No title

MARTIN DAVIS

Exponentia and Trigonometric Functions From the Book

fter the piecemeal manner in which we learn about the exponential and trigonometric functions, the view presented to us when we finally gain access to the standpoint of complex function theory is a revelation. It is like the panorama spread out before us after an ascent, in which various familiar landmarks are seen for the first time as part of a coher­ ent whole. In the few weeks before writing this note, I had been entertaining myself while lying in bed during spells of insomnia by thinking through how one could use the tools of complex variable theory to develop the properties of these functions, starting from scratch. How would they be treated in Paul Erdos's "book" of optimal proofs? It all turned out to be pretty easy and, I thought, rather elegant. What We Can Use

We have available the following tools: •

•

A power series defines a function analytic within its cir­ cle of convergence, and the derivative of that function can be computed by term-by-term differentiation. Ifj' (z) 0 in a domain CZ!J, thenjis constant in that do­ main. (We can see this by setting z = x + iy, f(z) = u(x, y) + iv(x, y), and noting that the hypothesis implies =

au ax •

=

au ay

=

av ax

=

av ay

=

0) ·

Cauchy's theorem and its corollaries. In particular, if

f is analytic in the simply connected domain CZ!J, then J�f(()d� is well defined for any a, b E CZ!J, indepen­ dent of the path joining them. Moreover, for a E CZlJ the function

F(z)

=

r f(()d� a

is analytic in CZlJ and F'(z) = f(z) in CflJ. On the other hand, the presence of the term 2m in Cauchy's integral formula serves as a warning that it and its corollaries will not be available to us. In particular, we can't use the fact that an analytic function can be expanded into a power series converging to the function.

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25. NUMBER 1. 2003

5

The Addition Formula

Using

We define exp(z) =

x

L

n=O

n

� n.

and we have at once exp'(z) = exp(z) for all z, and exp(O) = 1. Now let G(z) = exp(z + c) exp( - z).

cos(yl + Yz) + i sin(yl + Yz) = exp(i (Yl + Yz)) = exp(iy1) exp(iyz) = (cosy1 + i siny1)(cos Y2 + i sin Yz) = [cos y1 cos Y2 - sin Y1 sin Yzl + .i [siny1 cosy2 + cos Y1 sin Y2L we obtain the addition formulas

Then, a simple calculation yields G'(z) = 0 so that

cos(yl + Yz) = cos Y1 cos Y2 - sin Y1 sin Yz sin(Yl + Yz) = sin Y1 cos Y2 + cos Y1 sin Y2·

G(z) = G(O)= exp(c). Thus we have proved

Periodicity

exp(z + c) exp( - z) = exp(c). Setting c = 0, we have: exp(z) exp( - z) = 1.

(1)

Let the simply connected domain C!lJ consist of all z = x + iy for which x > 0 or y > 0 (or both). That is, C!lJ is obtained from the complex plane by excising the quadrant on the lower left, together with its boundary. For z E C!lJ, we defme log(z) =

This implies that exp(z) is never equal to 0, that exp( - z) =

1

z -z1 d?.

1

indicated above, it follows from Cauchy's theorem that this analytic function is well defined, and that log(z)' = liz. Also log(l) = 0. For z E C!!J, let As

exp(z) '

and that exp(z + c) = exp(z) exp(c).

(2)

an aside, we note that the power series implies that exp(x) > 0 for x � 0, and that (1) then implies the same for all real x. Finally, because exp'(x) is positive, we see that the exponential function is increasing for all real x.

As

The Trigonometric Functions

Writing z = x + iy, we have exp(z) = exp(x + iy) = exp(x) exp(iy). Here exp(x) is real, and we define sine and cosine by setting

G(z) =

exp(log(z)) z

Then, G'(z) = 0 so that G(z) = G(1)

=

1, i.e.

exp(log(z)) = z Next we consider the defining integral for the log func­ tion taken over the path consisting of an arc of the unit cir­ cle 1z l = 1 from 1 to z. For each such point z = x + iy, let r = log(z) so that exp(r) = z. Writing r = p + ia we have z = exp(p)(cos a + i sina). Thus

exp(iy) = cosy + i sin y. Setting y = 0, we get cos(O) = 1, sin(O) = 0. Using exp(iy)' = i exp(iy) = -siny + i cosy, we have

1 = l,z l = so that p

Y(exp(p) cos a)2 + (exp(p) sin a)2 = exp(p),

=

0 and cos a = x; sin a = y. In other words, z = (cosa + i sina) = exp(ia).

(sin y)' = cosy; (cosy)' = - siny.

If s is the length

Thus, (sin2 y + cos2y)' = 0, so that sin2 y + cos2y = sin2 (0) + cos2 (0)

=

Since exp(-iy) =

�· )

exp �y

1 cosy + i siny cosy - i siny cos2y + sin2y = cosy - i siny, we have cos( -y) = cosy; sin( -y) = -siny.

6

J

THE MATHEMATICAL INTELLIGENCER

1.

of the circular arc from 1 to z, then we have

ds2 = dx2 + dy2 = sin2 a daZ + cos2a daZ = da 2 . Since ds= da and s = 0 when a = 0, we see that the pa­ rameter a is just the arc length. Applying these considera­ tions to z = i , the length a is simply the length of a quad­ rant of the unit circle, i.e., rr/2 where, as usual, rr denotes the length of the unit semi-circle. Therefore, exp(i rr/2) = i ; cos( rr/2) = 0; sin(rr/2) = 1. From this it follows, using the addition formula, that exp(i rr) = -1 and exp(2m) = 1. Therefore the exponential function is periodic with period 2m. Finally, sin(x + 2rr) = sin x; cos(x + 2rr) = cos x.

AUTHOR

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Choose from an array of pecialties

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Ford Prizes by the Mathematical Association of America and

mation Science, Environmental

the Leroy P. Steele Prize by the American Mathematical So­

Science, Economics & Bu ines ,

ciety. Davis has been on the faculty of the Courant Institute

Engineering, Geoscience, Life

of Mathematical Sciences of New Yorl< University since 1965,

Science, Mathematics, Medicine,

was one of the charter members of the Computer Science

Physic , and Statistics.

Department founded in 1969, and is now Professor Emeritus. Since retiring, his book The Universal Computer, written for the general public, has been published. He lives with his wife

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Virginia in Berl<eley, California, where he is a Visiting Scholar at the University. They have two children and four grandchil­ dren.

Scientific Computing Dynamical Systems

Thank You

Operator Theory

door unobseNed. Jet Wimp may have thought he

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could leave us with the usual absence of fuss. He

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collecting and his other non-mathematical passions.

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VOLUME 25. NUMBER 1 . 2003

7

�?·fii•i§ufhi¥1Uii;J§.i, Colin

Adams,

E�itor I

A Difficult Delivery Colin Adams The proof is in the pudding.

(}pening a copy of The

Mathematical

Intelligencer you may ask yourself uneasily, "What is this anyway-a mathematical journal, or what?" Or

"Oh, my god, it hurts. " "It's okay honey. You're almost there. " "It's splitting me wide open!" "You can do it, honey, " said Jeff "What did I do to deserve this?, " screamed Karen.

you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta­ tion

is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Column editor's address:

Colin Adams,

Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01 267 USA e-mail: [email protected]

8

hey had met in the Math Lounge while grad students. Although Karen was an algebraic geometer, and Jeff was a number theorist, it didn't seem to matter. Their love transcended the bounds of their respective mathe­ matical specializations. But little was expected of the union. Dr. Sylvia Vit­ tle, Karen's advisor, had urged her to reconsider. In her Austrian accent, she said, "There are lots of strong algebraic geometers out there. Look at Brogan from UCLA. Or Stigglemeyer from Brown. Why settle for a number theo­ rist?" But Karen knew her heart, and the two were married three weeks af­ ter they both received their Ph.D.s. One morning, four months into the marriage, as they sat at the kitchen table sipping their morning coffee, Karen cleared her throat. Jeff looked up from his morning paper, "Zero-Free Regions for Dirichlet 1-Functions." "Urn, Jeff, there's something I want to tell you." "Yes, honey, what is it?" "Remember that night two months ago when we stayed up until 3:00 in the morning talking about jet bundles?" He smiled provocatively. "Who could forget it?" "Well, about three weeks later, I found myself having trouble sleeping at night."

T

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

"Yes?" "I just couldn't seem to get some of the ideas out of my head. I was waking up in the morning feeling lousy." "Uh huh." "So, I guess what I am trying to say is that I think I may be with theorem." "Oh my god," gasped Jeff as he reached for her hand across the table. "Really? You think so? How can we find out for sure?" "Well, I have a test that Dr. Vittle gave me. Just a set of possible coun­ terexamples. We can see if it with­ stands them." "Okay, defmitely, let's do it. Should I do anything?" "No, just wait here. I can do it in the study. It shouldn't take more than a half hour. " Karen pulled the belt of her bathrobe tight, picked up a pencil and a pad of paper and marched off to the study. After waiting impatiently for 20 minutes, Jeff knocked on the door. "Karen, are you doing okay?" "Just a few more counterexamples to try, honey. Shouldn't take much longer. " Fifteen minutes later, Karen threw open the door. Jeff leaped up from the kitchen table. "So?" She threw her arms around him. "It's true. I am with theorem." "Yahoo!," said Jeff. "We're going to be published!" The next day, they made an appoint­ ment with Dr. Vittle. "Well, yes it is unusual, but it is not unheard of. Look at the Atiyah-Singer Index theorem. That was a product of a topologist and an analyst. But these matches are risky. I want to put you on a strict regimen of ten pages of alge­ braic number theory a day, say Cohen's book. "And when it is time for the theorem to come, it is impmtant to be ready. Have you considered taking a Lemmas

class? They are good preparation men­ tally and physically for the big event." Karen woke in the middle of the night in a cold sweat. Calculations raced through her head. It appeared that the kernel of the Sowklitz opera­ tor was in fact a left R-module. She grabbed Jeffs arm. "Jeff, Jeff, wake up. I think it's here!" Jeff leaped out of bed, already dressed. "Okay, look. Stay calm. I'll call Dr. Vittle. We can meet her down at the university. You get dressed." Karen stopped by the side of the bed. "Oh goodness, that was a big idea. It's coming fast. We need to hurry."

Dumped Dr. Vittle like a sack of old conjectures. She swore she would never get involved with another num­ ber theorist for as long as she lives."

They arrived at the university and raced up to the faculty lounge. Dr. Vit­ tle was waiting for them there with sev­ eral clean pads of paper. "Here, she said to Karen, "you sit here." She turned to Jeff. "Are you going to be here through the whole process?" "It's as much my theorem as hers," he said. "Just like a number theorist," laughed Vittle. "You make one small contribution nine months ago, and you think you have done all the work." "Hey, that contribution nine months ago was key. Without it there would be no theorem. " "Yes, but I don't see you i n much pain right now." "You don't like me very much, do you?" "Don't take it personally. I don't like any number theorists. I am going to go down to my office, but I will check on you in a bit." "What's her problem with number theorists?", asked Jeff as soon as Vittle had disappeared down the stairs. "Don't you know? She was collabo­ rating with Smythe, and one day he saw a talk on wavelets, the hot new thing.

wrote another

Karen sat at the table. Jeff paced back and forth. Every once in a while she would say, "Ah, urn, I think, maybe, . . . maybe, . . . oh, no not yet." Then sud­ denly, she screamed, "Jeff, this is it. Quick, get Dr. Vittle." Symbols spilled out on the page. It was agonizing and amazing at the same time. Jeff flew down the stairs three at a time, and re-

She furiously half pag e of equatio n s , and then , awestruck, wrote QED. turned almost immediately with Dr. Vittle. Karen was writing furiously. Vittle peered over Karen's shoulder. "Ah, yes, things are going well. Looks like a big one." Karen tore page after page off the pad. Vittle turned to Jeff and pointed at one of the sheets of paper. "If you don't think it would be too difficult for you, perhaps you could clean up that lemma there." Jeff sat down and slowly began to write. Karen was scribbling like mad, as sweat dripped off her brow. She was breathing heavily. Suddenly she tensed. "Oh, my god," she screamed. "It's huge." "It's okay, " coached Dr. Vittle. "Re­ lax and just let it come out."

Karen tore filled sheets off the pad, one after another. She sketched dia­ grams and figures. Equations with subindices on superindices flowed from her pen. Then her eyes opened wide. "I see it," she gasped. "I see it all." She furiously wrote another half page worth of equations, stopped suddenly and then, awestruck, wrote QED at the bottom of the page. There was a mo­ ment of complete silence and then she collapsed on the table. "Are you all right?," Jeff gasped, grasping her by the shoulders. "Of course she is all right," said Dr. Vittle. "Let her rest. She has just given birth." Karen raised her head slowly from the table, a beatific smile on her lips. "Where is it? Can I hold my theorem?" "Of course you can," said Vittle. She scooped up the pages, pulling one from Jeffs grasp, and laid the pile in Karen's arms. Karen cradled the pages care­ fully. "It's really beautiful isn't it?," she said. Vittle nodded. "It's a healthy theo­ rem, probably 9 to 10 pages in 12 point type. What will you name it?" Karen looked at Jeff tentatively. "Well, we were thinking of calling it the Bounded Co-Generation theorem, but after what we have just been through, I was thinking maybe the Constrained Optimization theorem." Jeff smiled. "I think that's a won­ derful name, honey." "Well, I would like to keep an eye on it tonight. Make sure it's immune to counterexamples," said Vittle. "And then in the morning, we will send it to the Annals of Mathematics." "The Annals?," Jeff gasped. "Even in my wildest dreams, I didn't imagine . . . . Oh, Karen, I love you." The two hugged each other, cradling the theorem between them, and even Dr. Vittle smiled.

VOLUME 25, NUMBER 1 , 2003

9

IGOR PAK

On Fine's Partition Theorems, Dyson, Andrews, and Missed Opportunities

istory almost never works out the way you want it to, especially when you are looking at it after the dust settles. The same is true in mathematics. There are times when the solution of a problem is overlooked simply by accident, due to a combination of unfortunate circumstances. In a celebrated address [D4], Freeman Dyson described several "missed opportunities, " in particular his own advance glimpse of Macdonald's eta­ function identities. I present here the history of Fine's par­ tition theorems and their combinatorial proofs. As the reader will see, many of the results could and perhaps should have been discovered a long time ago. There was a whole string of "missed opportunities." The central event is the publication of a short note [Fl ] by Nathan Fine. To quote George Andrews, "[Fine] an­ nounced several elegant and intriguing partition theorems. These results were marked by their simplicity of statement and [ . . . ] by the depth of their proof." [A7] Without taking anything away from the depth and beauty of the results, I will show here that most of them have remarkably simple combinatorial proofs, in a very classical style. Perhaps that's exactly how it should be with important results! Even a reader who prefers analytic methods may find that here the combinatorial approach fits the problem well.

10

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

Fine's partition theorems can be split into two (over­ lapping) categories: those dealing with partitions into odd and distinct parts, a la Euler, and those dealing with Dyson's rank I shall separate these two stories, as they have relatively little to do with each other. The fortune and misfortune, however, had the same root in both stories, as you will see. Fine's note [Fl] didn't have any proofs; not even hints on complicated analytic formulae which were used to prove the results. It was published in a National Academy of Sci­ ences publication, in a journal devoted to all branches of science. Thus the paper was largely overlooked by subse­ quent investigators. The note contained a promise to have complete proofs published in a journal "devoted entirely to mathematics." This promise was never fulfilled. Good news came from a different quarter. In the sixties, George Andrews, while a graduate student at the Univer­ sity of Pennsylvania, took a course of Nathan Fine on ba-

sic hypergeometric series. As he writes in his mini biogra­ phy [AS], "His course was based on a manuscript he had been perfecting for a decade; it eventually became a book [F2 ] . " In fact, the book [F2] was published only in 1988, ex­ actly 40 years after the publication of [Fl]. It indeed con­ tained the proofs of all partition results announced in [Fl]. Meanwhile, Andrews kept the manuscript and used it on many occasions before [F2] appeared. Among other things, Andrews gave new analytic proofs of many results, found connections to the works of Rogers and Ramanujan, and, what's important for the subject of this paper, gave com­ binatorial proofs to some of the theorems. Much of the fame Fine's long-unpublished results now have is owing to An­ drews's work and persistence (see [Al-AS]). This is where the story splits into two. The rest of this article is largely mathematical, dealing separately with each of Fine's partition theorems. To simplify the presentation, I change their order and use different notation. I conclude the discussion with Dyson's proof of Euler's Pentagonal Theorem and a few more surprises. A few words about the notation. I denote partitions of n by A (;\ 1 , A2, . . . , At), and I write A 1- n, or ,A = n. Let A' be the conjugate partition to A. The largest part and the number of parts of A are denoted by a(;\) and €(;\), re­ spectively. Every partition A may be represented graphi­ cally by its Young diagram [A]; recall that one definition of A' is the condition that [A'] is the transpose of [A ] . See [A3] for standard references, definitions, and details. =

Partitions Into Distinct Parts and Franklin's Involution

THEOREM 1 (Fine) Let 5:?, and CiJJh b e the sets of partitions A of n into distinct parts, such that the largest part a(A) = A 1 is even and odd, respectively. Then -

12/lA

=

{

1, -1, 0,

if n = k(3k+l)/2 if n = k(3k - 1)/2 otherwise.

It is perhaps suggestive to compare Theorem 1 with the similar-looking Euler's Pentagonal Theorem, which can be stated as follows: THEOREM 2 (Euler) Let 229, and 92�, be the sets of partitions A of n into distinct parts, such that the number of parts t(A) = Ai is even and odd, respectively. Then

·22"0 1 0

_

122"11 .u

_

-

{

C- l t , 0,

if n k(3k otherwise. =

Proof Denote by 91;, 20� U 9£h. the set of all partitions into distinct parts. Let A E CiJJ11, and let [A] be the Young di­ agram corresponding to A. Denote by s(A) the length of the smallest part in A, and by b(A) the length of a maximal se­ quence of subsequent parts: a, a - 1, a 2, . . . , where a = a(A) = A 1 . One can views(;\) and b(A) as the lengths of the horizontal line and diagonal line of squares of [ A ] , as in Figure 1. Now, if s(A) s b(A), move the horizontal line to attach to the diagonal line. Similarly, if s(A) > b(A), move the diagonal line to attach below the horizontal line. If we cannot make a move, stay put. This defines Franklin's in­ volution a: C)) , � CiJJ , . Note that a changes parity of the number of parts, ex­ cept when A is a fixed point. Observe that the only fixed points of the involution are the Young diagrams where the lines overlap, and s(A) - b(A) is either 0 or 1 (see Fig­ ure 2). The number of squares in these diagrams are m(3m ± 1)/2, which are called pentagonal numbers. 92� is 0 unless n is a pentagonal number, Therefore, 192�! and is ± 1 in that case. This proves Theorem 2. Similarly, note that a changes parity of the largest part. '2/JA is 0 unless n is a pen­ Thus again, we infer that lrzv�l tagonal number, and is ± 1 in that case. This completes the proof of Theorem 1 . D =

-

The following result is straight from [Fl ] :

i21J�i

blance to the famous pentagonal theorem of Euler, but we have not been able to establish any real connection be­ tween the two theorems." In the Math. Reviews article [L], Lehmer reiterates: "This result parallels a famous theorem of Euler. " As I shall show, Theorem 1 has a proof nearly identical to the famous involutive proof by Franklin of Theorem 2. Franklin was a student of Sylvester at Johns Hopkins Uni­ versity, active in Sylvester's exploration of the "construc­ tive theory ofpartitions." He published his proof [ Fr] right before the publication of a celebrated treatise [Sl] by Sylvester (to which Franklin also contributed). These two papers laid the foundations of Bijective Combinatorics, a field which blossomed in the second half of the twentieth century. Of course, it is hard to blame Fine for not discovering the connection. In those days bijections were rarely used to prove combinatorial results. Since the late sixties, how­ ever, the method became popular again, with a large num­ ber of papers proving partition identities by means of ex­ plicit bijections. Franklin's involution was far from forgotten, and was used on many occasions to prove vari­ ous refinements of Euler's Pentagonal theorem [KP] , and even most recently to prove a new partition identity [ C ] . It is a pity that an application to Fine's theorem remained un­ noticed for so many years.

±

1)/2

Of course, this similarity was not overlooked. Fine him­ self acknowledged that Theorem 1 "bears some resem-

-

Figure 1. Young diagram [A] corresponding to a partition A = (9,8,7,6,4,3). Here s(A) = 3, b{A)

-

=

4, and a(,\)= (1 0,9,8,6,4).

VOLUME 25. NUMBER 1, 2003

11

2m-

2m

m

m

�/

...___ _ ......

m

m+l

7

Figure 2. Fixed points of Franklin's involution. Partitions Into Odd Parts and Sylvester's Bijection

Now I recall another famous theorem of Euler: that the number of partitions of n into odd numbers is equal to the number of partitions of n into distinct numbers. Here is an­ other gem from [F1 ] : 3 (Fine) Let O h and O g b e the sets of partitions A of n into odd parts such that the largest part a(A) is 1 and 3 mod 4, respectively. Then

THEOREM

,,O1n1- l'riAO �n l•

101' n

=

'qnnll,

if n is even, if n is odd.

Clearly Fine's Theorem 3 is a refinement of Euler's the­ orem. As we shall see shortly, the following result of Fine [F2] is an extension: 4 (Fine) For any k > 0, the number of partitions n into distinct parts such that a(f.L) = k, is equal to the number of partitions A 1- n 'into odd parts such that a(A) + 2€(,\) = 2k + 1 . THEOREM fL 1-

I n his early paper [A 1 ] , Andrews proved Theorem 4 com­ binatorially, but never noticed that it implies Theorem 3. The reason could be that Theorem 3 was coupled with The­ orem 1 in [F1 ] , while the proofs use two different classical combinatorial arguments. The proofs of Theorem 3 and Theo­ rem 4 follow from Syl­ vester's celebrated bi­ jection, sometimes called a fish-hook construc­ tion. This bijection is a map between partitions into odd and distinct numbers, and gives a combinatorial proof of Euler's theorem (see [A1,A3]). Sylvester's bijection is another fixture in the combina­ torics of partitions. It has been restated in many different ways (e.g., using Frobenius coordinates and 2-modular di­ agrams [A6,B,PP]), and was used to prove other refine­ ments of Euler's theorem [KY] . Had Theorem 3 been bet­ ter known and not omitted in [L] , the following proof could have been standard.

Proof Denote by On = Oh U 0� the set of all partitions into odd parts. Define Sylvester's bijection
This story started in 1944 with Dyson's paper [D1], which appeared in Eureka , a publication of mathematical students in Cambridge. Motivated by Ramanujan's identities for di­ visibility of the partition function, Dyson introduced the rank of a partition, which he conjectured would give a com­ binatorial interpretation of these identities. Still an under­ graduate, Dyson did not prove these conjectures. They were resolved in 1948 by Atkin and Swinnerton-Dyer [At], although their celebrated paper [AS] appeared some years later. Fortuitously, Dyson meanwhile had moved to the U.S. and published his conjectures as a short problem in the American Mathematical Monthly [D2]. Nathan Fine be­ came interested in the problem and devoted three theorems in [F1 ] to enumeration of partitions with given rank (see below). Taken out of context, his results seemed com­ pletely mysterious, and would have remained so if not for publication of the book [F2] and Dyson's paper [D3 ] . We know now that Fine's results were based on the third-order mock theta function identities due to Watson [W], a technique used in [AS] as well. Dyson's paper [D3] (see also [D6]) was aimed at finding a simple proof of the formula for a gen­ erating function for partitions with given rank This formula was used as a tool for practical calculations in [D 1 ] and later was established in [AS ] . Unaware of Fine's work, Dyson rediscovered one of Fine's then-unpublished equa­ tions, called it a "new symmetry," and proved it combina­ torially. He then deduced the desired formula, and obtained a new proof of Euler's Pentagonal Theorem (see below). I refer to [D5] for Dyson's personal and historical account of these discoveries.

N obody seems to have n oticed

that Dyson's map can be u sed

to g ive combi natorial p roofs of Fi ne ' s resu lts .

Figure 3. Sylvester's bijection (/): (7,5,3,3)---> (7,6,4, 1 ) .

12

THE MATHEMATICAL INTELLIGENCER

l l I I

...

f-'..... Figure 4. Dyson's map t/Jr:A--. JL, where A= (9,7,6,6,3,1) JL = (8,8,6,5,5,2) E '!l32+r,r-1• and

r=

By Theorem 5, and from the formula 7T(n) = h(n, 0) + g(n, 1) = h(n, 0) + h(n, -1), we deduce equation 1):

I l I I

2.

E

;Jt32,r+1,

Unfortunately, except for Andrews's paper [A5], nobody seems to have noticed that in fact Dyson's map, sometimes called Dyson 's adjoint [BG] , can be used to give combina­ torial proofs of Fine's results. Even Andrews did not seem to realize that Dyson's map proves two other theorems of Fine as well. I return to that Andrews paper in the next section. Define the rank of a partition A as r(A) = a (A)- € (A). Denote by �n,r the set of partitions of n with rank r, and let p(n , r) = 1�11.,.. Similarly, denote by 'Jen,r (Cfl11,,.) the set of partitions of n with rank at most r (at least r). Let h(n, r) 1£11,,., g(n, r) = Cfl11,,.:. Clearly, p(n, r) = h(n, r)- h(n, r - 1), and (by comparing A to A') g(n, r) = h(n, -r). Also, h(n, r) + g(n, r + 1) 1r( n), where 7T(n) = h(n, n) = I,. p(n, r) is the total number of partitions of n. =

=

THEOREM

5 (Fine) For all n > 0 , we have h(n, 1 + r) =

h(n + r, 1 - r). Proof I shall construct an explicit bijection if;,.: 'Jen,r+1--> C§n +r,r-l• which implies the result. Start with the Young diagram [A] corresponding to a partition A E € (A) squares. Add 1£11,,.+ 1. Remove the first column withe the top row with (€ + r) squares. Let [M] be the resulting Young diagram (see Figure 4.) Call the map if;,. : A - -> IL =

Dyson's map.

By assumption of A, we have r(A) = a (A) - e ::; r + 1, so a (f.t) € + r2 a(A) - 1 . Thus fL is a partition indeed. And the same inequality shows that the inverse map is defined. Clearly, 1f.t1 A -e + (€ + r) n + r. Also, r(/L) = a(/L) - f(M) = f(A) + r - (A2 + 1) 2r - 1 . There­ fore, fL = lj;,.(A) E C§n+r,r-l, which completes the proof. D =

=

=

I call the result of Theorem 5 the Fine-Dyson relations. The rest of the paper is built upon these relations and Dyson's map. First I prove the following four equations, which are listed in [F1] as one theorem as well. THEOREM

6 (Fine) We have:

7T(n + 1) - 7T(n) = (h(n + 1, 0) + h(n + 1, - 1)) - (h(n, 0) + h(n, - 1)) = (h(n + 1 , 0) - h(n + 1 , - 1)) + (h(n, 0) - h(n, -1)) + 2 h(n + 1 , -1) - 2 h(n, 0) = p(n + 1, 0) + p(n, 0) + 2(h(n - 1, 3) - h(n - 1, 2)) = p(n + 1, 0) + p(n, 0) + 2p(n- 1, 3). Equation 4) follows in a similar manner:

p(n - r - 3, r + 4) - p(n - r- 2, r + 3) (h(n - r- 3, r + 4) - h(n - r- 3, r + 3)) - (h(n- r - 2, r + 3) - h(n - r- 2, r + 2)) = h(n, -r - 2) - h(n 1, -r - 1) - h(n, -r - 1) + h(n - 1, -r) - (h(n, -r- 1) - h(n, -r - 2)) + (h(n- 1, -r) - h(n - 1, - r - 1)) = -p(n, -r- 1) + p(n - 1, -r) = -p(n, r + 1) + p(n - 1, r). =

-

=

This completes the proof. 0 Since the equations in Theorem 6 follow immediately from the Fine-Dyson relations, one can obtain combinato­ rial proofs for these equations as well, by separating terms with positive and negative signs and then using Dyson's map to obtain identical sets of partitions on both sides. I present a variation on such a proof in case of another the­ orem from [F1]. THEOREM

7 (Fine) For r2n - 3, we have 7T(n) - 7T(n- 1) =

p(n + r + 1, r). Proof Denote by 9Ji 11 the set of partitions A 1- n with the smallest parts(A)22.0bserve thati9Ji11, = 7T(n) - 7T (n-1). Indeed, one can always add a part (1) to every partition v 1- n - 1 to obtain all partitions of n, except for those in '!Fn· Now, to a partition A E 9Fn apply Dyson's map lj;,.+1: A--> f.t, corresponding to the rank (r + 1). We have /Lt = 1 + f(A) + r22 + (n - 3) = n - 1 . On the other hand, f.tz = A1 - 1 ::; n - 1 by construction. Therefore /Lt 2f.tz, and IL is a partition indeed. Because s (A) > 1, we know that f(M) € (A) + 1. Thusr(f.t) (f(A) + r + 1)- (£ (A) + 1) = r, so fL E �n+r+l,r· Since the map is clearly reversible, we ob­ tain the result. 0 =

=

The Iterated Dyson's Map

1) p(n + 1, 0) + p(n, 0) + 2p(n - 1, 3) = 7T (n + 1) - 7T(n), for n > 1, 2) p(n - 1, 0) - p(n, 1) + p(n- 2, 3) - p(n - 3, 4) = 0, for n > 3, 3) p(n - 1, 1) - p(n, 0) + p(n - 1 , 2) - p(n - 2, 3) 0, for n > 2, 4) p(n - 1, r) - p(n, r + 1) + p(n - r - 2, r + 3) - p(n r - 3, r + 4) O, jor n > r + 3. =

=

Proof Taking r = 0 and r = -1 in equation 4), and us­ ing p(m, r) = p(m, - r), we obtain 2) and 3), respectively.

mentioned before, Andrews in [A5] proved combinato­ rially the following theorem from [F1]:

As

8 (Fine) Let CZDn,r be the set of partitions fL E CZDn with rank r(/L) r. Let UUn,Zk+l be the set of partitions A E On, such that the largest part a(A) = 2k + 1. Then: THEOREM

=

,UUn, 2r+ll

=

:rzDn,2r +li + rzD n,Zrl·

One can view Theorem 8 as another refinement of Euler's theorem on partitions into odd and distinct parts. Andrews showed in [A5] that the theorem follows easily

VOLUME 25. NUMBER 1 . 2003

13

o-

IIIII-O±J1-

Figure 5. The iterated Dyson's map (:A---> J.L, where A = (5,5,3,3,1)

E

from the properties of Dyson's map !fi,.. It is unfortunate that Andrews's proof was published in a little-known jour­ nal and was never studied further. I will now present a di­ rect bijection between ()n and CZJJ, which is different from Sylvester's and Glaisher's bijections [A3], and which proves Theorem 8. Naturally, this construction is moti­ vated by [A5]. Let A= CA1, Az, ... , At) E ()n be a partition into odd parts. Consider a sequence of partitions vl, v 2, ..., ve, such that ve= (Ac), and vi is obtained by applying Dyson's map !fiA; to vi+I.Now let p.,= v1. Call the resulting map (: A� p., the iterated Dyson's m ap . See Figure 5 for an example.

l=Ef:Jitii­

II I I

�1117,5 and J.L = (8,6,2, 1) E ':£17,4•

Dyson's Proof of Euler's Pentagonal Theorem

I already mentioned that Dyson used his map to obtain a simple proof of Euler's Pentagonal Theorem, Theorem 2 above. He writes, "This combinatorial derivation of Euler's formula is less direct but per­ haps more illuminating, than the well-known combinator­ ial proof of Franklin. " [D3] Twenty years later he adds, "This derivation is the only one I know that explains why the 3 appears in Euler's formula." [D5] Here is how Dyson's proof goes. Let P(t) and G,.(t) be the generating functions for all partitions of n, and all par­ titions of n with rank 2: r :

Dyson used his map to

obtai n a proof of Euler ' s Pentagonal Theorem .

X

9 The iterated Dyson's map (defined above is a bijecti.on between 011 and CZJJ, . Moreover, �CUUn,2r+I)= CZl!n,2r U CZVn2r+ J, for all r 2: 0. Clearly, Theorem 9 implies Theorem 8. It would be in­

THEOREM

teresting to find further applications of the map ( to other partition theorems.

G,.(t)

=

X

P(t)= 1+

I

n�l

I

n =l

g(n, r) t" , X

7r(n)t"=

1

n

i�I

+

(1 - fi)

.

Write the relations h(n, r) + g(n, r 1)= 1r(n) and the Fine-Dyson relations h(n, 1 + r) h(n + r, 1- r) in terms of g( · ) alone: =

Proof First, note that ;vi = A;+ A;+ I+ ·+ A1. There­ fore !f..tl I v11= IAI= n, as required. Let us prove by induc­ tion that .,) is a partition into distinct parts, such that r( vi) is either A; or A; - 1. The base of induction, when i £ and vi= (A,), is obvious. Suppose the claim holds for vi+I, i.e., a(vi+1)- £(vi+ I) is either A; + I or Ai + I+ 1, depending on the parity. Since a(vi)= £(vi+I) + A;, we have ·

·

=

=

(vi) = a(vi) 1

2: (a(vi+I)- A;+I)+ A;> a(vi+I)- 1= (vi)2,

and this inequality is maintained ((vi)2- (vi)3= (vi+1 )1 (vi+1)2 > 0, and so on); this implies that vi is indeed a par­ tition into distinct parts. Now, observe that £(vi) = £(vi+I) or £(vi+I)- 1. We have

r(vi)= a(vi)- £(vi)= (£(vi+!)+ A;)- £(vi) E {A;,A;- 1) , which proves the induction step. Note that we never used the fact that A E 011• This be­ comes important in the construction of the inverse map C1. Define the map g-I by induction, starting with p.,= v1 and applying the inverses of Dyson's maps ljJ; 1.Clearly, the only freedom in the construction comes from the choice of r. But we need to have r= a( vi) - £(vi) or r= a(vi)- £(vi)- 1, and r has to be odd; this makes the choice of r unique. Therefore the map C 1 is well defined, and g is a bijection. The second part of the theorem is im­ mediate from the arguments above. This completes the proof. D 14

THE MATHEMATICAL INTELLIGENCER

g(n, r) + g(n, 1 - r)= 7r (n), g(n, r)= g(n - r- 1 ,-2 - r). In the language of generating functions, these relations im­ ply the following two equations:

1+ G,.(t)+ G I-r (t)= P(t),

G,.(t)= t"+1(1 + G-2-r(t)).

Here 1 in both equations comes from taking into account the "empty" partition. Thus we have G,.(t)= t"+1 P(t)- c·+l Gr+3 (t).

Iterating the above equation, we obtain: G,.(t)= tr+l P(t)- tr+l G r 3 (t) + = tr+l P(t)- t2r+5 P(t) + t2r+5 Gr+G(t) t2r+5 P(t) + t3r+12 P(t) = rr+ l P(t) t3r+12 G,.+g(t) _

_

x

=

I

Jrt=l

( -1)rn- l t

m(;1m

2

) -1

+

,.,

P(t).

Substituting this into P(t) - G0(t)- G (t)= 1, we deduce 1 Euler's Pentagonal Theorem:

(1

+

i

Jn=l

( -1)1'' t

m(:l

•�

.

1)

+i

tn=l

)

m •�-1 ) = 1.

(-1)"' t (:l

Dividing both sides by the product P(t) and equating the coefficients gives us Theorem 2. In fact, Euler [E] was interested in the recurrence rela­ tion for the number of partition ?T(n). The above formula implies

?T(n) = ?T(n - 1) + ?T(n- 2)- ?T(n - 5)- ?T (n- 7) + ?T(n - 12) + ?T (n 15) - . . . -

By analogy, Dyson [D5] obtained the following refinement of Euler's recurrence:

g(n, r) = ?T(n

-

r- 1) - ?T(n - 2r- 5)

+ 7T(n - 3r

-

1 2) -

Naturally, one is tempted to convert the above simple analytic proof into a bijective proof of both recurrences. This turns out to be possible. Denote by 'ZP11 the set of all partitions of n. Write Dyson's recurrence as follows:

'HII. -r = '!Pn-r-1 - gpll_2r-5 + 21'11-:!r-12 iC§n-r-1,-r-2 U Z�t'n-r-1,-r-:3! - 1C§II-2>·-5,-r-5 U Z�t'll-2r-:),-r-61 + ;ctln-:3r-12,-r-8 U Z�t'n-3r-12,-r-fl IC§n-r-1,-,·-d + (IZ�t'n-r-1,-r-31- lctin-2r-5,-r-5) - (Z�t'n-2r-5.-r-6 - 'C§n-3r-12.-r-s) + =

=

·

·

·

Now Dyson's map 1/J-r-1 gives a bijection between the left hand side and the first term on the right hand side of the equation. Similarly, maps 1/J-r-4, 1/J-r-7, etc., give bijections for the terms in the brackets. Thus we have a simple bijective proof of Dyson's recurrence. One can view the above bijection as a sign-reversing involution on the set of partitions A E ':ff11,-n or A� n - rm - m(3m - 1)/2, where

m ::::l .

Similarly, after combining two involutions for r = 0 and 1 , we easily obtain an involution 1' proving Euler's recur­ rence: 1':

U

m even

rPn-m(3m-l)/2 �

U

m odd

rPn-m(3m-1)/2,

where m on both sides is allowed to take negative integer values, and the map 1' (see Figure 6) is defined by the fol­ lowing rule:

if r (A) + 3m :s: 0, if r (A) + 3m > 0 . Now comes a final surprise. Bijection 1' is in fact well known! In this exact form it was discovered in 1985 by Bres­ soud and Zeilberger [BZ] , for the sole purpose of finding a simple proof of Euler's recurrence. The authors, completely unaware of Dyson's proof, managed to rediscover a version of Dyson's map anyway. It seems, the Fine-Dyson relations and Dyson's map ljf,. are simply so fundamental they resur­ face despite the "missed opportunities" . . . Final Remarks

There remains one last partition theorem of Fine [F1] with­ out a simple combinatorial proof. Let L(n) be the number of partitions A� n with odd smallest part s (A). The theo­ rem states that L(n) is odd if and only if n is a square. Shouldn't one look for an involution proving this result? Any interested reader may draw inspiration from an invo­ lutive proof of the Rogers-Fine identity [A2]. The conditions in Fine's theorems 6 and 7 are slightly changed in this paper, either to correct or simplify the re­ sults (so as not to define p(n, r) for n :s: 0). Dyson's map as defined here is the conjugate of the one in the literature. I find this version somewhat easier to work with. The iterated Dyson's map g appears to be new. It is ba­ sically a recursive application of Andrews's recurrence re­ lation for lcJU11,2k+1 and :rzn11,r (see [A5]). Whether this bijec­ tion between partitions into odd and distinct numbers has other nice applications or not, the map ? seems to give a natural proof of Theorem 8, just as Dyson's map gives a natural proof of Theorem 5. Unfortunately, the iterative construction of g is perhaps intrinsic. As Xavier Viennot once tole me, "Sometimes, a recursive bijection is the only one possible and one cannot do better." Note that Dyson's proof of Euler's recurrence relation [D3] produces a bijection almost immediately once one employs Dyson's map. A different bijection, based on Franklin's involution, was obtained by means of the invo­ lution principle by Garsia and Milne [GM]. These two "au­ tomatic" approaches challenge Sylvester's paradigm that bi-

...

m

Figure 6. Bijection 'Y proving Euler's Pentagonal Theorem.

VOLUME 25, NUMBER 1, 2003

15

jections "should rather be regarded as something put into the two systems by the human intelligence than an absolute property inherent in the relation between the two [sets)" [S2]. In a recent paper [BG], Berkovich and Garvan defined a 2-modular version of Dyson's map. They used this new map to give a combinatorial proof of Gauss's famous identity. It would be interesting to convert this proof into a fully bi­ jective proof of the identity and compare with Andrew's in­ volutive proof [A2 ] . Similarly, one can consider an iterated version of this map and try to find new partition theorems this construction may prove. This paper was motivated in part by the following quote: "[A5] seems to be the only known application of Dyson's transformation" [BG]. Let me add that had the preprint [BG] never been put on the internet, this paper might have never been written. That would have been another "missed opportunity" . . .

[D3] F. J. Dyson , A new symmetry of partitions, J . Combin. Theory 7 (1 969), 56-61 . [D4] F. J. Dyson, Missed opportunities, Bul l . Amer. Math. Soc. 78 (1 972), 635-652. [D5] F. J. Dyson, A walk through Ramanujan's garden, in Ramanujan revisited, Academic Press, Boston, 1 988, 7-28.

[D6] F. J. Dyson , Mappings and Symmetries of Partitions, J. Combin. Theory Ser. A 51 (1 989), 1 69-1 80 [E] L. Euler, lntroductio in analysin infinitorum. Tomus primus, Marcum­ Michaelem Bous-quet, Lausannae, 1 748. [F1 ] N. J. Fine, Some new results on partitions, Proc. Nat. Acad. Sci. USA 34 (1 948), 61 6-61 8. [F2] N. J. Fine, Basic hypergeometric series and applications, AMS, Providence, Rl, 1 988. [Fr] F. Franklin, Sure le developpement du produit infini (1

-

x)

(1 - x2)(1 - x3) . . . , C. R. Acad. Paris Ser. A 92 (1 881 ) , 448-450.

[GM] A. M. Garsia, S. C . Milne, A Rogers-Ramanujan bijection , J. Com­ bin. Theory Ser. A 31 (1 981 ) , 289-339.

Acknowledgments

I am very grateful to Oliver Atkin for sharing invaluable his­ torical comments on the discovery and the proof of Dyson's conjectures. I also thank Michael Noga of the MIT Science Library for obtaining a copy of [A5] not available in the li­ braries of the Commonwealth of Massachusetts. The au­ thor was partially supported by the NSA and the NSF.

[KY] D. Kim, A. J . Yee, A note on partitions into distinct parts and odd parts , Ramanujan J. 3 (1 999), 227-231 .

[KP] D. E. Knuth, M. S. Paterson , Identities from partition involutions, Fibonacci Quart. 16 (1 978), 1 98-2 1 2. [L] D. H. Lehmer, Math. Reve i ws 1 0,356d, and Errata 1 0,856 (on paper [F1]). [PP] I. Pak, A. Postnikov, A generalization of Sylvester's identity, Dis­ crete Math. 1 78 (1 998), 277-281 .

[S1 ] J. J. Sylvester, with insertions by F. Franklin, A constructive the­ REFERENCES

ory of partitions, arranged in three acts, an interact and an exodion,

[A 1] G. E. Andrews, On basic hypergeometric series, mock theta func­

Amer. J. Math. 5 (1 882), 251-330.

tions, and partitions

(II), Quart. J . Math.

1 7 (1 966), 1 32-1 43.

[A2] G . E. Andrews, Two theorems of Gauss and allied identities proved arithmetically, Pacific J . Math. 41 (1 972), 563-578.

[A3] G. E. Andrews, The Theory of Partitions, Addison-Wesley, Read­ ing , MA, 1 976. [A4] G . E. Andrews, Ramanujan's "Lost" Notebook. I. Partial (}-Func­

[S2] J. J. Sylvester, On a new Theorem 1n Partitions, Johns Hopkins University Circular 2: 22 (April 1 883), 42-43. [W] G. N. Watson, The final problem: An account of the mock theta­ functions, J . London Math. Soc. 1 1 (1 936), 55-80. AUTHOR

tions, Adv. Math. 41 (1 981 ) , 1 37-1 72.

[A5] G. E. Andrews, O n a partition theorem o f N. J . Fine, J . Nat. Acad. Math. India 1 (1983), 1 05-107. [A6] G . E. Andrews, Use and extension of Frobenius' representation of partitions, in "Enumeration and design", Academic Press, Toronto,

ON, 1 984, 5 1 -65. [A7] G. E. Andrews, Foreword to [F2] , 1 988. [AS] G. E. Andrews, Some debts I owe, Sem. Lothar. Combin. 42 (1 999), Art. B42a. [At] A. 0. L. Atkin, personal communication, 2002 . [AS] A. 0. L. Atkin, H. P. F. Swinnerton-Dyer, Some Properties of Par­

IGOR PAK

titions, Proc. London Math. Soc. (3) 4 (1 954), 84-106.

[BG] A. Berkovich, F. G. Garvan, Some Observations on Dyson's New Symmetries of Partitions,

preprint (2002) , available from

http://www.math.ufl.edu/�frank.

Department of Mathematics Massachusetts lnst1tute of Technology Cambridge, MA 021 39-4307 e-mail: [email protected]

[B] C. Bessenrodt, A bijection for Lebesgue's partition identity in the spirit of Sylvester, D iscrete Math. 132 (1994), 1-10.

[BZ] D. M. Bressoud, D. Zeilberger, Bijecting Euler's Partitions-Recur­ rence, Amer. Math. Monthly 92 (1 985), 54-55.

[C] R. Chapman, Franklin 's argument proves an identity of Zagier, El. J. Comb. 7 (2000), RP 54. [D1 ] F. J. Dyson, Some Guesses in The Theory of Partitions, Eureka (Cambridge) 8 (1 944), 1 0- 1 5 . [D2] F . J . Dyson, Problems for Solution: 4261, Amer. Math. Monthly 54 (1 947), 4 1 8. 16

THE MATHEMATICAL INTELLIGENCER

Igor Pak was bom in Moscow and educated at Moscow Uni­ versity and at Harvard (Ph.D. 1 997). After several temporary appointments, he had the good fortune to become Assistant Professor at MIT. In his free time he enjoys driving, bicycling, and roller­ blading. He is keenly conscious of his indebtedness to the in­ ventor of the wheel, and he hopes the future holds further de­ velopments in this direction.

l ,0, ffij.t§ @ih$11+J.JI.Irrll!,iih¥J ..

M a rj o r i e S e n e c h a l ,

E d itor

uch though I admired the Intelli­ account of the MASS pro­ gramme (vol. 24, no. 4, 50-56), grafting the best aspects of the formidable tra­ dition of the Russian school of mathe­ matics into the framework of US undergraduate mathematics training­ and much though I hope that students from my own alma mater will number amongst the participants in MASS courses-I was left thinking, Yes, this is a fine way to move into mathemat­ ics, but is it the unique best way? A for­ tuitous event had led me to look back at my own undergraduate training, which was very different. In October 2000 I received an invi­ tation to return to my undergraduate school, Smith College, in Northamp­ ton, Massachusetts, to give a talk at the first Smith College Alumnae Mathe­ matics Conference. It seemed a good idea at the time; I responded positively, with no greater expectations of the weekend than an opportunity to return for the first time in over twenty-five years to a place I remembered fondly. I certainly had no inkling that the ef­ fect of that weekend in April would be a comprehensive rewriting of my atti­ tudes about suitable undergraduate training for mathematicians.

Undergraduate MgenceT Training Revisited: Thoughts on an Unusua l Reunion by Marjorie Batchelor

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our 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 su bmissions to the Mathematical Communities Editor, Marjorie Senechal,

Department

of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail: [email protected] .edu

Smith College: A Promising Starting Ground for Working Mathematicians?

Those of us who were brought together at the Conference had had an under­ graduate experience quite unlike that at large universities or technical schools. That different experience which we shared, albeit at various pe­ riods during the last thirty years, is our common source, the defining property of our community. It appears to have given us all an approach to mathemat­ ics which has proved durable enough to see us through difficult bits, flexible enough to allow us to adapt to a wide variety of careers, and in its own way, solid enough to enable us to make our way in a competitive field. This is why

I

it seems important to write about it. So what is Smith? Smith College is a liberal arts college in western Massachusetts. The liberal arts degree course presents a consider­ able contrast to traditional undergradu­ ate mathematics courses offered at UK universities for example. British maths students expect to apply the great ma­ jority of their academic effort to math­ ematics. Their counterparts at a liberal arts college will be required to spend at least half their time studying courses (well) outside mathematics. Thus I stud­ ied, in my time, French poetry, history, Old English (Beowulf, all of it), chem­ istry, and Latin, amongst many other subjects. Whereas British students are expected to enter university with a sub­ stantial background in mathematics, in liberal arts colleges rank beginners are energetically encouraged to try mathe­ matics (you might like it), and the pos­ sibility for proceeding from humble be­ ginnings to a degree in mathematics is always available. Indeed I was exactly one such student, who signed up for a first course in calculus, expecting it to be my last. The wonderful thing about liberal arts courses is that surprising things can and do happen. A student ex­ pected to make a serious study of French literature can discover algebra instead. Smith, like many liberal arts col­ leges, is a teaching institution pre­ dominantly, and not a research insti­ tution. Like many others, it has a relatively small student body, and a correspondingly small faculty. The corollary to the "try it, you might like it" policy at Smith has been that while there has been a good number of stu­ dents majoring in mathematics, these students differ markedly in interests, ability, and ambitions. The student with the potential to become a work­ ing mathematician remains a rarity. The variety, level, and presentation of the courses offered obviously must aim to suit the demands of the major­ ity of the students.

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25. NUMBER 1 2003

17

Smith College is a women's college. Does this make any difference? Yes, and it possibly matters. The women­ only setting affects the style of teach­ ing, the patterns of collaboration be­ tween students and between students and staff, and probably also the pro­ portion of students with intentions of continuing in mathematics. From this background our "commu­ nity" dispersed, at various times, to fol­ low varied lives in mathematics, to re­ turn in April 2001. This is the story of how the reunion came about, and what we discovered over that weekend. The Idea of the Weekend

Happenstance brought four Smith alumnae and one (male) faculty mem­ ber together at a maths conference. Women mathematicians are not so few in number today that four women at a conference is noteworthy, but here were four from the same undergradu­ ate college, and one hardly widely recognised as a leader in supplying the world with mathematicians. At what point did the organisers de­ cide that talking over old times was such fun that they wanted to invite all Smith alums with PhDs in mathemat­ ics to come back for a weekend at Smith? They began counting. "All" turned out to be not so small a num­ ber: it surprised them, indeed, as it sur­ prised me: there are about thirty-five on the list. I have no idea how many the or­ ganisers imagined might actually make the trip back. While not accustomed to responding to reunion summonses, I found this invitation to return com­ pelling. These people, if any, were my colleagues. Although I knew only one other Smith mathematician, knowing where they started from, and how tough the path from there through a Ph.D. in mathematics and beyond can be, I was certain that my colleagues would have stories worth listening to. I went. So did about twenty-five of the thirty-five of us known to be work­ ing in maths. It was clear that others felt similarly drawn. Together with fac­ ulty, present students, and other inter­ ested parties, about fifty people gath­ ered for the conference. We certainly did have stories to tell.

18

THE MATHEMATICAL INTELLIGENCER

We discovered that we had each got­ ten from our perhaps unpretentious undergraduate mathematics experi­ ence something that well enabled us to meet the challenges that maths gradu­ ate students face. It began to appear, through the common themes in our stories, that perhaps we survived not so much in spite of our unusual un­ dergraduate experience, but because of it. That perhaps our ability to adapt to non-standard paths through study and employment, and our consequent satisfaction in employment, was also a strength gained through our particular undergraduate experience. If this is so, then it should be shouted out loud-at schools, to en­ courage students to consider this as a route through to the position of work­ ing mathematician, but also to the mathematical community at large. This

There had to be somethi ng i n o u r u n derg rad uate experience that gave us a solid start for our careers. route works. It offers great flexibility, and its graduates show resilience, durability, adaptability, and an uncom­ mon willingness to communicate. The Mechanics of the Weekend: The Making of a Community

The plans for the weekend were sim­ ple. Social gatherings on the Friday night. Sessions of twenty-minute math­ ematics talks throughout Saturday. "Round-table" discussion on women's issues in the mathematical community; a conference dinner. Further talks on the Sunday, including a panel discus­ sion on employment in computer sci­ ence; a brunch; then another session of talks before we went our various ways. It is a stiff task, to explain, to a room­ ful of people with only basic under­ graduate mathematics in common, the

work that is of interest to you, in no more than twenty minutes. The subject matter ranged from L-functions, to how best to design and run distance learn­ ing courses, to the launching of satel­ lites, with plenty of instalments of al­ gebra, graph theory, and operations research in between. The quality of the talks was the startling bit. Without ex­ ception the speakers demonstrated a desire to communicate and the ability to do a good job of it! Inevitably I re­ call most vividly those talks closest to my own field, but I remember with par­ ticular pleasure a twenty-minute invita­ tion to vertex algebras, which included a concise and appealing summary of Lie algebras and their representations in the first sixteen minutes! The next surprise was the round­ table discussion. Such sessions offer an opportunity to those in the grip of oppression to cry out against the in­ justices, to stir the sisters to arms against prejudice both deliberate and thoughtless. Not this time. I heard no crying. I did hear lots of laughter. The horror stories were there, all right. The ladies assembled had not been magically shielded from the common worries-two-body prob­ lems, combining children and careers, and insensitive colleagues. The tales were, however, of survivors, cheerful ones at that, who could find amuse­ ment in the absurd injustices; definitely not victims. The twenty-minute talks turned out to be an absolutely splendid method of introductions. What more essential could you know about someone than that she has a strong enthusiasm for launching rockets, or a passion for colouring graphs? By Sunday this group of people who had had only a pa­ per connection-that of having passed through the same place at various times-began to have a sense of a com­ munity. As particles of cosmic dust of sufficient quantities coalesce to form a star, by the time we left, we were a community. My Story

If any of us had been asked before the weekend, "How was it that you came to be a mathematician?" I suspect the most frequent answer would have been

some equivalent of "It was a complete fluke." Certainly it seemed so in my case: I was good at French in high school, it was assumed I would be a French student. But then I took calcu­ lus, with Michael Gemignani, decided to ask questions, and never stopped. So many flukes make no fluke at all. It cannot be coincidence that we sur­ vived, even flourished. There had to be something in our undergraduate expe­ rience that gave us a solid start for our careers. Smith, in my day, was not widely re­ garded as a suitable college for a tech­ nically minded young woman. (I was not regarded as a technically minded young woman.) Nor was it, by objective accounting, a suitable place for them. The breadth and depth of mathematical knowledge required for a Smith hon­ ours degree fell woefully short of that required of first-year graduate students. I spent much of my first year in gradu­ ate school borrowing notes from other students' undergraduate courses, trying to make up the background to under­ stand the first-year lectures. It was aw­ ful to keep hearing, "What, you don't know that?" It does not matter. Hang what I failed to learn at Smith! It is clear that what I did get from Smith was far more valuable. I learned that I liked mathematics, that I could do mathematics. I had the experience that all doors were open to me: every faculty member gave the im­ pression that he or she had any amount of time available to answer my ques­ tions. I had the experience of doing mathematics, not just learning it. I learned to talk mathematics both with my classmates and with my teachers. I developed the taste for thinking math­ ematics as a way of life, not just a course of study, or a career. Through talking, asking questions, doing maths I gained a confidence that I could do maths: perhaps I didn't know much, but that could be changed. I could work with what I knew, I liked the work, and I wanted to do it enough to take whatever steps were necessary to learn the trade. The weekend showed that I am not alone.

The most important experience was finally being able tofeelfree to express

myself without worrying that I was going to say something really stupid and embarrass myself (this was after say, my 181 year), and consequently developing my own voice. Just getting practice speaking out. I now speak out freely. I have a feeling that I make some professors uncomfortable and amaze other female students. The in­ teract'ions 'in classr-ooms at Smith have validated for me that fact that I have good things to say, so now I know that what I have the urge to say is probably worth saying.-Diane Christine Jam (9S, Rice University) What I learned at the weekend is that my experience seems also to have been the experience of the others. This is borne out in the e-mails and comments that have come back to me. It would appear that this is what matters. Per-

I had the experi ­ ence of d o i n g mathem atics , not j ust learn i n g it . haps we were not towed so far across the sea of mathematical knowledge as our contemporaries at more techni­ cally oriented universities, but we were equipped with j olly good paddles! Moreover, we discovered we were not alone: there were more than just one or two of us. That matters, too. What Is Special About Smith?

Through many different routes, against often considerable obstacles, the col­ leagues I met at the weekend had found their way into the greater math­ ematical community. This cluster of success stories is not simply a statisti­ cal aberration: if something at Smith enabled the successes, what are prin­ cipal features? Two stand out: an ag­ gressive "open door" policy, encourag­ ing questions outside as well as within the classroom; and the opportunity to do active mathematical research while still an undergraduate. If you visit the Mathematics Department at Smith now, you arrive at the "forum," an open area with comfortable chairs, tables, a

few computers, and space to spread books, work, and talk There are cof­ fee facilities in the corner. The place is designed to encourage discussion. Most math faculty offices open directly into this space. The only option for a faculty member wishing to avoid en­ countering students is to refrain from coming in at all. They come in. In my day, catching faculty mem­ bers was not so comfortable. My back has a permanent memory of the numb patch gained from hours spent sitting on the floor, leaning against the walls outside faculty doors, lying in wait against the arrival of my teachers, with the day's questions. It seemed natural to me: it was what they were there for, wasn't it? And not one of them ever gave me so much as a hint that my de­ mands on their time might be either ex­ cessive or inconvenient. I also felt, by the time I left Smith, that as far as research went, I was a seasoned campaigner. I had written an honours thesis, for which I had read a paper, and I had even, well, tried to prove something that might have been new. And I had spent a summer assisting in a research pro­ ject. I knew what research was about, I thought. And in essence, I wasn't far wrong.

I was offered a small part of a small part ofsornebody's research as a final pr-oject 'in a number theory class. It was the fi,rst problem I had ever seen that wasn't a homework pr-oblem. It was actually a problem to which no one knew the answer! I worked on it night and day for a few weeks andfell in love with research! It was exhila­ rating.-Debra Boutin (91 , Hamilton College) Perhaps the strong history of com­ binatorics and graph theory at Smith plays a significant part. Graph theory and combinatorics do have many prob­ lems that can be explained without the necessity for a great deal of advanced mathematical machinery. Could it be done in less accessible subjects, with greater difficulty perhaps? Before you are too quick to excuse yourself from the effort of trying, con­ sider: how much mathematical equip-

VOLUME 25, NUMBER 1 , 2003

19

ment do you really need to start to work out examples? I have had some strange experiences in recent years. The first involved teaching repre­ sentations of Lie algebras to a gradu­ ate class with wildly divergent, even inappropriate backgrounds. As an al­ ternative to yielding to catastrophic de­ spair, we settled into a routine of ex­ amples classes following the lectures, in which we got everyone decompos­ ing representations of sl(2) and sl(3), lots of them. I was quite impressed how the hands-on experience enabled even students whose understanding of vec­ tor space had started as a fistful of ar­ rows (to be added tail to head) to achieve a very satisfactory under­ standing of the theory of Lie algebras and their representations. The second experience has been a project with some local ten-year-olds in a state school, on orthogonality, dot products, leading up to Fourier analy­ sis of sounds. It is a challenge, indeed, to explain orthonormal bases to ten­ year-olds, but by no means impossible. They appear to have a pretty good grasp of the ideas. I did have one child in floods of tears-she was unable to add or multiply the fractions as re­ quired! (I now use decimals and a cal­ culator.) But it was not the ideas she was unable to grasp (and she has strong motivation to get to grips with the fractions). It's a bit like trying to tell a story without using the letter e, but it can be done (or, perhaps, you can do it). Quite apart from the benefit to the students participating in research, the discipline of having to explain some part of your subject to an undergraduate without the luxury of extensive mathematical baggage, can be a revealing exercise in its own right, leading to substantial simplifications in the theory. It is worth doing. Could It Happen Anywhere?

Why Smith? Clearly Smith offered the right climate for a certain type of po­ tential mathematician to flourish. The right climate again appears to depend on several factors. First, Smith does not have a gradu­ ate programme. It is perhaps paradox­ ical that this should be counted as a

20

THE MATHEMATICAL INTELLIGENCER

positive feature in the analysis of whether an institution is a suitable one for growing mathematicians, but it seems to play a critical role. In the ab­ sence of research students, the under­ graduates get the opportunity to take part in research and receive the direct attention of their teachers. Second, Smith manages to integrate the research interests of the faculty with their undergraduate teaching re­ sponsibilities. Courses labelled "Top­ ics in . . . " allow teachers to bring their interests into the classroom. This ap­ pears to blur the usual divide between teaching and research with apparent benefit to both aspects. This is of course contingent on the absence of a graduate school: where there is a grad­ uate programme, the "Topics in . . . " courses are aimed at graduate students (or, very often, visiting experts in the subject!).

I hadn't had as extensive a background in linear algebra, algebra, analysis as my peers when beginn'ing graduate school. I took upper-level undergradu­ ate courses to fill in my background. (This is not uncommon for first year graduate students from small liberal arts colleges at Cornell.) However, un­ like any ofmy peers, I had already seen Quantum Logic in David Cohen's top­ ics in analysis class, Groebner Bases in Patricia Sipe's topics in algebra class, and Quasi Crystals in Marjorie Senechal's Topics in Discrete Applied Mathematics class. As a graduate stu­ dent this exposure to cutting-edge mathematics raised my standing in the eyes of my peers, and in my own eyes.-Debra Boutin. Many schools have not got graduate programmes, but clearly not all such schools have had the same success in launching Ph.D. mathematicians. Per­ haps in this one respect the Smith suc­ cess story really is a fluke of personal­ ities. Probably every mathematician has been a part, at some time or an­ other, of a group that simply "worked." In such a group, there would be one or two people who set the style which others fit into. Over time the character of the group might change, but the past personality and strengths of the group

so influence the incoming leaders, that some of the original character would survive through complete changes of leadership. Smith was undoubtedly lucky in its early leaders: Neal McCoy and Alice Dickinson, to pick two. Neal McCoy retired in 1970, Alice Dickinson in 1980, but their style set the mould that stamps the practices and policy even now. Should It Happen Anywhere?

I have spent my working life at large schools with graduate programmes. They do an excellent job of training mathematicians, including at the un­ dergraduate level. I respect and admire these institutions. Nonetheless, the ev­ idence of the Smith weekend is that there is a role for other institutions of­ fering alternative routes to the status of PhD mathematician. Not only has Smith produced a sur­ prising number of mathematicians, but the unusual qualities of those mathe­ maticians-their adaptability, perse­ verance, and willingness to communi­ cate-suggest to me that this route into mathematics makes for particularly useful mathematicians at the end of their training. Therefore, this is a pathway which should be encouraged, studied, and adapted for use at other institutions. I do not believe that the principle is in any way gender-specific. Nor do I be­ lieve that beyond the circumstances mentioned above, there is anything so special about Smith that equally favourable conditions cannot be fos­ tered elsewhere. Those institutions which have those properties, an ab­ sence of graduate programme and a willingness to involve undergraduates in research, should recognise their po­ tential and set about making the most of the students they have got.

I wonder if teaching is different at Smith because methods that don't re­ ally workfor anybody, more obviously don't workfor women. I do think some women need more time to commit to mathematics. At some schools, you get a chance as a first-year student to join the team and you never really get an­ other chance. Smith, of course, is not like that.- Jim Henle

I believe this route in to mathematics has recruited, and will continue to re­ cruit students who would never other­ wise consider going into mathematics (myself amongst them!). I suspect also, observing my colleagues over the weekend, that those who follow this route will bring to the larger mathe­ matical community different strengths and expectations than those who fol­ low a more traditional route. I expect mathematics will benefit greatly from the added diversity. Where to Go from Here?

The Smith route works. However, it did not make for an easy time in the pas­ sage from our Smith degree through to the PhD. Let no one romanticise our struggles to overcome deficient back­ grounds and learn to compete with those "better prepared. " If this route into mathematics from the small liberal arts background is to become more commonly used, both the small col­ leges and the large graduate schools must consider the challenge of making the transition between undergraduate programmes and graduate training less rocky. One must wonder whether it is even conceptually possible to combine the virtues of the small liberal arts college with providing all those advanced courses. There are different policies small colleges can pursue. Smith has favoured encouraging a wide variety of students, allowing them to proceed to their own goals at their own rate. This provides Smith with a large number of mathematics majors, of whom only a minority will have interest in high-level courses. With the University of Massa­ chusetts within easy commuting dis­ tance, these ambitious students have the possibility of studying such courses there, although there are drawbacks for both student and Smith in doing so. Course scheduling can be complicated enough without having to schedule in commuting time, and Smith loses the presence of these students in its own courses. Some colleges adopt the strat­ egy of requiring of their mathematics students a commitment to acquiring a mathematical background comparable with what would be expected of po­ tential research students. Then few stu-

dents major in mathematics, and it is no easier to provide advanced level courses. Whatever strategies are followed by the smaller schools, graduate schools must share the responsibility for bridg­ ing the gap. A first positive step might be for admissions personnel to recog­ nise the value of training such as we received at Smith. Seek out some can­ didates who have come perhaps from smaller schools, who have been in­ volved in research, perhaps particu­ larly those who have tried other ca­ reers and yet wish to come back to do mathematics at graduate school. Ad­ mit, encourage, and value them. Help them catch up where their training falls short. The effort will be repaid. Even pure mathematics is no longer the introspective subject I studied as a graduate student. The larger mathe­ matical community could well benefit from an influx of adaptable students with well-developed abilities to com­ municate and a positive willingness to complement the strengths of whatever group they find themselves working in. If these virtues are indeed fostered by training such as we received at Smith, this is one more argument for a more imaginative and flexible concept of the "good undergraduate mathematical background."

ries are unusual, even knowing that there are others becomes important. As women in mathematics, we find ourselves with colleagues and neigh­ bours who mostly share too little with us to provide helpful understanding. The community of Smith alums in Mathematics can. We came from a common experience, we meet similar problems. Often it is enough to have someone else say, "Yes, I know, we had that problem too, and we have resolved it. Persevere. " Moreover, such a com­ munity can work in absentia. Knowing that other women, from the same un­ dergraduate background, have found their varied ways through graduate school, and have found a comfortable corner of mathematics to make a home in, is often sufficient to stifle doubts and counteract discouragement. And if that fails, they are there on e-mail.

AU T HOR

Smith Alums i n Mathematics: The Virtues of a Virtual Community

Before the weekend few of us knew each other: I would be surprised if any of us could have named more than four other Smith graduates working in mathematics. Now, more than a year after the weekend, we have returned to our routines. Some of us may never take part in a similar weekend, may never make further contact again. The pragmatic might wonder whether there is any reality in such a virtual commu­ nity: one that theoretically exists, which met once, which may or may not meet again. The answer is yes. The prime bene­ fit a community can give has got to be the support its members can provide each other, often simply by sharing sto­ ries. Where the members of the com­ munity are isolated, where their histo-

MARJORIE BATCHELOR

1 02 Mawson Road Cambridge CB1 2EA United Kingdom e-mail: mb1 [email protected]

Marjorie Batchelor, after her Smith years recalled here, was a graduate student at Warwick and then MIT (Ph.D. 1 g?8). Since then she has lec­ tured, in various capacities, at Cam­ bridge and Kings College London. Her research interests are comod­ ules, supermanifolds, and quantum groups. For the last ten years she has also, as an outgrowth of her hobby of chamber music, learned and prac­ ticed professionally the craft of re­ pairing and making violins. She is married, with three children.

VOLUME 25. NUMBER 1. 2003

21

i,¥, fflj.t§,@ih£il@j§#@ii,i,t§.iti

The M athematical Knight Noam D. Elkies Richard P. Stanley

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so

M i chael K l e t>e r a n d R a v i Vaki l ,

uch has been said of the affinity between mathematics and chess: two domains of human thought where very limited sets of rules yield inex­ haustible depths, challenges, frustra­ tions and beauty. Both fields support a venerable and burgeoning technical lit­ erature and attract much more than their share of child prodigies. For all that, the intersection of the two do­ mains is not large. While chess and mathematics may favor similar mind­ sets, there are few places where a chess player or analyst can benefit from a specific mathematical idea, such as the symmetry of the board and of most pieces' moves (see for instance [24]) or the combinatorial game theory of Berlekamp, Conway, and Guy (as in [4]). Still, when mathematics does find applications in chess, striking and in­ structive results often arise.

M

elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Please send all su bmissions to the Mathematical Entertainments Editor, Ravi Vakil,

Stanford University,

Department of Mathematics, Bldg. 380, Stanford, CA 94305-21 25, USA e-mail: [email protected]. edu

22

Introduction

This article shows several mathemati­ cal applications that feature the knight and its characteristic (2, 1) leap. It is based on portions of a book tentatively titled Chess and Mathematics, cur­ rently in preparation by the two au­ thors of this article, which will cover all aspects of the interactions between chess and mathematics. Mathemati­ cally, the choice of (2, 1) and of the 8 X 8 board may seem to be a special case of no particular interest, and indeed we shall on occasion indicate variations and generalizations involving other leap parameters and board sizes. But long experience points to the standard knight's move and chessboard size as felicitous choices not only for the game of chess but also for puzzles and prob­ lems involving the board and pieces, in­ cluding several of our examples. We will begin by concentrating on puzzles such as the knight's tour. Many of these are clearly mathematical prob­ lems in a very thin disguise (for in­ stance, a closed knight's tour is a Hamiltonian circuit on a certain graph C§), and can be solved or at least better

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

Ed i t o rs

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understood using the tem1inology and techniques of combinatorics. We also relate a few of these ideas with practi­ cal endgame technique (see Diagrams Iff., 10, 1 1). The latter half of the article shows some remarkable chess problems fea­ turing the knight or knights. Most "practical" chess players have little pa­ tience for the art of chess problems, which has evolved a long way from its origins in instructive exercises. But the same formal concerns that may deter the over-the-board player give some problems a particular appeal to math­ ematicians. For instance, we will exhibit a position, constructed by P. O'Shea and published in 1989, where White, with only king and knight, has just one way to force mate in 48 (the current record). We also show the longest known legal game of chess that is determined completely by its last move (discovered by Ri:isler in 1994)­ which happens to be checkmate by promotion to a knight. Algebraic notation

We assume that the reader is familiar with the rules of chess, but we assume very little knowledge of chess strategy. (The reader who knows, or is willing to accept as intuitively obvious, that king and queen win against king, or even against king and knight if there is no immediate draw, will have no diffi­ culty following the analysis.) The reader will, however, have to follow the notation for chess moves, either by visualizing the moves on the diagram or by setting up the position on the board. Several notation systems have been used; the most common one nowadays, and the one we use here, is "algebraic notation," so called because of the coordinate system used to name the squares of the board. In the re­ maining paragraphs of this introduc­ tory section we outline this notation system. Readers already fluent in alge­ braic notation may safely skip to the next section, A Chess Endgame.

Each square on the 8 X 8 board is uniquely determined by its row and column, called "rank" and "file" re­ spectively. The ranks are numbered from 1 to 8, the files named by letters a through h. In the initial array, ranks 1 and 2 are occupied by White's pieces and pawns, ranks 8 and 7 by Black's; both queens are on the d-file, and both kings on the e-file. Thus, viewed from White's side of the board (as are all the diagrams in this article), the ranks are numbered from bottom to top, the files from left to right. We name a square by its column followed by the row; for in­ stance, the White king in Diagram 1 be­ low is at d2. Each of the six kinds of chessmen is referred to by a single let­ ter, usually its initial: K, Q, R, B, P are king, queen, rook, bishop, and pawn (often lowercase p is seen for pawn). We cannot use the initial letter for the knight because K is already the king, so we use its phonetic initial, N for kNight. For instance, Diagram 1 can be described as: White Kd2, Black Ka1, Nf2, Pa2, Pc2. To notate a chess move we name the piece and its destination square, inter­ polating " X " if the move is a capture. For pawn moves the P is usually sup­ pressed; for pawn captures, it is re­ placed by the pawn's file. Thus in Dia­ gram 1 1 , Black's pawn moves are notated a2 and a X b2 rather than Pa2 and P X b2. We follow a move by " + " if i t gives check, and by " ! " or "?" if we regard it as particularly strong or weak. In some cases " ! " is used to indicate a thematic move, i.e. , a move that is es­ sential to the "theme" or main point of the problem. As an aid to following the analysis, moves are numbered consecutively, from the start of the game or from the diagram. For instance, we shall begin the discussion of Diagram 1 by con­ sidering the possibility l . K X c2 Nd3!. Here "1" indicates that these are White's and Black's first moves from the diagram; "KX c2" means that the White king captures the unit on c2; and "Nd3!" means that the Black knight moves to the unoccupied square d3, and that this is regarded as a strong move (the point here being that Black prevents 2.Kc l even at the cost of let­ ting White capture the knight). When

analysis begins with a Black move, we use " . . . " to represent the previous White move; thus " 1 . . . Nd3! " is the same first Black move. A few further refinements are needed to subsume promotion and castling, and to ensure that every move is uniquely specified by its notation. For instance, if Black were to move first in Diagram 1 and promoted his c2pawn to a queen (giving check), we would write this as 1 . . . c 1 Q +, or more likely 1 . . . clQ + ?, because we shall see that after 2.K x c l White can draw. Short and long castling are notated 0-0 and 0-0-0 respectively. If the piece and destination square do not specify the move uniquely, we also give the de­ parture square's file, rank, or both. An extreme example: Starting from Dia­ gram 9, "Nb1" uniquely specifies a move of the c3 knight. But to move it to dfi we would write "Ncd5" (because other knights on the b- and f-files could also reach d5); to a4, "N3a4" (not "Nca4" because of the knight on c5); and to e4, "Nc3e4" (why?).

natural try is l . K X c2, eliminating one pawn and imprisoning two of Black's remaining three men in the comer. But 1 . . . Nd3! breaks the blockade (Dia­ gram 2a). Black threatens nothing but controls the key square c l . The rules of chess do not allow White to pass the move; unable to go to cl, the king must move elsewhere and release Black's men. After 2.KX d3 (or any other move) Kb 1 followed by 3 . . . a1Q, Black wins easily. Diagram 2a

A Chess Endgame

We begin by analyzing a relatively sim­ ple chess position (Diagram 1). This may look like an endgame from actual play, but is a composed position-an "endgame study"-created (by NDE) to bring the key point into sharper focus.

Diagram 2b

Diagram 1

White to move

White, reduced to bare king, can do no better than draw, and even that with difficulty: Black will surely win if either pawn safely promotes to a queen. A

Returning to Diagram 1 , let us try in­ stead l . Kc l ! This still locks in the Black Ka1 and Pa2, and prepares to capture the Pc2 next move, for instance 1 . . . Nd3 + 2 . K X c2, arriving at Diagram 2a with Black to move. White has in effect succeeded in passing the move to Black by taking a detour from d2 to c2. Now it is Black who cannot pass, and any move restores the White king's ac­ cess to c l . For instance, play may con­ tinue 2 . . . Nb4 + 3.Kcl, reaching Dia­ gram 2b. Black is still bottled up. If it

VOLUME 25, NUMBER 1 , 2003

23

were White to move in Diagram 2b, White would have to release Black with Kd1 or Kd2 and lose; but again Black must move and allow White back to c2, for instance 3 . . . Nd3 + 4.Kc2 and we are back at Diagram 2a. So White does draw-at least if Black obligingly shuttles the knight be­ tween d3 and b4 to match the White king's oscillations between c 1 and c2. But what if Black tries to improve on this? While the king is limited to those two squares, the knight can roam over almost the entire board. For instance, from Diagram 2a Black might bring the knight to the far comer in m moves, reaching a position such as Diagram 3a, and then back to d3 in n moves. If m + n is odd, then Black will win since it will be White's tum to move. Instead of d3, Black can aim for b3 or e2, which also control c l ; but each of these is two knight moves away from d3, so we get an equivalent parity condition. Alter­ natively, Black might try to reach b4 from d3 in an even number of moves, Diagram 3a

to reach Diagram 2b with White to move; and again Black could aim for another square that controls c2. But each of these squares is one or three knight moves away from d3, so again would yield a closed path of odd length through d3. Can Black thus pass the move back to White? For that matter, what should White do in Diagram 3b? Does either Kcl or K X c2 draw, or is White lost re­ gardless of this choice? The outcome of Diagram 2a thus hinges on the answer to the following problem in graph theory:

Let C§ C§8,8 be the graph whose ver­ tices are the 64 squares of the 8 X 8 chessboard and whose edges are the pairs of squares joined by a knight's move. Does 'f) have a cycle of odd length through d3 ? =

Likewise White's initial move in Dia­ gram 3b and the outcome of this endgame comes down to the related question concerning the same graph C§:

What are the possible parities oflengths of paths on C§ from h8 to c1 or c2? The answers result from the following basic properties of C§:

LEMMA. (i) The graph 'fJ is connected. (ii) The graph is bipartite, the two parts comprising the 32 light squares and 32 dark squares of the chess­ board.

the same color as the one occupied by the knight. Our analysis would reach the same conclusions if the Black pawn on c2 were removed from Diagrams 1 and 3b; we included this superfluous pawn only as bait to make the wrong choice of c2 more tempting.

REMARK.

For which rectangular boards (if any) does part (i) or (ii) of the Lemma fail? That is, which C§m,n are not connected, or not bipartite? (All puz­ zles and all diagrams not explicated in the text have solutions at the end of this article.) Puzzle 1.

Knight's Tours and the Thirty-Two Knights

The graph C§ arises often in problems and puzzles involving knights. For in­ stance, the perennial knight's tour puz­ zle asks in effect for a Hamiltonian path on C§; a "re-entrant" or "closed" knight's tour is just a Hamiltonian cir­ cuit. The existence of such tours is classical-even Euler spent some time constructing them, finding among oth­ ers the elegant centrally symmetric tour illustrated in Diagram 4 (from [9, p. 1 9 1 ]): Diagram 4

Proof Part (i) is just the familiar fact

Diagram 3b

that a knight can get from any square on the chessboard to any other square. Part (ii) amounts to the observation that every knight move connects a light and a dark square. CoROLLARIES. (1) There are no knight cycles of odd length on the chessboard. (2) Two squares of the same color are connected by knight-move paths oj even length but not of odd length; two squares of opposite color are con­ nected by knight-move paths of odd length but not by paths of even length.

White to move

24

THE MATHEMATICAL INTELLIGENCER

We thus answer our chess ques­ tions: White draws both Diagram 1 and Diagram 3b by starting with Kcl . More generally, for any initial position of the Black knight, White chooses between c 1 and c2 by moving to the square of

A closed knight's tour constructed by Euler

The extensive literature on knight's tours includes many examples, which, when numbered along the path from 1 to 64, yield semi-magic squares (all row and column sums equal 260), some­ times with further "magic" properties, but it is not yet known whether a fully magic knight's tour (one with major di­ agonals as well as rows and columns

summing to 260), either open or closed, can exist. More generally, we may ask for Hamiltonian circuits on C§m, n for other m,n; that is, for closed knight's tours on other rectangular chessboards. A necessary condition is that C§ m ,n be a connected graph with an even number of vertices. Hence we must have 2)mn and both m,n at least 3 (cf. Puzzle 1). But not all Cf:!m ,n satisfYing this condi­ tion admit Hamiltonian circuits. For in­ stance, one easily checks that Cf:l3,4 is not Hamiltonian. Nor are Cf:l3,6 and C§3,s, but C§3, 1o has a Hamiltonian circuit, as does G 3,11 for each even n > 10. For in­ stance, Diagram 5 shows a closed knight's tour on the 3 X 10 board:

set of more than at least three squares must include two of the same color. Co­ cliques are more interesting: how many pairwise nonattacking knights can the chessboard accommodate? 1 We follow Golomb ([21], via M. Gardner [9, p. 193]). Again the fact that C§ is bipartite suggests the answer (Diagram 6): Diagram 6

32 mutually nonattacking knights x

10 board

There are sixteen such tours (ignoring the board symmetries). More gener­ ally, enumerating the closed knight's tours on a 3 X (8 + 2n) board yields a sequence 16, 1 76, 1536, 15424, . . . sat­ isfYing a constant linear recursion of degree 21 that was obtained indepen­ dently by Donald Knuth and NDE in April 1994. See [23, Sequence A070030]. In 1997, Brendan McKay first com­ puted that there are 132673644 10532 3 (more than 1.3 X 10 1 ) closed knight's tours on the 8 X 8 board ([ 19]; see also [23, Sequence A001230] , [26] ) . W e return now from enumeration to existence. After Cf:la,n the next case is Cf:l4,,. This is trickier: the reader might try to construct a closed knight's tour on a 4 X 1 1 board, or to prove that none exists. We answer this question later. What of maximal cliques and co­ cliques on C§? A clique is just a collection of pairwise defending (or attacking) knights. Clearly there can be no more than two knights, again because C§ is bi­ partite: two squares of the same color cannot be a knight's move apart, and any 1 Burt

Are Diagram 6 and its complement the only maximal cocliques? Yes, but this is harder to show. One elegant proof, given by Greenberg in [ 2 1 ] , invokes the existence of a closed knight's tour, such as Euler's Diagram 4. In general, on a circuit of length 2M the only sets of M pairwise nonadjacent vertices are the set of even-numbered vertices and the set of odd-numbered ones on the circuit. Here M 32, and the knight's tour in effect embeds that circuit into C§, so a fortiori there can be at most two cocliques of size M on Cf:l-and we have already found them both! Of course this proof applies equally to any board with a closed knight's tour: on any such board the light- and dark­ squared subsets are the only maximal cocliques. Conversely, a board for which there are further maximal cocliques can­ not support a closed knight's tour. For example, any 4 X n board has a mixed­ color maxin1al coclique, as illustrated for n 1 1 in Diagram 8. =

Diagram 5

A closed knight's tour on the 3

Puzzle 2 . What happens if m,n are both odd, or if m s 2 or n s 2?

Diagram 7

=

Diagram 8

A third maximal knight coclique on the 4 1 1 board

A one-factor in '§

It is not hard to see that we cannot do better: the 64 squares may be parti­ tioned into 32 pairs, each related by a knight move, and then at most one square from each pair can be used (Di­ agram 7). This is Patenaude's solution in [21]. Such a pairing of C§ is called a "one-factor" in graph theory. Similar one-factors exist on all C§m,n when 2:mn and m,n both exceed 2; they can be used to show that in general a knight coclique on an m X n board has size at most mn/2 for such m,n.

x

This yields possibly the cleanest proof that there is no closed knight's tour on a 4 X n board for any n. (According to Jelliss [ 14 ] , this fact was known to Euler and first proved by C. Flye Sainte-Marie in 1877; Jelliss attributes the above clean proof to Louis Posa.) WARNING: the existence of a closed knight's tour is a sufficient but not nec­ essary condition for the existence of only two maximal knight cocliques. It is known that an m X n board supports a closed tour if and only if its area mn is an even integer > 24 and neither m nor n is 1 , 2, or 4. In particular, as noted above there are no closed knight's

Hochberg jokes (in [1 1 , p. 5], concerning the analogous problem for queens) that the answer is 64, all White pieces or all Black: pieces of the sarne color can­

not attack each other' Of course this joke, and similar jokes such as crowding several pieces on a single square, are extraneous to our analysis.

VOLUME 25, NUMBER 1 2003

25

tours on the 3 X 6 and 3 x 8 boards, though as it happens on each of these boards the only maximal knight co­ cliques are the two obvious mono­ chromatic ones. More about C§: Domination Number, Girth, and the

We already noted that C§, being bi­ partite, has no cycles of odd length. (We also encountered the nonexis­ tence of 3-cycles as "C§ has no cliques of size 3".) Thus the girth (minimal cy­ cle length) of C§ is at least 4. In fact the girth is exactly 4, as shown for instance in Diagram 10.

pawn and giving White time for 4.N X a2 and a draw. On other Black moves from Diagram lOa White resumes con­ trol of a2 with 3.Nc l or 3.Nb4; for in­ stance 2 . . . Kc2 3.Nb4 + or 2 . . . Kc3 3.Ncl Kb2 (else Na2 + ) 4.Nd3 + ! etc. Note that the White king was not needed.

Diagram 1 0

NOTE TO ADVANCED CHESS PLAYERS: it might seem that the knight does need a bit of help after l.Nb4 Kb l !?, when either 2.Na2? or 2.Nd3? loses (in the latter case to 2 . . . a2), but Black has no threat so White can simply make a random ("waiting") king move. But this is not necessary, as White could also draw by thinking (and playing) out of the a2-b4-d3-cl-a2 box: l . Nb4 Kb l 2.Nd5! If now 2 . . . a2 then 3.Nc3 + is a new drawing fork, and otherwise White plays 3.Nb4 and resumes the square dance.

White to move draws

Construct a position where this Nd5 resource is White's only way to draw.

Knight Metric

Another classic puzzle asks: How many knights does it take to either occupy or defend every square on the board? In graph theory parlance this asks for the "domination number" of C§ . 2 For the standard 8 X 8 board, the symmetrical solution with 12 knights (Diagram 9) has long been known: Diagram 9

Diagram 10a

All unoccupied squares controlled Puzzle 3. Prove that this solution is unique up to reflection.

The knight domination number for chessboards of arbitrary size is not known, not even asymptotically. See [9, Ch. 14] for results known at the time for square boards of order up to 15, most dating back to 1918 [ 1 , Vol. 2, p. 359]. If we ask instead that every square, occupied or not, be defended, then the 8 X 8 chessboard requires 14 knights. On an m X n board, at least mn/8 knights are needed, since a knight defends at most 8 squares. Puzzle 4. Prove that mn/8 + 0(m + n) knights suffice. HINT: Treat the light and dark squares separately.

After 2 Nd3!

This square cycle is important to endgame theory: a White knight trav­ eling on the cycle can prevent the pro­ motion of the Black pawn on a3 sup­ ported by its king. To draw this position White must either block the pawn or capture it, even at the cost of the knight. The point is seen after l .Nb4 Kb3 2.Nd3! (reaching Diagram l Oa) a2 3.Ncl + ! , "forking" king and

Puzzle 5.

WARNING: This puzzle is hard and re­ quires considerably more chess back­ ground than anything else in this arti­ cle. The construction requires some delicacy: it is not enough to simply stalemate the White king, because then White can play 2.Na2 with impunity; on the other hand if the White king is put in Zugzwang (so that it has some legal moves, but all of them lose), then the direct 1 . . . a2 2.NXa2 K X a2 wins for Black Even more important for the prac­ tical chess player is the distance func­ tion on C§, which encodes the number of moves a knight needs to get from any square to any other. The diame­ ter (maximal distance) on C§ is 6, which is attained only by diagonally opposite corners. This is to be ex­ pected, but shorter distances bring some surprises. The table accompa­ nying Diagram 1 1 shows the distance from each vertex of C§ to a corner square:

2This terminology is not entirely foreign to the chess literature: A piece is said to be "dominated" when it can move to many squares but will be lost on any of them. (The meaning of "many" in this definition is not precise because domination is an artistic concept, not a mathematical one.) The introduction of this term into the chess lexicon is attributed to Henri Rinck ([1 2 , p. 93], [1 6, p. 1 51 ]). The task of constructing economical domination positions, where a few chessmen cover many squares, has a pronounced combinatorial flavor; the great composer of endgame studies G . M . Kasparyan devoted an entire book to the subject, Domination in 2545 Endgame Studies,

26

Progress Publishers, Moscow, 1 980.

THE MATHEMATICAL I NTELLIGENCER

Determine the knight dis­ tance from (0,0) to (m,n) on an infinite board as a function of the integers m,n. Puzzle 6.

5

4

5

4

5

4

5

6

4

3

4

3

4

5

4

5

3

4

3

4

3

4

5

4

2

3

2

3

4

3

4

5

3

2

3

2

3

4

3

4

2

1

4

3

2

3

4

5

3

4*

1

2

3

4

3

4

tZJ

3

2

3

2

3

4

5

Diagram 1 1

Further Puzzles

We continue with several more puzzles that exploit or extend the above dis­ cussion. Puzzle 7. How does White play in Di­ agram 12 to force checkmate as quickly as possible against any Black defense? Yes, it's White who wins, despite hav­ ing only king and pawn against 15 Black men. Black's men are almost paralyzed, with only the queen able to move in its comer prison. White must keep it that way: if he ever moves his king, Black will sacrifice his e2-pawn by promoting it, bring the Black army to life and soon overwhelm White. So White must move only the pawn, and the piece that it will promote to. That's good enough for a draw, but how to actually win? Puzzle 8. (See Diagram 13.) There are exactly 24 4! paths that a knight on dl can take to reach d7 in four moves; plotting these paths on the chessboard yields a beautiful projection of (the !­ skeleton of) the 4-dimensional hyper­ cube! Explain. =

White loses

The starred entry is due to the board edges: a knight can travel from any square to any diagonally adjacent square in two moves except when one of them is a comer square. But the other irregu­ larities of the table at short distances do not depend on edge effects. Anywhere on the board, it takes the otherwise ag­ ile knight three moves to reach an or­ thogonally adjacent square, and four moves to travel two squares diagonally. This peculiarity must be absorbed by any chess player who would learn to play with or against knights. One con­ sequence, known to endgame theory, is shown in Diagram 11, which exploits both the generic irregularity and the spe­ cial comer case. Even with White to move, this position is a win for Black, who will play . . . a2 and . . . a1 Q. One might expect that the knight is close enough to stop this, but in fact it would take it three moves to reach a2 and four to reach a1, in each case one too many. In fact this knight helps Black by block­ ing the White king's approach to a1!

Diagram 13

We saw that there is an es­ sentially unique maximal configuration of 32 mutually non-defending knights on the 8 X 8 board.

Puzzle 9.

i. Suppose we allow each knight to be defended at most once. How many more knights can the board then accommodate?

Diagram 12

White to play and mate as quickly as possible

The 4! shortest knight paths from d1 to d7

ii. Now suppose we require each knight to be defended exactly once. What is the largest number of knights on the 8 X 8 board satisfy­ ing this constraint, and what are all the maximal configurations? Puzzle 10. A "camel" is a (3, 1) leaper, that is, an unorthodox chess piece that moves from (x,y) to one of the squares (x :±:: 3, y :±:: 1) or (x :±:: 1 , y :±:: 3). (A knight is a (2, 1) leaper.) Because there are eight such squares, it takes at least mn/8 camels to defend every square, occupied or not, on an m X n board. Are mn/8 + 0(m + n) sufficient, as in Puzzle 4? Synthetic Games

The remainder of this article is devoted to composed chess problems featuring knights. A synthetic game [ 13] is a chess game composed (rather than played) to achieve some objective, usually in a minimal number of moves. Ideally the solution should be unique, but this is very rare. Failing this, we can hope for an "almost unique" solution, e.g. , one where the final position is unique, but not the move order. For instance, the shortest game ending in checkmate by a knight is 3.0 moves: l .e3 Nc6 2 .Ne2 Nd4 3.g3 Nf3 mate. White can vary the order of his moves and can play e4 and/or g4 instead of e3 and g3. The Black knight has two paths to f3. The biggest flaw, however, is that White could play c3/c4 instead of g3/g4, and Black could mate at d3. At least all 72 solutions share the central feature that White incarcerates his king at its home

VOLUME 25. NUMBER 1 , 2003

27

square. A better synthetic game in­ volving a knight is given in Puzzle 1 1 .

Diagram 1 4

Puzzle 1 1 . Construct a game of chess in which Black checkmates White on Black's fifth move by promoting a pawn to a knight.

Proof Games

very successful variation of syn­ thetic games that allows unique solu­ tions are proof games, for which the length n of the game and the final po­ sition P are specified. For the condi­ tion (P, n) to be considered a sound problem, there should be a unique game in n moves ending in P. (Some­ times there will be more than one so­ lution, but solutions should be related in some thematic way. Here we will only consider conditions (P, n) that are uniquely realizable, with the ex­ ception of Diagram 1 7.) The earliest proof games were com­ posed by the famous "Puzzle King" Sam Loyd in the 1890s but did not have unique solutions; the earliest proof game meeting today's standards seems to have been composed by T. R. Daw­ son in 1913. Although some interesting proof games were composed in subse­ quent years, the vast potential of the subject was not suspected until the fan­ tastic pioneering efforts of Michel Gail­ laud in the early 1980s. A close to com­ plete collection of all proof games published up to 1991 (around 160 prob­ lems) appears in [28] . Let us consider some proof games related to knights. We mentioned ear­ lier that the shortest game ending in mate by knight has length 3.0 moves. None of the 72 solutions yield proof games with unique solutions; i.e., every terminal position has more than one way of reaching it in 3.0 moves. It is therefore natural to ask for the least number n (either an integer or half­ integer) for which there exists a uniquely realizable game of chess in n moves ending with checkmate by knight, i.e., given the final position, there is a unique game that reaches it in n moves. Such a game was found in­ dependently by the two authors of this article in 1996 for n = 4.0, which is surely the minimum. The final position is shown in Diagram 14.

A

28

THE MATHEMATICAL INTELLIGENCER

Position after Black's 4th move. How did the game go?

Five other proof game problems in­ volving knights are presented in Puzzles 12 through 16. The minimum known number of moves for achieving the game is given in parentheses. (We repeat, the game must be uniquely realizable from the number of moves and final position.)

earliest of all proof games, while Dia­ gram 16 is considerably more chal­ lenging. Diagram 1 7 features a differ­ ent kind of impostor. Note that it has two solutions; it is remarkable how each solution has a different impostor. The complex and difficult Diagram 18 illustrates the Frolkin theme: the mul­ tiple capture of promoted pieces. Dia­ gram 19 shows, in the words of Wilts and Frolkin [28, p . 53] , that "the seem­ ingly indisputable fact that a knight cannot lose a tempo is not quite un­ ambiguous." Construct a proof game ending with mate by a pawn promoting to a knight without a capture on the mating move (6.0).

Puzzle 1 5 .

16. Construct a proof game ending with mate by a pawn promoting to a knight with no captures by the mat­ ing side throughout the game (7.0). Puzzle

Diagram 15

Construct a proof game without any captures that ends with mate by a knight (4.5).

Puzzle 1 2 .

13. Construct a proof game ending with mate by a knight making a capture (5.5).

Puzzle

14. Construct a proof game ending with mate by a pawn promoting to a knight (5.5). Puzzle

There is a remarkable variant of Puz­ zle 14. Rather than having the game de­ termined by its final position and num­ ber of moves, it is instead completely determined by its last move (including the move number)! This is the longest known game with this property.

After Black's 4th. How did the game go?

Diagram 1 6

Puzzle 1 4 ' . Construct a game of chess with last move 6.g x f8N mate.

The above proof games focus on achieving some objective in the mini­ mum number of moves. Many other proof games in which knights play a key role have been composed, of which we give a sample of five problems. Di­ agrams 15, 16, and 17 feature "impos­ tors"-some piece(s) are not what they seem. The first of these (Diagram 15) is a classic problem that is one of the

After Black's 1 2th. How did the game go?

Diagram 1 7

Diagram 20

After White's 1 3th. How did the game go?

Mate in one

should be dual-free, which means that Black has at least one method of de­ fending which forces each White move uniquely if White is to achieve his ob­ jective. The objective of checkmate can be combined with other condi­ tions, such as White having only one unit besides his king. The ingenious Di­ agram 2 1 shows the current record for a "knight minimal," i.e., White's only unit besides his king is a knight. For other length records, as well as many other tasks and records, see [20]. Diagram 21

Two solutions! Diagram 1 8

After White's 27th move. How did the game go?

history of the game. It is only assumed that the prior play is legal; no assump­ tion is made that the play is "sensible." Proof games are a special class of retro problems. We will give only one illus­ tration here of a retro problem that is not a proof game. It is based on con­ siderations of parity, a common theme whenever knights are involved. Dia­ gram 20 is a mate in one. A chess prob­ lem with this stipulation almost invari­ ably involves an element of retrograde analysis, such as determining who has the move. In a problem with the stipu­ lation "Mate in n," it is assumed that White moves first unless it can be proved that Black has the move in or­ der for the position to be legal.

Diagram 1 9 Length Records

After Black's 1 0th move. How did the game go? Retrograde Analysis

In retrograde analysis problems (called retro problems for short), it is neces­ sary to deduce information from the current position concerning the prior

In length records, one tries to con­ struct a position that maximizes the number of moves that must elapse be­ fore a certain objective is satisfied. The most obvious and most-studied objec­ tive is checkmate. In other words, how large can n be in a problem with the objective "mate in n" (i.e., White to play and checkmate Black in n moves)? Chess problem standards demand that the solution should be unique if at all possible. It is too much to expect, es­ pecially for long-range problems, that White has a unique response to every Black move for White to achieve his objective. In other words, it is possible for Black to defend poorly and allow White to achieve his objective in more than one way, or even to achieve it ear­ lier than specified. The correct unique­ ness condition is that the problem

Mate in 48 Paradox

The term "paradox" has several mean­ ings in both mathematics and ordinary discourse. We will regard a feature of a chess problem (or chess game) as paradoxical if it is seemingly opposed to common sense. For instance, com­ mon sense tells us that a material ad­ vantage is beneficial in winning a chess game or mating quickly. Thus sacrifice in an orthodox chess problem (i.e., a direct mate or study) is paradoxical. Of course it is just this paradoxical ele­ ment that explains the appeal of a sac­ rifice. Another common paradoxical theme is underpromotion. Why not promote to the strongest possible piece, namely, the queen? This theme is related to that of sacrifice, because in each case the player is forgoing ma­ terial. To be sure, underpromotion to knight in order to win, draw, or check­ mate quickly is not so surprising (and has even occurred a fair number of times in games), since a knight can make moves forbidden to a queen. Tim Krabbe thus remarks in [ 15] that

VOLUME 25, NUMBER 1, 2003

29

knighting hardly counts as a true "un­ derpromotion."3 Nevertheless, knight promotions can be used for surprising purposes that heighten the paradoxical effect. Diagram 22 shows four knight sacri­ fices, all promoted pawns, with a total of five promotions to knight. Diagram 23 shows a celebrated problem com­ posed by Sam Loyd where a pawn pro­ motes to a knight that threatens no pieces or checks and is hopelessly out of play. For some interesting com­ ments by Loyd on this problem, see [27, p. 403]. Diagram 22

other knights. Similarly the time-wast­ ing 5.h X g8N 6.Nh6 7.N X f7 of Diagram 19 seems paradoxical-why not save a move by 5.h X g8B and 6.b X f7 + ? Helpmate

In a helpmate in n moves, Black moves first and cooperates with White so that White mates Black on White's nth move. If the number of solutions of a helpmate is not specified, then there should be a unique solution. For a long time it was thought impossible to con­ struct a sound helpmate with the theme of Diagram 24, featuring knight promo­ tions. Note that the first obstacle to overcome is the avoidance of check­ mating White or stalemating Black The composer of this brilliant problem, Ga­ bor Cseh, was tragically killed in an ac­ cident in 2001 at the age of 26. Diagram 24

solution to #341 ) and called the method of "buttons and strings," is to form a graph whose vertices are the squares of the board, with an edge between two vertices if the problem piece (here a knight) can move from one vertex to the other. For Diagram 25, the graph is just an eight-cycle (with an irrelevant isolated vertex corresponding to the center square of the board). Diagram 26 is a representation of the problem that makes it quite easy to see that the minimum number of moves is sixteen (eight by each color), achieved for in­ stance by cyclically moving each knight four steps clockwise around the eight-cycle. If a White knight is added at b 1 and a Black knight at b3, then somewhat paradoxically the minimum number of moves is reduced to eight! A variation of the stipulation of Dia­ gram 25 is the problem presented as Puzzle 17, whose solution is a bit tricky and essentially unique. Diagram 25

White to play and win Exchange the knights in a minimum number of moves

Diagram 23

Diagram 26

c3

a2

1.&

Helpmate in 1 0 Piece Shuffle

Mate in 3

Note that the impostors of Diagrams 15-17 may also be regarded as para­ doxical, because we're trying to reach the position as quickly as possible, and it seems a waste of time to move knights into the original square(s) of

In piece shuffles or permutation tasks, a rearrangement of pieces is to be achieved in a minimum number of moves, sometimes subject to special conditions. They may be regarded as special cases of "moving counter prob­ lems" such as given in [2, pp. 769-777) or [3, pp. 58--68] . A classic example in­ volving knights, going back to Guarini in 1 5 12, is shown in Diagram 25. The knights are to exchange places in the minimum number of moves. (Each White knight ends up where a Black knight begins, and vice versa.) The sys­ tematic method for doing such prob­ lems, first enunciated by Dudeney [3,

3More paradoxical are underpromotions to rooks and bishops, but we will not be concerned with them here.

30

THE MATHEMATICAL INTELLIGENCER

cl

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"- b l

1.&

b3 al

�

a3

c2

The graph corresponding to Diagram 25

In Diagram 25 exchange the knights in a minimum number of move sequences, where a "move sequence" is an unlimited number of consecutive moves by the same knight. For some more sophisticated prob­ lems similar to Diagram 25, see [ 10, pp. 1 14-124) . The most interesting piece

Puzzle 17.

parity is a knight-move away from exactly one of the squares with co­ ordinates (2x,2y) with x == y mod 4. Intersecting this lattice with an m X n chessboard yields mn/16 + 0(m + n) knights that cover all odd squares at distance at least 3 from the nearest edge. Thus an ex­ tra 0(m + n) knights defend all the odd squares on the board. The same construction for the even squares yields a total of mn/8 + O(m + n).

shuffle problems connected with the game of chess (though not focusing on knights) are due to G. Foster [5, 6, 7, 8], created with the help of his com­ puter program WOMBAT (Work Out Matrix By Algorithmic Techniques). Puzzle Answers, Hints, and Solutions

The graph C!3rn,n is connected for m = n = 1 (only one vertex) and not connected for m = n = 3 (the central square is an isolated ver­ tex). With those two exceptions, C!3rn, n is connected if and only if m > 2 and n > 2. Every C!3m,n is bi­ partite, except Cf3u (empty parts not allowed); each non-connected graph C!3rn,n is bipartite in several ways except for CfJ 1,2 = Cf12, 1 · 2. If m = 1 or n = 1 then C§m,n is dis­ connected, so the maximal co­ clique is the set of all mn vertices. The graph 'fi2,n (or Cf3,, 2) decom­ poses into two paths of length Ln/2J and two of length ln/2l. It thus has a one-factor if and only if 41n, and otherwise has cocliques of size >n; the maximal coclique size is n + B where B E { 0, 1, 2) and n == :±: B mod 4. If m and n are odd in­ tegers greater than 1 then the max­ imal coclique size ofC§rn,n is (mn + 1 )/2, attained by placing a knight on each square of the same parity as a comer square of an m X n board. One can prove that this is maximal by deleting one of these squares and constructing a one­ factor on the remaining mn - 1 vertices of C!3m,n · 3. Each of the four 2 X 2 comer sub­ boards requires at least three knights, and no single knight may occupy or defend squares in two different subboards. Hence at least 4 · 3 = 12 knights are needed. For three knights to cover the {a1, b1, a2, b2 ) subboard, one of them must be on c3; likewise f3, f6, c6 must be occupied if 12 knights are to suffice. It is now easy to verify that Diagram 9 and its reflection are the only ways to place the remaining 8 knights so as to cover the entire chessboard. 4. ( [3, #319, p. 127]) On an infinite chessboard, each square of odd 1.

Diagram 27

White to move draws

One such position is shown in Di­ agram 27. Once the a-pawn is gone, the position is a theoretical draw whether Black plays fX g2 + (Black can do no better than stalemate against K x g2, Kh 1 , Kg2 etc.) or f2 (ditto after Ke2, Kf1 , etc.), or lets White play g x f3 and Kg2 and then jettison the f-pawn to reach the same draw that follows f X g2 + . But as long as Black's a-pawn is on the board, White can move only the knight since g x f3 would liber­ ate Black's bishop which could then force White's knight away (for instance l . Nb4 Kb 1 2.g X f3? g2 + ! 3.KXg2 Bd6 4.Nd5 Kb2) and safely promote the a-pawn. Black's pawn on f3 could also be on h3 with the same effect. 6. The distance is an integer, congru­ ent to m + n mod 2, that equals or exceeds each of !m'/2, ln'/2, and (im, + lni)/3. It is the smallest such integer except in the cases already noted of (m,n) = (0, :±: 1), ( :±: 1,0), or ( :±: 2, :±:2), when the distance ex5.

ceeds the above lower bound by 2. (adapted from Gorgiev) To win, White must promote the pawn to a knight, capture the pawns on b5 and c4, and then mate with N X b3 when the Black queen is on al. Thus N X b3 must be an odd-num­ bered move. Therefore l .h4, 2.h5, 3.h6, 4.h7, 5.h8N does not work be­ cause all knight paths from h8 to b3 have odd length. Since the knight cannot "lose the move," the pawn must do so on its initial move: l .h3!, followed by 6.h8N! , 7.Nf7, 8.Nd6, 9.N x b4, 10.Nd6, 1 l.N X c4. 12.Na5. At this point the Black queen is on a2, having made 1 1 moves from the initial position; whence the conclusion: 12 . . . Qa1 13. N X b3 mate. (We omitted from Gorgiev's original problem the initial move l . Kf2 X Ne 1 Qa2-a1, which only served to give Black his entire army in the initial position and thus maximize the material disparity; and we moved a Black pawn from c5 to b5 to make the so­ lution unique, at some cost in strategic interest.) 8. Recall that a knight's move joins squares differing by one of the eight vectors ( :±: 1 , :±: 2) or ( :±: 2, :±: 1), and check that to get some four of those to add to (0,6) we must use the four vectors with a positive ordinate in some order. Thus, to reach d7 from d1 (or, more generally, to travel six squares north with no obstruction from the edges of the board) in four moves, the knight must move once in each of its four north-go­ ing directions. Therefore each path corresponds to a permutation of the four vectors ( :±: 1 ,2) and ( :±: 2 , 1 ). The number of paths is thus 4! = 24, and drawing them all yields the image of the 4-cube under a pro­ jection taking the unit vectors to (:±: 1 ,2) and ( :±: 2, 1). Instead of d 1 and d7 w e could also draw the 24 paths from a4 to g4 in four moves to get the same picture. Not b2 and f6, though: besides the 24 paths of Diagram 13 there are other four­ move journeys, for instance b2-d3f4-h5-f6. 9. (i) The maximum is still 32 (though 7.

VOLUME 25. NUMBER 1 , 2003

31

there are many more configura­ tions that attain this maximum). To show this, it is enough to prove that at most 8 knights can fit on a 4 X 4 board if each is to be de­ fended at most once. This in tum can be seen by decomposing <§4. 4 as a union of four 4-cycles (Dia­ gram 28), and noting that only two knights can fit on each 4-cycle. (ii) Once again, the maximum is 32, this time with a new configu­ ration (Diagram 29) unique up to reflection! (But note that this con­ figuration has a cyclic group of 4 symmetries, unlike the elementary abelian 2-group of symmetries of the maximal coclique (Diagram 6).) That this is maximal follows from the first part of this puzzle. For uniqueness, our proof is too long to reproduce here in full; it proceeds as follows. In any 32knight configuration, each of the four 4 X 4 comer subboards must contain eight knights, two on each of its four 4-cycles. We analyze cases to show that it is impossible for two knights in different sub­ boards to defend each other. We then show that Diagram 29 and its reflection are the only ways to fit four eight-knight configurations into an 8 X 8 board under this con­ straint. 10. Yes, mn/8 + O(m + n) camels suf­ fice. The camel always stays on squares of the same color. The squares of one color may be re­ garded on a chessboard in its own right, tilted 45° and magnified by a factor of V2-in other words, multiplied by the complex number 1 + i. On this board, the camel's move amounts to the ordinary knight's move since 3 + i = (2 i)( l + i). We can thus adapt our solution to Puzzle 4. Explicitly, on an infinite chessboard each square with both coordinates odd is a camel's move away from exactly one square of the form (4x,8y). Thus camels at (4x + a, 8x + b) (a,b E {0, 1 }) cover the entire board without duplication, and the inter­ section of this configuration with an m X n board covers all but 0(m + n) of its squares. 32

THE MATHEMATICAL INTELLIGENCER

Diagram 28

Diagram Solutions

Diagram 14. (N. Elkies, R. Stanley, 1996) l.c4 Na6 2.c5 N X c5 3.e3 a6 4.Ne2 Nd3 mate. Diagram 15. (G. Schweig, Tukon, 1938) l.Nc3 d6 2.Nd5 Nd7 3.Nx e7 Ndf6 4.N x g8 N X g8. The impostor is the knight at g8, which actually started out at b8. Diagram 29

Solution of Puzzle 8(ii)

l.d3 e5 2.Kd2 e4 3.Kc3 e X d3 4.b3 d X e2 5.Kb2 e X dlN mate. White can play d4 instead of d3 (so Black plays e X d4) and can vary his move order, but the final position is be­ lieved to be unique. This game first appeared in [ 1 7]. 1 2 . (G. Forslund, Retros Mailing List, June 1 996) l.e3 f5 2.Qf3 Kf7 3.Bc4 + Kf6 4.Qc6+ Ke5 5.Nf3 mate. 13. (G. Wicklund, Retros Mailing List. October 1996) l.Nf3 e6 2.Ne5 Ne7 3.N x d7 e5 4.N Xf8 Bd7 5.Ne6 Rf8 6.N x g7 mate. 14. (P. Rossler, Problemkiste, August 1994 (version)) l .h4 d5 2.h5 Nd7 3.h6 Ndf6 4.hXg7 Kd7 5.Rh6 Ne8 6.g x f8N mate. 1 4 ' . See solution to Puzzle 14. 15. (G. Donati, Retros Mailing List, June 1996) l.h4 g6 2.Rh3 g5 3.Re3 g X h4 4.f3 h3 5.Kf2 h2 6.Qel hlN mate. 16. (0. Heimo, Retros Mailing List, June 1996) l.d4 e5 2.d X e5 d5 3.Qd4 Be6 4.Qb6 d4 5.Kd2 d3 6.Kc3 d2 7.a3 d iN mate. 1 7 . al-c2, cl-b3-al, c3-a2-c l-b3, a3-b l­ c3-a2-c l , c2-a3-b l-c3, al-c2-a3, b3c l . Seven move sequences. 11.

Diagram 16. (U. Heinonen, The Prob­ lemist 1991) l.c4 Nf6 2.Qa4 Ne4 3.Qc6 N X d2 4.e4 Nb3 5.Bh6 Na6! 6.Nd2 Nb4 7.Rcl Nd5 8.Rc3 Nf6 9.Rf3 Ng8 10.Rf6 Nc5 l l .f4 Na6 12.Ngf3 NbS. Here both Black knights are im­ postors, as they have exchanged places! For a detailed analysis of this problem, see [ 16, pp. 207-209]. Diagram 1 7. (D. Pronkin, Die Schwalbe, 1985, 1st prize) l.b4 Nf6 2.Bb2 Ne4 3.Bf6 e X f6 4.b5 Qe7 5.b6 Qa3 6.b Xa7 Bc5 7.ax b8B Ra6 8.Ba7 Rd6 9.Bb6 Kd8 10.Ba5 b6 l l.Bc3 Bb7 12.Bb2 Kc8 13.Bcl . l .Nc3 Nf6 2.Nd5 Ne4 3.Nf6 + e X f6 4.b4 Qe7 5.b5 Qa3 6.b6 Bc5 7.b Xa7 b6 8.aX b8N Bb7 9.Na6 0-0-0 10.Nb4 Rde8 l l.Nd5 Re6 12.Nc3 Rd6 13.Nb l. This problem illustrates the Phoenix theme: a piece leaves its original square to be sacrificed somewhere else, then a pawn promotes to ex­ actly the same piece which returns to the original square to replace the sacrificed piece. In the first solution the bishop at c l is phoenix, while in the second it is the knight at b l ! As if this weren't spectacular enough, Black castles in the second solution but not the first. Diagram 18. (M. Caillaud, Themes-64, 1982, 1st prize) l.a4 c5 2.a5 c4 3.a6 c3 4.aXb7 a5 5.Ra4 Na6 6.Rc4 a4 7.b4 a3 8.Bb2 a2 9.Na3 alN! 10.Nb5 Nb3 l l. c X b3 c2 12.Be5 clN! 13.Bb8 Nd3+ 14. e X d3 e5 15.Qg4 e4 16.Ke2 e3 1 7.Kf3 e2 18.Ke4 e X flN! 19.Nf3 Ng3+ 20.h X g3 h5 2 l. Re l h4 22.Re2 h3 23.Nel h2 24.Qh3 hlN! 25.g4 Ng3+ 26.fX g3 Ne7 27.Nd6 mate. An amazing four promotions by Black to knight, all captured! Diagram 19. (A. Frolkin, Shortest Proof Games, 199 1) l.g4 e5 2.g5 Be7 3.g6 Bg5 4.gXh7 Qf6 5.hX g8N! Rh4 6.Nh6

Re4 7.N X f7 K X f7 8.Nc3 Kg6 9.Na4 Kh5 10.c3 g6. If 5.h X g8B? Rh4 6.B X f7+ K x f7 7.Nc3 Re4 8.Na4 Kg6 9.c3 Kh5, then White must disturb his position before 10 . . . g6. A knight is able to "lose a tempo" by taking two moves to get from g8 to f7, while a bishop must take one or at least three moves. Diagram 20. (V. A. Korolikov, Schach, 1957) White's knights are on squares of the same color and hence have made an odd number of moves in all. The White rooks and the White king have made an even number of moves, and White has made one pawn move. No other White units (i.e., the queen and bishops) have moved. Hence White has made an even number of moves in all. Similarly, Black has made an odd number of moves. Since White moved first, it is currently Black's move, so Black mates in one with 1 . . . N X c2 mate. Diagram 2 1 . (P. O'Shea, The Prob­ lemist, 1989, 1st prize) l.Ne5 biN+ (the only defense to 2.Nf3 mate) 2.Ka2 Nd2 3.Kal Nb3+ 4.Kbl Nd2 + 5.Ka2. If Black moves either knight then checkmate is immediate, so 5 . . . a3 is forced. Now White and Black repeat the maneuver Kal, Nb3 + , Kb l, Nd2 + , Ka2 (any pawn moves by Black would just hasten the end): 8.Ka2 a4 l l . Ka2 a5 14. Ka2 a6. Then 15.Kal Nb3+ 16.Kbl a2 + 17.KXa2 Nd2. This maneuver gets re­ peated until all of Black's a-pawns are captured: 44.K x a2 Nd2 45.Kal Nb3 + 46.Kbl Nd2 + 47.Ka2. Finally Black must allow 48.Nf3 mate or 48.Ng4 mate ! Diagram 22. (H. M. Lommer, Szachy, 1965) White cannot allow Black's rook at e8 to stay on the board, but how does White prevent Black from being stalemated without releasing the sleeping units in the hl comer? l.fXe8N d3 2.Nf6 (not 2.Nd6? e X d6, and stalemate cannot be prevented without releasing the hl comer) g X f6 (capturing with the other pawn merely hastens the end) 3.g5 f xg5 4.g7 g4 5.g8N g3 6.Nf6 e X f6 7.Kg6 f5 8.e7 f4 9.e8N f3 10.Nd6 c X d6 l l.c7 d5 12.c8N d4 13.Nb6 a X b6 14.a7 b5

15.a8N and wins, as White can play 19 Nxf3 mate just after 18 . . . b lQ. For the history of this problem, see [25, pp. xxi-xxii] . Diagram 23. (S. Loyd, Holyoke Tran­ script, 1876) l . b X a8N! K X g2 2.Nb6, followed by 3.a8Q (or B) mate. Note that a knight is needed to prevent 2 . . . B X a7. A queen or bishop pro­ motion at move one would be stale­ mate, and a rook promotion leads nowhere. Normally a key move of capturing a piece is considered a se­ rious flaw because it reduces Black's strength. Here, however, the capture seems to accomplish nothing so it is acceptable. Loyd himself says " [i]f the capture seems a hopeless move . . . then it is obviously well con-

cealed, and the most difficult key­ move that could be selected" [ 18, p. 156]. For further problems by Loyd featuring distant knight promotion, see [27, pp. 402-403] . Diagram 24. (G. Cseh, StrateGems, 2000, 1st prize) l.hlN! Nd6 2.h2 Nf5 3.Ng3 + N X g3 4.hlN! Ne2 5.fX e2+ Kg2! (not 5 . . . K X e2?, since Black's tenth move would then check White) 6.flN! Rc7 7.Bg3 R X c3 8.B X e5 R X c2 9.Bg7 R X c l lO.b X clN! B X g7 mate. Four promotions to knight by Black. REFERENCES

[ 1 ] W. Ahrens, Mathematische Unterhaltun­ gen und Spiele, Teubner, Leipzig, 1 91 0 .

(Vo l 1 ) and 1 91 8 (Vol. 2). [2] E. R . Berlekamp, J. H . Conway, and R. K.

AUT H OR S

NOAM D. ELKIES

RICHARD P. STANLEY

Department of Mathematics

Department of Mathematics

Harvard University

MIT

Cambridge, MA 021 38

Cambridge, MA 021 39

USA

USA

e-mail: [email protected]

e-mail: [email protected]

Noam Elkies has been a mathematics

Richard Stanley received his Ph.D. from

olympiad winner and Putnam winner, and

Harvard University, though his thesis ad­

more recently has become an outstand­

viser was Gian-Carlo Rota at MIT. His

ing composer and solver of chess prob­

main field of interest is combinatorics, in­

lems, winning the title of Solving Grand­

cluding

master in

he is a

geometry, and other areas. Except for two

number-theorist at Harvard, working on

years at the University of California Berke­

Diophantine geometry and such related

ley, he has spent his entire career at MIT.

2001 .

Meanwhile,

its

connections

with

algebra,

fields as error-correcting codes. (When he

His two-volume Enumerative Combina­

was awarded tenure at Harvard in 1 993,

torics won the Steele Prize for Mathemat­

he was the youngest ever to achieve that

ical Exposition. Among his outside inter­

status, 26.)

ests is chess problems, and he hopes that

Elkies's main interest outside mathe­

the present paper will stimulate interest

matics is music, mainly classical piano

among

and

appreciated minor art form.

composition.

Recently

performed

mathematicians

in

this

under­

works include a full-length opera, Yossele So/ovey.

VOLUME 25, NUMBER 1 , 2003

33

Guy, Winning Ways , vol . 2, Academic Press, London/New York, 1 982. [3] H . E. Dudeney, Amusements in Mathe­ matics, Dover, New York, 1 958, 1 970

[1 2] D. Hooper and K. Whyld, The Oxford

no knight attacks any other?"), with solu­

versity Press, 1 984.

tions by R . Patenaude and R. Greenberg,

[ 1 3] G. P. Jelliss, Synthetic Games , Septem­ ber 1 998, 22 pp.

(reprint of Nelson, 1 9 1 7).

placed on a chessboard in such a way that

Companion to Chess, Oxford, Oxford Uni-

Amer. Math. Monthly 71 no. 2 (Feb. 1 964),

2 1 0-21 1 .

[4] N . D. Elkies, On numbers and endgames:

[1 4] G . P. Jelliss, Knight's Tour Notes : Knight's

[22] R . J . Nowakowski, ed. , Games of No

Combinatorial game theory in chess end­

Tours of Four-Rank Boards (Note 4a, 30

Chance, MSRI Publ. #29 (proceedings of

games. in [22], pp. 1 35-1 50.

November 2001 ), http://home.freeuk.net/

the 7/94 MSRI conference on combinato­

ktn/4a.htm

rial games), Cambridge, Cambridge Uni­

[5] G . Foster, Sliding-block problems, Part 1 , The Problemist Supplement 49 (Novem­

[1 5] T. Krabbe, Chess Curiosities, George Allen & Unwin Ltd. , London, 1 985.

ber, 2000), 405-407. [6] G. Foster, Sliding-block problems, Part 2, The Problemist Supplement 51 (March,

[1 6] J . Levitt and D. Friedgood, Secrets of Spec­ tacular Chess, Batsford, London, 1 995.

[ 1 7] C. D. Lacock, Manchester Weekly Times,

2001 ), 430-432. [7] G. Foster, Sliding-block problems, Part 3 ,

December 28, 1 91 2 .

The Problemist Supplement 5 4 (Septem­

[1 8] S . Loyd, Strategy, 1 881 .

ber, 2001 ), 454.

[ 1 9] B. McKay, Comments on: Martin Loeb­

versity Press, 1 996. [23] N. J. A. Sloane, The On-Line Encyclope­ dia of Integer Sequences , on the Web at

http ://www. research .att . com/� njas/se­ quences. [24] L. Stiller, Multilinear Algebra and Chess Endgames, in [22], pp. 1 51 - 1 92 . [25] M . A. Sutherland and H . M. Lommer, 1234

[8] G. Foster. Sliding-block problems, Part 4,

bing and lngo Wegener, The Number of

Modern End-Game Studies . Dover, New

The Problemist Supplement 55 (Novem­

Knight's Tours Equals 33,439, 1 23,484,

York, 1 968.

ber, 2001 ), 463-464.

294- Counting with Binary Decision Dia­

[9] M. Gardner. Mathematical Magic Show, Vintage Books, New York, 1 978. [1 O] J . Gik, Schach und Mathematik, MIR,

[26] G . Tornberg, " Knight's Tour", http://w1 .

grams, Electronic J. Combinatorics, http:/I

859.telia.com/�u85905224/knight/eknig

www.combinatorics.orgNolume_3/Com­

ht. htm .

ments/v3ilr5.html.

[27] A. C. White, Sam Loyd and His Chess

Moscow, and Urania-Verlag, Leipzig/Jena/

[20] J. Morse, Chess Problems: Tasks and

Berlin, 1 986; translated from the Russian

Records , London, Faber and Faber, 1 995;

reprinted (with corrections) by Dover, New

original published in 1 983.

second ed. , 2001 .

York, 1 962 .

[ 1 1 ] B. Hochberg, Chess Braintwisters , Ster­ ling , New York, 1 999.

[2 1 ] I. Newman, problem E 1 585 ("What is the maximum number of knights which can be

Problems , Whitehead and Miller, 1 91 3;

[28] G. Wilts and A. Frolkin, Shortest Proof Games, Gerd Wilts, Karlsruhe, 1 991 .

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

THE MATHEMATICAL 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.

34

T H E MATHEMATICAL INTELLIGENCER

AMY E. SHELL-GELLASCH

Reflections of My Adviser: Stories of Mathematics and Mathematicians ogician William Howard was my dissertation adviser at the University of Illinois at Chicago. During our work together, Bill told me many entertaining stories about his career and the mathematicians and logicians he knew. During the six weeks between submitting my dissertation to the Graduate School and graduation, we audiotaped these stories. Here I offer excerpts from those tapes, augmented with notes made by Bill. I hope you en­ joy these stories as much as I do. AS: How did you first become interested in mathematics? BH: I grew up among craftsman in Vancouver, British Colum­ bia. Tools were important: planes, screwdrivers, files, chisels, etc. When I was 9 or 10, my mother explained that by means of algebra, one could easily solve mathematical problems that are very hard to solve using only words. I asked what prob­ lems? She gave me her algebra book from high school and said I could read about them. A few days or weeks later I told my mother I could do all the problems in the chapter on systems of linear equa­ tions. I have a clear memory of discovering the method of substitution. Looking back now, perhaps we can say this was my first mathematical discovery. But the point is that algebra is an intellectual tool analogous to the carpenter's tools. With such tools, things could be done which had been impossible before. When I was about 1 1, I convinced my mother to send me to night school at Vancouver Technical High School to take a course in "electrical engineering," which was actually a course for people who wanted to become electricians. There I saw the first mathematical result that was mysterious to me. Someone from a previous class had left a mimeographed sheet containing a statement of the Pythagorean Theorem, but no proof. I was unable to prove it at the time, but no doubt

it was not long before I saw a proof in Hogben's Mathemat­ ics for the MiUion [1937]. Also in that night school course, I learned about trigonometry, and was intrigued by the fact that the voltage in standard power lines was a sine wave. By grade 1 1 or 12, I was reading calculus books. I had read in a model airplane magazine that the Davis airfoil had been designed by use of "the calculus," so I went down to the Vancouver Public Library to find out what this calcu­ lus was all about. When I entered the University of British Columbia I in­ tended to become a physicist, but in my final year it became clear that I would do better in mathematics. I guess the main reason was that mathematics is more inward. I asked one of the professors, Professor Derry, what the prospects were for earning a living as a mathematician. He said I wouldn't be­ come rich but I wouldn't starve. That was enough for me. So in the summer of 1948, I went to the University of Illinois at Urbana-Champaign to work on my Ph.D. AS: Bill, you mentioned that mathematics is very inward. Has that played an important part in your career and your life? BH: From childhood, I have been interested in the relation between the world of inner experience and the external world. I was a solitary child, the only child of a single par­ ent, with a rich fantasy life. So the relation between the world of inner experience and the external world has al­ ways been very significant to me, and I have spent a large amount of time and energy on the question. At one extreme,

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 1, 2003

35

a concern with this question was one of the main reasons for my foray into the transcendental meditation movement in the 1970s. AS: How did you end up doing your doctoral work at the University of Chicago? BH: I arrived at the University of Chicago in September 1949, having been very unhappy with the intellectual and cultural atmosphere at the University of Illinois Urbana-Champaign. The first or second night after arriving in Chicago, I went into Hutchinson Commons for supper (the structure is an imitation of some building at Cambridge), and at a large table at which I sat down, there was a group of five or six undergraduates. I became aware that they were talking about Aristotle and Plato. This contrasted with lunch or supper at the University of Illinois Student Union, where the topic of conversation amongst the undergraduates was almost entirely parties, football, or how much money their fathers made. So that was the difference. It was borne out during my time at Chicago. While in Urbana, I met Gerhard Hochschild. Quite a char­ acter in the math department. Hochschild and I got to know each other very well. He had bought a new car, a Dodge, and he liked to drive it at night. I used to go with him and we would drive all around and talk. He introduced me to the work of Bourbaki. He told me that I would be a lot happier at the Univer­ sity of Chicago. That Andre Weil was there, Mr. Bourbaki himself. That there was a student, Norman Hamilton, that I would get along well with. And indeed I became extremely good friends with Norman Hamilton. He was here at UIC until he died in 1996. We were always very close. At the end of spring semester 1949, we got into Hochschild's car; he wanted to see the United States. We put 10,000 miles on that car in three months and he never let me drive. We drove from Urbana to Chicago, and stayed overnight in Chicago. I slept in Irving Segal's apart­ ment on the floor. We picked up Norman Hamilton the next day, and drove from Chicago to Boulder, Colorado, in one long stretch. In those days that was quite a feat, since there were no superhighways. In Boulder we at­ tended the summer meeting of the American Mathemat­ ical Society. In Boulder we also picked up Shizuo Kakutani, who was a leading functional analyst at the time. At one stop in Col­ orado, Hochschild saw a mountain that he wanted to scale. At that time I was in extremely good shape, I had been do­ ing a lot of skiing and working in logging camps in British Columbia. So we went and climbed that mountain. We came back tired but happy three hours later. Meanwhile Kakutani stayed in the car. We proudly said we had got to the top of that peak. And he said he had proved a new the­ orem. So we were all pretty happy. We went way up to Seattle via Yellowstone National Park and the Columbia River Valley. It was quite a nice ex­ perience, driving down the Columbia River Valley at night. I was trying to figure out how to apply Zorn's lemma to some theorem in functional analysis, while Kakutani was saying, "No, no, no, you are not doing it the right way. What

36

THE MATHEMATICAL INTELLIGENCER

you have to do is" such and such. It was nice to learn math­ ematics that way. AS: How did you get into the University of Chicago? BH: At the Boulder meeting, Hochschild introduced me to Saunders Mac Lane, who was at that time chairman of the math department at Chicago. Mac Lane sat me down, and like L. E. Dickson, he gave me a problem. It involved com­ binatorial topology, which I didn't know much about. It was a problem basically in mechanics, the behavior of certain dynamical systems. I had enough intuition to see that there was a fixed-point. Afterwards Hamilton explained to me the Brouwer fixed-point theorem. Mac Lane had also given me a simpler problem that I was able to solve. So I passed. Mac Lane said he couldn't make any promises at that late date, but he would try to get me a teaching assistantship. He succeeded, and I arrived at the University of Chicago in September 1949. AS: What was the atmosphere like at Chicago at that time? BH: This can be illustrated by a little story about Adrian Al­ bert. When I arrived in Chicago I needed some money to tide me over for the first month. (In fact Hochschild had already lent me twenty dollars, I think, so I could accom­ pany him on his trip. Actually, it was about four years be­ fore I got enough money to repay him, I was living so close to the edge. And when I finally got enough money and mailed him a check, he was insulted and mailed it back to me. Here I was so proud that I could finally make restitu­ tion, and he was insulted. But I think it was an indication of his affection as well.) So there I was with practically nothing in my pocket. And I wasn't going to get a paycheck until the following month. I went to University National Bank, the local bank, on 55th Street, to take out a loan. I think it was for fifty dollars. I told the loan officer I was a graduate student at the University of Chicago in mathematics. He told me to go to my department chairman. I just looked at him. He said GO to your department chairman. So I went and saw Adrian Albert, the department chair­ man. I said that the loan officer said I should come to him. He asked me how much I needed. When I said fifty dollars, to my astonishment, Albert pulled a little metal box out of his desk, counted up the money and handed it over to me, without a word. I think he got a kick out of doing that. When I asked if he wanted me to sign anything he said just repay it when I was able. He then said, "When a person is hungry, it is hard to think about mathematics. We want our students to be free to think about mathematics." I was a rather bedraggled-looking character, and I'm not sure I would have lent me money at that point. Albert was well known for acts of kindness like that. AS: Your original advisor was Andre Weil. How did that come about? BH: After three or four months at the University of Chicago, I had the following view of the math department. At the top was the great Andre Weil, he was the archangel, or as Richard Kadison said, "Weil is god." So god was at the top, and there were some archangels: Saunders Mac Lane, An­ toni Zygmund. Later I understood that Zygmund was a god

in his own right. Siing-shen Chern was probably as high as Weil, not quite as broad, but certainly as high in geometry. I was interested in the top, so I started to associate with Weil. After the mountains around Vancouver, Chicago con­ crete was kind of hard on me. To get some relief, I used to walk in Jackson Park. Weil noticed that we would often pass each other going in opposite directions. So one day, probably in International House at lunch or supper, he said, "You walk in the park. I walk in the park. We should walk together." From that time on we walked together. We were about the only two people at the University of Chicago who liked to walk in the park. Two foreigners, one from Van­ couver, one from France. I would ask him about various parts of mathematics, his opinion of the educational system, the nature of creativity versus competence, anything. We had enough common in­ terests that he was happy to talk. By the summer of 1951, I had been at Chicago for two years, and it was time that I got to work on a thesis. That summer Weil lived in a little monastic cell in International House, while his family went to France; so I encountered him almost daily at lunch or supper. One day he put down his tray and said, totally out of the blue, that he had a thesis problem for me. I suspect that there had been a faculty meeting, and they thought it was time Bill Howard was put to work. This was a problem of Elie Cartan. Given one complex variable on a bounded domain, the group of one-to-one an­ alytic mappings of that domain onto itself is a finite-di­ mensional Lie group. This group of all isometries in the space of constant negative curvature is the Poincare group. The elder Cartan generalized this to two and three complex variables. Namely, he showed that the underlying metric is "symmetric" in the sense that locally, reflection through a point by means of geodesics is an isometry. Weil said I should try to prove the result for the general case of n com­ plex variables. And moreover, try to get a simple general proof for the case of two and three complex variables, so you wouldn't have to look at it case by case. The way he told it to me, I assumed that Cartan had thought the theorem was true, and I think Weil thought the theorem to be true in general for n dimensions. I spent sev­ eral months just acquiring the necessary background. I had to learn the theory of several complex variables, Lie groups, Riemannian geometry; it was too much. I was out of my depth but didn't know it. I had a good teacher for the geom­ etry, Professor Chern. By the spring of 1952 I thought I had solved the prob­ lem. I told Weil my proof and he thought it was okay. I pre­ sented it in a seminar; several high-powered visiting math­ ematicians were there. They thought what I had done was reasonable; they couldn't check every detail in seminar, but they thought I was using the proper methods. Weil told me to write it up and submit it with a proof of Lemma 2, which he was sure was true. I went home, and as I was falling asleep, I was think­ ing of how I would write out the proof of Lemma 2. Then I realized that I couldn't prove Lemma 2. Then I noticed

that there was a counterexample. Lemma 2 was false, so the whole thing imploded. I remember feeling that the bot­ tom had physically fallen out of the bed. That was a real sinking feeling. The next day, I told Weil. He said, "What do you mean Lemma 2 is false!" I showed him my coun­ terexample. He stormed out of the office. Ten minutes later he stormed back in and gave me his own coun­ terexample. He saw that I didn't look very happy, so he took me for a walk in Jackson Park. We walked along in silence for about ten minutes, and then he said, "Well, you've got enough results, bits and pieces, it's not what we had hoped. But you have enough there for a Ph.D. thesis, just write the bits and pieces up." I guess he must have been serious, thinking about what I had already done, and there was enough there. I said I wanted it all or nothing. No bits and pieces for me. We walked along in silence a little longer, and he said, "You have experience climbing mountains. You must know that you don't have to go to the top, to the peak. You can enjoy yourself going partway up. Just play around, you can see the other people going up." You can imagine what I felt like at that point. He was trying to make it better. He was trying to put himself in my place, and he realized that he wouldn't have accepted it ei­ ther. So what was coming out was sounding kind of strange. Suddenly out of the blue, he said, "Have you ever thought of becoming an administrator?" That was a real inspiration to him. I told him that knowing what he thought about ad­ ministrators, I considered that an insult. He said, "Take Wal­ ter Bartky" (who was the Dean of Physical Sciences). "Of course his mathematics is weak. But he's done some very worthwhile things. " I asked what h e had done that was worthwhile. And Weil responded, "Well, he brought me here!" AS: What did you do next? BH: I had applied for an ONR post-doctoral contract to spend a year at Columbia University, studying under his friend Claude Chevalley. I told Weil I would have to return the ONR contract. He replied that the contract was not con­ ditional on my receiving my Ph.D. So I received a post­ doctoral contract without having a doctorate. This was at a time when the ONR funding was very generous, under the leadership of Mina Rees. AS: You later finished your degree under Mac Lane? BH: After spending a year at Columbia, I returned to the University of Chicago in 1953 for another year on the ONR contract. By September, there were two young visiting faculty, leading recursion theorists, John Myhill and Jim Dekker. They were very gregarious. Raymond Smullyan, Stanley Tennenbaum, and the two young recursion theo­ rists had formed a very interesting group engaged in ex­ tremely lively discussions. Recursion theory was very new at that time; they were working on Post's problem. Ten­ nenbaum managed to rent a very nice house on Drexel St. belonging to a faculty member who had gone on sabbati­ cal for a year. So Tennenbaum ran a 24-hour salon on math­ ematics and philosophy. Moreover, he had John Myhill liv-

VOLUME 25, NUMBER 1 , 2003

37

ing in the basement, and Paul Cohen was up in the attic. Every once in a while Cohen would come down and heckle. It was overwhelming, and this was what I was really in­ terested in anyway: the relation between the inner world experience, and the external world. To me, that is what the foundations of science are all about, and the foundations of mathematics are part of that. I found this very exciting, so that is why I switched careers. There is a wonderful story about Raymond Smullyan from about this time. One day we were walking down the street, and some music was coming out of a window. Some­ body was practicing Mozart on the piano. He suddenly stopped and said, that is not the way you play Mozart. He got very distressed and he ran up and knocked on the door. A young woman came out, and he said he had to show her how to play that piece. He went up and played it the way he thought it should be played. I was just swept off my feet. The primary importance was art, everything was sacrificed to art. That is what Smullyan was like-and what life was like around the University at that time-it was intense. By the spring of 1954, I had developed lots of ideas in the areas of mathematical logic and the foundations of mathe­ matics, but I saw no chance that the Math Department would allow a thesis in these topics. I was resigned to the idea that I would write a thesis in ordinary mathematics, get my Ph.D., and then be free to work in mathematical logic and the foun­ dations of mathematics in a serious way. To my astonish­ ment, Anil Nerode told me one day that Mac Lane had agreed to allow us to write dissertations in mathematical logic un­ der him. At first, I did not believe him, but when I found that it was true, I was overjoyed. I made a couple of false starts, but by the summer of 1954 I had struck a real gold mine. I had read the work of Gerhard Gentzen and run across a paper by R6sza Peter connecting a segment of Gentzen's ordinals with Hilbert's primitive re­ cursive functionals of type-level 2. She had left a problem open, which I was able to solve. More importantly, I solved the problem by making a more fundamental connection be­ tween functionals and ordinals. Whereas Peter thought that the functionals of level 2 corresponded to the ordinals less than ww, I showed that they corresponded to the ordinals less w than w w . Unfortunately I was not able at that time to extend this to all primitive recursive functionals of finite type, though I did succeed eventually while at Penn State and published it in the Buffalo 1968 Conference volume. AS: Tell me about the Howard ordinal. BH: Here is the story of my ordinal [the Bachmann-Howard ordinal]. I have two big results for which I am known. One of them is this ordinal. I analyzed the functionals that in­ terpret a central part of Brouwer's mathematics, the so­ called Bar Theorem. I found the ordinal that compares to that. It has the same relation to Brouwer's mathematics as Gentzen's epsilon-zero has to Peano Arithmetic. So already it is beginning to sound like a good result. It is a much much larger ordinal than epsilon-zero. All these ordinals are con­ structive ordinals, but you actually use the uncountable car­ dinals to get to these countable ordinals. It is an interesting example of what Kurt Godel said, that

38

THE MATHEMATICAL INTELLIGENCER

to prove a result of a certain level of abstraction, you of­ ten go to much higher levels to get the result. The first ex­ ample of that is that one can get results in number theory by going to complex analysis. In fact, the solution to Fer­ mat's Last Theorem is an extreme example of that. This method of fmding constructive ordinals by going to uncountable cardinals was discovered by Oswald Veblen, in a 1908 paper. Paul Bachmann discovered this paper of Ve­ blen, and extended it in a beautiful way. This is called the Bachmann hierarchy of ordinals, or the Bachmann ordinals. At Penn State in 1962, Hilbert Levitz declared he wanted to write a thesis under me. I had never had any experience directing a thesis, and I didn't want to start at that point. I was too busy with my own research. So I thought I would fob him off onto Kurt Schutte who was a Visiting Professor at Penn State at the time. Levitz was taking a course by Schutte that I was auditing. Penn State had a lot of trees, and there was a little rather fragile building in the woods; Schutte gave his lectures there. It was quite charming. After the lecture we would walk back along a little trail. As we were walking along the trail, I asked Professor Schutte if he could suggest a the­ sis topic for Levitz. I thought Schutte would have to think. To my surprise he instantly said, "Yes: determine the relation be­ tween my ordinal notations and Takeuti's ordinal diagrams." This was a very hard problem, and I knew it was a hard problem; it would take a lot of work Levitz worked on it for the rest of the year, and didn't make much progress. Then he decided that he would follow Schutte back to Europe in 1963. First he went to the University of Kiel, on the North Sea. Apparently it was a very dreary place. Halfway through that year, Schutte got a position at the University of Munich. All the while I was getting letters from Levitz: he is working day and night and not making any progress at all. The letters sounded more and more dis­ couraged, when suddenly I received a letter stating, "I've made a breakthrough!" Levitz had simply, in a very schol­ arly way, read every single piece of literature that could possibly have any relevance to this problem. Ultimately he got to Bachmann's monograph on ordinals. Schutte and I had a constructive philosophy. We felt set theory didn't have anything to do with the external world, so we didn't think set theory could be relevant to develop­ ing constructive mathematics. But that is how it turned out. Levitz actually solved the problem very rapidly using Bach­ mann's work He came back to Penn State in the fall of 1964 very excited. I helped him put his thesis into final shape. The first day he was back I remember Uvitz said, "You have to see this, it is very interesting." He didn't know it had any bearing on problems I was working on. He said, "When I start explaining this, you're going to think, since I am talk­ ing about uncountable cardinals, that can't have anything to do with this problem. But actually you will see that that is the key to this problem." And it was. It was Veblen's dis­ covery, but Bachmann had extended it in a very substantial way. As soon as Levitz explained, I thought that this was probably also going to solve what I was working on. I obtained the ordinal for Brouwer's Bar Theorem in two steps, each equally important. I had already made the first

step of constructing it and making it into a problem about constructive transfinite recursion of finite type. Georg Kreisel had already shown Godel that work And Godel had said it was interesting, but not sufficiently mathematical. For Godel, ordinals had to be described in a more concrete way. I had gotten half way, but the Bachmann thing was a totally new idea. That was a big lesson for me because I had thought I could ignore set-theoretic reasoning because my goal was to understand the relation between mathe­ matics and the external world, and set theory was the wrong way to go. I remember writing to Schutte about the result, and he responded that the result was remarkable. So that is one of the basic ideas in the field, and the tech­ nique that I developed. In fact, now they are actually using it without my name. There is a certain collapsing process by which you get from the uncountable back to the count­ able. They say "by the collapsing process," but I take that as a compliment, it is so much a part of the work now they don't even have to mention my name. So that was my first good result, spring of 1965. AS: What is your other big result? BH: A year or so after that, I got my other first-rate result: the isomorphism between a typed A-calculus and the sys­ tem of proofs, or more precisely, derivations, in the corre­ sponding formal theory of intuitionistic mathematics. AS: Where were you at this point? BH: That was the summer of 1966, a year after I had arrived at UIC [University of Illinois at Chicago, formerly Chicago Circle]. AS: How did you get to UIC? BH: Let me back up. I received my degree from the Univer­ sity of Chicago in the fall of 1956. While at Chicago I worked at the Bartky Project on the stability of feedback servosys­ tems. Having been in school most of my life, I felt a need to do something real, so I decided to work in industry and physics. I accepted a job with Bell Labs at the Whippany [New Jersey] Lab for that fall. But I still had to go back to Chicago during the fall to take my oral exams. After a cou­ ple of years at Bell Labs I felt that I had had enough of in­ dustry and that I should get back into the academic world. In the spring of 1959 I saw Raymond Ayoub, whom I had known as a fellow student at Urbana. When I told him I was working at Bell Labs, he said I was the last person he would have expected to be working in the corporate world. I agreed. He then asked if I would like a position at Penn State. So I was at Penn state from 1959 to 1965. While at Penn State I started my correspondence with Georg Kreisel. This was an important part of my intellec­ tual life. Over the course of ten years, I received about one thousand pages of handwritten letters from him, and I wrote about one thousand pages in return. In 1965 I went to the newly founded University of Illinois at Chicago. That was where I was when I got my second big

1

1 Kreiseliana: About and Around Georg Kreisel

result. Kreisel felt it was important to develop a mathemati­ cal theory of Brouwer's notion of construction, and he was urging this problem on anyone who would listen. I remem­ bered something I had learned from Haskell Curry while at Penn State, the isomorphism between derivations and terms, mentioned above, and I developed it in a mathematical way. It did not solve the problem, though perhaps it was a contri­ bution. In any case, it has turned out to be useful in proof theory and theoretical computer science. AS: When were you at The Institute for Advanced Study? BH: In 1971 I mentioned to Kreisel that my first sabbatical year at Circle [UIC] was to be in the academic year 1972-73. I thought I would spend it in Denmark I had supervised the dissertation of Ib Axelsen, and he was going to Aarhus, which was quite a well-reputed place to go in logic, or physics for that matter. I liked the Danes and the other Scandinavians a lot. So I thought that would be a nice place to spend my sabbatical, except I understand the weather in Aarhus is kind of lousy, being near the North Sea. I was sure I could get accepted there. I told Kreisel, and his reply was, "Don't be ridiculous. You should spend your sabbatical at Princeton with Godel." I wrote back that I would never get accepted as a visiting member at the Institute for Advanced Study. He said, "Yes you will, I'll talk to him." I'm sure it was within a month that I got a letter from Godel to the effect that he had heard that I was interested in spending my sabbatical year at the Institute for Advanced Study. They would be glad to con­ sider me for membership. Right at that point I was pretty sure Godel had decided that I was going to go there. Sure enough, it was very easy, I sent the materials and it wasn't long before I got another reply from Godel offering me a visiting membership and asking what my current salary was, so they could provide the other half. I had just learned transcendental meditation in February of 1972. By the summer of 1972 I was really into it. I had been doing that meditation technique and going to weekend retreats and so on, doing the Hatha yoga, the postures, and the meditation. I had some very interesting changes of con­ sciousness, so I wanted to pursue this further. And the only way to get into the inner circle was to become a teacher. The higher-level knowledge wasn't available to the public. I took part in the first month of the teacher-training course, at Humboldt State College, in northern California. It is not right on the seacoast, but you can see the sea. Behind the campus is a redwood forest, and in the middle of this is the sports stadium, about a half-mile walk through the red­ wood forest. During the day we would be taking these courses, watching tapes, discussing the nature of transcen­ dental meditation and Vedic philosophy. Of course we would be doing a lot of meditating, a lot of yoga during the day, and that makes you quite high. So after supper, we would be really quite high, and we would walk through the forest in the twilight, to this sports stadium. There were a

contains articles, including one by Bill, for the occasion of Kreisel's 70th birthday. Ed. P. Odifreddi, published by A. K.

Peters, 1 996.

VOLUME 25, NUMBER 1, 2003

39

thousand of us, and there was Maharishi, up there in his white sheet, in the lotus position, with all the flowers around him. And he would take questions. Microphones throughout the audience, people would line up to ask their questions. That is how he transmitted his teaching, like the old way, two thousand years ago. It was a pretty good scene. I want to emphasize that this was, for me, not just a spir­ itual journey. Maharishi's teachings addressed, in a direct way, my central intellectual concern; namely, the relation be­ tween the world of inner experience and the external world. After a month of that, as you can imagine, I was just float­ ing. At the end of August, I flew back to Chicago and drove to the Institute. I remember the first day I arrived there, it was a nice day, it smelled grassy, the bumblebees were buzzing and the birds were tweeting-here I was, in Paradise. My friend Stanley Tennenbaum had spent quite a few years hanging around the Institute and had gotten to know Godel very well. After arranging where I was to stay and so on, I looked around for Tennenbaum and I found him that evening. I had an uncanny feeling that at last I had ar­ rived at the summit. I was now at the Institute. Tennen­ baum said, "No big deal." But I was very hesitant to get into contact with Gi:idel. I felt that first I had to prepare what I was going to say to him. My preparation went on for about three weeks. I got a telephone call from Gi:idel. "Hello, is this Profes­ sor Howard?" [Bill is talking in the slow, hesitant voice of Godel.] "Yes." "This is Gi:idel. Why haven't you come to see me?" "Oh, I am sorry. I thought I should prepare myself." He said, "No, that is not necessary. Just come and we will talk." So we arranged a time, at his convenience, early after­ noon. He had a very nice office, overlooking the pond, right by the library. But the problem is that you enter the office through the library. You go underneath the wall of the li­ brary, and up the stairs into his office. I didn't know that, so there I was walking around outside his office at the ap­ pointed time, wondering how I was going to get in. I went up onto the verandah that overlooked the pond, I saw a door with frosted glass. It was pretty clear that was not the main door, but it was a door. I knocked on the door. "One moment," and I see this image coming. He opens the door. At last: God! "Won't you come in? Did you have a nice trip? Are you rested? Can I get you some tea?" Very elegant Viennese, formal but warm, an interesting combination. First he said we must review the present status of my work and discuss what I would be doing. He wasn't going to tell me what to do, but he had some ideas. I remember in that first discussion, I felt as if the air was shimmering. That was the state of my brain. I had so much admiration, plus I had been doing all this meditating up in the woods. I just felt that I was up on a high glacier. We discussed the papers I had written. He said that my assignment of ordinals to primitive recursive functionals of finite type had to be simplified, it was not satisfactory. I

40

THE MATHEMATICAL INTELLIGENCER

said it was a proof. He said the proof had to be conceptual. I had worked on my proof for years, a number of people had tried it. I had finally succeeded by patience, but it was kind of like Rubik's Cube: I had moved the pieces around so much that it finally fell into place, but I didn't really know how I had gotten there. So I could understand what he was saying, but didn't know how to simplify it, and I didn't want to work on it anymore, actually. One theme throughout my stay at the Institute was that about once every three or four weeks, I'd get this telephone call. "Hello, is this Professor Howard?" "Yes." "This is Gi:idel. About the assignment of ordinals, I have an idea about how it can be done." And he would explain his latest idea. And I would say, "I tried that, Professor Gi:idel, I've tried everything." "No, I think you should have another try; this should work." So I would try, and it definitely would not work. I would write him a note to this effect. This went on all year. I told this story to Gaisi Takeuti once, and he said that everybody had that experience with Gi:idel. And in Takeuti's case, Gi:idel turned out to be wrong. Let me say, I had four meetings with him, one in the fall and three in the spring, but also there were quite a lot of these telephone calls, and two or three letters from me to him. But the pattern of our meetings during the spring was quite interesting. Each meeting was, on average, two hours. I feel good about this because he didn't talk to many peo­ ple that long. Usually the first hour was spent by my explaining to him my experiences with transcendental meditation. He was very, very interested, both in my experiences, and the whole outlook, how it was done. Then after about an hour of that, suddenly Gi:idel would look guilty and say that we had had our pleasure. That he was responsible for dis­ cussing my work with me, so we must do that. Then we would get down to my work on the foundations of mathe­ matics, and what he thought I should do. Almost everything he suggested that I do, I tried my utmost to do, but usually not very successfully. AS: What are your strongest memories of working with Kurt Gi:idel? BH: There are various items from our discussions of proof theory and the foundations of mathematics, but these would be of interest mainly to people who work in those fields. Here is a story from our discussions of TM and con­ sciousness. Because Gi:idel had repeatedly asked me to describe my experiences during meditation, I finally suggested that maybe he would like to learn how to do TM. He said no, and I asked why not. Gi:idel replied, "The goal of Mahar­ ishi's system of meditation is to erase thoughts, whereas the goal of German idealism is to construct an object." In saying this, he became quite forceful, holding out his hand-palm and fingers upward-as if he were grasping an object. I think what Gi:idel meant was that the goal was to

build a mental structure. I tried to explain that it was not the purpose of TM to erase thoughts; but his mind was made up. AS: In much of the literature about Godel, there is much about his purported paranoia. From the stories you have told me, Godel was a very private man, and tended to keep people at a distance, which may have led in part to this stereotype of him. It also sounds like he had a good, though subtle sense of humor. BH: Yes, for example, while Tennenbaum and I were at the Institute, we both heard a story that one of the other mem­ bers of the Institute claimed that Godel was irrationally scared of him, and hid behind trees when the other man was coming. Tennenbaum thought that maybe Godel ducked behind a tree because he simply didn't want to talk with this man. Tennenbaum told me a story once about sitting on a bench on the Institute grounds talking with Godel, when one of the young hotshots came up and asked Godel a highly technical question about large cardinals: whether it would be worthwhile to consider such and such. Godel re­ sponded that everything should be considered. This was the totality of Godel's reply. I gather that the young man was so disconcerted that he thanked Godel and left. AS: Godel seemed very interested in your experiences in transcendental meditation and other topics involving con­ sciousness. Do you have any more stories about that? BH: Yes, as I said earlier, we talked about TM often. Godel and I both were very interested in the external world ver­ sus the inner world. Once during a discussion about med­ itation and Maharishi's teachings, Godel asked me for my opinion about extrasensory phenomena and whether I had any such experience. I said that there was no evidence for extrasensory phenomena. Godel replied that he had once had the experience of thinking about a certain person whom he had not thought about for years, the telephone rang, and the caller was that very person. I replied that we could hardly take that as sci­ entific evidence for extrasensory phenomena. Godel re­ sponded that he agreed that there was no scientific evi­ dence! Well, he won that one. There was indeed evidence for extrasensory phenomena, just not scientific evidence. Af­ ter conversations like this, Tennenbaum commented that Godel was just teasing me. AS: Tell me your story about Godel and Abraham Robinson. BH: In March 1973, I saw Godel and Abraham Robinson in the cafeteria. Godel invited me to join them. While sitting there chatting with them, I noticed that Robinson seemed to have something on his mind that was making him quite nervous. Maybe it had something to do with the fact that

AUTH O R

AMY E. SHELL-GELLASCH

Department of Mathematical Sciences United States Military Academy West Point, NY 1 0996 USA e-mail: [email protected]

Amy Sheii-Gellasch, pictured here with William Howard, is an Assistant Professor at West Point and is in the last year of a post-doctoral fellowship. She received her D.A. from the Uni­ versity of Illinois at Chicago in 2000, with an historical thesis on the mathematician Mina Rees. Her current work is on the history of her own Department. She and her husband plan a three-year sojourn in Germany.

(as I learned a few years later) Godel wanted him to be his successor at the Institute. Or maybe he knew that there was something wrong with his health. A few months later, he was diagnosed as having pancreatic cancer, and he died not long after that. But back to March 1973. Robinson gave a talk at the In­ stitute on his nonstandard analysis. Godel sat in the back, writing in a little notebook. At the end of the talk, Godel stood up and, consulting his notes, gave a speech to the ef­ fect that, 300 years ago, mathematics had taken a wrong turning in ignoring the potentialities of in!mitesimals and nonarchimedian ordered fields, and that perhaps this ex­ plains why there has not been much progress in number theory during the past 300 years. This was a dig at Andre Weil, who had just given a series of lectures on "Three Hun­ dred Years of Number Theory." Weil just scrunched down in his chair. Godel went on to say that something like what Robinson had just described will be the analysis of the fu­ ture. At that point, it appeared to me that Robinson levi­ tated about a foot off the floor. But since that contradicts the laws of physics, I suppose that he just stood a little straighter and maybe puffed out his chest. In any case, Robinson must have been pleased. 2

2Bill is describing this event from memory. The transcript of G6del's remarks can be found on p. x of Robinson's Non-standard Analysis, Gbdel's Collected Works, Vol 2, Oxford University Press,

1 974 ,

also on p.

31 1

of

1 990 .

VOLUME 25, NUMBER 1 , 2003

41

JOHN HOLBROOK AN D SUNG SOO KIM

A Very M ean Va u e Theo rem

iven a continuous function f : [a, b] � � that is differentiable on (a, b), the MeanValue Theorem (MVT) of elementary calculus allows us to ''predict " a value of f ' (x). Knowing only the boundary values f(a) and f(b), we predict that ( f(b) - f(a))l (b - a) E f ' ((a, b)). May we not expect an analogous result for f : K � � when K is a compact, convex subset of IR1n? Figure 1 shows that the short answer is NO! There we see red, green, and blue "del­ toids" (these classic 3-cusped hypocycloids made an unan­ ticipated appearance in our construction!) that represent the ranges of the gradient V'f({x2 + y2 < 1 }) for three dif­ ferent continuous extensions f of the boundary function f(cos 0, sin 0) = sin 20. Since no point is common to all three of these deltoids, we certainly cannot predict a gra­ dient value from the boundary function alone. Presently we'll explain just how the boundary function sin 20 is var­ iously extended in Figure 1 , and comment on other features seen there. It is already clear, though, that the multivariate MVT is a very mean value theorem indeed. The long answer contains further surprises. For exam­ ple, the extensions of the boundary function sin 20 used in Figure 1 are continuously differentiable also on the bound­ ary of the unit disc, with the exception of a single point. Yet that exception is crucial: we'll see that for f(cos 0, sin 0) = sin 20 (and for many other boundary functions) we must have

0 E V'f({x2 + y2 ::s:

1 }),

wheneverf extends the boundary values to a function con­ tinuously differentiable on the whole closed disc []), Figure

42

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

2, to be described in more detail later on, illustrates this effect. Nevertheless, for sufficiently intricate boundary func­ tions g(O), it appears that the phenomenon of Figure 1 can occur even for extensions continuously differentiable on all of []). That is,

n

V'f ([]))

fla[)) -g

= 0,

where the intersection extends over continuously differ­ entiable f : []) � IR1 such that ('t/0) f(cos 0, sin 0) g(O). More generally, this discussion makes sense for compact, convex K C IR1n provided K is also regular, i.e., K is the clo­ sure of its interior K". In this case, what we mean by a con­ tinuously differentiable f : K � IR1 is a function f such that V'f is continuous on K" and V'f may be continuously ex­ tended to K. On the other hand, there are certain boundary functions for which we recover the full flavour of the one-variable MVT. For example, if the graph of the boundary function lies in a hyperplane, so that for some fixed vector v and constant c =

u E a[])

�

f(u)

=

u

· v

+ c,

[

4

(Vg'P(v(e, t)))* =

3

-sin (} cos (} - cos cp

cos (} sin e - sin cp

]- 1 [

]

g ' (O) g(O) - g(cp) '

(2)

where w* denotes the transpose of the vector w. Thus, for the functions g'P obtained by affine focussing, the range Vg'P([}0) is the same as the image of the circle: Vg'P({u(O) : e =I= cp}), which we expect to be a "one-dimen­ sional" object. More important is the observation that, with g fixed, each of the possible g'P has the same boundary values. Thus an example for which

2

0

-1

Vga({u(O) : (} =I= a }) n Vgfl ( l u(O) : e =1= 13 )) n Vg y({u(O) : e =1= y}) = 0

-2

� �--�--��--�--� -3 -2 -1 0 1 2 3 4

Figure 1. If the boundary function g(ll) is given by sin 211, NO gradi­ ent values can be predicted for continuous extensions to the disc.

we may adapt the familiar Rolle's theorem approach to the MVT to see that v E Vf([}0). This sort of multivariate MVT is treated by Furi and Martelli in [F-M95]. We should also note that more straightforward multi­ variate versions of the MVT are obtained if we want to pre­ dict the average value of the gradient, rather than a par­ ticular value. Consideration of such averages will arise in our detailed treatment of Figure 1 . Constructing Figure 1

Given a smooth 21r-periodic real function g(O), we shall de­ fine a family of functions g'P on the disc that extend the boundary values suggested by g via a procedure that we may call affine focus. We use the notation u(O) = (cos 8, sin 0). Every g'P has the same boundary values on []): g'P(u(O)) = g(O). The functions g'P differ only in the way they are (continuously) extended to the interior of the disc. For each real t and e, let g'P(tu(O) + (1 - t)u(cp)) = tg(O) + (1 - t)g(cp) .

(1)

The function so defmed is continuous over the whole disc [} and, by the assumed smoothness of g, g'P is smooth at every point but u(cp). Note that the graph of g'P is a "ruled surface," generated by the line segments joining (u(O), g(O)) to (u(cp), g(cp)). Differentiating with respect to e and t, we find that, with v(O,t) = tu(O) + (1 - t)u(cp),

- t sin OD�g'P(v(O, t)) + t cos e Dygiv(O, t)) = tg' (O), and (cos e - cos cp) D�'P (v(e, t)) + (sin e - sin cp) Dyg
(3)

shows that we cannot predict, in terms of boundary values alone, that any particular gradient vector will occur within the disc. Figure 1 displays an example of (3) where we have taken g(O) = sin 28. Each image Vg'P([}0) is indeed a one­ dimensional object, namely a deltoid curve (see Remark 3 below). Figure 1 shows a sequence of these deltoids cor­ responding to 20 values of cp equally spaced around the cir­ cle. Highlighted in red, green, and blue are the gradient im­ ages corresponding to three particular values of cp: a = 0, 21T, and y = 21T, and we see that (3) occurs. 13 =

2�

!�

REMARK 1. For any two values a, 13 the sets Vga([}0) and Vg/l([}0) must intersect. Indeed, iff1 and f2 have the same boundary values then "Rolle's theorem" implies that V(/i - f2)(v) = 0 for some point v inside the disc; hence, the gradient ranges will have at least one common point: Vfl(v) = Vfz(v). REMARK 2. Certain features seen in Figure 1 call for an explanation, although they are largely a result of our spe­ cial choice g(0) = sin 20 and are not central to our argu­ ment. When the "deltoids" (see Remark 3) first appeared in our simulations we could not immediately identify them, though they seemed somehow familiar. For the reader who is curious about mathematical "coinci­ dences," we include the following guide to the geometric phenomena seen in Figure 1 . A somewhat strenuous cal­ culation (based for example on (2)) reveals that (with g(O) = sin 20, of course)

(

Vg'P(u(O)) = u 2e + cp -

;) + 2u( ; - e) + u( ; - cp}

(4)

It is then easy to check that, if we set

(

V(O) = u 20 -

;)

+

2u

(; )

- e .

(5)

(; )

(6)

which is independent of cp, we have Vg'P(u(O)) = R'P13V(O + cp/3) + u

- cp .

where Ra denotes the operation of planar rotation (coun­ terclockwise) by a radians about the origin. Now (6) explains why, for this special choice of g, all the gradient ranges Vg'P([}0) are congruent, as Figure 1 suggests. They are con-

VOLUME 25. NUMBER 1 , 2003

43

gruent to the single object V(IR:); moreover, putting if! = 27T in (6) explains the three-fold synunetry seen in the object V(IR:), and explains why the sequence of 20 steps in Figure 1 returns the curve to its initial position with a twist of 120°. REMARK 3. In fact, the object V (IR:) is a classic curve, the three-cusped hypocycloid or deltoid (see, for example, the discussion in Rouse Ball and Coxeter [RB-C74], p 99+, which recalls the role of the deltoid in the history of Kakeya's problem). It is easy to use (5) to see that V(8) traces out the path of a point on a circle of radius 1 rolling (without slipping) on the inside of a circle of radius 3. REMARK 4. We may regard the secant slope (j(b) - f(a))l (b - a) in the elementary MVT as the average value ofj' (x) over the interval (a, b). This interpretation of the MVT has a straightforward generalization to the multivariate case. For example, if K is a compact, convex, regular subset of IR:n and f : K � IR: is continuous and differentiable on Ko, then the average value of Vf is determined by the bound­ ary values off and lies in the convex hull of Vf(K0). This generalizes the elementary MVT in view of Darboux's ob­ servation thatj'((a, b)) is convex (viz. , an interval). In gen­ eral, the average value of the gradient over � may be com­ puted from the boundary values off by applying a simple instance of Stokes's theorem:

where the boundary is oriented appropriately. In the case of the disc (K = IT:D), one form of this calculation yields, for the average value over [Do:

...!_ 7T

ff

2 2 X +y

<1

Vf(x, y)

dx dy = (Ax, Ay),

where

Ax =

� f 1 (!cVI=Y2, y) - f(- Vl=Y2, y) ) dy,

and

Ay

=

� f1 (

!ex,

�) - f(x, - �) ) dx.

Thus it is essential in examples such as we see in Figure 1 that the range Vf(K0) not be convex. Note also that for the example of Figure 1, (Ax, Ay) = (0, 0), and we see that the distribution of points within each deltoid changes so as to maintain (0, 0) as the centroid. The following propositions highlight some of the points made so far. PROPOSITION 1 . For certain boundary functions g(8) (for example, for the case g(8) = sin 28), there exist continu­ ousf1J2Js on [D such thatfk(u(8)) = g(8), each fk is con­ tinuously differentiable on IT:D0, and

On the other hand each pair of these sets must have points in common.

44

THE MATHEMATICAL INTELLIGENCER

2. Iff : K � IR: is continuous on the compact, convex, regular domain K C IR:n, J is differentiable on the (nonempty) interior K0, and the graph off restricted to the boundary aK lies in a hyperplane H, then there is some v E K0 such that the tangent plane to the graph off a t (v, f(v)) is parallel to H. PROPOSITION

Of course, in the elementary MVT (n = 1) the hyperplane condition has no force: we can always take H to be the "se­ cant line." For n ::::: 2 the hyperplane condition is very re­ strictive, but it is not easy to go beyond it. PROPOSITION 3. Let f : K � IR: be continuous on the com­ pact, convex, regular K C IR:n, where the interior� is non­ empty with (positive) n-volume �K0 1 , and let f be continu­ ously differentiable on K0• Then

..... A

=

1

IK O!

f

Ko

V'f

is determined by the boundary values l aK and lies in the convex hull of the gradient range Vf(�). In fact, A is the centroid of the distribution of Vf(v) relative to the uni­ form distribution of v over �. Smooth Boundary Values In contrast to the phenomenon presented in Figure

1, we shall see that smooth boundary values that oscillate in a simple fashion do allow us to predict a gradient value. We first ex­ amine situations where 0 may be predicted as a gradient value. The justification for these predictions may be seen as topological. Let us begin with the following result, which shows that the examples depicted in Figure 1 are possible only because the gradient there has a singularity on the boundary. 4. Let the real-valued function f be continu­ ously differentiable on the closed unit disc rr:D, and let f be an extension of the boundary function g(8) = f(u(8)), where g has a finite number of extrema satisfying

PROPOSITION

G ::::: L + 4;

( 7)

here G denotes the number of global extrema for g(8) (over a single period, e.g. , for 0 ::s 8 < 27T) and L the number of merely local extrema. Then f has a critical point some­ where in rr:D, i.e. ,

0

E

Vf(rr:D).

REMARK 5. In our earlier examples we considered g(8) = sin 28 where G = 4 and L = 0; nevertheless, Proposition 4 did not apply because the extensions considered were not smooth at every boundary point. Suppose we modify the affine focus construction so that the three extensions whose gradient ranges are displayed with primary colours in Figure 1 are focussed not on boundary points but rather at points just outside rr:D. Proposition 4 predicts that the re­ sulting gradient ranges must expand to include 0. This ef­ fect is seen in Figure 2. REMARK 6. We may extend our argument to cover the case where extrema occur over intervals; in G and L each such interval is counted as a single point. Note that if g(8) is con­ stant we may tum to Proposition 2.

4

2

3

1 .5

2

0

0

·

-1

-2

-1.5

-3 -4 -4

\( i\� \, · · · ;)., ·....

-0.5

-1

-3

-2

-1

2

0

3

4

Figure 2. With the same boundary function sin 211, every SMOOTH extension to the disc yields a gradient range including

o'.

REMARK 7. It seems to us that Proposition 4 has a converse, i.e., that the condition (7) characterizes those boundary functions such that every smooth extension has a critical point. We shall present later some examples supporting this position. PROOF OF PROPOSITION 4. We may assume that M = max g((}) and m = min g(O) are the extrema off over all of []), since there would otherwise be interior extrema, where neces­ sarily V'fvanishes. The points Pk where extrema (global or merely local) of g occur may be indexed so that Pk = u(Ok) where k = 1, 2, . . . , G + L and

We may also assume that V'f does not vanish on a[]); oth­ erwise the conclusion of the proposition is already in hand. We consider the winding number w of the loop y : [81, 81 + 217] � �2 defined by y(O) = V'f(u(O)). Our aim is to show that w =I= 0. Let a (O) denote a continuous argument for y(O). Let a(O) = e + p(O), so that p(O) measures the an­ gle between Vf(u(O)) and the outward-pointing direction at u(O). Now

w = (217 +

G +L

L

k� 1

!1pk)l217,

where !1pk denotes the increment in p over the interval [Okl Ok + d , with the understanding that 8a+ L+ 1 = 81 + 217. Consider the case where Pk is a global or local maxi­ mum: V'f(u(e)) must point precisely outward or inward at Pk and Pk + 1, whereas, since g(O) is nonincreasing on [Ok, ek + l ] , V'f(u(e)) must point within 17/2 radians of the clockwise-directed tangent to a[]); thus !1pk may be com-

· · ·� '"

41

/

'''

0.5

- �2L---_-1� 1.� 5--�2 -� 0----� 1 ---� .5---0.5 ----�--� -0.5----�

Figure 3. Showing the cases (i)-(iv) occurring in the proof of Propo­ sition 4, as we keep track of the increments to the argument of the gradient on a[]).

puted as o; + 8k+ 1 , where ;s; measures the angle ± 17/2 between 'Vf(u(Ok)) and the clockwise tangent direction, and similarly 8k+ 1 measures the angle ± 17/2 between V'f(u(Ok + 1)) and the clockwise tangent direction. In the other case, where Pk is a global or local minimum, so that g(O) is nondecreasing on [Ok, ek + I L o ; and i5k"+ I are deter­ mined by measuring the angle ± 17/2 between Vf(u(Ok)) and V'f(u((h+ 1)) (respectively) and the counterclockwise tan­ gent direction. It is easy to check (see Fig. 3) the follow­ ing four cases: (i) if Pk is a (global or local) maximum and V'f(u(Ok)) points outward, then i5k" = - 1712 and o; = - 17/2; (ii) if Pk is a (global or local) maximum and V'f(u(Ok)) points inward, then i5k" = 1712 and o; = 1712; (iii) if Pk is a (global or local) minimum and V'f(u(Ok)) points outward, then i5k" = 1712 and o; = 1712; (iv) if Pk is a (global or local) minimum and V'f(u(Ok)) points inward, then i5k" = - 1712 and o; = - 1712. Now we may compute

w = (217 + = (217 +

G +L

I

k�l

G+L

I

k� !

!1pk)l217 = (217 +

G+L

I

k�!

c;s;

+ ok" + 1))/217

(ok" + o;))/217.

(8)

If Pk is a global maximum w e must have case (i) (i.e., V'f(uOk)) must point outward), while if Pk is a global mini­ mum we must have case (iv). Thus at least G of the terms (i5k" + ot) in (8) must contribute - 17. In any of the four cases the contribution of (i5k" + ot) is at most 17. Thus

w

s

(217 - G17 + L17)1217 = 1 +

1 ,(L - G) 2

s

1 +

1

2( -4)

= - 1,

VOLUME 25, NUMBER 1 , 2003

45

so that the winding number cannot be 0. By a standard ar­ gument

'ilf must vanish somewhere

inside the disc. QED

REMARK 8. To see what may occur when the condition (7) = L + 2 (the case G L + 3 cannot

is relaxed, suppose G

=

occur, since the alternation of maxima and minima implies that G

+L

is even).

We present an example in which G no critical point: letf(x, 1.205 is adjusted so

c cos 8)

(2 - sin

ima (see Figure

'ilf(x, y)

=

we must have

3(cx)2

8)

= 4, L 2, andf has y) = ((cx)3 - cx)(2 - y) where c that g(8) = f(u(8)) = ((c cos 8)3 =

has 2 global maxima and 2 global min­

4). Because

((3c(cx)2 - c)(2 - y) , - (cx)3 + ex), ex =

'ilf = 0; but then y = 2. Thus f has no

0, ± 1 wherever

* 1 so that we must also have

critical point on the disc

[]), as Figure 5 makes clear.

Having examined a substantial array of special cases along with some suggestive but not entirely rigorous general tech­ niques, we believe that the example of Remark

8 is part of

a general phenomenon. CoNJECTURE. The converse of Proposition 4 is also correct; i.e., given a smooth boundary function g(8) with G < L + 4, there exists a continuously differentiable function f : [D � IR such that ('V8) f(u(8)) = g(8) and yet

0 Et: 'ilj([])).

If this conjecture holds, it follows that we have a very mean

g

such that for all amplitudes

'P the function

satisfies G

g(8; a, 'P) = g(8) - a cos(8

-

a

and

'P)

0 Et: Vf([)).

'ilf is never 0. Given v · q has gradient q and bound­ ary values of the form a cos( 8 - 'P) for some choice of a, 'P · Thus the function!(v) + v · q has boundary values given by g(8) but its gradient is never q. For such a g we would have ues specified by q

E IR2,

g(·; a, 'P)

such that

the linear function

n

'ilf(rr::D) =

.rla[ll -.q

0,

where the intersection extends over continuously differ­ entiable! : [D

value theorem even for functions smooth on all of rr::D . We need only choose

Figure 5. A contour plot of one extension of the function of Figure 4 to a function f : [) ..... IR with

A Smooth but Very Mean Value Theorem

phases

max

=

�

IR such that (V8).f(u(8)) = g(8). Since each

of the gradient ranges is compact, some such finite inter­ section is also null.

A suitable function g(8) must have a good supply of g(8; a, ip).

local extrema that survive the modifications to

4

< L + 4. The conjecture would imply that there

is a smooth function ! : [D

�

IR extending the boundary val-

1.5.--.----....--.---,

-1 .5 L----'----�--�----'------;-----;-----:! 0 Figure 4. Here the boundary function g(8) has G

=

4 and L

=

2 so that

� L---L---L---�-� 7 6 5 4 3 2 0 Figure 6. The boundary function g(8)

=

cos 26 + cos 68 is graphed in

a converse to Proposition 4 would suggest that there exist exten­

red. In this example it is modified by subtracting 1 .5 cos(8 - 4) (green

sions to [) with no critical points; one such extension is displayed in

graph) to obtain the function graphed in blue. In each such case we

Figure 5.

have G < L + 4; here in fact G

46

THE MATHEMATICAL INTELLIGENCER

=

2 and L

=

1 0.

One example is g(e) = cos 2e + cos 60. Figure 6 shows the graphs of g, a typical modifier a cos(fJ - cp), and the mod­ ified function g(fJ; a, cp). It is not hard to show that every modification has a < L + 4. Given any subinterval J + of length 7T, let /- denote [0, 27T]V+ (all operations to be interpreted modulo 2·77} Be­ cause g(e) has period 7T, every value in J+ also occurs in 1-. Thus if J+ is the interval where a cos(e - cp) is positive, the modified function h( e) = g(e; a, cp) cannot have a global maximum in J+ . Every maximum of h in J+ is local. Be­ cause maxima and minima alternate, we must have L + :2: a+ 1, where L + counts the local extrema in J+ and a+ counts the global extrema there. Similarly, because every minimum of h in /- must be local, we have L - ::::: a 1. Adding these inequalities, we see that a :::::: L + 2 . Given the Conjecture, an example such as this provides one approach to the following smooth but very MVT.

Other Domains K, and Dimensions

n

>2

In this note we have concentrated attention mainly on the simplest multivariate domain []). Some results, however, ex­ tend to domains with other shapes (domains diffeomorphic to [D, for example) and to domains of higher dimension. Proposition 4 appears to be related to the Poincare-Hopf theorem linking the degree of the gradient map on a bound­ ary to the zeroes within (see, for example, Milnor [M69], section 6). It may be harder, though, in higher dimensions, to express our hypotheses in terms of boundary values of the function itself.

-

-

PROPOSITION 5. If the Conjecture holds, there exist contin­ uously differentiable functions fk : [D � IR (k = 1, 2, . . . , N), for some finite N, such that all have the same values on a [D, yet N

n k=l

Acknowledgments

This work was supported in part by a Hanyang University research grant ( 1999) and by a research grant (now called a Discovery Grant) from NSERC of Canada, held at the Uni­ versity of Guelph.

REFERENCES

[F-M95] M. F uri and M. Martelli, A multidimensional version of Rolle's theorem. Amer. Math. Monthly 1 02 (1 995), 243-249.

V'fk([D) =

,

[M69] J. Mi lnor Topology from the Differentiable Viewpoint. University

0.

Press of Virg i n i a, 1 969.

Can we have N = 3 here, as in Proposition 1? This question, along with the Conjecture itself, will require further study.

[RB-C74] W. W. Rouse Ball and H . S. M. Coxeter, Mathematical Recre­ ations and Essays , 1 2th edition, University of Toronto Press. 1 974.

AU T HOR S

SUNG SOO KIM

JOHN HOLBROOK

Hanyang University

Mathematics and Statistics

Ansan, Kyunggi 425-791

University of Guelph

Korea

Guelph, Ontario N 1 G 2W1

e-mail: [email protected]

Canada e-mail: [email protected]

Sung Soo Kim (B.Sc. Hanyang University, Ph.D. Korea Advanced

been Visiting Professor 2001-2. Normally Kim works at Hanyang

Institute of Science and Technology) and John Holbrook (B.Sc.,

University at Ansan. Holbrook has been mainly at Guelph for many

M.Sc. Queens, Ph.D. Caltech) pursued mathematics separately

years but has also worked in Califomia, Venezuela, India, Slove­

and on opposite sides of the world until The Mathematical lntelli­

nia, and Hawaii. While their individual research topics range over

gencer (vol. 22, no. 4) brought them together. Their electronic col­

matrix analysis, special functions, statistical sampling, and image

laboration on the article "Bertrand's paradox revisited" led to a

analysis, the Kim-Holbrook joint work has centered more on math­

real-life collaboration at the University of Guelph where Kim has

ematical curiosities, paradoxes, and philosophical puzzles.

VOLUME 25. NUMBER 1 , 2003

47

l$ffll•i§u@h1¥11@%§4fJi•ipl§ljd

Prime Maze Dean Hickerson

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

M i chael K l e b e r and Ravi Vaki l ,

number-theorist set his car's odometer to zero and then went for a drive along the roads shown in the diagram, starting and ending in his home town. He noticed that his odome­ ter reading was a prime number each time he entered a town. Where did he live and how far did he drive?

A

(Notes

to

avoid

trick solutions:

Every town is at an intersection of roads and evecy such intersection has a town. Distances on the map are in miles and are exact. The odometer measures miles and is accurate. The driver didn't reset his odometer at any time after he

E d i t o rs

started. He never turned around outside of a town, but he did sometimes leave a town by the road on which he arrived. He didn't add extra distance by driving within a town or raising his car off the ground and spinning his wheels or get­ ting out of the car while someone else drove it. He didn't subtract distance by driving backward. He stayed on the roads shown here at all times, etc.) Department of Mathematics University of California at Davis Davis, CA 956 1 6, USA e-mail: dean@math. ucdavis.edu

5

Please send all submissions to the

13

Mathematical Entertainments Editor, Ravi Vakil,

Stanford University,

Department of Mathematics, Bldg. 380, Stanford, CA 94305-21 25, USA e-mail: [email protected]

48

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

The solution will appear in the next issue.

CHRISTIAN C. FENSKE

Extrema i n Case of Severa Variab es a

favourite topic of most calculus courses is the calculation of extrema. Every calculus student is confronted with the following:

Standard calculus result. Let n 2: 2 a nd J c !R; an open interval. Let f : J � IR; be n - 1 times differentiable on J and n times differentiable at some po·int a E J. Assume that jCk)(a) = O for k = 1 , . . . , n - 1 but fCn)(a) * 0. Then there is the following alternative:

(1) Either n is even. Then f has an isolated extremum at a, and that is a maximum in casefCn)(a) < 0 and

a minimum in case f(n)(a) > 0. (2) Or n is odd. Thenf does not attain a local extremum

at a. When the course proceeds to functions of more than one variable we meet this theorem again-but now only for second derivatives. But what about a function with the first five derivatives vanishing? Of course, there arises the ques­ tion of what precisely we mean by "vanishing" of an n-th derivative, and what we should use as a substitute for the conditionjCn)(a) being positive or negative. This again de­ pends on how we define differentiability for functions of several variables. In this paper, I first explain how the theorem would look for a low-brow approach. Then I will discuss briefly the modifications required for the high-brow approach where higher derivatives are viewed as multilinear forms. Of course, at first glance one suspects that the multi­ variable case should be well known, and I am pretty sure it is. Although I have looked into numerous calculus texts and asked at least as many colleagues, I have not been able to identify a source. Either there is a proof of this result in the literature, but I did not find it, or the result seemed plau­ sible to everyone who thought of it, but writing it down was not worthwhile. So I just present here a proof for reference purposes and maybe for use in calculus courses.

The Low-Brow Approach

In the low-brow approach a function f : U � !R; on an open set U C �H;rn is said to be n times continuously differentiable if all partial derivatives up to order n exist and are continu­ ous on U. (I write Di for the partial derivative with respect to the i-th variable.) Schwarz's theorem on the interchange­ ability of partial derivatives then tells us that for a E U and h(l), . . . , h(n) E [R;m , the map dnj(a) : !R;m X · · · X [R;m � [R; with dnj(a) (kC 1 ), . . . , h(n)) = L Dj, · · · DjJ(a)h ) · · · h ) (summation over all distinct n-tuples (j 1 , . . . , Jn ) with 1 :s: j1 :s: m) is a symmetric n-linear map. If h E �R;m we write dn f(a)(hn) : = d'1(a)(h, . . . , h). We then have

J�

)�

THEOREM. Let U be open in !R;m and let f : U � IR; be n times continuously differentiable. Let 2 :s: p :s: n, and assume that for some a E U and all h E �R;m, we have df(a)(h) = d2j(a)(h2) = · · · = dP-�a)(hP- 1 ) 0, but dPf(a)(hP) * 0 for some h E IR;m. Then the following holds: =

(1) A necessary condition forf to have a local extremum

at a is that p be even. Let then p be even. (2) A necessary condition for f to mum (minimum) at a is that for all h E !Rm. (3) A sufficient condition for f to mum (minimum) at a is that for all h E [Rm\{0}.

have a local maxi­ dPf(a)(hP) :s: 0 (2:0) have a local maxi­ dPf(a)(hP) < 0 (>0)

Proof To begin with, choose your favourite norm 11·11 o n !Rm . (1) Let p be odd. We have to show that f does not have a local extremum at a. By assumption there is an h such that dPf(a)(hP) * 0. Upon dividing h by its norm we may assume that llh ll = 1 . Since p is odd we may © 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25. NUMBER 1 . 2003

49

even assume that JL : dPf(a)(hP) < 0 (else we would replace h by -h). Now we set E : = - p./p! and use Taylor's formula ([J, Corollary 8. 17]) to find a 8 > 0 such that =

I f(a + k) - f(a) - � dPf(a)(kP) I :S !

fi lklfl'

whenever k E !Rm satisfies l lkll < 8. Let 0 < t < 8. We then have f(a + th) - f(a) :S _!!: :_ t P + �tP = J.LtP < 0. So there cannot be a local minp! 2 2p ! tmum at a. But we may as well choose an h with llhl l = 1 such that JL' = dPf(a)(hP) > 0. Let then E : = JL'/p!. We choose 8 and t as above and find that

' ' tP E 0 < _l!:__ tP = - - tP + 1!:__ :Sf(a + th) - f(a). 2p! 2p ! 2 So w e don't have a local maximum at a either. (2) Now assume that we have a local minimum at a (else we replace f with -f). Then there is an 17 > 0 such that f(a + h) - f(a) 2: 0 whenever llhll :S 11· Let E > 0. Again we apply Taylor's theorem, and find a positive 8 :S 17 such that

- EIIhl f :S j(a + h) - f(a) - _!_ dPf(a)(hP) :S EllhlfP p! whenever llhll < 8 . S o for 0 < llhll < 8 w e have

0 :S f(a + h) - f(a) :S _!_ dPf(a)(hP) + Ellhl f, p! hence 0 :S dPf(a)(hP) + p!EIIh l f. Now fix h. Choose a t i= 0 with llthll < 8, which implies dPf(a)((th)P) + p!EIIthlfP 2: 0. Because p is now assumed to be even we have tP > 0, hence dPf(a)(hP) + p!EIIhlfP 2: 0. Be­ cause this holds for each E > 0 we conclude that dPf(a)(hP) 2: 0. (3) We deal with the case where dPf(a)(hP) > 0 whenever h i= 0. Denote by S the unit sphere in !Rm. Since S is compact, the continuous map 4> : S � IR defined by cf>(h) : = dPf(a)(hP) attains its minimum. So there is a A > 0 with dPf(a)(hP) 2: A whenever llhll = 1, so dPf(a)(hP) 2: Allhl f for all h E !Rm. Let 0 < E < Alp!. In­ voking Taylor's theorem for a last time, we choose a dPf(a)(hP) I :S 8 > 0 such that if(a + h) - f(a) - _!_ pi EllhlfP whenever llhll :S 8. For these h we then have - � lhlf :Sf(a + h) - j(a) - _l_ CfP.f(a)(hP) :S �lhl f, hence p!

ll l

0 < - EIIhl f + � h f :Sf(a + h) - f(a). p! But this means that f(a + h) > f(a) as long a s 0 < llhl l < 8. 0 The High-Brow Approach

Of course, it is aesthetically unsatisfactory to have a condi­ tion such as dP.f(a)(hP) i= 0 where one would expect just dPf(a) i= 0. But this can be taken care of if we view higher derivatives as multilinear maps. So let E, F be finite-dimen­ sional vector spaces (over the reals), and choose norms in E, F. If D C E is open, a map f : D � F is said to be differ50

THE MATHEMATICAL INTELLIGENCER

entiable at a E D if there is a linear map T E L(E, F) and there is a function r : D � F continuous at a, such that r(a) = 0 and f(x)

=

f(a) + T(x - a) + llx - al l r(x)

for all X E D. One proves immediately that T in the above definition is uniquely determined and consequently writes df(a) : = T. One says thatfis continuously differentiable on D iff is dif­ ferentiable at each x E D and df : D � L(E, F) is continu­ ous. If E1 , . . . , Ek are vector spaces, the space of k-multi­ linear maps of E1, . . . , Ek to F is denoted by L(E1 , . . . , Ek ; F) . We endow this space with the obvious norm. If E1 · · · = Ek E this space is denoted Lk(E; F). If h E E we write T(hk) : = T(h, . . . , h). By L�(E; F) one denotes the subspace of symmetric mappings. We say that T E L1(E; IR) is positive (negative) semidefinite if T(hk) 2: 0 (resp. :S 0) for all h E E. Similarly, T is said to be positive (negative) definite if T(hk) > 0 (resp. T(hk) < 0)) whenever h i= 0. Returning to differentiation, we know from (multi-)lin­ ear algebra that there are natural isomorphisms =

=

L(E1 , L(E2 , . . . , Ek ; F)) = L (E1 , . . . , Ek-1 ; L(Ek, F) ) = L(Et, . . . , Ek ; F). If f is differentiable on D and df : D ---? L(E, F) is again differentiable at a E D, then d2f(a) : = d(df)(a) E L(E; L(E, F) ) = L2(E; F). Inductively, we define dkf(a) : = d(dk- 1 f)(a) E Lk(E; F) if dk- lj : D � Lk - 1 (E; F) is differ­ entiable at a. The Schwarz lemma then tells us that in fact dkf(a) E L}(E; F) providedf is k times continuously differ­ entiable. Our theorem then looks almost the same: THEOREM. Let U be an open subset of a finite-dimensional vector space E, and letf : U � IR be n times continuously differentiable. Let 2 :S p :S n, and assume that for some a E U we have dkf(a) = Ofor 1 :S k :S p - 1 but dPf(a) i= 0. Then the following holds:

(1) A necessary condition forfto have a local extremum

at a is that p be even. Let then p be even. (2) A necessary condition for f to have a mum (minimum) at a is that dPf(a) (positive) semidefinite. (3) A sufficient condition for f to have a mum (minimum) at a is that dPf(a) (positive) definite.

local maxi­ be negative local maxi­ be negative

The proof carries over almost verbatim from the low­ brow case. There is but one crucial point: We know that dPf(a) i= 0, so we know that there are h 1 , . . . , hp E E with dP.f(a)(h1 , . . . , hp) i= 0; but we need to know that this hap­ pens with h1 , . . . , hp all equal. It is here that we exploit the fact that dP.f(a) is symmetric. If p = 2 we could use the par­ allelogram identity to conclude that d::f(a) = 0 iff d 2f(a)(h2) = 0 for all h E E. For the general case we could appeal to the polarization identity [AMR, Proposition 2.2. 1 1 ] : PROPOSITION.

and k E

Let E, F be finite-dimensional vector spaces : E ---? F by A(h) : =

N. For A E Lk(E; F) define A

A(hk) and denote by Sk(E; F) the vector space {A A E Lk (E; F)} endowed with the norm I lA I : = sup{ //A (hk)/ 1 l lhll :S 1 } . Then I IAII :S !!A ll :S t i!Aii for A E Lk(E; F), and when t·e­ stricted to L.�(E; F) is an isomorphism. A

If you want to prove the theorem on extrema in a calculus course you might be reluctant to bother your students with too much multilinear algebra, so here is a simple and direct proof which I learned from my colleague Thomas Meixner [M]: LEMMA. Let E, F be vector spaces, n E N, and T E L�(E; F). If T(h") = 0 for all h E E, then T = 0. PROOF. We proceed by induction. If n = 1 there is nothing to prove. So assume the claim for n - 1 and let T E L�E; F) C L(E, L� - 1 (E; F)). Suppose T(h") = 0 for all h but T =!= Then there must be an a E E with 0 =!= T(a) E L�- 1 (E; F). Since we assume the claim for n - 1 there must be a b E E with T(a, b, . . . , b) =!= By our assumption, we must have that T((Aa + b)") = 0. Expanding by multilinearity and using the symmetry of T to collect common terms, we fmd that

0.

0.

0 = T((Aa + b)n)

for all A E 1R

II

= I mjT(a, . . . , a, b, . . . , b )}j with mi E j�o

=

n-1

I

j� 1

� n�

N\{0}

sis, and Applications, 2nd edition, Springer, 1 988, Appl. Math. Sci. 75.

T(b") = T(a") = 0.

[J] Jurgen Jost, Postmodern Analysis, Universitext, Springer, Berlin et

Now we put A = 1, . . . , n - 1 and we obtain the following equation:

( .�

n -

REFERENCES

[AMR] R. Abraham, J . E . Marsden, T. Ratiu, Manifolds, Tensor Analy­

mjT(a, . . . , a, b . . . , b)Ai,

because

(*)

Of course, we now might wish to remove the assump­ tion that our vector spaces be finite-dimensional. So now assume that E is a Banach space. Then we may copy the above arguments with almost no changes: We simply change our notation: we now denote by L(E, F) and Lk(E; F) the space of all continuous (k-)linear maps (in the finite-dimensional case (multi-)linear maps are automati­ cally continuous, so this is even consistent with our previ­ ous terminology). Moreover, I have deliberately chosen ref­ erences ([J] and [AMR]) that actually deal with the infinite-dimensional case, and Meixner's lemma does work in any vector space. There is but one point where we re­ ally needed a finite-dimensional vector space: in part 3) of the proof we used the fact that the unit sphere is compact. The best way out of this is to require just what we need: Let us call T E Lk(E; IR) strongly positive (negative) defi­ nite if there is a A > 0 such that T(hk) ::=: A//hllk (resp., T(hk) :S - A//hl/k) for all h E E. The high-brow theorem then continues to hold if we replace "finite-dimensional vector space" by "Banach space," provided we insert "strongly" before "negative (positive) definite" in 3).

1

X

(

m 1 T(a, b, b, . . . , b, b) m2 T(a, a, b, . . . , b, b) mn - 1 T(a, a,

�

•

.

.

.

, a, b)

) ()

2

AU T H O R

0

=

0

� .

We denote the matrix on the left-hand side by M, _ 1 and claim that det M,_ 1 =!= 0: Starting from the last column, mul­ tiply the (j - 1 )-st column by n - 1 and subtract the result from the j-th column. Thus the first column remains unal­ tered. This gives us 1

al. , 1 998. [M] Thomas Meixner, Personal communication.

CHRISTIAN C. FENSKE

2- n

2"- 2 (3 - n)

Mathematisches lnstitut Justus-Uebig-Universitat Giessen

det

n - 2 (n - 2)( - 1) 0 n- 1 Expanding the determinant according to the last row and collecting common factors we obtain

( - 1)"(n - 1)( - 1)"- 2 (n - 2)! det Mn - 2 = (n - 1)! det Mn - 2 · Because det M2 =!= 0, this shows what we needed. But then (*) has only the trivial solution. In particular, m 1 T(a, b, . . . , b) = which shows that T(a, b, . . . , b) = 0. Contra­ diction. D

0,

D-353g2 Giessen Germany e-mail: [email protected]

Christian Fenske, born in Germany in 1 g39, studied mathe­ matics and physics in TObingen and Bonn. Since 1 972 he has been at the Mathematical Institute in Giessen. He works on global analysis: topological fixed-point theory, and periodic or­ bits. Married with two children, he enjoys foreign languages and foreign travel.

VOLUME 25, NUMBER 1 . 2003

51

M a t h e ��n a t i c a l

E n t e rt a i n ��n e n t s

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegan� suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

M i c h ae l K l e b e r a n d Ravi Vaki l ,

Capitalism Overturned

29. 30. 32. 33.

Michael Kleber

1. 2. 3. 4. 5. 6. 8. 9. 13. 14. 15. 16.

Across BIKE

X DOCK (MODULO Sg)

REVISES YES TO BOURBAKI ___

WITHOUT LOSS OF GENERALITY

A GREEK

376/5 1 1 (BASE 2) CONFERENCE (SUFFIX)

PH.D. DIRTY ANGLE TRISECTOR, E.G. TOLD YOU SO!

2

IMPASSIVE ACCOUNT RECORD

ONE OF A KINE? ANTIPODAL TO SPANIARD

PC NOT AD REFERENCE WORK HAREM CONCUBINE SPHERE CONTACT AIN'T NOT OR

10, E.G., BUT NOT 1 1

PRONOUNCED ___

, AAH

! OF U - 1 ___

ARE ROUND

(CORNBREAD ARE SQUARE!)

SVP (EN ANGLAIS)

1

DARKENED

2

Down

Word boundaries are indicated by heavy lines. Answers extending past the right edge of the grid re-enter on the left, which may be somewhat dis­ orienting. A solution will appear in the next issue.

7. 10. 11. 12. 16. 1 7. 19. 21. 22. 23. 25. 27.

Ed i t o rs

3

4

5

18. 19. 20. 24. 25. 26. 28. 30. 31.

RULERS ALL FOR ONE, AND . . . THAT ONE

EUK TIP

MATH LANGUAGE REQUIREMENT PRE-JAN FARE STORAGE MEDIUM

6

7

8

11

10

13

12 15

14

17

16

Please send all submissions to the Ravi Vakil,

30

28

27

Stanford, CA 94305-2 1 25, USA e-mail: [email protected]

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

29 32

31

Stanford University,

Department of Mathematics, Bldg. 380,

52

26

25

23

22

21

Mathematical Entertainments Editor,

18

20

19

24

9

33

REUBEN HERSH

Th e B i rth of Ran d om Evo utions

t

he theory of random evolutions was born in Albuquerque in the late 1 960s, flourished and matured in the 1 9 70s, sprouted a robust daughter in Kiev in the 1 980s, and is to­ day a tool or method, applicable in a variety of "real-world " ventures.

"Random evolutions" are stochastic linear dynamical systems. "Random" means, not just random inputs or ini­ tial conditions, but random media-a random process in the equation of state. A year or two after I came to the University of New Mex­ ico, we hired a young probabilist, Richard Griego. Griego took his Ph.D. at Urbana and was well versed in stochas­ tic processes a la Doob. He showed me something I, a p.d.e. specialist, had never heard of at N.Y.U. or Stanford­ Brownian motion can solve Laplace's equation or the heat equation! We wrote this up in a popular article [ 132] for the

Scientific American.

Probabilistic methods worked for p.d.e.s of the para­ bolic or elliptic types. But I was a student of Peter Lax. I had learned to concentrate on hyperbolic equations, like the wave equation. "Can you do it for the hyperbolic case?" I asked. So far as Richard had heard, it seemed you couldn't. There was a vague impression that Doob had even proved it couldn't be done.

We organized a little seminar, with Bert Koopmans, a statistician, and Nathaniel Friedman, a young ergodicist. (Nat is now a sculptor, and organizer of mathematical art, in Albany, NY.) We also had the pleasure of interest on the part of Einar Hille, who had come to New Mexico after re­ tiring from Yale. By good luck, Richard discovered in Nat's bookcase a copy of Mark Kac's Magnolia Petroleum Company Lec­ tures in Pure and Applied Science [67]. These Lectures were not in the University of New Mexico's library, nor in many other libraries. Most of their contents had been pub­ lished elsewhere, but one section had never appeared in a journal. Kac considered a particle moving on a line at speed c, taking discrete steps of equal size, and undergoing "col­ lisions" (reversals of direction) at random times, according to a Poisson process of intensity a. He showed that the ex­ pected position of the particle satisfies either of two dif­ ference equations, according to its initial direction. With correct scaling followed by a passage to the limit, the dif­ ference equations become a pair of first-order p.d.e.s. Dif-

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 1 , 2003

53

ferentiating these and adding them yields the "telegraph equation":

( )

( )

!!:___ du + a du c _E_ du c = dt dt dt dx dx · �

(1)

This is an equation of hyperbolic type. (If you drop the lower-order term, it's just the one-dimensional wave equa­ tion.) A probabilistic solution of a hyperbolic equation! I was fascinated by Kac's little-known feat. Surely the telegraph equation couldn't be the only hyperbolic equa­ tion with a probabilistic meaning! Equally intriguing: if we can construct a probabilistic solution, we can exploit it. The central limit theorem was beckoning. If we could prove a central limit theorem for Kac's model, we'd have a limit theorem about p.d.e.s. As I contemplated the steps by which Kac constructed his probabilistic solution of the telegraph equation, I real­ ized that one could replace the group of translations at speed c by any group of operators! Suppose A is the generator of a group of linear operators, acting on a linear space 'lR>. In­ stead of translations moving randomly to the left and right at speed c, substitute time evo­ lutions according to generators A and -A. In place of a parti­ cle whose position at time t is the sum of random transla­ tions, we get a random element of 'lR>, the result of successive evolutions "forward" (generator A) and "backward" (gen­ erator -A). What is the expected value of this 'lR>-valued random process? Direct differentiation showed that in place of the classical telegraph equation, this expectation satisfies an operator differential equation. This can be ob­ tained by simply substituting, in the classical telegraph equation, the abstract generator A for c dldx, the genera­ tor of translation at speed c:

has a time coefficient tk equal to the occupation time in the kth state assumed by the n-state Markov chain. At first we called this random product of solution operators a "random semigroup." Peter Lax pointed out that it was not really a semigroup; he suggested the name "random evolution." If the generators Ak don't commute, Richard found that this random product should be written "backwards," last operator first; if we define the random evolution this way, u(t), the expected value of the random evolution, satisfies a simple ordinary differential equation:

du - = Z u + Q u. dt u(t) is an n-tuple of elements from the linear space 'lR>, in­ dexed according to each of n possible initial modes of evo­ lution. Z is a diagonal matrix of the operators AJ· Q is a real matrix, the transition matrix of the Markov chain which controls the switching among the Aj. In the classical case, the n semigroups are translations in IR3. Their generators, the di­ agonal elements of Z, are first­ order differential operators in the spatial variables. The equa­ tions are a hyperbolic system of first-order differential equa­ tions, coupled through Qu. So we called the general case, with abstract generators AJ, an "abstract hyperbolic system." In the special case of only two semigroups, each the negative of the other, with a symmetric mechanism of reversing direction, the abstract system of two equations is equivalent to the abstract telegraph equation written above. For this special case we proved a limit theorem. In prob­ ability language it's called a "central limit theorem," and in differential equations language it's a "singular perturbation theorem." When a lower-order term in a differential equa­ tion has a small coefficient, it's called a regular perturba­ tion. When a leading-order term has a small coefficient, it's called a singular perturbation. In the telegraph equation, the two leading terms have second order in x and in t. We can't throw away Uxx, for we'd be left with an ill-posed problem. But if we throw away uu, we get the heat equation: a well-behaved equation which has a probabilistic solution-Brownian motion. If we can make our random linear process approach Brownian motion, surely its expected value-which satisfies the tele­ graph equation-will go to Brownian motion's expected value-which satisfies the heat equation. We'll have a prob­ abilistic proof of a singular perturbation theorem for the heat and telegraph equations. How to put a small coefficient in front of the second time-derivative, to make our process look like Brownian motion? It's not hard to guess that we must speed up both the linear motion (translation) and the switching, to make the collisions frequent. The expected time between calli-

I n pro babil ity lang uag e

it ' s cal led a " central l i m it theo re m , " and in d iffer­ ential eq uations lan ­

g uage it ' s a "singu lar

pertu rbation theorem . "

( )

d du du - - + a - = A2u dt dt dt We called this the "abstract telegraph equation." For two speeds c and -c, switching at random times ac­ cording to a Poisson process is equivalent to defining the process as a two-state Markov chain. But clearly one could let the particle move at any n speeds, with the switch be­ tween speeds governed by an n-state Markov chain. In the abstract case, instead of switching among random speeds, you could switch among n semigroup generators Aj oper­ ating on a given linear space 'lR>. The result would be a ran­ dom process on 'lR>, which would in fact be a random prod­ uct of random solution operators, each generated by its randomly chosen generator. Each solution operator

54

THE MATHEMATICAL INTELLIGENCER

sions is the reciprocal of a, so to make the time between collisions small, multiply a by a large number R. To speed up the linear motion, multiply the speed c by another large number S. Then if you divide the equation by S 2 and choose R

=

(cS? , a

you get the telegraph equation (classical or abstract!), with u11 divided by S2. The mean free path is the speed Sc times the expected time between collisions, liRa. This reduces to liSe, going to zero as R and S go to infinity. Easy to guess now that as R and S go to infinity, the expected value u(t) goes to a solution of the heat equation. Since the funda­ mental solution of the heat equation is the distribution func­ tion of the normal random variable, it's no longer a surprise that this singular perturbation theorem for a p.d.e. can be proved by means of the central limit theorem of probabil­ ity. What happens in the abstract telegraph equation, where c d/dx is replaced by A? Then, of course, the solution con­ verges to the solution of an "abstract heat equation,"

Prof. Hille published an announcement of our results in the Proceedings qf the National Academy of Sciences [45]. Then we submitted the complete paper to the Journal of Functional Analysis. We were surprised and disappointed when it was rejected. However, the letter of rejection was twelve pages long, and contained sensible suggestions for improvement. The referee's references to "the book" left no doubt that he was William Feller. We took advantage of his advice, and the improved paper appeared in the A.M.S. Transactions [46]. I still regret that I never met Feller in person. By another piece of luck I stumbled across the M.I.T. thesis of Mark Pinsky. Pinsky had extended Kac's work on the telegraph equation. Instead of Kac's two velocities c and -c, Pinsky allowed n arbitrary velocities, switching ac­ cording to an ergodic n-state Markov chain. More impor­ tant, Pinsky proved a central limit theorem for this n-state process. The identity of the limiting equation in this gen­ erality is much less obvious than in the telegraph equation that Griego and I had treated. Mark spent two weeks in Albuquerque the following summer. My daughter baby-sat so that he and Joanna could sight-see. Mark and I undertook to extend his central limit theorem to abstract semigroups-or, what is the same thing, to extend the central limit theorem Griego and I had proved for the abstract telegraph equation to an arbitrary abstract hyperbolic system. We assumed that the generators were mutually com­ mutative. This was an undesirable restriction, but it was a significant step forward. It included the Kac and Griego­ Hersh papers, with switching between cA and -cA, and Pinsky's thesis, with switching between first-order spatial differential operators with constant coefficients. We showed that in the limit of an appropriately scaled

small parameter, all components of the solutions of the commutative abstract hyperbolic system converge to solu­ tions of an abstract heat equation:

du dt

=

Hu.

H is a certain quadratic expression in the n generators of the constituent semigroups. In the classical case (Pinsky's thesis), these generators are constant-coefficient first-or­ der spatial differential operators, and H is a second-order elliptic operator in IR3 with constant coefficients. Pinsky found an abstract characterization of random evolutions in terms of multiplicative functionals, which he presented in a book [ 104, 107]. After the first Griego-Hersh paper was published, and again after my paper with Pinsky was published, I received letters from Tom Kurtz. Kurtz had his own abstract-space version of singular perturbation theory. He used it to re­ prove both the abstract telegraph equation limit theorem (Griego-Hersh) and the commutative abstract hyperbolic systems limit theorem (Hersh-Pinsky). (He did the same thing to the non-commutative limit theorem I obtained later with George Papanicolaou.) It seemed that he was deter­ mined to find non-probabilistic proofs for all our proba­ bilistic theorems [86]. But as his career developed he be­ came committed to probabilistic problems and methods. He obtained first-order limit theorems [85] (laws of large numbers) for random evolutions, in addition to second-or­ der theorems (central limit theorems) of the type on which Griego and I had focused. Particularly interesting were his limit theorems for sequences of semigroups of nonlinear operators. The following year I had a sabbatical at my alma mater, the Courant Institute. I attended a probability seminar led by Monroe Donsker. There George Papanicolaou was re­ porting on recent work [81 , 82] of Khas'minskii. George sug­ gested we work together. Before I returned to New Mex­ ico we succeeded in proving limit theorems for random evolutions made of non-commuting semigroups. These proofs were more probabilistic than my earlier ones. Now we assumed that the process by which the semigroups switch has a state whose recurrence time has finite mean and variance. (This is automatically satisfied in case of an ergodic finite-state Markov chain.) Under this hypothesis, the evolution is a random product of random factors, each of which starts and ends at the distinguished ergodic state. These random factors are independent and identically dis­ tributed. We were able to calculate their common expected value, estimate its dependence on the small parameter, and find what happens when the parameter vanishes. At one point in our work, we were stuck A few words from S.R.S. Varadhan were sufficient to point toward the solution. As in the commutative case, all components of the "abstract hyperbolic system" which is satisfied by the expected val­ ues of the random process converge to solutions of a sin­ gle "abstract heat equation," with a single "abstract second­ order elliptic" operator on the right-hand side. This operator is again given by a quadratic expression in the gen-

VOLUME 25. NUMBER 1 . 2003

55

erators of the constituent semigroups. But these do not commute, so the quadratic expression is defined by non­ commutative algebra. It is given explicitly in our paper. George went on to write several more papers applying ran­ dom evolutions to physical problems. A specific, concrete example of non-commuting group generators is a collection of first-order differential opera­ tors in �3 with position-dependent coefficients. Each of these generates its group of translations along its family of curved trajectories in �3. The limiting equation is again sec­ ond-order parabolic, but now with variable (position-de­ pendent) coefficients. In translation semigroups, whether with constant or variable coefficients, the small parameter has a very sim­ ple physical interpretation. Just as in the case of the tele­ graph equation discussed above, it is proportional to the mean free path-the average distance traveled between collisions. (A simple de­ rivation of this fact is given in Hersh-Pinsky [51] .) Vanishing of the small parameter-the mean free path-means a drastic transformation of the physical process, from discrete particles in collision to smooth motion in a continuum. Our convergence theorem says that in the limit of small mean free path, the expected motion of a Newtonian par­ ticle goes over to diffusion. It may seem surprising that the limit depends only on shrinking the mean free path, not on increasing the density of particles. But physically the vanishing of the mean free path comes about by increasing the particle density. In­ creasing the density of particles results in a passage to the diffusion limit by making the mean free path go to zero. While in New York I spoke at Mark Kac's seminar at Rockefeller University. This was the first time I had met Kac. Fortunately, I was able to see him several more times before he died. During the seminar he listened closely, and challenged me twice. Is the square root of the Laplacian operator really the generator of a group? Was I right to call my formula for the limiting operator "explicit"? Both times I explained, and he yielded. In the telegraph equation (1) the term a du!dt is what makes energy dissipate, so I called this term the "parabolic part" of the equation. This term carries the stochastic part of Kac's solution of the telegraph equation-switching ac­ cording to a Poisson process with intensity a. If a is 0, there is no switching, the "parabolic term" drops out, and the equation reduces to the one-dimensional wave equation.

apparently unaware of the intimate connections between his measure and his own work on potential theory. It is now almost universally known that the generalized Wiener­ Perron capacitory potential of a closed set F (in Euclidean space of dimension 3 or higher) at a point p is the Wiener measure of the set of paths which originate from p and which at some time hit F. Moreover, Wiener's famous cri­ terion for regularity of boundary points has a most ap­ pealing interpretation in terms of his measure. The surprise is heightened if one recalls that Wiener's work on poten­ tial theory was almost simultaneous with that on Brown­ ian motion." Griego reminded me of this quotation because of the in­ structive fact that what Kac said of Wiener with regard to potential theory, he could have said of himself with regard to random evolutions. Kac was a great advocate of func­ tion space integrals as a tool for solving p.d.e.s-the Feyn­ man-Kac formula [68] . And Kac found a proba­ bilistic solution of the telegraph equation [67]. But he didn't connect his solution of the tele­ graph equation with his function-space inte­ grals. Had he done so, his use of an approxi­ mation by a finite difference equation would have become unnecessary, and he could have done the whole random evolution thing by him­ self-if he had cared for abstract spaces and operators! When I returned to New Mexico after my sabbatical leave, I was fortunate again. My colleague Bob Cogburn (one of Michel Loeve's three Ph.D. students) became in­ terested in random evolutions. With his mastery of proba­ bilistic estimation, it was possible to carry all earlier asymptotic results on random evolutions to their natural generalization [ 16]. The random process need not be finite or even discrete. The family of semigroups need not be fi­ nite or countable. The random process controlling the choice of semigroups need not be Markovian; it's sufficient that it satisfy a "mixing condition" (be almost independent when separated by long time intervals.) Shortly afterward, a comparable result was published by Papanicolaou and Varadhan [ 102]. Under some extra hypotheses they also got a rate of convergence. Also around the same time Richard Ellis and Walter Rosenkrantz published a paper in which Kac's particle is restricted to a bounded interval, with ap­ propriate boundary conditions [30, 3 1 ] . They obtained a central limit theorem for this case. In 1972 I was asked by the Rocky Mountain Mathemat­ ics Consortium to organize a summer meeting on stochas­ tic differential equations. This was part of a series of meet­ ings which had been supported by a grant of $25,000 a year from the National Science Foundation. The year I became responsible, the N.S.F. changed its policy, and gave no money at all. We had planned to meet in Santa Fe, NM. For­ tunately, the Consortium included a Canadian school-the

O u r convergence theorem

says that in the l i m it of smal l

m ean free path , the expected

m otion of a Newton ian particle g oes over to d iffusion .

The presence of the "parabolic term", a du!dt, is what makes possible a probabilistic solution of this hyperbolic equation. Kac leaned forward and exclaimed, "That's right!" A treasured and memorable moment for me. Later Richard Griego reminded me that in [68] Kac wrote of Norbert Wiener: "What is really surprising is that he was

56

THE MATHEMATICAL INTELLIGENCER

University of Alberta, in Edmonton. Jack Macki, Alberta's representative to the Consortium, convinced his university to contribute $10,000. There was also a small additional do­ nation from the Provincial Government of Alberta. Because of this Canadian support, we had our meeting in Edmon­ ton. Richard Griego took a lot of responsibility on this pro­ ject. The participants seemed glad to pay their own way to Edmonton. Mark Kac and Wendell Fleming were invited speakers. The proceedings became a special issue of the

Rocky Mountain Mathematics Journal. Two papers by Griego, one [47] with me and one [ 50] with Andrzej Korzeniowski, applied random evolutions to a seemingly remote area of pure mathematics: the spectral theory of elliptic operators. We were able to compute the asymptotics of the spectrum of a certain class of degener­ ate elliptic operators (merely non-negative definite, not strictly positive-definite) by solving an associated equation as a random evolution. Our students carried forward with random evolutions. We had limited ourselves to continuous evolutions, with jumps only in the derivatives (the "generator"). Mark Pin­ sky's student Bob Kertz obtained a central limit theorem for evolutions with discontinuities. Richard Griego's student Manuel Keepler was prolific. He wrote about duality and time reversals. David Heath, while still a grad student at Urbana, moved in quite an unexpected direction. He found a probabilistic model for the telegraph equation with space-dependent co­ efficients. Unfortunately, his thesis remains unpublished. Tom Kurtz's student Joe Watkins published impressive papers carrying the limit theorems further than before. Pinsky, Cogburn, and Papanicolaou went on to create different kinds of stochastic mathematics. Pinsky brought in differential geometry, and studied stochastic processes on manifolds (see his book [ 109]). Cogburn and his students did detailed analysis of dif­ ferent kinds of stochastic processes in different kinds of random environments. ("Random environment" means something rather similar to "random evolution.") George Papanicolaou published very extensively in ran­ dom media, and on "homogenization." In homogenization, the random variation takes place in space rather than in time. An example could be sound passing through a ran­ dom alternation of layers having different acoustic prop­ erties. If there are a great many very thin layers (in an ap­ propriate scaling, of course), the process is approximated by a "homogenized" medium. Other interesting contributions came from Koopmans's student Donald Quiring, Griego's student John Hagood, and applied mathematicians Steven R. Dunbar, G. A. Becus, and 0. Iordache. I felt that the joint paper with Cogburn had completed my romance with random evolutions, getting full general­ ity in both the operators and the random switching. (Un­ like some papers by later authors, the work reported here includes unbounded semigroups as well as bounded ones.) Unknown to me and my American collaborators, our work was noticed in Kiev. Vladimir Korolyuk, a distinguished

probabilist and academician, and his pupil Anatolyi Swishchuk, undertook a massive research into random evo­ lutions controlled by semi-Markov processes. In this gen­ erality they answered all the questions we had asked, and many others we had neglected. Semi-Markov processes, like Markov processes, "have no memory." The mode of evolution you jump into after a "collision" depends only on what mode you are in at the time of the collision. The difference is that, for Markov processes, the waiting time between collisions is expo­ nentially distributed. In the semi-Markov theory, this re­ striction is relaxed. The waiting time can be prescribed ac­ cording to the particular application in view. Swishchuk's two recent books [ 125, 126] give scores of references in physics, biology, and especially in modern finance. (Swishchuk is now at York University, in Toronto.) I only learned about this work in 1992, years after it had been going on. "Out of the blue" I received an invitation to the third ( ! !) conference on random evolutions, to be held on the shore of the Black Sea, in Katsively, where the Ukrainian Academy of Science has a beautiful resort. I took two weeks off from teaching to go to the Ukraine. First to Kiev, which to my surprise is a beautiful city on the shore of the magnificent Dnieper River; then to the stimu­ lating conference in beautiful Katsively (I had the difficult role of "important guest"); then back to Kiev. From there, Swishchuk took me south on a long bus and car trip, to the forest village of Butznivets, which my mother Malke had left in 1919. The big lake is there, just as my mother described it. We found two or three old ladies herding a few cows down the forest road. Swishchuk translated for me. "Do you remember Malke Shluger? Her father, Sholom Shluger? The store her grandmother kept? The Weinbergs? The Tobacks?" "There are no Jews here. The Germans killed them. A few ran away." More Ukrainian conversation between the ladies and Tolya Swischuk. "Those houses used to be Jew houses." I entered one of the houses. It had been burned out, and then used as a barn. I took pictures of the abandoned rooms. We drove away. REFERENCES

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VOLUME 25. NUMBER 1 . 2003

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1 01 . G. C. Papanicolaou, D. W. Stroock, and S. R. S. Varadhan, Mar­

1 22. A Swishchuk, Random evolutions: a survey of results and prob­

tingale approach to some limit theorems , Conf. on Statistical Me­

lems since 1 969, Random Op. & Stach. Eq. , VSP. no. 3 (1 991 ),

chanics, Dynamical Systems & Turbulence, M. Reed editor. Duke

(to appear).

U. Math. Series, 3, Durham, NC, 1 97 7 . 1 02 . G . C . Papanicolaou and S. R . S. Varadhan, A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations, Comm. Pure & Appl. Math. 26 (1 973), 497. 1 03. A A Pichardo-Maya, Brownian motion in a random environment, Dissertation, U. of New Mexico, 1 985. 1 04 . M. A Pinsky, Random evolutions i n Prob. Meth. in Diff. Eq . , Lec­ ture Notes in Math. 451 , Springer Verlag , New York, 1 975.

1 23. A Sw1shchuk, Semi-Markov random evolutions, Kluwer AP, Dor­ drecht (with V. S. Korolyuk). 1 24. V. S. Korolyuk and A Swishchuk, Evolution of Systems in Ran­ dom Media . CRC Press, Boca Raton , 1 995.

1 25 . A Swishchuk, Random Evolutions and their applications. Kluwer AP, Dordrecht, 1 997. 1 26. A Swishchuk, Random Evolutions and their applications. New Trends. Kluwer AP, Dordrecht, 2000.

1 05. M . Pinsky, Differential equations with a small parameter and the

1 27 . A Swishchuk and Jianhong Wu , Evolution of Biological Systems

central limit theorem for functions defined on a finite Markov chain,

in Random Media. Limit Theorems and Stability. Kluwer AP, Dor­

Z. Wahrsch. Gebiete 9 (1 968), 1 01 -1 1 1 .

1 06. M . Pinsky, Multiplicative operator functionals of a Markov process, Bull. A . M. S. 77 (1 97 1 ) , 377-380.

1 07 . M. Pinsky, Stochastic integral representation of multiplicative op­ erator functionals of a Wiener process, Trans. A. M.S. 1 67 (1 972), 89-1 04. 1 08. M . Pinsky, Multiplicative operator functionals and their asymptotic properties, Adv. in Prob. 3 (1 974), 1 -1 00. 1 09 . M. Pinsky, Lectures on random evolutions, World Scientific, Singapore, 1 991 . 1 1 0. D.

Quiring, Random evolutions on diffusion processes, Z.

drecht (submitted). 1 28. J. C. Watkins, A central limit problem in random evolutions, Ann. Prob. 1 2, no. 2 (1 984), 480-5 1 3 .

1 29 . J . C. Watkins, A stochastic integral representation for random evo­ lutions, Ann. Prob. 1 3 , no. 2 (1 985), 531 -557. 1 30. J. C. Watkins, Limit theorems for stationary random evolutions, Stach. Proc. & Appl. 19 (1 985), 1 89-224.

1 31 . J . C. Watkins, Limit theorems for products of random matrices; A comparison of two points of view, preprint, 1 986. 1 32 . R. J. Griego and R. Hersh, Brownian motion and potential the­ ory, Scientific American 220, no. 3 (1 969), 66-74 .

Wahrsch. Gebiete 23 (1 972), 230-244.

1 1 1 . R. Rishel, Dynamic programming and minimum principles for sys­ tems with jump Markov disturbances, SIAM J. Control 13 (1 975), 338-37 1 . 1 1 2 . S. I . Rosencrans, Diffusion transforms, J. Diff. Eq. 1 3 (1 973), 457-467.

A U THOR

1 1 3. G. Schay, Notices A . M.S. , no. 1 47 (1 973), Abstract 70-60-5. 1 1 4. A Schoene, Semi-groups and a class of singular perturbation problems, Indiana U. Math. J. 20 (1 970), 247-263. 1 1 5. K. Siegrist, Random evolution processes with feedback, Trans. A . M. S. 265, no. 2 (1 98 1 ) , 375-392.

1 1 6 . K. Siegrist, Harmonic functions and the Dirichlet problem for re­ newed Markov processes, Ann. Prob. 1 1 , no. 3 (1 983), 624-634. 1 1 7 . D. Stroock, Two limit theorems for random evolutions having non­ ergodic driving processes. Proc. Cont. Stach. D. E. and Appl. (Park City, Utah, 1 976), 24-253, Academic Press, New York, 1 977. 1 1 8. A Swishchuk, Markov random evolutions, IV Soviet-Japan Symp. on Prob. Th. & Math. Stat . , Abstracts. Tbilisi, 1 982, p. 39-40, (w ith V. S. Korolyuk, A F. Turbin). 1 1 9. A Swishchuk, Limit Theorems for semi-Markov random evolu­ tions in an asymptotic phase merging scheme. Dissertation, Kiev,

REUBEN HERSH

1 000 Camino Rancheros Santa Fe, NM 87501 USA e-mail: [email protected]

lnst. Math , 1 985, 1 1 6 pp. 1 20. A Swishchuk, Applied problems of theory of random evolutions,

Reuben Hersh has had many roles in his long career: poet (a

Znanie Publishing House, Ukrainian SSR, Kiev, RDENTP, 30 pp.,

half-century ago he won the Lloyd M. Garrison prize for po­

(with V. S. Korolyuk).

etry by an undergraduate two years in succession); machin­

1 2 1 . A Swishchuk, Semi-Markov random evolutions: a survey of the recent results, Cont. Trans. lith Prague conf. , 1 991 , 1 2 pp. (to appear).

60

THE MATHEMATICAL INTELLIGENCER

ist; mathematician; and in recent years especially, ated writer about mathematics.

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Hermann Weyl, the Reluctant Revolutionary David E. Rowe

Send submissions to David E. Rowe, Fachbereich 1 7 - Mathematik, Johannes Gutenberg University, 055099 Mainz, Germany.

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rouwer-that is the revolution!"-with these words from his manifesto "On the New Founda­ tions Crisis in Mathematics" [Weyl 192 1 ] , Hermann Weyl jumped headlong into ongoing debates concerning the foundations of set theory and analysis. His decision to do so was not taken lightly: this dramatic gesture was bound to have immense repercussions, not only for him but for many others within the fragile and politically frag­ mented European mathematical com­ munity. Weyl felt sure that modem mathematics was going to undergo massive changes in the near future. By proclaiming a "new" foundations crisis, he implicitly acknowledged that revo­ lutions had transformed mathematics in the past, even uprooting the entire edifice of mathematical knowledge. At the same time he drew a parallel with the "ancient" foundations crisis com­ monly believed to have been occa­ sioned by the discovery of incommen­ surable magnitudes, a finding that overturned the Pythagorean world view based on the doctrine "all is Num­ ber." In the wake of the Great War that changed European life forever, the zeitgeist appeared ripe for something similar, but even deeper and more per­ vasive. Still, revolutions cannot occur with­ out revolutionary leaders and ideolo­ gies, and these W eyl came to recognize in Egbertus Brouwer and his philoso­ phy of mathematics, which Brouwer originally called "neo-intuitionism" (in deference to Poincare's intuitionism, see [Dalen 1995]). Weyl had known Brouwer personally since 1912, and had studied his novel contributions to geometric topology as a prelude to writing Die Idee der Riemannschen Fldche [Weyl 1913]. But the Brouwer he and most others knew back then was the brilliant topologist, not the mystic intuitionist Dirk van Dalen ac­ quaints us with in his rich biography [Dalen 1999]. Weyl simply had not known the whole Brouwer, and proba-

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bly never did. True, he regarded him as a kindred philosophical spirit, but he seems never to have referred to Brouwer's Leven, Kunst en Mystiek (Life, Art, and Mysticism) or any of his other more general philosophical writings, presumably because he never read them (all were written in Dutch). If so, this surely precluded any chance of fully understanding the vision be­ hind Brouwer's views. Nevertheless, he was swept off his feet both by Brouwer's personality and by his revo­ lutionary message for mathematics. Weyl had been teaching since 1913 at the ETH in Zurich (on his career there, see [Frei and Stammbach 1992] ). His conversion experience took place in the summer of 1919 while vacation­ ing in the Engadin, where Brouwer, too, was staying. Their encounter was brief, lasting only a few hours, but long enough for Weyl to see the light. Af­ terward, Brouwer lent him a copy of his 1 9 13 lecture on "Formalism and In­ tuitionism," but Weyl returned it, com­ menting that he already had "a copy . . . from the old days," presumably an al­ lusion to the pre-revolutionary era. He further confessed that "at the time I did not pay attention to it or understand it. . . . " (Weyl to Brouwer, 6 May 1920, quoted in [Dalen 1999, p. 320]), a re­ mark befitting a new disciple of the faith. Discipleship played a crucial role in the social relations among the mathe­ maticians of this era, and no one felt this more keenly than Hermann W eyl when he studied under David Hilbert in Gottingen. Hilbert's aura as a youth leader-the "Pied Piper of Mathemat­ ics"-was perhaps the most distinctive quality that separated him from all his contemporaries. He must have felt a mixture of guilt and relief when, as he later described it, "during a short va­ cation spent together, I fell under the spell of Brouwer's personality and ideas and became an apostle of his in­ tuitionism" (Weyl Nachlass, Hs 9 1a: 1 7). Even young Bertus Brouwer was

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25. NUMBER 1, 2003

61

strongly attracted by Hilbert's alluring persona. He spent a considerable amount of time with him during the summer of 1909 when Hilbert was va­ cationing in Scheveningen, a seaside­ resort town near the Hague. The first personal encounter left a deep impres­ sion on Brouwer, as he related to his friend, the poet Adama van Scheltema: "This summer the first mathematician of the world was in Scheveningen; I was already in contact with him through my work, but now I have re­ peatedly made walks with him, and talked as a young apostle with a prophet. He was only 46 years old, but with a young soul and body; he swam vigorously and climbed walls and barbed wired gates with pleasure. It was a beautiful new ray of light through my life." (Brouwer to Adama van Scheltema, 9 November 1909, quoted in [Dalen 1999, p. 128]). Brouwer had already criticized Hilbert's axiomatic methods in his doc­ toral dissertation, submitted in 1907, where he concluded "that it has nowhere been shown, that if a finite number has to satisfy a system of con­ ditions of which it can be proved that they are not contradictory then the number indeed exists" (quoted in [Dalen 2000, p. 127]). For his part, Hilbert clearly recognized that the ax­ iomatic method could never show more than consistency, but he em­ phatically asserted that this was all a

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mathematician needed to prove in or­ der to assert that a mathematical ob­ ject exists. As van Dalen has observed, it would not have been like Brouwer to pass up this golden opportunity to ex­ plain his foundational ideas to Hilbert firsthand. Unfortunately, neither ap­ parently left any notes of what they talked about while strolling through the sand dunes of Scheveningen, but nearly twenty years later Brouwer did refer to these discussions while lamenting that Hilbert had in the mean­ time appropriated some of his key in­ tuitionist principles [Brouwer 1928]. In May 1920, the ink of his "New Cri­ sis" manuscript barely dry, Weyl sent it off to Brouwer along with the above­ cited letter in which he explained his motives. "It should not be viewed as a scientific publication," he informed his new-found ally, "but rather as a propa­ ganda pamphlet, thence the size. I hope that you will find it suitable for this pur­ pose, and moreover suited to rouse the sleepers . . . . " [Dalen 1999, p. 320] . That it certainly did. Weyl's provocative broadside caused the long-bubbling cauldron of doubts about set theory and analysis to boil over into what came to be known as the modem "foundations crisis, " a slogan taken di­ rectly from the title of this essay. Brouwer responded with almost glee­ ful delight: "your wholehearted assis­ tance has given me an infinite pleasure. Reading your manuscript was a con-

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62

THE MATHEMATICAL INTELLIGENCER

tinuous delight and your exposition, it seems to me, will also be clear and con­ vincing for the public . . . " [Dalen 1999, p. 32 1 ] . Arnong such delights was Weyl's use of politically inspired metaphor to con­ vey a heightened sense of urgency. The antinomies of set theory, he wrote, had once been regarded as "border con­ flicts" in "the remotest provinces of the mathematical empire" [Weyl 192 1, p. 143 ) . But now they could be seen as symptomatic of a deep-seated prob­ lem, till now "hidden at the center of the superficially glittering and smooth activity," but which betrayed "the inner instability of the foundations upon which the structure of the empire rests" (ibid.). Weyl likened the onto­ logical status of objects whose "exis­ tence" depends on proof by reductio ad absurdum to currency notes in a "paper economy," whereas true math­ ematical existence was surely a "real value, comparable to food products in the national economy." Nevertheless, "we mathematicians seldom think of cashing in this 'paper money.' The ex­ istence theorem is not the valuable thing, but rather the construction car­ ried out in the proof. Mathematics, as Brouwer on occasion has said, is more an activity than a theory" [Weyl 192 1, p. 157). With Bismarck's mighty German Empire now in shambles, Weyl clearly thought that the empire of modem analysis would soon fall, too. Its mighty fortress in Gottingen, led by the fear­ less and often ferocious Hilbert, had weathered all assaults up until now, but Weyl saw its walls cracking and prognosticated that they would soon come a crumblin' down, while the sage of Amsterdam stood ready to ride in and assume power. Weyl's defection to his intuitionist camp was clearly un­ dertaken in order to tip the scales in the Dutchman's favor, thereby prepar­ ing the overthrow of the old regime. His manifesto, penned during the pe­ riod of the abortive Kapp Putsch and its aftermath, reflected the mood of the times, when thoughts of revolution and counter-revolution abounded in Weimar Germany. Its principal target, of course, was his former mentor, Hilbert, who needed no rousing to see

armed and protected through Frege, Dedekind, and Cantor, it is doomed to fa ilure from the outset [Hilbert 1 922, pp. 159-1 60].

Hermann Weyl, circa 1 91 0, around the time he first became skeptical of Zermelo's ax­ ioms for set theory.

what was at stake. He struck back quickly, hard, and with plenty of polemical punch:

What Weyl and Brouwer are doing amounts, in principle, to a walk along the same path that Kronecker once fol­ lowed: they are attempting to estab­ l'ish the foundations of mathematics by throwing everything overboard that appears uncomfor-table to them and erecting a d ictatorship [Verbots­ diktatur] a la Kronecker. This amounts to dismembering our- science, which runs the t·isk of losing a large part of our most valuable possessions. Weyl a�nd Brouwer- ban the general concept of irrational number-function, . . . the Cantor-ian numbers of higher number cla.sses, etc. ; the theorem that among infinitely many whole numbers there is always a smallest, and even the log­ ical "Tertium non datur" . . . are ex­ amples offorbidden theorems and ar-­ guments. I believe, that just as earlier­ when Kronecker failed to do away with irrational number-s . . . so, too, today will Weyl and Brouwer not suc­ ceed; no: Brouwer- is not, as Weyl con­ tends, the revolution but rather only the repetition ofa Putsch attempt with old means. If earlier- it was earned out more sharply and still completely lost out, now, with the state so well

This tense encounter, pitting the all­ powerful Hilbert against his most gifted pupil, stands out as one of the more dramatic episodes of twentieth­ century mathematics. Yet despite all its high drama, Hermann Weyl's commit­ ment to Brouwer's intuitionist program soon lost its intensity. By the mid­ twenties, Brouwer had put an immense amount of energy into his program for revolutionizing mathematics, while in Gi:ittingen Hilbert and Paul Bemays were just as busy developing proof the­ ory as a bulwark of defense for classi­ cal analysis. By 1924, Brouwer had proved a series of results culminating in the theorem that every full function is uniformly continuous. Because these intuitionist findings had no counter­ parts in classical mathematics, Brouwer, who wasn't one to mince words, con­ cluded that "classical mathematics is contradictory" [Dalen 1999, p. 376 ] . S o where was Weyl? H e largely stood by and watched this lively action from the sidelines, albeit with consid­ erable interest. His flirtation with intu­ itionism seemed to many just that, a fleeting affair doomed from the start to end in disappointment. Weyl felt dif­ ferently. To understand why, it will be helpful to glance back at his earlier in­ terest not only in foundations of analy­ sis but in mathematical physics as well. It was hardly an accident that these two fields coincided with Hilbert's principal research interests after 1910, as both men shared high hopes for breakthroughs in these two realms. In Weyl's case these took concrete form in 1918 with the nearly simultaneous publication of Das Kontinuum [Weyl 19 18a] and Raum-Zeit-MateTie [Weyl 19 18b ] . On the Roots o f Weyl's Ensuing Conflict with Hil bert

Hilbert set out his early foundational views on a number of prominent occa­ sions, but for the most part he pre­ ferred to evade direct controversy [Rowe 2000] . After 1904, when he de­ livered a highly polemical address on

foundations issues at the Heidelberg ICM [Hilbert 1904 ] , he remained virtu­ ally mute about these matters for over a decade. He did not return in earnest to research in this field until the late war years. Still, this hardly meant that he had lost interest. As Volker Peck­ haus has described, Hilbert's long­ standing efforts on behalf of Ernst Zer­ melo, who held a modest position in Gi:ittingen teaching mathematical logic, as well as his support for the philosopher Leonard Nelson were part of Hilbert's long-term strategy aimed at providing institutional support for re­ search in set theory, foundations, and mathematical logic [Peckhaus 1990, pp. 4-22]. Hilbert, now at the height of his career, had emerged as Gi:ittingen's second great empire-builder. He did so, however, not so much by building on the groundwork Felix Klein laid in var­ ious branches of applied mathematics [Rowe 200 1 ] , but rather by promoting research that extended the territorial claims he himself had already staked out in analysis, number theory, foun­ dations of geometry, and mathematical physics. Compared with the Hilbertian production lines in these fields, Gi:it­ tingen research efforts in set theory and foundations resembled a mere cottage industry. Presumably Hilbert hoped that by delegating this research to specialists he could tum to other matters, in particular the foundations of physics, which dominated his atten­ tion after the death of Hermann Minkowski in 1909 [ Corry 1999 ] . Skuli Sigurdsson has addressed the theme of "creativity in the age of the machine" in connection with Weyl, who during his student days had close associations with Hilbert's "factory" for integral equation theory [Sigurds­ son 200 1, pp. 2 1-29] . Hermann Weyl clearly never wanted an ordinary job on this fast-moving assembly line, and he later downplayed the value of much that came off it. In one of his two obit­ uaries for his mentor, he wrote that it had been due to Hilbert's influence that "the theory of integral equations be­ came a world-wide fad in mathematics . . . producing an enormous literature of rather ephemeral value" [Weyl 1944, pp. 126--127] . Nevertheless, the young Weyl took a keen interest in the work

VOLUME 25. NUMBER 1 . 2003

63

of E. Schmidt, E. Hellinger, 0. Toeplitz, et al., and he also did a fair amount of mingling with his peers in the Gottin­ gen mathematical community. This gave him ample opportunity to partic­ ipate in discussions on set theory and foundations with Zermelo, whose proof of Georg Cantor's well-ordering theorem in 1904 led to an intense de­ bate regarding the admissibility of Zer­ melo's axiom of choice [Moore 1982] . Four years later, i n an effort t o quell this controversy, Zermelo presented his well-known system of axioms for set theory [Zermelo 1908]. Soon thereafter, Weyl began to take a serious and active interest in set the­ ory. Writing to his Dutch friend, Pieter Mulder, on 29 July 19 10, he character­ ized his standpoint as closer to that of Emile Borel and Henri Poincare than to Zermelo's views. But he also indi­ cated that he would have to think these matters through very carefully, espe­ cially because he feared the contro­ versy that typically ensued whenever issues in set theory and the founda­ tions of analysis were addressed. Re­ calling these times, Weyl would later write: "I grew up a stern Cantorian dog­ matist. Of Russell I had hardly heard when I broke away from Cantor's par­ adise; trained in a classical gymnasium, I could read Greek but not English" (Weyl Nachlass, Hs 9 1a: 1 7) . Eight years later, Weyl alluded to the diffi­ culties he encountered in trying to make sense of Zermelo's axiom system for set theory: "My investigations be­ gan with an examination of Zermelo's axioms for set theory. . . . Zermelo's ex­ planation of the concept 'definite set­ theoretic predicate,' which he employs in the crucial 'Subset'-Axiom III, ap­ peared unsatisfactory to me. And in my effort to fix this concept more pre­ cisely, I was led to the principles of definition of *2 [in Das Kontinuum]" [Weyl 19 18a, p. 48] . These principles were already enunciated in [Weyl 1910], a paper Solomon Feferman has discussed in connection with Alfred Tarski's ideas [Feferman 2000, p. 1 80 ] . Weyl described his initial orienta­ tion as similar to Dedekind's theory of chains, in that he sought to establish the principle of complete induction without recourse to the primitive no-

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THE MATHEMATICAL INTELLIGENCER

tion of the natural numbers. This quest "drove me to a vast and ever more com­ plicated formulation but, unfortu­ nately, not to any satisfactory result." He finally abandoned this as a "scho­ lastic pseudo-problem" after achiev­ ing "certain general philosophical in­ sights," presumably derived from read­ ing Edmund Husserl and distancing himself from Poincare's conventional­ ism. Nevertheless, he concluded that Poincare had been right regarding the status of the sequence of natural num­ bers as "an ultimate foundation of mathematical thought" [Weyl 19 18a, p. 48] . What prompted Weyl to reenter this arena in 1918, a move that took his good friend, Erich Heeke, by surprise? Probably he had several motives, but he surely kept a keen eye on his men­ tor's activities, about which he had first-hand knowledge. On 1 1 Septem­ ber 1 9 1 7, Hilbert delivered a lecture on "Axiomatic Thought" [Hilbert 1918] be­ fore a meeting of the Swiss Mathemat­ ical Society in Zurich. This gave the first clear signs that he was about to take up the foundations of mathemat­ ics once again. Probably no one in Hilbert's audience listened more atten­ tively than Hermann Weyl, who dis­ cussed this lecture in detail many years later. For Hilbert's talk offered a sweeping panorama of mathematical and physical ideas that stressed not only their mutual interdependence but the role of axiomatics in both realms (see [ Corry 1997] on Hilbert's back­ ground interests). Like many of his contemporaries, Hilbert regarded the growth of mathe­ matical knowledge as an essentially teleological process in which thought obeys higher, transcendental laws. As such, his positivism had something like an Hegelian flavor, only with the math­ ematician replacing the metaphysician as the highest human form of Reason. In his lecture, Hilbert described the manner in which axiomatization took place as part of a natural, organic process starting from an informal sys­ tem of ideas ("Fachwerk von Begrif­ fen"). These ideas, which arose spon­ taneously in the course of the theory's development, were merely provisional in nature. Only during the next stage,

when researchers attempted to pro­ vide deeper foundations for the theory, did axiomatization actually begin. By invoking architectonic imagery, Hilbert suggested how this process structured scientific thought:

Thus arose the actual, present-day so­ called axioms of geometry, arith­ metic, statics, mechanics, radiation theory, and thermodynamics. These axioms build a deeper-lying layer of axioms than the axiom layer that was earlier characterized by the funda­ mental theorems of the individual fields. The process of the axiomatic method . . . thus amounts to a deep­ ening of the foundations of the indi­ vidual fields just as it becomes neces­ sary to do with any building to the extent that one wants to make 'it se­ cure as one builds it outward and up­ ward [Hilbert 1 91 8}. In his concluding remarks, Hilbert mentioned two particularly pressing problems confronting the foundations of mathematics: proving the consis­ tency of his axioms for arithmetic (the second of Hilbert's 23 Paris problems), and doing the same for Zermelo's sys­ tem of axioms for set theory. He em­ phasized that both of these problems were wedded to a whole complex of deep and difficult epistemological questions of "specifically mathematical coloring": (1) the problem of the solv­

ability of every mathematical ques­ tion in principle, (2) the problem of the subsequent verification of the re­ sults of a mathematical investigation, (3) the question of a criterion for the simplicity for mathematical proofs, (4) the question of the relationship be­ tween content and form in mathemat­ ics and logic, and (5) the problem of the decidability of a mathematical question by means of a finite number of operations. Hilbert then summed up his position regarding all these com­ plex issues as follows: "All such fun­ damental questions . . . appear to me to form a newly opened field of re­ search, and to conquer this field-this is my conviction-we must undertake an investigation of the concept itself of the specifically mathematical proof, just as the astronomer must take into

account the movement of his position, the physicist must care for his appara­ tus, and the philosopher criticizes rea­ son itself" [Hilbert 1918, p. 155]. While conceding that, for the pres­ ent, these ideas remained but a sketch for future research, Hilbert retained his optimistic outlook for his program:

I believe that everything which can be the subject of scientific thought, as soon as it is ripe enough to constitute a theory, falls within the scope of the axiomatic method and thus directly to mathematics. By pursuing ever deeper-lying layers of ax·ioms . . . toe gain ever deeper i nsights into the essence of scientific thought itselj; and we become ever more conscious of the un·ity of our knowledge. In the name of the axiomatic method, math­ emal'ics appears called upon to as­ sume a leading role in all of science [Hi.tbert 1918, p. 1 56}.

to the approaching possib·ility, that out ofphysics in pri.nc'ipl.e a science simi­ lar to geometry w·ill a-rise: truly, the most glorious ja.me qf the axiomatic method, while here, as we see, the mighty instruments of analysis, namely the calculus qf vmiations and invariant theory, are taken into ser­ vice [Hilbert 1915, p. 407}. Tilman Sauer has recently noted how Hilbert, quite ironically, made only a vague allusion in his Zurich lec­ ture to this vision for a unified field physics [Sauer 2002 ] . Weyl could not have failed to notice that Hilbert sounded very subdued about these prospects on this occasion. He also knew very well what Einstein thought of Hilbert's methodological approach. In a letter from November 19 16, Ein­ stein confessed:

To me Hilbert's Ansatz about matter appears to be ch·ildish, just like an in-

This tour de force performance clearly signaled Hilbert's intentions to take up once again the foundations program he had sketched thirteen years earlier in his speech at the Hei­ delberg ICM. Indeed, his rhetorical flourishes clearly echoed themes Weyl and others would have recognized from Hilbert's even more famous ad­ dress at the Paris ICM in 1900. Just as striking, however, were the parallels with his concluding remarks from his first contribution to the general theory of relativity, in which he made similarly sweeping claims regarding the strength and resilience of the axiomatic method:

As one sees, the jew simple assump­ tions expressed in Axioms I and II suffice by sensible interpretation for the development of the theory: through them not only are our conceptions of space, time, and motion fundamen­ tally reformulated in the Einstein­ ian sense, but I am convinced that the most minute, till now hidden processes within the atom will become clarified through the fundamental equations herein exhibited and that U must be possible in general to refer all phys1:cal constants back to mathe­ matical constants-just as this leads

Einstein relaxing in h i s home office in Berlin. By 1918 he was carrying on an extensive sci­ entific correspondence.

fant who is unaware of the p i tfalls of the real world. . . . In any case, one cannot accept the m·ixture of well­ founded considerations arising from the postulate of general relativity and unfounded, risky hypotheses about the structure of the electron . . . . I am the first to admit that the discovery of the proper hypothesis, or the Hamil­ ton junction, of the structure of the electron is one of the most importan t tasks of the current theonJ. Th e "ax­ iomatic method, " however, can be of little use in this (Einstein to Weyl, 23 November 191 6 [Einstein 1 918a, p. 366}). Weyl took up these problems around this very time. In the summer semester of 1 9 1 7 he offered a lecture course on general relativity, and, on the advice of Einstein's close friend Michele Besso, he decided to adapt his notes into a book on special and gen­ eral relativity. This was published the following year by Julius Springer Ver­ lag as the first edition of Raum-Zeit­ Materie [Weyl 19 18b]; a second soon followed, and three substantially re­ vised editions appeared between 1919 and 1923. A few months before it came out, however, Weyl had proofs sent to both Einstein and Hilbert. Their re­ spective reactions reveal a good deal about both men. Einstein was euphoric: "it's like a symphonic masterpiece. Every word has its relation to the whole, and the design of the work is grand" (Einstein to Weyl, 8 March 1918 [Einstein 1918b, pp. 669-670]). A week earlier, Hilbert wrote also, but he merely acknowl­ edged receipt of the proofs (Hilbert to Weyl, 28 February 19 18, Weyl Nach­ lass, Hs. 9 1 : 604). Because he was on his way to Bucharest to attend a meet­ ing on space and time in physics, he had no time to read them. Still, he ex­ pressed regret that he would not be able to meet Weyl in Switzerland over the semester break, but hoped to do so during the summer or early the fol­ lowing year. He then added some re­ marks about the professorship in Bres­ lau recently offered to Weyl. Hilbert had been consulted during the deliberations over potential candi­ dates, and he had apparently recom-

VOLUME 25, NUMBER 1 , 2003

65

mended Weyl for the post. But he now

quickly added "unfortunately this again

Weyl received this letter in time to

counseled him against accepting the

creates a vacant mathematical position

make modifications in the text before

"im Interesse des Reichsdeutsch­ tum," because if he left Zurich this

in Switzerland that will be difficult to

the book went to press. He added a few

fill and unlikely so with a suitable per­

citations and brief remarks on Hilbert's

would probably leave some worthy

sonality."

first note, but these remained shadowy

offer

German mathematician without a job.

Miffed that W eyl had ignored his

features of his book compared with his

Hilbert took a very active role in this

wishes, Hilbert added some curt praise

own contributions and, of course, Ein­

game of mathematical musical chairs,

for Raum-Zeit-Materie: "I have looked

stein's. Hilbert could not have felt grat­

and he apparently thought that W eyl

more carefully at the proofs of your

ified by this, especially after reading

should pass on this round. He also

book, which gave me great pleasure,

the preface, which Weyl wrote while

thought

government

especially also the refreshing and en­

vacationing at his in-laws' home in

the

Prussian

would be receptive to this argument, so

thusiastic presentation. I noticed that

Mecklenburg. He could hardly have

that declining would not have unfavor­

you did not even mention my first Got­

failed to notice his protege's animus against his views on the axiomatization

able consequences for Weyl's future.

tingen note from 1 9 1 5 . . . . " He then pro­

Apparently Weyl was not inclined to

ceeded to rattle off a litany of com­

of physics. At the same time, Weyl an­

follow this advice; at any rate, in April

plaints

nounced that he had found a new av­

he accepted the chair in Breslau (al­

remarks that provide insight into what

though he would later turn it down for

Hilbert

health

achievements

reasons).

Hilbert,

having

re­

turned from Bucharest, wrote him on

paper

bearing himself on

in

on saw

this as

his

"Foundations

omission, the

enue to a truly unified field theory:

main

controversial of Physics"

22 April, sending "congratulations on

[Hilbert 1 9 1 5) . (For details, see the ac­

accepting the Breslau position," but

companying box.)

22 April 1918 Dear H<'rr Weyl, above all my congratulations on accepting tl1e Breslau position, tmfortunately this again creates a vacant mathematical position in Switzerland tilat will be difficult to fill and unlikely so with a suitable personality. Having returned from Bucharest I have looked more carefully at the proofs of your book which gave me great pleasure, especially also the refreshing and enthusiastic presentation. I noticed that you did not even mention my first Gottingen note from 1915 even though the foundations of the gravitational tileory, in particular the use of the Riemarmian curvature in the Hamiltonian

The Einsteinian theory in its present form ends with the duality of elec­ tricity and gravitation, ':field" and "ether"; these remain totally isolated and stand next to one another. Just now a promising path has opened fo-r the author for the derivation of both realms ofphenomenafrom a common source by an extension of the geomet­ rical foundations. Evidently the de­ velopment of the general theory of rel­ ativity had not yet been concluded. However, it was not the 'intention of this book to take the powerful, stirring life that stems frorn the field of phys­ ical knowledge to the point it has presently reached and transform it with axiomatic rigor into a dead mummy. [ Weyl 1918b, p. vi)

integral which you present on page 1 9 1 , stems from me alone, as does the the derivation of the

If Weyl saw general relativity as

Maxwellian equations, etc. Also the whole presmtation of Mie's th<'ory is pre­

opening new vistas for the geometriza­

separation of the

Hamiltonian function

in

H-L,

cisely that which I gaw for the first time in my fll"St note on the foundations

tion of physics, he also took up the chal­

of physics. For Einstein's earlier work on his definitive tileory of gra\itation

lenge Hilbert had laid down for founda­

appeared at the same tin1e as mine (nanwly in

tions of analysis. Indeed, immediately

owmber 1 9 1 5); Einstein's

other papers, in particular on elt'ctrodynamics and on H�m1ilton's principle

after Hilbert spoke about this in Zurich,

appeared however much later.

Weyl began a lecture course in the

aturally I am very gladly prepared to correct

any of my mistaken assertions. It also appears odd to me that where you do

win­

ter semester of 1 9 1 7-18 titled "Logical

take note of the fact tilat there are four fewer differential equations for grav­

Foundations of Mathematics." One of

Got­

his potential auditors, Paul Bemays,

wherea.c:; precisely this circumstance served as tile point

was unable to attend since Hilbert had

of departure for my invest igation and was especially enunciated as a tileorem

already snatched him up as his new as­

itation than tmknowns you only cite my second conm1unication in tile

linge-r Nadwi('hten,

whose consequences were pursued there. . . .

sistant in Gottingen. Thus began the col­ laboration that enabled Hilbert to push

With best greetings to you and your wife,

forward proof theory in earnest [Sieg

2000). In the meantime, Weyl used his Your Hilbert

lectures as a p latform for writing a booklet

Draft of letter from Hilbert to Weyl, NSC"B. Hilbert Naehlass 457, 1 7

on

The

Continuum

[ Weyl

1 9 18a). This aimed to salvage tile foun­ dations of analysis within the more

66

THE MATHEMATICAL INTELLIGENCER

modest scope of a constructivist pro­ gram that avoided the complexities of a full-blown axiomatic set theory a la Zem1elo. Its approach to the continuum thus clearly exposed the differences be­ tween Weyl's constructivist views and those of Hilbert. Weyl only gradually be­ came aware of this rift, but by 1918 he stood firmly on the other side of a deep abyss that separated his approach to foundations of analysis from conven­ tional wisdom on this subject. Even in Zurich, Weyl's ideas en­ countered strong opposition. His younger colleague, Georg P6lya, whom he deeply respected as an analyst, re­ acted incredulously, so much so that Weyl challenged him to a mathemati­ cal wager. He rounded up a dozen wit­ nesses who validated a document stat­ ing that:

Within 20 years P6lya and the ma­ jority of representative mathemati­ cians will admit that the statements 1. Every bounded set of reals has a

precise supremum 2. Every infinite set of numbers con­

tains a denumerable subset contain totally vague concepts, such as "number, " "set, " and "denumer­ able, " and therefore that their truth or falsity has the same status as that of the main propositions of Hegel's nat­ ural philosophy. Howeve1·, under a natural interpTetation (1) and (2) will be seen to be false. [P6lya 1972] Thus, already by this time, Weyl's heretical views were well known within Zurich's mathematical circles. They would become even better known after he completed the manu­ script of [Weyl 19 18a] and followed it up with [Weyl 1919], a note on the "vi­ cious circle" in conventional founda­ tions of analysis that appeared in the widely read JahTesbericht of the Ger­ man Mathematical Society. In Das Kontinuum, he described the programmatic features of his new treatment of the continuum as follows:

[Ijn spite of Dedekind, Cantor and WeieTstrass, the great task which has been facing us since the Pythagorean discovery of the irrationals remains

today as unfinished as ever; that is, the continuity given to us immediately by intuition (in the flow of time and in motion) has yet to be grasped mathematically as a totality of dis­ crete "stages" in accordance with that part of its content which can be in­ terpreted in an "exact" way. [Weyl 1918a, pp. 24-25j Weyl insisted on the need to avoid the problem of impredicativity which arose when the set of real numbers was defined by means of arbitrary Dedekind cuts. Hilbert had tried to finesse this problem with his Voll­ stiindigkeitsaxiom, but in later edi­ tions of his Grundlagen der Geometrie he was forced to fall back on Dedekind cuts to prove that his axiom system characterized analytic geometry over the field of real numbers. Weyl's gen­ eral philosophical orientation stood in sharp contrast to Hilbert's brand of ra­ tionalism, so it should not be surprising that he distanced himself from the fun­ damental tenets underlying his men­ tor's foundations research. Still, Das Kontinuum represented only a rela­ tively mild break with Hilbertian pre­ cepts. Unlike Brouwer's approach, which took the continuum to be an ir­ reducible given requiring the mathe­ matician to master the new techniques of choice sequences, Weyl's goal in [Weyl 1918a] was far more pragmatic. As an analyst, he was less concerned with capturing the "essence" of the con­ tinuum than he was in extracting from it a sufficiently rich arithmeticized sub­ structure which the "working mathe­ matician" could use to recover most of the results of classical analysis (for more on this, see [Feferman 2000]). Immediately after publishing Das Kontinuum and Raum-Zeit-Materie, Weyl began elaborating his gauge­ invariant approach to a unified field theory. Einstein called it "a first-class stroke of genius," but nevertheless a useless one for physics (Einstein to Weyl, 6 April 1918 [Einstein 1918b, p. 710]). Arnold Sommerfeld had no such reservations, at least initially:

What you are saying there is really wonderful. Just as Mie had grafted a gravitational theory onto his funda-

mental theory of electrodynamics that did not form an organic whole with it . . . Einstein grafted a theory of elec­ trodynamics . . . onto his fundamen­ tal theory of gravitation that had lit­ tle to do with it. You have worked out a true unity (Sommerfeld to Weyl, 7 July 1918, Weyl Nachlass, 91: 751). Apparently Weyl had no difficulty persuading Hilbert, either. As he had promised, his former mentor visited Weyl during September when he was vacationing in Switzerland. Whether or not they discussed foundational issues is unclear, but Weyl did report to Ein­ stein that Hilbert "gave me his unqual­ ified support" for the new unified field theory (Weyl to Einstein, 18 September 1918 [Einstein 1918b, p. 879]). Never­ theless, Einstein continued to express skepticism, so much so that Weyl wrote him in December: "I am now in a really difficult position; by nature so concilia­ tory that I'm almost incapable of dis­ cussion, I now must fight on all fronts: my attack on analysis and attempt to found it anew is encountering much more fierce opposition from the math­ ematicians who work on these logical things than my efforts in theoretical physics have received from you" (Weyl to Einstein, 10 December 1918 [Ein­ stein 19 18b, p. 966]). By now Weyl was grappling not only with the foundations of classical analy­ sis and mathematical physics but also with core issues that led to dramatic new developments in differential geom­ etry. Weyl described his research pro­ gram in 192 1 as follows:

It concerns, on the one hand, impulses to a new foundation of the analysis of the infinite, the present foundation of which is in my opinion untenable; on the other hand, in close connection with this [my emphasis] and in con­ nection with the general theory of rel­ ativity, a clarification of the relation of the two basic notions with which modem physics operates, that of field of action and of matter, the theories of which can at present by no means be joined together continuously [Weyl's

emphasis] [Dalen 2000, p. 31 1]. Erhard Scholz has emphasized the ex­ tent to which Weyl's mathematical

VOLUME 25, NUMBER 1 , 2003

67

A gathering of Zurich mathematicians during the visit of Josef Ki.irschSk (right front row), who came from Budapest along with Frederic Riesz (third from left). Pictured, from left to right, are Hermann Weyl, Louis Kollros, Riesz, Georg P61ya, Michel Plancherel, Jerome Franel, Mrs. Ki.irschak, (unknown), Andreas Speiser, Klirschak, and Rudolf Fueter. (From George P61ya, The P6/ya Picture Album: Encounters of a Mathematician, ed. G. L. Alexanderson, Boston: Birkhiiuser, 1 987, p. 75.)

work was intertwined with physical and epistemological problems [Scholz 200 1b ]. All three of these fast-converg­ ing interests were clearly very much on his mind when he first learned about Brouwer's intuitionism. Weyl's Emotional Attachment to Intuitionism

After meeting Brouwer in the summer of 1919, Weyl presented his provisional ideas on the "new crisis" the following December at three successive meet­ ings of the mathematics colloquium, which he co-chaired with P6lya. The topics he covered were identical with those eventually presented in his "pro­ paganda pamphlet," and apparently he made no attempt to tone down his en­ thusiasm for intuitionism when he de­ livered these lectures. Not surprisingly, Georg P6lya reacted just as vigorously in criticizing Weyl's philosophical claims in defense of intuitionism. Ac­ cording to notes taken by Ferdinand Gonseth, they exchanged words like these: You say that mathematical theorems should not only be true,

P6LYA:

68

THE MATHEMATlCAL INTELLIGENCER

but also be meaningful. What is meaningful? WEYL: That is a matter of honesty. P6LYA: It is erroneous to mix philo­ sophical statements in science. Weyl's continuum is emotion. WEYL: What P6lya calls emotion and rhetoric, I call insight and truth; what he calls science, I call symbol pushing (Buchstabenreiterei). P6lya's defense of set theory . . . is mysti­ cism. To separate mathematics, as being formal, from spiritual life, kills it, turns it into a shell. To say that only the chess game is science, and that insight is not, that is a re­ striction [Dalen 1999, p. 320). Weyl was not vituperative by nature, but he clearly hoped his ideas would create a stir. He especially wanted to arouse Hilbert from his dogmatic slum­ bers, something Brouwer had been un­ able to accomplish himself. Hard at work on his "New Crisis" paper, Weyl wrote Bernays in Gottingen (9 Febru­ ary 1920, Weyl Nachlass, 9 1 : 10) that he had returned to foundations of analy­ sis, and had "substantially modified [his] standpoint" thanks to Brouwer's

work. Referring to their meeting the previous summer, he wrote, "I was completely happy during the few hours we spent together." And he described Brouwer as a real "devil of a fellow (Mordskerl)" and a wonderfully intu­ itive person, adding that "if you get him to come to Gottingen as Heeke's suc­ cessor, then I envy you." The depar­ tures of Heeke and Caratheodory, two of Hilbert's most prominent proteges, reflected the fragile state of German academic life amid the surrounding political uncertainties. Brouwer turned down both chairs, but not before ne­ gotiating improved conditions in Ams­ terdam [Dalen 1999, 300-304]. Weyl's attraction to intuitionism and his feelings toward its founder were confirmed when he visited Brouwer at his home in Blaricum a few months af­ ter completing his "New Crisis" essay. Writing to Felix Klein, he proclaimed that "Brouwer is a person I love with all my heart. I have now visited him in his home in Holland, and the simple, beautiful, pure life in which I took part there for a few days, completely and totally confirmed the impression I had made of him" [Dalen 1999, p. 298]. Even after this passionate phase had passed, Weyl always extolled Brouwer's posi­ tive achievement, which he saw as clar­ ifying the essential tension between form and content, a dichotomy he took to be characteristic of mathematical thought. Yet Weyl was never an orthodox in­ tuitionist, not even when he presented himself as a Brouwerian in his "New Foundations Crisis" [Scholz 2000, pp. 199-200] . What appealed to him most about Brouwer's philosophy was its honesty and its human dimension. He shared Brouwer's aversion to Hilbert's overly sanguine, winner-take-all ap­ proach to mathematics that bordered on intellectual dishonesty, just as he deplored the tendency to reduce the quest for mathematical meaning and truth to a meaningless formalist game. Brouwer refused to accept carte blanche the paper currency of pure ex­ istence proofs unless it could first be demonstrated that this money could actually be used to purchase a genuine mathematical article. Intuitionism thereby threatened the weakest flank

in Hilbert's fortress by exposing the ontological assumptions of formalist dogma while advancing a vision for a new mathematics based on construc­ tive principles. For Brouwer, Hilbert's fom1alist doctrine amounted to an im­ poverished view of mathematical ac­ tivity, for it aspired merely to show that certain purely formal procedures could never produce a contradiction. Brouwer's views thus had a twofold significance for Weyl. First, they helped him focus on conflicts he had long been bothered by but had never quite confronted, including his general dissatisfaction with Hilbert's claims re­ garding the axiomatic method. Second, the clarity of Brouwer's critique gave Weyl the courage to articulate his own views in strong, provocative language that could not be ignored. In his eyes, intuitionism was an ideology that dig­ nified mathematical research as a hu­ man activity during a time of crisis. Still, it was only one element within the larger framework of ideas Weyl hoped to synthesize. Many intellectuals-Weyl, Brouwer, Hilbert, and Einstein included-were acutely aware that they were living through revolutionary times when the whole "civilized" world was seeking new leaders, structures, and answers. But Weyl alone, working in the quiet seclusion of Zurich, felt the need to pro-

mote a new intellectual reorientation that would encompass both mathe­ matics and physics. For Weyl, Ein­ stein's general theory represented nothing less than a revolution in human thought, a view shared by Hilbert. Yet Weyl went further; his pioneering work in differential geometry was motivated by the urge to explore the implications of space-time theories in order to reach a deeper conceptual understanding of the physical world. Moreover, he was convinced that this simultaneous revo­ lution in mathematics and physics was already underway, having burst into the public eye well before the sudden collapse of Imperial Germany that ended the Great War. Yet despite his sensitivity to these swirling intellectual currents, Weyl was not by temperament an iconoclas­ tic or revolutionary thinker. Moreover, he lacked the hard-nosed, combative streak that made both Hilbert and Brouwer such forceful personalities. By announcing that analysis was un­ dergoing a foundational crisis, Weyl sought to expose the hollowness of Hilbert's rhetoric and at the same time underscore his long-standing misgiv­ ings with Cantorian set theory, which Hilbert and Zermelo had sought to re­ tain and rigorize by means of axiomat­ ics. He was, however, more than happy to accomplish this by carrying Brouwer's

banner forward rather than holding up one of his own making. The political imagery in his "New Crisis" essay was all very apt, but the Oedipal dimensions of this conflict would seem even harder to ignore. For however strongly Weyl may have felt about Brouwer, he must have sooner or later realized that he had very little in common with him. In everything from their personalities and life styles to their philosophical views and math­ ematical tastes, he and Brouwer were opposites. Clearly, his feelings for Brouwer and his attachment to intu­ itionism were genuine, but just as clearly they were a means to gain some emotional distance from his tyrannical Doktor-vater, a man intent on directing the course of his career. Intellectually, he knew how much he owed Hilbert, and he largely shared his ambitions for the "new mathematics" as well as his vision for a unified field physics of gravitational and quantum phenomena. As the heir-apparent of the Gottingen master, Weyl must have felt almost pre­ destined to compete with him on these two major fronts. When seen from this vantage point, his brief flirtation with Brouwer's intuitionism appears less Olympian, more human, and in some sense more believable. REFERENCES

[Brouwer 1 928] "lntuitionistische Betrachtun­ gen

uber

den

Formalismus,"

Sitzungs­

berichte der PreuBischen Akademie der Wis­ senschaften, 1 928: 48-52.

[Corry 1 997] Leo Corry, "David Hilbert and the Axiomatization of Physics (1 894-1 905). " Archive for History o f Exact Sciences 5 1 :

83-1 98. --

[Corry, 1 999]

, "David Hilbert between

Mechanical and Electromagnetic Reduction­ ism (1 91 0-1 91 5), " Archive for History of Ex­ act Sciences 53: 489-527.

[Dalen 1 995] Dirk van Dalen, "Hermann Weyl's lntuitionistic Mathematics," The Bulletin of Symbolic Logic 1 (1 995), 1 45-1 69 .

[Dalen 1 999]

--,

Mystic, Geometer, and In­

tuitionist. The Life of L E. J. Brouwer, val. 1 : The Dawning Revolutio n. Oxford Clarendon

Press, 1 999. [Dalen 2000]

--

, "The Development of

Brouwer's Intuitionism, " in [Hendricks, Ped­ Weyl, far right, listening attentively to Edmund Landau, while Richard Courant stares at a book. Presumably taken in Gottingen around 1 930.

erson, J0rgensen 2000] , pp. 1 1 7-1 52. [Einstein 1 998a] Collected Papers ofAlbert Ein-

VOLUME 25, NUMBER 1. 2003

69

stein (CPAE) , Vol. SA: The Berlin Years: Cor­ 1 9 1 4- 1 9 1 7,

respondence,

Robert Schul­

gen aus dem Mathematischen Seminar der Hamburgischen

Universitat,

1

(1 922):

mann, et al. , eds. Princeton: Princeton Uni­

1 57-1 77 Reprinted in [Hilbert 1 932-35], vol.

versity Press.

3, pp. 1 57-1 7 7 .

[Einstein 1 998b] Collected Papers of Albert Ein­ stein (CPAE) , Vol. 8B: The Berlin Years: Cor­ 19 18,

[Hilbert 1 932-35]

Zeit-Materie and a Genera/ Introduction to his Scientific Work (DMV Seminar, 30). Basel/

Boston: Birkhauser Verlag, 2001 . [Scholz 2001 b]

--

, Gesammelte Abhand­

/ungen , 3 vols, Berlin: Springer.

--

, "Weyls lnfinitesimalge­

ometrie, 1 9 1 7-1 925," in [Scholz 2001 a] , pp. 48-1 04.

Robert Schulmann,

[Moore 1 982] Gregory H. Moore, Zermelo 's Ax­

[Sieg 2000] Wilfried Sieg, "Toward Finitist Proof

et al. , eds. Princeton: Princeton University

iom of Choice: Its Origins, Development and

Theory," in [Hendricks, Pederson, J0rgensen

Press.

Influence, New York: Springer, 1 982.

respondence,

2000] , pp. 95-1 1 5.

[Feferman 2000] Solomon Feferman, "The Sig­

[Peckhaus 1 990] Volker Peckhaus, Hilbertpro­

[Sigurdsson 200 1 ] Skuli Sigurdsson, "Journeys

nificance of Weyl's Oas Kontinuum , " in [Hen­

gramm und Kritische Philosophie, Stud1en

dricks, Pederson, J0rgensen 2000], pp.

zur Wissenschafts- , Sozial- und Bildungs­

in Spacetime," in [Scholz 200 1 a] , pp. 1 5-47. [Weyl 1 91 0] Weyl, Hermann, " U ber die Defini­

1 79-1 93.

geschichte der Mathematik, vol. 7, G6ttin­

tionen der mathernatischen Grundbegriffe,"

gen; Vandenhoeck & Ruprecht, 1 990.

Mathematisch-naturwissenschaftliche

[Frei and Stammbach 1 992] Gunther Frei and Urs Stammbach, Hermann Weyl und die

[P61ya 1 972] Georg P61ya, "Eine Erinnerung an

Mathematik an der ETH Zurich, 1 9 1 3- 1 930.

Hermann Weyl, " Mathematische Zeitschrift

Basel: Birkhauser, 1 992.

1 26 (1 972): 296-298.

[Hendricks, Pederson, J0rgensen 2000], Proof Theory, History and Philosophical Signifi­

the Storm: Hilbert's Early Views on Founda­

cance, ed. V. F. Hendricks, S. A. Pedersen,

tions," in [Hendricks, Pederson, J0rgensen

and K. F. Jmgensen. Dordrecht: Kluwer, 2000.

2000], pp. 55-94. [Rowe 200 1 ]

[Weyl 1 968], vol . 1 , pp. 298-304 . --

[Weyl 1 91 3]

[Rowe 2000] David E. Rowe, "The Calm before

Blat­

ter 7 ( 1 9 1 0): 93-95, 1 09-1 1 3. Reprinted in

, Die Idee der Riemannschen

Flache. Leipzig: Teubner, 1 91 3 .

[Weyl 1 9 1 8a]

--

, Oas Kontinuum . Leipzig:

Veil & Co. , 1 9 1 8. The Continuum, trans. S. Pollard and T. Bole. New York: Dover, 1 987.

--

,

"Felix Klein as Wis­

[Weyl 1 91 8b]

--

, Raum, Zeit, Materie , 1 st

[Hilbert 1 904] David Hilbert, " U ber die Grund­

senschaftspolitiker," in Changing Images in

lagen der Logik und der Arithmetik," Ver­

Mathematics. From the French Revolution to

handlungen des I l l . lnternationalen Mathe­

the New Millennium, ed. Umberto Bottazzini

matiker-Kongresses in Heidelberg 1 904, pp.

and Amy Dahan. London: Routledge, pp.

bericht der Deutschen Mathematiker- Vere­

1 74-1 85.

69-92.

inigung 28 (1 9 1 9) : 85-92.

[Hilbert 1 9 1 5] Physik

--

, "Die Grundlagen der

(Erste

Mitteilung), "

[Sauer 2002] Tilman Sauer, "Hope and Disap­

ed. , Berlin: Springer-Verlag, 1 9 1 8. --

[Weyl 1 91 9]

, "Der circulus vitiosus in der

heutigen Begrundung der Analysis , " Jahres­

[Weyl 1 92 1 ]

, " U ber die neue Grundla­

--

K6nigliche

pointments in Hilbert's Axiomatic ' Founda­

genkrise der Mathematik, " Mathematische

Gesellschaft der Wissenschaften zu G6tti­

tions of Physics, ' " in History of Philosophy

Zeitschrift 1 0 (1 92 1 ) : 39-79. Reprinted in

gen.

Mathematisch-physikalische

K/asse.

Nachrichten: 395-407 .

[Hilbert 1 9 1 8]

--

, "Axiomatisches Denken, "

of Science, ed . M . Heidelberger and F.

Stadler.

Dordrecht:

Kluwer,

2002,

pp.

225-237.

[Weyl 1 968], vol . 2 , pp. 1 43-1 80. [Weyl 1 944] Hermann Weyl, "Obituary: David Hilbert, 1 862- 1 943, " Obituary Notices of Fel­

Mathematische Annalen 7 9 (1 9 1 8): 405-41 5.

[Scholz 2000] Erhard Scholz, "Hermann Weyl

lows of the Royal Society 4 ( 1 944) : 54 7-553.

Reprinted in [Hilbert 1 932-35], vol . 3, pp.

on the Concept of Continuum," in [Hen­

Reprinted in [Weyl 1 968], vol. 4, pp. 1 20-1 29.

1 46-1 56.

dricks, Pederson, Jmgensen 2000], pp.

[Zerrnelo 1 908] Ernst Zermelo, "Neuer Beweis

[Hilbert 1 922]

--

,

"Neubegrundung der

Mathematik. Erste Mitteilung," Abhandlun-

70

THE MATHEMATICAL INTELLIGENCER

fUr

1 95-220. [Scholz 2001 a]

--

, Hermann Weyl's Raum-

die

M6glichkeit einer Wohlordnung, "

Mathematische Annalen 65: 1 07-1 28.

PAUL R. CHERNOFF

" Some of the People , AI of the Time" Crook, do you know I believe there are m en who want to take m:IJ life? And I have no doubt they will do i t . . I know no one could do it and escape alive. But if it is to be done, it is impossible to prevent i t. .

.

-Abraham Lincoln to bodyguard William H. Crook, April 14, 1865

he phone beeped insistently. Lisa Rentar, Professor of Experimental Mathematics at the University of California, Berkeley, reached over to answer it. ''Hello, Rentar here. Who is this? " "This is your wake-up call, Dr. Rentar. The time is 5: 30 A.M. " "Thank you, computer. Logout. " Lisa yawned and stretched. She had managed to get two hours of sleep, amazingly enough. She never suffered from insomnia-often enough she had insomnia, but she never suffered from it. As usual, she had used the wakeful hours to recheck her calculations one last time. And, as usual, there was no mistake. All was ready for the launch, sched­ uled in three hours at MRSI in the brown Berkeley hills. MSRI, the Mathematical Sciences Research Institute, fa­ miliarly known as "mis'ry" since its opening in 1982, had grown tremendously over the decades, and its large com­ plex of buildings now sprawled over 25 or 30 acres. (Early on, one Director had tried to get the pronunciation changed to "emissary," but "mis'ry" it remained.) Lisa's ancient red Jaguar tooled up Centennial Drive to­ ward the Hald Building, which housed the nerve center of Project Clio, the world's first time machine. Lisa Rentar had been awarded the Fields Medal-at the International Congress of Mathematicians in Kabul in 2022for her explicit solution of the finite Tipler model: a general­ relativistic universe with a long, but finite, massive rotating cylinder. Frank Tipler, the 20th-century "father of time travel," had shown in the early 1970s that associated with an infi­ nitely long rotating cylinder there were tirnelike geodesics connecting any two points of space-time outside the cylin­ der; in particular, there were retrograde geodesics. In other words, in that idealized but unrealizable situation, travel into the past was possible. It was much more challenging to deal

with the realistic case of a finite rotating cylinder, which Tipler suggested might exhibit similar phenomena and thereby function as a time machine. Proving this was extra­ ordinarily difficult. As was often true in mathematics, the in­ finite we can do now-the finite takes much longer. However, that was only half the problem. Rentar's ma­ jor breakthrough came when she succeeded in analyzing quantum effects. Numerical simulation using MSRI's femtoflop Ramanujan-1 729 hypercomputer showed that the creation of virtual Witten pairs in coherent Hawking radi­ ation made it possible to travel into the past beyond the time of creation of the finite Tipler cylinder. Admittedly, the probabilities were small, but another advance came with the replacement of the original cylinder model by a Fermat prime number of counter-rotating helices. That had won Rentar the Nobel, which, however, because of poor investments by the Bank of Sweden, was worth only about $600 U.S. Still, it bought her the University of California's highest honor: a free reserved parking space within sight of Evans Hall, the crenelated concrete cube which had housed Shing-Shen Chern, Ole Hald, Abraham Taub, and Frank Tipler himself: the eccentrically brilliant mathemat­ ical physicist had been one of Abe Taub's last postdoctoral students. As Taub is supposed to have remarked, "After Tipler, I don't need any more postdocs." Lisa was greeted at the main entrance of the Hald build­ ing by Dave Plankasky, the brilliant and beloved director

© 2003 SPRINGER·VERLAG NEW YORK. VOLUME 25. NUMBER 1. 2003

71

of MSRI, accompanied by Maurice Calvin Coolidge, the

cay, the cat should be in the state

jaunty, bow-tied nonagenarian Chair-for-Life of the Berke­

thing never observed by human beings, or, presumably, by

ley Mathematics Department, known as

MC2 for his

seem­

ingly boundless energy.

,alive > + ,dead > , some­

cats. Lisa, like many thoughtful scientists, had worried about this apparent split between the physics of microsystems and

"Well, Lisa, this is your big day-and Humankind's,"

that of macrosystems, but had never been able to resolve this

gushed Coolidge. "You bring honor upon Berkeley as well as

antinomy. The designers of the time machine, like most en­

yourself. I can't say how delighted I was that the Rational

gineers, happily ignored such foundational problems-and,

Science Foundation chose you for the mission instead of one

so far, they had gotten away with it.)

of those kid astronauts. Who better than the discoverer of

Finally, Lisa was sent back to spend two minutes in the

the mathematics underlying this mission? I am outraged that

year 2022, to emerge in her own mind and re-experience

some bureaucrats dared to say you're too old-why, you're

her joy and pleasure when she learned that she had been

barely half my age, and I wish I were going."

awarded the Fields medal.

No doubt Coolidge would have gone on and on, but Lisa

But now the serious mission had come, carefully se­

interrupted the flow. "Thanks for the kind words, Cool. I'm

lected by Clio's distinguished historians. The chosen date:

still amazed that a few squiggles on paper have turned into

Friday, November 22, 1963. The place: the Texas School

this massive mechanism. As Plato put it, God always

Book Depository in Dallas. The target: Lee Harvey Oswald.

geometrizes. And so do I."

It was 3 minutes before launch. Lisa planned to influence

Lisa strode down the main corridor of the Hald building.

Oswald's body to throw his Mannlicher-Carcano rifle from

There was a sharp odor of ozone in the air, coming from the

the Depository window, thus preventing what the Clio

terawatt Moore power drive, which had been switched on

group considered the greatest political tragedy of the past

at midnight to build up the morphogenetic field surrounding

century, the assassination of John Fitzgerald Kennedy.

the 17 rotating, 100-meter-long Tipler-Rentar helices, which

Millions shared the opinion of Clio's historians: Ameri­

resembled nothing so much as Bernini's beautiful columns

can politics had turned sour after the assassination of

in St. Peter's cathedral. The technicians strapped her into

J.F.K., largely because of the disastrous Vietnam war, and

the tiny capsule, then carefully wired the platinum circle sur­

the concomitant loss of faith in the veracity of the nation's

rounding her long, dark hair. Because the energy needed to

leaders. J.F.K.'s successor, Lyndon Baines Johnson, was, in

send a physical, massive object backwards in time was pro­

his own words, "not cut out to be Commander-in-Chief."

hibitive-comparable to several seconds of the solar out­

Worse, L.B.J. was followed by Richard Milhous Nixon, a

put-the machine worked by scanning the traveller's brain,

pathological liar. The course of American and world his­

then sending back the quantum wave function of her mind

tory had been grossly distorted. Johnson and Nixon-"two

along the retrograde geodesic. High energy was required to

ding-a-lings back to back," in the words of the conservative

keep the wave packet from spreading appreciably, or oth­

twentieth-century

erwise being distorted. The traveller's mind would then ma­

Many would amend this: Johnson was a tragic figure, while

terialize for a short interval-less than fifteen minutes in this

Nixon was no mere "ding-a-ling"-he was a psychopath.

case-in the brain of the target.

political

commentator

George

Will.

Now matters would be set right. Via the miracle of quan­

Project Clio's first experiments had shown that it was

tum amplification, the wave function of the universe would

possible to extend the lifetime of unstable particles, by con­

be caused to have evolved along a much happier path. Ac­

stantly pulsing their wave functions backwards in time.

cording to

Next, whimsical experimenters had placed the mind of a

Kennedy would serve two terms and be succeeded by two

cat 60 seconds earlier into the brain of a dog. It was most

normal men, Hubert Humphrey and Nelson Rockefeller, be­

the

best

estimates

of the

Clio historians,

amusing to see the dog eagerly lap up a bowl of milk But

fore the curvature of time returned to near its initial state

it drew protests from the Animal Liberation Front.

with a kinder, gentler Reagan-Bush administration. At least

(Naturally, this was known as the

Schrodinger's cat

e.:r­

that was the hope. It remained to be seen if such long-range

after the famous quantum mechanical "paradox."

intervention would work, but Lisa had every confidence

Although microsystems such as electrons and atoms can be

that it would. The calculations she had rechecked during

periment,

in any linear superposition of quantum eigenstates, large

her sleepless night showed that the probability of success

macroscopic systems such as a cat always seem to be in a

was at least 80 percent, and the probability of throwing his­

definite, well-defined state. Erwin Schrodinger had imagined

tory onto an even worse path than the existing one, was

the following demonic scheme: a live cat is sealed in a box

less than one part in a billion-a risk well worth taking. Af­

containing a radioactive atom whose unpredictable disinte­

ter all, Lisa mused, Manhattan Project scientists had cal­

gration would trigger a Geiger counter which, in tum, would

culated a certain tiny probability that the Trinity nuclear

close an electrical circuit, smashing a bottle of cyanide, killing

test might ignite a chain reaction in the Earth's atmospheric

the cat. The paradox was that the cat must always seem alive

nitrogen, but the chances of that were found small enough

or dead, while the controlling radioactive atom possesses

to dismiss. She was prepared to do the same.

more general quantum states, superpositions of "undecayed"

The final countdown had begun. Ten, nine (Lisa focused

and "decayed." Yet the state of the cat is completely corre­

her thoughts on Oswald) eight, seven, six. . . . Lights

lated with that of the radioactive atom, so that when the lat­ ter

72

is

in the state corresponding to a

THE MATHEMATICAL INTELLIGENCER

500Al

probability of de-

dimmed and air conditioners slowed all over the Bay Area as the Moore drive revved up for its final massive effort to

rocket the quantum wave function of Rentar's mind back into the past. She closed her eyes . . . Went through a moment of disembodied giddiness . . . , And, with a crushing headache, opened her eyes again "Tranquility base here; the Eagle has landed, " she mused groggily. But something was wrong. Instead of squatting with a rifle near a window, surrounded by cardboard cartons of books, she was standing in a dim alleyway near a brick building. As her mind cleared, she scanned the sky. It was evening, and in the distance was the black profile of a half­ completed enormous obelisk. On the alleyway wall of the building was posted a broadsheet: "Our American Cousin. The new comedy presented by the Edwin Booth acting troupe, April 1865. Ford's Theater." A wave of panic swept over her. Somehow the machine had focused on the wrong assassin, the wrong assassination. She reached into her breast pocket and pulled out a handkerchief, with the ini­ tials J.W.B. It was Good Friday, April 14, 1865, and Rentar was trapped in the mind of John Wilkes Booth. Her thoughts raced. She opined that there were two choices: to do nothing, or to try to prevent Booth from killing Lincoln. The former possibility went against her na­ ture. Mter all, why was she here? What had happened? There must be a reason for this pe­ culiar change in destination . . . and it wasn't the McKinley assassination or that of Garfield, after all. Lisa did not be­ lieve in God, but she did believe in the laws of Nature, and she believed that the wave function of the Universe carried mysteries beyond the ken of humankind. Because she had been given the opportunity, she would act. After all, a long string of weak or bad Presidents had succeeded Lincoln. Perhaps worst, in 1876 Rutherford Hayes, the Republican, had made a filthy deal for Florida's electoral votes in return for his promise to end Reconstruc­ tion. America would have been better off, she thought, with­ out Andrew Johnson, without the corrupt Grant administra­ tion, and without the venal Hayes. Better far for America's greatest President, Abraham Lincoln, to serve out his sec­ ond term, "with malice toward none; with charity for all." This was all very well, as sentiment, Lisa reflected as Booth's hand pushed open the stage door entrance. But what would the cool-minded Clio historians have advised? They had studied all the political assassinations in U.S. history, and the decision had been to fix on the Kennedy assassination. Besides, it was only seventy years in the past, not one hun­ dred and seventy. The time warp equations were highly non­ linear, and the Ramanujan-1729 had not been able to handle a serious computation concerning the effect of Lincoln's sur­ vival. As often in human affairs, Lisa reflected, calculation had to be superseded by judgment. And it was her judgment that the world would be better if Lincoln lived. It would be very simple. She concentrated hard, and Booth turned back towards the alley door. But he did not take a step. Against Lisa's will, he turned again, and began climbing the flight of stairs leading to the balcony. Lisa fought with all her energy, but apparently at a distance of 170 years from the

power source, the interaction Hamiltonian intertwining her mind with Booth's neurons was too small-and there was no way to boost it. She would have to try for a smaller action. Maybe she could get Booth to drop his pistol, maybe she could force him to croak out a warning yell. Nothing happened. Booth approached the entrance to the curtained Presi­ dential box. Sure enough, the sole bodyguard was missing from his post. John Wilkes Booth drew his pistol, a rictus distorting his face. His right hand trembled a bit, but that was all. He re­ mained silent. Lisa was desperate, but it appeared she could do nothing. Now Booth was in the box. A scream flashed through Lisa's mind as Booth fired point-blank into the back of Lincoln's noble head. Booth leaped over the balcony rail. Lisa struggled, but she managed only to catch Booth's spur in the large flag over the stage. Pain flashed as he landed, breaking his leg, just above the ankle. "Sic semper tyrannis!" Booth shouted. "Virginia is avenged!" He hobbled off the stage before the panicked actors and the stunned audience could do anything, rushed into the al­ ley, and painfully mounted his waiting horse. "Schrodinger's cat," Booth muttered as he rode into the dim Washington street. "Schrodinger's cat-now why should I think of a German cat?" he said, then vanished into history. REFERENCES

Tipler, Frank J . , Rotating cylinders and the possibility of global causal­ ity violation. Phys. Rev. 0(3) 9 (1 974), 2203-2206. MR 53 # 1 2496 AU T H OR

PAUL R. CHERNOFF

Department of Mathematics University of California Berkeley, CA 94720-3840 USA e-mail: [email protected]

Paul R. Chernoff, a Harvard Ph.D., has been at the University of California since 1 968. His field is unbounded operators and quantum mechanics. He has won his department's Distin­ guished Teaching Award several times. Hobbies include car­ tooning and writing limencks. (Formerly some of his limericks perturbed prudish visitors to his home page; a few of the tamer ones are still posted there.)

VOLUME 25. NUMBER 1, 2003

73

lil§ih4'·'?J

Osmo Pekonen ,

Editor

1

Noeuds: Genese d'une Theorie Mathematique by Alexei Sossinsky PARIS: EDITIONS DU SEUIL PAPERBACK 1 48 p p . :

<18.3,

1 999 ISBN

2-02-032089-4

REVIEWED BY OSMO PEKONEN

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: Agora Center, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: pekonen@mat h .jyu.fi

here are many good books on knots, perhaps too many. It is hardly possible any more to have read everything in the field. And yet there definitely still is room for the lovely book Noeuds: Genese d 'une Theorie

T

Mathematique (Knots: Genesis of a Mathematical Theory) by Alexei Sossinsky. This little gem of a book­ only 148 pages-starts from scratch, builds up many of the central concepts of knot theory through a historical ac­ count, and opens vistas onto the very latest research. Sossinsky, who is a member of the Russian Academy of Sciences, has a triple intellectual heritage: French, American, and Russian. He was born in Paris in 1937, studied in Paris and New York, then in Moscow where his thesis on knot theory was accepted in 1965. He is the author of several books and about 50 articles, and a collaborator of the magazine Kvant. His style seems to combine French clarity, American in­ formality, and Russian pathos. Sossinsky's book is written in a light conversational style. He starts with con­ crete examples and illustrates his state­ ments throughout the book with a total of 64 graphics, which are thoughtfully prepared and neatly executed. Applica­ tions of knot theory in biology, chem­ istry, and physics are discussed. Several calculations are concretely done to il­ lustrate the meaning of increasingly so­ phisticated knot invariants, but the reader never gets lost in the details. Al­ most without noticing, the reader, who may be an absolute novice to begin with, is gently guided ever deeper into the mysteries of knot theory. As we are

reassured by the author, the book can be read without studies at the Ecole Normale Superieure. The book is organized into eight chapters. It could be read as a collection of eight almost independent essays. Each of the first seven centers on some grand figure of knot theory whose con­ tributions were vital to the unfolding of the story. Sossinsky has chosen Lord Kelvin, James Waddell Alexander, Kurt Reidemeister, Horst Schubert, John Hor­ ton Conway, Louis Kauffman, and Vic­ tor Vassiliev as his heroes. Louis Kauff­ man could have equally well been replaced by Vaughan Jones, as the au­ thor admits, but he finds Kauffman's in­ terpretation of the Jones polynomial via statistical mechanics more accessible than Jones's original idea. A major omis­ sion in the early part of the story seems to be Henri Poincare; but the author does not want to spend time discussing homotopy groups, which do not pro­ duce particularly powerful invariants for knots. The eighth chapter is speculative: the respective grand innovator of the 21st century is merely identified as Xxx. Lord Kelvin's imaginative view of knots as entangled vortex tubes of ether is often cited as a school exam­ ple of a scientific flop. Yet, in its day, well before Mendeleev's periodic table of the elements, it appeared credible, and, according to Maxwell's verdict, satisfied "more of the conditions than any atom hitherto considered." Atiyah [ 1 ] summarizes the arguments in favor of Kelvin's theory as follows: 1. Stabi l ity. The stability of matter might be explained by the stability of knots (i.e., their topological nature). 2. Variety. The variety of chemical el­ ements could be accounted for by the variety of different knots. 3. Spectrum. Vibrational oscillations of the vortex tubes might explain the spectral lines of atoms. From a modern point of view, he adds a fourth argument: 4. Transmutation. The ability of atoms to change into other atoms at

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25. NUMBER 1. 2003

75

high energies could be related to cut­ ting and recombination of knots. Much more than a flop, Kelvin's atom theory sparked the development of knot theory in 19th-century Scotland. The first knot tables were compiled by Peter Guthrie Tait and Rev. Thomas Penyngton Kirkman. Their classifica­ tion of so-called alternating knots up to 10 crossings in the knot projection was printed in 1885, and, remarkably, it has been found to be completely correct. In our day, the knots have been completely classified up to 16 cross­ ings; there are 1 70 1 936 different ones (Hoste, Thistlethwaite, Weeks [2]). In the second movement of Sossin­ sky's symphony, braids are introduced. We are reminded of Alexander's theo­ rem of 1923: Each knot can be obtained as the closure of some braid. Vogel's braiding algorithm is described with an amusing Russian touch: we learn what kind of upheavals a perestroika may bring about in a heteroclitic empire . . . . In the third movement, Reidemeis­ ter's dream of untying a knot is con­ fronted with the Gordian knot due to Wolfgang Haken, and the reader is tempted to seize his or her sword in the manner of Alexander the Great. The fourth movement is Schubert­ ian. It ends with a line that one would like to sing in chorus with the exultant author: "le theoreme de Schubert n'a

pas besoin de corollaires, c'est de l'art pour l 'art de la plus haute volee. " In the fifth movement, John Horton Conway enters the scene to perform his aria of 1973. Some explicit exam­ ples of Conway polynomials are evalu­ ated. Then the choir of Hoste, Oc­ neanu, Millet, Freyd, Lickorish, Yetter, Przytycki and Traczik (abbreviated HOMFLYPT) joins in to produce even more polynomial invariants. But they still cannot distinguish between Die beiden Kleeblattschlingen (already dis­ cussed by Max Dehn in 19 14). Enter Vaughan Jones, movement six. But, as already noted, the author prefers to let Louis Kauffman perform his score. The proof of the invariance of the Kauffman brackets is compared by the author to the hymn An Die Preude of Schiller and Beethoven:

76

THE MATHEMATICAL INTELLIGENCER

Freude schoner Gotterfunken, Tochter aus Elysium, Wir betreten Feuertrunken, Himmlische, dein Heiligtum . . . And then we are made to believe that Kauffman's famous trick of introduc­ ing the writhe is already discernible in the designs of some 6000-year-old Celtic monoliths. The seventh movement is Russian. Infinitely many knot invariants of in­ creasing orders due to Victor Vassiliev, a student of Vladimir Arnold, are in­ troduced. According to a theorem of Maxim Kontsevich, the dimensions of vector spaces of Vassiliev invariants of order n can be computed to be 1, 0, 1, 1 , 3, 4, 9, 14, 2 7, 44, 80, 132, 232 for n 0, 1 , 2, 3, . . . , 12. The next three val­ ues are conjectured to be 383, 657, 1088 (Dror Bar-Natan, Harvard University). The final eighth movement, which is devoted to mathematical physics, has been composed in a more speculative mood. The author conjectures the ex­ istence of a deep relationship between the Artin relation of braid groups, the second Reidemeister move, and the equations of Yang and Baxter. Some names mentioned in the last chapter are the Fields medalists Vladimir Drin­ feld, Vaughan Jones, Maxim Kontse­ vich, and Edward Witten. The peren­ nation of the thought of Carl Friedrich Gauss emerges as a leitmotiv. Inspired by his study of electromagnetism, Gauss wrote down on the 22nd of Jan­ uary 1833 a beautiful double integral which yields the linking number of a pair of knots. The celebrated Kontse­ vich integral is a generalization, but its physical interpretation remains to be understood. I liked Sossinsky's book so much that I translated it into Finnish. The Finnish translation was titled Solmut: =

Eraiin matemaattisen teorian synty (Art House, Helsinki, 2002). Also an English translation from Harvard Uni­ versity Press (Knots: Mathema tics With a Twist) is in preparation by Giselle Weiss. Now, it is not customary to advertise one's own translations, but I do so with a purpose in mind. Indeed, I am a member of the committee for Raising Public Awareness of Mathe-

matics of the European Mathematical Society. Europe is a place of several small language areas, and the language barriers constitute an additional im­ pediment to the diffusion of mathe­ matics. Very few popular books on modem mathematics are available in the smaller European languages, and people prefer to pick only books pub­ lished in their own language for their leisure reading. Overcoming this diffi­ culty is a challenge for our committee. My translation, for instance, is the only book on the mathematical theory of knots available in Finnish, so I had to fix the entire Finnish terminology of the field. For those interested in curiosities, let me report that 'skein' (as in 'skein re­ lation') will be 'vyyhti' in Finnish, which is a literal translation. Amusingly, the English word 'skein' is used as such in knot theory books in several languages because translators assumed there was a mathematician named Skein! On the other hand, there is no unambiguous Finnish equivalent for the simple word 'flip,' so we had to use the English word Gust as in French). To introduce modem mathematics in a small language area like Finland, I have outlined the following policy: Translate one carefully selected popu­ lar book from every field-or, if you can, write one of your own. Carefully fix the vocabulary. Think of encultura­ tion (for instance, I found it useful to tone down a little bit the Slavic pathos of Sossinsky because the prevailing Finnish attitude towards mathematics is much more prosaic). Supplement your translation with an updated and extensive bibliography (besides tech­ nical books in English, I included sev­ eral books on the practical art of tying knots available in Finnish). The payoff: In Finland, books on mathematics often get reviewed in the literature columns of newspapers. Thus, in the field of knot theory, I warmly recommend Sossinksy's book to be translated into many more lan­ guages. However, a final word of warn­ ing is necessary. The proof-reading of Sossinsky's manuscript at Seuil has been sloppy. Several of the crucial for­ mulas are marred by tiny errors of sign conventions and other notational in-

congruities. The bibliography does not quite match with the text. And what do we make of the following "sequence of primes" which appears on page 80: 2, 3, 5, 7, 9, 1 1 , 12, 13, 1 7, 19, 23, 29, 31, 37, 4 1 , 43, 53, . . . As an act of penitence, Seuil should re­ vise and reprint the edition. REFERENCES

[1 ] Michael Atiyah . The Geometry and Physics of Knots, Cambridge: Cambridge University

Press, 1 990. [2] Jim Haste, Morwen Thistlethwaite, Jeff Weeks, The First 1 701 936 Knots, Math­ ematical lntelligencer 20 (1 998), no. 4,

33-48.

Fermat's Last Tango, a Musical MUSIC BY JOSHUA ROSENBLUM, BOOK BY JOANNE

SYDNEY LESSN E R , LYRICS BY LESSNER & ROSENBLUM,

A STAGE PRODUCTION FOR THE YORK THEATRE

COMPANY; VIDEOCASSETTE AND DVD PRODUCED BY THE CLAY MATHEMATICS INSTITUTE, CAMBRIDGE,

MA, 2001

REVIEWED BY MICHELE EMMER

n December 1996 Joshua Rosenblum read a review of Amir Aczel's book Fermat's Last Theorem [ 1 ] in The New York Times. He showed it to Joanne Sydney Lessner. "Is there a musical in here?" Lessner's answer was a definite yes. In spite of being math­ phobic, she immediately saw a com­ pelling detective story inherent in this three-hundred-year-old search for a proof. This was the birth of the idea of making a musical on Andrew Wiles's adventure and the proof of Fermat's theorem. In 1996, Simon Singh had made a film on this proof, and a few months later he published a book by the same title [2], [3] . Singh's idea, followed by John Lynch, the BBC producer of the series "Horizon" [4] , [5], was that the story of Wiles and his demonstration could be­ come great cinema. It was an adven­ ture in human emotion, loaded with un­ expected outcomes, a real drama. The success of Singh's film and

I

book was a necessary precondition to making a musical on the Wiles story. Many people knew the film and the book, so the musical's audience could easily follow the story. Joanne Lessner is the wife of pi­ anist, composer, and orchestra direc­ tor Joshua Rosenblum. She herself had written scripts and songs for other mu­ sicals. What did they want out of a mu­ sical called Fermat's Last Tango? In the author's notes in the video on the show (financed by The Clay Mathe­ matics Institute, an institute dedicated to increasing and disseminating math­ ematical knowledge), Lessner writes, "Our aim with Fermat's Last Tango is, above all, to entertain. We think it's a moving, timeless story that benefits from being told in a whimsical, some­ times irreverent fashion. If, after see­ ing it, you're inspired to take a crack at trying to find Fermat's simple, ele­ gant proof yourself, go right ahead." If even biography and film are nec­ essarily fiction-in that the author de­ cides what to include, emphasize or ne­ glect-a musical need not be judged by its historical truthfulness nor by the ac­ curacy of its references to so-called real life. Nevertheless, unlike most musical authors, the authors researched their subject. They had a head-start. Singh's film gives all the elements of the story, from the Shimura-Taniyama conjec­ ture to elliptic curves. Lessner and Rosenblum make the good choice of bringing in other fa­ mous mathematicians to form with Fermat a kind of chorus. As in Greek tragedy, they comment and intervene. They are the guardians of a mathe­ maticians' paradise which Keane (Wiles's name in the musical) would like to reach. They are Carl Friedrich Gauss (interpreted by Gilles Chiasson), Pythagoras (Mitchell Kantor), Euclid (Christianne Tisdale, a woman), Sir Isaac Newton (another woman, the tal­ ented Carie Wilshusen). The main char­ acters are obviously Daniel Keane (Chris Thompson) and Pierre de Fer­ mat (a very convincing Jonathan Rabb ). Anna Keane, the wife, is played by Edwardyne Cowan. The stage set is simple, the decor

down to a bare minimum: chairs, big numbers, a few books. The musical starts with an intro­ duction to the problem, as if it were a scientific foreword. Journalists are mobbing Keane. It is the evidently much-awaited moment of glory. They cannot help but comment, "He must be crazy"-a recurrent idea in films and books about mathematicians. Among other things, the journalists inquire about the meaning of the theorem. In Singh's film the students repeat a lim­ erick on Pythagoras's theorem. In the musical, naturally, they sing it. In the song "The Beauty of Num­ bers," Keane regrets having finished the proof-as in the film a Princeton mathematician asks who will ever pro­ vide a comparable problem. It is fine theatre. Some of the lines are good, even from a mathematical standpoint: "Perfection like pi and the golden sec­ tion." In an amusing song, Fermat makes fun of the complexity of Wiles's proof. Keane-Wiles come back with "Show me your marvelous proof." Fermat refuses: "My lips are sealed." Keane: "The world deserves to see." Fermat: "But if the mystery were re­ vealed I would lose my immortality; I take the answer to my death. " Keane has the last word: "You are al­ ready dead." Along with some of the quartets sung by the famous mathematicians, the Keane-Fermat duets are among the best parts of the musical. Keane's en­ counter with the famous four is also entertaining. Each of the four pro­ claims his own superiority. Keane rec­ ognizes a happy and grateful Gauss who exclaims: "Finally!" Then Euclid­ yes, the Euclid-is introduced to him. Toward the end, when Keane thinks he has secured a place among the famous mathematicians, Fermat, who is still keen on deflating him, tells him that in his proof there is a "big fat hole." Fer­ mat is brilliant, ironic. The mathemati­ cians burst out laughing. Fermat sings, "No one will share my fame. It is the only thing I have." In the most amusing scene of the musical, Keane is then scorned because, at almost 40, he is too old-too old to be considered for the

VOLUME 25, NUMBER 1 , 2003

77

congruities. The bibliography does not quite match with the text. And what do we make of the following "sequence of primes" which appears on page 80: 2, 3, 5, 7, 9, 1 1 , 12, 13, 1 7, 19, 23, 29, 31, 37, 4 1 , 43, 53, . . . As an act of penitence, Seuil should re­ vise and reprint the edition. REFERENCES

[1 ] Michael Atiyah . The Geometry and Physics of Knots, Cambridge: Cambridge University

Press, 1 990. [2] Jim Haste, Morwen Thistlethwaite, Jeff Weeks, The First 1 701 936 Knots, Math­ ematical lntelligencer 20 (1 998), no. 4,

33-48.

Fermat's Last Tango, a Musical MUSIC BY JOSHUA ROSENBLUM, BOOK BY JOANNE

SYDNEY LESSN E R , LYRICS BY LESSNER & ROSENBLUM,

A STAGE PRODUCTION FOR THE YORK THEATRE

COMPANY; VIDEOCASSETTE AND DVD PRODUCED BY THE CLAY MATHEMATICS INSTITUTE, CAMBRIDGE,

MA, 2001

REVIEWED BY MICHELE EMMER

n December 1996 Joshua Rosenblum read a review of Amir Aczel's book Fermat's Last Theorem [ 1 ] in The New York Times. He showed it to Joanne Sydney Lessner. "Is there a musical in here?" Lessner's answer was a definite yes. In spite of being math­ phobic, she immediately saw a com­ pelling detective story inherent in this three-hundred-year-old search for a proof. This was the birth of the idea of making a musical on Andrew Wiles's adventure and the proof of Fermat's theorem. In 1996, Simon Singh had made a film on this proof, and a few months later he published a book by the same title [2], [3] . Singh's idea, followed by John Lynch, the BBC producer of the series "Horizon" [4] , [5], was that the story of Wiles and his demonstration could be­ come great cinema. It was an adven­ ture in human emotion, loaded with un­ expected outcomes, a real drama. The success of Singh's film and

I

book was a necessary precondition to making a musical on the Wiles story. Many people knew the film and the book, so the musical's audience could easily follow the story. Joanne Lessner is the wife of pi­ anist, composer, and orchestra direc­ tor Joshua Rosenblum. She herself had written scripts and songs for other mu­ sicals. What did they want out of a mu­ sical called Fermat's Last Tango? In the author's notes in the video on the show (financed by The Clay Mathe­ matics Institute, an institute dedicated to increasing and disseminating math­ ematical knowledge), Lessner writes, "Our aim with Fermat's Last Tango is, above all, to entertain. We think it's a moving, timeless story that benefits from being told in a whimsical, some­ times irreverent fashion. If, after see­ ing it, you're inspired to take a crack at trying to find Fermat's simple, ele­ gant proof yourself, go right ahead." If even biography and film are nec­ essarily fiction-in that the author de­ cides what to include, emphasize or ne­ glect-a musical need not be judged by its historical truthfulness nor by the ac­ curacy of its references to so-called real life. Nevertheless, unlike most musical authors, the authors researched their subject. They had a head-start. Singh's film gives all the elements of the story, from the Shimura-Taniyama conjec­ ture to elliptic curves. Lessner and Rosenblum make the good choice of bringing in other fa­ mous mathematicians to form with Fermat a kind of chorus. As in Greek tragedy, they comment and intervene. They are the guardians of a mathe­ maticians' paradise which Keane (Wiles's name in the musical) would like to reach. They are Carl Friedrich Gauss (interpreted by Gilles Chiasson), Pythagoras (Mitchell Kantor), Euclid (Christianne Tisdale, a woman), Sir Isaac Newton (another woman, the tal­ ented Carie Wilshusen). The main char­ acters are obviously Daniel Keane (Chris Thompson) and Pierre de Fer­ mat (a very convincing Jonathan Rabb ). Anna Keane, the wife, is played by Edwardyne Cowan. The stage set is simple, the decor

down to a bare minimum: chairs, big numbers, a few books. The musical starts with an intro­ duction to the problem, as if it were a scientific foreword. Journalists are mobbing Keane. It is the evidently much-awaited moment of glory. They cannot help but comment, "He must be crazy"-a recurrent idea in films and books about mathematicians. Among other things, the journalists inquire about the meaning of the theorem. In Singh's film the students repeat a lim­ erick on Pythagoras's theorem. In the musical, naturally, they sing it. In the song "The Beauty of Num­ bers," Keane regrets having finished the proof-as in the film a Princeton mathematician asks who will ever pro­ vide a comparable problem. It is fine theatre. Some of the lines are good, even from a mathematical standpoint: "Perfection like pi and the golden sec­ tion." In an amusing song, Fermat makes fun of the complexity of Wiles's proof. Keane-Wiles come back with "Show me your marvelous proof." Fermat refuses: "My lips are sealed." Keane: "The world deserves to see." Fermat: "But if the mystery were re­ vealed I would lose my immortality; I take the answer to my death. " Keane has the last word: "You are al­ ready dead." Along with some of the quartets sung by the famous mathematicians, the Keane-Fermat duets are among the best parts of the musical. Keane's en­ counter with the famous four is also entertaining. Each of the four pro­ claims his own superiority. Keane rec­ ognizes a happy and grateful Gauss who exclaims: "Finally!" Then Euclid­ yes, the Euclid-is introduced to him. Toward the end, when Keane thinks he has secured a place among the famous mathematicians, Fermat, who is still keen on deflating him, tells him that in his proof there is a "big fat hole." Fer­ mat is brilliant, ironic. The mathemati­ cians burst out laughing. Fermat sings, "No one will share my fame. It is the only thing I have." In the most amusing scene of the musical, Keane is then scorned because, at almost 40, he is too old-too old to be considered for the

VOLUME 25, NUMBER 1 , 2003

77

[4] Simon Sing h , L'ultimo teorema di Fermat: il

Fields medal. The chorus sings, "Math­

dictable. Keane comes to Fermat's de­

ematics is a young man's game."

fense, saying he was a great mathe­

racconto di scienza del decennia, in M. Em­

matician and surely had a proof. That's

mer, ed. , Matematica e Cultura 2, Springer­

Keane is desperate. Like the real-life Wiles, he takes refuge in his family un­

not what the real Wiles says

in

Singh's

Verlag ltalia, Milano (1 999), p. 40-43.

film. On balance, though, the gamble of

[5] John Lynch, Alcune riflessioni sulla con­

Singh's film, where the

trying to produce an entertaining and

struzione del fil m , in M. Emmer and M. Man­

true Wiles is shown weeping with emo­

mathematically correct musical turned

aresi, eds . , Matematica, arte, tecnologia,

tion even six months later.

out a success.

til the enlightenment finally arrives. Dramatic as

in

cinema,

"You will always be there/ I think I will be a better husband/ It is often

REFERENCES

said, within your failure lie the seeds

[1 ] Amir Aczel, Fermat's Last Theorem, Four

of your success." This last phrase, by Keane's wife, cues an association al­

ltalia,

Milano

pear.

Walls Eight Windows. New York, 1 996. [2] Simon Singh ,

lowing him to complete the proof. Not a very convincing plot gimmick The

Springer-Verlag

(2002), p. 266-267; English edition to ap­

Fermat's Last Theorem,

1 997. --

[3]

press conference ending is also pre-

Dipartimento di Matematica Universita Ca' Foscari Dorsoduro 3825/E, Ca' Dolfin

& John Lynch, Fermat's Last Theo­

rem, Video, BBC (1 996).

301 23 Venice, Italy e-mail: [email protected]

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78

THE MATHEMATICAL INTELLIGENCER

1?1fi,I.C*i·h•i§i

Robin Wilson

Pedro Nunes, Portuguese Mathematician and Cosm ographer Nuno Grato

ive stamps commemorate Pedro Nunes ( 1502-1578), the foremost Portuguese mathematician and cos­ mographer during the Age of Discovery. Nunes wrote extensively on cosmogra­ phy, spherical geometry, astronomy, navigation, and algebra, and in his time was one of the most well-known math­ ematicians and scientists in Europe. Two of the stamps were issued in 1978 and commemorate the 400th anniver­ sary of his death; the other three were issued on 6 March 2002, the 500th an­ niversary of his birth. His two important discoveries pre­ sented on the stamps shown here are the loxodromic curve and the nonius. Loxodromic curves on a globe, also called rhumb lines, are spirals that con­ verge to the poles. They are lines that maintain a fixed angle with the merid­ ians: a ship following a fixed ideal com­ pass direction (other than the north­ south and east-west directions) travels along a loxodromic curve. Nunes dis­ covered the loxodromic lines and ad­ vocated the drawing of maps in which loxodromic spirals appear as straight

F

lines. This led to the celebrated Mer­ cator projection that is constructed on these principles. Nonius scales are concentric scales that allow a precise reading of the height of stars on a quadrant. They were used and perfected by Tycho Brahe, Ja­ cob Kurtz, and Christopher Clavius, among others. In 1630, following their improvements, Pierre Vernier con­ structed a device with a sliding scale over a fixed one, which proved to be the most practical nonius scale and is still in use today. For some centuries, this device was called a nonius, but during the 19th century, especially in France, it began to be called a vernier. The in­ strument with nonius scales repro­ duced on one of the stamps is now in the Museum for the History of Science in Florence. No portraits or paintings of Pedro Nunes survive. Department of Mathematics, ISEG Technical University of Lisbon R. Miguel Lupi 20 1 200, Lisbon, Portugal e-mail: : [email protected] . pt

Commemoration of the 500th Anniversary of the Birth of Pedro Nunes

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

The Open University, Milton Keynes, MK7 6AA, England e-mail: r.j .wilson@open .ac.uk

80

Non ius

T H E MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG N E W YORK

Pedro Nunes