y important. I finally designed the new system, which is very simple and which conveys Bill Thurston's philosophy. I h a v e n ' t written it up v e r y well, I've only w r i t t e n one b r i e f p a p e r a b o u t it, b u t that situation is going to be c h a n g e d soon. It's a l r e a d y on its w a y to b e c o m i n g the stand a r d n o t a t i o n for m a t h e m a t i c i a n s . There's also a g o o d c h a n c e that I can r e a c h t h e so-called arty community, that p a r t o f the art c o m m u n i t y t h a t ' s interested in m a t h e m a t i c s . It'll t a k e a long time to get t h r o u g h to the crystallographers, t h e genuine chemists a n d p h y s i c i s t s w h o have to use a little bit o f this stuff, and I d o n ' t see m u c h p o i n t in trying, b u t we'll publish s o m e p a p e r s . I have a young colleague at p r i n c e t o n , Daniel Huson, w h o is the p e r s o n w h o m o s t h e l p e d to c o m p l e t e the r e - e n u m e r a t i o n of s p a c e groups. He is a y o u n g m a n and he n e e d s p u b l i s h e d p a p e r s to a d v a n c e his career, to say w h a t he has b e e n doing for the last y e a r o r so, so he's very k e e n to get t h e s e things published. The t h r e e - d i m e n s i o n a l thing d e p e n d s on the t w o - d i m e n s i o n a l thing. We w r o t e the t h r e e - d i m e n s i o n a l p a p e r k n o w i n g that w e ' d have to write---paying a h o s t a g e to fortune---the twod i m e n s i o n a l paper. In t h e last few w e e k s b e f o r e I left

12


Princeton, w e w r o t e the t w o - d i m e n s i o n a l paper. We have a plan to w r i t e a m u c h longer paper, lavishly illustrated with arty pictures, and a d d r e s s it to a m u c h w i d e r community. I also have a plan to write a b o o k on these things. That's a real t r o u b l e for me. So m a n y o f the things I do a r e e l e m e n t a r y that publishing p a p e r s is n o t the right w a y to do it. I w a n t to r e a c h a w i d e r audience, I w a n t to re-found s o m e subjects. That d e m a n d s writing a book, but writing a b o o k is s u c h a big hassle. There are a b o u t five b o o k s I ought to write s o m e time. IH: Does it b o t h e r y o u that physicists talk m o r e a b o u t broken s y m m e t r i e s t h a n symmetries? J C : It d o e s w o r r y m e a b o u t the Universe. If we d e p e n d on the b r e a k i n g o f symmetry, it's not as nice as it w o u l d b e if s y m m e t r i e s w e r e there. It d o e s s e e m to be w h a t the Universe does. I d o n ' t fault the p h y s i c i s t s for talking a b o u t w h a t ' s true. I u n d e r s t a n d that, and it h a p p e n s also on a very e l e m e n t a r y level. In Aristotelian p h y s i c s t h e r e was a concept of "down," t h a t was invariant. The direction d o w n is different from t h e direction up. If y o u ' r e p r e p a r e d to jump, in o t h e r words, if y o u go to high e n o u g h energies, up bec o m e s rather m o r e similar to sideways. This is a very simple instance o f s y m m e t r y breaking. If you really want to travel as easily upwards, rather than horizontally, you need a t r e m e n d o u s a m o u n t of energy and to build yourself a rocket. This is a paradigm; this is something all over the place. If you w a n t to see the s y m m e t r y b e t w e e n space and time, you have to travel at s p e e d s close to the speed of light. So I recognize that the symmetry breaking has actually happened. I deliberately chose some e x a m p l e s that are easier and prior to the examples w o r k e d out b y the physicists. IH: Would y o u c a r e to tell us s o m e t h i n g a b o u t y o u r background? J C : I w a s b o r n in Liverpool in n o t a terribly well-off district. My father w a s a l a b o r a t o r y a s s i s t a n t who also did s o m e m i n o r t e a c h i n g at the school w h e r e t w o of the Beatles w e n t too. I w a s i n t e r e s t e d in m a t h e m a t i c s from a very young age. My m o t h e r always u s e d to s a y that she found me reciting the p o w e r s of two w h e n I w a s four. I t e n d e d to be top o r n e a r l y top in m o s t s u b j e c t s until I b e c a m e an adolescent, w h e n I w e n t d o w n and got i n t e r e s t e d in o t h e r things. But s o m e h o w m a t h e m a t i c s w a s always there. The interest in o t h e r s u b j e c t s was also a l w a y s there, b u t I d o n ' t call m y s e l f a "somethingelsist." When t h e y couldn't t e a c h me anything n e w at school, I d e c i d e d to b e c o m e a lightning calculator. That's a little h o b b y that I'm getting b a c k to now. Tell m e the d a t e w h e n you w e r e born. IH: August 11, 1941. J C : OK. That w a s a Monday. Now, give m e a three-digit number. IH: 999. J C : That's t h r e e t i m e s three times t h r e e times thirty-seven. IH: H o w did y o u develop this ability? J C : I p r a c t i c e d it during the six m o n t h s w h e n I w a s still in Liverpool after I'd b e e n a c c e p t e d to go to Cambridge as a s t u d e n t on a scholarship. Then I w e n t to Cambridge. I found it very h a r d b e c a u s e m o s t of the s t u d e n t s w e r e from r a t h e r p o s h homes, well off, had b e e n to public [i.e., private]

s c h o o l s a n d I w a s a p o o r boy. However, I sort o f gradually a d a p t e d to the life. One thing that did h a p p e n was that t h e r e w e r e o t h e r p e o p l e w h o w e r e i n t e r e s t e d in m a t h e m a t i c s there, w h a t e v e r their b a c k g r o u n d s . Then I got m a r r i e d at quite an early age and had four d a u g h t e r s by m y first wife. My p e r s o n a l life has b e e n d e c i d e d l y unhappy. I h a d t w o b o y s b y m y s e c o n d wife. Over the b r e a k - u p of m y s e c o n d m a r r i a g e I w e n t suicidal a n d I a t t e m p t e d suicide a n d I w a s in hospital for a w e e k after it. This w a s about five y e a r s ago. It's t a k e n me a long time to r e c o v e r from that. N o w I'm getting b e t t e r and hoping to m a r r y again w h e n everything c a n b e s o r t e d out. IH: Did y o u b e c o m e an insider in Cambridge society? J C : My old college, Caius, in C a m b r i d g e m a d e m e an hono r a r y fellow last year. That w a s very nice. But still I k n o w that I w o n ' t u s e this fact v e r y much, b e c a u s e I still feel faintly uneasy; I don't feel that I b e l o n g in this p a r t i c u l a r social grouping. IH: Is t h e r e an intellectual social life in Princeton? J C : I've a t t e n d e d a few d i n n e r p a r t i e s in Princeton and a few p a r t y p a r t i e s w h e r e things happen, and there is p l e n t y o f intellectual discussion going on there. I've always lived in s o m e intellectual c e n t e r like this since I grew up. It's nice w h e n the n e w s p a p e r s are saying s o m e t h i n g a b o u t s o m e n e w d i s c o v e r y in astronomy, to be able to ask m y neighb o r who is a famous a s t r o n o m e r a b o u t it. Something I didn't k n o w until v e r y recently is that I'm r a t h e r well k n o w n in Princeton. I w a s trying to get P r i n c e t o n to buy t h e s e f a m o u s m a n u s c r i p t s o f A r c h i m e d e s that w e r e up for sale at Sotheby's. This involved going a r o u n d v a r i o u s d e p a r t m e n t s soliciting opinions. I s a w a n u m b e r o f people in the Classics Department, Hellenic Studies, and others, a n d I found t h a t a large n u m b e r o f t h e s e p e o p l e k n e w m e o r k n e w o f me. I w o u l d ' v e e x p e c t e d this o f the m a t h e m a t i c i a n s o r the physicists, m a y b e a few chemists, b u t to find that the classicists k n o w w h o I a m was a surprise. So I a m p a r t of the s o c i e t y there although I still don't feel like a Princetonian. When I b e c a m e p r o f e s s o r in Cambridge, and it m e a n s a lot m o r e t h a n it d o e s in the States, I w a s r a t h e r hoping that s o m e b o d y w o u l d a p p r o a c h m e and say something like, "Excuse me, Professor," and p e o p l e w o u l d look a r o u n d to see w h o this god-like figure was. It n e v e r happened. The s t u d e n t s w e n t on j u s t calling m e "you" o r "John," a n d that was that. But w h e n I w e n t to Princeton, p e o p l e s t a r t e d calling m e P r o f e s s o r and the s t u d e n t s did, a n d then I found it r a t h e r annoying b e c a u s e it d i s t a n c e d them. One of the secretaries got it absolutely beautifully right. If I c a m e in b y m y s e l f in the morning, she said, "Hi, John." If I c a m e in with s o m e o n e else, she said, "Good Morning, P r o f e s s o r Conway." I've also c h a n g e d m y a p p e a r a n c e a bit. My hair u s e d to be longer a n d m y b e a r d u s e d to b e longer. After I got m y h a i r c u t I w e n t into the local ice c r e a m shop n e x t d o o r a n d

the girl said, "Oh, y o u l o o k a lot younger." The s e c r e t a r i e s in the d e p a r t m e n t said the s a m e thing. IH: So you c a r e w h a t o t h e r p e o p l e say. J C : I always t h o u g h t that I didn't care a b o u t a p p e a r a n c e , and I didn't c a r e until recently, but I a m getting a bit worried a b o u t getting old. IH: You have said t h a t you no longer h a d ambition. Aren't you looking f o r w a r d to something? J C : I don't think I am. What's there at t h e end, death, and I don't like that v e r y much. I a m thinking a b o u t h o w m u c h time is there to go. I d o n ' t w a n t to g r o w old. I d o n ' t feel old in m y mind. On t h e o t h e r hand, I see m y s e l f behaving in various w a y s I w o u l d n ' t have b e h a v e d w h e n I w a s twenty. Growing old is a bit upsetting. This is o n e of the r e a s o n s I'm taking up this lightning calculation again. I envisage myself in twenty y e a r s time hobbling in with a stick, sitting d o w n painfully. In an a c a d e m i c environment y o u ' r e always s u r r o u n d e d b y young, v e r y bright people, a n d I envisage one of t h e m looking o v e r at this old fool, saying, "Oh, yes, he did some interesting stuff once." But n o w he mentions the date he was b o r n and I instantly say it w a s a F r i d a y and I do this even though m y physical frame is so fragile, and he thinks, "There m u s t be something in t h e r e still working." IH: Are you vain? J C : Very. I w o u l d like to think that I d o n ' t care w h a t o t h e r p e o p l e think, b u t it's n o t true, as the haircutting episode showed. I do c a r e w h a t p e o p l e think. But I d o n ' t care v e r y much. The conventions a b o u t the w a y s y o u act o r d r e s s d o n ' t i m p r e s s m e at all. I j u s t p r e f e r to be comfortable. I'm p r e p a r e d to go to s o m e length to defend this. I s o m e t i m e s deliberately think, "What w o u l d C o n w a y do here?" and t h e n do it. By behaving in s o m e u n e x p e c t e d w a y you give y o u r s e l f the right to b e h a v e in an u n e x p e c t e d way. That's very, very nice. H e r e ' s a little thing I r e m e m b e r . I att e m p t e d suicide and, in fact, I usually think to m y s e l f I comm i t t e d suicide b u t d i d n ' t quite succeed; I w o k e up in a hospital and then I w a s v e r y glad. But then c a m e the p r o b l e m o f coming b a c k to life. I w a s r a t h e r worried, I didn't w a n t p e o p l e whispering b e h i n d m y back. I thought, "What w o u l d C o n w a y do here?" C o n w a y w o u l d m a k e it p e r f e c t l y obvious that he knew. So I b o r r o w e d from Neff Sloane a T-shirt he had, which said "SUICIDE" in very large letters and then "Rock" u n d e r n e a t h it. It indicated that he h a d c l i m b e d this r o c k "Suicide," he is a v e r y k e e n r o c k climber, a n d "Suicide" is the s e c o n d m o s t difficult r o c k to climb in the United States. So I r e - e m e r g e d in society after m y suicide a t t e m p t a n d w e n t a r o u n d for t h r e e days in this shirt. I j u s t m e t the p r o b l e m head-on, splash. That w a s a c a s e w h e n I actually thought a b o u t it b e c a u s e it w o u l d ' v e b e e n painful for m e to ignore this p r o b l e m . I also r e m e m b e r w h e n starting lecture c o u r s e s at v a r i o u s times, I've felt, "What can w e do to j u s t s h a k e t h e s e students?" So I'd j u s t b u r s t into the r o o m with a big s c r e a m o r a jump. It's the s a m e s o r t of thing.

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


13

IH: You obviously do care. J C : I really do love m y subject, and that includes the teaching o f it. It's not j u s t d e v e l o p i n g the subject, hut the teaching is as important. I often s a y that I c o n s i d e r m y s e l f a t e a c h e r m o r e than a mathematician. I s p e n d a lot of time thinking h o w to teach, I really do. Are w e d o n e ? IH: Is there a m e s s a g e ? J C : There is something. It's h o w I feel a b o u t m a t h e m a t i c a l discovery. You're w a n d e r i n g up and down, it's like wandering in a strange t o w n with beautiful things. You turn a r o u n d this c o m e r a n d y o u d o n ' t k n o w w h e t h e r to go left o r right. You do something o r o t h e r and then, suddenly, y o u h a p p e n to go the right way, a n d n o w you are on t h e P a l a c e steps. You see a beautiful building a h e a d of you, a n d y o u d i d n ' t k n o w that the P a l a c e w a s even there. T h e r e ' s a certain wonderful p l e a s u r e y o u get on discovering a mathem a t i c a l structure. It h a p p e n e d to m e t r e m e n d o u s l y w h e n I d i s c o v e r e d the surreal numbers. I had no i d e a t h a t I w a s going to go in there at all. I h a d no i d e a of w h a t I w a s doing. I thought I was studying games, and s u d d e n l y I found this t r e m e n d o u s infinite w o r l d of numbers. It h a d a beautiful simple structure, and I w a s j u s t lost in a d m i r a t i o n of it, a n d in a kind of s e c o n d a r y a d m i r a t i o n of m y s e l f for having f o u n d it. F o r a b o u t six w e e k s I just w a n d e r e d a b o u t in a p e r m a n e n t daydream. What h a p p e n s after t h a t is t h a t I'm vainly trying to re-create that in the p e o p l e I'm trying to t a l k a b o u t it, trying to s h o w w h a t this w o n d e r f u l thing is like a n d h o w amazing it is---that you can r e a c h it b y studying something else. I'm p e r e n n i a l l y fascinated b y mathematics, by h o w we can a p p r e h e n d this amazing w o r l d that a p p e a r s to be there, this m a t h e m a t i c a l world. H o w it c o m e s a b o u t is not really p h y s i c a l anyway, it's n o t like t h e s e conc r e t e buildings or the trees. No m a t h e m a t i c i a n b e l i e v e s that t h e m a t h e m a t i c a l w o r l d is invented. We all believe it's discovered. That implies a certain Platonism, implies a feeling t h a t there is an ideal world. I don't really believe that. I d o n ' t u n d e r s t a n d anything. It's a perennial p r o b l e m to und e r s t a n d w h a t it can be, this m a t h e m a t i c a l w o r l d w e ' r e studying. We're studying it for years a n d y e a r s a n d years, a n d I have no idea. But it's an amazing fact that I c a n sit h e r e without any e x p e n s i v e equipment and find a world. It's rich, it's got u n e x p e c t e d properties, you d o n ' t k n o w w h a t y o u ' r e going to find. I can't c o m p r e h e n d h o w this can be. I d o n ' t k n o w w h a t it means. I don't k n o w w h e t h e r there is such an a b s t r a c t world, a n d I tend not to believe there is and to believe that w e are fooling ourselves. We u s e d to think that the e a r t h is flat and it w a s inconceivable that it could b e round. It w a s only s o m e very painful facts that eventually f o r c e d us to believe that the e a r t h is roughly spherical. What's h a p p e n e d continually in the p h y s i c a l sciences is t h a t the truth w a s not one o f the possibilities that w a s c o n s i d e r e d and then rejected, not even that. It w a s one o f the possibilities that c o u l d n ' t even b e c o n s i d e r e d b e c a u s e it w a s so obviously impossible.

14


In m a t h e m a t i c s our d e v e l o p m e n t h a s c o m e a little bit later, b u t t h e s a m e sort of thing h a p p e n e d with G6del's theo r e m a n d s o on. What we thought w a s the truth w a s j u s t a kind of a p p r o x i m a t i o n of the truth. N e w t o n i a n dynamics is an a p p r o x i m a t i o n of relativistic dynamics, and it's not literally true if y o u go to high s p e e d s a n d high energies; if y o u go to small d i s t a n c e s it d o e s n ' t quite w o r k either, a c c o r d ing to the q u a n t u m theory. In m a t h e m a t i c s w e have t h e s e beliefs that t h e r e are infinitely m a n y integers and so on. Any belief like t h a t a b o u t the n a t u r e of things arbitrarily far a w a y has t u r n e d out to be false in physics. I think it is false in m a t h e m a t i c s too. I think that eventually we'll find s o m e t h i n g w r o n g with the integers a n d then the classical integers will b e j u s t an a p p r o x i m a t i o n . That's a big puzzle for me. I d o n ' t quite believe in this artificial m a t h e m a t i c a l world. There a p p e a r s to be a w o n d e r f u l consistency a b o u t it, which m e a n s that I can think o f s o m e t h i n g in s o m e w a y and s o m e o n e else can think a b o u t it in a different way, a n d w e b o t h c o m e to the s a m e conclusion. If w e don't, t h e r e m u s t b e a m i s t a k e - - a t least it has b e e n so, so far. But I don't see w h y t h e r e should be this c o n s i s t e n c y in a w o r l d that I d o n ' t really believe exists. So to m e it's a sort of fairy tale, and fairy tales don't have to b e c o n s i s t e n t b e c a u s e t h e y are h u m a n c r e a t i o n s ultimately. But this m a t h e m a t i c a l w o r l d is c o n s i s t e n 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,-1A l e x a n d e 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

Shen,

Editor

Constant-Sum Figures

]

(Fig. 3) the sum o f squares of their sides is constant.

Li C. Tien

and fascinate. We welcome, encourage, and frequently publish contributions from readers--either new notes, or

o n s i d e r t w o squares (shown in gray) inside an isosceles triangle (Fig. 1):

C

replies to past columns.

Figure 3.

0 2

-~- b 2 = const,

To see w h y it is t h e case, let us d r a w t w o equal right triangles of sides a and b (Fig. 4). This is p o s s i b l e since the bott o m sides of the s q u a r e s form a segm e n t of length a + b that can be div i d e d into s e g m e n t s o f length b and a.

Figure 1. o + b = const.

When one of the squares b e c o m e s bigger, the o t h e r b e c o m e s smaller. It is easy to verify that the sum of their sides (a + b) is a constant. The s a m e is true for two tangent semicircles inside an isosceles triangle:

Figure 4. Applying Pythagoras theorem: 02

-(-

b 2 = r 2.

The h y p o t e n e u s e s o f the t w o triangles are equal to ~ + b 2, and t h e y are orthogonal. Therefore the c o m m o n vert e x of these triangles is the c e n t e r o f the circle and its r a d i u s is V ~ a2 + b 2. N o w w e c o m e to o u r last and probably m o s t interesting example. (It is a p p a r e n t l y n e w in t h e literature.) Here t w o small s e m i c i r c l e s a r e o f equal size a n d their centers lie on the d i a m e t e r of the big semicircle. The gray circles fit tightly. It turns out that dl + d2 = const.

Figure 2. dl + o+2 = const.

Please send all submissions to the Mathematical Entertainments Editor, A l e x a n d e r S h e n , Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia e-mail:[email protected]

Here the s u m o f d i a m e t e r s (dl + d2) remains the same. (Note that the c e n t e r s of the circles lie on the triangle's base.) This is also an e a s y exercise. A bit m o r e difficult is to s h o w that for two squares inside a semicircle

Figure 5. dl + d2 = const.


15

Why? There is a nice p r o o f of this fact. Let r be the radius of the small semicircles and R be the radius of the big semicircle. These radii are fLxed, while radii r l = dl/2 a n d 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 a n d construct three equal triangles whose sides are a = r + rl, b = rl § r2, a n d c = r + r2 (Fig. 6). Then draw circles of radii r, r b r2 a n d again r, centered at the vertices of these triangles, except for the c o m m o n central vertex (Fig. 7). It is easy to check that the radii and sides are chosen in such a way that

Figure 6. Three equal triangles.

these four circles touch each other (except 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 b e t w e e n centers are exactly 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 m y claim. I have proved that if r + r l + r2 = R, then the configuration with touching circles is possible, b u t n o t vice versa. The following argument completes the proof. Imagine that for touching circles we have r + rl + r2 r R. T h e n we change (say,) r l and find some r~ such that r + r~ + r2 = R. We have proved that there is a configuration formed b y circles of radii r, r~', r2 and r inside a circle of radius R. Now we have two configurations where all the circles are the same except for one, a n d this is impossible

16


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 relation is apparently new. In the Mathematical Gazette, from 1937 to 1967 there were several n o t e s 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, Mathematics in School discussed similarly formed rectangles. I haven't found the general case in the literature. Details and generalizations of constant-sum figures will probably be published elsewhere. Acknowledgment: I would like to thank Mr. Martin Gardner for his encouragement. Li C. Tien 4412 Huron Drive Midland, MI 48642 USA e-mail: [email protected]

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 time and deprive you of your health, peace of m i n d and happiness in life. - - W o l f g a n g Bolyai urging his son Jfinos Bolyai to give up w o r k on hyperbolic g e o m e t r y

In June o f 1997, Daina was in a w o r k s h o p watching the l e a d e r of the workshop, David, helping the participants study ideas of hyperbolic geometry using a paper-and-tape surface in m u c h the s a m e w a y that one can study ideas of spherical g e o m e t r y by using the surface o f a physical ball. David's hyperbolic plane was then so tattered and fragile that he w a s afraid to handle it much. Daina immediately began to think: "There m u s t be some w a y to m a k e a durable model." David m a d e his first p a p e r h y p e r b o l i c plane in the summ e r o f 1978, while on c a n o e trip on the lakes of Maine, using the scissors on his Swiss A r m y knife. He h a d j u s t l e a r n e d h o w to do the c o n s t r u c t i o n from William T h u r s t o n at a w o r k s h o p at Bates College. This crude p a p e r surface w a s u s e d in David's g e o m e t r y c l a s s e s and w o r k s h o p s (becoming m o r e and m o r e t a t t e r e d ) until 1986, w h e n s o m e high s c h o o l t e a c h e r s in a s u m m e r p r o g r a m that David w a s leading c o l l a b o r a t e d on a new, larger p a p e r - a n d - t a p e hyp e r b o l i c surface. This s e c o n d p a p e r - a n d - t a p e h y p e r b o l i c surface (used in classes a n d w o r k s h o p s for the n e x t 11 y e a r s ) w a s the one that Daina w i t n e s s e d in use. Daina e x p e r i m e n t e d with knitting (but the result w a s n o t rigid enough) and then settled on crocheting. She p e r f e c t e d h e r technique during the w o r k s h o p a n d c r o c h e t e d h e r first small h y p e r b o l i c plane; then, while camping in the forests o f Pennsylvania, she c r o c h e t e d more, and w e s t a r t e d exploring its uses. In this p a p e r w e s h a r e h o w to c r o c h e t a h y p e r b o l i c plane (and m a k e r e l a t e d p a p e r versions). We also s h a r e h o w w e have u s e d it to i n c r e a s e o u r o w n und e r s t a n d i n g of hyperbolic geometry. (What are h o r o c y c l e s ? Where d o e s the a r e a f o r m u l a qrr 2 fit in hyperbolic geometry?) We will also p r o v e that the intrinsic g e o m e t r y of t h e s e

surfaces is, in fact, (an a p p r o x i m a t i o n of) hyperbolic geometry. But, Wait! y o u say. Do not m a n y b o o k s state that it is impossible to e m b e d the hyperbolic p l a n e isometrically (an isometry is a function that p r e s e r v e s all distances) as a c o m p l e t e s u b s e t of the Euclidean 3-space? Yes, they do: F o r p o p u l a r l y w r i t t e n examples, s e e Robert O s s e r m a n ' s Poetry of the Universe [9], page 158, a n d David Hilbert and S. Cohn-Vossen's Geometry and the Imagination [6], page 243. F o r a d e t a i l e d discussion and proof, s e e Spivak's A

Comprehensive Introduction to Differential Geometry [10], Vol. III, p a g e s 373 and 381. All of the r e f e r e n c e s are implicitly assuming surfaces e m b e d d e d with s o m e conditions of differentiability, and refer (implicitly o r explicitly) to a 1901 t h e o r e m by David Hilbert. Hilbert p r o v e d [5] that there is no real analytic isometric e m b e d d i n g of the hyperbolic p l a n e onto a c o m p l e t e s u b s e t of 3-space, a n d his a r g u m e n t s also w o r k to s h o w that there is no i s o m e t r i c e m b e d d i n g w h o s e derivatives up to o r d e r four are continuous. Moreover, in 1964, N. V. Efimov ([2] Russian; d i s c u s s e d in English in Tilla Milnor's [8]) e x t e n d e d Hilbert's result b y proving that there is no isometric e m b e d d i n g defined by functions w h o s e first and s e c o n d derivatives are continuous. However, in 1955, N. Kuiper p r o v e d [7] that there is an i s o m e t r i c e m b e d d i n g with continuous first derivatives of the h y p e r b o l i c plane onto a c l o s e d s u b s e t of 3-space. F o r a m o r e d e t a i l e d discussion of these ideas, see T h u r s t o n [11], pages 51-52. The finite surfaces d e s c r i b e d h e r e can a p p a r e n t l y b e e x t e n d e d indefinitely, but they a p p e a r always not to b e differentiably emb e d d e d (see Figure 12).


17

C o n s t r u c t i o n s of H y p e r b o l i c P l a n e s

We d e s c r i b e several different isometric c o n s t r u c t i o n s of t h e hyperbolic plane (or a p p r o x i m a t i o n s of the h y p e r b o l i c p l a n e ) 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 m a y be constructed as follows: Cut out m a n y identical annular strips of radius p, as in Figure 1. (An annulus is the region b e t w e e n two concentric circles, and w e call an annular strip a portion of an annulus cut off by an angle from the center of the circles.) Attach the strips together by attaching the inner circle of one to the outer circle o f the other or the straight ends together. (When the straight ends of annular strips are attached together you get annular strips with increasing angles, and eventually the angle will be m o r e than 2~r.) The resulting surface is of course only an approximation of the desired surface. The actual annular hyperbolic plane is obtained by letting 8--~ 0 while holding p fixed. (We show below, in several ways, that this limit exists.) Note that because the surface is constructed the s a m e everywhere (as 8--~ 0), it is homogeneous (that is, intrinsically and geometrically, every point has a neighborhood that is isometric to a neighborhood of any other point). We will call the results of this construction the annular hyperbolic p/ane. We urge the r e a d e r to try this by cutting out a few identical annular strips and taping t h e m together as in Figure 1. How to crochet the annular hyperbolic plane

If y o u tried to m a k e y o u r a n n u l a r hyperbolic p l a n e from p a p e r annuli you certainly realized that it t a k e s a lot of time. Also, later you will have to p l a y with it carefully b e c a u s e it is fragile and tears and c r e a s e s e a s i l y - - y o u m a y w a n t j u s t to leave it sitting on y o u r desk. But there is a w a y to get a s t u r d y m o d e l of the h y p e r b o l i c plane which you can w o r k a n d p l a y with as m u c h as y o u wish. This is the c r o c h e t e d h y p e r b o l i c plane. To m a k e the c r o c h e t e d h y p e r b o l i c plane, y o u n e e d j u s t a few v e r y basic c r o c h e t i n g skills. All you n e e d to k n o w is

Figure 2. Crochet stitches for the hyperbolic plane.

h o w to m a k e a chain (to start) and h o w to single crochet. See Figure 2 for a picture of these stitches, which will be d e s c r i b e d further in the n e x t paragraph. Choose a y a r n that will not s t r e t c h a lot. Every yarn will stretch a little, b u t you n e e d one that will keep its shape. That's it! N o w y o u are r e a d y to start the stitches: 1. Make y o u r b e g i n n i n g c h a i n s t i t c h e s (Figure 2a). (Topologists m a y recognize that as the stitches in the Fox-Artin wild arc!) A b o u t 20 chain stitches for the beginning will b e enough. 2. F o r the f i r s t s t i t c h i n e a c h r o w , insert the h o o k into the 2nd chain from the hook. T a k e y a r n over and pull through chain, leaving 2 l o o p s on hook. Take yarn over and pull t h r o u g h both loops. One single c r o c h e t stitch has b e e n c o m p l e t e d . (Figure 2b.) 3. F o r t h e n e x t N stitches, p r o c e e d e x a c t l y like the first stitch, e x c e p t insert the h o o k into the n e x t chain (ins t e a d o f the 2nd). 4. F o r t h e ( N + 1 ) s t s t i t c h , p r o c e e d as before, e x c e p t insert the h o o k into the s a m e loop as the Nth stitch. 5. R e p e a t S t e p s 3 a n d 4 until y o u r e a c h the end of the row.

6. A t t h e e n d o f t h e r o w , before going to the n e x t row, do one e x 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 w a n t , you can stop, j u s t b y pulling the y a r n t h r o u g h the last loop. Be sure to c r o c h e t fairly tight and even. That's all you n e e d from c r o c h e t basics. Now you can m a k e y o u r hyperbolic plane. You have to increase (by the a b o v e p r o c e d u r e ) the n u m b e r of s t i t c h e s from one r o w to the n e x t in a c o n s t a n t ratio, N to N + 1 - - t h e ratio d e t e r m i n e s the radius of the hyperbolic p l a n e ( c o r r e s p o n d i n g to p in the former construction). You can e x p e r i m e n t with different ratios, but not in the s a m e model. You will get a h y p e r b o l i c plane only if you i n c r e a s e the n u m b e r of stitches in the s a m e ratio all the time. Crocheting will t a k e s o m e time, b u t later you can w o r k with this m o d e l without worrying a b o u t destroying it. The c o m p l e t e d p r o d u c t is d e p i c t e d in Figure 3. A polyhedral annular hyperbolic plane

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

18


A p o l y h e d r a l v e r s i o n of the a n n u l a r h y p e r b o l i c plane can be c o n s t r u c t e d o u t of equilateral triangles b y putting 6 triangles t o g e t h e r at half the vertices and 7 triangles t o g e t h e r

Figure 3. A crocheted annular hyperbolic plane.

at the others. (If we were to p u t 6 triangles t o g e t h e r at every vertex, then w e w o u l d get the E u c l i d e a n plane.) The precise c o n s t r u c t i o n can be d e s c r i b e d in three different (but, in the end, equivalent) ways: 1. C o n s t r u c t p o l y h e d r a l annuli as in Figure 4, and then t a p e t h e m t o g e t h e r as with the a n n u l a r hyperbolic plane. 2. You can c o n s t r u c t t w o annuli at a time by using the s h a p e in Figure 5 and taping one to the next b y joining: a----> A, b---->B, c--> C. 3. The quickest w a y is to start with m a n y strips, as pictured 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 additional triangles. Next, add a n o t h e r strip every place there is a vertex with 5 triangles and a gap (as at the m a r k e d vertices in Figure 6b). Every time a strip is added, an additional vertex with 7 triangles is formed. The c e n t e r of each strip runs p e r p e n d i c u l a r to each annulus, a n d you can s h o w that e a c h of these curves (the c e n t e r lines of the strip) is geodesic bec a u s e t h e y all have global reflection symmetry. This m o d e l has the a d v a n t a g e o f being constructible m o r e easily than the t w o m o d e l s above; however, one c a n n o t m a k e b e t t e r and b e t t e r app r o x i m a t i o n s b y decreasing the size o f the triangles. This is true b e c a u s e at each sevenfold v e r t e x the c o n e angle is (7 x 60 ~ = 420 ~ no m a t t e r w h a t the size of the triangles, and the r a d i u s of the polyh e d r a l annulus will d e c r e a s e b e c a u s e it is about 1-1 2 t i m e s the side length of the triangles (see Figure 4), w h e r e a s the hyperbolic p l a n e locally looks like the E u c l i d e a n plane (360~

Figure 4. Polyhedral annulus.

Figure 5. Shape to make two annuli.

Figure 6a. Strips.

Figure 6b. Forming the polyhedral annular hyperbolic plane.


19

T h e hyperbolic s o c c e r ball

Polyhedral models of the hyperbolic plane can also be constructed 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 radians ( = 128.57...~ the vertices of this construction have cone angles of 3 6 8 . 5 7 . . . o and thus are m u c h s m o o t h e r than the {3,7} and {7,3} polyhedral constructions. The finished product has a nice appearance if you make the heptagons a different color from 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

Figure 7. The hyperbolic soccer ball.

20


(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 . 0 4 2 . . . ) 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 construction 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 centimeter rule to indicate the scale. Note that as p increases the hyperbolic plane b e c o m e s flatter and flatter (has less and less curvature). For both the sphere and the hyperbolic plane, as p goes to infinity they b e c o m e 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/p2. How Do We K n o w 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 8a. Hyperbolic plane with

p == 4

cm.

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

four main w a y s of describing the h y p e r b o l i c plane; we h o p e one o f t h e s e is y o u r favorite: 1. A h y p e r b o l i c plane satisfies all the p o s t u l a t e s o f E u c l i d e a n g e o m e t r y e x c e p t for Euclid's Fifth ( o r

Parallel) Postulate. 2. A h y p e r b o l i c plane has the s a m e local (intrinsic) geometry as the pseudosphere. 3. A h y p e r b o l i c plane is a simply c o n n e c t e d c o m p l e t e

Riemannian manifold with constant negative Gaussian curvature. 4. A h y p e r b o l i c plane is d e s c r i b e d b y the upper halj:plane model. The italicized t e r m s will be e x p l a i n e d as we deal with e a c h d e s c r i p t i o n in the s e c t i o n s t h a t follow. But first w e c o n s i d e r n a t u r a l c o o r d i n a t e s that w e will find useful. Intrinsic geodesic coordinates

Let p b e the fixed inner radius of the annuli, and let H~ b e the a p p r o x i m a t i o n of the a n n u l a r hyperbolic plane constructed, as above, from annuli of r a d i u s p and t h i c k n e s s 8. On H~ p i c k the inner curve o f any annulus, calling it the b a s e curve, p i c k a positive direction on this curve, and p i c k any p o i n t on this curve a n d call it the origin O. We can n o w c o n s t r u c t an (intrinsic) c o o r d i n a t e s y s t e m x~ : R 2 ---) H~ b y defining x~(0, 0) = O, x~(w, 0) to b e the point on the b a s e curve at a d i s t a n c e w from O, and x~(w, s) to be the p o i n t at a d i s t a n c e s from x~(w, 0) along the radial (along the radii o f e a c h annulus) curve t h r o u g h xs(w, 0), w h e r e the positive direction is c h o s e n to be in the direction from o u t e r to inner curve of each annulus (see Figure 9). The r e a d e r can easily c h e c k that this c o o r d i n a t e map is one-to-one and onto (if you were to c r o c h e t indefinitely). Let x = lim x8 : R 2 ~ H 2, the annular h y p e r b o l i c plane. ~~ that each c o o r d i n a t e m a p x s induces a metric, ds, on R 2 b y defining d~(p, q) to b e the (intrinsic) distance bet w e e n x~(p) and xs(q) in H~. Those r e a d e r s w h o desire a m o r e formal d e s c r i p t i o n of the limit can check that, in the limit as 8 --->0, the m e t r i c s ds c o n v e r g e to a metric d on R 2, and this defines the annular h y p e r b o l i c plane a s R 2 with a special metric. In fact, this p r o c e s s also defines a

Riemannian metric, b u t this will be e a s i e r to s e e after w e s h o w the c o n n e c t i o n s with the u p p e r half-plane model. What can we experience

about hyperbolic

geodesics and isometries?

The following facts w e r e o b s e r v e d b y o u r s t u d e n t s during one class p e r i o d in which, working in small groups, they e x p l o r e d for the first time the c r o c h e t e d h y p e r b o l i c plane. T h e r a d i a 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 s y m m e t r y . The radial curves (curves that run radially across each annulus) have intrinsic reflection s y m m e t r y in each H~ b e c a u s e of the s y m m e t r y in each annulus and the fact that the radial curves intersect the bounding curves at right angles. These reflection s y m m e t r i e s c a r r y over in the limit to the annular hyperbolic plane. Such bilateral s y m m e t r y is the basis of our intuitive notion of straightness (see Chapters 1 of references [3] a n d [4] for m o r e details), and thus we can conclude that t h e s e radial curves are g e o d e s i c s (intrinsically straight curves) on the annular hyperbolic plane and that reflection through t h e s e curves is an isometry. T h e r a d i a l g e o d e s i c s are a s y m p t o t i c . Looking at our hyp e r b o l i c surfaces, w e s e e the radial g e o d e s i c s getting closer and closer in one direction and diverging in the other direction. In fact, let A a n d fr be two of the radial geodesics in H~. The distance b e t w e e n t h e s e radial g e o d e s i c s changes by p/(p + 8) every time they cross one annulus. (Remember, the annuli all have the s a m e radii.) If w e c r o s s n strips, then the distance in Hs b e t w e e n A and tz at a d i s t a n c e c = n6 from the b a s e curve is:

( \ cp / )n~ ~ _I _ p~ _ Now take the limit as 8---) 0 to s h o w that the d i s t a n c e bet w e e n A and tz on the annular h y p e r b o l i c p l a n e is:

d exp(-c/p).

(1)

A s y m p t o t i c g e o d e s i c s never h a p p e n on a E u c l i d e a n plane o r on a sphere.


21

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 t h e a n n u l i . Because reflections through radial geodesics are isometries that preserve each annulus, the composition of two such reflections m u s t also be an isometry that preserves each annulus. A brief consideration of w h a t h a p p e n s on a given annulus should convince us that this isometry shifts the annulus along itself. In the plane we would call such an isometry a rotation (about the center of the annulus). But, on the annular hyperbolic plane, an annulus has no c e n t e r and the isometry has no fLxed point because the radial geodesics (which are perpendicular to the annulus) do n o t intersect. Also, w e do not w a n t to call this isometry a translation b e c a u s e there is no geodesic that is preserved by the isometry. So, this is a type of isometry that we have n o t m e t before on the plane. Such isometries are traditionally called horolations, and annular curves are traditionally called horocycles. Horolations can b e thought of as rotations a b o u t a point at infmity (since the radial geodesics are asymptotic), 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 c a n b e f o u n d in a p p r o x i m a t e itive ways.

intu-

9 Hold two points of the h y p e r b o l i c surface b e t w e e n the i n d e x finger and t h u m b on y o u r two hands. N o w pull gent l y - - a geodesic s e g m e n t (with its reflection s y m m e t r y ) should a p p e a r b e t w e e n the two points. This is using the p r o p e r t y that a g e o d e s i c is locally the s h o r t e s t path. 9 F o l d the surface to a c r e a s e with bilateral symmetry. 9 You can lay a (straight) r i b b o n on the surface a n d it will follow a geodesic. This Ribbon Test for g e o d e s i c s on surf a c e s is discussed further (with proofs) in r e f e r e n c e [3], P r o b l e m s 3.4 and 7.6.

22


The following p r o p e r t i e s of g e o d e s i c s can be easily exp e r i e n c e d by playing with the a n n u l a r hyperbolic plane. These p r o p e r t i e s can be rigorously c o n f i r m e d later by using the u p p e r half-plane model. G1. Every p a i r 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 r o m that point, the geodesic approaches being perpendicular to the annuli that it crosses on the w a y to infinity. C o n n e c t i o n s to Euclid's postulates

Euclid's five p o s t u l a t e s in m o d e r n w o r d i n g are: P1. A (unique) straight line m a y be drawn f r o m any point to any other point. P2. Every limited straight line can be extended indefinitely to a (unique) straight line. P3. A circle m a y 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

F

cross-section of surface P

P

Figure 10. Hyperbolic surface of revolution.

We will have to w a i t for t h e a n a l y t i c p o w e r o f the upp e r half-plane m o d e l to c o n f i r m r i g o r o u s l y t h e s e p r o p e r ties for t h e a n n u l a r h y p e r b o l i c plane, b u t w e c a n give intuitive a r g u m e n t s now. It is e a s y to c o n v i n c e y o u r s e l f t h a t t h e first t h r e e p o s t u l a t e s a r e t r u e b y playing w i t h the ann u l a r h y p e r b o l i c plane, b u t the o t h e r t w o t a k e s o m e m o r e thought. All r i g h t angles are equal. What does this p o s t u l a t e m e a n ? H o w is it p o s s i b l e to imagine right angles that a r e n o t equal? To see this w e m u s t l o o k at Euclid's definition of "right angle":

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

(Playfair's) Parallel Postulate: G i v e n a line I and a p o i n t P not on l, there is a u n i q u e line through P that is parallel to I. Since any t w o g e o d e s i c s (great circles) on a sphere intersect, it is clear that Euclid's Fifth P o s t u l a t e is true on a s p h e r e while Playfair's Postulate is not true, c o n t r a r y to the s t a t e m e n t s in m a n y b o o k s that the t w o a r e equivalent. The correct s t a t e m e n t is that they are equivalent i n the presence o f all the other postulates. Connection to the pseudosphere

When a s t r a i g h t line intersects another s t r a i g h t line s u c h that the adjacent angles are equal to one another, then the equal angles are called right angles. By this definition, t h e right angles at a v e r t e x o f a polyh e d r o n a r e less t h a n 90 ~ a n d t h u s a n y p o l y h e d r o n can n o t satisfy E u c l i d ' s F o u r t h Postulate. To s h o w t h a t t h e annular h y p e r b o l i c p l a n e satisfies this postulate, c o n s i d e r a right angle a at the p o i n t P d e f i n e d b y the lines 1p and mR a n d a n o t h e r right angle/3 at Q d e f i n e d b y IQ and m v . Then, b y reflecting R in the p e r p e n d i c u l a r b i s e c t o r (see G3) of the line s e g m e n t PQ (see G1), t h e p o i n t P is t a k e n to t h e p o i n t Q; o n e o r two m o r e r e f l e c t i o n s t h r o u g h the b i s e c t o r s (see G4) o f the angles d e f i n e d b y the s i d e s of R ( a ) a n d fl will e v e n t u a l l y bring t h e r e f l e c t e d image o f a into c o i n c i d e n c e with ft. Euclid's F i f t h Postulate. C o n s i d e r two radial g e o d e s i c s i n t e r s e c t e d b y the geodesic 1 d e t e r m i n e d by i n t e r s e c t i o n s of t h e s e radial geodesics with a given annulus curve. The radial g e o d e s i c s do not intersect, even though it is clear that t h e y m a k e angles on the s a m e side of I that are e a c h less t h a n a right angle. Thus Euclid's Fifth Postulate d o e s n o t hold on the annular h y p e r b o l i c plane. In m a n y t r e a t m e n t s of a x i o m a t i c geometry, Euclid's Fifth P o s t u l a t e is r e p l a c e d b y

Take the annulus w h o s e inner edge is the b a s e curve and e m b e d it isometrically in the x - y plane as a complete annulus with center at the origin. Now attach to this annulus portions of the other annuli as indicated in Figure 10. Note that the s e c o n d and subsequent annuli form t r u n c a t e d cones. Let the vertical axis be the z-axis; t h e n at e a c h z w e have the picture in Figure 11. Thus AR Az

_

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

In the limit as 6 (and AR and h z ) go to zero, w e get dR dz

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

(2)

We can get the s a m e differential equation by using (1) above, which implies that the circle at height z has circumference 27rpe -s/p, w h e r e s is the arc length along the surface from (0, r) to (z, R(z)). We can solve this differential equation explicitly for z: z = h / p2 _ R2 _ p ln

P+

~ R -~ ~R2 "

Here z is a c o n t i n u o u s l y differentiable function of R and the derivative (for z r 0) is never zero, h e n c e R is also a


23

curvatures of generating 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 p e r p e n d i c u l a r to the surface, and thus is also (+_) n o r m a l curvature. The curvature of the circle is 1/R(z), which m u s t b e p r o j e c t e d onto the direction p e r p e n d i c u l a r to the surface, giving the n o r m a l c u r v a t u r e as 1

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

Figure 12. Crocheted pseudosphere.

c o n t i n u o u s l y differentiable function of z. B e c a u s e R is n e v e r zero, we can c o n c l u d e that this h y p e r b o l i c s u r f a c e o f revolution is a s m o o t h s u r f a c e (traditionally called the pseudosphere). Thus, T H E O R E M : The p s e u d o s p h e r e is locally i s o m e t r i c to the a n n u l a r hyperbolic plane. We can also c r o c h e t a p s e u d o s p h e r e b y starting with 5 o r 6 chain stitches and continuing in a spiral fashion, inc r e a s i n g as when c r o c h e t i n g the hyperbolic p l a n e (see Figure 12). Note that, w h e n y o u c r o c h e t b e y o n d the annular strip that lies flat and forms a c o m p l e t e annulus, the s u r f a c e begins to form ruffles and is no longer a s u r f a c e o f revolution. In fact, it a p p e a r s that it is not even differentiable w h e r e the ruffles start, for the "top ridge" o f the ruffles (see Figure 12) a p p e a r s to be straight and thus n o t tangent to the plane of the c o m p l e t e annulus. Connections to R i e m a n n i a n manifolds with c o n s t a n t negative Gaussian curvature

If a s u r f a c e is differentiably e m b e d d e d into 3-space b y an i s o m e t r y w h o s e first a n d s e c o n d derivatives a r e continuo u s (C2), t h e n the s u r f a c e is said to be a R i e m a n n i a n m a n ifold. At a given p o i n t P on t h e surface, call the n o r m a l d i r e c t i o n one o f the t w o d i r e c t i o n s that are p e r p e n d i c u l a r to the s u r f a c e at t h a t point. The n o r m a l c u r v a t u r e at a p o i n t o f a curve on the s u r f a c e is d e f i n e d to b e the c o m p o n e n t of the c u r v a t u r e o f t h e curve that is in t h e n o r m a l direction. The c o l l e c t i o n o f all n o r m a l c u r v a t u r e s o f all t h e ( s m o o t h ) curves t h r o u g h P has a m a x i m u m a n d a mini m u m value. These e x t r e m a l v a l u e s of the n o r m a l curvat u r e a r e the p r i n c i p a l c u r v a t u r e s (and can be s h o w n to b e the normal curvatures of two curves that are perpendicu l a r at P). The G a u s s i a n c u r v a t u r e of the s u r f a c e at P is d e f i n e d to be the p r o d u c t o f t h e s e two p r i n c i p a l curvatures. The p s e u d o s p h e r e is a Riemannian surface, a n d at e a c h p o i n t [z, R(z), 0] the p r i n c i p a l curvatures are the n o r m a l

24


We do n o t have a formula for R, b u t w e do have a f o r m u l a (2) for R ' ( z ) . The Gaussian curvature is then the p r o d u c t of these t w o n o r m a l curvatures, w h i c h you can check [using (2)] is -1/p2; the minus sign o c c u r s b e c a u s e the t w o n o r m a l c u r v a t u r e s are in o p p o s i t e directions. Thus, the p s e u d o s p h e r e h a s constant negative G a u s s i a n curvature. Gauss's f a m o u s Theorema E g r e g i u m states that the Gaussian c u r v a t u r e is i n d e p e n d e n t o f the (C 2) embedding, h e n c e is an intrinsic p r o p e r t y of the surface. Thus, since the annular h y p e r b o l i c plane is locally isometric to the p s e u d o s p h e r e , w e can say it also h a s c o n s t a n t negative Gaussian curvature. Most differential g e o m e t r y texts give intrinsic m e t h o d s for determining the Gaussian curvature, which can b e a p p l i e d directly to the a n n u l a r hyperbolic plane (see [3], P r o b l e m 7.7, for t w o s u c h methods). Note that in the c r o c h e t e d p s e u d o s p h e r e (Figure 12) there are points that a p p a r e n t l y have no t a n g e n t p l a n e s and thus no n o r m a l direction, and therefore it is n o t p o s s i b l e to define (at t h e s e p o i n t s ) the principal curvatures. In addition, the result of N. V. Efimov [2] a l r e a d y d i s c u s s e d s h o w s that, no m a t t e r h o w the annular hyperbolic p l a n e is p l a c e d in 3space, if it is e x t e n d e d enough it c a n n o t be C 2 e m b e d d e d (and thus c a n n o t have principal c u r v a t u r e s at all points). Connection to the upper half-plane model

As s h o w n above, t h e c o o r d i n a t e m a p x p r e s e r v e s (does n o t distort) d i s t a n c e s along the (vertical) 2nd c o o r d i n a t e curves, b u t at x ( a , b) the d i s t a n c e s along the 1st c o o r d i n a t e curve are d i s t o r t e d b y the factor of e x p ( - b / p ) when comp a r e d to the d i s t a n c e s in R 2. To b e m o r e 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 m e t r i c space to another, and let t ~ A(t) be a curve i n A. Then, the distortion o f y along A at the p o i n t p = A(0) is defined as: lim x-~0

a r c length along y(A) from y[A(x)] to y[A(0)] a r c length along • from A(x) to A(0)

We s e e k a change of c o o r d i n a t e s t h a t will distort dist a n c e s equally in b o t h directions. The r e a s o n for seeking this change is t h a t if distances are d i s t o r t e d to the s a m e degree in b o t h c o o r d i n a t e directions, t h e n the m a p will preserve angles. (We call such a m a p conformal.) We c a n n o t h o p e to have zero d i s t o r t i o n in both coordinate directions (if t h e r e were no d i s t o r t i o n then the chart w o u l d be an isometry), so w e try to m a k e the distortion in the 2nd c o o r d i n a t e direction the s a m e as the distortion in

the 1st c o o r d i n a t e direction. After a little experimentation, w e find that the desired change is z(x, y) = x[x, p in(y/p)] with the d o m a i n of z being the u p p e r half-plane R 2+ ~- {(x, y) E R21y > 0} w h e r e x is the geodesic c o o r d i n a t e m a p defined above. This is the usual u p p e r half-plane model of the hyperbolic plane, thought of as a m a p of the h y p e r b o l i c plane in the s a m e w a y that w e use p l a n a r m a p s of t h e spherical surface of the earth. L E M M A : The distortion o f z along both coordinate curves x ~ z(x, b) and y ---* z(a, y)

at the p o i n t z(a, b) is p/b. P R O O F . We n o w focus on the pin(b/p)). Along the first c o o r d i n a t e x(x, pin(b/p)), the arc length from Ix - a I e x p ( - c / p ) by (1) above. Thus, distortion:

point z(a, b ) = x ( a , curve, x --->z(x, b) = x(a, c) to x(x, c) is w e can calculate the

lira Ix - a] e x p [ - [ p ln(b/p)]/p] x~a ix _ al = p/b. Now, l o o k at the s e c o n d c o o r d i n a t e curve, y --->z(a, y) = x(a, p ln(y/p)). Along this c o o r d i n a t e curve (a radial geodesic) the s p e e d is not constant; but, since the s e c o n d coo r d i n a t e o f x m e a s u r e s arc length, the arc length from 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 a n d the distortion is

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

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

In the a b o v e situation, w e call t h e s e distortions the distortion o f the m a p z at the p o i n t p and denote it d i s t ( z ) ( p ) . Thus,

dist(z)((a, b)) = p/b

and we can s h o w directly that they a r e g e o d e s i c s with bilateral symmetry. In particular, w e can s h o w directly that inversion in a s e m i c i r c l e c o r r e s p o n d s to a reflection isometry in the a n n u l a r hyperbolic plane. D E F I N I T I O N : An inversion w i t h respect to a circle F is a t r a n s f o r m a t i o n from the e x t e n d e d p l a n e (the plane with 0% the p o i n t at infinity, a d d e d ) to itself that t a k e s C, the center of the circle, to oo and vice v e r s a and t h a t takes a point at a d i s t a n c e s from the c e n t e r to the p o i n t on the s a m e r a y (from the center) that is at a d i s t a n c e o f r2/s from the center, w h e r e r is the radius of the circle (see Figure 13). We call (P, P') an inversive p a i r because (as the r e a d e r can check) they a r e t a k e n to e a c h o t h e r b y the inversion. The circle F is called the circle o f inversion. Inversions have the following well-known properties (see reference [1], Chapter 5, and reference [4], Chapter 14): 9 Inversions are conformal. 9 Inversions t a k e circles not passing t h r o u g h the c e n t e r of inversion to circles. 9 Inversions t a k e circles passing t h r o u g h the c e n t e r of inversion to straight lines not through t h e c e n t e r of inversion. If f is a t r a n s f o r m a t i o n taking the u p p e r half-plane R 2+ to itself, then c o n s i d e r the diagram

H 2 g.~ H 2

z-iS

t R2 + ~

z

R 2+

f We call g = z 9 f 9 z - t the t r a n s f o r m a t i o n o f H 2 that corr e s p o n d s to f. We will call f an i s o m e t r y o f the u p p e r halfplane model if the c o r r e s p o n d i n g g is an i s o m e t r y of the annular h y p e r b o l i c plane. T H E O R E M : Let f be the inversion i n 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. P R O O F . (Refer to Figure 14.) 1. Note that e a c h o f the m a p s z, z - I , f is c o n f o r m a l and has at each p o i n t a (non-zero) distortion that is the s a m e

Hyperbolic Isometries and Geodesics We have seen that there are reflections in the annular hyperbolic p l a n e a b o u t the radial geodesics, but w e s a w the e x i s t e n c e o f o t h e r reflections a n d g e o d e s i c s only a p p r o x i mately. However, we were able to s e e that non-radial geod e s i c s a p p e a r to b e tangent to one annulus and then in b o t h directions from that p o i n t to a p p r o a c h being p e r p e n d i c u lar to the almuli. To assist us in looldng at t r a n s f o r m a t i o n s of the a l m u l a r hyperbolic space, w e use the u p p e r halfplane model. As the almuli c o r r e s p o n d to horizontal lines in the u p p e r half-plane model, g e o d e s i c s should t h e n b e curves that start and end p e r p e n d i c u l a r to the b o u n d a r y xaxis. Semicircles with c e n t e r s on the x-axis are such curves,

pt

Figure 13. Inversion with respect to a circle.

VOLUME 23, NUMBER2, 2001

25

F i g u r e 14. Hyperbolic reflections correspond to inversions.

for all curves at that point. Using the definition o f distortion, the r e a d e r can easily c h e c k that dist(g)(p) = dist(z)((f 0 z-1)(p)) x dist(f)(z-l(p)) x dist(z-1)(p). 2. If z(a, b) = p, then, using (1), dist(z

1)(I9 ) = 1/[dist(z)((z-l(p))]

=

b/p.

in t h e a n n u l a r h y p e r b o l i c p l a n e ) h y p e r b o l i c r e f l e c t i o n s . Thus t h e i m a g e s o f the s e m i c i r c l e s in t h e u p p e r half-plane have b i l a t e r a l s y m m e t r y a n d so a r e i n t r i n s i c a l l y s t r a i g h t (geodesics). We have established that the annular hyperbolic plane is the same as the usual upper half-plane m o d e l of the hyperbolic plane. The usual analysis of the hyperbolic plane can

3. Let r be the radius o f the circle C w h i c h defines f, and let s be the distance from the c e n t e r of C to (a, b) = z - l ( p ) . The inversion being conformal, the d i s t o r t i o n is the s a m e in all directions. Thus, w e n e e d only c h e c k the distortion along the r a y from C, the c e n t e r of circle, t h r o u g h p. The r e a d e r c a n c h e c k that this d i s t o r t i o n d i s t ( f ) ( ( a , b)) = r2/s 2. One w a y to do this is to n o t e that, in this case, the dist o r t i o n is the s p e e d (at s) o f 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. Putting everything together, w e n o w have ps 2 r 2 b _ dist(g)(p)-

1.

b r 2 s2 P

Since this is true at all p o i n t s p, the m a p g m u s t b e an isometry o f the annular h y p e r b o l i c plane. We call t h e s e i n v e r s i o n s t h r o u g h s e m i c i r c l e s w i t h cent e r on t h e x - a x i s ( o r t h e 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

26


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

(-1,o)

(1,o)

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

Given a g e o d e s i c triangle with i n t e r i o r angles/3i and exterior angles ai, we e x t e n d the sides o f the triangle as indic a t e d in Figure 15. The t h r e e e x t r a lines are g e o d e s i c s that are a s y m p t o t i c at b o t h e n d s to an e x t e n d e d side of the triangle. It is traditional to call the region e n c l o s e d b y these three e x t r a geo d e s i c s an ideal triangle. In the a n n u l a r hyperbolic p l a n e t h e s e are n o t actually triangles b e c a u s e their vertices a r e at infinity. In Figure 15 we s e e t h a t the ideal triangle is div i d e d into the original triangle a n d t h r e e "triangles" that have t w o of their vertices at infinity. We call a "triangle" with t w o vertices at infinity a 2~3-ideal triangle. You c a n use this d e c o m p o s i t i o n to d e t e r m i n e the a r e a of the hyp e r b o l i c triangle. First w e m u s t d e t e r m i n e the a r e a s o f ideal and 2/3-ideal triangles. It is i m p o s s i b l e to picture the w h o l e of an ideal triangle in an a n n u l a r hyperbolic plane, b u t it is easy to picture ideal triangles in the u p p e r half-plane model. In the u p p e r halfp l a n e m o d e l an ideal triangle is a triangle with all t h r e e vertices either on the x-axis or at infinity (see Figure 16). At first glance it a p p e a r s that t h e r e must be m a n y different ideal triangles; however: T H E O R E M : All ideal triangles on the s a m e hyperbolic plane are congruent. PROOF OUTLINE: P e r f o r m an inversion (hyperbolic reflection) t h a t t a k e s one of the vertices (on the x-axis) to infinity a n d thus t a k e s the two sides from that v e r t e x to vertical lines. Then apply a similarity to the u p p e r halfplane, taking this to the s t a n d a r d ideal triangle with vertices ( - 1 . 0 ) , 0, 1), and ~ (see Figure 16). T H E O R E M : The area o f an ideal triangle is 7rp2. (Remember, this p is the r a d i u s o f the annuli, and equal to ~L---1/K, w h e r e K is the G a u s s i a n curvature.) PROOF: By (3), the distortion d i s t ( z ) ( a , b) is p/b, and thus the d e s i r e d a r e a is

(1,0)

Figure 17. 2/3-ideal triangles in the upper half-plane model.

n o w be considered as analysis of the intrinsic geometry of the annular hyperbolic plane. We give only one example here b e c a u s e it results in the interesting formula ~rr 2. Area of Hyperbolic Triangles

o./

(-1,o)

S:,

:,.

We n o w p i c t u r e in Figure 17 2~3-ideal triangles in the u p p e r 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)p2. Show, using inversions, that all 2/3-ideal triangles with angle 0 are c o n g r u e n t to the s t a n d a r d one at the right of Figure 17 and t h u s that the a r e a is the d o u b l e integral:

Combining t h e s e t h r e e t h e o r e m s and Figure 15 w e 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, NJ: Prentice Hall, 1998. [4] Henderson, D. W. Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. Upper Saddle River, NJ: 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, NJ: Princeton University Press, 1997.


27

MOVING? We need your new address so that you do not miss any issues of

THE M A T H E M A T I C A L INTELLIGENCER. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485 U.S.A. Please give us six weeks notice.

28


I Colin

Adams,

Editor

J

U-Substitution his is c o m p l e t e and utter humiliation in its n a s t i e s t form, thought the coach, as he glanced up at the scoreboard. His t e a m was still in the single digits while their o p p o n e n t s were a b o u t to b r e a k fifty. He l o o k e d d o w n the b e n c h at his d e m o r a l i z e d team. It w a s n o t a bright m o m e n t for the Valparaiso Variables. What h a d the o w n e r J a m e s S t e w a r t b e e n thinking when he m o v e d the t e a m up to the big leagues? They w e r e n ' t in any w a y comp a r a b l e to the Indianapolis Integrals, the t e a m that w a s currently scoring at will as he w a t c h e d from the sideline. f e i ~ d x stole the ball from h a p l e s s z and effortlessly s c o r e d again. His players j u s t c o u l d n ' t k e e p up. And not that t h e y n e e d e d it, b u t the Integrals h a d r e c e n t l y signed f l / ( 1 + x 2) dx. Here w a s one of the m o s t fam o u s integrals on the planet, with end o r s e m e n t d e a l s galore. You couldn't turn on y o u r television without seeing f l / ( 1 + x 2) d x biting into a hot dog, o r hawking graphing calculators. He w a s scoring at will. The Variables w e r e s c a r e d of coming within 10 feet of him. As the c o a c h l o o k e d d o w n his bench, all the p l a y e r s s t a r e d d o w n at their feet, afraid to m e e t his eye. They didn't w a n t to be p u t into the g a m e j u s t to be humiliated. All e x c e p t that skinny kid tan u, sitting at the end. That kid's got guts, thought the coach. He'd b e e n hustling all semester. He had two left feet, b u t he w a n t e d to play so bad. A n d n o w he was looking at the c o a c h with d e s p e r a t e h o p e in his eye. What the hell, thought the coach, this game is lost anyway. "Okay tan u, y o u ' r e going in for x. You c o v e r the big integral." Tan u l e a p e d off the bench. The coach signaled the referee. "I'm doing a u-substitution," he said. "I'm replacing x with tan u." The rest of the b e n c h l o o k e d up,

T

The proof is in the pudding.

Opening a copy of T h e M a t h e m a t i c a l I n t e l l i g e n c e 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 disorientation 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]

startled to h e a r t h e call. x r a c e d 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," s a i d the coach, as he t h r e w him a towel. f l / ( 1 + x 2) d x l a u g h e d w h e n he s a w the s c r a w n y p l a y e r that w a s covering him. "Hey l o o k at this," he said. "They did a u-substitution." His t e a m m a t e s guffawed. As play r e s u m e d , f l / ( 1 + x 2 ) d x c a m e d o w n the c o u r t fast, with little tan u trailing behind. But as f l / ( 1 + x 2) d x m a d e a move to the right, tan u slipped to the inside. As everyone w a t c h e d in a m a z e m e n t , the 1 § x 2 bec a m e 1 § tan2u. A s t u n n e d s e c o n d later, 1 + tan2u b e c a m e sec2u. The d x b e c a m e sec2u du. The c r o w d leapt to its feet. The sec2u's cancelled, and all that was left of the mighty f l / ( 1 + x 2) dx was f du. The o t h e r integrals s t o o d dumbfounded. The c o a c h w a s waving a towel over his head. The entire Variable b e n c h w a s up screaming. Do it, do it! The f du b e c a m e j u s t u + C. P a n d e m o n i u m e r u p t e d all o v e r the stadium. "Okay, x, finish it off," said the coach, grinning from e a r to ear. A s h e e p i s h x w e n t b a c k in for t a n u, and the u + C b e c a m e a r c t a n x + C. The building r e v e r b e r a t e d with cheers. The Variables lifted s c r a w n y t a n u on their s h o u l d e r s and p a r a d e d a r o u n d the c o u r t as the c r o w d chanted, "Tan u, t a n u." F a n s m o b b e d t h e m from all sides. The Variables had w o n the game. Later in the l o c k e r room, after all the r e p o r t e r s had c o m e and gone, and all the c h a m p a g n e h a d b e e n s w a l l o w e d o r d u m p e d on heads, the c o a c h gathe r e d the t e a m together. "Well, I didn't think w e could do it, but thanks to tan u, w e beat the Indianapolis Integrals. And I want you to savor this victory. You deserve it. But don't get carried a w a y with it, either. Next w e e k 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

29

SPRINGER FOR MATHEMATICS S.K. JAIN, Ohio University, Athens. OH; A. GUNAWARDENA, Camegie Mellon University, Pittsburgh, PA and P.B. BHATTACHARYA, (deceased) formerly, University of Dehli, India

LINEAR ALGEBRA WITH MATLABo Interactive Text with CD

'[

., ."'.,'h.',. [• l

This introduction to linear algebra is suitable for a one-semester course and can be used in either a computer lab or a traditional classroom setting. MATLAB is used as the interface to practice problems incorporat-

~:t:~d:a~ehdca~a;~~o~~~r~:~~~~ s~:~:~:~~

linear equations. The text is packaged with a CD-ROM that contains the entire contents S.K.Jaln ~ of the book neatly organized into an elecA.D. Gunawa11lena tronic text. In addition, the CD also contains '.... concept demos, MATLAB drills, solutions, projects and chapter tests. System requirements: IBM or compatible, 486 or higher PC, Windows 3.x, 95, or NT, CD drive, and MATLAB software. 2001/336 PP., WITH CD-ROM/SOFTCOVER/$45.00 (TENT.) ISBN 1-930190-16-6 KEY COLLEGE PUBLISHING

•

M.R. MURTY, Queen's University, Kingston, ON, Canada

PROBLEMS IN ANALYTIC NUMBER THEORY This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to give a rapid introduction to how analytic methods are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic numbertheory.

MARIAN FABIAN, Czech Academy of Sciences, PETR HABALA, Czech Technical University, PETR HAJEK, Czech Academy of Sciences, all, Prague, Czech Republic, VICENTE MONTESINOS SANTALUCIA, Polytechnic University of Valencia, Spain; JAN PELANT and VACLAV ZIZlER, both, Czech Academy of Sciences, Prague, Czech Republic

FUNCTIONAL ANALYSIS AND INFINITE DIMENSIONAL GEOMETRY This book introduces the reader to the basic principles of functional analysis and to areas of Banach space theory that are close to nonlinear analysis and topology. The presentation is self-contained, including many folklore results, and the proofs are accessible to students with the usual background in real analysis and topology. The second part covers topics in convexity and smoothness, finite representability, variational principles, homeomorphisms, weak compactness and more. The text can be used in graduate courses or for independent study. 2001/480 PP./HARDCOVER/$79.95/ISBN 0-387-95219-5 CMS BOOKS IN MATHEMATICS, VOL. 8

J.N. CEDERBERG, St. Olaf College, Northfield, MN

ACOURSE IN MODERN GEOMETRIES Second Edition This book is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary schooL It begins with several finite geometries in an axiomatic framework and introduces Euclid's geometry and ideas of non-Euclidean geometry. Later chapters present symmetry and hands-on explorations of isometries, A new chapter uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history.

2000/452 PP-/HARDCOVER/$49.95/ISBN 0-387-95143-1 GRADUATE TEXTS IN MATHEMATICS, VOL. 206

2000/456 PP., 151ILLUS./HARDCOVER/$59.95/ISBN 0-387-98972-2 UNDERGRADUATE TEXTS IN MATHEMATICS

J. ARNDT, Bayreuth, Germany and C. HAENEL, Eichenau, Germany

T.W. GAMELIN, University of California-Los Angeles, CA

PI-UNLEASHED

COMPLEX ANALYSIS

In the 4,OOO-year history of research into pi, results have never been as prolific as present. This book describes, in easy-to-understand language, the latest and most fascinating findings in the field of pi. Attention is focused on new methods of high-speed computation. The accompanying CD-ROM contains the source code of all programs described, algorithms for pi computation, instructions for testing hardware with programs for pi computation, and complete high-precision libraries. System requirements: HTML document in a standard CD-ROM file system (ISO 9660), platform-independent.

The book provides an introduction to complex analysis for students with some familiarity with complex numbers. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle, the Schwarz lemma and hyperbolic geometry, the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph_D_ qualifying exams in complex analysis. 2000/464 PP., 180 ILLUS-/SOFTCOVER $49.95/ISBN 0-387-95069-9

2000/250 PP., WITH CD-ROM/SOFTCOVER $29.95 " ISBN 3-540-66572-2

ALSO IN HARDCOVER: $79.95/ISBN 0-387-95093-1 UNDERGRADUATE TEXTS IN MATHEMATICS

iLvAl~'t||[=]t=l'=i|[,-~-Ii[="J[,]==l|=lml=n||[~--li M a r j o r i e

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

Senechal,

Editor

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 completing 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 abbreviation 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-yearolds 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.

I

I

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 J e w - - b y religion and nationality-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 nationalities 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 lectured in Sarajevo. Many of these guests stayed in our home. These visits, besides helping us to keep in touch with


31

the great outside world of mathematics, also formed a base of contacts for later activities. The Mathematical Gymnasium in Sarajevo, where Lejla and Alica attended 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 Michigan 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 Matemati6ki. (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

32

THE MATHEMATICALINTELUGENCER

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 University of Sarajevo presented twentyfour research talks. In addition there were seminars in Algebra, Harmonic Analysis, Real Analysis, Differential Equations and Dynamical Systems, Differential Algebra and Algebraic Differential Equations, and Differential geometry. The University of Banja Luka had three Colloquium talks, and the University 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 Communications" (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 University 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, SemiGroups 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 engineering education in Zagreb and personally 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 committed 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 populace 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 extraction 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 spokesperson 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

e d u c a t i o n in Zagreb. He m o v e d to Sarajevo in the 1960s and b e c a m e the leading algebraist in Bill. Although he h a d a very rewarding l i f e - - a n d w a s well r e w a r d e d - - i n Bosnia, he supported, without hesitation, the doctrine o f greater Serbia at the e x p e n s e o f C r o a t i a and Bosnia. Foolishly, like m a n y Serbs and Montenegrins he t h o u g h t that with the Yugoslav National Army acting on their b e h a l f t h e y w o u l d achieve their territorial a m b i t i o n s quickly and with little effort. Let's do a little m o r e n a m e - d r o p ping. Biljana Plavsi5 w a s the Dean of the Natural Sciences and M a t h e m a t i c s F a c u l t y in Sarajevo during the p e r i o d j u s t b e f o r e the w a r in Bosnia. Biljana is an attractive, u r b a n e biologist, w h o s e Ph.D. father w o r k e d at the M u s e u m o f Natural History in Sarajevo. We w e r e close to her, thinking that she s h a r e d o u r p a s s i o n to p u t Bill "on the map" scientifically (and she d i d - - a t the time). In the 1990s, Biljana's life t o o k a s h a r p c h a n g e - - f r o m s c i e n c e to politics. She w a s elected to the P r e s i d e n c y of Bill (the 7-member ruling b o d y that c o n s i s t e d of two Serbs, t w o Croats, t w o Muslims, and one i n d e p e n d e n t ) as a m e m b e r of the Serbian D e m o c r a t i c Party, along with Nikola Koljevid. Biljana u n d e r w e n t an amazing transf o r m a t i o n from a m o d e r n civilized liberal citizen of the w o r l d into a supp o r t e r of the G r e a t e r Serbian Policy, w h i c h w a s b a s e d on ethnic cleansing a n d u s e d rape, murder, a n d t e r r o r as i n s t r u m e n t s for executing policies. She

o p e n l y e m b r a c e d ( r e c o r d e d for posterity on v i d e o ) the m o s t vicious ethnic c l e a n s e r Zeljko Re2njatovi5 ( b e t t e r k n o w n as A r k a n ) j u s t after his b a n d of b a r b a r i a n s c o m m i t t e d a series of atrocities in E a s t e r n Bosnia in 1992. H o w cultured, civilized, well-educated "normal" p e o p l e t u r n into p s y c h o p a t h s und e r the i n t o x i c a t i o n of r a b i d nationalism is one o f the mysteries of this war, and a n o t h e r e x a m p l e of a s p e c i e s o f h u m a n b e h a v i o r that is all too well docu m e n t e d in the twentieth century. J u s t as s e i s m i c political u p h e a v a l s can p r o d u c e h o r r e n d o u s changes in people, w a r a n d tragic events are often the b a c k d r o p for acts of great c o u r a g e and c o m p a s s i o n . It was a great pleasure for us to b e c o m e acquainted with the then little-known CNN reporter, Christiane A m a n p o u r , early in the Bosnian war. I w a s walking t o w a r d m y h o m e one d a y w h e n her driver p o i n t e d m e out to her, saying, "He's t h e American." I t o o k Ms. A m a n p o u r to o u r home, and t h e r e at the d o o r a s c e n e was p l a y e d o u t that will always r e m a i n vivid in m y mind. Naza, standing 10 stairs a b o v e Christiane, in a v e r y serious voice, declared, "If you w a n t to do a serious j o b a n d explain w h a t ' s really h a p p e n i n g here, t h e n you are w e l c o m e in o u r h o m e with all that w e can offer you, b u t if y o u w a n t to do a threeminute p i e c e to a m u s e A m e r i c a n h o u s e w i v e s a b o u t h o w a b u n c h of savages are m u r d e r i n g each o t h e r in the Balkans, t h e n p l e a s e leave and let us fight and die in dignity." My p r i d e in m y

wife at that m o m e n t was off the Richter scale. Christiane slowly, also with dignity, replied, "I e x a c t l y w a n t to do the former. I w a n t to k n o w a n d explain to o t h e r s w h a t is happening." F a t e h a d b r o u g h t t o g e t h e r two courageous, very strong, women. The CNN c r e w s p e n t t w o full d a y s following us a r o u n d in our daily activities, collecting materials for t h e s t o r y they aired in the s u m m e r of 1992. Christiane, a h a t e d figure in Serb e x t r e m i s t circles, r i s k e d h e r life r e p o r t i n g from all over Bosnia. We love h e r dearly. Tragic events can p r o d u c e humorous moments. We w e r e also interv i e w e d by Bob Simon of CBS news, w h o w a s b a s e d in Tel Aviv at the time. Bob's c r e w v i d e o t a p e d us as w e ran from o u r h o m e to the Bosnia A c a d e m y of Science, a d i s t a n c e of a b o u t 4 blocks. A c o u p l e o f r o u n d s of s n i p e r fire w e n t off n e a r us as w e a p p r o a c h e d the Academy. We w e r e h a p p y to get inside. After w e c a u g h t o u r b r e a t h Bob a n n o u n c e d that t h e s o u n d had b e e n off as he r e c o r d e d t h e last 100 y a r d s o f o u r dash. He a s k e d us to go out and r e p e a t the fmal h u n d r e d o r so y a r d s of our s p r i n t - - s o t h e y c o u l d r e t a p e it with the s o u n d on! J u s t as w e finished our second sprint, s h o t s w e r e heard, and I said, "I feel like Paul Newman, only these bullets a r e real." The f i l m - - a n d that q u o t e - - w e r e aired on the CBS evening news. N e e d l e s s to say, m y mother, seeing it in Chicago, was less than enthusiastic! Naza's b o o k a b o u t o u r w a r experiences, Testimony of a Bosnian, will be p u b l i s h e d by t h e T e x a s A&M Press early in 2001. R e a d e r s interested in the historical b a c k g r o u n d o f Bill s h o u l d also consult Noel Malcolm's A Short

History of Bosnia.

Party of the Faculty of Natural Sciences and Mathematics, 1990. Standing at left, Prof. Biljana Pav.~id; standing first at right, the author.

Because of t h e c a t a s t r o p h i c effect of the w a r and t h e d i s p e r s i o n of m a n y of our m a t h e m a t i c i a n s abroad, it t o o k s o m e time to get Radovi MatematiSki b a c k on its feet after the Dayton agreement. The first p o s t - w a r Chronicle of m a t h e m a t i c a l activities in Bill app e a r e d in vol. 8, no. 2. Due to the aggression against o u r c o u n t r y and its consequences, this Chronicle covers the p e r i o d 1992-1998. The r e p o r t includes scientific a n d a c a d e m i c activities of m a t h e m a t i c i a n s from Bosnia


33

Harry and Naza Miller, in January 1998, on a bridge near their home in Sarajevo,

a n d Herzegovina, at h o m e a n d abroad, with w h o m c o o p e r a t i o n w a s maint a l n e d during and after the war; a certain n u m b e r of m a t h e m a t i c i a n s have c h o s e n to stop all c o n t a c t with their f o r m e r institutions in o u r country, and so w e w e r e unable to r e p o r t on their activities. We are a w a r e that this reView is therefore incomplete, and w e will try to include m i s s e d d a t a in our n e x t Chronicle. During the p e r i o d 1992-1998 we have r e c o r d e d 56 res e a r c h p a p e r s p u b l i s h e d from mathem a t i c i a n s from Bill; a l m o s t all of t h e m a p p e a r e d in foreign journals. During the s a m e period, the m a t h e m a t i c i a n s of Bill p r e s e n t e d 91 lectures on their res e a r c h activities. B e c a u s e o f the cons t a n t shelling of Sarajevo during the w a r the regular Colloquium of the D e p a r t m e n t of M a t h e m a t i c s of the University of Sarajevo w a s interrupted. It is n o w being reactivated: 14 talks were presented during 1996-1997. The activities of the s e m i n a r s m e n t i o n e d p r e v i o u s l y (i.e., in the 1991 Chronicle) c e a s e d with the beginning o f the w a r - April 5, 1992. The fifth p o s t - w a r numb e r of the j o u r n a l has j u s t b e e n issued. A f t e r the war, seminars, in the form

34


of a s e r i e s o f lectures, w e r e p r e s e n t e d in Algebra, P h i l o s o p h y of M a t h e m a t ics, C o m p l e x Analysis, M a t h e m a t i c a l Physics, Numerical Mathematics, a n d Applied Mathematics. In 1992 t w o students, one Chinese, one Bosnian, received their Ph.D. degrees in Mathematics from the University of Sarajevo. In 1993 three Bosnians got their Ph.D.s at Michigan State and the Universities o f Wisconsin a n d Michigan, respectively. In 1994 o n e Bosnian s t u d e n t got h e r

Ph.D. at the University of Michigan. In each of the y e a r s 1995 and 1996 one Bosnian s t u d e n t got his Ph.D. from the University of Sarajevo. Between 1993 and 1997 five s t u d e n t s got their M.S. degrees, three in Sarajevo, one in Tuzla, and one at the University of Washington. During the 1996-97 school y e a r o u r graduate s c h o o l r e s u m e d its work. Fifteen students, mainly teaching assistants at the Universities of Bihad, Mostar, Sarajevo, a n d Tuzla, registered. They were offered courses in four areas: Functional Analysis, Harmonic Analysis, Algebra, a n d A p p l i e d Mathematics. During the p e r i o d from 1992 to 1997 six m e m b e r s of the Bosnian m a t h e m a t i c a l c o m m u n i t y w e r e killed or p a s s e d away. The latest Chronicle p r e s e n t s the extensive list o f a c a d e m i c activities o f the Bosnian m a t h e m a t i c a l c o m m u n i t y since the beginning of the war. It includes a list of the latest t e x t b o o k s and lecture n o t e s w r i t t e n by Bosnian authors, as well as a list of educational articles a n d activities of the Bosnian Mathematical Society. We m e n t i o n with p r i d e that from the s u m m e r of 1993, w h e n B o s n i a and Herzegovina was r e c o g n i z e d a s a sovereign state, until today, t e a m s o f young Bosnian m a t h e m a t i c i a n s have c o m p e t e d in the International High School Mathematics Olympics. In the s u m m e r of 2000, in South Korea, o u r team, for the first time, s c o r e d higher than Croatia. Under the a u s p i c e s 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 magazine Triangle, for elementary and high school students and teachers, has appeared in print quarterly since 1997. The Math Department at the Natural Sciences and Mathematics Faculty of the University of Sarajevo is spic-andspan, in better shape than before the war, including a new fully equipped computer lab. The reconstruction and improvement efforts were made possible with the help of Western monies. Naza and her colleagues report the post-war students are eager beavers, anxious to work hard and get on with a normal life. Many of Bosnia’s most productive intellectuals have settled and are pursuing careers in Europe and North America. About 100 Bosnian medical doctors, aged 30 to 45, are currently in the United States. This represents a tremendous loss to BiH. We ase asking the government of the Bosnian Federation to take a constructive approach in dealing with talented Bosnians who are living outside of the country; we are suggesting that, as a first step, the Prime Minister compile a database of these people and write to them to express the government’s pride in their successes abroad. By taking a positive approach, it is hoped that talented scientists from Bosnia will contribute in their own ways to the development of the new country. We hope that academic and scientific personnel permanently settled abroad will spend sabbaticals or scientific visits in their homeland and in general be connected and aid scientific development in BiH. Paul Erdiis was not physically present in Hungary for most of his life after the age of twenty-two, but what a contribution he made to Hungarian mathematics! In my optimistic moments, I envision the future of the territory of the former Yugoslavia developing over the next decade or two in concentric circles. Bosnia, and particularly Sarajevo, are at the geographic center of the former Yugoslavia. It is the region where the tradition of multicultural living was

strongest and lasted longest. If the Western powers show more (much more!) resolve in resettling refugees in their homes and remain steadfast in their support of a single unified BiH (they get a D, in my opinion, so far, in this area) over all of its territory, a viable democratic civil state will emerge. With the death of Tudjman and the election of Mesii: as President of Croatia, we are witnessing a 180-degree change in Croatian policy toward BiH. MesiC has firmly and unilaterally ended the Tudjman-MiloSeviC formula of dividing BiH into Croatian and Serbian parts. It remains to disassemble the Serbian entity, Republika Srpska-the monstrous construction of Richard Holbrooke at Dayton-and restore BiH into a single unified civil state. In Serbia we have seen the end of the MiloSeviC regime. Eventually, one hopes, the Serbian people will renounce, once and for all, as morally wrong, the concept of “Greater Serbia,” which was used to justify the land grab at the expense of its neighbors in the last decade. We hope that, 20 years down the road, these three states (and others) will live in a loose (economic) confederation to everyone’s mutual advantage. Oh, that this could have happened in the late 198Os, and avoided four wars, hundreds of thousands of deaths, and billions of dollars of destruction! Most of the people of BiH yearned for such a solution, but in 1992 the negative forces of history pushed past reason and ignited a holocaust against the non-Serbian citizens of BiH. My personal plans are much easier to state. I started as a docent, at age 30, in 1969, at the University of Sarajevo. In for a dime; in for a dollar. Thirty years later, it’s back to Sarajevo. Naza’s plan to make an academic impact in her homeland still seems like the right thing to do. And we have our work cut out for us: with the assistance of many people from Bosnia and around the world, Naza and I are trying to found an international, private, English-language university-AUS, the American

University of Sarajevo. Despite its excellent record in producing future Ph.D.s, the existing university organization is outdated and inefficient. The AUS, the first international academic center for higher learning and research in BiH, will be a model for educational reform. We also hope that the AUS will contribute to a faster recovery of the academic, economic, and social life in BiH and will help reduce the alarming brain drain. Being situated in a city unique for the heritage of different civilizations, the AUS will attract foreign students and scholars and help rebuild peace and stability in this region.

VOLUME 23. NUMBER 2, 2001

35

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 from the north. Lauren and Greg's admonitions, "Dad, don't forget

J

t : brsring us pres en ts fro m yo ur trip tothe Wild Wes t, " are s till ring i n g in m y

Seattle The s k y is intensely blue, unusually so for the Pacific N o r t h w e s t ' s rainy season; j u s t a few cumulus c l o u d s scatt e r e d h e r e and there. Two J a p a n e s e teenage girls, continuing to Osaka, are craning t h e i r necks, n o w to t h e right, where, b e y o n d the glimmering Puget Sound, one can s e e a s p r a w l i n g Olympic Range, n o w to the left, w h e r e the lonely 14,000-foot volcanic c o n e o f Mount Rainier d o m i n a t e s the l a n d s c a p e . The s n o w b l a n k e t s the mountains, b u t Seattle b e l o w us is gray with the late winter grime. Only the Douglas firs justify the p r o u d n i c k n a m e the E v e r g r e e n State. After renting a c a r and checking in at the University Hotel, I call the n u m b e r w h i c h P r o f e s s o r B i r n b a u m em a i l e d to me. It is a b o u t 1 p.m. A female voice with a slight G e r m a n a c c e n t calls, "Bill, t h a t ' s for you." I suggest that I

stop by to p i c k him up if he'd give m e directions, but he will have n o n e o f that. "I'll m e e t y o u at y o u r hotel in half an hour. I k n o w e x a c t l y w h e r e it is b e c a u s e w e lived in that a r e a for m a n y y e a r s after we s o l d o u r h o m e in the Hilltop Community. Our a p a r t m e n t w a s on the 24th floor of the highrise y o u s h o u l d be able to see from y o u r r o o m ' s window." His v o i c e is quiet but d e t e r m i n e d a n d energetic, his instructions polite b u t firm. "Yes, I'll d r i v e . . . No, p l e a s e do not w a i t b y t h e curb. I'll w a l k up the s t e p s to y o u r room. See you soon." We s p e a k English, a n d t h e r e is not a s h a d e of d o u b t t h a t this is a natural choice o f the language for o u r conversation. This has its c o n s e q u e n c e s : Polish affairs d i s c u s s e d in a different language acquire a n e w detachment. Medium is the message. I'm a little e m b a r r a s s e d , b e c a u s e B i r n b a u m will celeb r a t e his 94th b i r t h d a y in O c t o b e r a n d the three-story 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 author appreciates comments from Monroe Sirken and Ron Pyke on the first draft of the English version.

36

THE MATHEMATICALINTELLIGENCER9 2001 SPRINGER-VERLAGNEWYORK

W. Birnbaum in his University of Washington office, February, 1997.

tel has no elevator. Energetic k n o c k i n g on the d o o r rem o v e s m y misgivings. Birnbaum is slight, and his smallb o n e d figure, s e e n against the light while I guide him in, r e m i n d s m e o f Steinhaus from 1970. His p o s t u r e is erect, and his high f o r e h e a d is c r o w n e d with a w r e a t h of hair, n o t a l t o g e t h e r grey. Thick lenses in horn f r a m e s add to the air of intensity. " . . . Probably, very s o o n I will have to go t h r o u g h an eye s u r g e r y , . . , y o u know, c a t a r a c t s . . . . Hilde w e n t t h r o u g h it a b o u t a m o n t h ago a n d n o w feels like new." He is w e a r i n g a long and heavy o v e r c o a t covering a b r o w n j a c k e t a n d d a r k - c o l o r e d plaid flannel shirt. The suggestion is that w e do n o t sit d o w n but j u s t r e p a i r to his university office. "I k e e p all m y p a p e r s and d o c u m e n t s there. This w a y I can c o n s u l t t h e m w h e n m y m e m o r y falls. It's j u s t a couple o f b l o c k s from h e r e . . . Yes, we'll drive m y c a r because you do n o t have the p a r k i n g permit." The w e a t h e r returns to normal. It drizzles w h e n w e b o a r d a t w o - d o o r F o r d E s c o r t with a u t o m a t i c transmission. He drives carefully, and close to the curb; the traffic is heavy a n d the s t r e e t s busy, filled with the usual university crowd. We p a r k on the t e r r a c e d p a r k i n g lot behind the building h o u s i n g the M a t h e m a t i c s Department. Lake Washington is s p r e a d in front of us, b u t the mist p r e v e n t s us from seeing the far s h o r e o r the downtown. " . . . This is a g o r g e o u s p a r t of the country. When w e w e r e y o u n g e r w e u s e d to hike in the m o u n t a i n s and c a m p out quite a bit." B i r n b a u m arrived here in 1939, w h e n the city w a s still small and provincial and the University consisted of a few buildings on the w e s t e r n shore of Lake Washington. The

Lake e m b r a c e s Seattle from the east and p r e s s e s it against the island- and inlet-filled Puget Sound, w h i c h s p r e a d s itself on the w e s t side o f the city. It's a sailor's p a r a d i s e and Navy submarines' haven. His choice of d e s t i n a t i o n was alm o s t accidental, the only alternative w a s p r e s e n t e d by an Australian visa for w h i c h he had a p p l i e d while still in Poland, but which r e a c h e d him only after his arrival in America. The title o f A s s i s t a n t P r o f e s s o r a n d the $2,000 annual salary w e r e n o t impressive, but infinitely b e t t e r than the two-year u n c e r t a i n existence in the M a n h a t t a n cauld r o n - n o w as a s p e c i a l c o r r e s p o n d e n t of the Polish Illustrated Daily Courier, n o w as Felix B e r n s t e i n ' s assistant at New York University, studying c o r r e l a t i o n s b e t w e e n the p r o g r e s s o f f a r s i g h t e d n e s s and longevity, n o w as an ind e p e n d e n t c o n s u l t a n t equipped with a well-used Monroe electric calculator a n d a r e n t e d d e s k in an old office building in the n e i g h b o r h o o d o f the New York Public Library. "I advertised m y services in the j o u r n a l Science, and all kinds of p e o p l e w o u l d s h o w up on my office d o o r s t e p s . A certain gentleman b r o u g h t m e s o m e data, the origin t h e r e o f w a s not k n o w n to me. He w o u l d not v o l u n t e e r their proven a n c e either, and a s k e d m e to do simple r e g r e s s i o n on them. A w e e k later he returned, c h e c k e d m y graphs, comparing t h e m against the light with curves that he had brought along, gave m e s o m e m o r e data, a n d p r o m i s e d to c o m e b a c k soon. The s c e n e r e p e a t e d itself a few m o r e times. He was paying m y fees without delay, so I did not a s k any probing questions. However, after a few w e e k s and a n o t h e r inspection o f m y graphs, he s h o o k his h e a d and resignedly allowed, 'After all, this is unlikely to work.' It t u r n e d out that he p l a y e d the s t o c k m a r k e t relying on m y predictions. After that e x p e r i e n c e I p r o m i s e d m y s e l f never to analyze d a t a o f an u n k n o w n origin." The Lw6w Student

Birnbaum's g r a n d f a t h e r was born in the Russian zone of p a r t i t i o n e d Poland, b u t the p r o s p e c t of being a draftee in the tsarist a r m y drove him to find shelter in Galicia, which w a s then in the A u s t r i a n zone. There the y o u n g b o y was a d o p t e d by the B i r n b a u m family. He quickly b e c a m e independent. Initially he t o o k to logging in E a s t e r n Galicia's Carpathian Mountains; the logs were t h e n floated on the San and Vistula rivers all the w a y to Gdatisk on the Baltic Sea, following an a n c i e n t route. It was b a c k - b r e a k i n g physical labor. In due time he h a d m a d e g o o d m o n e y on raftsm a n s h i p and b o u g h t a few t h o u s a n d a c r e s o f logging forest in the Pysznica t o w n s h i p of Nisko county. The business w a s good and the family p r o s p e r e d , a s s u m i n g the life-style o f the local Polish gentry. Of his ten children, four settled in the capital city o f Lw6w. One of t h e m w a s Birnbaum's father Izak. A c c o r d i n g to the r e c o r d s of the local Israelite Registry Office, Zygmunt Wilhelm B i r n b a u m w a s b o r n O c t o b e r 18, 1903, in the family of Izak a n d Lina, n6e Nebenzahl. The family w a s not p a r t i c u l a r l y religious and s p o k e Polish at home. " . . . Now we are m e m b e r s o f a synagogue," c o m m e n t s B i r n b a u m a b o u t his c u r r e n t Seattle life, "but in A m e r i c a this has a totally different, m o r e social s i g n i f i c a n c e . . . " . Izak ran the family sawmill and o t h e r


37

businesses. They lived in a c o m f o r t a b l e five-room apartm e n t at 1 St. Ann's Street. In the school y e a r 1913/14 Wilek, as the family called young Wilhelm, was enrolled in the first g r a d e o f the private G y m n a s i u m run by Dr. Niemiec. But Wilhelm's e d u c a t i o n a n d the s m o o t h family life w e r e i n t e r r u p t e d by the w a r o f 1914. The family sought s h e l t e r in Vienna, where Wilhelm s o o n found himself a n n o y e d and b o r e d by the low level o f t h e school he was s e n t to, and r e f u s e d to continue in it. F o r the r e m a i n d e r of World War I he p r o c e e d e d b y i n d e p e n d e n t study, with a daily o n e - h o u r s e s s i o n with a hired tutor. In the consecutive y e a r s 1915, 1916, 1917, and 1918, he p a s s e d special e x a m s a d m i n i s t e r e d for h o m e - s c h o o l e d s t u d e n t s b y the Viennese State E x a m i n a t i o n Commission. With the w a r over, it w a s time to r e t u r n to the Lw6w h o m e s t e a d , his r e p a t r i a t i o n to the r e b o r n Poland m a d e e a s i e r b y the formal d o c u m e n t testifying that the family w e r e Pysznica landowners. I n d e p e n d e n t e d u c a t i o n was, however, p o s s i b l e only up to a point. It was well u n d e r s t o o d that, after a certain stage, it w a s advisable to get a c q u a i n t e d with the habits a n d stand a r d s of t e a c h e r s w h o w e r e going to c o n d u c t the f e a r e d c o m p r e h e n s i v e Matura examination, the m a n d a t o r y conc l u s i o n of Gymnasium (High School) education. University e n r o l l m e n t was not p o s s i b l e without a successful p a s s i n g of the Matura. So, for g r a d e s VII and V I i i - - t h e last t w o y e a r s o f the G y m n a s i u n l - - W i l h e l m was enrolled in the H. Sienkiewicz State G y m n a s i u m No. X in Lw6w. By s h e e r coincidence, m a t h e m a t i c s w a s taught there b y a y o u n g doct o r a l student, w h o s e e n t h u s i a s m for set t h e o r y a n d topology e x c e e d e d his fear of the principal's r e p r i m a n d for violating the official syllabus. He did not p a y t o o m u c h att e n t i o n to the logarithmic tables and trigonometry, and t a l k e d in class a b o u t m a t h e m a t i c s in a w a y that left a perm a n e n t i m p r e s s i o n on Wilhelm. Come v a c a t i o n time, usually s p e n t in the region o f the family sawmill in the eastern Carpathian mountains, the serious t e e n a g e r w a s lugging along college m a t h t e x t b o o k s . In 1921 he w a s a w a r d e d the Matura certificate "with distinction." Majoring in m a t h e m a t i c s w a s not an option. The family, w h o s e b u s i n e s s e s faltered as a result of the Great War, w a s d e c i d e d l y in favor of a m o r e p r a c t i c a l direction of study. The initial effort to enroll in the faculties of Medicine a n d Engineering failed in view of the n u m e r u s clausus, then c o m m o n at m a n y E u r o p e a n and American universities, w h i c h restricted the p e r c e n t a g e of Jewish s t u d e n t s in s o m e d e p a r t m e n t s . In the Law a n d Political Science F a c u l t y of the J a n Kazimierz University, luck was on Wilhelm's side and, for the full c o u r s e o f twelve t ~ n e s t e r s , he successfully s t u d i e d law, s u p p o r t i n g himself with the Polish gove r n m e n t scholarship. The high p o i n t of these legal s t u d i e s w a s his s e m i n a r p r e s e n t a t i o n of George P61ya's mathem a t i c a l p a p e r on election systems. Birnbaum g r a d u a t e d with the degree of Magister Utriusque Juris (i.e., m a s t e r o f b o t h lay and c a n o n laws) in 1925, and for a y e a r he app r e n t i c e d as a "koncypient," a legal clerk, in the L w 6 w law firm of his p a t e r n a l uncle, Dr. Henryk Birnbaum. Although legal e d u c a t i o n s e r v e d him well on m a n y a l a t e r occasion, it was not s o m e t h i n g that Wilhelm w a n t e d

,38


to do as an occupation. He r e p e a t e d l y r e t u r n e d to mathematics during his law studies, and for a d e c a d e was an active p a r t i c i p a n t in the intellectually electrifying and uno r t h o d o x events that led to the c r e a t i o n of the Lw6w m a t h e m a t i c a l school. His first university m a t h e m a t i c s ins t r u c t o r w a s Stanislaw Ruziewicz. The well-read Wilhelm was confident that he k n e w everything that n e e d e d to b e k n o w n a b o u t real numbers, and suffered a s h o c k w h e n Ruziewicz p r e s e n t e d an e x p o s i t i o n of D e d e k i n d cuts. Then c a m e Zylinski's a l g e b r a lectures, w h i c h w e r e entered as the first record, d a t e d March 10, 1922, in his Indeks, the official grade b o o k t h a t all s t u d e n t s w e r e required to c a r r y with t h e m a n d s u b m i t for a p r o f e s s o r ' s e n t r y w h e n e v e r t h e y t o o k an examination. The grade, like all the others in the m a t h e m a t i c a l subjects, was "excellent." An invitation to p a r t i c i p a t e in the group t h e o r y s e m i n a r run by the s a m e p r o f e s s o r followed. Formally, for an additional twelve trimesters, from 1925 to 1929, he w a s a s t u d e n t in the faculty o f Mathematics and Natural Sciences, supporting himself from a n o t h e r Polish g o v e r n m e n t scholarship. It w a s the s t u d e n t generation of Juliusz Schauder, Mark Kac, Stanislaw Ulam, Wladystaw Orlicz, Marceli Stark, Henryk Auerbach, Ludwik Sternbach, Stanislaw Mazur a n d Julian Schreier. " . . . I was a p a r t of that generation. Mathematics in that group of infatuated young p e o p l e w a s kind of a fever. We w o u l d get t o g e t h e r at all times of d a y and night, talking incessantly mathematics. In a r o o m which served as a c o m b i n a t i o n of the s e m i n a r r o o m a n d a small library t h e r e w a s a large tile stove, with one side a t t a c h e d to the wall. I r e m e m b e r long hours of freezing winter nights w h e n w e w e r e standing glued to the w a r m tiles of the t h r e e available stove sides, talking a r o u n d c o r n e r s a b o u t mathematics." Initially, he w a s interested in c o m p l e x function t h e o r y and influenced b y Steinhaus and Banach; he t o o k classes with b o t h o f them. The Indeks entry d a t e d O c t o b e r 24, 1924, r e c o r d s the p r o b a b i l i t y theory c o u r s e in which Steinhaus used Markov's t e x t b o o k . B i e b e r b a c h ' s m o n o g r a p h on functions of c o m p l e x variables h a d j u s t a p p e a r e d , and the second v o l u m e c o n t a i n e d a c h a p t e r on univalent functions. They w e r e the s u b j e c t of Birnbaum's d o c t o r a l dissertation d e f e n d e d in 1929. "Around 1926 I w r o t e m y first p a p e r in c o m p l e x function theory. It was a simple r e m a r k on the Cauchy formula. I initially s h o w e d it to Banach w h o p r o m p t l y k i c k e d m e out of his office with a succinct comm e n t 'This is nonsense.' But the n e x t day he called m e b a c k to his office, apologized, and a s k e d m e to p r e p a r e the p a p e r for p u b l i c a t i o n . . . . During the Rigorosum, a Ph.D. qualifying examination, m y n o n m a t h e m a t i c a l subject was astronomy. Dean Ernst, w h o w a s s u p p o s e d to be m y exa m i n e r in that subject, fell ill, a n d t w o o t h e r examination c o m m i s s i o n m e m b e r s , Banach and Steinhaus, had been ins t r u c t e d to c o n d u c t the e x a m i n a t i o n in his absence. The only question a s k e d was: 'Could you n a m e at least one ast r o n o m i c a l r e s e a r c h instrument b e s i d e s the t e l e s c o p e ? ' . . . Banach's a n d Steinhaus's lecture styles w e r e very different. Banach i m p r e s s e d one with the striking and a l m o s t b r u t a l p o w e r of his pure but cold intellect. Steinhaus w a s

warm, a g o o d motivator, with broad, active interests stretching from p o e t r y to medicine." Teacher The p r a c t i c a l side of life, however, w a s not to b e forgotten. " . . . My family could not s u p p o r t m e financially. I lived with m y p a r e n t s but h a d to c o n t r i b u t e to daily expenses. 9 B i r u b a u m shows m e a four-page (written in tiny calligraphic handwriting) DIPLOMAfor high school t e a c h e r s iss u e d June 10, 1929. It brings b a c k m e m o r i e s of a similar d o c u m e n t issued to m y mother, w h o a few y e a r s later p a s s e d similar e x a m i n a t i o n s in W a r s a w u n d e r Dickstein a n d SierpiAski's tutelage. It starts generically and in the s t a n d a r d officialese: "Mr Zygmunt Wilhelm Birnbaum w a s a d m i t t e d [ . . . ] on F e b r u a r y 26, 1926, to the e x a m i n a t i o n for a t e a c h e r o f mathematics as the m a i n subject, and civil e d u c a t i o n as an additional subject, in high s c h o o l s with Polish as the language of instruction." But the r e m a i n d e r of the d i p l o m a is an illuminating document, w o r t h quoting at length, especially in the c o n t e x t of o u r own c u r r e n t and p e r e n n i a l discussions on the quality of the school education at the e n d of the t w e n t i e t h c e n t u r y in general, and high s c h o o l s in particular. I p r e s e r v e the original's punctuation. The Lw6w State E x a m i n a t i o n Commission for candid a t e s to the p r o f e s s i o n of high s c h o o l t e a c h e r c o n s i s t e d o f university p r o f e s s o r s Kazimierz Chylifiski, StanisIaw Rdziewicz a n d Kazimierz Ajdukiewicz. "As h o m e assignm e n t the C o m m i s s i o n [ . . . ] a c c e p t e d a s e m i n a r paper, s u b m i t t e d b y the candidate, entitled 'On w e a k c o n v e r g e n c e o f s e q u e n c e s o f functions of a real variable', w h i c h memoir w a s j u d g e d to be 'very good'." "The s u p e r v i s e d written examination: a) in mathematics. May 25, 1926. Topics: '1) Prove that the derivative of a continuous function, if it exists in e a c h p o i n t of a certain interval, is c o n t i n u o u s in the s e n s e of D a r b o u x [ . . . ]. Is the c o n v e r s e true: is every function continuous in the sense of D a r b o u x the derivative of a certain function? [ . . . ] 2) D e m o n s t r a t e that the odd p e r f e c t numb e r (if it exists) does not c o n t a i n in its d e c o m p o s i t i o n into p r i m e factors any factors o f the form 4k + 3 in o d d p o w e r [ . . . ] 3) Calculate the volume o f a tri-axial ellipsoid.' Score: very good. May 25, 1926. Topics: '1) The relation y + e xy = 0 defines y = y(x); in w h a t x interval d o e s y = y ( x ) exist? Calculate dy/dz for x = 0. 2) Prove that if Z n~= 0 a n = A , n=0 bn = B, and Z n=0 ~ Cn = C w h e r e Cn Z #k=n =0 an-kbk, then AB = C.' Score: good. b) in civil education. May 26, 1926. Topics: '1) Position of an individual in a m o n a r c h y and in a republic---differences in standing 2) P a r l i a m e n t ' s p o s i t i o n in a m o n a r c h y a n d in a r e p u b l i c - - d i f f e r e n c e s in standing.' Score: good." But that was not the end of it. The oral m a t h e m a t i c s e x a m t o o k p l a c e June 24, 1926, and in civil education June 19. The m a t h e x a m c o m m i t t e e w a s chaired by P r o f e s s o r

Chylifiski, and Ruziewicz and Steinhaus served as examiners. "The c a n d i d a t e was a s k e d a b o u t the following matters: Sets of the first a n d s e c o n d Baire category, The p r o b l e m of measure. Lebesgue measure. N o n m e a s u r a b l e sets. Euler's and F e r m a t ' s theorems. Proofs of the t h e o r e m about existence of infinitely m a n y prime numbers. P r o o f of the Weierstrass t h e o r e m a b o u t the existence of a c o m p l e x analytic function with p r e s c r i b e d zeros. The fundamental theo r e m of algebra. Bernoulli-Laplace theorem, Bayes's rule. Cauchy's t h e o r e m in c o m p l e x function theory. P r o o f of the transformation o f f f f d x dy dz (over the ball) into spherical coordinates. Geodesics. The surface c r e a t e d by lines tangent to a spatial curve. Curvature lines. Orthogonal systems. Orthogonalization.' The a n s w e r s w e r e j u d g e d very good. During the e x a m the candidate d i s p l a y e d a sufficient k n o w l e d g e o f the German language." C o m m e n t s Birubaum: "Probably I would flunk this e x a m today." "In civil education [ . . . ] the following issues were a s k e d about: 'The voyvodship selfrule. Voyvodships of the E a s t e r n Malopolska. Ethnic minorities. The school-language bill. State budget. Legitimism. Agrarian reform. A r i s t o c r a c y and oligarchy. P r o p o r t i o n a l elections in general, and t h e i r organization in Poland, in particular. The m a n n e r o f s e l e c t i o n of the s u p r e m e c o m m a n d e r . ' " The last question, a s k e d a few w e e k s after Marshal PiIsudski's May coup, could have h a d a special hidden significance. In the 1926-27 s c h o o l y e a r Birubaum taught m a t h e m a t ics in a private g y m n a s i u m (grades 4-12) "with the rights to serve the public" run b y Dr. A d e l a Karp-Fuchsowa, and, during the n e x t t w o years, in the private c o e d u c a t i o n a l gymnasium of the Lw6w Evangelical community; in the spring t r i m e s t e r o f 1928 his i n c o m e w a s s u p p l e m e n t e d by teaching in the J a n K o c h a n o w s k i State G y m n a s i u m No. IX. "Initially I w a s a s s i g n e d to a c o e d u c a t i o n a l class o f twelveyear-olds. I w a s c o n v i n c e d that they w o u l d b e enthusiastic a b o u t m a t h e m a t i c s , and, following the e x a m p l e o f my distinguished university professors, I s p o k e in a low voice facing the b l a c k b o a r d , concentrating on developing ideas, stopping to c o n t e m p l a t e the n e x t m a n e u v e r w h e n the p r o o f did not evolve smoothly. Already during m y first lesson I was a target of p a p e r airplanes and spitballs; laughter and c o m m o t i o n b e h i n d m y b a c k o v e r w h e l m e d m y scholarly discourse. I w a s facing the so-called discipline problem. Somehow, I survived it though. I m o v e d to a n o t h e r school w h e r e an o l d e r a n d m o r e e x p e r i e n c e d t e a c h e r t o o k m e aside and e x p l a i n e d a few basic principles: 'Always prep a r e y o u r class in a d v a n c e so that you w o n ' t have to dep e n d on thinking in front of y o u r students,' 'Never turn y o u r b a c k on y o u r e n e m y ' . . . . It t o o k a while, b u t I m a n a g e d to s h a k e off the initial shocldng e x p e r i e n c e a n d later was j u d g e d to be a c o m p e t e n t t e a c h e r . . . . When, in 1974, I r e a c h e d the m a n d a t o r y r e t i r e m e n t age I w a s m o s t u p s e t by the fact that the event t o o k a w a y m y regular c o n t a c t with undergraduates. Fortunately, m y Professor E m e r i t u s po-

Mathematics in that group of infatuated young people was kind of a fever.


39

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 pedagogy 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~,

40

THE MATHEMATICALINTELUGENCER

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. G6ttingen

and the "Birnbaum-Orlicz

spaces"

In October 1929, G6ttingen still 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 konjugierten Funktionen" by Birnbaum and Orlicz, published in the 3rd volume of Studia Mathematica. The almost 70page 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 Lp 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 accepted 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 accepted 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 mentioning Birnbaum, although because of my own family connections with pre-World-War-II mathematics, all kinds of

historical details were d i s c u s s e d from time to time, including the role of Mr. H e n r y k Majko, the famed c u s t o d i a n of the W a r s a w Mathematical Seminar (in Wroclaw after the war) w h o "befriended" and c o r r e s p o n d e d with several p r o m i n e n t mathematicians from Henri Lebesgue to John von Neumann. But Birnbaum was never p r e s e n t in those reminiscences. I myself did not p r e s s the issue. Perhaps, like p r o b a b l y m a n y others, I subconsciously assumed that, like m a n y Polish mathematicians, he p e r i s h e d in the Holocaust. The question "What h a p p e n e d to the first half of BirnbaumOrlicz?" was put away on the b a c k shelf of m y subconsciousness a n d disappeared from m y intellectual radar. Since their a p p e a r a n c e in 1930-31 the L r s p a c e s f o u n d b r o a d a p p l i c a t i o n s in h a r m o n i c analysis, w h e r e f r o m t h e y in a s e n s e arose, in functional analysis, in partial differential a n d integral equations, a n d in probability theory. Besides t h e Lw6w "functional thinking," two influences on the Birnbaum-Orlicz p a p e r s are evident: W.H. Young's 1912 article "On classes of s u m m a b l e functions and their F o u r i e r series" p u b l i s h e d in v o l u m e 87 o f the Proceedings of the Royal Society, which contains a p r o o f o f the inequality for p a i r s of conjugate functions, and an earlier 1907 E d m u n d Landau p a p e r from the G6ttinger Nachrichten. HSlder's inequality i m m e d i a t e l y implies that if r = lug, ~ ( u ) = lulq, 1/p + 1/q = 1, and b o t h series Zi ~P(ai), Zi T ( b i ) < ~, t h e n the series Zi aibi converges. Landau p r o v e d a sort o f c o n v e r s e to the effect that if the series Zi aibi c o n v e r g e s for e a c h s e q u e n c e bi for w h i c h the series Zi xi~(bi) < ~, then the series Zi dP(ai) < oo. A generalization o f this theo r e m to the case of arbitrary conjugate-in-the-sense-ofYoung c o n v e x functions w a s one of t h e main results of the B i r n b a u m a n d Orlicz papers. The influential Antoni Zygmnnd monograph on trigonometric series, p u b l i s h e d in 1935, c o n t a i n s the first systematic e x p o s i t i o n o f the t h e o r y of L(P-spaces, although it n e v e r calls t h e m Orlicz spaces. But t h e two-volume 1959 C a m b r i d g e edition a l r e a d y h o n o r s t h a t name. As the b a s i c s o u r c e Zygmund quotes Orlicz's p a p e r "Ober eine gewisse Klasse von R i u m e n v o m Typus B" p u b l i s h e d in 1932 in the C r a c o w Bulletin International de l'Acaddmie Polonaise, and the e a r l i e r Birnbaum-Orlicz m e m o i r quoted a b o v e is only m e n t i o n e d in the c o n t e x t o f the a b o v e generalization o f L a n d a u ' s theorem, which is given as a simple corollary to the Banach-Steinhaus theorem. B a c k to 1997. We are sitting in a c o r n e r office in the n o n s t a n d a r d shape of a n o n c o n v e x hexagon, w h i c h B i r n b a u m s h a r e s with the r e t i r e d topologist E r n e s t Michael. A shelf is filled with m o r e than sixty v o l u m e s of Probability and Statistics, an A c a d e m i c Press series o f m o n o g r a p h s a n d t e x t b o o k s , w h i c h Birnbaum has e d i t e d (jointly with Eugene Lukacs, from the beginning until the latter's d e a t h ) for the last few d e c a d e s . The d e s k t o p comp u t e r ' s s c r e e n is frozen on the e-mail window; " . . . w e a r e trying, b e t w e e n Hilde, the c o m p u t e r , a n d myself, to run a friendly mdnage ~ t r o i s . . . ," he c o m m e n t s . This is t h e present, but w e r e t u r n to the old days. "I dev e l o p e d the i d e a to w o r k on t h o s e L r p a p e r s during m y

GOttingen visit, a n d m y conversations with Landau had something to do with it. I s h a r e d the t h o u g h t with Orlicz, who, during the first y e a r of m y stay there, also r e s i d e d in G6ttingen. F r o m t h a t m o m e n t o n w a r d s w e w o r k e d on that p r o j e c t together, a n d in principle the w o r k w a s finished before he left G6ttingen in the Spring o f 1930. Our motivation was a p u r e l y intellectual curiosity, w e h a d n o c o n c r e t e applications in mind. Myself, having w r i t t e n the s e c o n d pap e r on a p p r o x i m a t i o n in the mean, I c o m p l e t e l y a b a n d o n e d the area. Orlicz, with w h o m I h a d since lost contact, obviously c o n t i n u e d to w o r k on the subject, a n d f o u n d e d the w h o l e s c h o o l in Poznafi concentrating on r e l a t e d topics. More recently, m y Seattle colleague E d w i n Hewitt emb a r k e d on a c a m p a i g n to a d d m y n a m e to the L % s p a c e s n o m e n c l a t u r e ; . . , in 1978, I.M. Bund p u b l i s h e d in Silo Paulo a m o n o g r a p h Birnbaum-Orlicz Spaces." "At the beginning o f m y s e c o n d y e a r in G6ttingen, Landau offered m e an assistantship, b u t s i m u l t a n e o u s l y s o b e r l y a d v i s e d m e n o t to t a k e it. He h a d c o r r e c t premonitions. A few y e a r s later he himself w a s unceremoniously, in the middle o f a semester, fired from his c h a i r by the Nazis. Simply, in lieu of Landau, his f o r m e r assistant, W e r n e r Weber, s h o w e d up at one o f his lectures. He brought with him L a n d a u ' s lecture notes, r a i s e d his h a n d in Heil Hitler and a n n o u n c e d that from n o w on he would t a k e over the lecture and continue at a new, higher level g u a r a n t e e d by NSDAP philosophy. I vividly r e m e m b e r him from a j o i n t d i n n e r of the G6ttingen M a t h e m a t i s c h e F a c h s c h a f t at der Gasthof zum Kehr l o c a t e d on top of a w o o d e d hill rising b e h i n d G6ttingen, as he e n t e r t a i n e d a p r e t t y female s t u d e n t u n d e r the p r e t e x t of a d i s c u s s i o n on

algebraische Fldchen. "I t o o k a d v a n t a g e o f Landau's advice a n d s p e n t m y second y e a r in G6ttingen working on an actuarial d i p l o m a within the n e w p r o g r a m established b y Felix Bernstein, w h o was also a f o u n d e r of the Institut ftir M a t h e m a t i s c h e Stochastik on LStze Strasse w h i c h still exists today."

Actuary The n e w career. The p o s i t i o n of an a c t u a r y in the Viennese insurance company, Phoenix, w a s t o o attractive to turn down. The salary w a s excellent. "I w a s t a k e n a b a c k w h e n P r o f e s s o r Berger, w h o b e s i d e s the Vienna Polytechnic chair also o c c u p i e d the position o f the chief a c t u a r y at Phoenix, w e l c o m e d m e with a s t a t e m e n t that he was familiar with m y j o i n t p a p e r s with Orlicz a n d h a d a probabilistic i n t e r p r e t a t i o n for our t h e o r e t i c a l results. My imm e d i a t e s u p e r v i s o r w a s E d w a r d Helly, w h o was a wonderful boss, warm, c o m p a s s i o n a t e , a n d with sunny disposition. I l e a r n e d from him a lot a b o u t h o w m a t h e m a t i c s can be of service in evaluation of actuarial transactions. He, of course, e s t a b l i s h e d his m a t h e m a t i c a l r e p u t a t i o n by proving f u n d a m e n t a l t h e o r e m s in real a n d functional analysis, but h a d no l u c k in the a c a d e m i c career. Throughout the rest o f his life he r e m a i n e d j u s t a Privat Dozent and w a s never p r o m o t e d to a well d e s e r v e d Professor Extraordinarius--in t h o s e days J e w s h a d a h a r d time in Vienna. My direct c o w o r k e r - - o u r d e s k s a b u t t e d and w e


41

f a c e d e a c h o t h e r - - w a s E u g e n e Lukacs . . . . " A f t e r the w a r L u k a c s had a distinguished a c a d e m i c c a r e e r in the United S t a t e s a n d a u t h o r e d a w e l l - k n o w n m o n o g r a p h on characteristic functions; his n a m e s u r f a c e d in this story before, he w a s Birnbaum's c o - e d i t o r on the A c a d e m i c P r e s s b o o k series. "The everyday actuarial w o r k did not have any signific a n t m a t h e m a t i c a l o r statistical component. Mainly it cons i s t e d of legal and c o m p u t a t i o n a l chores. My Vienna mathe m a t i c a l c o n t a c t s w e r e minimal. I did m e e t Richard y o n Mises, w h o a c t e d s u r p r i s e d w h e n I told him t h a t I w e n t in detail through his treatise on the foundations of p r o b a b i l ity theory; he used to b o a s t tongue-in-cheek that he w a s the only p e r s o n w h o h a d r e a d the b o o k with any understanding. Later, in 1933, I w r o t e with Schreier a small pap e r for Studia Mathematica on the law of large n u m b e r s , s h o w i n g h o w von Mises's RegeUosigkeits Axiom c o u l d b e r i g o r o u s l y e s t a b l i s h e d within the Kolmogorov axiomatics." At that time the Polish c o m m i s s i o n e r of the State Office of I n s u r a n c e Control had i s s u e d an executive o r d e r that the Polish subsidiary of the P h o e n i x Corporation be s p u n off into an i n d e p e n d e n t entity. The chief a c t u a r y p o s i t i o n in t h e n e w l y f o r m e d firm with t h e Polonized n a m e F e n i k s w a s offered to Birnbaum, w h o i m m e d i a t e l y a c c e p t e d it with joy, a n d returned for the seco n d time in his life to his native LwSw, w h e r e the F e n i k s h e a d q u a r t e r s w a s to b e located. The y e a r was 1932, a n d for the n e x t five y e a r s B i r n b a u m led the life of an e s t a b l i s h e d a n d affluent businessman. " . . . during t h o s e y e a r s I t o o k p a r t in the Scottish Caf6 a n d s e m i n a r life only sporadically, f r o m t i m e to t i m e , . . , the i n t e r e s t remained, b u t t h e r e w a s s i m p l y no time for r e s e a r c h work." But "business as usual" in the m a n a g e m e n t o f the F e n i k s did n o t last long. Something strange started h a p p e n i n g in t h e Viennese center; as ff s o m e b o d y tried to hide the firm's funds from the Nazis. T h e r e w a s p r e s s u r e on the Lw6w s u b s i d i a r y to c o o k the b o o k s a n d the annual financial report. " . . . I had a c o w o r k e r t h e n b y the n a m e o f Ludwik Sternbach, and the Vienna central office d e m a n d e d t h a t he a n d I turn our financial b o o k s o v e r to them. Before shipping t h e d o c u m e n t a t i o n the t w o of us p h o t o g r a p h e d t h e w h o l e thing on glass negatives. Sure enough, after a few m o n t h s the p a p e r s w e r e r e t u r n e d to Lw6w b u t obviously the n u m b e r s had b e e n d o c t o r e d . Soon t h e r e a f t e r the firm filed for b a n k r u p t c y a n d t h e Polish insurance commiss i o n e r a p p o i n t e d a liquidation commission. I w a s one o f the c o m m i s s i o n ' s m e m b e r s . As the external e x a m i n e r the g o v e r n m e n t n a m e d Zbigniew {~omnicki, an a c t u a r y at the PKO B a n k in W a r s a w a n d t h e n e p h e w of Antoni {,omnicki, p r o f e s s o r of m a t h e m a t i c s at the Lw6w Polytechnic (the latt e r is b e s t r e m e m b e r e d b y m a t h e m a t i c i a n s as a p e r s o n w h o offered the first a c a d e m i c p o s i t i o n to a diploma-less Stefan Banach). The affair of d o c t o r e d b o o k s i m m e d i a t e l y surfaced, b u t our glass negatives h e l p e d the c o m m i s s i o n to rec o v e r the truth. The principal legal counsel and the m a i n b o o k k e e p e r e n d e d up in jail. In the process, I h a d discov-

e r e d that, miraculously, our run-down family p r o p e r t i e s in the Pysznica t o w n s h i p e n d e d up r e c o r d e d in the firm's financial b o o k s as Feniks, Inc.'s a s s e t s v a l u e d at one million gold U.S. dollars. To this day I have no i d e a h o w this happened. By that t i m e Pysznica h a d a l m o s t no value, all the t i m b e r having b e e n logged a long time before. "That p e r i o d also w i t n e s s e d m y b a p t i s m by fire as a lab o r activist. After Feniks, Inc. filed for b a n k r u p t c y , the gove r n m e n t a l a d m i n i s t r a t o r of leftover a s s e t s fired all the employees. In r e s p o n s e , we organized a cell of the Labor Union of Whitecollar Workers a n d I w a s elected the shop steward. F a c i n g n o p r o g r e s s in n e g o t i a t i o n s with the management, w e d e c l a r e d an o c c u p a t i o n a l strike of our thirdfloor offices. T h e r e were a b o u t thirty strikers, and the food supplies w e r e delivered by o u r families on r o p e s l o w e r e d through the windows. After a few d a y s the g o v e r n m e n t t h r e w in the t o w e l and s u m m o n e d m e for negotiations in Warsaw. That w a s my first airplane flight. We k e p t our j o b s b u t n o t for long. It was clear that b o t h my local position a n d the general a t m o s p h e r e in E u r o p e were n o t promising .... "I left Lw6w on May 3, 1937, a n d w a s n o t to see m y parents n o r m y sister, Franciszka,

Orlicz spaces are

Birnbaum-Orlicz spaces,

42


seven y e a r s m y junior, again. They w e r e s e e n for the last time

in the Bergen-Belsen concentration c a m p . . . . An A m e r i c a n visa was n o t e a s y to obtain. People w o u l d wait for t h e m for ten y e a r s after filing an application. However, m y cousin Ludwik Rubel, w h o was the editor-in-chief of the C r a c o w p a p e r Ilustrowany Kurjer Codzienny, f a m o u s in P o l a n d u n d e r the n i c k n a m e Ikac, t o o k m e to t h e A m e r i c a n consulate and i n t r o d u c e d m e as his n e w s p a p e r ' s reporter. My j o u r n e y t o o k m e to Vienna w h e r e I b a d e farewell to m y relatives and acquaintances, and to Paris, w h e r e with m y rep o r t e r ' s c r e d e n t i a l s I m a n a g e d to see the World Exhibition a few days b e f o r e its official opening. The then F r e n c h premier, L6on Blum, s u r r o u n d e d b y a flock o f officialdom, h a d j u s t b e e n given a t o u r of the exhibition grounds. I t o o k s o m e p i c t u r e s a n d excitedly hurried to Le Monde to have t h e m developed, sensing the o c c a s i o n as a b r e a k t h r o u g h in m y j o u r n a l i s t i c career. However, n o b o d y s h o w e d any int e r e s t in using m y services . . . . I left for A m e r i c a from Le Havre on the liner M.S. Georgia o f the Cunard White Line. Two y e a r s l a t e r she had the doubtful distinction of being the first allied p a s s e n g e r vessel s u n k b y the G e r m a n UBoats." Statistician The New York c a r e e r of the Ilustrowany Kurjer Codzienny r e p o r t e r did not last long. "Shortly after m y arrival in New York, I w a s strolling in C o l u m b u s Circle w h e r e a certain individual, p e r c h e d on the p r o v e r b i a l soapbox, k e p t abusing F r a n k l i n Delano R o o s e v e l t with the m o s t disgusting a d j e c t i v e s - - b y the way, that p e r f o r m a n c e significantly e n r i c h e d m y English vocabulary. A fat policeman, looking p e r f e c t l y the role of a New York Irish cop from the A m e r i c a n s u s p e n s e movies, s t o p p e d briefly in front of the

speaker, t u r n e d around, s p a t a d i s t a n c e that I still believe to be a w o r l d record, and then j u s t w a l k e d away. That w a s m y first, s h o c k i n g lesson in the A m e r i c a n c o n c e p t of freed o m o f s p e e c h . . . . This did not m e a n that everything w a s hunky-dory. F o r a while I r o o m e d at Columbia University's International House and b e f r i e n d e d an attractive b l a c k female s t u d e n t from one o f the C a r i b b e a n Islands. Soon I was w a r n e d that our w a l k s t o g e t h e r were not r e c e i v e d kindly by others. That w a s also a s h o c k i n g experience, b u t of a different kind." B i r n b a u m r e n e w e d his G6ttingen acquaintances. Courant a n d Felix Berustein w e r e a l r e a d y in New York, a n d the latter offered him an assistantship in the B i o m e t r i c s D e p a r t m e n t at New York University. The soft m o n e y did n o t last, however, and he h a d to l o o k for n e w employment: the c a r e e r o f a statistical c o n s u l t a n t in private p r a c t i c e w a s n o t very p r o m i s i n g either. "A suggestion that I s u b m i t a j o b a p p l i c a t i o n in Seattle c a m e from Harold Hotelling w h o s e statistical s e m i n a r at C o l u m b i a University I had s t a r t e d to attend: 'Do s e n d an application, t h e y n e e d s o m e b o d y to t e a c h statistics. But do not m e n t i o n that it was m y idea, I'm on the blacklist there . . . . ' My r e c o m m e n d a t i o n letters w e r e from Richard Courant, E d m u n d Landau, a n d Albert Einstein. I w a s i n t r o d u c e d to the l a t t e r during a s h o r t formal visit. I could not go for an interview to Seattle, so it w a s c o n d u c t e d in New York b y t w o gentlemen. One w a s the P r e s i d e n t of the New School for Social Research; the second, the chief executive of the Sun Oil Company. A p p a r e n t l y I m a d e a g o o d impression, b e c a u s e s o o n thereafter I r e c e i v e d a written offer from P r o f e s s o r Carpenter, c h a i r m a n o f the M a t h e m a t i c s D e p a r t m e n t in Seattle. He also a t t a c h e d an a d m o n i t i o n that I p o l i s h up m y English before m y arrival at the University of Washington to the p o i n t that even f r e s h m e n could u n d e r s t a n d it. I t h e r e f o r e s p e n t the rest o f the S u m m e r of 1939 at the University of Vermont t a k i n g intensive language c o u r s e s and swimming in Lake Champlain." B i r n b a u m r e a c h e d Seattle in O c t o b e r 1939. The c a r he was traveling in from San F r a n c i s c o r a n into a ditch on a s n o w y m o u n t a i n p a s s in the Siskiyous, and the trip h a d to be c o m p l e t e d b y train. "In the D e p a r t m e n t of Mathematics of the University of Washington the statisticians a n d applied m a t h e m a t i c i a n s were v i e w e d as if t h e y had grease und e r their fingernails. But w e s t u b b o r n l y d e v e l o p e d the m a t h e m a t i c a l statistics program, and in 1948 1 f o u n d e d the Statistical R e s e a r c h L a b o r a t o r y which, for the n e x t 25 years, w a s f u n d e d by the Office of Naval Research. At a certain p o i n t w e had twelve full-time p r o f e s s o r s of statistics. In 1979, t h r e e years after m y retirement, the university finally a g r e e d to e m a n c i p a t e statisticians and c r e a t e a s e p a r a t e Statistics Department. "My r e s e a r c h w a s then c o n c e n t r a t e d in the a r e a of probabilistic inequalities, in p a r t i c u l a r t h e two-dimensional C h e b y s h e v inequality. When I p r e s e n t e d it to George P61ya at Berkeley, the previous e x p e r i e n c e with Banach w a s repeated. P61ya's initial r e a c t i o n w a s 'but this is trivial,' b u t after a day's reflection, he invited m e to publish it. I was also i n t e r e s t e d in the effects of p r e s e l e c t i o n on multivari-

ate distributions. One o f the first questions w a s h o w constraints on one c o m p o n e n t affect the j o i n t distribution o f the r a n d o m vector. Here, the p r o c e s s o f a d m i s s i o n of stud e n t s to a university, o r of catching fish in a n e t with a fLxed m e s h size, a r e g o o d examples. This w a s a continuation of Felix B e r n s t e i n ' s ideas. "I met J o h n n y von N e u m a n n at an IBM symposium. I m e n t i o n e d to him t h a t Kolmogorov h a d found an asymptotic distribution o f t h e so called Kolmogorov-Smirnov statistics. It was the y e a r 1950 and von N e u m a n n was abs o r b e d b y c o n s t r u c t i o n of the first digital computers. I suggested that t h e y be u s e d to calculate the distributions o f these statistics for s a m p l e s of finite size. He h e l p e d m e to obtain funds for t h e s e calculations, w h i c h w e r e done in the Western C o m p u t e r Center of the National Bureau of S t a n d a r d s at the University of California at Los Angeles. My interest in distribution-free statistics l a s t e d much longer and during m y stay at Stanford, j o i n t l y with H e r m a n Rubin, w e f o r m u l a t e d the c o n c e p t u a l f o u n d a t i o n s for such statistics and p r o v e d a few c h a r a c t e r i z a t i o n theorems." A c o l l a b o r a t i o n with the Boeing Airplane Co., with its h e a d q u a r t e r s a n d r e s e a r c h labs l o c a t e d in Seattle, origin a t e d in 1958 and l a s t e d for a decade, m o s t l y in the a r e a of testing m a t e r i a l fatigue and reliability o f s y s t e m s with m a n y components. "Boeing's interest w a s s p a r k e d by a series of c r a s h e s of the n e w British C o m e t jets. Boeing was j u s t gearing up for t h e p r o d u c t i o n of its first 707 model. The British f o r m e d a Royal Commission to s t u d y the causes of the accidents, a n d after careful s t u d y a r r i v e d at the conclusion that the r e a s o n w a s material fatigue: the sharp angles of the r e c t a n g u l a r w i n d o w openings, c o m b i n e d with the alternating p r e s s u r i z a t i o n and d e p r e s s u r i z a t i o n cycles, w e r e a deadly combination. Of course, Boeing did n o t w a n t to c o m m i t the s a m e errors. It is interesting that fifty years earlier a n o t h e r British Royal C o m m i s s i o n found a similar c a u s e for initially u n e x p l a i n a b l e rail accidents. Boeing initially r e t a i n e d m e as a consultant for t h e i r r e s e a r c h organization, the Boeing Scientific R e s e a r c h Laboratories, to direct basic r e s e a r c h - - ' f u t u r e work', as t h e y called i t - - o n material fatigue and on the reliability of c o m p l e x structures. With a t e a m o f mathematicians, s o m e of t h e m m y f o r m e r students, w e f o r m u l a t e d basic c o n c e p t s of the m a t h e m a t i c a l t h e o r y of reliability o f m u l t i - c o m p o n e n t systems, and o b t a i n e d m a n y fundamental results. "I have also s e r v e d as a consultant to the U.S. National Center for Health Statistics, developing n e w multiple-sampling designs a n d studying biases and v a r i a n c e s in infant mortality rates. This w o r k led to p u b l i c a t i o n s with Monroe Sirken, a n o t h e r f o r m e r student. F u r t h e r m o r e , NCHS supp o r t e d m y w o r k on t h e m a t h e m a t i c a l t h e o r y of c o m p e t i n g risks, which resulted in a monograph. The j o i n t w o r k with Monroe Sirken initiated a sequence of further p a p e r s that have a p p e a r e d over m a n y years. "I b e c a m e an activist in the Institute of Mathematical Statistics (IMS) b y accident. At one of the annual meetings o f the society one o f the topics d e b a t e d w a s the racial segregation e n c o u n t e r e d at meetings organized in the Southern states. One o f the shocking racial incidents in-


43

v o l v e d David Blackwell, P r o f e s s o r of Statistics at the University of California, Berkeley. He w a s not p e r m i t t e d to lodge at the host d o r m i t o r i e s u s e d by p a r t i c i p a n t s o r dine at c o m m o n facilities. Harold Hotelling, w h o b y t h e n h a d alr e a d y m o v e d from C o l u m b i a to the University o f North Carolina at Chapel Hill, a p p e a l e d for calm and n o t r o c k i n g t h e boat; p e o p l e here a r e civilized, the racial r a p p o r t is improving, and we should w o r k h a r m o n i o u s l y within the existing structures. I strongly disagreed. As a result of this inc i d e n t I was a p p o i n t e d c h a i r m a n of the e u p h e m i s t i c a l l y n a m e d Committee on Physical Facilities for Meetings, which, for the n e x t few years, was s u p p o s e d to verify w h e t h e r the universities offering to host IMS m e e t i n g s satisfied certain m i n i m u m antisegregation conditions. There w e r e c a s e s w h e n o u r c o m m i t t e e r e j e c t e d c o n f e r e n c e prop o s a l s b e c a u s e w e felt t h a t t h e guarantees w e r e n o t suffic i e n t . . , my s u b s e q u e n t election to the IMS p r e s i d e n c y w a s p a r t i a l l y related to m y activities on that committee. "I w a s the Editor-in-Chief o f the Annals of Mathematical Statistics for t h r e e years, from 1967 to 1970. T o w a r d s the e n d of m y term, w h i c h I a c c e p t e d with g r e a t relief, I m a d e a r e c o m m e n d a t i o n t h a t the Annals be split into two series: Annals of Statistics and Annals of Probability. This p r o p o s a l has b e e n i m p l e m e n t e d b y m y successors, I n g r a m Olkin and Ron Pyke. "Life is a stochastic p r o c e s s , and in m a n y instants the t r a j e c t o r y of m y life c o u l d have d e v e l o p e d in a totally different direction . . . . T h e s e d a y s I'm o c c u p i e d with the problem o f characterization of b i m o d a l distributions w h i c h are m i x t u r e s of two u n i m o d a l populations. A m o n g t o p i c s I w o r k e d on I cannot r e m e m b e r even one w h i c h c o u l d be called 'real applied statistics.' My interests a l w a y s lay in the mathematics of statistics. Certainly it h a d s o m e t h i n g to do with m y Lw6w b a c k g r o u n d . I always get m o s t satisfaction from doing nontriviai mathematics. Perhaps, m y c o m p u t a t i o n s of correlations, done while I w a s trying m y h a n d at consulting in N e w York City, w e r e c l o s e s t to the definition of true applications." Hilde Saturday. We are eating lunch in the p o s h dining r o o m of the I d a Culver House on G r e e n w o o d Avenue, w h e r e the B i r n b a u m s m o v e d a c o u p l e o f y e a r s ago. Their small apartment, filled with w o r k s of art and antique furniture inherited from Hilde's parents, is p a r t of the large a n d u p s c a l e c o m p l e x for affluent retirees. Beyond w i n d o w s that c o v e r t h e w h o l e w e s t e r n wall of the dining room, one c a n s e e below, p a s t the d e n s e w o o d s o f the coastal State C a r k e e k Park, p a l e in the drizzle, t h e w a t e r s of the Puget Sound. B i r n b a u m and I o r d e r R e u b e n sandwiches, he will finish only half of his, a n d Hilde eats a light s e a f o o d s a l a d s e r v e d b y a w a i t e r in a fancy uniform. I have m y o w n a f t e r n o o n talk at the University of Washington to l o o k f o r w a r d to, w h i c h leaves a b o u t t h r e e h o u r s for our conversation. Hilde is d r e s s e d with great attention to detail, a n d radiates energy and intellectual vitality. In h e r p r e s e n c e the c o n v e r s a t i o n naturally t u r n s to h e r and Bill's c o m m o n interests, of which t h e r e are many, s o m e quite nontrivial. Her


father, Dr. R i c h a r d Merzbach, w a s a well-known l a w y e r in Frankfurt-on-Main, and she g r e w up t h e r e and studied law. " . . . as a student, I spent the S u m m e r o f '31 in London, as an intern at a law firm. The n e w s p a p e r headlines announcing the first electoral s u c c e s s e s o f the Nazis, w h e n their Reichstag r e p r e s e n t a t i o n j u m p e d from 5 to 112 delegates, caught m e during a midnight stroll through the Piccadilly Circus . . . . I b u r s t into t e a r s right there on the street. The n e x t d a y I wrote to m y p a r e n t s that as a w o m a n and a J e w I c o u l d never p r a c t i c e law in Germany, and that I'd like to settle in England. There w a s a good c h a n c e that I could r e m a i n e m p l o y e d at the law firm at which I was doing m y internship. My father tried to convince me to return. The family let m e k n o w that t h e y believed that m y o u t l o o k w a s t o o pessimistic. They c o u l d n o t see Hitler coming to power. So, I returned to G e r m a n y a n d g r a d u a t e d with a law d e g r e e in 1932. According to the rules governing the legal p r o f e s s i o n in Germany, I h a d to p r a c t i c e for a while as a c o u r t clerk. "But a y e a r l a t e r Hitler did c o m e to power, and April 1, 1933, w a s d e c l a r e d the day of the anti-Jewish boycott. It was no F o o l ' s Day joke. On March 31 I w a s informed t h a t there w a s no n e e d for m e to s h o w up at w o r k the next day. At midnight w e left with m y sister for England. In the middle of the D e p r e s s i o n it was n o t e a s y to obtain a w o r k permit. But the l a w y e r I w o r k e d for t w o y e a r s earlier h a d a client w h o n e e d e d a London representative. The p o s i t i o n w a s to b e f u n d e d by a Swiss c o n g l o m e r a t e . In this fashion I b e c a m e their r e p r e s e n t a t i v e a n d m a n a g e d their three production facilities and shops. I w a s n o t v e r y fond of that j o b but I did all right. "Almost f r o m the beginning o f m y stay in England I w a s trying to c o n v i n c e m y family to emigrate. But in 1933 the m a j o r i t y of p e o p l e in G e r m a n y a n d England did not think it r e a s o n a b l e to a b a n d o n the security of the family dwelling for an u n c e r t a i n existence of the exile. I often served as a c o u r i e r moving m o n e y and o t h e r m a t e r i a l s from G e r m a n y to England. I usually traveled during holidays as the bord e r c r o s s i n g s w e r e m o r e c r o w d e d a n d detailed c u s t o m c h e c k s less likely . . . . My p a r e n t s h a d their r o o t s in F r a n k f u r t a n d o w n e d c o n s i d e r a b l e p r o p e r t y there. The thought o f a b a n d o n i n g the h o m e t o w n w a s for t h e m unb e a r a b l e . . . . In 1936, I even t r a v e l e d to the Middle East investigating a possibility of obtaining for t h e m a visa to Palestine, w h i c h in those days w a s a British mandate. The Nazis p e r m i t t e d J e w s emigrating to Palestine to t a k e 80~ of their p r o p e r t i e s , w h e r e a s the J e w s leaving for o t h e r destinations c o u l d t a k e only a small fraction of their belongings. But m y p a r e n t s w o u l d n o t agree to that either. They left j u s t a few w e e k s before the KrystaUnacht . . . . In 1938 they j o i n e d m y sister, who had earlier m a r r i e d and settled in Seattle . . . . The g o v e r n m e n t decree, p u b l i s h e d in the n e w s p a p e r s b y the Nazis, d e c l a r e d m e an e n e m y o f the Reich, s t r i p p e d m e of citizenship, a n d confiscated m y w h o l e p r o p e r t y . I b e c a m e stateless. After that I s t o p p e d visiting G e r m a n y b u t did not give up on England, w h e r e I felt comfortable. However, in S e p t e m b e r of 1939 the atmos p h e r e in L o n d o n b e c a m e very tense, the civil defense o r -

Hilde Birnbaum in the Birnbaums' Seattle apartment, 1997.

ganized e x e r c i s e s in gas masks, t r e n c h e s were being dug in the City . . . . I was offered a p o s i t i o n of d i r e c t o r of o u r c o m p a n y ' s Brazilian subsidiary, b u t after m a n y e m o t i o n a l t r a n s a t l a n t i c t e l e p h o n e c o n v e r s a t i o n s with m y p a r e n t s - quite u n u s u a l in t h o s e d a y s - - I d e c i d e d to emigrate to the United States 9 "I m e t Bill through m y brother-in-law . . . . F o r six m o n t h s he tried to m a k e a m a t h e m a t i c i a n out of me, but to no avail. C o m p e t i n g with the h u s b a n d in the a r e a in which he w a s so m u c h s t r o n g e r did not m a k e any s e n s e . . . . F r o m the v e r y beginning w e had m a n y c o m m o n interests: law, social issues such as the terrible state o f health insurance in America, and, in particular, the distressing p o s i t i o n of blacks, c o n s u m e r protection, liberal politics. I enrolled in the e c o n o m i c s p r o g r a m at the University of Washington; the G e r m a n law d i p l o m a is n o t easily exported. After graduation I b e c a m e a teaching a s s i s t a n t with a p r o s p e c t for a p e r m a n e n t p o s i t i o n in the D e p a r t m e n t of Economics 9 Shortly thereafter, however, I r e c e i v e d a letter from the University's p r e s i d e n t informing m e t h a t the fact that b o t h Bill and I hold p o s i t i o n s at the University violates University's n e p o t i s m rules. If Bill h e l d only a part-time position t h e n p e r h a p s s o m e t h i n g c o u l d be arranged for me. Our c h o i c e w a s obvious. Bill's c a r e e r h a d to have the priority. We h a d a small child and the s e c o n d was on its way. 9 After t h e children g r e w up I taught e c o n o m i c s at one of the local colleges and w a s the h e a d of the E c o n o m i c s d e p a r t m e n t t h e r e for 10 years." The lunch is a l m o s t over; w e o r d e r decaffeinated coffees and desserts. Hilde continues, "In 1946 1 was one of the f o u n d e r s of the Washington State b r a n c h of A m e r i c a n s for

Democratic Action, a n d later on w o r k e d as a Washington State lobbyist for the A m e r i c a n F e d e r a t i o n of T e a c h e r s . . . . Doctors h a d in t h o s e days an a l m o s t unlimited f r e e d o m of action, including the right to refuse t r e a t m e n t to uninsured patients . . . . Bill a n d I h e l p e d to organize the G r o u p Health Cooperative of Puget Sound, taking o v e r the p r a c t i c e of a small group of p h y s i c i a n s who during the w a r h a d a contract with the, then v e r y active, Seattle shipyards. Bill's exp e r i e n c e in i n s u r a n c e affairs w a s very helpful. Initially he served on the B o a r d of Trustees b u t r e s i g n e d w h e n we left for his s a b b a t i c a l at Stanford. After o u r return, I was e l e c t e d to the B o a r d w h e r e I served for 23 y e a r s including four y e a r s as its chairperson. We fought p i t c h e d battles with the A m e r i c a n Medical Association, w h i c h w a s trying to b o y c o t t us. Many physicians w o u l d a g r e e to w o r k for us only on the c o n d i t i o n o f anonymity. We built several hospitals and clinics a n d today, with 700,000 subscribers, w e are the largest health i n s u r e r in the Pacific Northwest. Bill is still a m e m b e r o f the Grievance C o m m i t t e e . . . . Of course b y n o w this kind o f health m a i n t e n a n c e organization has b e c o m e c o m m o n p l a c e all over the country. "In 1946 Bill w a s a p p o i n t e d c h a i r m a n of the c o m m i t t e e to analyze the University's r e t i r e m e n t a n d health insurance system, and p r e p a r e d a draft of state legislation which is current to this day. In 1955 he p l a n n e d a n d s u p e r v i s e d the r e f e r e n d u m on merging the University s y s t e m into the national Social Security System. "In 1947 we, t o g e t h e r with a group of friends, b o u g h t 70 a c r e s of land on the hills b e t w e e n the t o w n s of Bellevue and Renton, s o u t h e a s t of Lake Washington. In t h o s e days it was j u s t a 20-year-old forest slated for logging 9 We built s o m e forty h o m e s a n d f o u n d e d a t o w n s h i p w h i c h w e called the Hilltop Community. Until this day it is an i n d e p e n d e n t c o r p o r a t i o n in spite o f the a n n e x a t i o n efforts o f the neighboring towns. We lived t h e r e for 27 y e a r s a n d o u r t w o children grew up t h e r e - - b o t h settled in S e a t t l e - - r a r e luck for p a r e n t s in A m e r i c a . . . . But the n a m e of B i r n b a u m is quite p o p u l a r in the United States outside Seattle as well. P e r h a p s it has s o m e t h i n g to do with the fact t h a t Bill's oldest p a t e r n a l uncle h a d eighteen children." We talk a b o u t travels a n d Birnbaum reminisces: "In 1960 I s p e n t m y s a b b a t i c a l in Paris. We lived in a t w o - r o o m penth o u s e a p a r t m e n t in a small hotel at 29 Rue C a s s e t t e on the Left Bank. Initially w e p l a n n e d to stay at the h o t e l only for a few days, b u t Hilde d i s c o v e r e d b y a c c i d e n t a n e w s p a p e r a d informing the public that the hotel w a n t e d to let the p e n t h o u s e a p a r t m e n t on a long-term basis. We had no kitchen, b u t Hilde t o o k advantage of o u r e a r l i e r camping e x p e r i e n c e s in the Seattle a r e a and organized a m a k e s h i f t kitchen in the b a t h r o o m using a small electric stove. She even m a n a g e d to p r e p a r e roast b e e f there 9 I w o r k e d a little bit on finishing m y b o o k and m e t with Paul L~vy. But Paris itself was v e r y absorbing 9 We s p e n t a lot of time at the m u s e u m s a n d visiting with E u r o p e a n friends . . . . The Polish A c a d e m y o f Science invited us in 1963, a n d we visited Warsaw, Wroclaw, and Cracow 9 We also t r a v e l e d to the m o u n t a i n r e s o r t o f Z a k o p a n e to see Steinhaus. It was a very e m o t i o n a l r e u n i o n . . . .


45

clientele, m a i n l y international t y c o o n s and H o l l y w o o d types. But Hilde m a d e a great i m p r e s s i o n on the o w n e r w h o t u r n e d out to b e a German-born Italian baroness. The contessa w o u l d bring us a daily b r e a k f a s t to o u r r o o m and aft e r n o o n s w e p a r t i c i p a t e d with o t h e r guests in lively wine parties." The lunch is over and w e bid e a c h o t h e r good-byes. "Please, s t o p in Seattle and visit us again when you travel to Tokyo in June. It's on the way."

"I still r e m e m b e r an evening at the a p a r t m e n t of one of the W a r s a w mathematicians. That was the time o f a political t h a w in Poland, a n d Stan Mazur, w h o held a high position in the C o m m u n i s t Party, confided to m e that telling political j o k e s w a s n o w O.K. People h e a r d that I w a s suing t h e United States G o v e r n m e n t and w o r r i e d w h e t h e r it w a s safe for us to r e t u r n h o m e and if our j o b s w o u l d still b e waiting for us on o u r return. I e n t e r e d the litigation j o i n t l y with the A m e r i c a n Civil Liberties Union, in w h i c h I'm active until this day, trying to get the Loyalty Oath required of all University o f Washington e m p l o y e e s d e c l a r e d unconstitutional. The c a s e e n d e d up before the U.S. S u p r e m e Court, and m y d e p o s i t i o n was the only one quoted in t h e highest c o u r t ' s opinion; the decision w a s in o u r favor. "During m y n e x t s a b b a t i c a l w e s p e n t four m o n t h s in Rome. At the beginning w e lived far from the city's center. One day, strolling on a side s t r e e t in the n e i g h b o r h o o d o f the Spanish Steps, Hilde n o t i c e d a discreet, s m a l l p e n s i o n e . The l a d y o w n e r was s u r p r i s e d w h e n w e e n t e r e d asking for a room, and inquired w h o r e c o m m e n d e d us. It t u r n e d out that the pensione w a s r e s e r v e d for a carefully s e l e c t e d

46


Cleveland Sunday. My flight h o m e leaves a r o u n d noontime. Light rain and foggy. The day is mercilessly s h o r t e n e d b y the time zones b e t w e e n the West and E a s t Coasts. During the p a s t 48 h o u r s B i r n b a u m had b e c o m e a n a t u r a l and obviously missing p a r t o f m y own tradition. My private "rediscovery" of B i r n b a u m was, in a sense, serendipitous. Two y e a r s ago at the s p e c i a l c o n f e r e n c e in h o n o r o f Kazimierz Urbanik, m y Ph.D. advisor, I ran into R o m a n Duda, a topologist a n d p r e s i d e n t o f t h e Wroclaw University, w h o a s k e d m e if I w o u l d be willing to write s o m e t h i n g a b o u t Birnbaum for the p l a n n e d Centennial issue o f the Annals of the Polish Mathematical Society. I reluctantly a g r e e d but p l e a d e d m y i g n o r a n c e o f the subject. Besides, as far as "after-dinner" literary p r o j e c t s w e r e concerned, I a l r e a d y had on m y d e s k the English v e r s i o n of B a n a c h ' s biography. At the s a m e time I felt u n d e r p r e s s u r e to oblige, as I h a d a d e b t to repay. A few d e c a d e s ago, w h e n b o t h o f us served his E x c e l l e n c y R e c t o r Marczewski as s e c r e t a r i e s of the Colloquium Mathematicum, R o m a n d i d the overwhelming majority of the work. The writing w a s p u t on a faster t r a c k w h e n Chris Burdzy and Richard Bass invited me to give a talk at the Pacific Northwest Probability Seminar organized b y the Oregon, British Columbia, a n d Seattle probabilists. I c a n n o t r e s i s t looking at the B i r n b a u m story t h r o u g h the glasses o f m y own experiences. Toute proportion gardde--I did n o t found any cities, a n d did not sue the A m e r i c a n G o v e r n m e n t in the Court o f Law. There is also this trifling 40-year difference in o u r ages. But the parallels and p o i n t s o f t a n g e n c y are u n m i s t a k a b l y there: the LwSwW r o c l a w t r a d i t i o n and fascination with Banach and Steinhaus, a m a t h e m a t i c s Ph.D. b u t only after a d i p l o m a in a n o t h e r discipline, the Birnbaum-Orlicz spaces, GSttingen, Paris, a n d a s t u b b o r n struggle for m a t h e m a t i c s that is not an i s o l a t e d enterprise, but m a t h e m a t i c s in the foreground always, teaching as one o f the great j o y s in life, creation of n e w statistics p r o g r a m s , social and editorial activities,... In Cleveland the sky is ink-black, freezing, the visibility excellent. We land from the n o r t h o v e r the partly frozen Lake Erie. On t h e left I s e e k the lights o f Shaker Hights. Lauren a n d Greg are impatiently waiting with the usual "Dad, it's g o o d to have you b a c k home." I have for t h e m s h r i n k - w r a p p e d s a m p l e s of the m i n e r a l s from the Wild West b o u g h t at the last minute at the Seattle airport: obsidian, quartz, sulphur, t e r r a c o t t a , . . .

lld,[=-aa~'jIR'i~l,z:-]ii~lilzr

Kepler in Eferding Karl Sigmund

Dirk Huylebrouck,

Editor

f you follow the summer crowd of tourists biking along the Danube, you may discover, close to Linz, a short detour leading through shady woods to Eferding. This is a quiet little Upper Austrian town, offering the usual sightseems' fare: castle, church, and marketplace. The first house on the main

I

J

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? If 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 commemorating 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

47

Measuring barrel contents. The left barrel is measured in the Austrian way, by help of a gauging rod. The other barrel's volume is determined by pouring its content into vessels of specific volume. This is from the title page of a book on analysis, which appeared in 1980. Kepler in 1620, age forty-nine. According to his friend, the poet

The drawing is from the title page of a treatise by Johann Frey, which

Lansius, this is how Kepler did not look. Lansius jocularly put the

was published in 1531 in Nuremberg. The formula (in white) is

blame on the motion of the Earth (a heresy at that time, as Galileo

Kepler's barrel rule. (From the cover of Analysis 1, 126, I.)

came to learn): if the Earth had stood still, the artist's hand would have been steadier.

had broken off his theological studies in Ttibingen to b e c o m e professor of mathematics at a college in Graz. Later, he joined the astronomer Tycho Brahe in Prague, as one of the m a n y scientists, astrologers, and alchemists attracted there by Rudolf II, the oddest of all Habsburg emperors, a dreamer suffering from fits of madness, who ended as a prisoner in his o w n castle. Kepler, who furnished his fair share of horoscopes, eventually held the job of Imperial Mathematician under three Habsburg rulers. Each was more nfilitantly 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 ignominously away from their land. Having gone through an unhappy first marriage, Kepler took great pains to avoid all mistakes on his second matrimonial 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 temporising, 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 modest, thrifty, and devoted. He had met her in the household of a friend with a ringing 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 sympathy 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 h a d no money, but on the o t h e r hand, no in-laws either. She was a w a r d o f Baron E r a s m u s ' s wife Elisabeth, w h o p a t r o n i z e d an institution for the upbringing of i m p o v e r i s h e d young ladies. After having t a k e n the plunge, Kepler never m e n t i o n e d his s p o u s e again in all his c o p i o u s c o r r e s p o n d e n c e , e x c e p t on the seven o c c a s i o n s w h e n she gave birth. B a s e d on this, all b i o g r a p h e r s agree that t h e marriage i n d e e d w a s a h a p p y one. The Eferding wedding p l a y s a curious role in the p r e h i s t o r y of calculus. In Kepler's words: After m y r e m a r r i a g e in N o v e m b e r o f last year, at a time w h e n b a r r e l s o f wine f r o m Lower Austria w e r e s t o r e d high on the s h o r e s of the D a n u b e n e a r Linz after a c o p i o u s vintage, on offer for a r e a s o n a b l e price, it w a s the duty of the n e w husb a n d a n d d e v o t e d family-head to p u r c h a s e the drink n e e d e d for his household. F o u r days after several b a r r e l s h a d b e e n brought to the cellar, the wine-seller c a m e with a r o d w h i c h he u s e d to m e a s u r e the content o f all barrels, irrespective of their form a n d without any further r e c k o n i n g o r computation. The metallic e n d of the gauge-rod w a s i n t r o d u c e d through the bung-hole till it r e a c h e d the o p p o s i t e p o i n t on the b o r d e r of the b a r r e l s ' s bottom. I w a s a m a z e d that the diagonal t h r o u g h the half-barrel could yield a m e a s u r e for the volume, and I d o u b t e d that the m e t h o d could work, since a m u c h l o w e r b a r r e l with a s o m e w h a t b r o a d e r b o t t o m and h e n c e m u c h less content could have the s a m e rod-length. To m e as a newlywed, it did not s e e m inopportune to investigate the mathematical principle behind the precision of this practical and w i d e s p r e a d measurement, and to bring to light the underlying geometrical laws.

The fact that their c o n t e n t was measured b y m o r e c o m p l i c a t e d m e a n s in o t h e r countries, for instance on the Rhine, r e n d e r e d him suspicious. But a few days sufficed to convince him of the validity of w h a t he t e r m e d the Austrian method. He w r o t e a s h o r t note, and d e d i c a t e d it, as a New Year's gift, to Maximilian von Liechtenstein and H e l m h a r d Jhrger, t w o of his generous supporters. He n e x t tried to pub-

lish the l e a f l e t - - a t t h a t time, an onerous enterprise that required, for starters, buying the n e c e s s a r y r e a m s of paper. Actually, Kepler h a d even to c o n v i n c e a printer, first, to set up shop in Linz. The inevitable delays, which t o o k a l m o s t two years, offered him the o p p o r t u n i t y to e x t e n d his results considerably. His Nova stereometria dol i o r u m v i n a r i o r u m g r e w to a fullfledged book. The first p a r t d e a l s with

9

Title page of the Nova Stereometria. When well-meaning experts told Kepler that a mathematical 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,

Posterity did n o t r e c o r d w h a t Susanne m a d e out of this. Kepler, w h o s e father h a d b e e n an i n n k e e p e r w h e n not a b r o a d as a s o l d i e r of fortune, m u s t have b e e n on familiar t e r m s with w i n e - c a s k s of all shapes.

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 integral text of his "Dream" during his lifetime, but it raised rumors of black magic which surfaced during the nearly fatal witchcraft trial that his mother had to undergo in her last years9 Kepler seems to have been the first to see science as the cumulative effort of successive generations.


49

c u b a t u r e s in general, and in p a r t i c u l a r with the volumes of solids o f revolution. The s e c o n d p a r t d e a l s with barrels. F o r Kepler, these w e r e s o m e t i m e s cylinders, s o m e t i m e s t h e y c o n s i s t e d of t w o t r u n k s of a cone, a n d s o m e t i m e s t h e y w e r e w h a t he t e r m e d "lemons" ( o b t a i n e d by rotating a s e m i c i r c l e ' s arc a r o u n d its chord) w h o s e t o p a n d bott o m h a d b e e n sliced off. The third part o f his b o o k dealt with p r a c t i c a l probl e m s in measuring the c o n t e n t of totally o r partly filled casks. Kepler tried to avoid all algebra, and w r o t e in the style of Greek geometers.

But the content of his b o o k was not at all classical. In a remarkable display o f intuition, he anticipated parts o f calculus, arguing about infmities with a nonchalance quite foreign to the rigor of the exhaustion m e t h o d of Archimedes (who is invoked a great deal). F o r instance, Kepler c o n s i d e r s the a r e a o f the circle as being m a d e up of infinitely m a n y triangles having one v e r t e x in the center, and the o p p o s i t e base, r e d u c e d to a point, on the circumference. If the circle rolls along a line for one full turn, the b a s e l i n e s o f the triangles c o v e r 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 number 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.

50


and the circle's c e n t e r for a vertex, h a s the s a m e a r e a as t h e circle. The s a m e w o r k s for the full sphere: it is m a d e up of infinitely m a n y p y r a m i d s w h o s e vertices m e e t in the center; their b a s e s reduce to p o i n t s on t h e surface of t h e sphere. In a n o t h e r vein, since a t o r u s is o b t a i n e d b y rotating a circle a r o u n d a line that lies in the circle's plane (but does not t o u c h the circle), its v o l u m e is the p r o d u c t o f t h e a r e a of the circle times the c i r c u m f e r e n c e d e s c r i b e d by rotating its c e n t e r a r o u n d that axis. Indeed, the t o m s is m a d e up of infinitely m a n y thin discs, w h o s e v o l u m e s have to be a d d e d up. Kepler admits that since such a disc is m o r e like a wedge, he m a k e s an e r r o r in assuming that it has uniform thickness; two errors, actually, since the o u t e r part of the w e d g e is thicker, and the inner part thinner. But these errors cancel each other. The arguments run fast a n d loose, and a few of the results are wrong. But they c a m e d e c a d e s before B o n a v e n t u r a Cavalieri, Rend Descartes, a n d Pierre de Fermat, and they display in their blind groping t o w a r d calculus an u n c a n n y sense of direction. Kepler m a y well be the forem o s t e x a m p l e of w h a t Arthur Koestler t e r m e d a scientific "sleep-walker." The Nova Stereometria's main result c o n s i s t e d in finding, among all cylinders i n s c r i b e d in a sphere, t h o s e with the m a x i m a l v o l u m e (today an easy e x e r c i s e for first-year students). This implied that a m o n g all cylinders having the s a m e "measure" given b y the rod-length, t h o s e have m a x i m a l volume w h o s e height is equal to the dia m e t e r of the b o t t o m multiplied by square r o o t of two. Kepler a d d e d judiciously that his result was still app r o x i m a t e l y valid for barrels of close to cylindrical shape: indeed, "whenever there is a transition from smaller to larger and b a c k to smaller again, the differences are always insensible, to a deg e e . " This anticipates an argtnnent explicitly m a d e only d e c a d e s later by Fermat: close to a maximum, changes are small; i.e., optima 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, Austrian barrels h a d (and still have) a height that is equal to

drinkable stuff m a y b e a r o u n d in copious quantities, Et cum p o c u l a mille mensi erimus, C o n t u r b a b i m u s illa, ne sciamus." Some d e d i c a t e d t e a c h e r s tried for five years to t e a c h m e s o m e Latin, but I cannot help you w i t h the translation. It's a b o u t wine a n d science, though. EDITOR'S NOTE: A b o u t wine and ignorance, rather! A s c h o l a r informs us that the two lines, a learned allusion to Catullus's p o e m "Vivamus, m e a Lesbia," mean, "and if w e have m e a s u r e d each other a t h o u s a n d vessels, w e will confuse them, in o r d e r not to know." REFERENCES The best figure of all. Wine barrels in the rundown entrance of the Keplerhof Inn. In

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.

T h e r e are several g o o d Kepler sites on the net, for starters s e e www. kepler, arc. nasa. gov/j ohannes, hmtl www.es.rice.edu/ES/humsoc/Galilieo/ Files/kepler.html www.roups.dcs.st-and.ac.uk/ history/Mathematics/Kepler.html Institut for Mathematik Universit~t Wien Strudlhofgasse 4 1090 Vienna Austria e-mail: [email protected]

German textbooks, Kepler's name is associated with the so-called barrel rule of numerical integration (Simpson's 1/3 rule). In spite of Kepler's praise of Austrian barrels, the district deputies of Upper Austria decided in 1616, after a formal scrutiny of all his publications, to dispense with his services. However, influential friends made sure that this decision was never put into effect.

three t i m e s the radius of their bottom. The fact that 1.41422 is close to 1.50000 sufficed to p e r s u a d e Kepler t h a t Austrian b a r r e l s "had the b e s t figure o f all" (figurae omnium optissimae): in fact, he i n c l u d e d this p r o u d claim in the full title, w h i c h covers half of the frontispice of his book. Kepler g o e s on to ask, "Who will deny t h a t n a t u r e can t e a c h g e o m e t r y to h u m a n s t h r o u g h a vague feeling for form, w i t h o u t any r e c o u r s e to rational arguments?" He toys with the possibility that o n c e u p o n a time, s o m e pree m i n e n t g e o m e t e r could have t a u g h t the rule to Austrian barrel-makers; b u t then he d i s c a r d s it, with the a r g u m e n t that in this case, o t h e r wine-growing countries w o u l d also have a d o p t e d the s a m e rule. Kepler ends his t r e a t i s e with a h e a r t y p r a y e r that "our spiritual a n d material g o o d s m a y be preserved, a n d


51

lllli[:-IIJ~lEiii[~inl'~-Ii[,-l':llil,l!ldlL-tlil

The Bolzano House in

Prague P. Maritz

5 ~)

Dirk Huylebrouck,

B

Editor

ernardus Placidus Johann Nepomuk Bolzano was born in Prague, Bohemia (now part of the Czech Republic), on 5 O c t o b e r 1781. His h o u s e o f birth d o e s n o t exist any more, b u t it w a s on the site o f the p r e s e n t Maria Square (Marihnsk~ nfim~st0 in the Old Town. His mother, Caecilia Maurer, w a s a d a u g h t e r o f a h a r d w a r e t r a d e s m a n in Prague; at the age of twenty-two she m a r r i e d the e l d e r Bernard Bolzano, a n Italian i m m i g r a n t who e a r n e d a m o d est living as an art dealer. Both p a r e n t s w e r e p i o u s Christians. Bernardus was the fourth of twelve children, t e n of w h o m died b e f o r e reaching adulthood. He grew up with a high m o r a l c o d e and a belief in holding to his principles. It w a s this b a c k g r o u n d that a t t r a c t e d him to the c h u r c h and the p r i e s t l y life. F r o m 1791 to 1796 he w a s a pupil in the Piarist Gymnasium, a n d in 1796 he e n t e r e d the philos o p h i c a l faculty at Charles University ( e s t a b l i s h e d b y Charles IV [ 1315-1378], Holy R o m a n E m p e r o r and B o h e m i a n King in 1348), w h e r e he f o l l o w e d c o u r s e s in philosophy, physics, a n d m a t h e m a t i c s . Bolzano's i n t e r e s t in m a t h e m a t i c s was stimulated b y reading A. G. Kfistner's Anfangsgri~nde der Mathematik, mainly b e c a u s e Kfistner t o o k c a r e to prove s t a t e m e n t s that w e r e c o m m o n l y u n d e r s t o o d as e v i d e n t in o r d e r to m a k e clear the a s s u m p t i o n s on w h i c h t h e y d e p e n d e d [12, p. 273]. After having finished his studies in phil o s o p h y in 1800, Bolzano e n t e r e d the theological faculty and was o r d a i n e d a Catholic p r i e s t in 1804 [4]. In 1805 E m p e r o r Franz I o f the Austro-Hungarian Empire, of which B o h e m i a w a s then a part, d e c i d e d that a chair in the philosophy o f religion would be established at each university. The r e a s o n s for this were mainly political. The empire was comprised of m a n y different ethnic groups that w e r e p r o n e to nationalistic movements for independence. The e m p e r o r feared the fruits of Enlightenment e m b o d i e d in the F r e n c h Revolution. The authorities con-

THE MATHEMATICAL INTELUGENCER 9 200t SPRINGER-VERLAG NEW YORK

I

sidered the Catholic Church to be conservative and h o p e d it would control the liberal thinldng of the time in Bohemia. Bolzano was called to the new chair at Charles University in 1805 [12, p. 273]. Bolzano, t h o u g h a priest, spiritually belonged to the Enlightenment. He w a s a "free thinker"; his a p p o i n t m e n t w a s received in Vienna with suspicion a n d was not a p p r o v e d until 1807. F o r the next 14 y e a r s Bolzano taught at the university, lecturing mainly on ethics, social questions, a n d the links b e t w e e n m a t h e m a t i c s a n d p h i l o s o p h y [4]. In 1815 he b e c a m e a m e m b e r of the K6niglichen B 6 h m i s c h e n Gesellschaft der Wissenschaften and, in 1818, Dean of the p h i l o s o p h i c a l faculty. However, the Austro-Hungarian authorities bec a m e d i s p l e a s e d with his liberal views. On 24 D e c e m b e r 1819, he was susp e n d e d from his professorship, forbidden to publish, a n d p u t u n d e r p o l i c e supervision. Bolzano refused to b a c k down, and in 1825 t h e action c a m e to an end through t h e intervention o f the influential nationalist l e a d e r J o s e f Dobrovsk~ (1753-1829). The latter h a d b e e n e d u c a t e d for the R o m a n Catholic p r i e s t h o o d and d e v o t e d himself to scholarship after t h e 1773 dissolution of the Jesuit Order. He was an important Enlightenment figure, and his textual criticism of the Bible led him to study Old Church Slavonic and subsequently the Slavic languages as a group. F r o m 1823 on, Bolzano s p e n t summers on the estate o f his friend J. Hoffmann, n e a r the village of T~chobuz in Southern Bohemia. He lived there permanently from 1831 until the death o f Mrs. Hoffmann in 1842. He then ret u r n e d to Prague w h e r e he c o n t i n u e d his m a t h e m a t i c a l a n d philosophical studies until his d e a t h on 18 December, 1848. A small pension, and the generosity of Count Leo Thun-Hohenstein (1811-1888, the B o h e m i a n a r i s t o c r a t and later Austrian s t a t e s m a n ) relieved him of all m o n e t a r y c a r e [7]. In Prague, Bolzano and his b r o t h e r s t a y e d in the h o u s e that had b e l o n g e d

Figure 1. The house at 25 Celetnd Street, showing the Bolzano plaque centered over the keystone arch.

to their p a r e n t s at 25 Celetmi Street (Celetn~ ulice 25) n e a r Old T o w n Square (StaromSstsk~ n ~ n ~ s t 0 . The p h o t o g r a p h s (Figs. 1, 2, t a k e n by the author) are of the h o u s e and its Bolzano plaque. This h o u s e is o w n e d by the City o f Prague, and all the s p a c e in the h o u s e is filled with apartments. A r o u n d the turn o f the n i n e t e e n t h century, E u r o p e a n m a t h e m a t i c i a n s w e r e mainly c o n c e r n e d with the status of Euclid's parallel p o s t u l a t e a n d with the p r o b l e m o f providing a solid foundation for m a t h e m a t i c a l analysis. Bolzano t r i e d his h a n d at b o t h p r o b lems. In 1804 Bolzano p u b l i s h e d a t h e o r y of parallel lines, which a n t i c i p a t e d Adrien Legendre's well-known theory. It is c o m m o n l y a c c e p t e d that Bolzano in his m a n u s c r i p t Anti-Euklid, w a s the first to state the t h e o r e m ( n o w k n o w n as the "Jordan Curve Theorem") that a simple c l o s e d curve divides the p l a n e into t w o p a r t s [12, p. 274]. The introduction of infmitesimals b y Isaac Newton and Gottfried Leibniz in the seventeenth century had m e t with violent resistance from philosophers and mathematicians. To o v e r c o m e the difficulties p r e s e n t e d by infmitesimals, Joseph-Louis Lagrange p r o p o s e d to b a s e analysis on the existence o f B r o o k Taylor's e x p a n s i o n for functions, while J e a n d ' A l e m b e r t p r o p o s e d to found differential calculus on the notion o f limit.

Among the first to d o u b t the rigor of Lagrange's e x p o s i t i o n of the calculus were Abel Btirja (1752-1816) of Berlin, the Poles J. M. Ho~n6-Wron3(si (17761853) and J. B. Sniadecki (1756-1830), and Bolzano [3, p. 258], who d e v o t e d his m a n u s c r i p t R e i n analytischer Bewe/s (1817) to a p r o o f of the "Bolzano I n t e r m e d i a t e Value Theorem." Bolzano argues that a s o u n d p r o o f of this theorem requires a s o u n d definition of continuity. His definition is the first that does not involve infinitesimals. The definition as it w a s f o r m u l a t e d in

Volume I of his m a n u s c r i p t Functionenlehre reads: If F ( x + hx) - F(x) in a b s o l u t e value b e c o m e s less t h a n an arb i t r a r y given fraction 1/N, if o n e takes Ax small enough, and r e m a i n s so the s m a l l e r one t a k e s Ax, the function F(x) is said to be c o n t i n u o u s in x [12, p. 275]. Bolzano's definition of continuity was r e p l a c e d in 1821 b y Augustin-Louis C a u c h y ' s elegant a n d generally acc e p t e d definition. Also in his p r o o f of the "Intermediate Value Theorem," Bolzano uses a l e m m a 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 n o n e m p t y set. His 1817 manuscript 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 a w a r e of the difficulties involved. A fairly c o m p l e t e t h e o r y of real functions is c o n t a i n e d in Bolzano's Functionenlehre, including m a n y of the fundamental results that w e r e rediscovered in the s e c o n d half of the nineteenth century through the work of Karl Weierstrass (1815-1897) and others. Bolzano proved in this manuscript that a function that is u n b o u n d e d on a closed interval [a, b] cannot be continuous on [a, b] [12, p. 275]. In proving this, Bolzano used the so-called BolzanoWeierstrass Theorem, that a b o u n d e d infinite 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.


53

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 constructs 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 appointed 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 grandson, 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 hospitable 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 authorities of Austria if he communicated with the compromised Bolzano. An autobi-

54


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 unpublished 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 accompanied 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 i e r u n g , . . . ) , 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. Freudenthal and H. Sinaceur clarified the differences 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 coincidental. 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 investiture 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 exiled 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 rediscovered 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 Cemetery (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 Figure 3 was issued by Czechoslovakia in

Karel Segeth, Director of the Mathematical Institute of the Academy of Science in Prague, and the assistance received from Professor Marcel Wild, Mathematics Department, University of Stellenbosch.

[7] Leimkuhler, M. Bernard Bolzano, The Catholic Encyclopedia.

Volume II. Robert

Appleton Company, 1907. Transcribed by Thomas J. Bress. Retrieved June 3, 2000 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

REFERENCES [I] Belhoste, B. Augustin-Louis Cauchy. A Biography. Springer-Verlag, New York,

d’&forie d e s S c i e n c e s 2 6

1991. [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

Edition. Chelsea Publishing Company, New York, 1980.

commemorating

the

mathematician’s

birth.

1981 to commemorate the 200th anniversary of his birth [lo, p. 621, SG 25681.

97-l 12.

Volume 1, 2000

Edition. Foreign Countries, A-J. Stanley Gibbons Ltd, London, 1999. [ll] Struik, D.J. and R. Struik. Cauchy and

[4] Golba, P. Bolzano, Bernard (1781-1848). R e t r i e v e d J u n e 3,200O

(1973),

[I 0] Stanley Gibbons Simplified Cafalogue, Stamps of the World.

Wiley and Sons, Inc., New York, 1968. [3] Cajori, F. A Histo~ of Mathematics. Third

Figure 3. The 1981 Czechoslovakian stamp

des Sciences 15 (1962) 163-l 64. [9] Sinaceur, H. Cauchy et Bolzano, Revue

from the World Wide

Web: http://www.shu.edu/html/teaching/

Bolzano in Prague, lsis 11 (1928) 364366. [12] Van Rootselaar, B. Bolzano, Bernard. In: Dictionary of

math/reals/history/bolzano.html

Scientific Biography. Volume

[5] Kline, M. Mathematics. The Loss of

II, pp. 273-279. C. C. Gillispie (Ed.).

Certainty. Oxford University Press, New

Charles Scribners Sons, New York, 1970.

York, 1980. [6] Kudryavtsev, L. D. Bolzano-Weierstrass

Department of Mathematics,

Selection Principle: Bolzano-Weierstrass

University of Stellenbosch

Theorem. In: Encyclopeadia

Private Bag X 1

of Mafhe-

ACKNOWLEDGMENTS

mafics.

The author gratefully acknowledges the information supplied by Professor

Hazewinkel (Ed.). Reidel, Kluwer Aca-

7602 South Africa

demic Publishers, Dordrecht, 1988.

e - m a i l : [email protected]

Volume 1,

pp.

419-420.

M.

Matieland

Ild,,[~V~l~-'lil|[~l=,lr~-ll|[*"~-llll[*]!|d~ll

The Kentucky Vietnam Veterans Memorial John W. Dawson, Jr.

Dirk Huylebrouck,

Editor

I

rankfort, Kentucky, lies j u s t n o r t h of Interstate highway 64, about midway b e t w e e n Lexington and Louisville. Often b y p a s s e d by motorists hastening b e t w e e n those metropolises, it should be on the itinerary of visiting mathematicians, for it is the site of a remarkable public monument w h o s e innovative design, based on horological principles k n o w n from antiquity, evokes a powerful emotional response. The Kentucky Vietnam Veterans Memorial is a massive horizontal sundial I c o n s t r u c t e d on a hilltop overlooking the state capitol (Figure 1). Its stainless-steel gnomon, over 5.3 meters in length, rises a little m o r e t h a n 4.3 m a b o v e a rectangular plaza, s o m e 21.6 m wide and 27.1 m long, that is c o m p o s e d o f 327 b l o c k s o f granite weighing a b o u t 195 metric t o n s (Figure 2). As in all such dials, the g n o m o n is aligned with the celestial pole, so t h a t the sun's r a y s are p e r p e n d i c u l a r to it at the equinoxes. On those days, the tip of the g n o m o n ' s s h a d o w t r a c e s a straight line a c r o s s the plaza. On all

other days, its p a t h is a hyperbolic arc. But this s h a d o w is n o t e m p l o y e d to indicate the h o u r o f the day. Its p u r p o s e is rather to c o m m e m o r a t e the 1065 Kentuckians w h o w e r e lost during t h e Vietnam conflict. Inscribed along radial lines extending o u t w a r d from in front of the gnom o n are the n a m e s of all those killed in action. They are p o s i t i o n e d in s u c h a w a y that the s h a d o w cast by the tip o f the g n o m o n f a 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 i n d i v i d u a l ' s death. The n a m e s o f those missing in action or held as p r i s o n e r s of w a r are also i n s c r i b e d on the plaza, in the region south of the g n o m o n w h e r e t h e s h a d o w n e v e r falls. 2 Figure 3 s h o w s the layout of one o f the granite slabs, w h i c h are a r r a n g e d in s e c t o r s c o r r e s p o n d i n g to the y e a r s of the conflict, from 1962 through 1975. The s e c t o r s a r e i n t e r s e c t e d b y curves marking the equinoctial s h a d o w line and the solstitial s h a d o w trajectories, and are s e p a r a t e d b y w a l k w a y s analogous to the h o u r lines on a traditional sundial (Figure 4). The c o n c e n t r a t i o n

F

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.

U.S. 60 I

dl 676 EAST-WEST CONNECTOR

.,,j

Library& Archives

TO LOUISVILLE

Figure1. Locationof the memorial. 56

THE MATHEMATICALINTELLIGENCER9 2001 SPRINGERVERLAGNEWYORK

~

"~'~ i-64

State Capitol

I'~ETN V

9 AM VETERANS MEMORIAL

O~ O

TO LEXINGTON

of n a m e s in certain of the s e c t o r s vividly illustrates the m a j o r engagem e n t s of the war, especially the Tet offensive o f 1968. The p o s i t i o n r e a c h e d by the s h a d o w at 11:11 a.m. on Veterans' Day, Novemb e r 11, is i n d i c a t e d b y a special m a r k e r that the s h a d o w t r a v e r s e s during the minute of silence o b s e r v e d at that time, a n d a r o u n d the b a s e of the gnom o n the w o r d s o f E c c l e s i a s t e s are engraved: "For everything there is a seas o n . . , a time to b e born, and a time to d i e . . , a time to kill, and a time to h e a l . . , a time for war, and a time for peace." The p l a n for the m e m o r i a l w a s conceived b y Lexington architect Helm Roberts, w h o s u b m i t t e d his i d e a to a design c o m p e t i t i o n s p o n s o r e d by the Kentucky Vietnam Veterans Memorial Fund. The j u d g e s u n a n i m o u s l y selected his design a s t h a t which b e s t exemplified the criteria established for

the monument, a m o n g w h i c h were that it " n o t . . . imitate o t h e r m o n u m e n t s " a n d that it "evoke an e m o t i o n a l rem e m b r a n c e whilst being aesthetically authentic." They e x p r e s s e d s o m e doubts, however, regarding its feasibility. To determine the precise azimuth and elevation of the sun for various dates and times the project employed the astronomical computer program ACEcalc (Astrosoft Computerized Ephemeris). Drawings of the individual slabs w e r e p r o d u c e d using AutoCAD, a n d the acc u r a c y of the calculations was so good that the names could b e engraved in the stone before the slabs were set in place. Only the equinoctial and solstitial lines w e r e incised i n s i t u . 3 C o m p l e t e d in 1988 at a c o s t of over one million dollars, the m e m o r i a l was d e d i c a t e d on N o v e m b e r 12 of that y e a r a n d has b e e n o p e n to visitors a r o u n d t h e c l o c k every d a y since.

Figure 2. Gnomon and plaza: view from the southeast.

3For further details concerning the design and construction of the monument, see [1].

Figure 3. Slab detail (AutoCAD drawing. Courtesy of Helm Roberts)


57

Figure 4. Overhead plan of the memorial, with superimposed shadow trajectories. Radial bars indicate placement of names. (Courtesy of Helm Roberts.)

A r m c h a i r travelers c a n view pictures o f the memorial, a s well as architectural drawings o f it, online at http://www.helmr.com/ky.htm; o t h e r p h o t o s m a y be v i e w e d at http://grunt. s p a c e . s w r i . e d u / k y m e m . h t m . Information and a d o w n l o a d a b l e b r o c h u r e are a v a i l a b l e at h t t p : / / w w w . s t a t e . k y . us/agencies/khs/museums/military/ vietnam.


Acknowledgments

I a m indebted to Mr. J a m e s Halvatgis, s u p e r i n t e n d e n t of the memorial, a n d to Mr. Helm Roberts, its designer, for inf o r m a t i o n a n d illustrations. 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 e-mail: [email protected]

m'.ir:l~-lw'-.,[.~- J e r e m y Gray, Editor

Symbols and Suggestions: Communication of Mathematics in Print* Jeremy Gray

I

a t h e m a t i c i a n s are creators (or discoverers, if you wish). They are also c o m m u n i c a t o r s , and receivers. In all t h e s e roles t h e y are p e o p l e with s o p h i s t i c a t e d w a y s o f assembling and re-assembling ideas. F o r several centuries, the m o s t effective communication m e d i u m has b e e n print. Historians of m a t h e m a t i c s s o m e t i m e s u s e d to claim that w i t h o u t this or that p i e c e of notation s o m e i d e a w a s unthinkable. This d o e s not s e e m entirely satisfactory, especially to m a t h e m a t i c i a n s w h o k n o w very well t h a t n e w s y m b o l s are easy to devise, b u t t h e claim has s o m e merit. It is m o r e p r o f i t a b l e to argue that m a t h e m a t i c a l notation, like any language, is torn b e t w e e n syntax and semantics and often p r o c e e d s by relying on tacit u n d e r s t a n d i n g s a b o u t meanings. Two e x a m p l e s will be c o n s i d e r e d here, one d r a w i n g on r e c e n t w o r k a b o u t the i m p l i c a t i o n s o f ratio and equality, and one on e x a m p l e s of logical notation.

M

Equality and Proportion in Geometry and Algebra Franqois Vi~te, w h o w a s the leading F r e n c h m a t h e m a t i c i a n of the turn of the s e v e n t e e n t h century, w r o t e his alg e b r a with a latent a p p e a l to geometry, which survives in t h e balancing of the terms, so that e a c h m o n o m i a l in an equation has the s a m e weight o r dimension. This is to p r e v e n t the nonsense of adding a side to an area. There is also a c o n v e n t i o n a b o u t w h a t letters stand for. It's not the m o d e m one: here vowels stand for the unknowns. This could catch on; it b a r e l y needs to be said and s e e m s so natural that it doesn't travel with an explanation. Such tacit understanding is an i m p o r t a n t p a r t o f printed c o m m u n i c a t i o n , and it implies the existence of a c o m m u n i t y in on the secret. The c o n v e n t i o n t h a t did catch on is, of course, due to Descartes. He intro-

d u c e d l o w e r c a s e letters, r a t h e r than capitals, with x and y for unknowns. He argued his w a y a r o u n d t h e dimensionality convention b y showing how a length times a length can b e thought of, n o t as an area, but as a n o t h e r length. B e c a u s e all his m a g n i t u d e s have the s a m e dimension, the d i m e n s i o n conc e p t d o e s no w o r k a n d can b e forgotten, which is w h a t h a p p e n e d . F o r Descartes, p r o b l e m s s t a r t e d in geometry, w e r e translated into algebra, w h e r e t h e y w e r e solved, a n d then r e t u r n e d to geometry. 1 So a curve for him was a truly geometric object. This gave him a p r o b l e m : What e x p r e s s i o n s truly define curves? D e s c a r t e s ' s criterion for a curve to be truly g e o m e t r i c was (roughly, b e c a u s e D e s c a r t e s could not b e m o r e precise) that it be an algebraic curve defined by a p o l y n o m i a l equation in t w o variables. Curves o f double motion, as he called them, including r o u l e t t e s like the cycloid, w e r e not defined exactly enough. In the 1660s, D e s c a r t e s ' s greatest f o l l o w e r was the y o u n g I s a a c Newton, b u t b y the 1680s the middle-aged N e w t o n had t u r n e d against the great F r e n c h m a n on m a t t e r s of physics, theology, and m a t h e m a t i c s . He did not s h a r e the implied p r e f e r e n c e for algeb r a o v e r geometry, o r his criterion for simplicity (roughly: d e g r e e o f the defming equation). What, N e w t o n protested, w a s m o r e simple than t h e cycloid? And as he turned b a c k to the o l d Greek m e t h o d s of geometry, he f o u n d that g e o m e t r y was the n a t u r a l language for his physics. Here is one of the k e y p a s s a g e s in N e w t o n ' s Principia, w h e r e he add r e s s e s J o h a n n e s Kepler's law that p l a n e t s s w e e p out equal a r e a s in equal times. Crucially, the p r o g r e s s in the dia g r a m is to be e n a c t e d b y the reader, as the a c c o m p a n y i n g t e x t m a k e s clear. At e a c h m o m e n t the a r e a actually s w e p t out is specified, s h o w n to be

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

*Based on a paper presented to the Math ML Conference, Urbana-Champaign, October 19-21, 2000. 1As Bos carefully describes (Bos 1981).

9 2001 SPRINGER-VERLAGNEWYORK,VOLUME23, NUMBER2, 2001 59

Soit p a r e x e m p l e A B l'vnit6, & qu'il faille multiplier BD p a r BC, ie n'ay qu'a ioindre les p o i n s A&C, puistirer DE parallele a CA, & BE eft le p r o d u i t de cete Multiplication. Oubien s'il faut diufer BE p a r BD, a y a n t ioint les poins E&D, ie tire AC p a r a l l e l e a DE, & BC eft le p r o d u i t de cete dinifion. Figure 1. Descartes's proof that multiplying a line by a line gives a line, from his La Geometrie.

physical ones (position, velocity, force), which evolve in interrelated w a y s as time goes b y - - s a y , as a p l a n e t m o v e s r o u n d the Sun, n o w close in, n o w further away. They w e r e naturally e x p r e s s e d m a t h e m a t i c a l l y as geometric quantities, although Newton was brilliant in his blending of g e o m e t r y and dynamics. But thinking and writing ratios b e t w e e n variables, as Newton did, is n o t the s a m e as thinking and writing a b o u t functions, and this is a central p o i n t in any study of the relation b e t w e e n notation, ideas, a n d understanding.

equal to another, and so t h e y are all the same. Finally, the r e a d e r is to imagine t h a t the impulsive action o f gravity as it is d e p i c t e d here (happening in disc r e t e intervals of time) b e c o m e s continuous, b y shrinking the time intervals. This is seductive. N e w t o n typically wrote his conclusions in the language of ratio and proportion, but with a significant modification. His immediate p r e d e c e s s o r and to s o m e extent model, Christian Huygens, did not appreciate this change, and w h e n he read the Principia he tried w h e n e v e r possible to put the hill apparatus of ratios back. 2 Huygens's geometrical language cannot treat Galileo's law of falling bodies, that the distance a b o d y has travelled is proportional to the square of the elapsed time, s ~ t 2, except as a comparison of two ratios: to be written as sl : s2 :: t 2 : t22. It s e e m s that Newton m a d e the move to thinking of

statements of proportion like statements of equality only during the writing of the Principia. It has the signal benefit of involving only two terms, not four, and so allowing terms to b e c o m e more obviously variables. Although N e w t o n ' s language is det e r m i n e d l y geometric, he did n o t use the full p a r a p h e r n a l i a of p r o p o r t i o n theory, with its b u r d e n s o m e symbolism and its restriction on w h a t can b e said, or rather, written. His guide h e r e w a s Descartes, although N e w t o n did not go the final stage and i n t r o d u c e coordinates. F o r Newton, the f u n d a m e n tal o b j e c t s w e r e quantities, chiefly

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

THEMATHEMATICALINTELUGENCER

Newton, Calculus, and Mechanics We have arrived at the calculus. F o r Newton in the 1680s this was a set of techniques applied to curves. He did not, as a discredited r u m o u r has it, write his Principia using the calculus and then rewrite it for p o p u l a r consumption. He did occasionally resort to the method of infinite series w h e n he had to. But the Principia is a geometry book, marvellously adapted to mechanics. What happ e n e d in the years after Newton was that m o r e and more mathematicians, especiaUy Continental ones who had rela-

b o t h this and that. Parallel to classes are statements: we can a s s e r t this and a s s e r t that, and a s s e r t either this or that, o r b o t h this and that. G e o r g e Boole was an o b j e c t man: I p i c k at r a n d o m this expression, x(1 - y) + y(1 - x),

tively easy access to the Leibnizian calculus, found that they wanted to use it to understand, then to replace, and then to extend Newton's geometric arguments. But what does the calculus apply to? Most efficiently, to formal expressions, fimctions (however they are defmed). It says things like this:

finally, b e c a u s e of its efficacy in the calculus, it replaced the old ratio idea alm o s t entirely. As it did so the quantities being equated were allowed to b e c o m e variables, and gradually, as that happened, the equations w e r e taken to exp r e s s functional dependence.

Logic, Set Theory, and Language if y = f ( x ) t h e n

dy = ~ dx.

So w h e n L e o n h a r d Euler set a b o u t rewriting all o f p u r e and applied mathematics in the mid-eighteenth century, he cast it all in the language o f functions r a t h e r than of g e o m e t r y a n d proportions. In short, the i d e a that "this ratio of variables is equal to that" is not the s a m e as "these variables are functions of those." Equality has its origins in simple p r o b l e m s that can be written a s - note the w o r d - - e q u a t i o n s . It p u s h e d its way into geometry, and into mechanics;

The delightfully v e x e d history of logic offers a different d o m a i n of mathematics, one with a rich, ongoing history, that might suggest the sorts of p r o b l e m s future w r i t e r s of mathematics m a y encounter, indicative of t h o s e g e n e r a t e d b y o t h e r p r o f e s s i o n a l groups that use m a t h e m a t i c s b u t do not consider themselves mathematicians. There a r e two parallel families of ideas in early m o d e r n logic. T h e r e is the i d e a that among a class o f o b j e c t s there are e l e m e n t s with this o r that property, a n d we can form the class of all elem e n t s that are either this o r that, or are

w h i c h he glossed as things that are x ' s b u t n o t y ' s or y ' s but n o t x's. 4 The exp r e s s i o n 1 - y stands for n o t y's, bec a u s e Boole had a universal set, which he d e n o t e d 1, and c o m p l e m e n t s he den o t e d b y analogy with subtraction. Similarly for 1 - x. The p r o d u c t o r juxtaposition he interpreted as w e do intersection, so x(1 - y) is all objects (in the universal set) that are both x ' s and not y's. The p r o b l e m c o m e s with the "or," as w e shall see shortly, b e c a u s e Boole's was the disjunctive "or"; for him a o r b m e a n t either a or b but not both. Boole was insistent that his Laws of Thought obey, so far as possible, the rules o f arithmetic. In his b o o k The Laws of Thought (1853) he p o i n t e d out that t h e symbolic rules he found to apply w e r e very nearly t h o s e that also applied to arithmetic. The only p o i n t at w h i c h the laws of logic a n d of arithm e t i c differed, he felt, w a s the g e r m o r seminal principle of logic: the a s s e r t i o n that x 2 -- x. He set great store b y the a l g e b r a i c nature of his equations and his ability at solving them. F o r example (p. 105), he e x p r e s s e d the statem e n t "No m e n are perfect" as y = v(1 - x), w h e r e y r e p r e s e n t s men, v is an indefinite class, and x p e r f e c t beings. By eliminating v he t u r n e d this into 0

1-x=y+-~(1-y), "Imperfect beings are all m e n with an indefinite r e m a i n d e r o f beings, which are n o t men." Note the n e w s y m b o l 0 0 w h i c h he i n t r o d u c e d to allow for the s y m b o l i c e x p r e s s i o n o f the i d e a of "some." In Boole's day, a n d for s o m e y e a r s before, m a n y logicians felt that if

3See Gray [1994]. 4Boole, p.105,


61

t h e r e was any w a y in w h i c h the study o f logic could s u r p a s s the state to w h i c h Aristotle h a d b r o u g h t it, it w o u l d be in "quantifying the predicate," a reso n a n t p h r a s e that m e a n t being able to talk a b o u t "all," "some," a n d "none." Boole's exclusive "or" c a u s e d problems with the algebra. Not many years later, William Stanley Jevons replaced it with the inclusive "or," thus abolishing the p r o b l e m of understanding expressions of the form x + y w h e n x and y have elements in c o m m o n (particularly a c u t e in the case w h e n x + x is to be understood). An eloquent example of h o w much he simplified the p r o c e s s of elimination was his d e m o n s t r a t i o n that it could be mechanised. After some ten y e a r s of work, he exhibited a logical machine at the Royal Society of London in 1870 (a description was also published in his The Principles of Science, 1874). It b e c a m e k n o w n as Jevons's logical piano.

Peirce and the Logic of Relations A little later, in the late 1870s and early 1880s, leadership in logic p a s s e d to C. S. Peirce and the s t u d e n t s he briefly g a t h e r e d a r o u n d him at J o h n s Hopkins. P e i r c e began by e x t e n d i n g Boole's operations for classes to b i n a r y relations. He i n t r o d u c e d the relative p r o d u c t o f t w o relations, w h i c h c o n t a i n s existential s t a t e m e n t s implicitly. If I is a relation (Peirce s u g g e s t e d "lover of") and w is a class ("woman"), t h e n lw s t a n d s for "lover of woman." This m u s t be unp a c k e d . With Peirce, let l(i, 3) stand for "i is a lover of j " a n d w(?) for ':?' is a woman," then lw(i) m e a n s "i is a lover o f a woman," or, m o r e formally, "There is a w o m a n j such that i is a lover of j . " E x p o n e n t i a t i o n w o r k s like this: lW(i) m e a n s "i is a lover o f e v e r y woman," o r m o r e formally, "For e v e r y j such that j is a woman, i is a l o v e r o f j . " Peirce's group, in line with o t h e r authors, u s e d + to d e n o t e the inclusive "or." Signs were i n t r o d u c e d to assert existence, of which P e i r c e ' s w a s - - < . He wrote, G r i f f m - - < b r e a t h i n g fire "to m e a n that every griffm (if there be s u c h a creature) b r e a t h s fire; that is, no griffin not breathing fire exists," and Animal - - < Aquatic

62

THE MATHEMATICALrNTELLIGENCER

"to m e a n that s o m e animals are not aquatic, o r that a non-aquatic animal d o e s exist." In 1885, Peirce m a d e the b r e a k t h r o u g h to the productive i n t r o d u c t i o n o f quantifiers into logic. He u s e d the s y m b o l s E and H, s o m e t i m e s with subscripts, as he put it, "in o r d e r to m a k e the n o t a t i o n as iconical as possible, w e m a y use E for some, suggesting a sum, and II for all, suggesting a p r o d u c t . . . . If x is a simple relation, I]iHjxij m e a n s that every i is in this relation to a j , ZiIIjxij t h a t to e v e r y j s o m e i o r o t h e r is in this relation, ~,iEyxij that s o m e i is in this relation to s o m e j " (in Brady, p. 187). His reasoning was that if the xij w e r e either 0 or 1, and one w r o t e Z~ZF~J = 1 to indicate that the corres p o n d i n g s t a t e m e n t w a s true, t h e n the s t a t e m e n t "ZiZjxij = 1 w a s true precisely w h e n the s u m w a s n o t zero. A n a l o g o u s arguments dealt with the use o f the p r o d u c t symbol. He r e f e r r e d to his s y m b o l s as quantifiers, a n d the indices as pronouns, thus bringing out a linguistic analogy. Note that t h e quantifiers, like functions in m a t h e m a t i c s , m u s t h e r e b e r e a d from right to left. Peirce t h e n s h o w e d h o w to e x t e n d this analysis to deal with several r e l a t i o n s at o n c e a n d d r o p p e d the r e d u n d a n t exp r e s s i o n " = 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 b o o k he was writing before finishing Part II. In the event, the three published volumes of the Algebra der Logik (some of which were published posthumously) cover the calculus of classes ("the connection of ideas"), the calculus of propositions ("the connection and relation between judgements"), and the calculus of relatives. The fundamental relation in the first is that of inclusion, which SchrSder did not denote C , but

and which has t h e s e properties: 1. a C a (reflexivity) 2. If a C b, a n d b C c, then a C c (transivity) He defined equality this way: a = b if and only if a C b and b C a. A feature of his p r e s e n t a t i o n is its e m p h a s i s on duality, so that dual constructions are p r e s e n t e d in parallel columns (as w a s often the case in cont e m p o r a n e o u s projective g e o m e t r y books). The identically null or nothing, d e n o t e d 0, a n d the identically one, o r all, is i n t r o d u c e d as satisfying 0 C_ a and a C 1 for all d o m a i n s a, making t h e m initial and terminal obj e c t s in l a t e r terminology. What he called identical multiplication and identical addition he defined this way: for all d o m a i n s a, b, c: i f c C a a n d c C b then c C a b and if a C_ c and b C c then a + b C c. SchrSder t h e n e s t a b l i s h e d the distributive laws, m u c h as Peirce had done, without use o f t h e c o n c e p t of negation, and in the form ab + ac C a(b + c) and a(b + c) C a b + ac. He introduced models of s y s t e m s satisfying these a x i o m s to s h o w that the t w o distributive laws were i n d e p e n d e n t of each other. Finally, he defined the negation of a domain a to b e a d o m a i n a l such that

aal C_ 0 and 1 C a + al. Frege In the w o r k of Boole, Peirce, a n d SchrSder, m a t h e m a t i c a l notation w a s a d a p t e d to the n e e d s of the logician, in particular o f the inventive or creative logician. Notation w a s a d o p t e d with an eye to its m e a n i n g in the old domain; w h a t w a s w a n t e d w a s a calculus, and the n e w i d e a s t h a t w e r e e x p r e s s e d were partly mathematical, partly philosophical, p a r t l y linguistic, partly logical. More radical i d e a s a b o u t language w e r e held b y a m a n SchrSder could n o t abide (and w h o s u b j e c t e d s o m e o f his ideas to his c u s t o m a r y destructive criticism): Gottlob Frege. Frege w r o t e

B

for w h a t he c u m b e r s o m e l y e x p l a i n e d w a s the a s s e r t i o n ( m a r k e d b y the t h i c k vertical line) that the j u d g e m e n t h o l d s that it is n o t the case that A is d e n i e d and B holds. A little w o r k allows us to r e a d this a s "A or not B," or, if you prefer, "B implies A." Denying a j u d g e m e n t he w r o t e this way:

[

,

[ I

~

~]-----

A

I(

F (o)

f (Y,Z)

Figure 4. A Frogean diagram, from his Begriffsschrift nr. 87, p. 65. Matters r a p i d l y got complicated, because F r e g e ' s w h o l e aim was to produce p r o o f s b a s e d on logic without appeal to facts, a n d in p a r t i c u l a r to found our c o n c e p t o f n u m b e r on such proofs. He t h e r e f o r e set himself the t a s k of preventing "anything intuitive from penetrating h e r e unnoticed," and "in attempting to c o m p l y with this requirem e n t in the s t r i c t e s t p o s s i b l e way," he wrote, "I found the inadequacy of language to b e an obstacle." The s e a r c h

F(A)

m e a n t that for all a it is the c a s e t h a t isn't too difficult to see that

va

8 F (a)

f (x,a) 'y f (x~, yz)

F(a). It

'

(z)

a f(&a)

Since he w a s trained as a m a t h e m a t i cian and w o r k e d at Halle, w h e r e G e o r g Cantor was, Prege had no p r o b l e m s with function notation. So e x p r e s s i o n s like

!'

F

F()

m e a n s that F(a) is n o t the c a s e for all a, o r that t h e r e is s o m e a that d o e s n o t have p r o p e r t y F.

for precision led him to this ideograp h y (whence the title o f his book, Begriffsschrift, literally, i d e a script), w h i c h he also r e f e r r e d to as a formula language for p u r e thought. He comp a r e d it to the m i c r o s c o p e , an instrum e n t generally inferior to the h u m a n eye but for select p u r p o s e s a vast (and indispensable) i m p r o v e m e n t . Frege w a s optimistic t h a t his i d e o g r a p h y c o u l d be used to s t u d y the foundations o f the calculus and, with less work, of geometry. It was, in any case, he reckoned, an a d v a n c e in logic. As a language, it did n o t catch on. One has to rewrite it before one can understand it. But this e x a m p l e (Figure 5) from Noam Chomsky's Aspects should remind us of the complexities of natural (and indeed formal) languages. Peano and International Languages

But t h e r e is a n o t h e r a s p e c t of Frege's w o r k that I should like to mention. The p e r i o d a r o u n d 1900 w a s the h e y d a y

#-S-#

(3S) NP

I

N [ + A ~

Predicate-Phrase Aux Present

VP Copula

Predicate Compar

John

Adjective

A.

clever

Predicate-Phrase

NP

I N

VP

iux

I

Present Bill

Copula

Predicate I Adjective

7X

clever Figure 5. A Chomskyian diagram, from his Aspects, p. 178.


63

o f international l a n g u a g e s such as Esperanto. 5 Frege w a s n o t averse to them, at least for scientific purposes, a n d Schr6der was keen. He h o p e d to s h o w that Peirce h a d a l r e a d y defined e n o u g h s y m b o l s for all of mathematics. A n o t h e r enthusiast w a s Giuseppe Peano, to w h o m w e o w e further adv a n c e s in m a t h e m a t i c a l symbolism. He i n t r o d u c e d N for "and" a n d U for "or," 9 for "is" (as in a 9 K for a is a class). But his aim, realised in his j o u r n a l Rivista and his e n c y c l o p a e d i c Formulario, was to eliminate natural language in m a t h e m a t i c a l p a p e r s entirely in favour of a heavily s y m b o l i s e d pres e n t a t i o n (often a c c o m p a n i e d in practise by translations into Italian, or perh a p s P e a n o ' s f a v o u r e d international language, Latino sine F l e x i o n e o r Interlingua). These a t t e m p t s failed. 6 They remind us that m a t h e m a t i c a l s y m b o l i s m cannot always b e held a p a r t f r o m other symbolisms, including t h o s e of languages themselves. The reasons for this failure are interesting. Prior to Esperanto, there had b e e n an international language called Volaptik. It had not lasted very long before it collapsed into schism, split bet w e e n users who w i s h e d to allow the language to evolve and simplify and others, grouped around its inventor, who w i s h e d it to stay as it was. The failure weighed heavily on the m i n d s of the Esperantists, who d e c i d e d instead that there should be no h e a d office or central committee 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 betrayal 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 r o b l e m all over a g a i n - - a n d uns e e m l y behaviour among proselytisers for internationalism and its benefits und e r m i n e d their cause corrosively. It w o u l d s e e m that the efforts to c r e a t e an international language (at least for m a t h e m a t i c s and p e r h a p s for general use) serve to indicate the n e e d for co-operation, standardisation, a n d inter-translatability. P r e s e n t a t i o n s s u c h as F r e g e ' s Begriffsschrift are t o o narr o w to c a t c h on. But an-embracing p r o p o s a l s for a truly scientific international artificial language have also n e v e r s u c c e e d e d . Language, including m a t h e m a t i c a l language, m a y b e t o o hum a n a m a t t e r for c o n s e n s u s 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).

5See Gray [2001] forthcoming. 6Although new proposals circulated a while back for something not too different.


Frege, G. 1879 Begriffsschrift, eine der arithmetischen 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 intrinsic geometry. In The Intersection of History and Mathematics: Proceedings of the Tokyo Symposium in the History of Mathematics, 1990, ed. C. Sasaki, Birkhauser, Basel (1994), 39-50. Gray, J. J. Languages for mathematics and the language of mathematics in a world of nations. 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, Cambridge (1999). Mathematics Unbound: The Evolution of an InternationalMathematical Community, 18001945, 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, 180202. Peirce, C. S. 1883 Studies in Logic. By Members 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).

I I - - , ~. . . .9. [ ~ ,

~--~ J e t

Wimp,

Editor

I

Geometry from Africa: Mathematical and Educational Explorations MATHEMATICAL ASSOCIATION OF AMERICA, WASHINGTON D.C., 1999, ix + 210 pp. ISBN: 0-88385715-4. 2.

Le Cercle et le Carr L'HARMA'FI-AN, PARIS, 2000, 301 pp. ISBN: 2-7384 9235-5

by Paulus Gerdes REVIEWED BY D O N A L D W.

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

Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA

CROWE

lthough Paulus Gerdes has lived for d e c a d e s in Mozambique, a country k n o w n to m o s t of us mainly for its earlier long-standing civil w a r with its resulting land-mine hazard, and in the y e a r 2000 for devastating floods, but until n o w not for its intellectual achievements, he has p r o d u c e d a b o d y of w o r k that is remarkable for its depth and inventiveness. Unfortunately, until recently most of it was not readily accessible to those w h o only read English, in part b e c a u s e m u c h of it was published in Portuguese, German, or French, and in part b e c a u s e m a n y of his English publications, while attractively presented, were published (several with financial support from Sweden) in Mozambique. Some of these are listed at the end of the present review. Now, however, the publication by the Mathematical Association of A m e r i c a o f Geometry f r o m Africa has given him a chance to present in readily accessible form material from many of his earlier publications, extended in p a r t b y n e w problems and exercises. The general t e r m for the activity in which G e r d e s h a s b e e n engaged is "ethn o m a t h e m a t i c s " (a t e r m p o p u l a r i z e d by Ubiratan D'Ambrosio), and he might legitimately be c o n s i d e r e d to be the leading active r e s e a r c h e r in this field. But, in the m i n d o f this reviewer, in its original f o r m u l a t i o n this t e r m had polemical c o n n o t a t i o n s w h o s e absence is an a s s e t in the p r e s e n t books. These t w o b o o k s e m p h a s i z e the dis-

A

c o v e r y of w h a t G e r d e s has aptly called the "hidden" or "frozen" m a t h e m a t i c s (geometry, in this c a s e ) in cultures, o r cultural activities, that m o s t o f us imagine have no m a t h e m a t i c a l significance o r component. This idea that there is unrevealed mathematical insight in areas outside of formal mathematics, though it has bec o m e more w i d e s p r e a d in recent times beginning with Claudia Zaslavsky's Africa Counts (1973) and continuing with Marcia Ascher's Ethnomathematics (1991) and n o w with Paulus Gerdes's Geometry from Africa, turns out not to be completely new. Several years ago R. L. Wilder called my attention to a quotation that simultaneously expresses this idea quite eloquently (with apologies for the w o r d "savage"!) and reminds us that the recognition of it dates back at least a hundred years: A careful s t u d y o f all w o m a n ' s w o r k in basketry, as well as in weaving and embroidery, r e v e a l s the fact that b o t h in the w o v e n and in the s e w e d o r coil w a r e e a c h stitch t a k e s up the very s a m e a r e a of surface. When w o m e n i n v e n t e d basketry, therefore, they m a d e art possible. Along with this fact, t h a t e a c h stitch on the s a m e b a s k e t m a d e of uniform material o c c u p i e s the s a m e n u m b e r of square millimetres, goes a n o t h e r f a c t - - t h a t m o s t savage w o m e n can count to ten at least. The p r o d u c t i o n o f geometric figures on the surface o f a b a s k e t or blanket, therefore, is a m a t t e r of counting. If the enumeration is c o r r e c t e a c h time the figures will be uniform. Now, m a n y o f the figures on savage b a s k e t r y c o n t a i n e d intricate series of numbers, to r e m e m b e r which c o s t much mental effort and use of numerals. This constant, every day and h o u r use of n u m e r a l s d e v e l o p e d a facility in them, and, coupled with form in ornament, m a d e 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-VERLAGNEW YORK, VOLUME 23, NUMBER 2, 2001

~5

In Le Cercle et le Carrd G e r d e s pres e n t s the beginning o f a c a t a l o g of shall o w w o v e n b a s k e t s a n d trays from various p a r t s of the world, m a d e from a v a r i e t y of materials b y b o t h w o m e n a n d m e n (conveniently in F r e n c h "vann i t r e s et vanniers") d e p e n d i n g on the locality. Most of t h e s e are from Africa a n d South America, the t w o parts o f the w o r l d G e r d e s k n o w s best, b u t there a r e several e x a m p l e s from Vietnam, Indonesia, China, the Philippines, and Tonga. Almost all of t h e s e are initially w o v e n in the s h a p e of a square, and t h e n s e w n and t r i m m e d to a circular rim, s o that the final s h a p e is circular. Particularly attractive a n d familiar exa m p l e s are m a d e by the Y e k u a n a in the u p p e r r e a c h e s of the Orinoco in southe r n Venezuela, as s h o w n in Figure 1 (Planche 9.8). This example, like several others from the s a m e source, app e a r s at first glance to have 90 ~ rotational symmetry, b u t in fact there is only half-turn symmetry. (In the b l a c k - w h i t e s c h e m a t i c o f Figure 1, t h e r e is also reflection s y m m e t r y in t w o p e r p e n d i c u l a r axes. But in the original weave t h e r e is directionality, w h i c h destroys the reflection symmetry. G e r d e s provides d e t a i l e d drawings in w h i c h the directionality is s h o w n in detail.) It is aesthetically d e s i r a b l e that the c e n t e r of the square be at the cent e r o f the circle, and he gives a detailed d e s c r i p t i o n of w a y s w e a v e r s throughout t h e world vary the w e a v e o r c o l o r in o r d e r to m a k e the c e n t e r of the square i m m e d i a t e l y obvious, often by clearly indicating the two diagonals of the square as lines o f symmetry.

Figure 1. Yekuana.

66


Figure 2. Making a knot.

Geometry f r o m Africa goes m u c h further t h a n m e r e l y cataloging the variety of African designs and p a t t e r n s with a d e s c r i p t i o n of the m a t h e m a t i c a l principles i n h e r e n t in them, t h o u g h its C h a p t e r 1 d o e s consist of an o v e r v i e w of suggestive African images. Many c o n t e m p o r a r y examples, s u c h as t h e g e o m e t r i c h o u s e paintings of s o u t h e r n Africa o r the multi-colored kente cloth from Ghana, a r e relatively familiar, b u t the intricate c o m b i n a t o r i c s o f old (eleventh c e n t u r y to fifteenth c e n t u r y ) Tellem w o v e n material from Mali will be n e w to most. The remaining t h r e e c h a p t e r s d e v e l o p the underlying mathe m a t i c a l principles into p e d a g o g i c a l tools in the form of n e w g e o m e t r i c a l and c o m b i n a t o r i a l problems. These a r e the c h a p t e r s w h e r e w e find the unique c o n t r i b u t i o n s o f the author. C h a p t e r 2 s h o w s h o w v a r i o u s designs c a n l e a d to finding the Pythagorean Theorem. Among these is a simple k n o t u s e d in Mozambique b a s k e t r y w o v e n o f t w o s t r a n d s as the beginning o f a b u t t o n for closing a basket, as in Figure 2a. [Gerdes Figure 2.2]. When pulled tight, as in Figure 2b [Gerdes 2.4a] a d i s s e c t i o n of a square appears,

which can easily b e t r a n s f o r m e d to yield a familiar d i s s e c t i o n p r o o f o f t h e Pythagorean Theorem, as in Figure 2c [Gerdes 2.6]. Many o t h e r e x a m p l e s follow. The last s e c t i o n of Chapter 2, "From m a t weaving p a t t e r n s to Pythagoras, a n d Latin a n d magic squares," s h o w s h o w an "over 4, u n d e r 1" m a t weaving p a t t e r n in a C h o k w e m a t l e a d s in a natural w a y to a 5 x 5 Latin square, from which a set of five pairwise orthogonal Latin s q u a r e s is derived, from which a 5 x 5 magic square is constrncted. It will require a persistent a n d patient t e a c h e r to i m p l e m e n t this in the classroom, b u t the s t e p s and p i c t u r e s are clearly p r e s e n t e d . C h a p t e r 3 d e a l s mainly with basketry. This includes the p l a n a r w e a v e with h e x a g o n a l h o l e s in c o m m o n use throughout the world, for e x a m p l e in chair-caning. W h e n b a s k e t - m a k e r s w a n t to convert s u c h a p l a n a r w e a v e into an actual container, they realize that s o m e holes s m a l l e r than hexagons, often pentagons, m u s t be introduced. If a closed ball is desired, the intuitive discovery is m a d e that 12 p e n t a g o n a l holes will b e needed. Such a ball, with the fewest p o s s i b l e n u m b e r of hexago-

Figure 3. Woven ball (left), soccer ball (right).

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 internal reflecting barriem on an imagined billiard table before starting the billiard ball at a 45 ~ angle to the sides and tracing 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 projections, 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.

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 combinatorially the truncated icosahedron, now known to the chemists as buckminsterfullerene. Gerdes has already elaborated on this theme in his paper "Molecular modeling of fullerenes with hexastrips" in the Mathematical Intelligencer (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 chemical 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-

Figure 4. The 7 x 7 design.

Figure 5. Sipatsi strip patterns belonging to each of the seven symmetry classes.


67

a

5. Jablan, Slavik, Mirror generated curves, Symmetry." Culture and Science, Vol. 6, No.2,275-278 (1995). (Followed by an electronic presentation at http://members.tripod.corn/- modularity/mir.htm.)

--

I

Q

9

D

9

9

B ~ D

B

9

t

LD

t

9

9

LP

B

;

9

! 9

tl

Q

9

9

9

9 t

b

Figure 6. Mirror design (top), corresponding mirror curve (bottom).

As m e n t i o n e d at the beginning, m u c h o f this material h a s a p p e a r e d in G e r d e s ' s less a c c e s s i b l e publications, s o m e t i m e s in greater detail. In the unlikely event that the r e a d e r d o e s not find enough here to k e e p occupied, furt h e r r e w a r d s await, b o t h t h e r e and in future work, which is s u r e to appear. ERRATA: An unfortunate editorial slipup led to the mislabeling o f m o s t of the figures (Planches) in C h a p t e r 2 of Le Cercle et le Carrd. The r e a d e r can corr e c t t h e s e labels as follows: 2.1 to 2.10 should be 2.6 to 2.15; 2.11 to 2.25 should b e 2.18 to 2.32; 2.26 to 2.30 should b e 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, Universidade Pedag6gica, Maputo (1996). 3. ., On Lunda-Designsand Some of their Symmetries, VisualMathematics (electronic journal: http://members.tripod.com/ vismathh, Vol. 1, No. 1 (1999). 4. Gerdes, Paulus, and Bulafo, Gildo, Sipatsi: Technology, Art and Geometry in Inhambane, Universidade Pedag6gica, Maputo (1994).


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 A z z o u n i CAMBRIDGE: THE UNIVERSITY PRESS, 1994. x + 2 4 9 pp. US $54.95, ISBN: 0-521-44223-0 REVIEWED

BY DIEGO BENARDETE

y t h e 1930s, there w e r e t h r e e m a i n a p p r o a c h e s to the p h i l o s o p h y o f m a t h e m a t i c s : logicism, intuitionism, and formalism. In J o d y Azzouni's book, algorithmic p r o c e d u r e is t a k e n to b e the h a l l m a r k o f mathematics, a n d it can b e c o n s i d e r e d as a s u c c e s s o r to formalism. By careful attention to h o w m a t h e m a t i c s is done, the b o o k att e m p t s to m e e t the usual o b j e c t i o n that r e c i p e s for m e r e l y making m a r k s on a s h e e t o f p a p e r cannot a c c o u n t for the beauty, truth, and usefulness o f mathematics. It s o m e t i m e s s e e m s that mathematicians have acquired a h e r e d i t a r y wariness o f philosophy, arising from encounters s u c h as those r e p o r t e d by Plato b e t w e e n the distinguished ancient m a t h e m a t i c i a n s T h e o d o r u s and Theaetetus and that respectful b u t tricky dialectician Socrates. T h e o d o r u s p l e a d s his old age as an e x c u s e to avoid a n s w e r i n g the question "What is knowledge"; the young T h e a e t e t u s is finally f o r c e d to confess that all the ans w e r s to that question w h i c h he has c o m e up with in r e s p o n s e to S o c r a t e s ' s p r o d d i n g are "mere wind-eggs." M a t h e m a t i c i a n s m a y fear that t h e i r vital intuitions about the n a t u r e of their

B

subject, w h e t h e r true or false, will b e stifled or ridiculed b y the m i s a p p l i e d logical rigor o f the philosophers. An exa m p l e is the following r e m a r k of J e a n Dieudonnd m a d e in r e s p o n s e to a question at the end o f a lecture [D]: On f o u n d a t i o n s w e believe in the reality of m a t h e m a t i c s , but of c o u r s e w h e n p h i l o s o p h e r s a t t a c k us with their p a r a d o x e s w e rush to hide behind f o r m a l i s m and say, "Mathematics is just a combination of meaningless symbols," and then we bring out Chapters 1-3 of Bourbaki's Set Theory a n d so a r e left in p e a c e to go b a c k to o u r m a t h e m a t i c s a n d do it as we have a l w a y s done, with the feeling t h a t e a c h m a t h e m a t i c i a n h a s that he is w o r k i n g with something real. This s e n s a t i o n is p r o b a b l y an illusion, b u t is v e r y convenient. That is Bourbaki's attitude t o w a r d foundations. The c u r r e n t g e n e r a t i o n of m a t h e maticians is even m o r e r e m o v e d from the p h i l o s o p h y o f m a t h e m a t i c s of o u r day than the f o u n d e r s of B o u r b a k i were from theirs. S o m e think that the erection of formal b a r r i c a d e s has b e e n a c c o m p l i s h e d with Zermelo-Frankel set theory, while o t h e r s share in the general waning o f interest in foundations for science. Nonetheless, s o m e familiarity with the s u b j e c t might n o t only help us reflect on our own activity, but also e n a b l e us to s p e a k m o r e effectively in c u r r e n t d e b a t e s a b o u t the e d u c a t i o n a l a n d r e s e a r c h roles o f mathematics. After all, m a t h e m a t i c s held its p l a c e in the curriculum in p a r t b e c a u s e o f its d i r e c t applicability, b u t also b e c a u s e of its p r e s u m e d merit as a training in abstract, rigorous thinking. During the last half-century the m o s t important and influential A m e r i c a n p h i l o s o p h e r o f s c i e n c e and m a t h e m a t ics has b e e n Willard Quine. This influence arises not so m u c h from his technical c o n t r i b u t i o n s to symbolic logic. The great c h a r m o f Quine's w o r k r a t h e r is h o w he m a n a g e s to c o m b i n e a non o n s e n s e b e l i e f t h a t c o n t e m p o r a r y science offers o u r b e s t a c c o u n t of the universe t o g e t h e r with a willingness to revise prevailing p h i l o s o p h i c views on

h o w to articulate, justify, a n d e l a b o r a t e that belief. We can certainly note t h e s e Quinean c h a r a c t e r i s t i c s in his studies of the being o f m a t h e m a t i c a l entities. Roughly speaking, universal terms, such as c o m m o n nouns like "dog" o r "chair," have b e e n c o n s i d e r e d in t h r e e ways. F o r a realist, these t e r m s refer to extra-physical, extra-mental entities, s o m e t i m e s picturesquely p l a c e d in w h a t is called with dubious historical accuracy "Platonic heaven." F o r the mentalist, the terms refer to ideas in the mind; for the nominalist, they are merely n a m e s which denote many objects as o p p o s e d to p r o p e r 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 m a n y thinkers a term such as "three" is not considered to be a universal term but rather a p r o p e r noun denoting the n u m b e r three. Quine argues that w e m u s t a s c r i b e reality to t h o s e entities w h o s e exist e n c e is affirmed in natural science, o u r b e s t t h e o r y of the universe, unless such affirmations can b e successfully p a r a p h r a s e d a w a y [Q1, Q2]. Natural s c i e n c e relies on several b r a n c h e s of m a t h e m a t i c s such as n u m b e r t h e o r y a n d real analysis. These disciplines in their usual formulations are r e p l e t e with existential c o m m i t m e n t s to various n u m b e r s , sets, functions, a n d s p a c e s o f functions. In a 1947 article, Quine and Nelson G o o d m a n a t t e m p t e d the first s t e p s t o w a r d s a nominalist reformulation of these s u b j e c t s t h a t w o u l d free t h e m of c o m m i t m e n t s to such a b s t r a c t entities. Quine did n o t see a w a y to c o m p l e t e this project, a n d with s o m e r e l u c t a n c e settled into the realist c a m p in the p h i l o s o p h y o f mathematics. One thinks of the old j o k e a b o u t the t o w n rationalist w h o w h e n a s k e d a b o u t the rabbit's foot in his wallet replied that it w o r k e d even if one didn't believe in it. A p o k e r - f a c e d Quine w o u l d not allow such shenanigans, such t r i c k y bookkeeping. Quine's h a s b e e n called a "pragmatic" r e a l i s m b e c a u s e it r e g a r d s o u r existential c o m m i t m e n t s as d e v i c e s u s e d to simplify our u n d e r s t a n d i n g o f on o t h e r w i s e confusing h u b b u b of sensory input. It was o t h e r w i s e with t h e distinguished logicim-I Kurt GOdel [Wl,

W2]. While one c o u l d not decide using Zermelo-Frankel set t h e o r y w h e t h e r there w a s a set with a cardinality bet w e e n that of ~0 a n d that of the p o w e r set oflr G0del t h o u g h t that there w a s a c o r r e c t a n s w e r to this question because the sets involved existed indep e n d e n t l y of any o f o u r theories regarding them. This correct a n s w e r would ultimately b e found using not only c o n s i d e r a t i o n s of simplicity and scientific applicability b u t also a kind of intuition. In fact, G0del's p o s i t i o n is s h a r e d by m a n y mathematicians, though few of t h e m p u r s u e the issue with G0del's tenacity. F o r example, over wine and c h e e s e at a r e c e p t i o n following a lecture at Hunter College in New York, A n d r e w Wiles told the story about being p e s t e r e d by a correspondent who w a n t e d to know w h e t h e r he c o n s i d e r e d m a t h e m a t i c s to b e discovered (realism) or invented (formalism or p e r h a p s mentalism). When a s k e d for his reply, Wiles said, "Discovered, of course. Doesn't every m a t h e m a t i c i a n believe that?" These beliefs o f m a n y p h i l o s o p h e r s and m a t h e m a t i c i a n s were challenged in two very influential articles b y Paul B e n a c e r r a f [B]. In the first article, "What n u m b e r s c o u l d not be," which a p p e a r e d in 1965, B e n a c e r r a f r e a s o n e d that for b o t h p r a c t i c a l and mathematical p u r p o s e s t h e r e w a s nothing to c h o o s e b e t w e e n the von N e u m a n n c o n s t r u c t i o n o f the n a t u r a l n u m b e r s as the set of sets [}, [{]}, [{{}}},..., w h e r e {} is the e m p t y set, and the Zermelo c o n s t r u c t i o n of t h e s e n u m b e r s as the set of sets [}, {{]}, {{}, [{}}}, . . . . Because any such iteratively constructed sequence could serve the purpose, he c o n c l u d e d that the natural n u m b e r s could n o t be identified with any set of sets. The m a t h e m a t i c a l realist w o u l d have to l o o k e l s e w h e r e for the natural n u m b e r s o r p e r h a p s c o n c e d e that t h e y did n o t exist. In the s u b s e q u e n t p a p e r "Mathematical truth" [B], w h i c h a p p e a r e d in 1973, B e n a c e r r a f a c c e p t e d a position of Quine that our t h e o r y of k n o w l e d g e could go so far b u t no further than assuming all the c o n c l u s i o n s of natural science. Such a naturalized epistemology w a s thought to require that a causal c o n n e c t i o n e x i s t e d b e t w e e n the

k n o w e r and the thing known, as for exa m p l e was p r o v i d e d b y t h e reflection o f light in the s i m p l e r c a s e of sight. However, even if one g r a n t e d to the realist the e x i s t e n c e o f v a r i o u s extranatural m a t h e m a t i c a l beings, it w o u l d be impossible to explain in a natural w a y h o w these beings c o u l d causally influence and thus b e c o m e k n o w n to animals such as ourselves. In the following decades, naturalized theories of k n o w l e d g e no longer insisted on the n e e d for causal connections, but the p r o b l e m of h o w natural beings could k n o w a b s t r a c t o b j e c t s remained. A related problem, w h i c h is p e r h a p s m o r e basic, is h o w o u r m a t h e m a t i c a l language can successfully refer to mathem a t i c a l objects. The s t a t e m e n t that the Eiffel T o w e r is on the right b a n k of the Seine is false, b u t w e have still succ e e d e d in referring to this famous structure. Some s o r t of c o n n e c t i o n bet w e e n the u s e r o f language a n d the referred-to o b j e c t s e e m s to be necessary. Such c o n n e c t i o n s a p p e a r to be a b s e n t in the c a s e o f m a t h e m a t i c a l objects, at l e a s t as they are c o n c e i v e d by the realist. Various r e s p o n s e s to B e n a c e r r a f s t w o articles have s h a p e d the ensuing d e c a d e s of p h i l o s o p h y of m a t h e m a t i c s IF, K, M1, M2, S]. J o d y Azzouni's thinking is in s o m e w a y similar to that of the thinkers gathered in T y m o c z k o ' s collection New Directions in the Philosophy of Mathematics [T] a n d vigorously e x p o u n d e d in the m a t h e m a t i c i a n Reuben Hersh's n e w b o o k What is Mathematics, Really? [He]. Like them, Azzouni believes that one should closely observe the p r a c t i c e s of mathematicians. But while t h e y wish to sideline the c o n c e r n with foundational questions, he believes that he can provide the c o r r e c t a n s w e r to the foundational questions and, b e t t e r yet, witho u t requiring revision o f s t a n d a r d a c c o u n t s of truth o r reference. An a p p r o a c h like his (which, while d r a w i n g on e l e m e n t s o f all three app r o a c h e s to the f o u n d a t i o n s o f mathematics, m o s t obviously r e s e m b l e s formalism) can easily a r o u s e the indignation of m a n y mathematicians. They believe that it c o u l d only be held b y s o m e o n e w h o d o e s n o t have the exp e r i e n c e s of a real mathematician.


69

However, Azzouni believes that his acc o u n t allows for t h e s e experiences. He might differ on h o w to i n t e r p r e t them. The b o o k has three parts. In Part 1, Azzouni p r e s e n t s the puzzles that involve k n o w l e d g e of, and r e f e r e n c e to, m a t h e m a t i c a l objects. In P a r t 2, he gives his solution to t h o s e puzzles. In P a r t 3, he discusses o t h e r t o p i c s including the applicability of m a t h e m a t ics. The A p p e n d i x p r e s e n t s a formal t h e o r y that includes a truth predicate. My review will not follow this order, b u t r a t h e r go directly to a r e c o n s t r u c tion o f the b o o k ' s positive doctrine. F o r Azzouni, the k e y ingredient of m a t h e m a t i c s is algorithmic practice. "One sets down an arbitrarily given set o f axioms, an arbitrary set o f inference rules, and then derives t h e o r e m s from them" (page 80). He is certainly a w a r e that at the core o f c u r r e n t m a t h e m a t i c s t h e r e are sets o f a x i o m s that are i m p o r t a n t b e c a u s e o f their applicability to n a t u r a l science a n d to other a r e a s of mathematics. However, if I interpret him correctly, he believes that the n e o p h y t e w h o tries all s o r t s of odd variations on the axi o m s o f a group o r o f a ring is still doing mathematics, though p e r h a p s of a v e r y insignificant kind. He d o e s not require that this p r a c t i c e b e c a r r i e d out in a formal language. The a x i o m s and inference rules n e e d only to be sufficiently explicit that one can d e t e r m i n e w h a t counts as a d e d u c t i o n in the system. This gives enough latitude so that E u l e r ' s manipulations with infinite series a r e still m a t h e m a t i c s even though t h e y ignore questions o f convergence. N e e d l e s s to say, he i n c l u d e s within mathematics various n o n s t a n d a r d practices, such as intuitionism, constructivism, and multi-valued logics. Howe v e r formalist this all might seem, it is i m p o r t a n t to realize that m a t h e m a t i c s is n o t c o n s i d e r e d to be a p r o c e s s of m a k i n g m a r k s on a s h e e t o f paper. R a t h e r it is c o n s i d e r e d to b e a series of s e n t e n c e s e x p r e s s e d either in a natural language such as English or in a formal language such as first-order p r e d i c a t e calculus. In fact t h e r e is no essential n e e d for the m a t h e m a t i c a l d i s c o u r s e to b e w r i t t e n down at all. A p o s i t i o n like Azzouni's will be as-

70


s o c i a t e d in m a n y minds with the form a l i s m o f David Hilbert. In fact, P a r t II of this b o o k b e a r s as an epigraph the following quote from a letter t h a t Hilbert w r o t e to Gottlob Frege: "If arbitrarily given axioms do n o t contradict one a n o t h e r with all their consequences, t h e n they are true a n d t h e things d e t e r m i n e d by the a x i o m s exist." Many m a t h e m a t i c i a n s believe that Hilbert's a p p r o a c h was fatally w e a k ened b y the famous i n c o m p l e t e n e s s results o f Kurt G6del. This c o n c l u s i o n is drawn, for example, in a w o n d e r f u l b o o k b y H o w a r d Eves a n d Carroll N e w s o m [E, p a g e 305]. The p r o b l e m is twofold. Hilbert believed that his formal a p p r o a c h to the transfinite p a r t s o f

"If axioms do not contradict one another,.., then they are true and the things determined by the axioms exist." m a t h e m a t i c s required there to be a proof, n o t j u s t a belief, that the s y s t e m w a s consistent, i.e., that the "arbitrarily given a x i o m s do not c o n t r a d i c t one another." However, G6del s h o w e d t h a t any formal s y s t e m strong e n o u g h to contain e l e m e n t a r y arithmetic is n o t strong e n o u g h to prove its o w n consistency. F u r t h e r m o r e , it s e e m s that such a formal system contains sent e n c e s w h i c h are unprovable b u t true. This s e e m s to indicate that m a t h e m a t ical truth c a n n o t be r e d u c e d to a matter of d r a w i n g c o n s e q u e n c e s in a formal system. Azzouni s c a r c e l y replies explicitly to the first objection, i.e., to the l a c k of a consistency proof. It is likely that his implicit a r g u m e n t is that all of us are in the s a m e boat, w h a t e v e r o u r p h i l o s o p h y of mathematics. If tom o r r o w a c o n t r a d i c t i o n should be derived within Zermelo-Frankel set theory, b o t h p l a t o n i s t and formalist w o u l d be equally obliged to tinker with the ax-

ioms in o r d e r to e s c a p e the dilemma. As for the s u p p o s e d l y true but unprovable sentences, Azzouni points out that v i e w e d in a strictly syntactic w a y all G6del s h o w s is t h a t if we a s s u m e the c o n s i s t e n c y o f arithmetic t h e n there is a s e n t e n c e o f arithmetic s u c h that neither it n o r its negation is provable (page 134). This sentence is "true" only if we s t e p o u t s i d e our formal system and c o n s i d e r its terms to refer to the natural n u m b e r s u n d e r s t o o d in the usual way. As w e shall see below, Azzouni is all in favor of our transcending any given formal system a n d expanding it o r linking it to others. However, s u c h linkages do not require any belief in m a t h e m a t i c a l objects existing a p a r t from f o r m a l systems. What then is the m o d e of e x i s t e n c e enjoyed b y m a t h e m a t i c a l objects? According to Azzouni they are "posits" (Section II.6). He is here accepting a notion of Quine, w h o also c o n s i d e r s m a t h e m a t i c a l o b j e c t s to be posits [Q3]. F o r Quine, a p o s i t is an entity that w e introduce to simplify our u n d e r s t a n d ing of the c o m p l e x i t i e s of experience. These include the sets of mathematics, but they also include theoretical entities from p h y s i c s s u c h as electrons and quarks. In fact, even familiar e v e r y d a y objects such as t a b l e s and chairs can be v i e w e d as p o s i t s i n t r o d u c e d to simplify the w e l t e r of s e n s e - d a t a in which w e are engulfed. F o r Quine all t h e s e p o s i t e d entities have equal claim to existence since t h e y are ineliminable parts of science, w h i c h is our b e s t account of reality. There is, however, a crucial difference b e t w e e n Quine and Azzouni. F o r the latter, p o s i t s c o m e in t h r e e "widths": thick, thin, and ultrathin. Entities such as e l e c t r o n s and quarks are thick posits, for w e believe that w e can have causal i n t e r a c t i o n s with t h e m by m e a n s of v a r i o u s e x p e r i m e n t a l procedures. He r e s e r v e s the t e r m "thin posit" for p o s i t s w h i c h have the simplifying utility w h i c h Quine insists on, but which a r e n o t involved in t h e s e causal relationships. He is hard p r e s s e d to name a specific example, but considers certain "properties" to be candidates for thinness. P e r h a p s he has in mind a p r o p e r t y such as electric

charge. Finally, ultrathin p o s i t s have no o t h e r being than that of t e r m s int r o d u c e d in s o m e kinds of s t r u c t u r e d discourse. A g o o d e x a m p l e is the king in chess. One d e s c r i b e s a game in w h i c h there a r e several kinds of p i e c e s that can m o v e and c a p t u r e in specific w a y s on a certain grid. The king is one such piece. Mathematical entities a r e included a m o n g the ultrathin posits. But w h a t is a posit? What m o r e c a n w e s a y a b o u t its m o d e of being? After all, w e usually c o n s i d e r physical obj e c t s to exist w h e t h e r or not t h e r e a r e any m i n d s perceiving them. Are p o s i t s also m i n d - i n d e p e n d e n t ? In an important f o o t n o t e at the end o f his b o o k (page 213), Azzouni states that t h e r e can b e no p o s i t s without positors. I a m not sure w h e t h e r this is m e a n t to apply to p h y s i c a l posits such as e l e c t r o n s j u s t as m u c h as to the posits o f mathematics. In a n y event, it s e e m s to leave t h e s e latter p o s i t s as being mind-dep e n d e n t in an i m p o r t a n t way. In t e r m s of m y earlier classification of t h i n k e r s as realists, mentalists, or nominalists, one w o u l d have to say that Azzouni is at least in p a r t a mentalist. Let m e p a u s e here to note that with their notion o f posit, both Quine a n d Azzouni have a n s w e r e d the B e n a c e r r a f challenge as to h o w the a b s t r a c t entities of m a t h e m a t i c s can be known. F o r b o t h t h e s e thinkers, the m a t h e m a t i c a l o b j e c t s have no n e e d of causally impinging on h u m a n m i n d / b o d i e s in ord e r to be known. They have simply b e e n posited. F o r Quine they are n e e d e d auxiliaries to the s m o o t h running of natural science. F o r Azzouni, they have b e e n brought into existence by the mathematician engaged in axiomatic algorithmic procedures. We must now see h o w these seemingly evanescent ultrathin posits are r o b u s t enough to a c c o u n t for all of pure and applied m a t h e m a t i c s both p a s t and present. One m a j o r difficulty that arises is that m a t h e m a t i c a l t e r m s are identified a c r o s s m a n y different algorithmic systems, b o t h p a s t and present. Azzouni gives m a n y examples, and it is e a s y to supply m a n y m o r e (Sections 1.4,1.5, 1.6, II.6, II.7). Egyptian p a p y r i are t a k e n to refer to the n u m b e r 12 even t h o u g h the Egyptians did n o t e m p l o y the P e a n o

axiomatization of t h e natural numbers. Mathematicians s u c h as Euler are taken to have g o t t e n c o r r e c t the s u m of the infinite g e o m e t r i c series 1 + i/2 + 1/4 + . . . even though t h e y did not e m p l o y our c u r r e n t definitions of convergence. Similarly Newton was able to integrate t h e function x 2, even though he did n o t have the benefit of the t h e o r y of R i e m a n n or Lebesgue integration. Also, often w e do not distinguish b e t w e e n the n u m b e r 1 unders t o o d as a natural number, an integer, a rational number, o r a real number, even though e a c h o f t h e s e c a s e s involves a different s y s t e m of axioms. Finally, the a x i o m s o f a group are t a k e n to apply to a w h o l e 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 integers m o d u l o n to the set of c o n t i n u o u s h o m e o m o r p h i s m s of a topological space. All t h e s e e x a m p l e s suggest that m a t h e m a t i c a l entities, if they are posits, a r e n o t local posits. That is t h e y are n o t r e s t r i c t e d to a particular a x i o m a t i c system. It w o u l d s e e m that a m a t h e m a t i c a l realist w o u l d have little difficulty with these examples. The realist w o u l d s a y that the ancient E g y p t i a n and the mode m A m e r i c a n are a b l e to refer to the s a m e n u m b e r s (e.g., 12) in m u c h the s a m e w a y as t h e y are able to refer to the s a m e celestial b o d i e s (e.g., the sun). In b o t h c a s e s the entities have an objective e x i s t e n c e which, not surprisingly, can be d i s c e r n e d with greater o r lesser clarity by different thinkers. A formalist such as Azzouni n e e d s to c o m e up with an e x p l a n a t i o n o f his own. He argues t h a t t h e r e is, in fact, no absolutely c o m p e l l i n g n e e d for us to identify entities a c r o s s the b o u n d a r i e s

of m a t h e m a t i c a l systems. However, he e l a b o r a t e s and a c c e p t s the m a n y reas o n s which induce m a t h e m a t i c i a n s a n d historians of m a t h e m a t i c s to do j u s t that. He t a k e s t h e s e identifications to b e simply a m a t t e r of "stipulation" (page 121). We s i m p l y identify one t e r m with another. F o r example, w h e n constructing the real numbers, w e identify the natural n u m b e r 1 with a certain Dedekind cut. T a k e n m o r e generally, it s e e m s to m e that this business of "identification" is a familiar part of m a t h e m a t i c a l practice. It is informally t a l k e d a b o u t w h e n e v e r s o m e quotient o b j e c t is being c o n s t r u c t e d in, say, t o p o l o g y or group theory. Of course, in c o n t e m p o r a r y m a t h e m a t i c s , which is usually m o d e l e d in set theory, these c o n s t r u c t i o n s are e l a b o r a t e d as equiva l e n c e classes d e t e r m i n e d b y equival e n c e relations. Azzouni's identifications have a similar flavor, b u t they are a c c o m p l i s h e d by identifying bits of s y n t a x a c r o s s systems. The m a t t e r of group theory, which I a l l u d e d to above, is s o m e w h a t m o r e c o m p l i c a t e d . Such t h e o r i e s have m a n y n o n - i s o m o r p h i c m o d e l s n o n e of w h i c h is c o n s i d e r e d standard. He t a k e s t h e s e to be collections of s e n t e n c e s c h e m a r a t h e r than collections of s e n t e n c e s (Section II.7). A n o t h e r usual o b j e c t i o n to a conventionalist a c c o u n t o f m a t h e m a t i c s s u c h as Azzouni's is that it c a n n o t explain the u n c a n n y effectiveness of applied mathematics. The s t r u c t u r e s of m a t h e m a t i c s do not s e e m to b e m a t t e r s of a g r e e m e n t or stipulation, b u t r a t h e r p a r t and p a r c e l of the formal b e d r o c k o f the universe. In his a c c o u n t of these matters, Azzouni begins from the obs e r v a t i o n that there are m a n y formal g e o m e t r i c a l systems, a n d that it is a m a t t e r of empirical s c i e n c e to decide w h i c h one b e s t fits the cosmos. So, for example, if a model of the universe is t a k e n t o be a Lorentz manifold with underlying topological s p a c e S 3 x R, then the identifications b e t w e e n 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 s o m e mathematical theories as sentence s c h e m a t a which


71

can be turned into sentences by fixing s o m e abstract or empirical interpretation. Azzouni has a similar understanding of all other applied mathematics including applied arithmetic. He takes the s u p p o s e d "unreasonable effectiveness of mathematics" to be not so unreasonable. It is simply a m a t t e r of our evolutionary adaptation to the world in which w e live (Sections 1].3, III.6). A final difficulty is that m a t h e m a t i cians for the m o s t p a r t do n o t think of t h e m s e l v e s as operating in the w a y that Azzouni suggests. This he acknowledges. F o r example: "this is n o t h o w E u l e r s a w w h a t he w a s doing; a n d generally no m a t h e m a t i c i a n s e e s things in this way. M a t h e m a t i c i a n s invariably have a semantic i n t e r p r e t a t i o n in mind. AS a result they have no c o m p u n c t i o n a b o u t augmenting s y s t e m s at any time on the basis of this interpretation" (page 84). He d o e s n o t view this obj e c t i o n as fatal, for, as his title proclaims, he is trying to b e faithful to the p r a c t i c e of m a t h e m a t i c i a n s and not n e c e s s a r i l y to their o w n u n d e r s t a n d i n g o f this practice. C o n s i d e r t h e following strong statement. "In general, realism is n e v e r n e e d e d to explain w h y a group o f p r o f e s s i o n a l scientists have the p r a c t i c e s they have: The collective d e l u s i o n that w h a t t h e y are studying exists can do j u s t as well" (page 138). He d o e s not dismiss the non-formal motivations, insights, a n d intuitions t h a t guide mathematicians. They j u s t b e c o m e part of the p s y c h o l o g y and sociology of the subject. "In practice, one p r o d u c e s proofs in somewhat the s a m e w a y that humans play chess: we see patterns, p r o o f patterns, that we then sketch out informally. Seeing these things, this way, is not a mechanical matter: w e do not w o r k within algorithmic systems algorithmicaUy" (page 137). Lest we be alarmed by the comparison with chess, Azzouni soon concludes that "mathematics is a m o r e creative and intriguing subject than anything afforded by a game" (page 139). Broadly spealdng, Azzouni gives a conventionalist account of mathematics. Today, when the so-called "science wars" are raging and threatening to spill over to mathematics, and when conventionalist accounts of science are d a m n e d

'72


as pernicious for their political consequences, it m a y be appropriate for me to discuss his position 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 p r o c e e d to mathematics, I think it is interesting to try to apply Azzouni's account to a famous literary confrontation of politics and mathematics. Many of us recall reading in O r w e l r s novel 1984 about the terrifying experiences in R o o m 101 by which the Inner Party m e m b e r O'Brien induced the

I remain puzzled. However, this is not necessarily a bad thing. Outer Party m e m b e r Winston to abandon his view that 2 § 2 = 4. The objective truth of mathematics is here presented as one of the last bulwarks of freedom against totalitarianism. H o w might Azzouni deal with O'Brien's arguments? (This wicked man c o m b i n e d discourse with torture as means of persuasion.) Well, if it is a matter of pure mathematics, and if O'Brien accepts the axioms and rules of inference of P e a n o arithmetic, then the denial that 2 + 2 = 4 is simply a mistake. If it is a m a t t e r of applied mathematics, Azzouni a c c e p t s 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 destruction of Big Brother's tyranny. We see, therefore, that O'Brien's triumph remains the triumph of a bully, with no laurels conferred by Azzouni's philosophy. Some mathematics texts claim little in the w a y o f formal prerequisites b u t do state a n e e d for that all-important m a t h e m a t i c a l sophistication. However, this b o o k d o e s p r e s u p p o s e a certain sophistication in the p h i l o s o p h y o f mathematics. Even without such p r e p a r a tion, m a n y s e c t i o n s are a c c e s s i b l e a n d

interesting. These include those w h i c h p r e s e n t the c o r e of his teaching (Sections II.2, II.3, II.6, II.7, III.6) a n d others w h i c h a r e d e n s e with mathematical e x a m p l e s (I.4, 1.5, 1.6). However, the n e o p h y t e w h o wishes to get the m o s t out of it w o u l d do well to r e a d first the t w o B e n a c e r r a f articles [B], and then follow with Maddy's Realism in Mathematics [M1, Hi]. This is an i m p o r t a n t b o o k a n d should be in every college library. As far as I can tell t h e r e are no e r r o r s in mathematics, logic, or philosophy. Furthermore, it s h o w s s o m e familiarity with the w a y s o f mathematicians. Azzouni's b o o k n o t only has m a n y wellc h o s e n historical examples, but also contains s o m e of t h o s e informal rem a r k s which are m a d e as we chat o v e r coffee and scribble on napkins. F o r example, "a p r o f e s s o r of m a t h e m a t i c s once told m e that his opinion of s o m e one's m a t h e m a t i c a l ability would d r o p sharply if he l e a r n e d that he o r she h a d r e a d a m a t h e m a t i c a l t e x t from c o v e r to cover" (page 168, n o t e 27). However, the k e y r e a s o n for my reco m m e n d a t i o n is t h a t this is the b e s t c o n t e m p o r a r y d e f e n s e of a minimalist position in the p h i l o s o p h y of mathematics that I a m familiar with. Why is such m i n i m a l i s m i m p o r t a n t ? Let us listen to the p o e t Wallace Stevens, w h o in Credences of Summer [St] a s k s us to "Exile d e s i r e / F o r w h a 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 w e m u s t j e t t i s o n the impossible, that w h i c h can no longer be believed, a n d try to get by on the possible, and even try to find a n e w c h a r m in such l o w a n d humble ground. Most m a t h e m a t i c i a n s believe that t h e y are uncovering objective truths a b o u t s o m e p r e e x i s t i n g reality. But, as Reuben Hersh p o i n t s o u t [He, page 11], w h e n a s k e d to e x p l a i n h o w this kind of being and this kind o f k n o w l e d g e is p o s s i b l e they quickly change the sub-

ject. Now everyone would grant the little that Azzouni presupposes for our cognitive powers. While one may quibble on the details, it seems plausible that mathematics could be reconstructed 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 glorious 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 Fundamental 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 Mathematics, Oxford Univ. Press, 1990. [M2] Penelope Maddy, Naturalism in Mathematics, Oxford Univ. Press, 1997. [Q1] W. V. Quine, "On what there is," in From a Logical Point of View, (second edition, revised), Harvard Univ. Press, 1961. Also in Philosophy of Mathematics: selected readings (first edition), Paul Benacerraf and Hilary

Putnam (editors), Prentice Hall, 1964. [Q2] W. V. Quine, "Truth by convention," in Philosophy of Mathematics: selected readings (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 Mathematics: 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 e-mail: [email protected]

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 J E T W l M P

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 performance 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

T

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 abstruse moral dicta than on rich and easily accessible human concerns. Proof was such a critical and popular success at its Manhattan Theater Club offBroadway run that it was soon moved to Broadway. Catherine, the young daughter of a brilliantly successful but tragically unbalanced University of Chicago mathematician, is, at the play's beginning, about to bury her father. She has spent six onerous years caring for him, nursing 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 cartons of old notebooks, thereby bolstering 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


73

o u t of a playwright's bag o f tricks. The p l o t twists g r o w logically out of the events and the c h a r a c t e r s the playwright has w o r k e d v e r y h a r d to put before us. Nevertheless, the surprises m a k e it difficult to d i s c u s s the p l a y without indulging in, to u s e Netnews film discussion parlance, "spoilers." It is not giving a w a y too much to reveal that at the a p e x of the d r a m a a noteb o o k is discovered that contains a revolutionary result in n u m b e r theory in a p r o o f occupying s o m e 40 pages. This resuit has been sought by mathematicians for centuries, and it p r o m i s e s to have substantial implications for the future of mathematics. The questions p o s e d for the audience are two: First, is the proof, which utilizes the m o s t m o d e r n mathematical t e c h n i q u e s - - a l gebraic geometry and representation theory are m e n t i o n e d - - r e a l l y a proof, or is it just 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, b o t h of these questions are answered, subtly and sat-

isfyingly. Several s c e n e s stick gratifyingly in m y mind. One, a flashback, opens with the father, a p p a r e n t l y emerging from his long struggle with insanity, scribbling in a n o t e b o o k with childlike enthusiasm. What he is writing, he volubly assures his d a u g h t e r as he wrings his h a n d s in self-satisfaction, is g o o d stuff; it signals the beginning of his ret u r n to p r o f e s s i o n a l m a t h e m a t i c s . He s h o w s the n o t e b o o k to Catherine, but she s o m e h o w is c o n c e r n e d only that the p o r c h is t o o chilly for him, and begs him to c o m e inside. He insists she r e a d t w o lines of w h a t he h a s done, and she again implores him to c o m e inside. Finally, red-faced a n d raging, he dem a n d s she r e a d from the proof. As she begins to r e a d slowly a n d tonelessly, w e are shockingly c o n f r o n t e d with the

'74


d i m e n s i o n s of the father's tragic illness. All the a c t o r s deserve the g r e a t e s t plaudits. Mary-Louise Parker, w h o has h a d an a b u n d a n t c a r e e r on stage and in film, gives a wonderfully n u a n c e d a n d highly p r a i s e d p e r f o r m a n c e as Catherine: at first abrasive, b u t vulnerable and fearful of the d e g r e e to w h i c h h e r father's fate m a y have infected her, she spits h e r lines with a laughing snarl l a c e d with au courant college-age patois as she quaffs cheap c h a m p a g n e right from the bottle. Yet during the c o u r s e o f the d r a m a she is r e v e a l e d to b e a p e r s o n of imposing spiritual and intellectual resources. Larry Bryggman as the f a t h e r is equally impressive. Soap fans will recognize him as the c h a r a c t e r Dr. J o h n Dixon, w h i c h he has p l a y e d in "As The World Turns" since 1956. With his reso n a n t voice and urgent physicality, he c o n v e y s with c o m p l e t e conviction b o t h the father's i m m e n s e love for his d a u g h t e r and also his d e s p a i r w h e n he realizes, as in the a b o v e - m e n t i o n e d scene, the irreparable d e v a s t a t i o n of his mind. J o h a n n a Day a n d Ben S h e n k m a n provide great support, respectively, as Catherine's v e r y concerned but unabashedly achievemento r i e n t e d sister and as the g r a d u a t e student whose ambiguous character-is this a knight in shining a r m o r o r the g r a d u a t e student from h e l l ? - - i s y e t ano t h e r tribute to the p l a y w r i g h t ' s skill. The m a t i n e e a u d i e n c e w a s n o t a typical one. A b o u t half w e r e high s c h o o l students, but, m o u s e quiet, t h e y att e n d e d to the play raptly, s o m e t i m e s r e s p o n d i n g with a unified gasp o r sigh at narrative high points. My disapp o i n t m e n t was with the b e h a v i o r of s o m e o f the oldsters w h o h a d failed to h a r n e s s their electronic a c c e s s o r i e s - listening devices, beepers, cell telephones. Twice the t h e a t e r m a n a g e r had

to r e m i n d us t h a t these i m p l e m e n t s m u s t be controlled. The discomfiture of the a u d i e n c e at these incursions w a s extreme. The e n d of this p r o b l e m is n o w h e r e is sight, and soon I e x p e c t w e will r e a d in o u r morning p a p e r s of the first incident of B r o a d w a y t h e a t e 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 w e mathematicians are both blessed and bedeviled by our penchant for abstraction, that our gift both bonds us and separates us from the rest of manldnd, and the critic expresses surprise that the playwright has managed to reach into the forbidding terrain of intellectual pursuit and discover "good people." 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 n o t m u c h m a t h e m a t i c s as such in the play, only that required to establish context. I think a little m o r e would have e n h a n c e d m y r e s o n a n c e with the p l a y ' s central mystery. So m a n y of the g r e a t unsolved p r o b l e m s in n u m b e r t h e o r y can be communicated easily to a lay audience. Was the p r o o f that of the the twin p r i m e conj e c t u r e ? G o l d b a c h ' s conjecture? But the use o f m a t h e m a t i c s as a m e t a p h o r for the creative e x p e r i e n c e is unerring and totally dramatic. The play has m u c h to s a y a b o u t the corrosive effect of a c a d e m i c competitiveness, a n d its rueful o b s e r v a t i o n that truth often c h o o s e s as its h u m a n vessels t h o s e t o o fragile to c o n t a i n it is forcefully made. I can e x t e n d to this w o r k no g r e a t e r tribute t h a n to say that at the play's end, w h e n a c h a r a c t e r delivers t h e line, "This is a s t a n d a r d n u m b e r - t h e o r e t i c result," the hair on the b a c k of m y n e c k s t o o d up. A n d I'll b e t yours will too.

SPRINGER FOR MATHEMATICS K. JANICH, Universitiit Regensburg, Regensburg, Germany

ROBIN WILSON, The Open University, Oxford, UK; and JEREMY GRAY, The Open University, Milton Keynes, UK

MATHEMATICAL CONVERSATIONS Selections from the Mathematical Intelligencer

Since its first issue, The Mathematicallntelligellcer has been the main forum for exposition and debate between some of the world's most renowned mathematicians, covering not only history and history-making mathematics, but also including many controversies that surround all facets of the subject. This volume contains forty articles that were published in the journal during its first eighteen years. The selection exhibits the wide variety of attractive articles that have appeared over the years, ranging from general interest articles of a historical nature to lucid expositions of important current discoveries.

VECTOR ANALYSIS Classical vector analysis deals with vector fields; the gradient, divergence, and curl operators; line, surface, and volume integrals; and the integral theorems of Gauss, Stokes, and Green. Modem vector analysis distills these into the Carlan calculus and a general form ! ofStokes's theorem. This essentially modern text carefully develops vector analysis on manifolds and reinterprets it from the classical viewpoint (and with the classical notation) for three-dimensional Euclidean space, then goes on to introduce de Rham cohomology and Hodge theory. The material is accessible to an undergraduate student with calculus, linear algebra, and some topology as prerequisites. The many figures, exercises with detailed hints, and tests with answers make this book particularly suitable for anyone studying the subject independently.

I

2000/288 PP., 114 ILLUS./HAROCOVER/$34.95 ISBN 0387-98649-9 UNDERGRADUATE TEXTS IN MATHEMATICS

2000/488 PP./HAROCOVER/SS9.9S/ISBN 0387·9868&3

S. ABBOTI, Middlebury College. Middlebury, VT

UNDERSTANDING ANALYSIS

MARTIN AIGNER, Freie Universitiit and GUNTER M. ZIEGLER, Technische Universitiit, both, Berlin, Germany

PROOFS FROM THE BOOK Second Edition

The (mathematical) heroes of this book are "perfect proofs": 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-I inatorics, and Graph Theory. Thirty beautiful exampIes are presented here. They are candidates for The Book in which God records the perfect proofs - : according to the late Paul Erdos, who himself suggested many of the topics in this collection. The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background. For . this second edition several chapters have been revised and expanded, and three new chapters have been added.

I

2000/224 PP./HAROCOVER/S29.95/ISBN 3-540678654

l. BADESCU, Romanian Academy, Bucharest, Romania

ALGEBRAIC SURFACES

This book outlines an elementary, one-semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathemmatical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set ofpoints where a function is discontinuous be arbitrary? Can the rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it. 2001/264 PP./HARDCOVER/$39.9S/ISBN 0387-950605 UNDERGRADUATE TEXTS IN MATHEMATICS

Order Today! CAll: 1-8OO-SPRINGER • FAX: (201) 3484505

The main aim of this book is to present a completely algebraic approach (due essentially to Zariski, Mumford, and Bombierei, but inspired by the ideas of the classical Italian algebraic geometers) to the Enriques' classification of smooth projective surfaces defined over an algebraically closed field of arbitrary characteristic. This algebraic approach is one of i the novelties of this book among the other modern I textbooks devoted to this subject. Two chapters on surface singularities are also included. The book can be useful as a textbook for a graduate course on surfaces, for researchers or graduate students in Algebraic ' Geometry, as well as those mathematicians working in Algebraic Geometry or related fields. 2000/272 PP.,lSILLU5./HARDCOVER/$39.9S/ISBN 0387-98668-5 UNIVERSITEXT

WRITI: Springer-Verlag New York, Inc., Dept. 52532,

po Box

2485, Seaucus, NJ 07Q96.2485

VISIT: Your local technical bockstore.

E-MA~ [email protected] INS1RIJCTORS: Call or write for info on textbook exam copies

Your 30

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

No title

Recommend Documents