No title

i:C§ifJ

H u man ism and H u man R espo nsib i I ity MARIA GEORGIADOU

ith the text that follows, the editorial board of Mathema­ tische Annalen, the most renowned mathematical journal of that epoch, announced Constantin Cara­ theodory's death in volume 1 2 1 of 1949/50. Constantin Caratheodory died in Mu­ nich at the age of 77, on 2 February 1950. The news of his death will be received with sorrow in the whole in­ tellectual world. With his multifaceted brilliant achievements in mathematics and neighbouring disciplines, he had become an unforgettable master of our science. With his breadth of learn­ ing such as graces the noblest hu­ manist spirits, with his natural patri­ cian kindness, and with his surprising skill in mastering so many languages as his mother tongue, he was a Eu­ ropean in the best possible sense of this word. In addition, he was in the fortunate position of using the high standing he enjoyed within intellec­ tual circles of so many countries to create strong bonds of cooperation with mathematicians of many nations. He was active for more than four decades as a collaborator of the Math­ ematischen Annalen. During the dif­ ficult period of World War I he was responsible for editing it. His initiative and his sound advice were invaluable to the board. He helped us keep the gates open to the world in critical times. He was still contributing to the editing of the most recent issues. This was a time when German aca­ demic institutions, after their denazifica­ tion, were in a process of reconstruction. An objective of this reconstruction was to restore Germany's position within the in­ ternational academic community. It is thus not by chance that, in the above obitu­ ary, emphasis is put on Caratheodory's standing as an international scholar and mediator between cultures. More, his ca­ reer is presented in terms of an uninter­ rupted progression, and there is no com­ ment on the Nazi era, a period when any person, especially a person of hu­ manistic culture, was confronted with es­ sential questions of political and moral responsibility. In January 1 95 1 , one of Cara-

theodory's friends and colleagues at the University of Munich insisted on a more honest examination of the political atti­ tude of German academics during the Third Reich. The mathematicians of the Bavarian Academy of Sciences were about to elect new academicians. In a three-page letter to this group , Oskar Perron put the mourning of Cara­ theodory to one side and focussed on human responsibility: Dear colleagues, Since the reopening of the Acad­ emy in 1946, I have each year taken the view that the time has not yet come to elect foreigners as corre­ sponding members. Unfortunately, things develop so slowly that even to­ day I am not yet willing to say that the waiting time has ended. I would like here to explain the reasons; and if I cannot on principle support the proposed elections, I ask you at the same time not to see this as a dig at either the proposing or the proposed [members], and not to take it as an ar­ gument against van der Waerden, whom I proposed and whom I do not count as a foreigner, though certainly you esteem van der Waerden just as highly as I esteem the gentlemen named by you. I hope that even if you do not share my point of view, you will still understand and respect it. The Dutch mathematician Bartel L. van der Waerden, editor of val. 1 2 1 of Mathematische Annalen, had been up to 1945 a full professor of mathematics at the University of Leipzig. After the war, in 1948, he became professor at the University of Amsterdam. In 1 9 5 1 he took up a full professorship at the Uni­ versity of Zurich and was proposed as a corresponding member of the Bavar­ ian Academy of Sciences. In his letter, Perron went on to say that, despite friendly contacts with in­ dividuals and organisations abroad, Germans as a whole were still looked at with a certain contempt, or at best with condescension. He was, on prin­ ciple, against naming foreign members to the Academy as long as the Germans were considered by other nations as collectively guilty. Perron wished to avoid the possibility of a foreign scien-

Maria Georgiadou is the author of Constantin Caratheodory: Mathematics and Politics in Turbulent Times (Springer-Verlag, 2004)

© 2007 Spnnger Science + Business Media, Inc., Volume 29, Number 4, 2007

5

tist refusing membership, or of the blocking of an election by the Ameri­ can Military Government, then politely styled the High Command. He empha­ sized that he would feel such a rejec­ tion as shameful. This does not mean that Perron did not accept the burden of guilt that fell upon the academicians. On the contrary: And now, about the emigrants! It is before them especially that we can­ not present ourselves as an illustrious company. For we have let them all down. We should not try to elevate ourselves before them, but rather crawl away and hide in shame. I feel personally ashamed myself, and that is my point. This is why I advocate utmost restraint and modest flourish­ ing in seclusion. For, what did we, what did the Academy do during the Third Reich to save the persecuted scholars? Nothing. We adopt the com­ fortable excuse that we didn't have the power, that we were incapable of it. In reality, however, we failed to make any serious attempt. Every one of us simply shrank from the risk, per­ haps before the conflicting ethical motive of looking out for his own family. We were no heroes. But the emigrants had, for the most part, no

choice at all; they had to take the risk together with their families and they suffered infinitely. This is why they stand, in the world's esteem, a con­ siderable step above us, who silently let it happen. Hermann Weyl, to be sure, is willing to forget (not only be­ cause as a star of the highest order he got through relatively well, but also because he is a generous soul); he has shown by coming here that he wishes well to all of us. But I still do not want to magnify myself in front of him, and I'm afraid that for instance Einstein, his neighbour and colleague, will say to him, "Oh, I see, the Bavarian Academy! They have nothing to be so conceited about; I offer condolences. By the way, those gentlemen already invited me too, but I smacked them in the face. " This whole tragic situation can be fun damentally changed only when Germany has full recognition and equality in the world again. Only then will the German academies also regain their old standing. No one can predict how long this will take; perhaps today's heavily burdened generation (the burden, by the way, is not only on us) will have to die out first.

For van der Waerden, things are entirely different. He spent most of his scientific life in Germany, so in particular he experienced with us, to a certain extent as a fellow prisoner, the Third Reich at its source. For me he does not count as a foreigner, any more than, for instance, Caratheodory. We also do not need to feel shame in his presence. He did what we all did, that is, detest Nazism and keep his clenched fist in a sack. That is why he under­ stands us and we understand him. This is a quotation from Oskar Perron's letter, typed in German, of January 195 1 (Deutsches Museum. Archiv NL 89, 012), published in vol. 2, pp. 647-649 of Eckert, Michael, and Marker, Karl (eds.),

Arnold Sommeifeld-- Wissenschaftlicher Briefwechsel (Berlin, Diepholz, Munchen,

Deutsches Museum, Verlag fur Geschi­ chte der Naturwissenschaften und der Technik, 2000, 2004). The letter was translated by Spyridon Georgiadis; the obituary was translated by the author.

Schlosserstr. 1 48 D-70 1 80 Stuttgart Germany e-mail: heuristic@s. netic.de

Erratum In The Mathematical Intelligencer, val. 29, no. 2, the article The Hexagonal Parquet Tiling, k-Isohedral Monotiles with Arbitrarily Large k, by Joshua E. S. Socolar, appeared on pp. 33-38. Inadvertently it was not the complete and final version. The final version is posted on Marjorie Senechal's website, http://math.smith.edu/-senechal/. -The Editors

6

THE MATHEMATICAL INTELLIGENCER

The AM-G M I neq ua l ity MICHAEL D. H IRSCHHORN

The AM-GM (

=

arithmetic mean-geometric mean) inequality, (X I+...+ x, )"

n

2: XI ... Xn

(all Xk> 0)

is notoriously difficult to prove. Let me present a simple proof. Observe that

(n + l)x+ n = (X- 1)2(xn-I + 2x"-z

xn+l -

Now suppose x1,

•

•

·

,

Xn+I >

0, and let a=

·

·

·

X] + . .. + Xn+ I

+ n) 2: 0 for X> 0.

, b=

n+ 1

X] + ... + Xn

We have: (a)n+I -

b

( n + 1)

(a)

b

+n

2:

n

and x=

a

b

.

0,

or, an+I

or, (X ]+.

•

•

2:

((n + 1)a- nb) b11,

+ Xn+I)n+I >

n+ 1

-

Xn+I

(X]+

•

�· + Xn )n·

The AM-GM inequality follows by induction on n. School of Mathematics and Statistics, UNSW Sydney 2052, Australia e-mail: [email protected]

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4 , 2007

7

A M i l l io n­ do l lar Proof AARON ABRAMS

To joe Gallian: Happy 65'h birthday!

came up with the main theorem of this paper during the summer of 1992 while I was an undergraduate student at Joe Gallian's REU at the University of Min­ nesota, Duluth. Until recently, I neither shared it with anyone nor fully ap­ preciated its value. But now Fermat's Last Theorem has a proof, and it seems the Poincare Conjecture does too, and with ever more attention directed toward prob­ lems like these, I thought I'd better get this in print. Here is the theorem:

I

THEOREM

1

N=

1.

The proof i s based on what I thought a t the time was a surprising observation, namely the following identity:

(1) Now, this is almost surely false, since for one thing the terms on the right are larger than the corresponding terms on the left. Nonetheless there are many proofs of this result, most involving evaluating both sides of the equation. (In most proofs I 've seen, the common value is 2.) In fact, one need not evaluate either side to prove the result; moreover, one needs no words:

.........

· · ·-

1/16 ···

1/16

1/8

· · · ···

· · · · ·· · · · · ·· ·

1/16

1/8

1/16

··········

!-------. 1/4

···························

1/8

t------------, 1/2

I

-4ll�-�--�----�-----------�1/16

Figure I .

_

· · · ·······

1/4

1/8

1 + �2 + �4 + �8 + J_ + 16

1/2 ·

·

·

= �2 + �4 + l8 + ..i. + ...2... + 16 32

·········

•

• • -

1/115 1/8

1/4

1/2 I

(Note that the

x-axis scale is different from the y-axis scale.)

Anyway, beginning with (1), I was naturally tempted to cancel the sigmas and the denominators and thereby conclude that N = 1 . This didn't seem quite rigor­ ous, however. Luckily Eric Wepsic was around to help me formalize the argu­ ment . Here's the proof:

PROOF. Define the function j(x) = By (1), j(l)

= j(N).

Thus

N= 1.

I 2:,. Note the crucial fact that fis one-to-one.

N�O

I couldn't help but notice the corollary:

COROLlARY

1

P

= NP.

Department of Mathematics & Computer Science Emory University Atlanta, GA 30322 USA e-mail: [email protected]

8

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media. Inc.

Improving the Fundamental Theorem of Algebra JOSEPH SHIPMAN

he "algebraic part" of the Fundamental Theorem of Algebra says that under certain purely algebraic hy­ potheses, a field of characteristic 0 must be algebraically closed. In this article I will give a best possible ver­ sion and extend the theorem to characteristic p. I will also give an algorithm for determining all finitary implications between "degree axioms" of the form "every polynomial of degree n has a root."

The Fundamental Theorem The "Fundamental Theorem of Algebra" is the usual name for the theorem that the field of complex numbers is alge­ braically closed. However, all proofs of this fact involve, in addition to algebra, a certain amount of analysis, topology, or complex function theory. The less algebra there is in the proof, the more of other kinds of mathematics there must be. The more algebra there is in the proof, the more gen­ erally applicable it is and the easier the non-algebraic part of the proof is. The book [FR] is an excellent summary of the known proofs of this theorem, which provides an illuminating in­ troduction to many branches of modern mathematics. Hun­ dreds of articles on the Fundamental Theorem of Algebra have been published, almost all of which involve new proofs or variations on old proofs. Despite all this attention, I have something entirely new to demonstrate. I am going to improve, not simply a proof of the theorem, but the theorem itself: assuming less, and concluding more. Most of the proofs in existence apply only to the com­ plex number field, and contain very little actual algebra; some writers have therefore suggested that the theorem is mis-

named. However, Gauss's 1815 "second proof'' of the theo­ rem [Gl, which was the first entirely rigorous proof, justifies the name. In this proof, Gauss showed by purely algebraic reasoning that every real polynomial resolves into factors of the first and second degree. A modernized and simplified version of Gauss's proof (due to E. Artin) is given by van der Waerden [vdW], who states the theorem as follows: If in an ordered field K every positive element possesses a square root and every polynomial of odd degree at least one root, then the field K(i) obtained by adjoining i is alge­ braically closed. That the real field satisfies these conditions is a very easy piece of analysis; the algebra required is much harder, but as a reward the theorem is applicable to all "real closed" fields, not just the real and complex numbers. An examination of the proof in [vdW] shows that it does not need K to be ordered, only that every element of K have a square root in K(i) (which is an easy consequence of K 's being ordered and having square roots for positive elements). The proof also implicitly uses that K has char­ acteristic 0 (which follows from the original restriction to ordered fields), by applying the Primitive Element Theo­ rem. We may therefore restate the theorem more generally: If a field K has characteristic 0, if all odd-degree poly­ nomials in K[x] have roots in K, and if all elements of K have square roots in K(i), then K(i) is algebraically closed. In this form, the theorem applies to fields which are not necessarily ordered, and we have the simple corollary:

© 2007 Springer Science +Business Media, Inc., Volume 29, Number 4, 2007

9

If K has characteristic 0, and if all polynomials whose degree is 2 or an odd number have roots, then K is alge­ braically closed. But we have not gone far enough towards finding the "algebraic essence" of the Fundamental Theorem of Alge­ bra. The hypotheses actually needed for a field to be al­ gebraically closed are much weaker; I shall optimize them.

"Degree Axioms" Gauss proves the theorem by induction on the number of factors of 2 in the degree of the polynomial. Given a real polynomial j(x) of even degree d, Gauss constructs another real polynomial of degree C 1 ) = d(d - 1 )/2, which has one fewer power of 2, such that the new polynomial has a root in the complex numbers only if f does. Through repetition of the process, a polynomial of odd degree is eventually obtained, from a root of which we may obtain a root for f by solving a sequence of quadratic equations. From the ex­ istence of complex roots to real polynomials, we may ob­ tain roots for any complex polynomial g(x) via the real poly­ nomial g (x)g (x) , where g' is the "complex conjugate" of g. The only properties of the real numbers that Gauss used were the existence of roots for equations of odd degree, and the existence of square roots for non-negative num­ bers. This "algebraic" proof is more useful than the proofs involving analysis or topology, because it applies to many more fields. Artin and Schreier's theory of "real closed fields" is built on this foundation. A field K is said to be "formally real" if - 1 is not a sum of squares. Such K can be ordered, and have characteristic 0. K is "real closed" if every odd­ degree polynomial has a root in K and every positive ele­ ment has a square root. (The definition still applies to fields with no defined order relation, if -1 is not a sum of squares and every element is a square or the negative of a square.) These assumptions are all expressible in the first-order language of fields . It follows from the work of Tarski (T] that all real closed fields satisfy the same first-order sen­ tences, and the following axiomatization characterizes real closed fields: Group i) AOF: The conjunction of the standard axioms for ordered fields. Group ii) Axiom about existence of square roots: '

Group iii) Degree Axioms (one for each odd integer) : (1): VXJ 3xt ((XJ + x1) = 0) [3]: VXJ Vx1 Vxz 3x3 ((XJ + Cx3 Cx1 +(X:3 (xz + x�)))) = 0) [5] : VXJ Vx1 Vx2 Vx3 Vx4 3X; ((.xo +(X; Cx1 + Cxs 0) Cx2 + Cxs Cx3 + (X; Cx4 + X;))))))))) Etc. •

•

If K has characteristic 0, and if all polynomials whose degree is 2 or an odd n umber have roots, then K is alge­ braically closed. This leads to a complete axiomatization for algebraically closed fields of characteristic 0 (all of which satisfy the same sentences as the complex numbers): Group i) AF: The conjunction of the standard axioms for fields. Group ii) Axioms for characteristic 0 (one for each prime): C02 : �(1 + 1 = 0) C03 : �(1 + 1 + 1 = 0) C05 : �(1 + 1 + 1 + 1 + 1 0) C07 : �(1 + 1 + 1 + 1 + 1 + 1 + 1 = 0) C011 : �(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 0) Etc.

a PhD in Logic. In the years since then, he has worked in a variety of

fields-biomedical research, software development and more-while

always continuing mathematical research pro bono. At present he is Di­ rector of Data Management at ALK Technologies of Princeton, NJ. He is also devoted to chess, to singing in his church choir, and to

joseph Shipman 20 M ontgomery Avenue Rocky Hill, NJ 08553 USA e-mail: [email protected]

10

THE MATHEMATICAL INTEWGENCER

•

=

•

Each "degree axiom" asserts the existence of roots for all polynomials of a given degree. Note that the first de­ gree axiom [1] merely restates the existence of additive in­ verses and is true in all fields. Note also that the degree axiom [n) implies (d) for any d dividing n, because we can construct a rootless polynomial of degree n by taking a power of a rootless polynomial of degree d. Since -1 is not a square in an ordered field, the poly­ 2 nomial (x + 1) has no roots. If there were a polynomial of odd degree d> 1 with no roots, then we could multi­ ply it by powers of (x2 + 1) to construct rootless polyno­ mials of degrees d + 2, d + 4, etc. Therefore, ANY infinite subset of Group iii) suffices to axiomatize real closed fields (together with the axioms AOF and SR). This is as far as we can weaken the assumptions for an ordered field to be real closed. But the situation is much more interesting when we start with a field which is not necessarily "real." In the preceding section, we saw that Gauss's proof, as adapted by Artin and van der Waerden, has the corollary

JOSEPH SHIPMAN was educated at MIT and Brandeis, where he got

his family: wife and four children.

•

•

=

Group iii) Degree Axioms

([2]

and one for each odd integer)

[2]: 'ii.'I:Q 'ilx1 3xz ((XQ + (Xz Cx1 + Xz))) 0) [3]: 'il.xo 'ilX1 'ilXz 3x3 ((X{J + (x3 Cx1 + (x:l =

•

(Xz

+

X0)))))

•

=

0)

•

•

=

*

•

•

X;)))))))))))))

•

=

•

•

•

•

•

•

=

•

The key observation for improving the Fundamental The­ orem of Algebra is that each degree axiom [d] , when d is an even number >2 , is a consequence of finite�y many of the degree axioms Hil1 i 2 or an odd integerl, together \Vith AF and the axioms for characteristic 0. This follows (nonconstructively) from the Compactness Theorem for first-order logic, but Gauss's proof provides an explicit re­ duction: [d ] follows from [2] and [( � )] [d (d- 1)/2]. Thus, we can prove [6] from [2] and [15]. To prove [8] we can use [2] and [28], and to get [28] we use [2] and [378], and to get [378] we use [2] and [71253]. We will find a necessary and sufficient condition for a set of degree axioms to imply another degree axiom. This will allow us to find an optimal axiomatization of algebraically closed fields, where each axiom is inde­ pendent of the others. As a bonus, it will turn out that the strengthened theorem is true in fields of all charac­ teristics. =

=

Degree Axioms and Galois G roups

Fix a field K. For now. require K to be of characteristic 0. For every polynomial j(x) in K [x], there is an associated splitting field L and an associated finite Galois group G. Suppose f has degree d and roots r1, r2, . . . , rc1 (multiple roots appearing the appropriate number of times with dif­ ferent labels). G acts on the set h, r2, . . . , rc�), and this action has a fixed-point iff j has a root in K. If the degree axiom [d ] is true, then subgroups of Sc1 which act without fixed-points on the roots are ruled out as possible Galois groups for polynomials of degree d. On the other hand. if [d ] is false, then there is a poly­ nomial j(.x) of degree d, with irreducible factors . h, .fz, . . . , of degrees d1, d1, . . . , with each d; > 1 and d1 + d2 + d. Since degrees of irreducible polynomials corre­ spond to degrees of field extensions, there is a sequence of extension fields K1, K2, . . . which correspond to sub­ groups G1 • G2, . . . of the Galois group G of j, where d; is the degree of K; over K and also the index of G; in G. This restricts the possible G to groups such that d can be ex­ pressed as the sum of indexes of proper subgroups of G. Denote by the additive semigroup gen­ erated by the positive integers a,b,c, . . . . For any finite group G, let < G> denote the additive semi­ group generated by the indexes in G of its proper subgroups. We are now ready for a sufficient condition for impli­ cations between "degree axioms. " ·

=

is true in all fields of characteristic 0 if for every subgroup G of S, which acts without fixed­ points on {1,2, . . . , n), semigroup < G> contains one of the i1. Note that the condition in (**) is obviously computable.

0)

Etc.

·

The statement

(**)

•

[9]: 'V:x:o 'ilx1 Vx2 'Vx3 V.x4 'Vx5 'il:xo 'Vx7 'tlxp, 3-MJ ((Xo + (.MJ Cx1 + (.MJ (.xz + (.MJ * C.x3 + (.MJ (x4 + (.MJ Cxs + (.MJ (.x<, + (.MJ (x7 + (.MJ (.xp, + .MJ))))))))))))))))) 0)

·

1

(* )

•

[5]: 'ilXo 'ilX1 'ilXz 'Vx3 'VXc, 3xs ((XQ + Cxs Cx1 + Cxs (xz + Cxs Cx3 + Cxs (x4 + Xs))))))))) 0 [7]: 'il:x:o 'Vx1 'ilxz 'Vx3 'Vx:, 'Vxs 'iiX(, 3x7 ((XQ + (x7 Cx1 + (.>..7 (xz + (.x7 * (x:� + (x7 (x, + C.x7 C.xs + (.x7 * (X(, + •

THEOREM

P ROOF. Assume the condition (**) is true for i 1,i2, . . . im,n. Choose a field K of characteristic 0, and suppose there is a polynomial f in K[x] of degree n with no roots in K; since degree axiom [n ] fails, we now need to falsify one of the de­ gree axioms [�]. The Galois group of the splitting field off over K acts without fixed-points on the roots of j (if Jhas multiple roots, we add extra copies of the roots of j to the set G is acting on to get a fixed-point-free action on a set of size n). Every subgroup of index h corresponds to a field extension of K of degree h; in characteristic 0, these extensions have prim­ itive elements, so we can get irreducible polynomials of all those degrees, and multiply them together to get rootless polynomials of all degrees contained in the semigroup < G>. By assumption, � is in the semigroup for some j in {1, . . . m), and the corresponding rootless polynomial coun­ terexemplifies [�], as required. 0

G

CoROLLARY 1 [n] follows from the conjunction oj [p] for primes dividing n with [m] for any sufficientry large m. PROOF. For any finite group G, for each prime p dividing j G j , G 's Sylow-p subgroups have indexes not divisible by

p.

iG!

If is not a prime power, then the gcd of these in­ dexes is 1, and < G> contains all sufficiently large integers; only finitely many G are relevant, so any sufficiently large m causes (**) to be satisfied for all those G. If is a fr group and p doesn't divide n, then can't act without fixed-points on {1, . . . , n }, because all orbits must have size 1 or a power of p, so (**) is vacuously satisfied. If p does divide n, then we already have [pl on the left-hand side of (**); this suffices, because frgroups have subgroups of index p, so < G>

. 0

G

G

=

The Fundamental Theorem of Algebra, Improved These ideas make possible much better versions of the Fun­ damental Theorem of Algebra: not only do fields of char­ acteristic 0 no longer need degree axioms for composite degrees, but the theorem now applies to fields of all char­ acteristics.

COROLLARY 2 .if a field K has characteristic 0, if all odd­ prime-degree porynomials in K [x] have roots in K, and if all elements of K have square roots in K(i), then K(i) is alge­ braicalry closed. PRooF. We are able to replace "odd" with "odd prime" by applying Corollary 1: for any odd composite [d], the primes dividing dare odd and there is a sufficiently large odd prirae.

© 2007 Spnnger Science+ Business Media, Inc., Volume 29, Number 4, 2007

11

For completeness, I give an argument which does not depend on the proof in [vdW] . Assume K (i) has square roots for all elements and K has roots for polynomials of odd prime degree. Applying Corollary 1 , all odd-degree polynomials have roots. If f in K [x:l has even degree, its Galois group has order 2rm for m odd. Corresponding to the 2-Sylow subgroup, which has index m, is an exten­ sion of degree m; but there are no irreducible polynomi­ als of odd degree, so m 1 and 2r. Since p-groups have subgroups of index p, we can build a chain of ex­ tensions of degree 2 to reach the splitting field of f; but since K (i) has square roots for all elements, each extension comes from a degree-2 polynomial with coefficients in K, so f splits into linear and quadratic factors. Any polynomial in K (i )[x] can be multiplied by its "conjugate" to get a poly­ nomial in K [x], and from the resulting factorization into lin­ ear and quadratic factors we can get a complete split into linear factors in K (i )[x] . 0

G

lei

=

=

THEOREM 2 Any field which satisfies [ p] for allprimesp sat­ isfies [n] for all natural numbers n. PROOF. If the field K has characteristic 0, this follows di­ rectly from Corollary 1 and the existence of infinitely many primes. The only place where the assumption of charac­ teristic 0 was needed in the proof of Theorem 1 was to ob­ tain primitive elements for algebraic extensions of K; but we have [p] for all primes p, so every element of K has a p-th root in K; this holds in particular for the characteristic of the field, so K is a perfect field, and all algebraic ex­ tensions are separable and they have primitive elements anyway. 0 Theorem 2 allows us to delete all axioms [ n] for com­ posite n from our axiomatization of algebraically closed fields. Can we go further? No!

THEOREM 3 Theorem 2 is not trne if we omit any single prime from the hypothesis. PRooF. Let K be the field generated by all algebraic num­ bers whose degree over Q is not divisible by a given prime p. This K contains no numbers of degree p over Q, because we can write K as an expanding union of fields of finite de­ gree over Q, where each field is obtained from the previous one by adjoining the "next" algebraic number whose degree is not divisible by �at each stage we have a finite exten­ sion whose degree over Q is not divisible by p, so no num­ ber of degree p can ever get in. Therefore there are polyno­ mials of degree p in Q [x] (and so also in K [x]) with no roots inK For any other prime q, every polynomial in K [x] of de­ gree q has an irreducible factor of degree not divisible by p, and so has a root r whose degree over K is not divisible by p. But r has the same degree over the subfield of K gener­ ated by the coefficients of its irreducible polynomial, which has a finite degree over Q that is not divisible by p; so r also has such a degree and is therefore in K by construction. 0 We have thus obtained an "optimal" axiomatization for algebraically closed fields: ACF {AF, [2], [3], [5], [7], [ 1 1 ] , =

12

THE MATHEMATICAL INTELLIGENCER

. . . I, where each axiom is independent of the others. Adding the axioms {COz, C03, COs, C07, . . . ) gives an op­ timal axiomatization for algebraically closed fields of char­ acteristic 0, while adding the single axiom �cop gives an optimal axiomatization for algebraically closed fields of characteristic p. However, omitting any set of primes is no worse than omitting one, as long as we still have infinitely many "good degrees" for which all polynomials have roots:

THEOREM 4 For any field K, if there are arbitrarily large "good degrees" d such that all polynomials of degree d have roots, then either K is algebraically closed, or there is exact�y one "bad prime" which is the degree of a rootless polynomial, and a degree is ''good" ifand only if it is not a multiple of that prime. PRooF. We know there can be at most one "bad prime," because if two primes were bad then all sufficiently large degrees could be expressed as a sum of those primes and so would have a rootless polynomial, contradicting the as­ sumption of arbitrarily large "good degrees." Corollary 1 im­ plies that if infinitely many primes are "good degrees" then any number only divisible by "good primes" is a "good de­ gree." If there are no bad primes, the proof goes through to show that K is algebraically closed. D

Sufficiency for Characteristic p

Theorem 1 gives us the best possible version of the Fun­ damental Theorem of Algebra, but it can itself be made stronger: the sufficient condition is also necessary, and the characteristic 0 assumption can be dropped. First, let's look at some examples. Suppose n is odd. We know the alternating group An is a possible Galois group, and it contains subgroups of index n, G), ('_3), . . . , (d), where d (n- 1)/2. These subgroups are intransitive and arise from partitioning { 1 , . . . , n} into two pieces. When n = 2k is even, there is also a transitive imprimitive sub­ group of index (�)/2 containing those even permutations which permute { 1 , . . . , kl and {k + 1 , . . . , n } indepen­ dently OR switch the two blocks. It is not difficult to prove (see [DM, section 5 .2]) that, with a few small exceptions where n < 10, any other subgroup of An is smaller than these or is contained in one of them. What degree axioms do we need to ensure [ 1 5]? The largest subgroups of A15 have indexes 1 5 , 105, 455, 1365, 3003, 5005, 6435. The semigroup is therefore gen­ erated by these numbers plus some others larger than 6435. However, it is not hard to see that < 1 5 , 455, 3003> in­ cludes 1 0 5 , 1 365, 5005, 6435, and all larger indexes of sub­ groups of A15, so = < 1 5 , 4 5 5 , 3003>. This means that to derive the degree axiom [ 1 5] , we will need either [ 1 5kl for some k, or at least [4551. And [455] by itself isn't enough, because it only eliminates the possibility of A15 as a Galois group, but we also need to get rid of the prime 3. It turns out (I omit the details of the derivation from The­ orem 1) that [ 1 5] follows from any set of degree axioms where the degrees include a multiple of 3, a multiple of 5, and an element of the semigroup <15, 455, 3003> (of which 3533 is the first prime) . =

Now let's see if the proof of Theorem 1 can fail in char­ acteristic p. If a "degree implication" (i1 ]& . . . &[iml ==> [ n] holds in characteristic 0, we know that it holds in charac­ teristic p also if p divides one of the ii, because the proof fails only in the case of "inseparable extensions," which can­ not occur in characteristic p when every element has a p-th root. But if p does not divide any of the �, it doesn't divide n either, for in the preceding section, "The Fundamental The­ orem of Algebra Improved," we constructed a characteristic0 field in which [ n] was true iff n was not a multiple of p. So we may assume p does not divide n. If n divides any of the �, the degree implication is trivially true, so we may rule out this possibility. Purely inseparable extensions have degrees that are powers of the characteristic, which means we may assume there is an irreducible polynomial of de­ gree pr for some r; furthermore, pr must be < n if we are going to have a degree-n polynomial give an inseparable extension. So if there is a counterexample, we have root­ less polynomials of degree p r and degree n. This means we can construct rootless polynomials of all degrees in <pr, n>, and since p doesn't divide n, this semigroup includes all sufficiently large degrees, in particular, all degrees r(n - 1) or greater. If n is even, then pr is odd, and <pr, n> includes n(n - 1)/2 as well, because n ( n - 1)/2 = (n/2)•(n - 1) = (n/2)•(n - 3) + n (n/2)•(n - 5) + 2n = = (n/2)•Pr + ((n + 1 - pr)/2) • n. But we saw above that, for n > 9, the smallest element of that is not a multiple of n is n(n - 1)/2, if n is even, and for odd n is at least CD= n( n - 1)(n - 2)/6, which is greater than n ( n- 1 ) since n > 9. Therefore, <pr, n> contains the entire semigroup , so at least one of the & must be in <pr, n> and there is a rootless poly­ nomial of that degree. Thus we can't get a counterexam­ ple to our degree implication, because one of the degree axioms on the left-hand side must fail. We can deal with the remaining cases n < 10 by direct calculation. When n is prime, the only valid degree impli­ cations have a multiple of n on the left-hand side, and they are trivially valid in all characteristics. For n 4, 6, 8, 9 , we calculate the following semigroups: =

·

·

·

Begin by constructing fields K and L such that L is the splitting field over K of a polynomial j(x) of degree n, with Galois group Gal(UK) G. (This can be done so K and L are both algebraic over Q.) Let z be a primitive element for this extension, so L = K(z) and z satisfies an irreducible poly­ nomial of degree G over K Let Krnax be a maximal alge­ braic extension of K with the property that Lmax = Kma:x:(z) has degree over Kmax- (We can construct this by succes­ sively adjoining algebraic numbers that don't kill any of G, because there is an enumeration of the algebraic numbers.) Since we haven't disturbed G, j(x) still has G as its Ga­ lois group, and no roots in Kmax, but any further algebraic extension of Kmax will fail to extend Lmax by the same de­ gree-that is, for any new algebraic number y, Kmax(y, z) = Lmax(y) has a degree over KmaxCY) that is smaller than We need to show that all the degree axioms (hl, . . . [iml are true for Kmax-then, since j(x) is still rootless, [n] is false and thus (* ) is also false, as required. So suppose that we have a polynomial g (x) of degree � over Kmax, where by assumption i1 is not in the semi­ group < G> . g is a product of irreducible polynomials, and at least one of these must not have a degree in < G> (for if they all did, their product would). So we now have an irreducible polynomial h(x) whose degree i is not in < G > . Let y be a root o f h . Then KmaxCy) has degree i over Kmax, since h is irreducible. Consider the intersection M of KmaxCy) and Lmax = KmaxCz). Let d1 be the degree of this field over Kmax· Since M is a subfield of Lmax, the subgroup of G fix­ ing it must have index d1 , so either d1 = 1 or d1 is in < G> . =

I l

lei

lei.

Lmax(y)=Kmax(Y ,z)

=

< � > = < 3, < � > = <6, < 8, <9 , = =

4> 10, 1 5 > 1 5 , 28, 35> 84, 280>.

In each case, for any prime power pr less than n and not dividing n, the generators of the semigroup (and so the whole semigroup) are in <pr, n>, so we can't get a coun­ terexample to the degree implication. Therefore the char­ acteristic 0 assumption in Theorem 1 can be eliminated.

COROLLARY 3 ([3]&(10]) COROLLARY 4 ([2]&[15])

==>

[6] is true in all fields.

==>

(8]

is

1

L

K(z)

true in all fields.

PRoOF OF NECESSITY. Reversing the direction of Theorem 1 is trickier. Suppose ( ) is false, so we have G acting on < 1 , . . . , n> with none of the fs in < G> . We need to falsify (*) , so we must construct a field where [id, . . . [inJ **

are true but [ n] is false.

K

Fields defined in proof of Theorem 5

© 2007 Spri nger Science+Bus1ness Media, Inc., Volume 29, Number 4, 2007

13

But d1 also divides i because M is a subfield of Kmaly), which means we must have d1 = 1 , because we know i is not in < G > . Thus M= Kmax: the extension fields KmdY) and KmaxCz) have only Kmax in common.

( ** ) for every subgroup G of Sn which acts withoutfixed-points on {1 ,2, . . . , n), the semigroup < G > contains one of the ii . (Compared with Theorem 1 , Theorem 5 eliminates the characteristic 0 hypothesis and works in both directions.)

way, and led ultimately to the formulation of Theorem 1 (which is not hard to prove once it is formulated just right!). Although Theorem 5 may appear definitive, there are several directions for further investigation. The algorithm implicit in (**) is slow, but it can be sped up by making certain assumptions about permutation groups; however, verifying these assumptions will require careful analysis of the O 'Nan/Scott Theorem on maximal subgroups of An (see [DM]) and the Classification of Finite Simple Groups. There is also a rich theory for several kinds of weak­ ened degree axioms, such as [ n] ' : "all polynomials of degree n are reducible," or [ nk] : "all polynomials of degree n have a factor of de­ gree li' (when k = 1 this is the standard degree axiom [n]). These weakened axioms are still expressible in the lan­ guage of field theory, but they translate differently into the language of Galois groups. Finally, the "finite choice axioms'' deserve further inves­ tigation. The great progress in finite group theory over the last 35 years ought to make it easier to calculate the rela­ tionships between these axioms, including weakened ver­ sions which identify subsets or partitions of {1 , . , n) in­ stead of elements.

Conclusion

ACKNOWLEDGMENTS

Theorems 2 and 3 establish the minimum algebraic condi­ tions necessary for a field to be algebraically closed, and they can therefore be said to "optimize" the Fundamental Theorem of Algebra. But each specific "degree implication" is a first-order consequence of the axioms for fields, and could have been discovered two centuries ago; the exis­ tence of these finitary relationships appears to have been unsuspected by practically everyone, with one important exception. The inspiration for Theorem 1 was the work done by John H. Conway on "Finite Choice Axioms" in 1 970, de­ tailed in [Co] . Conway, building on earlier work of Mostowski and Tarski, identified a necessary and suffi­ cient condition for effective implications between axioms of the form "Every collection of n-element sets has a choice functio n . " Conway's group-theoretic condition is very similar to ( * * ), the difference being that one could use the semigroup < H > for any subgroup H of a group G acting fixed-point-freely on { 1 , . . . , n), rather than re­ quiring G = H. The present article also borrows some ter­ minology, notational conventions, and proof ideas from Conway's work. Theorem 2 was originally proved by a difficult combi­ natorial argument that generalized Gauss's original proof. Corollaries 3 and 4 emerged during discussions with Con-

I am grateful to Dan Shapiro, Alison Pacelli, Harvey Friedman,

But this means that every automorphism of Lmax fixing Kmax extends to an automorphism of LmaxCy) fixing KmaxCy), because it doesn't matter which of the conjugates of z we use when forming Kma.xCy,z) = KmaxCz,y) = Lma.xCy) . Therefore the Galois group of Lmax(y) over Kma.x(Y) is still G; but we constructed Kmax so that any algebraic extension would collapse some of G. Therefore KmaxCy) is not really an extension: y must already be in Kmax. which means that h(x) is of degree 1 , and g(x) has a root, as was to he shown. We have now established Theorem 5 .

I Gl

THEOREM 5. The statement ([il)&[izl& . . . &[inJ)

(*)

=>

[ n]

is true in all fields if!

14

THE MATHEMATICAL INTELLIGENCER

Frank Morgan, Simon Kochen, Noam Elkies, and Jonathan Co­ hen for verifications, suggestions, and encouragement. I would especially thank Professor John Conway for many instructive and enjoyable conversations over the last 20 years, as well as for his inexhaustibly inspiring writings and per­ sonality. REFERENCES

[Co] Conway, John H., "Effective Implications between the 'Finite ' Choice Axioms, " in Cambridge Summer School in Mathematical Logic (eds. A. R . D. Mathias, H . Rogers), Springer Lecture Notes in Math­ ematics 337, 439-458 (Springer-Verlag, Berlin 1 971 ). [DM] Dixon, John D . , and Brian Mortimer, Permutation Groups, Springer Graduate Texts in Mathematics 1 63, Springer-Verlag, 1 996.

[FR] Fine, Benjamin, and Gerhard Rosenberger, The Fundamental The­ orem of Algebra, Springer-Verlag, New York 1 997.

[G] Gauss, Carl Friedrich, Werke, Volume 3, 33-56 (In Latin; English translation available at http://www.cs.man.ac.uk!�pt/misc/gauss­ web.html). [T] Tarski, Alfred, A Decision Method for Elementary Algebra and Geom­ etry, University of California Press, Berkeley and Los Angeles, 1 951 .

[vdW] van der Waerden, B. L. , Algebra (7th edition, Vol. 1 ), Frederick Ungar Publishing, U .S.A. , 1 970.

MatheiTII atically Bent

The proof is in the pudding.

Colin Adam s , Editor

North North Wester n State U n iver sity M athe mat ics Departme nt Safety M anua l COLIN ADAMS

Opening a copy of The Mathematical Intelligencer you may ask yourself

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

Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267 USA e-mail: [email protected]

elcome to North North Western State University (NNWSU). We here at the Office of Office Safety (OOS) are happy to be one of the first offices on campus to welcome you to the Department of Mathematics. This manual is but one of a pile of documents that you have just received as a new member of the fac ulty. BUT IT IS THE MOST IMPORTAN71 Because safety is our number-one concern here at NNWSU. You might have thought it was education, or research, or bringing in grant dollars, or showing support for our surprisingly large football team. But, no, it is safety. Of course, new faculty members might not associate safety issues with a mathematics department. They might think that safety concerns should be rel­ egated to chemistry departments, where an exploding beaker can send shards of glass streaking toward an unprotected eyeball, or physics departments, where an errant laser can b urn holes in the seats of pants and what they contain. New teachers may believe that they need not fear for their physical safety when work­ ing in a mathematics environment. But some once new fac ulty members are no longer employed at NNWSU. In fact, some of them are n ot employed any­ where, because they no lnnger walk this earth! So please read this docu­ ment carefully. It could save your life!

1. What is the number-one safety con­ cern in a mathematics department? This is an exceUent question. We made it up, but it is an excellent ques­ tion nonetheless. The number-one safety concern is eye strain. Did you know that? We bet you didn't. Strained eyes cause more lost work days than any other single mathematics office in­ j ury. Often, we see faculty members dri­ ving erratically, on their way home af­ ter a debilitating eye strain injury. They are pulled over by police officers who believe they are intoxicated, and who ask them to walk a straight line. And often they fail, because of eye strain. Then it's off to the pokey for them. Don,t let this happen to you!

2.

How do eye strain injuries occur?

Another excellent question. And yes, again, we made it up. There are three main categories of eyestrain inj ury. A. Eye Fatigue Syndrome (EFS): Just as we can strain a leg muscle from overexertion, we can strain our eyes by staring in one direction too long, say at a c omputer screen or at a par­ ticularly enchanting fractal poster. What can be done to prevent or al­ leviate EFS? Here are some Eye Strain Prevention Exercises (ESPE): 1 . Look away from the computer screen, say, at the corridor outside your office door. Cup the palms of your hands over your eyes and stare for 60 seconds, counting out l oud. Then slowly twist your wrists to alternately cup your eyes and create blinders while staring at anyone looking in from the cor­ ridor. Continue for 60 seconds. Then cup with one hand while making a blinder with the other. Alternate hands back and forth for another minute. 2. Close one eye . Moving the op­ p osite hand in a clockwise circle of diameter one foot, follow the index finger with the open eye. Do this for three revolutions and then change eyes. If anyone is staring at you through your open door, scan from their feet up to their head and then back down to their feet. Repeat six times. Then return to work.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

15

B. Sudden Eye Movement Injury (SEMI): This injury may occur while you are in your office working at your computer and a student comes to the open door. "Excuse me pro­ fessor, but I don't think I should have lost this point on the home­ work . " Startled, you jerk your eyes from the screen to the doorway and feel an immediate explosion of pain caused by tearing of the muscles that control eyeball direction. What can be done to prevent SEMI? 1. Keep the office door closed. 2. When there is a knock at the door, do not swivel the eyes quickly to the door, but rather, move them slowly in that direction, taking time to peruse the items you see along the way. Avoid jerky eye movements at all times. C. Repetitive Eye Roll Injury (RERI): This injury typically occurs at de­ partment meetings. Excessive rolling of the eyes can cause severe fatigue of the muscles under the upper eye­ lid, and leave one incapable of look­ ing anywhere but down without pain. (The common misconception that mathematicians are shy is a di­ rect result of this phenomenon.) If, however, you are a representative of the faculty senate, you have had ample opportunity to condition your eyes to rolling. So roll on!

3. Is it safer to work with a pencil or with a pen? Here at Double-North Western, pen injuries exceed pencil injuries by six to one. Yes, six to one! We have had three pen injuries and about half a pen­ cil injury. How to explain the discrep­ ancy? We don't know, but there it is . You choose pen and you may be writ­ ing your own obituary. 4.

Must I wear loose-fitting clothes?

Yes, you must. Constricting clothing can be a constant distraction, causing you to lose focus on safety issues, and risk calamity. But not so loose that they fall off, causing a distraction for your fellow workers as they are handling sharp implements. 5.

What is the most dangerous object in a mathematics department?

Great, great question. This is an easy one, too. It is the cups of coffee. Many

16

THE MATHEMATICAL INTELLIGENCER

people have claimed that styrofoam cups were the greatest invention in history. We here at the Double N Double-D SU Dou­ ble 0 S beg to differ. The styrofoam cup is perhaps the most dangerous invention ever, as measured by the Steiner Hot Cof­ fee Burn Index. They are tippy and they retain the heat in the coffee. This is an obvious recipe for disaster. More mem­ bers of the Mathematics Department have been treated for coffee burns than for any other single kind of burn, with the ex­ ception of rug burns, which really don't deserve to be called burns at all. We actually keep spare pairs of pants at the OOS for members of the Mathe­ matics Department who spill coffee on themselves. We have men's cuffed tan pleated slacks, in a size 34 waist, and women's gabardine plaid slacks in a size 6, in case you want to plan ahead. Well, actually, we don't have the men's slacks right now. We are waiting to get them back from a certain someone who we suspect is purposely spilling coffee on himself just to get the use of the slacks. 6.

Why is mathematics so dangerous?

Mathematics is perhaps the most ab­ stract of subjects. To study and do mathematics, you must remove your mind from the real world around you. In the process, you lose touch with re­ ality. You don't see those stairs that you are approaching. You don't see the open file cabinet drawer. You don't see the students milling around after that exam with angry looks on their faces. 7. But isn 't mathematics goodfor your brain development?

Is all exercise good for you? Is it good for your back to lift heavy boxes filled with safety goggles for hours at a time, day in and day out, for no apparent pur­ pose? I think you know the answer is no. Well, it's the same with brain exer­ cise. You do difficult problems day in and day out, you could blow out your medulla oblongata, rip your brainstem , o r split your hemispheres permanently. Then where would you be? And even if that didn't happen, you might overdevelop your brain, and it could end up looking like Arnold Schwarzennegger used to look, when he was dressed in a speedo and slathered in oil, glistening in the bright lights. Forgetting about the fact he mar­ ried Maria Shriver and is now the Gov-

ernor of California, be honest: Is that how you want your brain to look? So, after doing your share of math, kick back at the end of the day. Forget about that lemma that's been driving you crazy. Go home, have a soda and watch some reality 1V. You'll be glad you did. 8. Why is safety the most important is­ sue at NNWSU?

Each college and university strives to be the best it can be. Those of us at Northie have realized that we don't have a chance in hell of being the best in any academic discipline. So, instead, we have decided to focus on safety. Our goal is nothing less than to be the safest educational institution in the country. Better than Harvard. Better than MIT. Better than Bluebonnet Hill Community College, that pretender to the safety crown right down the road. Remember when you were a candi­ date for a faculty position in mathemat­ ics here? You probably thought you were being evaluated on the basis of your abil­ ity to teach and do research. Nothing could be farther from the truth. In fact, you were being evaluated solely on the basis of your previous safety record and your future safety potential . During your job talk, we had a checklist. Low-heeled shoes? Laces tied in a double knot? Zip­ per up? Pens carefully capped? You must have satisfied all the criteria on the check­ list, as otherwise, you would not be read­ ing this document. But don't think you can rest on your laurels. Tenure and pro­ motion are also contingent on your at­ tention to safety procedures. Well, that concludes this initial dis­ cussion about safety in an arithmetic en­ vironment. We hope we haven't scared you with all of this talk of the dangers of mathematics. If you approach math­ ematics with an eye to safety, you may find it productive, and yes, perhaps even enjoyable. We look forward to meeting you per­ sonally at our mandatory weekly math safety seminars, which begin soon. Look for our multi-colored notice com­ ing in your mailbox shortly. But in the meantime, remember. Safety: It's not just a word anymore.

This document brought to you compli­ ments of the North North Western State University Office of Office Safety (NNWSUOOS).

M a t h e m atical C o m m u n ities

Re l ig io us H er esy and M at he mat ica l Cr eat iv ity i n Russ ia LOREN G RAHAM AND JEAN-MICHEL KANTOR

Ibis 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.

uring the last several decades we have frequently gone to the So­ viet Union and, after its collapse, to Russia, where we have often worked with scientists. Gradually, a remarkable story about the relationship of science and religion in Russia has emerged from conversations with Russian colleagues. The story helps explain the birth of the Moscow School of Mathematics, one of the most influential modern movements in mathematics. The conflict at the cen­ ter of the story persists today and raises fundamental questions about the nature of mathematics, not only in Russia, but throughout the world. Since the history of the issue dates to the early years of the last century, we must begin with a single event from that time. This event kicked off a movement that is still alive. Early in the m orning of July 3, 1 91 3 , two ships from the Imperial Russian Navy, acting on orders from Tsar Nicholas II, steamed into the azure wa­ ters surrounding the holy mountain of Mt. Athas in Greece, a center of Or­ thodox Christianity for a thousand years. The ships, the Donets and the Kherson, anchored near the Pantalei­ mon Monastery, a traditional center of Russian Orthodoxy and residence of h undreds of Russian monks. Small boats loaded with armed Russian marines made their way to the dock, where the men disembarked. The marines pro­ ceeded to the cathedral of the monastery, at that moment nearly empty. There, the officer in charge met with several of the religious ascetics and told them that they were to inform all their brethren to leave their cells and assemble in the cathedral. When the other monks learned of the order, they barricaded the doors of their cells with furniture and boards. Inside they fell to their knees and began crying "Lord, Have Mercy!" ( Gospodi pomiluz) and many of them launched into a unique prayer,

one causing controversy in the Church, called "The Jesus Prayer. " (We will say more later about the Jesus Prayer.) The Russian officer demanded that the monks come out. When he was ig­ nored, he ordered his marines to tear down the barricades and aim water from fire hoses at the men inside. The marines flushed the recluses from their cells and herded them into the cathe­ dral. There, the officer announced to the soaked and terrified monks that they must either renounce their heretical be­ liefs or be arrested. Only a few stepped forward and promised to obey. The oth­ ers remained obstinate, crying that the marines represented the "Anti-Christ. " The officer commanded the marines to force the recalcitrant crowd onto the waiting ships, which took them to the Ukrainian-Russian city of Odessa, on the Black Sea. In all, approximately 1000 monks were detained in this fashion. (The sources differ on how violent this operation was; according to some, the marines at one point used a machine gun and killed several monks; official ac­ counts deny this, but it was certainly a bloody affair, with many wounded.) In Odessa the religious believers were told that the Holy Synod in St. Pe­ tersburg-the highest a uthority of Rus­ sian Orthodoxy-had condemned them as heretics for engaging in the c ult known as "Name-Worshipping . " They were forbidden to return to Mt. Athas or to reside in the major cities of St. Pe­ tersburg and M oscow. They were also warned that they m ust not practice their deviant religious beliefs in Russian Or­ thodox churches on penalty of excom­ munication. Otherwise they were free to go. The unrepentant monks dis­ persed all over rural Russia, where they often lived in remote monasteries, far from central a uthorities, and continued there to practice their heresy and to propagate their religious faith.

Please send all submissions to Marjorie Senechal, Department

of Mathematics, Smith College, Northampto n , MA 01063 USA e-mail: senechal@min kowski.smith.edu

The authors are writing a book on this subject titled Naming God, Naming Infinities and scheduled for publi­ cation by the Harvard University Press. The article also draws heavily on Loren Graham and Jean-Michel Kan­ tor, "A Comparison of Two Cultural Approaches to Mathematics: France and Russia, 1 890-1930," /sis, Vol.

97 (No. 1 , 2006). pp. 56-74.

© 2007 Springer Science + Business Media, Inc., Volume 29, Number 4, 2007

17

Instead of dying out, as the tsarist a uthorities obviously hoped that it would, the heresy continued to spread surreptitiously. With the outbreak, a year later, of World War I the attention of the tsarist government shifted else­ where. The practice of Name-Worship­ ping quietly increased in strength, grad­ ually moving from the countryside to the cities, where it attracted the atten­ tion of the intelligentsia, especially mathematicians, some of whom be­ lieved it contained profound insights for their field. Among the leading mathe­ maticians who became interested in Name-Worshipping were Dmitri Egorov ( 1 869-1931) and Nikolai Luzin ( 1 8831 950), later the founders of the Moscow School of Mathematics. In seeing c on­ nections between mathematics and Name-Worshipping, they were aided by a heretical priest, Father Pavel Floren­ skii (1882- 1937), a former mathematics student at Moscow University, where both Luzin and Florenskii studied un­ der Egorov. While at the university, Flo­ renskii and Luzin served, one after the other, as secretary of the Student Circle of the Moscow Mathematical Society, of which their professor, Egorov, was later president. In subsequent years they car­ ried on an 1 8-year correspondence, of­ ten about mathematics and religion. Both the Russian Orthodox Church and the new Communist regime perse­ c uted Name-Worshipping after the Rev­ olution of 1 9 1 7, but the practice never died out. Following the collapse of the S oviet Union it has been enjoying a small resurgence in Russia. B ut even n ow it remains a "heresy, " equally op­ posed by intellectual camps so differ­ ent that their followers usually agree on very little: Marxists, the leaders of the Russian Orthodox Church, and secular rationalists. What was " Name-Worshipping" and h ow could this religious movement have anything to do with mathematics? B oth mathematicians and religious be­ lievers try to grasp concepts that seem inexpressible, ineffable, or even incon­ ceivable. The history of mathematics demonstrates a number of such mo­ ments. "Infinity" was first denoted by the Greeks as apeiron ("endless, un­ limited mass , " "primal chaos"), irrational numbers ("alogoi , " absence of logos) were unspeakable or unthinkable at the time of Pythagoras, and imaginary n um­ bers were only reluctantly accepted in 18

THE MATHEMATICAL INTELLIGENCER

Seated: N. N. Luzin (1883-1950); standing on right: D. F. Egorov (1869-1931); standing on left: W. Sierpinski, well-known Polish mathematician.

Plaque which is now located on the building on Anbat St. in Moscow where Luzin lived.

the Henaissance . In modern times "ideal theory'' began with numbers that were only supposed to exist "ideally. " In the period 1 H90-1930 a great de­ bate was occurring among mathemati­ cians over the new field of set theoty, a controversy that became connected in the minds of some leading Russian mathematicians to Name-Worshipping. A "set" is a collection of objects shar­ ing some property and given a "name. ·· For example, the set of all giraffes in South Carolina could be named "SCG" for "South Carolina Giraffes." This set obviously has a finite number of ele­ ments . The set 1 , 2,3, . . . of whole num­ bers has an infinite number of elements; Georg Cantor names it � 0 . The birth o f set theory at the end of the nineteenth century brought with it new debates about the nature of "in­ finity . .. Is the "infinity" of points in a line segment just another description for �0• or is it an infinity of another type? A new theory of infinities was born in De­ cember 1873 when Cantor proved that these are different: one cannot "count" the number of points on a line. He then defined an infinity of infinities, the alephs. and another infinity of other numbers corresponding to ordered sets, and he gave new names to all these in­ finities. for example � 0 and � 1 . A cru­ cial point here is the idea of naming. In retrospect, we can see that after Can­ tor assigned different names to differ­ ent infinities. these infinities seemed to take on a reality that they earlier had not possessed. A new world of transfi­ nite numbers was being created. More­ . over, the concept of ''naming, . as we will see, became the link between reli­ gion and mathematics. Even many leading mathematicians were reluctant to accept this new world. How do we define these new infinities? Is it possible to postulate the existence of a mathematical entity before it is de­ fined? According to most monotheistic reli­ gions "God'' is also beyond the com­ prehension of mere mortals, and can­ not be defined. Is it possible to postulate the existence of a deity he­ fore it is defined? If God is, in princi­ ple, beyond human comprehension (and in the Christian and Jewish scrip­ tures there are many such assertions). how. in complete ignorance of his na­ ture, can human beings worship him? What does one worship? Traditionally,

religious believers have side-stepped this question through the use of sym­ bols: prayers, names, rituals, music, relics. scents. tastes, etc. Symbolism is the term given to a perceptible object or activity that represents to the mind the semblance of something which is not shown but is realized by associa­ tion with it. And the importance of sym­ bols both to religion and mathematics is one of the many bonds that brought mathematicians and religious believers together in Russia in the early decades of the last century. Both mathematicians and religious believers use symbols they do not fully master. Names are symbols, and the signifi­ cance of assigning names to objects has been controversial throughout the his­ tory of philosophy and religion. One of the great theological disputes of the mid­ dle ages, that over nominalism, revolved around it. When one invents a name. does one at the same time create some­ thing new, or does one merely give a label to an existing thing? For example, we might ask. ''I s the term 'virtual real­ ity,' so commonly used in computer sci­ ence, a human construction or a tag at­ tached to something already existing?" The issue goes back to the begin­ ning of human thought. In Genesis we are told, "God said, ·Let there be light. ' and there was light.'' He gave the thing a name before he created it. The an­ cient Egyptian God Ptah is described in Memphite theology as creating with his tongue that which he first conceived in his head. Naming God is forbidden in the Jewish tradition, and in the mysti­ cal Kabbala (Book of Creation, Zohar) a large role is assigned to language in the act of creation. In the first verse of the gospel according to St. John we read, "In the beginning was the Word, and the Word was with God, and the Word was God . " Words are names, and one of the leaders of the Russian Name­ Worshippers. the monk Ilarion, said "the name of God is God!'' ("Imia Bozhie est ' sam Bog"). Intellectual and artistic Russia at the end of the nineteenth century and in the first decades of the twentieth was seized with the question of the signif­ icance of symbols. The Symbolist Movement affected ballet, music, liter­ ature, art, and poetry, as the names Di­ aghilev, Stravinsky, Belyi, Stanislavsky, Nemirovich-Danchenko, and Meyer­ hold remind us. Now we should add

the mathematicians Egorov and Luzin to such lists. Indeed, there was even a connection between the literary and mathematical movements. Andrei Belyi, the symbolist poet, was the son of a Moscow mathematician, and he ma­ jored in mathematics at Moscow Uni­ versity, where he studied under Egorov and together with Luzin. He was fa­ miliar with Name-Worshipping. Belyi once wrote an essay called "The Magic of Words, " in which he claimed, "When I name an object with a word, I thereby assert its existence. " We can ask, " Does this apply both to mathematics and to poetry? If the object is a new type of infinity, does that infinity exist just af­ ter you name it?'' At the heart of the Name-Worship­ ping cult was the "Jesus Prayer" Ciis­ usovaia molitua), a religious practice with ancient roots. In the Jesus Prayer the religious believer chants the names of Christ and God over and over again, hundreds of times, until his or her whole body reaches a state of religious ecstasy in which even the beating of the heart and the breathing cycle, are sup­ posedly in tune with the chanted words "Christ" and "God." According to Name­ Worshippers, the proper practice of the prayer brings the worshipper to a state of unity with God through the rhythmic pronouncing of his name. Franny ob­ served in J D. Salinger's novel Fran ny and Zooey that in this state of ecstasy "you get an absolutely new conception of what everything's about. " The Jesus Prayer has always been part of the Russian Orthodox tradition, but it took on an unusual prominence in the late nineteenth century after the publication in 1 884 of a hook titled The Way qf the Pilgrim, later translated into many languages, in which the potency of the prayer was acclaimed. The prayer became popular throughout Russia. Ac­ cording to some sources , the Empress Alexandra and her notorious advisor Rasputin sympathized with the heresy and unsuccessfully tried in 1 9 1 3 to stay the hand of Tsar Nicholas II in arrest­ ing the heretical monks in Athos. But the establishment Orthodox Church won out with its view that the Name­ Worshippers were pagan pantheists who confused the symbols of God with God Himself. Church officials advised Nicholas to squelch the heresy before it hopelessly split the faith and the na­ tion. Since that time the position of the

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

19

Church on Name-Worshipping has re­ mained the same. On the question of whether more than one kind of infinity exists, each of which can be given a name, not all mathematicians agreed with Cantor. For some of them, set theory can not apply to the line, what they called "the con­ tinuum. " The debates became very complex and sometimes very heated. French and Russian mathematicians were leaders in this debate. The French who wrestled with set theory included Emile Borel (187 1-1 956), Rene Baire (1874--1 932), and Henri Lebesgue (18751 944); they were the inheritors of a great and powerful mathematical tradition, and at first they taught the Russians more than they learned from them. Both Egorov and Luzin traveled repeatedly to Paris to talk with their French col­ leagues. They usually lived in the aca­ demic heart of the city, in the Hotel Parisiana, near the Pantheon. The concierge of the building remembered many years later both the devotion of the Russian visitors to their studies and their religious piety. The French tended to be skeptical of set theory, or at least the furthest exten­ sions of it into discussions of new types of infinities. A few of them, such as Borel, were at first attracted to it but gradually became more hesitant. The old French establishment of mathematics, represented by Emile Picard, stoutly re­ sisted. Picard acidly remarked, "Some be­ lievers in set theory are scholastics who would have loved to discuss the proofs of the existence of God with Saint Anselm and his opponent Gaunilon, the monk of Noirmoutiers. " Picard thought that he could dismiss set theory by link­ ing it to discussions of religion, exactly the way the Russians thought they could strengthen it. The French worked within the tradition of Cartesian rationality; the Russians were speculating within the tra­ dition of Russian mysticism. A contrast between the cold logic of the French and the spirituality of the Rus­ sians is not new in the history of cul­ ture. Leo Tolstoy, in War and Peace, compared Napoleon's Cartesian logic in his assault on Russia with his opponent Kutuzov's emotional religiosity. After the critical battle of Borodino, the novelist described the Russian general Kutuzov kneeling in gratitude before a holy icon in a church procession while Napoleon rationalized his "miscalculation." Tolstoy 20

THE MATHEMATICAL INTELLIGENCER

Emile Borel

Rene Baire

saw Borodino as a victory of Russian spirit over French rationalism. Eventually the French mathematicians lost their nerve and yielded the field to their Russian colleagues. The French could not stomach the thought that new infinities could be created simply by naming them, and that these new in­ finities then became legitimate, and even necessary, objects of study by mathe­ maticians. Some of the French actually feared that one could lose one's mind pursuing the problems of set theory ap­ plied to these infinities. They noticed that the founder of the field, Georg Cantor, had a series of attacks of depression af­ ter 1884. Baire, who already had some digestive problems, fell seriously ill in 1898, as if being punished for his flirta­ tion with the new ideas; eventually, he killed himself. Borel, after referring to the illnesses of Cantor and Baire, told his friend Paul Valery that he had aban­ doned set theory "because of the fatigue it caused him, which made him fear and

set theory was to Florenskii a brilliant example of how naming and classifying can bring mathematical breakthroughs. To him a "set" was simply a naming of entities according to an arbitrary mental system, not a recognition of real objects existing in nature. When a mathemati­ cian created a "set" by naming it, he was giving birth to a new mathematical be­ ing. Mathematicians who created sets by naming them, according to Florenskii, were performing an intellectual and re­ ligious act similar to what Name-Wor­ shippers did when they named and wor­ shipped God.

foresee in himself serious illness if he persisted in that work." The Russians did not have these problems. They rejoiced in what they saw as the fusion of mathematics and religion. At the time of the Russian Rev­ olution in 1917 Father Florenskii was liv­ ing in a monastery town near Moscow and he translated the religious ideas of Name-Worshipping into mathematical parlance. He stated his goal as creating a "synthesis between religious and sec­ ular culture. " He expounded the view that "the point where divine and human energy meet is 'the symbol', which is greater than itself." The development of

Henri Lebesgue

When Egorov, Luzin, and their stu­ dents created a new set, they often called it a "named set, " in Russian imen­ noe mnozhestvo. Thus the root word imia (name) occurred in the Russian language in both the mathematical terms for the new types of sets and the reli­ gious trend of imiaslavie ("Name­ Praising," or "Name-Worshipping"). In Luzin's personal papers in the Moscow archives, the historian can see today how obsessed he was with "naming" as many subsets of the continuum as he could. Roger Cooke studied Luzin's pa­ pers and noted that he "frequently stud­ ied the concept of a 'nameable' object and its relationship to the attempted cat­ alog of the flora and fauna of analysis in the Baire classification . . . . Luzin was trying very hard to name all the count­ able ordinals. " At one point Luzin scrib­ bled in infelicitous but understandable French "nommer, c'est avoir individu" ("naming is having individuality"). 1 The circle of eager students at Moscow University which formed around Egorov and Luzin at about the time of the beginning of World War I and continued throughout the early twenties was known as "Lusitania. " This group caused an explosion of mathe­ matical research that still affects the world of mathematics. Lusitania was at first a small secret society, and the place of religion in that society is illustrated by the names the members gave one another; Egorov was "God-the-father," Luzin was "God-the­ son'' and each of the students in the so­ ciety was given the monastic title of "novice. " They all went to Egorov's home, an apartment not far from the university, three times a year: Easter, Christmas, and Egorov's Name-Day (again the emphasis on "names. ") But how long could such a religiously oriented group exist in the Soviet Union, where the campaign against religion was gathering force? In their effort to com­ bat religion, the Communists made no distinction between orthodox believers and heretics. The three men most in­ volved in the effort to link religion and mathematics followed different paths in responding to this threat. Florenskii was the most defiant, refusing to take off his priest's robe, causing the Soviet leader Trotsky to inquire at a meeting they both

attended, "Who is that?" Egorov also continued his religious practices and worked closely with Florenskii in in­ spiring the "True Church" movement aiming at a religious revival in Russia despite the Soviet efforts to suppress re­ ligion. Luzin was much more cautious, refusing to attend meetings of the Name­ Worshippers, and concealing his reli­ gious convictions. Meanwhile, the Moscow School of Mathematics flourished. It grew until it included dozens of young mathemati­ cians, many of them now prominent in the history of mathematics (e. g . , Andrei Kolmogorov, Pavel Aleksandrov, Alek­ sandr Khinchin, Mikhail Lavrent'ev, Lazar Lyusternik, Petr Novikov) . It was inevitable that as the group increased in size it would lose its earlier ethos. Some of the students of Egorov and Luzin were out of sympathy with their teachers' religious impulses. A few were even members of the Young Commu­ nist League. Divisions, rivalries, and ide­ ological disputes began to develop among Moscow mathematicians. In 1 930 Ernst Kol'man, a militant Marxist mathematician who was never a member of Lusitania himself, attacked Florenskii and Egorov in an address to mathematicians, castigating their use of "mathematics in the service of religion, " "mathematics i n the service of priest­ craft." He continued the attack in pub­ lished articles, saying "Diplomaed lack­ eys of priestcraft right under our noses are using mathematics for a highly masked form of religious propaganda. " Responding t o such denunciations, starting around 1 930 the Soviet authori­ ties moved heavily against the Name­ Worshippers. Fortunately, the most im­ portant mathematical work had already been done. They arrested Father Floren­ skii, the main ideologist of mathematical Name-Worshipping, and eventually sent him to a labor camp in the Solovetsky Islands, far north in the Arctic Ocean, where he continued to do scientific work. On December 8, 1937, he was executed by firing squad. In one of his last letters to his grandson, who lives in Moscow to­ day, Florenskii wrote, "Above all I think about you, but with worry. Life is dead." All Florenskii's voluminous writings were removed from Soviet libraries, and even mentioning his name was forbidden.

Tombstone of Egorov in Kazan.

Dmitrii Egorov, president of the Moscow Mathematical Society, was ar­ rested in 1 930 and exiled to a camp near Kazan, on the Volga River. There he went on hunger strike because the prison guards would not permit him to practice his religious faith. Near death, Egorov was sent to a local hospital, where he was recognized by a physi­ cian, the wife of a mathematician named Nikolai Chebotaryov. The two Chebo­ taryovs did everything they could to save Egorov's life, but it was too late. We are told that he died in the arms of Dr. Chebotaryova . Egorov's name, like Florenskii's, was not to be mentioned in Soviet society. The Name-Worshippers became the object of name censorship. The most talented of the mathemati­ cians connected with the religious movement, Nikolai Luzin, was subjected to a show trial, known even today as the "Luzin Affair. " One of the ideologi­ cal charges against him was that he "loved" capitalist France, where he of­ ten worked, and was a friend of the French mathematician Emile Borel. Borel was at that moment Minister of the Navy in the French government, and therefore was obviously a "militarist" ea­ ger for aggression against the Soviet Union. In a great act of heroism, one of the most famous physicists in the Soviet Union, Peter Kapitsa, wrote a confiden­ tial letter to the Soviet leaders Molotov

' Roger Cooke, "N. N. Luzin on the Problems of Set Theory," unpublished draft, January 1 990. pp. 1 -2, 7. Luzin's notes are held in the Archive of the Academy of Sci­

ences of the USSR, Moscow, fond 606, op. 1 , ed. khr. 34.

©

2007 Springer Science +Business Media, Inc., Volume 29, Number 4 , 2007

21

Emile Borel

and Stalin, pleading for mercy for "one of our greatest mathematicians, known throughout the world." Luzin was repri­ manded but miraculously saved, and he continued mathematical work until his death in 1 950, although no longer in set theory but instead in applied mathe­ matics, and no longer in communication with his French friends. The persecution of the Name-Worshippers continued throughout the Soviet period, with ar­ rests as late as the 1 980s up to the time of the Gorbachev years, starting in 1 985. In the summer of 2004 Loren Gra­ ham met with a prominent mathemati­ cian in Moscow known to be in sym­ pathy with Name-Worshipping. The mathematician implied he was a Name­ Worshipper without stating it outright. His a partment was decorated with the symbols of Name-Worshipping, includ­ ing photographs of its leaders. His li­ brary was filled with rare books and ar­ ticles on Name-Worshipping. Graham asked if it would be possible for him to witness a Name-Worshipper in the je­ sus Prayer trance. "No," replied the mathematician, "this practice is very in­ timate, and is best done alone. For you to witness it would be considered an intrusion. However, if you are looking for some evidence of Name-Worship­ ping today I would suggest that you visit the basement of the Church of St. Tatiana the Martyr. In that basement is a spot that has recently become sacred to Name-Worshippers. " Graham knew o f this church; forty­ five years earlier, as an exchange stu­ dent, he had attended a student dance in the building after the church itself had been eliminated by Soviet authorities 22

THE MATHEMATICAL INTELLIGENCER

and converted into a student club and theater. Now, in the post-Soviet period, it has been restored as the official church of Moscow University, as it had been be­ fore the Revolution. It is located on the old campus near the Kremlin, in a build­ ing next to the one that housed the De­ partment of Mathematics when Egorov and Luzin dominated that department. It is the church where they often prayed. Graham asked the mathemati­ cian, "When I go into the basement, how will I know when I have reached the sa­ cred spot?" The mathematician replied, "You will know when you get there." The next day Graham went to the Church of St. Tatiana the Martyr, and made his way to the basement. There he found a particular corner where the photographs of Father Florenskii and Dmitri Egorov, founders of mathemati­ cal Name-Worshipping, faced each other, and he knew that he was in the place where Name-Worshippers liked to come, alone, to practice the jesus Prayer. But six months later, in Decem­ ber 2004, he visited the basement again and found that the sacred spot had been eliminated by the Church, which had fi­ nally realized that Name-Worshippers were coming to the basement to cele­ brate their "heresy. " Now an official chapel of the Church occupies the base­ ment, with a priest guarding it and en­ suring the orthodoxy of all worshippers. The Jesus Prayer is not practiced there any more. Thus, the struggle over Name­ Worshipping continues today. This story is a tragic and dramatic one, like many stories about Stalinist Russia, but this one also contains a deep philosophic question about the nature of mathematics. Where do the concepts and objects used by mathematicians come from? Are they invented in the brains of mathematicians, or are they in some sense discovered, perhaps in a platonistic world? Florenskii, Egorov, and Luzin believed that the objects of mathematics are invented not through analysis but through mystical inspiration and naming. They thought that French mathematicians like Baire, Borel, and Lebesgue were mistaken in their com­ mitment to Cartesian rationalism. We were trained in the tradition of Western rationalism, and we do not share the mysticism of the Russian founders of the Moscow School of Mathematics. We would point out that naming is not identical with creating.

We can name "unicorns·· but that does not make unicorns real. We also note that the basic idea behind Name-Wor­ shipping is not new; there are many similarities between Name-Worshipping and other types of religious and medi­ tation practices, including variants of Hinduism, Buddhism, Judaism, and Is­ lam. The practice of "'talking in tongues·· of Protestant evangelicals is also related. The endpoint, as in Name-Worshipping, is a state of glottal ecstasy. We do not see this state as one usually conducive to scientific creativity. But the reason that this episode in Russian history is different is that in this case mysticism may actually ha\'e helped science. In the early twentieth century, mathematicians truly differed among themselves about the existence of vari­ ous infinite sets. The French, with their secular, rationalist worldview, had neither the courage nor the motivation to enter the frightening world of the hierarchy of infinities. The French feared what the Russians exalted. And in the hands of the Russians what earlier seemed like fanci­ ful unicorns became useful mathematical objects. (A similar situation may have oc­ CI.med more recently in string theory, when Anglo-Saxon and Russian mathe­ maticians and physicists were ahead of French scholars.) If we had been mathematicians in the period 1 900-1 930 we surely would have hesitated along with the French mathe­ matical establishment, constrained by our rationalism. The Russians, however, be­ lieved that they had absolute freedom to invent mathematical objects and to give their inventions names. Following their approach, the Russians created a new field, descriptive set theory, at a time when mathematicians elsewhere faltered. And the Moscow School of Mathematics, founded by Egorov and Luzin, still exists today. And the significance of their achievement is still with us. Loren Graham Program in Science, Technology, and Society Room E5 1 - 1 28 Massachusetts Institute of Technology Cambridge, MA 02 1 39 Jean-Michel Kantor lnstitut de Mathematiques de Jussieu, Case 247, 4 Place Jussieu 75252 Paris Cedex, France [email protected]. fr

I.�Fjj.i§,@hi$11@%§[email protected],1§.id

Th e H eptago n to the Sq uar e, and Ot her W i l d Tw ist s G REG N . FREDERICKSON

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.

M i c hael Kleber and Ravi Vaki l ,

geometric dissection is a cutting of a geometric figure into pieces that can be rearranged to form another figure [ 1 2 , 22]. Such visual demonstrations of the equivalence of area span from the times of the ancient Greeks [3, 7] to the flowering of Arabic­ Islamic mathematics [ 1 , 4, 27, 29] to the heyday of mathematical puzzle columns in newspapers and magazines [8, 9, 24, 25] to the appearance of articles on the \\'orid-wide web [31]. It has long been known that two polygons of equal area can be dissected in a finite number of pieces [5, 18, 23, 32]. During the last 100 years. the emphasis has generally been on minimizing the number of pieces for any given dissection. As dissection methods have become more sophisticated, attention has also been paid to special properties. Most notable is the property that all pieces of a dissection be connected by hinges, so that when the pieces are swung one way on the hinges, they form one fig­ ure. and when swung the other way on the hinges, they form the other figure.

Editors

A hundred years ago, Henry Dudeney demonstrated a hinged dissection of an equilateral triangle to a square [ 10]. Since then, there have been an increas­ ing number of such dissections [2, 6, 13, 14, 20, 2 1 , 30, 33] culminating in a whole book on the subject [ 1 5] . An open problem is whether for any two polygons of equal area there is a swing­ hinged dissection in a finite number of pieces. Other types of hinges have also drawn attention. A twist hinge has a point of rotation on the interior of the line segment along vvhich two pieces touch edge-to-edge. It allows one piece to be flipped over relative to the other, using 1 80° rotation through the third di­ mension. Pieces A and B (with exag­ gerated thickness) are twist-hinged to­ gether in Figure 1 . The twist-hinged dissection o f a n el­ lipse to a heart (Figure 2) is a direct ap­ plication. We mark any piece that ends up turned over with an ·· * ·· on one side and a " * " on the other. A few isolated dissections [ 1 1 , 26, 28] were the only

Contributions are most welcome.

·

� A '

-

,

- · -

·

- · · - · - ·

-

Figure I . A twist hinge for pieces A and B .

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford Univers ity,

Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-m a i l : [email protected] nford.edu

Figure 2. Twist-hinged dissection o f a n ellipse to a heart.

© 2007

Springer Science+Business Media, Inc., Volume 29. Number 4, 2007

23

examples of twist-hinged dissections prior to a more concerted search [13, 14, 1 5] . It is also open whether for any two polygons of equal area there is a twist-hinged dissection in a finite num­ ber of pieces. In this article we shall concentrate on extending the range of twist-dissec­ tions of one polygon to another. Com­ pletely unanticipated are our 10-piece twist-hinged dissection of a regular hep­ tagon to a square and our 6-piece (!) twist-hinged dissection of a regular hexagon to a square. Our other new twist-hinged dissections are surprising as well, using a variety of techniques that allow us to produce the first twist­ hinged dissections for certain pairs of figures or to reduce the number of pieces compared with previous twist­ hinged dissections. We shall go well be­ yond the approach of converting a swing-hinged dissection that is "hinge­ snug" to be twist-hinged [15]. Taken to­ gether, these wild dissections should make you flip! In the next section we shall review the technique of crossposing strips to produce dissections with some swing hinges and also a technique to replace swing hinges with twist hinges. In the remaining sections, we shall identify twist-hinged dissections of certain reg­ ular polygons, first to squares, then to equilateral triangles, and finally to reg­ ular hexagons. These dissections were first presented in March 2006 at the Sev­ enth Gathering for Gardner (G4G7), for which the twist-hinged dissection of the heptagon to the square was particularly apropos. Animations from that pres­ entation are posted on the webpage: http://www . cs.purdue.edu/homes/gnf/ book2/rni_anims.htrnl

Crossposing Strips and Converting Swing Hinges Let's first review two fundamental tech­ niques for creating swing-hinged and twist-hinged dissections. An effective method for dissecting one polygon to another is the strip technique [12, 22]. We first cut each polygon into pieces that we can rearrange to form a strip element. We then fit copies of this element to­ gether in a regular fashion, forming a strip that stretches infinitely in two op­ posite directions. We then crosspose the strip for one polygon on top of the strip for the other polygon, so that the com-

24

THE MATHEMATICAL INTELLIGENCER

,

/

/

'

"'

'

/

/

'

'

/

/

v

/

/

""

/

/

'

/

'

, "'

/

'

/

/

/

/

/

/

/

/�'

/ '

'

'

' '

">' "'

/

/

/

/

/

/

/

/

Figure 3. Crossposition of triangles and squares.

Figure 4. Henry Dudeney's swing-hinged dissection of a triangle to a square.

Figure 5. Stealing isosceles triangles in the dissection of a triangle to a square.

Figure 6. Twist-hinged dissection of a triangle to a square.

Figure 7. Intermediate configurations for a twist-hinged triangle to a

square.

Figure 8. Heptagon to square.

mon area is precisely the area of one of the two polygons, or is double that area. When the common area is double, then crossposition should have rotational symmetry about certain points. We see the strip technique applied to an equi­ lateral triangle and a square in Figure 3. The crossposition leads to the swing­ hinged dissection in Figure 4, which Henry Dudeney first described [10] . The pieces are swing-hingeable because every line segment in one strip that crosses a line segment in the other strip is either on the boundary of the strip or has the crossing at a point of rotational symmetry in its strip. The small dots in Figure 3 identify points of rotational symmetry where such crossing occurs. Two pieces connected by a swing hinge are hinge-snug if they are adjacent along different line segments in each of the figures formed, and each such line segment has one endpoint at the hinge [1 5). This property enables us to replace a swing hinge by two twist hinges, by stealing an isosceles triangle from each piece, unioning the two pieces together, and attaching the twist hinges to the new piece. We call this technique hinge con­ version . Using it on the swing-hinged dissection in Figure 4, we steal isosceles triangles in Figure 5, where the dashed edges indicate the bases of the triangles. We then glue the isosceles triangles to­ gether to produce, in Figure 6, a 7-piece twist-hingeable dissection of an equilat­ eral triangle to a square [15]. Intermediate configurations are in Figure 7. On the left, we see the lower left corner of the triangle flipped up, us­ ing a pair of twists. Then on the right, we see the lower right corner of the tri­ angle similarly flipped up. Flipping the right corner of what results will then give us the square. With the fundamental techniques of crossposition and hinge conversion as our base, we are now set to introduce additional techniques that will help to handle a variety of challenging problems.

Regular Polygons to Squares

Figure 9. Twist-hinged heptagon to square.

We shall first consider dissections of various regular polygons to squares, and immediately tackle a dissection that should be rather challenging: a heptagon to a square. There is an unhingeable dis­ section (solid lines in Figure 8) due to Gavin Theobald which uses 7 pieces [12, Figure 1 1 .30]. At first it doesn't look too

© 2007 Springer Science + Business Media,

Inc., Volume 29, Number 4 , 2007

25

promising, since there are just two po­ sitions where we can place swing hinges: at the end of a shared edge between pieces E and F, and at the end of a shared edge between pieces C and E. Amazingly, we can convert this dis­ section to a 1 0-piece twist-hinged dissec­ tion. First, enlarge piece G by annexing an isosceles triangle from piece F. Next, we carve an isosceles trapezoid out of piece B. We twist-hinge this trapezoid with pieces A and C and with what re­ mains of piece B. Using the trapezoid al­ lows pieces A and C to be interchanged. Dotted lines indicate the isosceles trian­ gle and the isosceles trapezoid. Next, convert the swing hinge be­ tween pieces E and F. Then convert the swing hinge between pieces C and E . A remarkable byproduct of this latter conversion is that we can twist-hinge piece D with the little right triangle cre­ ated by the conversion. The resulting dissection is shown in Figure 9. We next attempt to find a twist­ hingeable dissection of a regular hexa­ gon to a square. There are any number of 5-piece dissections of a hexagon to a square, and even more 6-piece swing­ hingeable dissections. For inspiration we turn to the previous dissection of a heptagon, in which piece B is adjacent to pieces A and C, allowing those pieces to switch positions. Most promising is Gavin Theobald's 5-piece dissection (solid lines in Figure 10) [31]. Although no piece is swing-hinge­ able with any other, we can add twist hinges rather easily. Because piece B is an isosceles triangle, we can twist-hinge it with pieces A and C. Furthermore, we can cut an isosceles trapezoid out of piece D , and twist-hinge this piece with both pieces C and E. Finally, we can then twist-hinge the isosceles trapezoid with what remains of piece D , produc­ ing the 6-piece twist-hinged dissection in Figure 1 1 . With the heptagon and the hexagon being so cooperative, can we use sim­ ilar techniques to find a twist-hingeable dissection of a pentagon to a square? We have found a 7-piece unhingeable dissection (Figure 1 2) of a pentagon to a square that is similar in some ways to Theobald's previous dissections. Actu­ ally, it's difficult to see the seventh piece, piece B, which is a long and very

26

THE MATHEMATICAL INTELLIGENCER

Figure I 0. Partially twist-hingeable hexagon to a square.

Figure I I . Twist-hinged hexagon to a square.

Figure 1 2. Pentagon to square.

Figure 1 3 . Pentagon to a rectangle.

Figure 1 4. Twist-hinged pentagon to rectangle.

\

\ \ \

\ \

\

\ \" \

\ \ \

\ \ \

Figure I 5. Crossposition of a

\ \

\

{ 1 2/2} to a square.

Figure 1 6. Swing- and twist-hingeable

Figure 1 7. Twist-hinged

\ \ \ \ .A

\

{ 1 2/2} to a square.

{ 1 2/2} to a square.

thin right triangle that is to the right of pieces A and C in both the pentagon and the square. The long leg of the tri­ angle has length equal to the sidelength of the square, and the length of the short leg of the triangle is .002789 of the length of the side of the square. We can convert this dissection into an 1 1piece twist-hinged dissection, much the way that we did in Figure 9 . T o b e able t o visualize the process that produces the twist-hinged dissec­ tion, we will dissect a pentagon to a rectangle whose length is 1 . 1 559 times its width, as shown in Figure 1 3 . Pieces A and D can readily be swing­ hinged together, as can pieces C and E , and pieces E and F. Furthermore, there is an isosceles triangle that we can re­ move from piece F and combine with piece G that would allow us to twist the new piece G and what remains of piece F. Next, we can carve an isosce­ les trapezoid out of piece B, which we can twist-hinge with pieces A, C, and what remains of piece B. The isosceles triangle and the isosceles trapezoid are indicated by dotted lines in Figure 1 3 . Finally, w e convert the three swing hinges, producing the 1 1 -piece twist­ hinged dissection in Figure 1 4 . The cor­ responding twist-hinged dissection of the pentagon to the square is com­ pletely analogous. We will tackle one final dissection to a square, namely that of a { 1 2/2}, which is a 1 2-pointed star with every second vertex connected. This will involve somewhat different techniques from the previous three dissections. The swing­ hingeable dissection in [ 1 5 , Figures 1 1 .53 and 1 1 .54] is not hinge-snug. However, we can create a strip element as a result of doing certain twists, and from its crossposition (Figure 1 5) with a square strip get a 1 2-piece swing- and twist-hinged dissection that is hinge­ snug (Figure 16). The six pieces A through F are con­ nected by swing hinges, and we can convert each of the five swing hinges to a twist hinge. This would give a 1 7piece twist-hinged dissection. We can do better by cutting a zigzag piece out of pieces C and D by taking a rectangle from C and a rectangle from D and combining them. We use this one piece to twist-hinge pieces B, C , D, and E together. As before, we finish by con-

© 2007 Springer Science+ Business

Media. Inc., Volume 2 9 , Number 4, 2007

27

verting the swing hinge between pieces A and B, and the swing hinge between pieces E and F. We thus get a 1 5-piece twist-hinged dissection (Figure 17).

Regular Polygons to Equilateral Triangles Let's next attempt twist-hinged dissec­ tions of regular polygons to equilateral triangles. Gavin Theobald [31] gave an unhingeable dissection of a regular hep­ tagon to an equilateral triangle which is based on a crossposition very similar to that shown in Figure 18. The resulting 8-piece dissection is given in solid lines in Figure 19. Note that piece F is an equilateral triangle of one quarter the area of the given equilateral triangle. We can convert this dissection to a 1 5-piece twist-hinged dissection. First, enlarge the small isosceles triangle (piece B) by annexing an isosceles triangle from piece A. Next, convert the swing hinge between pieces A and C and then con­ vert the swing hinge between pieces C and E. As we have seen previously, a byproduct of the latter conversion is that we can twist-hinge piece D with the lit­ tle right triangle created by that conver­ sion. Then convert the two swing hinges between piece F and pieces E and G. To handle piece H in the heptagon, we swing-hinge it to piece E and then convert the swing hinge. To make room for this piece in the full-size equilateral triangle, we cut an obtuse triangle out of piece A. Finally, we use a rectangu­ lar piece to twist-hinge the new obtuse triangle onto piece F. The new obtuse triangle takes the original place of piece H in the equilateral triangle. We see the resulting 1 5-piece twist-hinged dissec­ tion in Figure 20. We find the next dissection, of a reg­ ular pentagon to a triangle, a bit easier. Lindgren [22] observed how to use a crossposition to derive Goldberg's dis­ section of a pentagon to a triangle [19]. We see a slight variation of that cross­ position in Figure 2 1 . The triangle strip is what Lindgren called a T-strip, and small dots indicate the points of 2-fold rotational symmetry. Edges of pieces in the strips cross only at strip boundaries or at points of symmetry. Thus pieces A, B , C, and D in Figure 22 can be swing-hinged. Pieces E and F can be swing-hinged as well. To connect all of the pieces with

28

THE MATHEMATICAL INTELLIGENCER

'

'

'- ' ' '

- ­

'

'

'

' ...

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

Figure 1 8. Crossposition of a heptagon to triangle.

Figure 1 9. Heptagon to triangle.

Figure 20. Twist-hinged heptagon to triangle.

'

I I A: / ' ' I ' I , I ' ' I ' I , I ' ' I

,''�

I I

,

---

'

--

'

'

I I I I I I I '

'

'

'

'

---

'

'

'

'

I

_ ;:, I

I I I I I I

Figure 2 1 . Crossposition for a pentagon to a triangle.

Figure 2 2 . Partially-hingeable pentagon to a triangle.

Figure 23. Twist-hinged pentagon to a triangle.

hinges, we would like to have piece E connect with the top of piece D, rather than touch the bottom of piece D. We can accomplish this by cutting an isosceles trapezoid from D , which in­ terchanges top and bottom of the left side of piece D, using a twist hinge. Besides the twist hinge, there are five swing hinges, each of which converts

to an additional piece plus two more twist hinges. This produces the 1 2-piece twist-hinged dissection in Figure 23.

Regular Polygons to Regular Hexagons Finally, let's attempt twist-hingeable dis­ sections of regular polygons to regular hexagons. Gavin Theobald [31] gave an

8-piece unhingeable dissection of a reg­ ular heptagon to a hexagon. To be able to produce a twist-hinged dissection, we shall use his heptagon strip with a hexagon strip in the crossposition in Figure 24. This gives the 9-piece par­ tially-hinged dissection (solid lines) in Figure 25. We can convert this dissection to a 1 5-piece twist-hinged dissection. First, enlarge the small isosceles triangle (piece G) by annexing an isosceles tri­ angle from piece F. Next, convert the swing hinge between pieces E and F and then convert the swing hinge be­ tween pieces C and E. Again, a byprod­ uct of the latter conversion is that we can twist-hinge piece D with the little right triangle created by that conversion. Finally, convert the four remaining swing hinges, between pieces A and B , B and C, F and H , and H a n d I . W e see the resulting 1 5-piece twist-hinged dis­ section in Figure 26. We next dissect an {8/3} to a hexa­ gon. (An {8/3} is an 8-pointed star with every third point connected.) There is a 10-piece strip dissection of an {8/3} to a hexagon. We use the same strip for the hexagon as we did in the preced­ ing dissection. We form the strip for the {8/3} by cutting off four of its points and nestling them between the other four. The crossposition (Figure 27) treats the hexagon strip as a T-strip and produces a partially-hingeable dissection (solid lines in Figure 28). Note that pieces A and B swing-hinge together, as do pieces B and C, and C and D . Also, pieces D and I twist-hinge together, as do pieces B and ] . Assuming we have positioned the hexagon strip appropriately, we can then steal small isosceles right triangles from pieces E and F and attach them to piece B, allowing us to twist-hinge what remains of E and F onto B. We can steal the same-size isosceles right triangles from pieces G and H. We need an in­ termediate piece that results from merg­ ing the two isosceles triangles with a rectangle from piece D . This new piece effects the appropriate shift of pieces G and H relative to piece D . We get a n 1 1-piece swing- and twist­ hinged dissection that is still hinge-snug with three swing hinges. Converting the the three swing hinges, we get a 14-piece twist-hinged dissection (Figure 29).

© 2007 Springer Science+ Business Media, Inc . . Volume 29, Number 4, 2007

29

I I I

I I I

Figure 24. Crossposition for a heptagon to a hexagon.

Figure 2 5 . Partially-hingeable heptagon to a hexagon.

Figure 26. Twist-hinged heptagon to a hexagon.

30

THE MATHEMATICAL INTELUGENCER

....

....

....

....

'

'

,

'

,

'-::: ....- - -< ....

....

....

I

I

I

,,

\ .... \ \

I

I

I

....

....

....

....

,'L - - � .... ....

'

'

....

'

....

.... , .... .... \

\

\ .... \ ...

....

'

Figure 27. Crossposition for an

'

{8/31

'

{8/31

Figure 28. Partially-hingeable

Figure 29. Twist-hinged

'

....

,

,...._

_

....

....

-{ ....

I

I

\ \ .... \ ....

I

,

to a hexagon.

{8/31

I

I

\

....

....

....

>- - - �..... .... ' I

'

'

,

....

....

....

....

to a hexagon.

to a hexagon.

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 4, 2007

31

Our last dissection is of a regular pentagon to a hexagon. There is a swing-hinged dissection of a hexagon to a pentagon in [ 1 5 , Solution 1 1 .2] which uses 10 pieces and is hinge-snug. A direct conversion would give a 19piece twist-hinged dissection. However, we can do better if we use the cross­ position from [22, Appendix D], as in Figure 30. We thus get the 7-piece dissection (solid lines) in Figure 3 1 . We then need to hinge piece G (from the pentagon) onto the rest of the pieces. We do this as follows: We split an isosceles trape­ zoid from piece F and place a twist hinge between the trapezoid and what remains of piece F. If piece G is swing­ hinged to the trapezoid, then we are able to bring piece G from its position in the pentagon to the appropriate po­ sition in the hexagon. Noting that pieces A and B form an isosceles trapezoid in the hexagon, we flip these pieces over in the hexagon and connect piece B to piece C with a twist hinge. There are five swing hinges, between pieces A and B, C and D, C and E, E and F, and G and the trapezoid. Each of the five swing hinges con­ verts to an additional piece plus two additional twist hinges, producing a 13piece twist-hinged dissection (Figure 32). Note that the new triangle between pieces C and E is so small that there is no room to mark one side with an "*" and the other side with a " * " .

'

'

\

\

\

\

I

'- - - - - �

\

'

'

'

'

'

'

I

I

I

'

' \

1 I I I

v '

'

'

'

� - - - -

'

'

'

'

'

'

I

I

I

I

'

'

'

'

'

'

'

'

�----,

Figure 30. Crossposition for a hexagon to a pentagon.

\

\

\

\

'

'

Figure 3 I . Partially-hingeable hexagon to a pentagon.

Conclusion We have identified some surprising new twist-hinged dissections, using special­ purpose techniques. Further examples and adaptations of these techniques appear in a companion paper [17]. All of these dissections, and a few others, are summarized in Table 4 . 1 of [16, chap. 4], in which it is discussed how to convert them into yet another type of hinged dissection, namely, piano­ hinged dissections. REFERENCES

Figure 32. Twist-hinged hexagon to a pentagon.

[1 ] Abu'I-Wata.' ai-BOzjanT. Kitab fTma yahtaju al-sani' min a' mal al-handasa (On the

deney

Geometric Constructions Necessary for the

Akiyama, M . Kano, and M. Urabe, editors,

Artisan). Mashdad: Imam Riza 37, copied

Discrete

in the late 1 Oth or the early 1 1 th century.

Japanese Conference, JCDCG'98, LNCS,

Dublin, 1 889. [4] Anonymous. FT tadakhul al-ashkal al-mu­

Persian manuscript.

volume 1 763, pages 1 4-29. Springer-Ver­

tashabiha aw al-mutawafiqa (Interlocks of

lag, 2000.

Similar or Complementary Figures). Paris:

[2] Jin Akiyama and Gisaku Nakamura. Du-

32

THE MATHEMATICAL INTELLIGENCER

dissection and

of

polygons.

Computational

In

J.

Geometry,

[3] George Johnston Allman. Greek Geometry from Thales to Euclid. Hodges, Figgis & Co. ,

Bibliotheque Nationals, ancien fonds. Per­ san 1 69, ff. 1 80r-1 99v.

[5] Farkas Bolyai. Tentamen juventutem. Typis Collegii Reformatorum per Josephum et Simeonem Kali, Maros Vasarhelyini, 1 832. [6] Donald L. Bruyr. Geometrical Models and Demonstrations.

J. Weston Walch, Port­

land, Maine, 1 963. [7] Moritz

Cantor.

[1 5] Greg N. Frederickson. Hinged Dissec­

column in Sunday edition of Philadelphia

University Press, New York, 2002.

Inquirer, October 23, 1 898-1 901 .

[1 6] Greg N . Frederickson . Piano-hinged Dis­

Vorlesungen

Ober

[26] Ernst Lurker. Heart pill. 7 inch tall model

sections: Time to Fold. A K Peters, Welles­

in nickel-plated aluminum, limited edition

ley, Massachusetts, 2006.

of 80 produced by Bayer, in Germany,

[1 7] Greg N. Frederickson. Unexpected twists in geometric dissections. Graphs and Combi­

Geschichte der Mathematik, volume 1 . B .

[25] Sam Loyd. Mental Gymnastics. Puzzle

tions: Swinging and Twisting. Cambridge

1 984. [27] Alpay Ozdural. Mathematics and arts: Connections between theory and practice

natorics, 23 (Suppl): 245--258, 2007.

in the medieval Islamic world. Historia

[1 8] P. Gerwien. Zerschneidung jeder beliebi­

G . Teubner, Stuttgart, third edition, 1 907.

Mathernatica, 2 7 : 1 7 1 -201 , 2000.

gen Anzahl von gleichen geradlinigen

[8] Henry E. Dudeney. Perplexities. Monthly

Figuren in dieselben StUcke. Journal fOr

puzzle column in The Strand Magazine from

die reine und angewandte Mathematik

forming geometric shapes. U . S . Patent

May, 1 9 1 0 through June, 1 930.

(Grelle 's Journal), 1 0:228-234 and Taf. I l l ,

4,392,323, 1 983.

[9] Henry E. Dudeney. Puzzles and prizes. Col­

[29] Aydin Sayili. Thabit ibn Ourra's general­

1 833.

umn in the Weekly Dispatch, April 1 9 ,

ization of the Pythagorean theorem. Isis,

[1 9] Michael Goldberg . Problem E972: Six

1 896-March 27, 1 904.

51 :35-37, 1 960.

piece dissection of a pentagon into a tri­

[1 0] Henry Ernest Dudeney. The Canterbury Puzzles and Other Curious Problems. W.

Heinemann , London, 1 907.

angle. American Mathematical Monthly,

adapted to define a plurality of objects or shapes. U.S. Patent 4,542,63 1 , 1 985. [ 1 2] Greg N . Frederickson. Dissections Plane & Fancy. Cambridge University Press,

New York, 1 997 .

[30] H . M. Taylor. On some geometrical dis­ sections.

59: 1 06-- 1 07, 1 952. [20] Anton Hanegraaf. The Delian altar dissec­

[ 1 1 ] William L. Esser, Ill. Jewelry and the like

tion. Elst, the Netherlands, 1 989.

Mathematics,

HTMUindex . html.

Australian Mathematics Teacher, 1 6:64--65,

[32] William Wallace, editor. Elements of Geom­

etry. Bell & Bradfute, Edinburgh, eighth edi­

[22] Harry Lindgren. Geometric Dissections. D .

tion, 1 831 . [33] Robert C. Yates. Geometrical Tools, a

Jersey, 1 964.

tions that swing and twist. In Discrete and

[23] (Mr.) Lowry. Solution to question 269,

Computational Geometry, Japanese Con­

[proposed] by Mr. W. Wallace. I n Thomas

ference,

Leybourn, editor, Mathematical Reposi­

Lecture Notes in

of

http://home.btconnect.corn/GavinTheobald!

[2 1 ] H . Lindgren. A quadrilateral dissection . 1 960.

Messenger

35:81 -1 01 ' 1 905. [3 1 ] Gavin Theobald. Geometric dissections.

Van Nostrand Company, Princeton, New

[1 3] Greg N. Frederickson. Geometric dissec­

JCDCG'OO,

[28] Erno Rubik. Toy with turnable elements for

Computer Science, LNCS 2098, pages

tory, volume I l l , pages 44--46 of Part I. W.

1 37-1 48. Springer-Verlag, 2000.

Glendinning, London, 1 8 1 4.

Mathematical Sketch and Model Book.

Educational Publishers, St. Louis, 1 949.

Department of Computer Science Purdue University

[1 4] Greg N. Frederickson. Geometric dissec­

[24] Sam Loyd . Weekly puzzle column in Tit­

tions now swing and twist. Mathematical

Bits, starting in Oct. 3, 1 896 and contin­

USA

lntelligencer, 23(3):9-20, 2001 .

uing into 1 897.

e-mail: [email protected]

West Lafayette, Indiana 47907

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

THE MATHEMATICAL INTELLIGENCE& Please send your old address (or label) and new address to:

Springer Joumal Fulflllment Services P.O. Box 2485, Secaucus, NJ 07094-2485 U.S.A.

Please give

us

six

weeks notice.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

33

Ricc i Flow and the Poincare Conjecture SIDDHARTHA GADGIL AND HARISH SESHADRI

he field of Topology was born out of the realisation that in some fundamental sense, a sphere and an el­ lipsoid resemble each other but differ from a torus. A striking instance of this can be seen by imagining water flowing smoothly on these. On the surface of a sphere or an ellipsoid (or an egg), the water must (at any given in­ stant of time) be stationary somewhere. This is not so in the case of the torus. More formally, in topology we study properties of (classes of) spaces up to certain equivalence relations. For instance, one studies topological spaces up to homeomor­ phism, or smooth manifolds up to diffeomorphism. A fun­ damental problem in topology is thus to classify a class of topological spaces, say smooth manifolds of a given di­ mension, up to the appropriate equivalence relation. The first interesting case is that of dimension 2, i.e., sur­ faces. In the case of surfaces (more precisely closed sur­ faces), there are two infinite sequences of topological types. The first sequence, the so-called orientable surfaces, con­ sists of the sphere, the torus, the 2-holed torus, the 3-holed torus, and so on (see figure 1). The non-orientable surfaces are obtained from the sphere by removing interiors of dis­ joint discs and gluing Mobius bands to the resulting bound­ ary components-with the surfaces differing according to how many discs have been replaced by Mobius bands. One would like to have a similar classification in all dimensions. However, due to fundamental algorithmic issues, it is im­ possible to have such a list in dimensions 4 and above. Manifolds of dimension 3 are also too complex to be re­ duced to such a list. Nevertheless, one may hope that some features of the classification of surfaces continue to hold in higher di­ mensions. I n particular, there is a simple way to charac­ terise the sphere among surfaces. If we take any curve on the sphere, we can shrink it to a point while remaining on

T

34

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Mecia. Inc.

the sphere. A space with this property is called simply con­ nected. A torus is not simply connected, as a curve that goes around the torus cannot be shrunk to a point while remaining on the torus. In fact, the sphere is the only sim­ ply connected surface. In 1 904, Poincare raised the question whether a similar characterisation of the (3-dimensional) sphere holds in di­ mension 3. This has come to be known as the Poincare conjecture. It can be formulated as follows.

CONJECTURE (POIN CARE) Any closed, simply connected, smooth 3 -manifold is diffeomorphic to the 3 -dimensional sphere s3. Note that by a theorem of Moise, every 3-manifold has a unique PL (piecewise-linear) structure, that is, it can be homeomorphic to only one complex of 3-dimensional poly­ topes, up to equivalence. Further, work of Kervaire-Milnor, Munkres, Hirsch, Smale, and others gives a very good un­ derstanding of the relation between PL and smooth struc­ tures, which in particular implies, using the theorem of Moise, that every 3-manifold has a unique smooth struc­ ture. Hence the above conjecture is equivalent to the state­ ment that every closed, simply connected topological 3manifold is homeomorphic to the 3-dimensional sphere 53. As topology exploded in the twentieth century, several attempts were made to prove this (and some to disprove it). However, at the turn of the millennium this remained

Figure I . The first three orientable surfaces.

unsolved. Surprisingly, the higher-dimensional analogue turned out to be easier and was solved by Smale and Freed­ man. For a brief history of the Poincare conjecture, see [8] . In 2002-2003, three preprints ([10] , [ 1 1] , and [ 1 2]), rich in ideas but frugal with details, were posted by the Rus­ sian mathematician Grisha Perelman, who had been work­ ing on this in solitude for seven years at the Steklov Insti­ tute. These were based on the Ricciflow, which had been introduced by Richard Hamilton in 1 982. Hamilton had de­ veloped the theory of Ricci flow through the 1 980s and 1 990s, proving many important results and developing a programme [6] which, if completed, would lead to the Poin­ care conjecture and much more. Perelman introduced a se­ ries of highly original ideas and powerful techniques to complete enough of Hamilton's programme to prove the Poincare conjecture. It has taken three years for the mathematical community to assimilate Perelman's ideas and expand his preprints into complete proofs. Recently, a book [9] containing a com­ plete and mostly self-contained proof of the Poincare con­ jecture has been posted. An earlier set of notes which filled in many details in Perelman's papers is [7]. Another article [2] regarding the proof of the geometrisation conjecture (see below) has also appeared (see also its erratum [3] and a note from the editors in the same issue). In this article we attempt to give an exposition of Perel­ man's work and the mathematics that went into it. ACKNOWLEDGMENTS

It is our pleasure to thank Kalyan Mukherjea for several help­ ful comments that have considerably improved the exposition, and Gerard Besson for inspiring lectures on Perelman's work. We also thank C. S. Aravinda, Basudeb Datta, Gautham Bha­ rali, and Joseph Samuel for helpful comments.

Why the Poincare Conjecture Is Difficult Both the plane and 3-dimensional space are simply con­ nected, but with an important difference. If we take a closed, embedded curve in the plane (i.e., a curve which does not cross itself), it is the boundary of an embedded disc. However, an embedded curve in 3-dimensional space may be knotted (see figure 2). As we deform a knotted curve to a circle, along the way it must cross itself.

Figure 2. A knotted curve.

Thus, an embedded curve in a simply connected 3-man­ ifold M may not bound an embedded disc. Furthermore, such a curve may not be contained in a ball B in M. While embedded discs are useful in topology, immersed discs (i.e. , discs that cross themselves) are not. I t is this which makes it hard to use the hypothesis of simple connectivity, and thus to prove the Poincare conjecture (in dimension 3). The analogue of the Poincare conjecture in dimensions 5 and above is easier than in dimension 3 for a related rea­ son. Namely, any (2-dimensional) disc in a manifold of di­ mension at least 5 can be perturbed to an embedded disc, just as a curve in 3-dimensional space can be perturbed so that it does not cross itself. What made Perelman's proof, and Hamilton's pro­ gramme, possible was the work of Thurston in the 1 970s, where he proposed a kind of classification of 3-manifolds, the so called geometrisation conjecture [ 1 3] . Thurston's geometrisation conjecture had as a special case the Poin­ care conjecture, but being a statement about all 3-mani­ folds, it could be approached without using the hypothe­ sis of simple connectivity. However, most of the work on geometrisation in the 1 980s and 1 990s was done by splitting into cases, so to prove the Poincare conjecture, one was still compelled to use the simple connectivity hypothesis. An exception to this was Hamilton's programme. Interestingly, Perelman found a nice way to use simple connectivity within Hamilton's programme, which simplified his proof of the Poincare con­ jecture (but not of the full geometrisation conjecture). To introduce Hamilton's approach, we need to refor­ mulate the Poincare conjecture as a statement relating topol­ ogy to Riemannian geometry: namely, that a compact, sim-

SIDDHARTHA GADGIL received his PhD from

HARISH

the California Institute of Technology and then

work at the Indian Institute of Technology Kan­

taught at SUNY Stony Brook before returning

pur, and received his PhD from SUNY Stony

to India. His main area of research is low­

Brook, USA He works in differential geometry.

dimensional topology. Department of Mathematics Ind ian I nstitute of Science

SESHADRI did

undergraduate

He is now an assistant professor at the Indian In­ stitute of Science. Department of Mathematics

Bangalore 5600 1 2

I ndian Institute of Science

India

Bangalore 5 600 1 2

e-mail: [email protected]

his

India e-mail: [email protected] isc.ernet.in

© 2007 Springer Science+ Business Media, Inc., Volume 2 9 , Number 4, 2007

35

ply connected 3-manifold admits an Einstein metric. To make sense of this we need a summary of some Riemannian geometry.

Some Riemannian Geometry Intrinsic differential geometry and curvature

I n intrinsic differential geometry, we study the geometry of a space M in terms of measurements made within the space M This began with the work of Gauss, who was involved in surveying large areas of land where one had to take into account the curvature of the earth. Even though the earth is embedded in 3-dimensional space, the measurements we make do not take advantage of this. Concretely, one has the question whether one can make a map of a region of the earth on a flat surface (a piece of paper) without distorting distances (allowing all distances to be scaled by the same amount). This is impossible, as can be seen by considering the area of the region consist­ ing of points with distance at most r from a fixed point P on the surface M. The area in case M is a sphere can be seen to be less than 7T'r2, which would be the area if we did have a map that did not distort distances. In fact, for r small the area of the corresponding region on any surface is of the form 7T'r2(1

-

.!!_ r2 12

+

. . . ) , and K is

called the Gaussian curvature a t P. Intrinsic differential geometry gained new importance because of the general theory of relativity, where one stud­ ies curved space-time. Thus, we have manifolds with dis­ tances on them that do not arise from an embedding in some !R n . This depended on the higher-dimensional, and more sophisticated, version of intrinsic differential geome­ try developed by Riemann. Today, intrinsic differential geometry is generally referred to as Riemannian geometry. To study Riemannian geometry, we need to understand the analogues of the usual geometric concepts from Eu­ clidean geometry as well as the new subtleties encountered in the more general setting. Most of the new subtleties are captured by the curvature. Tangent spaces

Let M be a k-dimensional manifold in !R n and let p E M be a point. Consider all smooth curves y : ( - 1 , 1) � M with y(O) p. The set of vectors v y'(O) for such curves y gives the tangent space TpM. This is a vector space of dimension k contained in fR n . For example, the tangent space of a sphere with centre the origin at a point p on the sphere con­ sists of all vectors perpendicular to the radius ending at p. If a particle moves smoothly in M along the curve a(t), its velocity V(t) = a'(t) is a vector tangent to M at the point a(t) , i . e . , V (t) E Ta(t)M =

=

Riemannian metrics

I f a : (a, b) � [Rn is a smooth curve, then its length is given by l(a) J!; lla ( t) ll dt. In Riemannian geometry we consider manifolds with distances that are given in a similar fashion in terms of inner products on tangent spaces. A Riemannian metric g on M is an inner product speci­ fied on TpM for each p E M. Thus, g refers to a collection =

36

'

THE MATHEMATICAL INTELLIGENCER

of inner products, one for each TpM. We further require that g varies smoothly in M. For a point p E M and vectors V, W E TpM, the inner product of V and W corresponding to the Riemannian metric g is denoted g( V, W). A Riemannian manifold (M,g) is a manifold M with a Riemannian metric g on it. Recall that near any point in M, a small region U C M can be given a system of local coor­ dinates x1 , . . . , Xk· At every point p in U we denote the corresponding coordinate vectors by el ' ek; then the inner product on TpU is determined by the matrix giJ g(e1, e1). This is a symmetric matrix. The first examples of Riemannian manifolds are mani­ folds M C !R n, with the inner product on TpM taken to be the restriction of the usual inner product on !R n. This met­ ric is called the metric induced from fR n. A second important class of examples are product met­ rics. If (M,g) and (N,h) are Riemannian manifolds, we can define their product (M X N, g E9 h) . The points of M X N consist of pairs (x,y), with x E M and y E N The tangent space 1ix,y)(M X N) of the product consists of pairs of vec­ tors ( U, V) with U E TxM and V E TyN The inner product (g E9 h) is given by 0

0

0

'

=

(g E9 h)(( U, V), ( U', V')) = g( U, U') + h ( V, V ) '

We can identify the space of vectors of the form ( U,O) (respectively (0, V)) with TxM (respectively, TyN). Distances and isometries

Given a pair of points p, q E M in a Riemannian manifold (M,g), the distance d(p,q) between the points p and q is the minimum (more precisely the infimum) of the lengths of curves in M joining p to q. For p E M and b > 0, the ball of radius r in M with cen­ tre p is the set Bp( r) of points q E M such that d(p,q) < r. Note that this is not in general diffeomorphic to a ball in Euclidean space. Two Riemannian manifolds (M,g) and (N, h) are said to be isometric if there is a diffeomorphism from M to N so that the distance between any pair of points in M is the same as the distance between their images in N In Rie­ mannian geometry, we regard two isometric manifolds as the same. Geodesics and the exponential map

Geodesics are the analogues of straight lines. A straight line segment is the shortest path between its endpoints. A curve with constant speed that minimises the distance between its endpoints is called a minimal geodesic. More generally, a geodesic is a smooth curve with con­ stant speed that locally minimises distances; i.e., it is a smooth function y : (a, b) � M such that II'Y'(t)ll is constant and having the following property: for any p y(to), there is an E > 0 so that the segment of the curve y from time to E to to + E has minimal length among all curves join­ ing y(to E) to y(to + E). Let p E M be a fixed point. Then we can find r > 0 such that if d(p,q) < r, then there is a unique minimal geodesic y joining p to q. This follows from the fact that a curve is ge­ odesic if and only if it satisfies a certain second-order (non­ linear) ordinary differential equation (ODE), as explained in =

-

-

Appendix A. The existence and uniqueness of solutions for ODEs, together with some geometric arguments, then give us the corresponding statements for geodesics. We can parametrise y (i. e . , choose the speed along y) so that y(O) = p and '}'(1) = q. Then the initial velocity y'(O) gives a vector in TpM with norm less than r. This gives a one-to-one correspondence between points q in M with d(p,q) < r and vectors V E TpM with norm less than r. The point that corresponds to the vector V is denoted by expp( V), and this correspondence is called the exponential map. As an example, consider the exponential map at the north pole of the 2-sphere p. This map is one-to-one on BoC7r) and it maps I3o(7r) to the sphere minus the south pole. The supremum of the values of r for which there are unique geodesics as above is called the injectivity radius at p E M. This term comes from its equivalent description as the largest r such that expp is injective on Bp(r). Sectional, Ricci, and scalar curvatures

Let p E M be a point and let g C TpM be a 2-dimensional subspace. Choose an orthonormal basis { U, V) of g and con­ sider the following family of closed curves in M:

Cr(8)

=

expp (r cos(8)

U+

r sin(8) V), 8 E [0,27Tl.

It can be proved that the length of expansion:

Cr has the following

REMARK: It is important to note that in local coordinates these curvature quantities can be expressed in terms of giJ and its first and second derivatives. This follows from the more standard way of defining curvature in terms of the Levi-Civita connection associated to g. See Appendix A for details. The fact that curvature can be so expressed provides a link between Riemannian geometry and partial-differential equations. We consider some examples. This is just !Rn with the usual inner product. In this case, all the sectional curvatures are zero. Hence so are the Ricci tensor and the scalar curvature.

(1) Euclidean space. (2)

Sphere sn(r) of radius r with the metric induced from

jR n+ l . In this case, all sectional curvatures are equal

( n - 1)r - 2g(U, V), and R(p) point p. Here g( · ; , ) is (the restric­ tion of) the standard inner product in !Rn (3) There is an analogue of Example 2 , called hyperbolic space, for which the sectional curvature is - r- 2. The underlying manifold can be taken to be !Rn. We will not describe the metric since we won't need it. r-2, Ric(U, V) ( n - 1) r-2 for any to

=

=

We have the following important converse of the above examples: Let (M,iJ be a simply-connected complete Rie­ mannian manifold of constant sectional curvature k. Then M is isometric to Euclidean space, the sphere of radius Vi/k, or hyperbolic space according as k 0, k > 0, or k < O. =

We define the sectional curvature of (M,lf) along g to be the number KCp,g) above. Other notations for sectional curvature include Kg(p,g) to clarify what metric we consider, and K(p, U, V> to indicate that g is the linear span of U and V In the latter notation, we put K(p, U, V> 0 if U and V are linearly dependent. We often omit the point p in the notation if it is clear from the context. Averaging all the sectional curvatures at a point gives the scalar curvature ap). More precisely, let {E1 , . . . , En) be an orthonormal basis of TpM. Then we define =

ap) =

L i,j

K(E;,Ej).

There is an intermediate quantity, called the Ricci ten­ which is fundamental in our situa­ tion. The Ricci tensor Ric(U, V) at a point p E M depends on a pair of vectors U and V in TpM. Further, it is linear in U and V and is symmetric (i.e., Ric(U, V) Ric(V, U)). It is

sor or Ricci curvature

=

defined as follows: If U is any unit vector in TpM, then we extend U to an orthonormal basis { U, E , . . . , E ) and define

2

n

By linearity, for a general vector aU, with U a unit vector, a2Ric(U, U). Further, by linearity and symme­ try, if U and V are any two arbitrary vectors in TpM, then we put Ric(U, V) = (Ric(U+ V, U+ V) - Ric(U- V, U- V)) in analogy with the formula

Ric(aU, aU)

=

±

(a + b)2 - (a - b)2 = 4ab.

(4) A product Riemannian manifold (M X N, g g1 EB gz): If g is a plane in Tp(M X N) that is tangent to M (re­ spectively, N), then K(p,g) K1(g) (respectively, K2 (g)). Here K1 and K denote the sectional curvatures with 2 respect to g1 and g . On the other hand, if g is the span 2 =

=

of a vector tangent to M and one tangent to N, then 0. (5) As a special case of the above, consider a surface M which is the product of two circles, possibly of differ­ ent radii, with the product metric. Then the tangent plane at any point is spanned by a vector tangent to the first circle and one tangent to the second circle. Hence the sectional curvature of M at any point is zero. (6) Another example of a product metric that we need is that on M 52 X R In this case, the sectional curva­ ture K(x,g) is 1 if g is the tangent plane of 52 and 0 if g contains the tangent space of R

K(g)

=

=

Manifolds with non-negative sectional curvature

We have defined sectional curvature in terms of the growth of lengths of circles under the exponential map. In other words, sectional curvature measures the divergence of ra­ dial geodesics. In particular, if a Riemannian manifold has non-negative curvature, geodesics do not diverge faster than in Euclid­ ean space. This has strong consequences for the geometry and topology of these manifolds. In fact, if a simply con­ nected 3-manifold (M,lf) has non-negative sectional curva­ ture, it has to be diffeomorphic to one of IR3, S3, and

52 X IR.

© 2007 Springer Science+ Business Media, Inc., Volume 2 9 , Number 4, 2007

37

Scaling and curvature

Suppose (M,g) is a Riemannian manifold and c > 0 is a con­ stant. Then the sectional curvature K' of the Riemannian manifold (M, elf) is related to the sectional curvature K of (M,g) by

for every point p E M and every tangent plane g C TpM at that point. Note that if c is large, then K' is small. Hence, given a compact Riemannian manifold (M,g), we can always choose c large enough so that (M, cg) has sectional curvatures ly­ ing between - 1 and 1 .

Einstein Metrics and the Poincare Conjecture An Einstein metric is a metric of constant Ricci curvature. More precisely, g is said to be an Einstein metric if, for all p E M and U, VE TpM, we have

To get a feeling for the analytical properties of this equa­ tion, we first consider the simpler case of the heat equa­ tion, which governs the diffusion of heat in an insulated body. The heat equation is

au = l:, u. at The temperature i n an insulated body becomes uniform a s time progresses. Further, the minimum temperature o f the insulated body increases (and the maximum temperature decreases) with time. This latter property is called a max­

imum principle. To see the relation of the Ricci flow with the heat equa­ tion, we use special local coordinates called harmonic co­ ordinates (i. e . , coordinates {xi} such that the functions xi are harmonic: /:,xi = 0) . We can find such coordinates around any point in a Riemannian manifold M In these co­ ordinates we have

Ric( U, V) = a g( U, V), for some a E IR. In general relativity, one studies an action functional on the space of Riemannian metrics called the Einstein-Hilbert action, which is the integral of the scalar curvature of a metric. Einstein metrics are the critical points of this functional among Riemannian metrics on a manifold with fixed volume. To relate Einstein metrics to the Poincare conjecture, one notes that an Einstein metric g on a 3-manifold necessarily has constant sectional curvature (in all dimensions metrics of constant sectional curvature are Einstein metrics). Hence, by the remark after Example (3) above, one concludes that if (M,g) is closed, simply connected, and Einstein, then (M,g) is isometric to S3 with a multiple of the usual met­ ric. Note that we can rule out Euclidean and hyperbolic space because they are not closed. In particular, M is dif­ feomorphic to S3. Hence the Poincare conjecture can be formulated as say­ ing that any closed, simply connected 3-manifold has an Einstein metric. More generally, Thurston's geometrisation conjecture says that every closed 3-manifold can be de­ composed into pieces in some specified way so that each piece admits a so-called locally homogeneous metric. This means that any pair of points in the manifold have neigh­ bourhoods that are isometric. Metrics of constant sectional curvature are locally homogeneous.

Hamilton's Ricci Flow In the 1980s and 1 990s Hamilton built a programme to prove geometrisation, beginning with a paper [5] where he showed that if a 3-manifold has a metric with positive Ricci curvature then it has an Einstein metric. By positive Ricci curoature we mean that if p E M and if U E TpM is non­ zero, then Ric( U, U) > 0 . Hamilton's approach was t o start with a given metric g and consider the 1-parameter family of Riemannian metrics g(t) satisfying the Ricci flow equation (1)

"*

=

-2 Ric(t),

g(O) = g,

where Ric(t) is the Ricci curvature of the metric g( t).

38

THE MATHEMATICAL INTELLIGENCER

where Q is an expression involving g and the first partial derivatives of g, and where RiciJ = Ric(e i,e1). Hence the Ricci flow resembles the heat flow

agiJ = l:,gif, at

leading to the hope that the metric will become symmetric (more precisely, the Ricci curvature will become constant) as time progresses. However, there is an extra term (tg, a g) of lower order. Such a term is called the reaction term and equations of this form are known as reaction-diffusion equations. In order to understand such an equation, one needs to understand both the nature of the reaction term and conditions that govern whether the reaction or the dif­ fusion term dominates. Let us consider some examples: If g is the induced metric on the sphere s3 of radius 1 , then g(t) (1 - 4 t)g is the solution to ( 1 ) . Note that the radius of (S3, g(t)) is =

V1

-

4t and the sectional curvatures are

1

t � 4 , these curvatures blow up.

1

� 4t .

As

More generally, if g(t) is an Einstein metric, the Ricci flow simply rescales the metric. In fact, if Ric = ag, then g(t) = (1 2at)g satisfies (1). Note that (M,g(t)) shrinks, ex­ pands, or remains stationary depending on whether a > 0, a < 0, or a = 0. On the other hand, if the metric is fixed up to rescaling by the Ricci flow, then it is an Einstein metric. Let (M1 X M2, g1 EB g2) be a product Riemannian mani­ fold. Then the Ricci flow beginning at g1 EB g2 is of the form g(t) = g1(t) EB g2(t) , where g1(t) and gz(t) are the flows on M1 and M2 beginning with g1 and g2: Ricci flow pre­ serves product structures. In particular, the flow beginning with the standard product metric g0 EB g1 on S2 X IR is g(t) = (1 - 2 t)g0 EB g1 , i . e . , the S2 shrinks while the IR direction does not change. This example is crucial for understand­ ing regions of high curvature along Ricci flow. We now consider some analytical properties of Ricci flow. One of the first results proved by Hamilton was that, given any initial metric g(O) on a smooth manifold M, the -

Ricci flow equation has a solution on some time interval [O,E) with E > 0. Furthermore, this solution is unique. It fol­ lows that a solution to the equation with initial metric g(O) exists on some maximal interval [0, T), with T either finite or infinite. Further, if T is finite then the maximum of the absolute value of the sectional curvatures tends to oo as we approach T The main idea of Hamilton's programme is to evolve an arbitrary initial metric on a closed simply connected 3-man­ ifold along the Ricci flow and hope that the resulting met­ ric converges, up to rescaling, to an Einstein metric. Hamil­ ton showed that this does happen when g has positive Ricci curvature. It is convenient to analyse separately the cases where the maximal interval of existence [0, T) is finite and infinite. It turns out that if the manifold is simply connected, then this time interval is finite. In particular, the curvature blows up in finite time on certain parts of the manifold. The central issue in Hamilton's programme was to un­ derstand, topologically and geometrically, the parts of the manifold where curvature blows up along the Ricci flow.

Curvature Pinching The first major steps in understanding the geometry near points of large sectional curvature were due to Hamilton and Ivey, using maximum principles. In the simple case of the classical heat equation, the maximum principle implies that if the temperature is ini­ tially greater than a constant a at all points in the mani­ fold, then this continues to hold for all subsequent times. In the case of the Ricci flow, we have a similar maximum principle for the scalar curvature. This is because the scalar curvature also satisfies a reaction-diffusion equation with the reaction term positive. Indeed, the evolution equation for scalar c urvature is

where /':,. denotes the Laplacian with respect to the metric (see Appendix A) and IRiq denotes the norm of the Ricci tensor. An immediate consequence of this equation is that if R > 0 on M at t = 0, then R > 0 at any subsequent t E (0, T]. This is seen as follows: Let Xm(t) E M be such that RCxm(t),t) = minxEM R(x, t) , where R(x, t) denotes the scalar curvature of (M,g(t)) at x. Assume, for the sake of brevity, that Xm(t) varies smoothly in t. We then see, from the evolu-

-

aR . . cXmc t),t) � 0. This implies that the t10n equat10n, th at at

positivity of R is maintained. More generally, scalar curva­ ture R is bounded below, along the Ricci flow. Hamilton also developed a maximum principle for ten­ sors. Using this, Hamilton and Ivey independently obtained an inequality for sectional curvature, which we mention and use in the next section. A consequence of the Hamilton­ Ivey inequality is that if, at a point p E M, there is a 2-plane E TpM for which the absolute value is large, then Hp) is large. Furthermore, the Hamilton-Ivey inequality im­ plies the following crucial fact: At a point of high curva-

g

IKCp,g)l

ture, there are 2-planes of positive sectional curvature much bigger than any negative sectional curvature. All these maximum principles amount to showing and using positivity properties of the reaction term.

Blow-up and Convergence of Riemannian Manifolds To study points of high curvature, we use a version of a classical technique in PDEs called blow-up analysis. Roughly speaking, this involves rescaling manifolds near points of high curvature. We then study the limit points, if any, of the rescaled manifolds. For this approach to be fruit­ ful , we need a theorem guaranteeing the existence of the limit points, i . e . , a compactness theorem. Such a theorem, addressing convergence of Riemannian manifolds, was proved in the 1 970s by M . Gromov, R. Greene, H . Wu, and others, following the pioneering work of J. Cheeger. First we need to define the distance between Riemann­ ian manifolds with respect to which we consider conver­ gence. This is the so-called Lipschitz distance, on the space of all Riemannian manifolds of a given dimension with given basepoints. It is defined as follows. Let (M,g) and (N, h) be Riemannian n-manifolds and let x E M and y E N be chosen as basepoints. Then the dis­ tance between M and N is the infimum of real numbers E > 0 such that there is a diffeomorphism f from the ball Bk of radius 1/E in M to the ball B of radius 1/E in N with j(x) = y so that for p, q E B�;, -e

< log

(

d([C[iJJCq))

d(p,q)

)

< E.

Note that our notion of distance, hence limits, depends on the choice of basepoints. We call the manifold (M,g) with basepoint x the pointed manifold (M,g,x). Consider the set o f pointed Riemannian manifolds o f a fixed dimension n, equipped with the above notion of dis­ tance. One would like to understand when a given se­ quence (M;, g;, p;) in this set has a subsequence which con­ verges to a pointed Riemannian manifold of the same dimension. Let us note two necessary conditions: First, as the example in Figure 3 shows, if the curvature of (M;, g;, p;) is not bounded, then the limit may not be a smooth manifold. Second, the injectivity radius at p; should be bounded below by a constant not dependent on i: A sequence of manifolds with bounded curvature need not have limiting manifolds (of the same dimension), as the manifolds may collapse to lower dimensions. For example, let M; = 51 X 51 be the 2-torus, g; = r 1g0 E9 g0 and p; = Cp,q), where g0 is the usual metric on the circle. Observe that (M;, g;) is the torus, with the product metric obtained by viewing the

Figure 3. A sequence without bounded curvature with the

limit singular.

© 2007 Springer Science+ Business Media. Inc. . Volume 29. Number 4. 2007

39

@00

Figure 4. An example of collapsing.

torus as a product of a circle of radius 1/ i with a circle of radius 1 . In this case the sectional curvature of (M;, gf) is zero for any i. On the other hand, the limit of this sequence of metrics is the degenerate metric 0 EB g0. Hence the limit of the Riemannian manifolds (in the appropriate sense) is a circle (see Figure 4). It turns out that these two conditions are also sufficient to guarantee convergence: For K E IR and r > 0, the space .M(n,K,r) of c� pointed Riemannian manifolds (M,g,x) with sectional curvature bounded above by K and injectivity ra­ dius at x bounded below by r is pre-compact in the topol­ ogy given by the Lipschitz distance. More precisely, any se­ quence of pointed manifolds in .M( n,K, r) has a subsequence which converges to a manifold with a Riemannian metric g which is C1 . Note that the limit may not be in .M(n,K, r), since the metric may not be C"'. We sketch briefly a key idea in the proof of the com­ pactness result. Suppose that we have both a lower bound on the injectivity radius and an upper bound on the cur­ vature of a Riemannian manifold (M,fi). As mentioned ear­ lier in this article, we can choose harmonic coordinates x1, . . . , Xn , i.e., coordinates such that each Xk is a har­ monic function, near each point in M. The bounds on cur­ vature and injectivity radius guarantee that these coordi­ nates exist on balls of fixed radius. Furthermore, the bound on the curvature gives a bound on the C1·a norm, for any a < 1 , of giJ. Hence, by the Arzela-Ascoli theorem, a sub­ sequence of the giJ and their first derivatives converge. The limiting local metrics can be patched to give a global C1 metric. Now let us return to the case of a Ricci flow. Suppose that (M,g(t)) is a Ricci flow on a closed 3-manifold whose maximal interval of definition is a finite time interval [0, T). Since T < oo, we know that limt -. T lemax(t) = oo, where kma.lJ) = suPiKCx,g)l is the maximum of the absolute values of sectional curvatures of (M,g(t)). Choose a sequence t; � T We rescale g by km.a:f.. tf) to get manifolds (M, kma.£tf)fi) with bounded sectional curva­ ture (see "Scaling and curvature" above and figures). In or­ der to apply the compactness theorem to this sequence, one needs to know that the injectivity radius at p is bounded below (independent of i). One of the major results of Perel­ man was that, for these manifolds, there is indeed a lower bound on the injectivity radius (Perelman's non-collapsing theorem). Thus, by the compactness result, some subse­ quence of the manifolds has a limit. In order to extract special properties of the limit, recall that the Hamilton-Ivey pinching estimate implied that at any point of high curvature on (M, g(t)), there are 2-planes of positive sectional curvature much bigger than any negative sectional curvature. This implies that the limiting manifold is non-negatively curved.

40

THE MATHEMATICAL INTELUGENCER

So far we have been considering limits of the Riemann­ ian manifolds (M, kmaol.tJ/i). In fact, as explained in Ap­ pendix B, one can rescale not only the metric but the en­ tire Ricci flow at these times and at suitable points, and consider the convergence of manifolds with Ricci flows. A compactness theorem for flows, similar to the compactness theorem above, was proved by Hamilton, and using this, one gets a limit manifold with a limit 1 -parameter family of metrics. In fact, as a result of the smoothing properties of heat-type equations, one has a stronger conclusion than the general compactness theorem. Namely, one can deduce that the limiting 1 -parameter family is C"' and again satisfies the Ricci flow equation. The nonnegativity of curvature along with Perelman's non-collapsing result shows that the flow for the limiting manifold is what Perelman calls a K-solution. Perelman proved that points in a K-solution have canonical neigh­ bourhoods (which we explain below). Furthermore, he proved a technical result giving a bound on the derivative of curvature for K-solutions, which was crucial in under­ standing behaviour near points of high (but not necessar­ ily maximum) curvature.

The Canonical Neighbourhood Theorem By considering limiting manifolds as above, it follows that small neighbourhoods of the points of maximum curvature are close to being 'standard' . However, this procedure does not work if we want to understand points with high cur­ vature which are not the maximal curvature points. The problem is that rescaling with respect to these points does not give metrics with curvature bounded independent of i. A surprising and remarkable result of Perelman's, which overcomes this difficulty and can be considered to be one of the central results in his proofs, is the canonical neigh­ bourhood theorem. This says that either M is diffeomorphic to S3!G, with G a finite group acting freely, or every point of high scalar curvature has a canonical neighbourhood which is an E-neck or an E-cap. An E-neck is a Riemann­ ian manifold which is, after rescaling, at distance less than E to the product of a sphere of radius 1 and an interval of length at least

.!. .

An E-cap is either an open ball or the E complement of a ball in the real projective 3-space, with a metric such that the scalar curvature is bounded and every point is contained in an E-neck on the complement of a compact set. In case M is simply connected, we must have M = S3 in the first case and M an open ball in the third. This result is surprising in many ways. Normally, by the kind of rescaling argument sketched above, we can study a neighbourhood of a point of maximal curvature. How­ ever, one expects that near points of high (but not maxi­ mal) curvature, there are nearby points where the curva­ ture is much higher. This means that the curvature can be fractal-like, and the resulting system has behaviour at many scales (as happens with complex systems). To study a neighbourhood of a point of high scalar cur­ vature, Perelman used the bounds on the derivative of the curvature of standard solutions in an ingenious inductive ar­ gument (which proceeds by contradiction) to show that the curvature of the appropriate rescaled metric is bounded near

C><)

Limiting manifold

i� i: Figure 5. Sequences of manifolds and their blow-up limits.

the point. After refining this, using geometric arguments (based on the theory of Alexandrov spaces), Perelman showed that one can construct blow-up limits at points of high (but not necessarily maximum) curvature. Hence the results mentioned in the previous section can be used to construct canonical neighbourhoods for all points of high curvature.

Ricci Flow with Surgery The canonical neighbourhood theorem allows one to un­ derstand regions where the curvature becomes very large. However, if the curvature remains bounded on some region of the manifold, one cannot deduce much about the topol­ ogy of the manifold. One would like to continue the Ricci flow in regions with bounded curvature, while using the canonical neighbourhood to study regions with high curva­ ture. This is accomplished by a process known as Ricciflow with surgery. This process involves modifying the manifold, geometrically and topologically, at regions of high curvature at a time close to T The resulting manifold has bounds on curvature that allow the process to continue beyond time T Consider the subset OP of M where the scalar curvature is bounded by a large number p for all t E [0, T), i.e., let OP {x E R(x, t) ::::; p for all t}. We choose p large enough that points of scalar curvature greater than p have a canon­ ical neighbourhood. For a time t close to T, the canonical neighbourhood theorem holds for the complement N of the interior of OP. Thus, every point in this complement has a neighbourhood that is a neck, a cap, or diffeomorphic to a sphere (if the initial manifold is simply connected). Putting these neigh­ bourhoods together, we get either a sphere or a manifold diffeomorphic to 5 2 X [ - 1 , 1) (which is a union of several necks), perhaps with a cap attached at one or both ends. Topologically, in each of these cases we obtain a sphere, =

Mj

Neck with Cap

Neck

Figure 6. Surgery performed on manifold.

a ball, or 5 2 X [ - 1 , 1 ] . It follows in particular that the bound­ ary of OP consists of 2-spheres. If OP is empty however large we choose p, in other words, if the curvature blows up on the entire manifold M, then the above implies that M is diffeomorphic to 53. We then replace M by the empty manifold. Otherwise, we remove the interior of the set N = M- OP and we attach balls to each of the boundary spheres of OP to get a Riemannian manifold. This operation is called surgery. Now we continue to evolve the manifold, which in gen­ eral has several components, by the Ricci flow. Repeating the above procedure for each of the components, we can inductively define Ricci flow with surgery (see Figure 6) . Note that if the curvature becomes high at all points in all components of M, then the manifold after surgery is empty. In this case, we say that the manifold has become extinct. One can deduce from the canonical neighbourhood theorem that, in this case, the manifold just before surgery was a collection of 3-spheres. We need technical results that say that all the properties that we have for the ordinary Ricci flow hold for Ricci flow with surgery. We also need a result saying that in any fi­ nite time interval only finitely many surgeries are required to show that Ricci flow with surgery can be defined for all positive times. The essential reason for this is that every time one performs surgery the volume is reduced by a def­ inite amount, i.e., by an amount independent of time. On the other hand, one can show, by the evolution equation for volume along the Ricci flow, that volume cannot go to infinity in a finite time period. These two facts together im­ ply that one needs to perform surgery only finitely many times in a finite time interval. To achieve these results one needs to choose the pa­ rameter p carefully, in general depending on the time T

Outline of the Proof We are now in a position to outline the proof of the Poin­ care conjecture. Consider a simply-connected 3-manifold M with a Riemannian metric on it. We evolve this using the Ricci flow with surgery. A result of Perelman (for which an alternative proof was provided by Colding and Minicozzi [4]) says that if the man­ ifold Mis simply connected, then the Ricci flow with surgery becomes extinct in finite time. This is proved by consider­ ing a geometric quantity called the waist and showing that it goes to zero in finite time. Consider Ricci flow with surgery up to the time when it becomes extinct. If we view the process backwards from the extinction time, we see that either spheres are created (the opposite of extinction) or two components are con­ nected by a tube (the opposite of surgery). Note that when

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4 , 2007

41

two spheres are connected by a tube, the result is still a sphere. As a result, when each surgery is viewed back­ wards, we see spheres either being created or being merged with other spheres. Thus at each time the manifold we see is a collection of spheres. In particular, as the manifold M we started with is connected, it must have been a sphere.

Concluding Remarks The value of a mathematical theorem in science and engi­ neering often lies not just in its statement but in the ideas that are developed in the course of proving the theorem. In this respect, Perelman's (and Hamilton's) work is very rich in ideas which, when digested, may have consequences in a wide range of subjects outside mathematics. Further, those techniques and ideas applicable to Ricci flow in all dimensions may be widely applicable to complex systems, while those special to dimension three may help us un­ derstand when a complex system is well behaved.

Appendix A: The Levi-Civita Connection and Curvature Let (Mn, fi) be a Riemannian manifold, and let x(M) denote the set of C"" vector fields on M, regarded as a vector space over IR. Associated to g, there is a bilinear map

V:

x(M)

X

xCM) � x(M),

denoted by (X, Y) � V x Y, called the Levi-Civita connec­ tion. V possesses the following properties, and it is the unique map from x(M) X x(M ) to x(M) possessing them: (i) V;xY = JV xY (ii) vx(/Y) = JVX y + dj(X) y (iii) VxY - 'V yX = [X, Y] (iv) X(g(Y,Z)) = g(V xY,Z) + g( Y,VxZ) It is clear that V can be described in local coordinates by functions r� defined by (2) It follows from (i) to (iv) that the in terms of the metric g by

C3)

r� = .l gk' 2

(� a�

+

rt can

§_gg_ a�

be expressed

�). a�

Conversely, V can be defined (locally) by (2) and (3). It turns out that a smooth curve y : (a, b) � M is a geo­ desic if and only if, in local coordinates, y = ( y1, . . . , y,) satisfies the ODE system n

(4)

YZC t) + .L r� yiC t)yJC t) = iJ� l

o,

k = 1,2, . . . ' n.

The Riemann curvature tensor R is defined by (5) R(X, Y, Z, W)

= g(V xV yZ -

V y'VxZ - V [x, YJZ, W),

where X, Y, Z, W are vector fields. It should be noted that R is indeed a tensor, i . e . , the value of R(X, Y, Z, W) at p depends only on the values of X, Y, Z, W at p. The sectional curvature KCD of a 2-plane g C TpM is then given by

42

THE MATHEMATICAL INTELLIGENCER

K(g)

(6)

= R(X, Y, Y,X),

where {X, Yl is an orthonormal basis of f Since V can be expressed in terms of f � (which can be expressed in terms of giJ and its first derivatives), it follows from (5) that the curvature tensor R can be expressed in terms of giJ and its first and second derivatives. Finally we mention the Laplacian operator L associated to a metric. L acts on functions and in local coordinates is given by

Lf=

1 v

�

1

det(gij)

L

a

-

k, l axk

(

-

Vdet(giJ)gk1

aF

.:::L

ax/

)

,

where f: M � IR is a smooth function. This Laplacian has many of the familiar properties of the Laplacian on !Rn. For instance, at a local minimum p, Lj(p) 2:: 0.

Appendix B: Some Ideas in the Proof In this appendix we sketch some of the techniques and ideas involved in proving the results mentioned. Perelman's entropy functional

Before the work of Perelman, it was not even known that there were no non-trivial periodic orbits for the Ricci flow. Periodic orbits can occur for some smooth flows, but not for gradient flows of a function, as the function must de­ crease along the gradient flow. One of the first achievements of Perelman was the con­ struction of an entropy functional, such that the Ricci flow is the gradient flow of the negative of this entropy (up to change of coordinates, i . e . , up to a diffeomorphism). The entropy and its extensions were used to prove many re­ sults, including the non-collapsing results. To construct the entropy, one starts with a manifold M with a fixed volume element (technically, a smooth mea­ sure) dVo on it. In local coordinates, this means that we are given a positive density function p0 so that the volume of a region is defined to be the integral of p0 on the region. Any Riemannian metric g has a volume element dVg as­ sociated with it, whose density in local coordinates is det(giJ) . We can find a function f such that dV = efdV0. Given this, Perelman defines an entropy functional by

f R + IIV!112 . As mentioned above, the Ricci flow is, up to change of coordinates, the gradient flow of the negative of the entropy. Thus entropy increases along the Ricci flow, ruling out pe­ riodic orbits. Note that as we have to rule out orbits peri­ odic up to scaling, some additional arguments are needed. Further conclusions can be obtained by using the free­ dom in choosing the measure dV0. For instance, we can take dV0 to be concentrated near a point x in M. If (M,g(t)), rescaled to make the curvature bounded at x, collapses near x, then one can show that the entropy must go to zero. But this contradicts the result that the entropy is increas­ ing, proving non-collapsing. There is plenty of speculation as to the meaning of en­ tropy, but we confine ourselves to quoting some percep­ tive remarks of Mike Anderson. Observe that there are Ricci

flows along which the geometry of the manifold does not change, namely, the Ricci solitons. Further, for so-called gradient Ricci solitons, the metric along the Ricci flow is fixed up to scaling and volume-preserving change of co­ ordinates. Thus, the entropy must be constant for such flows ; i . e . , these must be critical points for the entropy. A calculation shows that the the critical points of the above functional are precisely the gradient Ricci solitons. Parabolic rescaling and asymptotic solitons

Consider now the situation where the curvature blows up at time T We have seen that by rescaling by the maximal scalar curvature we can get limiting manifolds of positive curvature. One can do better than this-by rescaling time as well, we get a new solution to the Ricci flow equation on a large interval. By shifting in time, we can take this to be an interval of the form [- A,O] , with A large. This process is called parabolic rescaling. More precisely, let Eft) be a Ricci flow for t E [0, T). Given a scaling factor A > 0 and a time T E [0, T), we can define a new Ricci flow on the interval [-AT, A(T- T)] by

( �}

g(t) = A g T +

For the rescalings that we consider, A is the maximum of sectional curvature at time T, and [0, T) will be the maxi­ mal interval of definition of the flow. Since T< oo, we know that A � oo. Hamilton proved a convergence theorem for manifolds with Ricci flows similar to the theorem guaranteeing the ex­ istence of limits of Riemannian manifolds. In particular, we can consider limit points of the sequence of parabolically rescaled flows. These limits will then be what Perelman calls K-solutions, i.e., solutions to the Ricci flow equation defined on ( -oo,O] of non-negative curvature which satisfy a non-collapsing condition. In order to understand K-solutions (M, h(t)), Perelman associated an asymptotic flow solution to such a solution, as follows: Take an appropriate sequence of points qk and times 1 tk � -oo and parabolically rescale (M, h(t)) by A = t;; . This classified then these rescaled flows, which leads to the re­ quired understanding of K-solutions. As a consequence, one obtains a canonical neighbour­ hood theorem for K-solutions, i.e., one obtains a neigh­ bourhood U of x and a time interval (t1, t ) such that the 2 Ricci flow on U X (t1, t2) is close to the usual flow on 5 2 X IR or 53 or an E-cap. Further, Perelman obtains a control of the scalar curva­ ture for such canonical neighbourhoods. This plays a cru­ cial role in proving the canonical neighbourhood theorem for points of high (but not maximal) curvature. Points of high scalar curvature

For points of maximal scalar curvature, we can paraboli­ cally rescale to get a limiting K-solution. By the above, these points have canonical neighbourhoods. To deal with points of high curvature which may not be maximal curvature points, Perelman uses an ingenious in­ ductive argument. A point is said to be good if the canon­ ical neighbourhood theorem is valid for it. Perelman con-

siders a sequence of 'maximally bad points' y;, more pre­ cisely, bad points y; with scalar curvature R; so that any point with scalar curvature at least 2R; is a good point. Then he considers a ball of some fixed radius p in the metric g; rescaled so that the scalar curvature at y; becomes 1 . As y; is in general not a point of maximal curvature, there may be points where the curvature is larger than 1 in the rescaled metric. However, Perelman gives an elegant argument to show that the curvature is bounded in balls of fixed size p (with respect to g;) centred at y;. One sees that any bad point has curvature bounded by 2, while the cur­ vature in the neighbourhood of a good point does not os­ cillate by much. So if p is small, neither good points nor bad points have curvature above 4 in the rescaled metric. Having a bound on the curvature, one can consider the limit of the Riemannian metrics on some ball of size p. Fi­ nally, Perelman considers the largest such p and shows that it must be infinite, i . e . , we have a parabolic limit as with points of maximal curvature, allowing one to prove the canonical neighbourhood theorem. Here he uses the the­ ory of Alexandrov spaces with curvature bounded below. REFERENCES

Note that [7]. (9], [1 0].

[1 1 ] , [1 2] refer to the website arXiv.

[1 ] M . T. Anderson, Geometrization of 3-manifolds via the Ricci flow, Notices Amer. Math. Soc. 51 (2004), no. 2, 1 84-1 93 .

[2] H.-D. Cao, X . - P . Zhu, A Complete Proof o f the Poincare and Geometrization Conjectures-Application of the Hamilton-Perel­ man Theory of the Ricci Flow, Asian Journal of Mathematics 1 0 (2006), no. 2 , 1 85-492. [3] H . - D . Cao, X.-P. Zhu, Erratum to "A Complete Proof of the Poin­ care and Geometrization Conjectures -Application of the Hamil­ ton-Perelman Theory of the Ricci Flow", Asian Journal of Mathe­

4, 663. [4] T. H. Golding, W. P. Minicozzi I I : Estimates for the extinction time matics 1 0 (2006), no.

for the Ricci flow on certain 3-manifolds and a question of Perel­ man, Journal Arner. Math. Soc. 1 8 (2005), no. 3, 561 -569. [5] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Georn. 1 7 (1 982), no. 2, 255-306.

[6] R. S. Hamilton, The Formation of Singularities in the Ricci Flow, Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1 993),

7-1 36, l nternat. Press, Cambridge, MA, 1 995. [7] B . Kleiner, J . Lott, Notes on Perelman's papers, math.DG/0605667. [8] J. Milnor, Towards the Poincare conjecture and the classification of 3-manifolds, Notices Arner. Math. Soc. 50 (2003), no. 1 0, 1 226--1233. [9] J. Morgan, G . Tian, Ricci flow and the Poincare conjecture, math.DG/0605667.

[1 0] G . Perelman, The entropy formula for the Ricci flow and its geo­ metric application, math. DG/021 1 1 59 .

[1 1 ] G . Perelman, Ricci flow with surgery on three-manifolds, math. DG/02 1 1 1 59 . [1 2] G . Perelman, Finite extinction time for the solutions t o the Ricci flow on certain three-manifolds, math.DG/02 1 1 1 59 . [1 3] W . P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1 982), no. 3, 357-38 1 . [ 1 4] P. Topping, Lectures on the Ricci Flow, www.maths.warwick.ac.ukl topping/RFnotes.html

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

43

Pr i nce of Samar qand Star s ALAIN JUHEL

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cajf3 where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

min Maalouf, a French-Lebanese author, wrote a novel titled Samarqand. One of the main characters was Omar al-Khayyam (1048-1 1 22), the famous poet and mathematician. Today, however, the Mathematical Tourist will find no refer­ ences to his presence in this town: the buildings of that time no longer exist; moreover, he worked mostly in Isfahan, Iran. On the other hand, many places in Samarqand evoke the large and bril­ liant team of astronomers and mathe­ maticians assembled by the inspired fif­ teenth-century ruler, Ulugh-Beg, the "Prince of the Stars," a distinguished member of the brotherhood himself. The last scientific sparks of a Medieval Islamic civilization blazed several cen­ turies ahead of Europe, but political troubles brought intellectual life of the region to a very rough and irreversible halt. Founded in the fifth century B . c . , Samarqand is one of the oldest cities in Central Asia. It was a flourishing stop­ ping place on the mythical Silk Road (used from the fifth century to the twelfth century), and it was plundered by many invaders. It had hardly recov­ ered from being sacked and destroyed by Gengis-Khan in 1 220, but a second Turkish-Mongolian conqueror, Timur­ Leng (1336-1405), made it his capital in 1370. In time, the city became the most beautiful in the whole Eastern world, as

A

a testimony to Timur's power and wealth. In 1 394 he had a grandson who would become his favorite: Ulugh-Beg, born in Soltanieh (now in Iran). The boy was as fond of science and a peace­ ful existence as his grandfather was of wars and fighting. He was only 1 5 when his father, Shah Rhuk (1377-1447), came to power, but as the elder son he was at once appointed viceroy in Samarqand. His position was an im­ portant one, for the capital had moved to Herat, in the Khorasan province (now in Afghanistan). Ulugh-Beg came to the throne in 1447, but his reign as emperor was short: barely two years: on October the 27, 1449, he was murdered in a con­ spiracy by fundamentalist Muslims led by his own son, Abdul Latif. Perhaps the courtiers could no longer endure his philosophical, sometimes very ad­ vanced concepts (many of them deeply religious). He was also accused of star­ gazing rather than ruling the Empire. Nevertheless, he had a very strong and long-lasting political and scientific in­ fluence in Samarqand: as Viceroy, he had ruled the city and made astronom­ ical observations for over 30 years. He managed to get both the funds and the time necessary to complete his most ambitious project, in creating the great­ est observatory in the Muslim world, combined with a large research and ed­ ucation centre.

KAZAKHSTAN

TURKMENISTAN

Please send all submissions to M athematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,

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

44

Samarqand, today in Uzbekistan (formerly the USSR).

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+Business Media, Inc.

Ulugh-Beg's statue, outside the Observatory, at the center of the sundial (left), and his Medressa (right). Many monuments related to the Ulugh-Beg epoch can still be seen in Samarqand. They are highlighted in every street guide, and every sightsee­ ing tour in Uzbekistan includes them. Here is a brief overview.

Ulugh-Beg's Medressa The Registan square forms the core of the city, and this image of Central Asia will probably remain engraved in the visitor's memory. However, it is not rep­ resentative of the epoch we are talking about. At that time, only Ulugh-Beg's medressa on the west side had been built; the other two would be erected in the seventeenth century, to complete the design.

The rectangular, 56 m wide and 81 m long, building was erected between 1417 and 1420. The minarets are 35 m high. The star-spangled design of the pishtak (entrance portal) echoes the prince's chief interest. There used to be a bazaar in front of it, and at the north end, a caravanserai. These features im­ ply that the monument was in the very heart of the city, not far from the noisy open market. It is an important structure because it shows one of Ulugh-Beg's rev­ olutionary architectural innovations: an open-work brick balustrade that re­ placed the traditional medressa wall. Traditionally, a medressa, an Islamic academy of that era, was surrounded by a closed wall, cutting off the scholars

from the working-class uproar. As a mat­ ter of fact, it was to remain a secret place, inaccessible to all but the elite. Ulugh­ Beg, however, wanted to suggest a new social model, in which the elite could keep the working class at a distance but not be completely isolated. He envi­ sioned a technical solution-asking the architect to design a brick portal that could be regarded as a symbol of the "open college. " It offered the opportunity to look inside, but still served as a pro­ tective device so that the indispensable serenity within would not be troubled by the marketplace. Nothing similar seems to have been designed elsewhere, and the uninformed visitor could easily overlook this rather daring idea.

Open-work brick balustrade with hexagonal pattern at the portal to the medressa. In 1 636, the Sher Dar (Tigers) medressa replaced the ancient market.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

45

As already mentioned, in Ulugh­ Beg's time a medressa was an Islamic academy, where not only theology, but also literature, poetry, mathematics, and astronomy were taught. Their appear­ ance and modern-day function do not suggest what they used to be. They were quite like the early European uni­ versities, depending more or less heav­ ily on the Church and serving as cen­ ters for intellectual growth. Also, as with the dawn of science in Europe, science and religion were heav(en)ly linked: re­ ligion set out the problems and science got the mission to solve them. However, these problems were slightly different in the East and the West: in Islam, three great questions in­ terlaced astronomy and trigonometry with religious observances. 1 . Defining prayer time; the very first tables were computed by Al­ Khowarizmi in person! 2. Determing the visibility of the New Crescent Moon. 3. Establishing Quibla, that is, the di­ rection to Mecca (and as a conse­ quence the right way to face when saying one's prayers). Qadi-Zada-al­ Rumi wrote a significant treatise that has come down to us. Before the Observatory was completed, all astronomical observations were made at the medressa, but afterwards, the medressa was solely devoted to academic tasks.

A Research Team From 1 408 to 1437, as many as 70 as­ tronomers were working with the Viceroy, Ulugh-Beg. But there were two outstanding members of the team, astronomer Qadi-Zada-al-Rumi (13641 436), and the famous mathematician, AI-Kashi. The former came to Samar­ qand in 1 4 1 0 and met the then 17-year­ old Ulugh-Beg. This was the beginning of a long friendship. He was to become the young prince's mentor and scien­ tific adviser: he had the medressa built, he would be its chairman and, perhaps most important, it was his suggestion to recruit the man he thought was the best mathematician of the time: Al-Kashi. Born in Kashan, Iran, in 1380, Al-Kashi joined the group in about 1420, and the "principal" would stay until his death in 1429. Through the study of astronomy, Al­ Kashi became an engineer of astro­ nomical devices and an authority in numerical analysis. He computed n lit­ erally to horsehair precision, for a cir­ cle whose diameter is 600,000 times the Equator; that is, his estimate for the limit of the universe: "the equator on the sphere of fixed stars." That equa­ tor equaled 8,000 parasanges, and 1 parasange 12,000 cubits and 1 cu­ bit = 24 thumbs, whereas 1 thumb = 36 hairs of a horse. Thus, the Equator is readily seen to be 4.97 . . . X 1 0 16 horsehairs thick, a number between the =

9th and 1Oth power of 60! Therefore, he needed an accurate computation up to ten sexagesimal places. AI-Kashi ran Archimedes's method, using an 8,050,306,368-sided polygon approxi­ mation. In a sexagesimal representation, he got:

pi =

3. 08,29,44,00,47,25,53,07,25.

He converted this to decimals

pi =

3. 1 4 1 592 653 589 793 2 5 .

The computation i s right u p t o the 1 6th place (the 1 7th should be 4, not 5). He was the first to go further than 1 0 dec­ imal places, a symbolic frontier. His record places him ahead of Zu Chongzhi's 6 decimal places (China, 480) and Romanus's 1 5 decimal places (1 593). Even more noteworthy is Al-Kashi's computation of sin(l 0), again with an accuracy of 10 sexagesimal places, but anticipating the method of successive approximations about 200 years ahead of Kepler ( 1 6 1 8) . He also described the scientific life in Samarqand in a corre­ spondence with his father, who had re­ mained in Kashan. Every letter is care­ fully dated and teaches us many historical details from a first-hand wit­ ness, about progress in the building of the observatory and the strength of the first-rate work in the medressa: His Majesty himself takes pleasure in the sciences, and scholars are legion here . . . . One can find the cream of Intelligence in town. . . . In a quite modern way, the university where the actual teaching was done was not separate from the research seminar. It was the place where all kinds of unsolved questions were dis­ cussed. Very often, however, the de­ bates were limited to three outstanding protagonists: only Ulugh-Beg and Qadi­ Zada-al-Rumi were qualified to engage in a discussion with Al-Kashi. A group of statues is located in one of the iwans, a vaulted hall walled on three sides, so typical of Persian architecture and, later, of Islamic architecture in general. The statues represent Ulugh­ Beg together with his team of as­ tronomers, and they remind us of the scientific goals of this building.

The Observatory AI-Kashi and Qadi-Zada-al-Rumi (Observatory Museum)

46

THE MATHEMATICAL INTELLIGENCER

To the disappointment of the visitor, the foundations and the buried part of the

Ulugh-Beg and his astronomers (inside the medressa). Note the patterns on the rear wall, a mix of pentagons, hexagons, and enneagons. Moreover, inside the nine-sided polygonal stars, seven light blue points form a regu­

lar heptagon, and this pattern is repeated several times, unmistakably.

marble observation quadrant are the only vestiges of the original observa­ tory. Nevertheless, these elements pro­ vide a good idea of how big it must have been. The vertical quadrant had a 40-m radius and the building itself was cylindrically shaped. Its circular basis, some 48 m in diameter, is quite distinct. The plans in the Observatory Museum allow estimating of the height of this three-storied building: about 45 m. These plans were drawn by Russian ar­ chaeologist Vladimir Vyatkin, who led the excavations in 1 908. Completed in 1 428, the observatory was built according to the plans of Maragha, an earlier observatory de­ signed and ruled by Nasir al-din al-Tusi ( 1 201-1 274), which remained active un­ til 1 3 16, but was ruined in 1350. Ulugh­ Beg was said to have visited it. The half-buried quadrant principle had been used in Rayy (near Tehran): such a device allowed a maximal size for the arc of the circle while the build­ ing did not become excessively high. Carefully oriented along the meridian line, the arc had been scaled minute by

minute to allow the most accurate mea­ surement of the altitude of the sun from the horizon. More generally, the altitude of a star or a planet could be read at the moment when its trajectory crossed the meridian plane. At the site, infor­ mation boards remind the visitor that Ulugh-Beg was able to get from this in­ strument-the largest ever built-such results as: 1. The duration of the year. 2 . The sidereal period of planets (the table below gives the measure of the arc described by a planet during one year on Earth) : 3. The obliqui�y qf the Ecliptic, measur­ ing the angle between the Earth's axis and the normal vector to the ecliptic plane (that is, the plane in

which the Earth moves around the Sun, or, in accordance with the geo­ centric viewpoint of those days, the plane of apparent movement of the Sun). From top to bottom are the val­ ues discovered by Eratosthenes, Hip­ parchus, Ptolemy, Al Battani, Al Sufi, Abu al Wafa, Al Quhi, Ibn Yunus, AI Tusi, and at last Ulugh-Beg himself, together with the date of this dis­ covery (BC in the first two lines). No better result was achieved until the observations of Gian Domenico Cassini between 1 655 and 1669. The naked-eye observation quadrant would still be in use up to Tycho Brahe in the late sixteenth century. It was the in­ strument of discovery for Kepler's three famous laws, a milestone in the long

Ulugh-Beg's Planet

value

Present value

Difference

Saturn

1 2° 1 3' 39"

1 2° 1 3' 36"

3"

Jupiter

30° 20' 34"

30° 20' 3 1 "

5"

Mars

1 9 1 ° 1 7 ' 1 5"

1 91 ° 1 7 ' 1 0"

5"

Venus

224° 1 7 ' 32"

229° 1 7 ' 30"

Mercury

53° 43' 1 3"

53° 43' 3"

2" 1 0"

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

47

Foundations of rhe Observatory, a portal (a modern fancy) , and the buried part of the quadrant.

The Observatory: plan and spatial (virtual) reconstruction as shown in the Observatory Museum.

1437 The comparative table i n the museum for the obliquity o f the Ecliptic.

48

THE MATHEMATICAL INTELLIGENCER

.1�� �� -"'*''�(.rl Cj� -'/'
L·O S

T A B V L .£

C. A C

I. ·' T

S T E. LL A IUI M F I X A ll. V M '

H O B S £ 1.

A T I O H !.

ULUGH B EI GH I, T A M E R L A N I S Magni Ncpo.... R r ionum u!tu cru�qucq f l ll'V J( (r.O •r) Prrncrpn rottnti!frmi. ..

I \. . '

v-

��;����:�/ Treatise shown in the exhibition.

struggle between the theories of Coper­ nicus and Ptolemy. The exhibition shows treatises and astronomic tables by Ulugh-Beg and a few pages of his celebrated Zij-i Sul­ tani or Catalogue of Stars, compiled in 1 437. It contains a register of 992 stars and their complete coordinates, to­ gether with comparative catalogues in which the results are compared to those of other famous astronomers. This work was completed by the first British Royal astronomer, John Flamsteed (16461719), and published posthumously, in 1725. The works of Ulugh-Beg and his astronomers reached a large interna­ tional audience and acquired interna­ tional fame, as shown in a few allegor­ ical engravings made in Europe. They show Ulugh-Beg surrounded by the most famous astronomers ever, such as Hipparchus, Ptolemy, Tycho Brahe, and Copernicus. Outside the Observatory Museum, a recently built sundial evokes the plan­ etary system that held such a prominent place in Ulugh-Beg's thinking. It is a classical "table-top sundial" device, on which the gnomon is the hypotenuse of a right triangle standing in a vertical North-South plane. The dial itself has a circular shape, and shows the planetary orbits as concentric circles. Despite the artist's good intentions, there are two anachronisms: 1 . As the observations table shows,

only 5 planets were known in Ulugh-Beg's epoch, and Uranus was only to be discovered in 1 78 1 ; Nep­ tune, in 1846. 2. In Ulugh-Beg's mind, as in the minds of his predecessors in the Islamic world, the planetary model in use was the Ptolemaic geocentric one, not the Copernican heliocentric one, which became known in 1 5 1 4 . Yet, the latter was chosen for this mon­ ument.

Final Tribute As a sign of the high regard in which he held his friend, Ulugh-Beg placed

�·

the grave of Qadi-Zada-al-Rumi in the princely necropolis of Shar-i-Zindah, where the members of the Timurid dy­ nasty and a few high dignitaries are buried. This grave presents a problem for archaeologists, however, because the only remains found there is the skeleton of a forty-year-old woman. There is no doubt, however, about where Ulugh-Beg is buried: he lies in the golden dome in the Gur-Emir, the Timurid Mausoleum, at his grandfather's side. The harmony of the proportions in the building and the extravagant or­ namentation make it worth a visit, but a mathematical tourist will, of course,

A European allegorical engraving depicting Ulugh-Beg in the "Hall of Fame" of astronomy (3rd from left).

© 2007 Springer Science+Business Media,

Inc., Volume 29, Number 4, 2007

49

The "Qadi-Zada-al-Rumi" grave in Shar+Zindah.

The dome in the Gour-Emir.

The Gour-Emir, the burial place of Ulugh-Beg.

Ulugh-Beg's gravestone.

make a special stop in front of the tomb­ stone of the " Prince of Stars. "

the eighth century, had launched the "al-jebr revolution. "

Afterword

REFERENCES

When the Russian archaeology team led by Mikhail Gerasimov opened the tomb of Ulugh-Beg in 194 1 , they got confir­ mation of the circumstances of his death. He was decapitated, and then buried, dressed in his clothes, follow­ ing the funeral rite for martyrs in Islamic tradition. His observatory was pulled down and leveled to the ground. Sci­ entists became martyrs in this land of Islam, as science fell down from its highest peak to ignominy and a sudden standstill: Al-Kashi and Ulugh-Beg were the last scientists of the brilliant era commenced by Al-Khwarizmi, who, in

50

THE MATHEMATICAL INTELLIGENCER

tory of Arabic Science, volume 1 : Astron­ omy- Theoretical and Applied. Routledge,

New York, 1 996.

[ 1 ] F.

Beaupertuis-Bressand,

[7] R. Rashed (editor), Encyclopedia of the His­ Ouloug

Beg,

Prince des Etoiles. Ulysse, no65, April 1 999.

[2] J-L. Chabert (editor), A History of Algo­ rithms: From the Pebble to the Microchip.

Springer-Verlag, New York 1 999. [3] A. Djebbar, Une Histoire de Ia Science Arabe. Seuil, Paris, 2001 .

[4] J. L. E. Dreyer, A History of Astronomy from Thales to Kepler. Dover Books, New York,

tory of Arabic Science, volume 2: Mathe­ matics and the Physical Sciences. Rout­

ledge, New York, 1 996. [8] Website of the Samarqand Observatory: http//home.nordnet.fr/-ajuhei/Obs_Samar kand/observatoire. html [9] Special issue dedicated to pi in the Series "Petit Archimede, " Palais de Ia Decouverte, Paris, no 64-65, May, 1 980.

1 953. [5] A. Maalouf, Samarqand, translated from French by Russell Harris. Interlink Books, New York, 1 996. [6] R. Rashed (editor), Encyclopedia of the His-

209, rue Frederic Chopin 59850 Nieppe France e-mail: [email protected]

M athematics and M athemati ca STAN WAGON

M:

athematica, a software package designed by Stephen Wolfram and his team at Wol­ fram Research (WRI) in Champaign, Illinois, first came on the scene in 1 989. Since then, it has gained sophistication and has totally changed the way one integrates computation and mathematics. Indeed, it has changed the way I view mathematics: because it has allowed me to visualize things even as abstract as the Banach-Tarski Paradox [W]. I find myself less interested in mathematical objects that do not have a computational real­ ization. Not every new version of software has big changes. Mathematica 's version 3 was a big improvement, with typesetting integrated into the environment. Now with version 6, the han­ dling of graphics and animations has been totally changed and so the abilities for visualiza­ tion go even farther. And this version has many other big new things, such as fast access to databases in various fields. In this report I will summarize the new developments and also comment on some things that were available even pre-version 6 . This i s not a review, but rather a report from a true enthusiast. I have worked closely with the company for many years, teach an annual private course about Mathematica, and have become an expert programmer. But many of the new things in version 6 can be done with very little code and in this article I hope to convey the many reasons why a mathe­ matician should be aware of these capabilities. For specific problems there might well be other software that can do a task better than Mathematica; for example, the Concorde package (publicly usable at [C]) can provide prov­ ably correct solutions to many instances of the Traveling Salesman Problem. But a great virtue of Mathematica is its breadth. When working on, say, an optimization problem, one has ac­ cess to all the functions and capabilities of the package: from the special functions library to map and economic data for countries of the world, to spectacular 3-D graphics capabili­ ties, to a comprehensive interval arithmetic environment Here's a short summary of what is new; these points are discussed in greater detail in this article. 1 . Real-time rotation of 3-dimensional images. This has been a long time coming, and greatly enhances the viewing of surfaces, polyhedra, knots, and so on. 2 . New algorithms for computing 3-dimensional surfaces that are adaptive in nature. This al­ lows us to visualize things that are nearly impossible to see using a rectangular grid. The redevelopment of surface graphics also allows great flexibility in adding curves to surfaces or restricting the plotting region, both of which allow nicer images. 3. A new approach to animations that makes use of dynamic output. The place to start is the Manipu l a t e command, which allows one to set up a live panel with sliders that control parameters. Move the sliders and the output in the panel, which can be graphic, symbolic, numeric, or a combination of all three changes. This is much more flexible than making the images of an animation that change only as one parameter changes.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

51

4. A public web page (demonstrations.wolfram.com) that contains about 2000 illustrations of dynamic output.

5. Large databases containing information on knots, polyhedra, graphs, lattices, socio-economic­ political data on countries of the world, English words, and objects from the financial world. Also data on about 160,000 cities of the world, and astronomical, chemical, and physical objects. Some of this data is downloaded from a Wolfram web site when it is called. 6. Additional capabilities in algebra, Diophantine equations, and optimization. 7. Miscellaneous new programming constructs that experienced users will appreciate.

Three-Dimensional Graphics This area has been totally redone. Formerly, graphs of j(x,y) were generated by a rectan­ gular grid: the domain was subdivided into congruent rectangles. Obviously, there are se­ vere limitations when /changes rapidly. The grid has to be made super-fine and that is both computationally wasteful and never really perfect. In version 6 surfaces are plotted adap­ tively, with extra work done in the regions that require it. Further, one can superimpose curves defined in a general way on top of the surface. The function x2y!(x4 + y) illustrates the value of the adaptive method. This function is discontinuous at the origin: the limit along any straight line through the origin is 0, but along the parabola y c x2 the limit is c I (1 + c-2). The rectangular grid approach makes a mess of the region near the origin. But the image in Figure 1 , created by using P l o t 3 D together with some options to add the projections of lines and parabolas onto the surface and also exclude a small circle about the origin from the plot domain, shows the picture quite dearly. And one can use the mouse to grab and rotate this image in real-time, making its shape very clear indeed. =

Plot 3 D

[ X;:� ,

{ x , 0 . 0 0 1 , 2 } , {y, 0 . 0 0 1 , 2 } ,

Mesh � { Tan [ Range [ 0 . 0 7 , 1 . 5 6 , 0 . 2 ] ] , 15 } , MeshFunctions � { #2 /#1 &: , #3 &: } , MeshStyle � { Thickness [ 0 . 0 0 7 ] , Thicknes s [ 0 . 0 0 0 4 ] } , MaxRecursion � 5 , PlotPoint s � 1 5 0 , RegionFunct ion � ( Norm [ { # 1 , #2 } ] > 0 . 0 1 &: ) , AxesEdge � { { - 1 , - 1 } , { - 1 , - 1 } , { 1 , - 1 } } , ViewPoint � { - 0 . 7 , - 2 , 0 . 5 } , Boxed � False

]

Figure I . A surface whose limit at the origin is 0 along any straight

line (the thick black lines), but is discontinuous at the origin (the parabolas shown approach different limits).

Here are two other new graphics features. 1 . The ability to generate the convex hull of a set of points in IR3, useful when teaching lin­ ear programming. 2. The inclusion of a Reg i onPl o t function that can show the region in the plane corre­ sponding to a system of inequalities. I find this extremely useful when looking at differ­

ential equations, since, for an autonomous system x' = j(x, at the regions between the nullclines (the curves f= 0 or g

y), y'

=

52

THE MATHEMATICAL INTEWGENCER

=

g(x, y),

one can look

0) in the phase plane. In the

region where, say, both f and g are positive, the motion is entirely to the northwest. Thus one learns a lot about the system by examining such a plot. Here is one example. Consider the system (this is from [BCB]) given by x'(t) x - y 2 cos y and y'(t) = x sin x - y. It is useful to see, in the phase plane (the x-y plane), the region where f and g are both posi­ tive, and also both negative, and also f positive and g negative, and also the reverse. Here is how, in version 6, one can get all these regions with minimal programming effort. Fig­ ure 2 shows the result. One can then superimpose a vector field or actual solution curves for a comprehensive view of the orbits. =

f

=

x - y2 Cos [ y ] ; g

=

x Sin [x] - y;

RegionPlot [ { f > 0 &:&: g > 0, f < 0 &:&: g > 0 , f > 0 &:&: g < 0, f < 0 &:&: g < 0 } , {x, - 1 0 , 1 0 } , {y, -10, 1 0 } , MaxRecurs ion � 6 , BoundaryStyle � Thickness [ 0 . 0 0 5 ] PlotPoints � 1 0 0 ]

-

1

0

-s

o

s

10

Figure 2. Coloring the regions between the null­

clines in an autonomous system of two differen­ tial equations captures areas where the flow is in the same rough direction. These enhancements-adaptive plotting, restricting the plot domain, adding curves to the surface, varying the coloring method, and more-also apply to the more sophisticated surface­ plotting functions such as Parame t r i c P l o t 3 D, which shows the parametric surface (x(u,v), y(u,v), z(u, v)), and ContourPl o t 3 D, which can generate the surface corresponding to j(x, y, z) 0. Taken together this is a great improvement in Matbematica's ability to gen­ erate useful, informative, and pretty surfaces. And, finally, there is real-time rotation of 3-di­ mensional graphics, so that one can just grab the image with the mouse and spin it. For one more example, Figure 3 shows a very complicated surface that gives a new look to Pascal's Triangle. Some years ago, David Fowler [Fo] had the idea of making a plot of <;) using the gamma function for the factorials. It was not easy to get a good picture because of the singularities of f(z). But adaptive plotting as well as the new Exc l u s i ons option that, in this example, allows us to eliminate x E { - 1 , - 2 , - 3 ) , yields a quite nice picture of � , where f(z+ 1) is used for z!. The use of the mouse to rotate the image allows z= us t 6' 7J. �J'a ize it in a way that is difficult to show in print. =

( x!

\

l

Plot3D [ Binomial [ x , y] , {x, -3 . 5 , 1 . 5 } , { y , - 2 . 5 , 1 . 5 } , Mesh � { Range [ - 6 , 6 , 1 ] } , MeshFunctions � ( #3 &: ) , MeshStyle � Brown, Boxed � False, Cl ippingStyle � Fa l s e , PlotRange � { - 6 , 6 } , View­ Point � { 0 . 7 , - 2 . 3 , 0 . 5 } , AxesLabel � { x , y , z } , AxesEdge � { { - 1 , - 1 } , { 1 , - 1 } , { - 1 , - 1 } } , Exc lus i ons � { x

==

-1, x

==

-2 , x

==

-3 } ]

Manipulations One new feature is the ability to create a window with sliders that control parameters that directly affect the contents of the window. For example, the window might show a graph, in two or three dimensions, that depends on the parameters, or the result of a symbolic com© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

53

5

0

y

0 Figure 3. The graph of z . . 1 coeff"ICJent. nom1a

=

( x). y

X

using f(z + l) for the factorials in the bi-

putation. Here is a simple example. As one moves the slider the value of the exponent n changes and the polynomial factorization is updated instantaneously.

Manipulate [Factor [x" - 1] , {n, 10 } , 1 , 5 0 , 1 } 1

n ---{] 10

:I

0

l l-1 � I:+I I�I�I EJ

On the one hand this is an improvement on the generation of animations in earlier versions. More important, it allows one to set up sliders for any number of parameters, as opposed to having just one parameter change in a traditional animation. And it is also easy to include "locators" in the image so that one can just move them with the mouse. Thus, for example, the locator could be the initial value of a differential equation's orbit in the phase plane. The possibilities are limitless, and the company has set up a web site at < demonstrations.wol­ fram.com> which has about 2000 examples of how Manipu l a t e can be used, in a variety of fields throughout mathematics. A free downloadable product called MathPlayer allows non-Mathematica users to access and manipulate these examples. The simple example above only scratches the surface of what can be done: There can be multiple sliders, the output can be symbolic, numeric, graphic, or combinations of all. In short, it is very flexible and totally integrated into the Mathematica environment. Here are some examples to illustrate the value of all this. 1 . Morley's Theorem on the equilateral triangle formed by trisections . I was going to program this myself-a bit of work is

needed to get the trisections-when I thought of checking the site. Of course, someone had done it already. 2. Country data and Benford's law . This demo allows one to select from a variety of country-based data sets (land area, miles of unpaved roads, and so on) and examine a plot of leading digits to see if they follow the logarithmic prediction of Benford's Law. Some do, some don't. Elec­ tricity consumption (in kwh) does (see Fig. 4, where the slider allowing one to choose the data set to display are not shown). This example shows the value of having complex data sets included with Mathemat­ ica. Another example: if one wanted to know the ratio of unpaved airports to total air-

54

THE MATHEMAnCAL INTELUGENCER

1

2

3 4 5 6 7 8 9

30.% 15.% 14.% 10.% 7.5% 5.6% 8.5% 4.7% 4.7%

60 50 40 30 20 10 0

2

3

4

5

6

7

8

9

Figure 4. A histogram of the leading digits based on electricity consumption in

kwh for all countries.

ports for India and the USA, that is easy to find as follows . It is 29% for India, 66% for the USA.

{Nr Nr

countryoata [ " India" , " UnpavedAirport s " 1 CountryData [ " Indi a " , "Airport s " ] countryData [ " US " , "UnpavedAirport s " 1 CountryData [ " US " , " Airport s " ]

{ 0 . 2 873 9 , 0 . 655472 }

l}

l

·

4. Comprehensive data sets such as GraphDa ta, KnotDa t a , and PolyhedronDa ta (search for these terms at the demonstrations site). These show how one can access and visual­ ize some data sets of special interest to mathematicians. For example, the following code produces a knot known as 8 (see Fig. 5). 3

Figure 5. An example using

Kno tData. In the Alexander­

Briggs notation, this is knot 83.

Graphics 3 D [ { Orange, Specularity [White, 7 0 ] , KnotData [ { 8 , 3 } , " ImageData " ] } , Boxed � Fal s e , ViewPoint � { 0 , 0 . 1 , 5 } ]

Here is an example that shows how one can use the new GraphPl o t functionality to relate countries in an unusual way. In this graph (Fig. 6), two countries are connected if they have 50 shared city-names. This makes use of the Ci tyData database which has information for 163428 cities. These data are not actually in Mathematica, but are downloaded from a Wol­ fram web site when called for. The three lines of code for generating this example are in the C i tyData documentation. There are no great surprises in this graph-the USA and Spain have the largest degrees-but it shows how one can examine complex data in imaginative ways.

© 2007 Springer Science +Business Media, Inc., Volume 29, Number 4, 2007

55

�

\I

Figure 6. Countries are adjacent if they share more than

50 city-names.

truncated tetrahedron

cuboctahedron

truncated octahedron

truncated cube

small rhombicuboctahedron

great rhombicuboctahedron

icosidodecahedron

truncated dodecahedron

truncated icosahedron

nub cube

8

26

32

14

26

38

14

32

62

92

Figure 7. The thirteen Archimedean solids, ordered by the number of faces.

THE MATHEMATICAL INTELUGENCER

32

small rhombicosidodecahedron great rhombico idodecahedron

snub dodecahedron

56

14

62

Another example: the ten most popular city-names. Take [ SortBy [ Tally [ First /@ CityData [ ] ] , Last ] , - 1 0 ]

{ {Georgetown, {SanAntonio, { SanFrancisco, {Washington, {Clinton, {Franklin, {Salem, {SanJose, {SanMiguel, {SantaCruz, PolyhedronDa ta. 27} ,

27 } ,

27} ,

27} ,

28} ,

28} ,

28} ,

28} ,

28} ,

28} }

As a final example of the use of the databases, consider An Archimedean solid is a convex polyhedron whose faces consist of at least two regular polygons, whose structure at each vertex is the same, and whose symmetry group is the same as the sym­ metry group of one of the five regular polyhedra (thus excluding certain prisms and an­ tiprisms) . Here is how to show the 13 Archimedean solids in an array with labels giving names and numbers of faces (Fig. 7). polydata = Sort [ PolyhedronData [ " Archimedean " , { " FaceCount " , " Image " , " Name " } ] ] ; GraphicsGrid [ Part i t ion [ ( Graphi c s 3 D [ # [ [ 2 , 1 ] ] , PlotLabel � StringForm [ " " \ n " " , # [ [ 3 ] ] , # [ [ 1 ] ] ] , Boxed � Fal s e ] &: ) /@ polydata, 4 , 4 , { 1 , 1 } , { } ] ]

Programming and Discovery Mathematica itself is a programming language, and because it gives the programmer access to functions in so many areas of mathematics, it allows one to design programs of great ver­ satility. Here is a personal example that led to some surprises. On April Fools Day, 1975, Martin Gardner published a planar map as a hoax, stating that it required five colors if adjacent countries were to be colored differently. One year later Ap­ pell and Haken proved the four-color theorem! A few years ago I wondered about algorith­ mically coloring planar maps and set about to do it using Mathematica. This required cre­ ating planar map objects, writing a program to convert them to planar graphs where edges encode map-adjacency, and then figuring out an algorithm to 4-color the graph. For the al­ gorithm we used a small modification to Kempe's old method which, though flawed as a proof of the theorem, does in fact lead to a quite efficient algorithm. Thus it is only a con­ jecture that the modification we found [HW] will always work; yet it has never failed and so seems to be a perfectly reasonable practical approach. The program had no problem 4-coloring the Gardner map. But why put in so much effort to obtain a 4-coloring that one knows exists by the Appel-Haken Theorem? Well, there was an unexpected side benefit. I was asked if my code could be used to 4-color a Penrose tiling us­ ing rhombuses. There is nothing difficult about the associated map, so I encoded the Penrose rhombs as a map and did it. To my surprise, the algorithm never used more than three colors. Was there a theorem here? I soon learned that this observation had been made by others, but was still an unsolved problem, for both rhombs and kites-and-darts. I became very interested in the problem and was able, with Tom Sibley, to work out a short proof that Penrose rhomb tilings are 3-colorable [SW]. The kite-and-dart case was handled later by others [B] and is more complicated. While computation did not play a direct role in our proof, it was critical to inter­ esting me in the problem in the first place and providing experimental evidence for its truth. The point of the preceding example is that computational investigations often lead to discovery, and Mathematica makes it easy to make such investigations. Another example of this arises from the amazing Bailey-Borwein-Plouffe [BBP] formula 7T

(

1

X

=ln=O 16 n

4 8n+ 1

2

-

8n+4

-

1 8n+5

-

1 8n+6

)

.

Using Matbematica, Victor Adamchik and I were able to investigate hundreds of other se­ ries and eventually found one for

7T

that is a little simpler. Here is a sketch of how one can

discover such a formula using version 6. Step 1 : Find a closed form for a family of alternating, near-geometric series. The family was chosen on a hunch. sum =

b

oo

L

n• o

(-1)n

---

4n

(

a 4n

+

1

+

b 4n

+

2

+

C

4n

+

d

3

+ --4n + 4

)

ArcTanl � ] + a HypergeometricPFQl{ � , vl}, { ! } - � ] + � c HypergeometricPFQl{ � , 1}, {:}, - �] - d Log[4] + d Log [5] ,

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 4, 2007

57

Step 2 : Replace the special function with arccot and log, and simplify. sum = Fu l l Simpl i fy [ TrigToExp [ FunctionExpand [ sum] ] l

8 ( (a + 2 c) ('7T+ 4 ArcCot [3 ] ) + )8 (bArcCot [2] + [:]) (a - 2 c [ 2 5

]:_

d Log

l

) Log

+

Step 3: Eliminate one arccot and collect the transcendentals.

�; [ -4 (a + 2 c) -8 ( 8b - 4 (a 2 c) ) ArcCot 2] +

sum = Collect sum I . ArcCot [ 3 ]

1

1T +

1

- ArcCot [ 2 ] , { _ArcCot , _Log } [

+

d Log

=

[ 5 ] (a - 2 c) 4 8 -

+

1

-

]

Log

[ 2 5]

It is now elementary to find values of a, b, c, d that annihilate all transcendentals except for 1T, which remains with a coefficient of 1 : just let a b 2 , c = 1 , d 0. Starting over with those coefficients, out pops a nifty new formula for 1T. =

Fu l l S imp l i fy 1T

[ � =

( -1) n

---

4n

(

2

4n

+

1

+

2

4n

+

2

+

1

4n

+

3

=

)]

Optimization This field sends out tentacles in many directions: 1 . Linear programming to optimize linear functions. 2. Integer linear programming (ILP), where all variables are restricted to integers. 3. Algebraic optimization with constraints, of the sort taught in elementary calculus or that use Lagrange multipliers, but not only those. 4. Purely numerical optimization, where one gets an approximation to the optimum, not a proved-correct value, by some sort of randomized heuristic search. 5. Special cases, such as the Traveling Salesman Problem. 6. Local optimization, where one seeks a local optimum from a given starting value. Mathematica has functions for all of these, including simulated annealing (a random move­ ment method where the conditions allowing movement change as time passes) and differ­ ential evolution (a type of genetic algorithm) for (4); an approximation algorithm for the TSP; and state-of-the-art LP and ILP methods with extensions, in the case of (1), to the case that the objective function is quadratic or a ratio of linears. For one example, we could look at the most difficult problem of the 10 problems in the SIAM-100 digit challenge [BLWW] . Find the cubic polynomial that provides the best approxi­ mation to 1/r(z) on the unit circle in the sup norm; that is, minimize the maximum error over the circle. It is hard to know where to start with this, but once one has the objective function defined (using (6) with a bunch of seeds to get the max error for a specific cubic), differen­ tial evolution zooms into the correct answer in a very impressive way. More precisely, one first defines an objective function that for a cubic defined by a, b, c, d determines the largest er­ ror. Then one uses to seek the values of a, b, c, d that minimize the error. can use various heuristics and it is not always clear which is the best choice. But it turns out that differential evolution works very well on this problem. For more details see [BLWW] . The point is that this was the most difficult problem in a modern computational con­ test, and Mathematica had no trouble with it, once the proper method was identified. Here is an example related to recent work on the Frobenius problem [ELSW] . Given pos­ itive integers a, b, c, and d, we seek the smallest positive integer multiple of b that is con­ gruent modulo a to a nonnegative linear combination of c and d. It is clear that the multi­ ple a · b works, so we add the condition k :s a.

imize

NMinimize

NMin­

{a , b , c , d } = { 1 0 6 , 2 0 0 3 4 7 5 4 1 , 5 9 9 854 1 1 1 , 9 4 5 1 2 1 2 3 3 } ;

Minimi ze [ { k , k b - y c - z d == m a && { k , m, y, z } E Integers && 1 :::S k :::S a && 1 :::S m && O :::S y && O :::S z } , { k , m, y, z } ]

{493 , {k� 493 , m� 3349 , y� 22 , z � 87 } }

493 and the coefficients on c and d be­ This ILP problem is solved instantaneously with k ing 22 and 87. Indeed 493 b and 2 2 c + 87 d both end in 337713 and so are congruent mod 1 06 , but we learn much more here in that 493 is minimal. And this works for much larger numbers with no problem. The work involved in incorporating this sort of thing into Math­ ematica was related to current research on the Frobenius problem and led to inclusion of =

58

THE MATHEMATICAL INTELLIGENCER

algorithms for the Frobenius number [ELSW] . The Frobenius number g (A) of a set of posi­ tive integers A is the largest number not in the semigroup generated by A over the natural numbers; e.g., g( {2, 5}) 3. Here is a much larger example. =

FrobeniusNulnber [ { 1 0 1 0 1 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 98 7 6 5 4 3 3 3 3 3 3 3 3 1 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 1 9 9 9 9 9 9 9 9 9 9 111 1 1 1 1 1 1 } ]

5 646 419 728 710 886 407

And here is an algebraic optimization that appeared in a problem journal recently. The max­ imum of the first expression is 5/4, and it occurs when each variable is 1/2. Maximi ze [ { a b + b c + c d + d a + a c + b d - 4 a b c d 1 a > 0 1 b > 0 1 C

> 0 1 d > 0 I a 2 + b2 +

{ � {a � 2_ 4 '

2

' b�

2_ ' 2

c

C2

+d

�

2

2_ ' 2

::::;

d

1}

{ aI b1

I

� 2_ }}

C

1 d}]

2

Some problem journals seem to like to pose these problems, which take a page or more of algebra to work out. I have written uncountably many letters claiming that such work is now best left to machine, but some editors and problemists still subscribe to the view that it is instructive to work these out by hand. To me, such problems appear to be in the same class as computing the tenth decimal digit of V39. This problem appeared in [D] where the 5/4 was given as the bound. But Mathematica does not need that hint, since solves the problem and produces the upper bound without breaking a sweat. One can raise the interesting and relevant question: "Can we certify correctness?" Such certification is possible with integrals and many symbolic sums, but is not available with the sort of algebra illus­ trated here. Still, there are many empirical ways to validate the answer if one is skeptical.

Maximize

Numerics Mathematica's adaptive precision allows one to get accurate results without knowing in ad­

vance the precision needed. For example, if one wants 20 digits of the sine of 1 05°, one would need about 70 digits of precision. But by asking for 20 digits the amount of needed precision is computed internally and is invisible to the user. N [ Sin [ 1 0 5 0 ]

I

20]

-0 . 78 9 67249342931008271

This idea can be used to get accurate plots of difficult functions. The degree-200 Maclaurin approximation to cos x is numerically unstable (because of the alternating signs) and a plot using only machine precision is a mess. But now one can add an option of the form 2 0 0 and get an accurate plot. And as noted above, this can also be done without knowing in advance that 200 digits are sufficient. Just turn each input to the plot into a rational and evaluate the polynomial at that rational to 20-digit accuracy using adap­ tive precision. Here is the code, with the result in Fig. 8.

Work­

ngPrecision� rr

Plot N

10 0

i

•O

n

( -1 ) n x (2

2

n) !

n

I.

{ y l 0 1 8 0 } 1 PlotRange

X� �

1 Rationalize [y I --3 -] 10

{-1� 2 }

I

5

J

I

Two other problems from the SIAM 1 00-Digit Challenge showed off Mathematica's nu­ merical capabilities well. One asked for the solution of a sparse linear system of 20,000 2.0 1.5 1.0

{\

0.5

{\

A

II

�

2

-0.5 - 1 .0

1\

v

v

1\

80

6( v

v

v

v

v

Figure 8. A graph of the degree-200 Maclaurin poly-

nomial of cos x.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

59

equations . The new-at-the-time Spa r s eArray technology allowed this to be done eas­ ily, once the appropriate method was chosen; details in [BLWW] . Another asked for

f�

cos

C0! x) d x; this integrand is extremely nasty, with infinitely many unbounded os­

cillations near 0. Nint egra t e handles it cleanly once one makes a transformation to the interval (0, oo) using Lambert's W-function; that is, substitute u - (log x)lx, which corre­ sponds to x W( u)!u. =

=

Nintegrate [ Co s [ u ] La.mbertw' 1 [ u ] 1 {u l 0 1 oo } ]

0 . 32 3 3 67

Miscellaneous New Things Here are some miscellaneous new things of interest. Inclusion of an algorithm to compute the inverse of the Euler 0 1 x1 Integers ]

ll

ll

ll

ll

X = = 7 0 0 1 x = = 7 1 7 l x = = 7 2 7 9 x == 7 7 1 1 x == 8 8 7 5 1 1 X = = 1 4 0 0 2 I x = = 1 4 3 4 2 x = = 1 4 5 5 8 l lx == 1 5 4 2 2 x = = 1 7 7 5 0

I

ll

ll

We can do a quick computation related to the Carmichael Conjecture and the Sierpinski-Ford Theorem. The first says that the number of solutions to c/J(x) n-the Euler multiplicity-is never 1 ; the second says that this multiplicity can be anything else. So here are the multi­ plicities that arise as n varies from 1 to 100. One can see that 1 does not show up and the other numbers do: going farther will yield 1 2 , 13, and so on, and it is now a proved theo­ rem of Ford [F] that every integer other than 1 can be an Euler multiplicity. =

union [ Table [ Length [ Reduce [ EulerPhi [x] == n

&:&:

X > 0 I XI Integers ] ]

I

[nl 1 0 0 } ] ]

{ 0 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 101 11 1 17}

The treatment of Diophantine equations is quite nice. One can get the symbolic solution to the Pell equation, or a list of solutions less than some bound. Less familiar are algorithms to solve other special types; here is a binary quadratic equation. The workhorse function here is Redu c e . Reduce [ 1 + 12

x + 2

< x == -4 && y == - 1

l

II

x2 + 7

y + 5

x y+ 2

y2 == 0 1

{xl

y} l

I I < x = = 4 && y == - 9 l

< x == 2 && y == -3 l

Integers ]

And here is a Thue equation; Reduce finds all solutions in under a second. Reduce [x3 - 4 X y2 + r == 11 {XI y} I Integers ]

( x = - 2 && y = 1 )

II

II

< x = 2 && y = 1

( x = O && y = 1 )

l

II

II

( x = 1 &&

< x = 5 0 8 && y = 2 7 3

l

y= O)

II

( x = 1 && y = 4 )

Here is a difficult puzzle: Write 31 as a sum of four cubes. The reader might think about how one would develop a reasonable algorithm to solve this. Using a somewhat obscure option, one that is explained in the advanced documentation on Diophantine equations, F i ndins t ance can find an instance in a second. SetSystem<>pt ions [ " ReduceOptions " � { " S ieveMaxPoint s " � 1 0 7 } ] ; Findinstance [x3 + y3 + z3 + t3 == 3 1 1 { x 1 y1 z 1 t } 1 Integers]

{ {x � 1 0 3 1 y � 3 4 1 z � - 6 5 � t � -9 5 } }

There is no doubt it is correct: 1033 + 343 - 653 - 953 31. Experienced users will appreciate the following additions. ( 1 ) The ability to add borders around rectangles, polygons, and disks via EdgeForm; formerly one would have had to man­ ually program the L i n e or C i rc l e object that forms the border. (2) Some improvements to Table and Do that allow constructions such as Do [ s omething { i 1 { 1 5 7 9 } } ] which causes the iterator i to run through the objects (not necessarily numbers) in the list =

I

I

I

I

I

that follows it; this is more flexible than the traditional situation where the iterator has the form { i ffi 1 n } and i takes on the values from m to n. Here's a cute one: Qui e t [ expr J evaluates expr ignoring any messages or warnings that might be generated. I

Things That Could Be Better Here are some features that could be improved. One hopes that some of these points will be addressed in the not too distant future. 60

THE MATHEMATICAL INTELLIGENCER

1 . The treatment of graphs needs to be integrated. Having GraphPlot as a Mathematica ker­ nel function is inconsistent with having Graph objects and the ShowGraph function appearing only in the Combinatorica package. And an interactive method of inputting graphs would be most welcome. Also it would be nice if graph drawings could have piecewise linear edges, not just straight-line edges; then one could handle planar graphs in a more flexible way. 2 . Sometimes one has monotonic data and wants an interpolating function that is also mo­ notonic. There are well-known methods for achieving this and it would be nice if this ca­ pability were added to the general interpolation environment. On a related note, some more support for Bezier curves would be welcome. 3. Triangulation of polygons is one of the most basic functions of computational geometry and must be added to the kernel to make its computational geometry capabilities comprehensive. 4. While not built-in, one can use existing functions to get all the roots to f= 0 in a given interval. There are ways to handle the 2-dimensional version of this, and one would like to see that added. That is, I want a built-in way to get all the roots to a system of two equations (j= 0, g = 0) that lie in a given rectangle. An important application is that one could get the set of equilibrium points to an autonomous 2-dimensional system of DEs; this corresponds to the curve intersections in Figure 2 .

Conclusion The impact of computing on mathematics in the last 20 years has been broad and deep, on both the theoretical and applied sides. There are lots of software packages available and for just about any computational task, there is a tool that can handle it. But in terms of cover­ age of the diverse fields of mathematics, nothing comes close to the newest version of Math­ ematica. And the company's outlook that mathematics is used throughout modern science, economics, and engineering has led to a program of unmatched breadth. REFERENCES

[AW] V. Adamchik and S. Wagon, A simple formula for

71',

American Mathematical Monthly, 1 04 (1 997) 852-855.

[B] R. Babilon, 3-Colourability of Penrose kite-and-dart tilings, Discrete Mathematics , 235 (2001 ) 1 37-1 43. [BBP] D. H . Bailey, P. B. Borwein, and S. Plouffe, On the rapid computation of various polylogarithmic constants, Mathematics of Computation, 66 (1 997) 903-9 1 3. [BCB] R. I. Borelli, C. Coleman, and W. E. Boyce, Differential Equations Laboratory Workbook, Wiley, New York, 1 992. [BLWW] F. Bornemann, D. Laurie, J . Waldvogel, and S . Wagon, The SIAM 1 00-0igit Challenge, SIAM, Philadelphia, 2004. [C] The NEOS server for Concorde, < http://www-neos.rncs.anl.gov/neos/solvers/co: concorde!TSP.html> [D] G. Dospinescu , Problem 3059, Crux Mathematicorum, 32 : 6 (Oct. 2006) 399. [ELSW] D. Einstein , D. Lichtblau, A. Strzebonski, and S. Wagon, Frobenius numbers by lattice point enumera­ tion, INTEGERS 7 (2007) #A1 5 . [F] K. Ford, The distribution o f totients, Electronic Research Announcements o f the American Mathematical Society. 4 (1 998) 27-34.

[Fo] D. Fowler, The binomial coefficient function, American Mathematical Monthly 1 03 (1 996) 1 -1 7 . [HW] J . P. H utchinson and S. Wagon, Kempe revisited, American Mathematical Monthly, 1 05 (1 998) 1 70-1 74. [SW] T. Sibley and S. Wagon, Rhombic Penrose tilings can be 3-colored, American Mathematical Monthly, 1 06 (2000) 2 5 1 -253. [W] S. Wagon, The Banach-Tarski Paradox, from The Wolfram Demonstrations Project

STAN WAGON Stan Wagon's enthusiasms for mathematics and mountaineering are

expressed separately, but are also combined in frequent efforts in mathematical snow sculpture. Readers may recall his article (with fellow-sculptors) in The lntelligencer 22(2000), no. 4, 37-40. More recently, his team took second prize in the Brecken­ ridge International Snow Sculpture Competition in January 2007. Department of Mathematics Macalester College

St Paul, MN 55 1 OS USA e-mail: [email protected]

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4 , 2007

61

A C lash of M athe matica l T itans i n Austi n : H arry S . Vand iver and Robert Lee M oore ( 1924-1974) LEO CORRY

Send submissions to David E. Rowe, Fachbereich 08-lnstitut fUr Mathematik,

he mathematical scene at the Uni­ versity of Texas was dominated from the mid-1920s to the late 1960s by two towering, yet very differ­ ent figures: Robert Lee Moore (18821974), and Harry Schultz Vandiver (1882-1973). Starting in the late 1930s, these two giants entered into a conflict that grew to mythic proportions and lasted for more than three decades. Though this affair permeated all aspects of departmental life, and even spilled over into the wider arena of academic affairs in Austin, it became most visible in 1 945 when Vandiver-whose re­ search focused exclusively on number theory and associated algebraic fields­ was transferred to the Department of Applied Mathematics and Astronomy. In this unlikely setting, the alienated east­ erner and the feisty southerner carried on their own private cold war that echoed the politics of the post-war era. In retrospect this conflict may seem rather preposterous. In fact, eye wit­ nesses at Austin have never been able to say precisely when and how the en­ mity began, though many could later remember the icy non-relations be­ tween Moore and Vandiver. After the departments of pure and applied math­ ematics were joined in the early fifties, Moore and Vandiver made sure that their offices in UT's new Benedict Hall not only were on different floors but also could be reached by separate stair­ ways.1 Vandiver's son, Frank (19262005), a highly respected historian of the American Civil War and president of Texas A&M University, remembered Moore pointing a loaded gun at him when he was a child:2 I was . . . walking home from school one day, . . . and this car pulled up by me on the curb, and Dr. Moore was in it. I thought he was going to offer me a ride home which I was willing happily to accept. Instead of

T

1 [Greenwood 1 988, 47]. 2Frank Vandiver, interview with Ben Fitzpatrick and Albert C. Lewis, June 30, 1 999 (Oral History Project, The Legacy of R. L. Moore, Archives of American Mathematics, Center for American History, The University of Texas

Johannes Gutenberg Un iversity,

at Austin).

055099 Mainz, Germany.

3[Corry 2007].

62

that, he pointed this pistol at me, and said, "Ah ha, what do you think of this?" I was absolutely terrified. I thought he was actually going to shoot me. I don't remember what I said. . . . I realized that Moore and Daddy were not friends, and I had the feeling that maybe he was go­ ing to kill me, but I think it was sort of a grim joke he was playing. The gun was loaded, that I could tell, so I was not enamored of that moment. In R. L. Moore: Mathematician and Teacher, John Parker devotes an entire chapter to this legendary feud, fittingly entitled "Clash of Titans. " Here I offer a fresh view of this rather bizarre episode in the history of American mathematics against the background of the portrait of Vandiver-a somewhat forgotten figure-presented in my arti­ cle in the last issue of this magazine.3 There, the focus was on Vandiver's life­ long pursuit of Fermat's Last Theorem (FLT); now I turn to broader themes in his career, many of which reflect on­ going conflicts at the University of Texas, as well as the particular antag­ onism that existed between him and Moore. Some of the main elements of this story appear in Parker's book, but I emphasize Vandiver's perspective and complement the picture with some in­ teresting unpublished documents from the latter's archive in Austin. It is also important, of course, to con­ sider this conflict in context and pro­ portion. There are undoubtedly many such stories of local feuds in mathe­ matics departments or of local figures who single-handedly dominated de­ partmental life. Still, this dispute had a special intensity and tone, heightened no doubt by the stature of both men in the American context at the time. Moore was certainly a much respected figure in the American community; he served as mentor to several students who went

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media, Inc.

on to positions of prominence. An as­ sessment of Vandiver's standing in the community is a more complex matter, as I pointed out in my previous article. Personal differences were no doubt a central factor in igniting and then sus­ taining and exacerbating this conflict. The gun incident with young Frank Vandiver was just one extreme exam­ ple of Moore's often aggressive behav­ ior. In 1 944, for instance, a heated dis­ cussion in the mathematics department reportedly ended up in a fistfight be­ tween Moore and Edwin Ford Becken­ bach (1902-1982), an associate profes­ sor at Austin at the time. 4 As Albert C. Lewis has pointed out, "in Texas, at least, the successful use of nonverbal language need not detract from one's reputation. In fact, for an established male scholar it adds a cachet which can probably only help one's reputation 5 outside the scholarly world. " In his younger years, Moore trained inten­ sively in boxing, and his rather aggres­ sive personality could occasionally slip into physical intimidation and even as­ sault.6 Still, Moore was hardly a singu­ lar case; his colleague and life-long friend H. ]. Ettlinger was involved, in his youth, in physical incidents (one in response to an anti-Semitic insult), and later "was accused of using less violent but still physical tactics in departmental controversies of subsequent years . "7 This rough-and-tumble Texas atmo­ sphere was not congenial to Vandiver's naturally reticent personality. He would sometimes isolate himself for days to do research and listen to his large collec­ tion of classical records. Vandiver was "hardly the athletic type," and in the winters he worked in a top coat with a portable electrical heater warming his feet and legs.8 Moore, on the other hand, was a dynamo. A strongly au­ thoritarian personality, he was directly involved in, and made great efforts to shape, every detail of departmental life for decades. Vandiver always kept him­ self at a safe distance from any kind of administrative duties. He was famous

Figure

I . Harry S. Vandiver (Creator:

Walter Barnes Studio (HSV).

for taking frequent leaves of absence, drawing on the financial support of var­ ious foundations in order to visit other departments both in the United States and abroad. The clash between these two mathe­ matical titans thus operated at a variety of levels, including cultural and political issues that were charged with tense emo­ tions. As I will show, personal differ­ ences by no means tell the whole story. This once-famous feud deserves closer attention because of its deeper, under­ lying dimensions, which reflect how each of the protagonists saw himself as a researcher and a teacher. Moreover, the contrasting opinions and attitudes of Vandiver and Moore also had ramifica­ tions for their respective mathematical activities. As we shall see, Vandiver took a very different approach from Moore's when it came both to mathematical re­ search and mathematics education.

Two Mathematicians, One University, Two Departments Soon after it opened in 1883, the Uni­ versity of Texas at Austin appointed George Bruce Halsted ( 1 8 1 0-1936) its first professor of mathematics. Leonard

Eugene Dickson (1874-1 954) was the most prominent among the relatively few mathematics students in those early years. After completing an M.A. degree in 1894, Dickson moved to Chicago to become one of the first doctoral stu­ dents of Eliakim H. Moore ( 1 862-1932). In 1899 Dickson accepted a three-year appointment at Texas, but soon left again for Chicago, this time for good. One of the students in his calculus course during his brief tenure at UT was Robert L. Moore, who also took courses with Halsted. R. L. Moore later went to Chicago for doctoral studies as well, working on foundations of geometry between 1903 and 1 905.9 Always outspoken and critical, Hal­ sted eventually got into trouble with the Board of Regents, and at the end of 1 902 he was dismissed from his post. Math­ ematical leadership at UT devolved to Milton Brockett Porter ( 1869-1960) and Harry Yandell Benedict ( 1869-1937), both of whom had studied at Austin and later completed Ph.D . degrees at Har­ vard. As university regulations then al­ lowed for only one professor in each department at UT, Benedict was ap­ pointed professor in applied mathemat­ ics. These regulations were later to change, but the division into two de­ partments would remain, and the rela­ tionships between them remained a source of ongoing administrative trou­ bles. 10 The increase in student popula­ tion in the USA in the period following WWI heightened the demand for math­ ematics teachers across the country, Austin included. During the war, Goldie Prentis Horton (1887-1972) had worked with Porter and in 1 9 1 6 became the first recipient of a doctoral degree in math­ ematics granted by the University of Texas. Soon after graduating she joined the Austin faculty; she and Porter mar­ ried in 1934. Porter's aim was to raise research standards at UT by hiring mathe­ maticians of proven quality; he was obviously undeterred by unconven­ tional personalities. R. L. Moore was

4[Greenwood 1 983, 53]. This incident has been confirmed to me in a personal communication by Richard Kelisky, one of Vandiver's students. 5[Lewis 1 989, 225]. 6[Parker 2005, esp. 84-6] . 7[Lewis 1 989, 224]. 8Robert Greenwood, "The Benedict and Porter Years, 1 903-1 937," unpublished oral interview (March 9, 1 988) (MOHP), p. 26. 9For historical information on mathematics at UT, I rely on various sources, and especially on [Greenwood 1 983, 1 988], [Lewis 1 989], [Parker 2005]. 10[Lewis 1 989, 232].

©

2007

Springer Science+ Business Media, Inc., Volume

29,

Number 4,

2007

63

appointed assistant professor in 1 920 after a decade at the University of Penn­ sylvania. Moore's mathematical capa­ bilities had been recognized while he was just an undergraduate at Austin. In 1 90 2 he succeeded in sharpening Hilbert's analysis of the axioms of geom­ etry (in an early edition of Grnndlagen der Geometrie) by pointing out a re­ dundancy. This research was part of a new trend of enquiry known as postu­ lational analysis, which emerged in the United States in the first decade of the twentieth century_ll Three years later he completed his dissertation at Chicago under the supervision of E. H . Moore and Oswald Veblen (1880-1960) on "Sets of Metrical Hypotheses for Geometry," a study that followed the same approach but focused on topo­ logical questions. For the remainder of his career-which began with brief appointments at Tennessee, Princeton, and Northwestern, prior to his 1 0-year­ stay at the University of Pennsylvania­ Moore continued his research on these same topics. Thus, in an important pa­ per from 1 9 1 5 he investigated separa­ tion properties in a strikingly innovative way. By the time he returned to his Alma Mater in 1920, Moore had published 1 7 research papers i n a field whose name he had coined: point-set topology. Four years later Vandiver arrived in Austin, having taught for five years at Cornell. A high-school dropout, Van­ diver had studied some college-level mathematics in Pennsylvania but never took a college degree. In 1900 he be­ gan submitting solutions to problems posed in the American Mathematical Monthly, some in collaboration with the young George David Birkhoff ( 18841 944). After spending more than ten years as a customs house broker, Van­ diver obtained the position at Cornell in 1 9 1 9, thanks in part to Birkhoff's en­ dorsement. That same year he collabo­ rated with Dickson (then at Chicago) in the preparation of the latter's book on the history of the theory of numbers, es­ pecially the chapter on FLT. Dickson be­ came Vandiver's main source of inspi­ ration in all aspects of mathematical

activity. Dickson also did much over the following years to promote Vandiver's career. In 1914 Vandiver published his first article on FLT and for many years continued to present short communica­ tions to the AMS on that topic. In 1920 he published his first truly substantial contributions to FLT, for which he be­ gan to receive recognition. During his early years in Texas he continued this research, which led to a landmark pa­ per in 1929. He was subsequently awarded the first AMS Cole prize for out­ standing research in number theory. 1 2 Moore's first ten years at Texas were similarly productive. In 1929 he pre­ sented a summary of his work in the Colloquium Lectures Series of the Amer­ ican Mathematical Society. Published in 1932, his Foundations of Point Set The­ ory came to be regarded as Moore's magnum opus. 1 3 Other members of the department of pure mathematics at the time included John William Calhoun (1871-1947), Edward Lewis Dodd (18751 943), Paul Mason Batchelder (1886-1971), and Hyman Joseph Ettlinger (1889-1986). In 1925 Renke G. Lubben (1898-1980) was the first of Moore's stu­ dents to join the faculty at Austin. Thus Porter's efforts led to the consolidation of a respectable graduate faculty, with Moore and Vandiver as its central pillars. It seems that relations between Moore and Vandiver began on reason­ ably friendly terms. As an outsider and a later arrival in Austin, Vandiver was in a less advantageous position. He was also without formal academic training; but in Porter's view "the mere posses­ sion of a doctoral degree (or any other degree) was small indication of abil­ ity." 1 4 Moore presumably felt the same way. But later, when Vandiver became recognized world-wide for his research and was elected to the National Acad­ emy of Sciences only shortly after Moore himself, the latter took such mat­ ters of status very seriously. Moore seems to have been especially irritated when in 1946, at the height of their feud, an Honorary Doctorate of Science was conferred on Vandiver by the Uni­ versity of Pennsylvania, an institution

that Moore always saw as his second academic home. In view of the deep differences in background and personality between the two men, one can hardly be surprised that Vandiver and Moore did not develop a strong friendship. Moreover, a glance at the trajectories of their respective ca­ reers does suggest reasons why they be­ came such fierce rivals. Beginning around 1930, Moore's research output gradually declined, both in numbers and in impact. Throughout the 1930s he pub­ lished only five research papers, choos­ ing instead to devote most of his time and efforts to teaching. By now he was also supervising large numbers of grad­ uate students, several of whom would become distinguished researchers. The Moore school flourished in no small part because the Texas topologist knew how to use his influence effectively when it came to landing key positions for his former students. Vandiver, by contrast, would remain fully devoted to research for decades to come. At the same time, he never distinguished himself as a lec­ turer and attracted relatively few stu­ dents. Instead, he worked with a faith­ ful circle of collaborators, most of them from outside Austin. He met with them often, especially during his frequent leaves of absence. Whereas Moore ex­ celled in the classroom, Vandiver fa­ vored scholarship. His expository papers and authoritative accounts related to FLT and the theory of cyclotomic fields were widely read. These striking differences between Moore and Vandiver went to the core of their respective identities as mathe­ maticians, and there can be little doubt that those differences contributed to the mutual animosity that developed be­ tween them. An anecdote from many years later is telling: In 1963, at the age of 8 1 , Vandiver submitted his final pa­ per to be published in the Proceedings

of the National Academy of Science. 1 5

On this occasion, Edwin Wilson (18791964) wrote to express his delight that Vandiver was still working at an age when "most have had enough. " To this, Vandiver replied that if people stop

1 1 See [Corry 2004, 1 72-182]. 12For details see [Corry 2007]. 13[Moore 1 932]. The revised edition of 1 962 also contains many acknowledgements of results obtained by his students.

14[Greenwood, et a/. 1 973, 1 0929]. 15[Vandiver 1 963].

64

THE MATHEMATICAL INTELLIGENCER

publishing at an old age it is not be­ cause they have had enough, but in many cases, because "they permit teaching duties and certain other acad­ emic pursuits to take up so much of their time that it is impossible to pre­ pare any original mathematical paper of their own."16 Vandiver evidently took pride in saying that he had never let this happen to him. He did not need to add that the other Texas titan, who was still teaching in Austin, had given up re­ search decades earlier.

Moore's Method and Vandiver's Lack Thereof Nothing better signifies the stark con­ trast between the personalities of Moore and Vandiver than their respective atti­ tudes toward teaching. The subtitle of Parker's biography very aptly captures the essence of Moore's character: "Math­ ematician and Teacher." If a reliable bi­ ography of Vandiver is ever written, the word "Teacher" will most certainly not appear in its title. Even his closest col­ laborators and friends stressed his poor abilities as a lecturer. His intellectual and personal energies were never di­ rected toward teaching or supervising graduate students. Nor did he maintain close relations over the years with the few he did supervise (one in 1941 and four in the 1950s). Parker emphasizes the centrality of teaching throughout Moore's entire ca­ reer, including the development and in­ fluence of the famous Moore Method: The 50 students he guided to their PhDs can today claim 1 ,678 doctoral descendants. Many of them are still teaching courses in the style of their mentor, known universally as the Moore Method, which he devised. Its principal edicts virtually prohibit stu­ dents from using textbooks during the learning process, call for only the briefest of lectures in class and de­ mand no collaboration or conferring between classmates. It is in essence a Socratic method that encourages students to solve problems using

their own skills of critical analysis and creativity. Moore summed it up in just eleven words: 'That student is taught the best who is told the leastY To be sure, a precise definition of the Moore Method is not a straightfor­ ward matter. Moreover, given the quan­ tity and quality of mathematicians who came under Moore's direct and indirect influence, one must presume that many developed their own versions of this teaching method.18 Parker gathers a large number of testimonials from grate­ ful and admiring students who went on to successful careers; many pointed to the training they received from Moore as the single most decisive factor in the consolidation of their mathematical out­ looks and scientific personalities. One distinguished pupil, Raymond L. Wilder (1 896-1982), offered this vivid account of his former teacher's methodology:19 He started the course with an infor­ mal lecture in which he supplied some explanation of the role to be played by the undefined terms and axioms. But he gave very little intu­ itive material-in fact only meager indication of what "point" and "re­ gion" (the undefined terms) might refer to in the possible interpreta­ tions of the axioms . . . . The axioms were eight in number, but of these he gave only two or three to start with; enough to prove the first few theorems. The remaining axioms would be introduced as their need became evident. He also stated, without proof, the first few theo­ rems, and asked the class to prepare proofs of them for the next session. . . . In the second meeting of the class the fun usually began. A proof of Theorem 1 would be called for by asking for volunteers. If a valid proof was given, another proof dif­ ferent from the first might be of­ fered. In any case, the chances were favorable that in the course of demonstrating one of the theorems that had been assigned, someone would use faulty logic or appeal to

a hastily built-up intuition that was not substantiated by the axioms . . . . The course continued to run in this way, with Moore supplying the­ orems (and further axioms as needed) and the class supplying proofs. . . . Moore put the students entirely on their own resources so far as supplying proofs was concerned. Moreover, there was no attempt to cater to the capacities of the "aver­ age" student; rather was the pace set by the most talented in the class. Not everyone, of course, shared this enthusiasm for the Moore Method, which was roundly criticized by stu­ dents as well as established mathe­ maticians from the time the master first began to promote it. Vandiver was by no means an overt critic, but he also clearly showed no sympathy for such a radical approach. Nor was he willing to invest a similar amount of time and energy in teaching and supervision, and he remained essentially sceptical that any didactical method, including Moore's, could systematically turn out outstanding research mathematicians. Vandiver also disliked Moore's aggres­ sive tactics when it came to hunting down promising students in UT's en­ tering classes. In this manner, Moore gained indirect control over many of the best talents, including those who received financial aid, while depleting funds that might have gone to students associated with Vandiver and other, more passive, colleagues. With regard to the training of grad­ uate students, Vandiver's views were close to those of another Dickson pro­ tege, Eric Temple Bell (1883-1960). In­ deed, Vandiver and Bell had much in common, beginning with their mutual interests in number theory, though Bell's research never attained the level of Vandiver's. Like his Texas counter­ part, Bell took a dim view of certain of his colleagues at Caltech who were constantly hunting for brilliant new stu­ dents.20 Nor did Bell ever distinguish himself as a lecturer, 21 though he was

1 6Wilson to Vandiver, March 1 8, 1 963; Vandiver to Wilson March 27, 1 963. Like other letters cited in this article, this one is kept in the Vandiver Collection, Archives of American Mathematics, Center for American History, The University of Texas at Austin (hereafter cited as HSV). Letters are quoted by permission. 1 7[Parker 2005, vii] . 18For information on Moore's students as teachers, see [Parker 2005, 1 44-1 59], [Zitarelli & Cohen 2004]. 19[Wilder 1 959]. 2o[Reid 1 993, esp. 261 -265]. 21 For a devastating criticism of Bell's didactic abilities voiced by a former student, Clifford Truesdell, see [Reid 1 993, 284] .

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

65

much more active than Vandiver when it came to supervising doctoral stu­ dents. Still, Bell's antipathy toward teaching is apparent from a letter sent to Vandiver in 1933 in which he bluntly expressed his views about the futility of training researchers. Concerning Moore's avowed ability to produce original research mathematicians, he wrote: I don't blame you for getting away from the damned students. The more I see of them, the more I am convinced that trying to train peo­ ple to do research is a waste of time. What few ideas a trainer has left af­ ter ten years of it are too precious to be thrown away. A man who is worth a damn will train himself.22 Raymond Wilder's account of Moore's classroom technique, cited above, highlights another aspect of de­ cisive importance, namely the close connection between the subject matter taught, point-set topology, and the di­ dactical approach taken. As noted ear­ lier, R. L. Moore's mathematics was part of the new trend of research in postu­ lational analysis through which he emerged as a central figure in Ameri­ can mathematics. His didactical method thus arose as a natural concomitant of this new research orientation. In contrast, for Vandiver, axiomatic analysis was of very limited interest. For one thing, axiomatics simply were not needed for the kinds of problems he was pursuing in number theory and the theory of cyclotomic fields. In fact, his stance toward modern, structural alge­ bra was ambivalent at best. Vandiver's mathematical strengths lay in very dif­ ferent directions, and because didacti­ cal concerns were not high on his math­ ematical agenda, he did not develop a systematic approach to teaching that could be related to axiomatics. This was evident even in his occa­ sional attempts to imitate Moore's method in his own teaching. According to one of Moore's prominent students, Richard D. Anderson, describing a course in 194 1 : Vandiver didn't realize that Moore had a very carefully organized struc-

ture sequence in his questions, with prompts in between so he didn't just send us off and tell us to see what we could do. He was definitely lead­ ing students towards more and more sophisticated thinking, towards re­ search with the goal of developing research mathematicians, people who were really creative. Vandiver, on the other hand, would just come in sort of casually and ask things and eventually gave up on that and went to reading books, chapters from Albert's Algebra and from Vandiver's own books.23 To the extent that Vandiver did adopt any pedagogical principles, these re­ flected a reliance on classical mathe­ matical literature (preferably read in to­ tal isolation) . This approach he had learned from Dickson, as he repeatedly explained in later years: [Dickson] had an office adjoining the Mathematical Library, which fine li­ brary was very quiet, a fact, of course, which helped him in con­ centrating on any matter at hand. Also, if he wished to consult or re­ view any mathematical article, all he had to do was walk a few steps to locate it . . . . This situation may have had a great deal to do with the fact that as far as the publication of orig­ inal mathematical articles is con­ cerned, D ickson was probably the most prolific mathematician of his time.24 It is therefore interesting to notice that back in the 1920s Dickson had been among the early critics of Moore's then-emerging pedagogical views. Moore himself reported that in the early twenties, during a summer visit to Chicago, he discussed effective meth­ ods of teaching mathematics with E. H. Moore and Dickson. R. L. Moore ex­ plained the approach he had been de­ veloping at the University of Pennsyl­ vania: posing questions or theorems for students and insisting that they settle them on their own. Assistance of any sort, including conversations with fel­ low students and searching in books,

22Bell to Vandiver: November 1 , 1 933 (HSV). Emphasis in the original. 230uoted in [Parker 2005, 1 82]. Actually, Vandiver published no book of his own. 24[Vandiver 1 960, 50]. 25[Traylor 1 972, 92]. 26[Lewis 1 989, 236].

66

THE MATHEMATICAL INTELUGENCER

were strictly forbidden. Students should rely on their own capabilities. Dickson "tended to quickly deride that ap­ proach, but E. H. Moore, as was his wont, said little. He customarily gave some thought to new ideas before re­ acting to them. "25 Vandiver was work­ ing in close collaboration with Dickson at that time, especially on the latter's History ofthe Theory ofNumbers, which Dickson saw as highly important for both teaching and research in mathe­ matics. Whether or not Vandiver ex­ plicitly heard Dickson speak critically about Moore's didactical method, he certainly shared a similarly critical atti­ tude toward it.

From Mounting Tension to Open Clash ( 1937-1952) The interwar period was one of thriv­ ing expansion for the departments of pure and applied mathematics at Austin. Some faculty members, above all Porter, did not think Texas was a truly first­ class university or that the atmosphere there was conducive to its becoming one but, arguably, the two mathemat­ ics departments came closer than any others at the time to meeting Porter's high standards. 26 This was, above all, due to the combined presence of Moore and Vandiver, both of whom were as­ sociate editors of leading mathematical publications. Both were elected to the National Academy of Sciences (in 1931 and 1934, respectively), and both had received the distinction of being named as AMS Colloquium Lecturers, as well as, respectively, President 0937-1938) and Vice-president 0933-1935) of the AMS. Toward the end of the 1930s, how­ ever, when political tensions were mounting in distant Europe, and Texas politics entered a tumultuous period that eventually swept UT into its midst, the personal clash between the two mathematical figures reached its height. The first concrete evidence dates to 1937 when Moore was nominated "Dis­ tinguished Professor" at UT. This re­ cently created status was not only an academic honor reserved for "nationally

distinguished" faculty members, it also came with a substantial increase in salary. Moore was among the first three recipients of that honor to be elected by the entire graduate faculty. In 1939, Vandiver sent his long list of publications and grants to his col­ league Calhoun, now acting president of UT, arguing that his reputation might be damaged were he not to be consid­ ered sufficiently distinguished. 27 His im­ pressive credentials notwithstanding, Vandiver would not be named a Dis­ tinguished Professor until 1947. Even then, the title he received was Distin­ guished Professor of Applied Mathe­ matics and Astronomy, in accordance with the name of the department to which he had recently been transferred. Vandiver sarcastically commented to a friend that "he was the only distin­ guished professor of applied mathe­ matics and astronomy in the world who knew not a damn thing about either one."28 And indeed, the rather ridicu­ lous transfer of Vandiver to applied mathematics in 1945 came as a conse­ quence of the by then unbearable rela­ tions between UT's two mathematical titans. The broader background leading up to these events was marked by mount­ ing general tension in Texas during the midst of the Great Depression. Texas governor W. Lee O'Daniel (1890-1969) was elected in 1938 on a Democrat ticket. After reneging on several cam­ paign promises, he became an outspo­ ken critic of the New Deal, especially after Franklin Delano Roosevelt's elec­ tion to a third term in 1 94 1 . O'Daniel was particularly disgusted by the price­ fixing policies that affected the Texas oil industry, but he also loathed Eleanor Roosevelt's support for legislation aimed at racial desegregation. Soon af­ ter his appointment in 1939, UT Presi­ dent Homer P. Rainey (1896-1985) became a major target of O'Daniel's at­ tacks against New Dealers. Rainey had openly challenged accusations of al­ leged un-American activities at UT, claims aired by Texas Congressman

Martin Dies, who chaired the recently founded House Un-American Activities Committee (HUAC). Dies warned of Stalinist and Marxist cells operating at the university under Rainey's nose. Cap­ italizing on this hysteria, O'Daniel nom­ inated his own conservative supporters to UT's Board of Regents. These new appointees were expected to carry out his policies for getting rid of "subver­ sives, Communists, and homosexuals," but also to enforce tighter budget con­ trols and to influence academic life in general. And indeed, the Board of Regents did its best to please the governor. Be­ tween 1941 and 1945 the Regents un­ dertook a series of aggressive steps to strengthen its control over academic matters. Rainey was ordered to fire pro­ fessors of economics who espoused New Deal views, and the board sought to ban the study of literature they deemed subversive and perverted, works such as John Dos Passos's USA trilogy. The Board also attempted to weaken tenure conditions and ordered the cancellation of research funds for the social sciences. The peak of the cri­ sis came on November 1 , 1944, when the Regents fired Rainey for his liberal policies and his lax attitude regarding racial issues. Students protested this ac­ tion and academic organizations ex­ pressed their dismay. The American Association of University Professors (AAUP) put the University of Texas on its blacklist, where it remained for the next nine years, and The Southern As­ sociation of Colleges and Secondary Schools also put UT on probation.29 The situation at UT initially made na­ tional headlines and attracted consider­ able attention, but of course the events in Austin were quickly overshadowed by the far more dramatic events taking place overseas. Press coverage of local affairs, like the one at UT, quickly faded, but the events that shook Austin in 1944-1945 were hardly forgotten. In or­ der to understand the respective reac­ tions of Moore and Vandiver to this crit­ ical situation, some information about

their political views is needed, bearing in mind the difficulty of judging their actions in the absence of documentary evidence. Moore's politics-as Parker succinctly put it-"were firm and outspoken, and still steeped in the Southern principles by which he was raised. He would have no truck with American left-wingers. " 3° This certainly applied to his active op­ position to New Deal policies, but it also reflected his general views on the ero­ sion of states' rights by those who ad­ vocated an expansion of the powers of the federal government. Clearly, Moore never equivocated when it came to is­ sues like the right to bear arms. He was also far from enthusiastic about the ar­ rival of large numbers of European emi­ gres who were offered positions in mathematics departments at American universities. Concerning Jews, Moore was outwardly respectful of their math­ ematical abilities, and he had close per­ sonal relations with Ettlinger (who was well-known also as a Roosevelt sup­ porter) . But Moore explicitly opposed an open-door policy for Jewish mathe­ maticians. Above all, on the issue of segregation, Moore's record is unam­ biguous: he was firmly reluctant to ac­ cept African-American students into his courses. Moore once told Walker E . Hunt, "you are welcome t o take my course but you start with a C and can only go down from there . "31 As else­ where in the South, the process of in­ tegration was exceedingly slow in Texas. Following a Supreme Court de­ cision, UT would open its doors to black students in 1 95 1 , but only to those accepted by the law school or the grad­ uate school. Seen in this light, Moore's traditional Southern outlook was in no way outside the mainstream. And while his flamboyant style and prominence no doubt made his positions more visible than those of other UT colleagues, his views were not exceptional for the time. Vandiver was less outspoken when it came to politics, so one can only spec­ ulate about his views. He worked for many years at a segregated university,

27See [Greenwood 1 983, 20], [Lewis 1 989, 235-236] . 28[Frank Vandiver, interview. Also quoted in [Parker 2005, 227]. 29See [Parker 2005, 1 94-205] for additional details on this story. 30[Parker 2005, 1 65]. 31 Scott W. Williams, Professor of Mathematics at Buffalo, maintains a website called: "R. L. Moore, racist mathematician unveiled," with information on this matter. See http://www.math .buffalo.edu/mad/speciai/RLMoore-racist-math.html.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

67

apparently without qualms. Although we have no direct testimonies of any initiatives he took to address the injus­ tice embodied by institutionalized seg­ regation, nevertheless, a letter that Van­ diver wrote in 1951 suggests that his political sympathies were essentially very different from Moore's: You speak of visiting Austin again next Christmas. The situation here is such that I wish that matters were re­ versed and that I was coming to Princeton next fall. After you left here the Texas legislature really went af­ ter our institution and the present in­ dications are that the appropriations for next year will be cut 40o/o below what it was for the last biennial. They also demanded that one of our eco­ nomic professors be investigated on suspicion of being a Socialist and so the place is in somewhat of a tur­ moil. Perhaps their next move will be to have a faculty member fired for being unkind to dumb animals.32 Vandiver's close friendship with Emma and Derrick Henry (Dick) Lehmer may perhaps also be taken as an important indicator of his political inclinations, or at least his tolerance of leftists. Dick Lehmer was among nine­ teen faculty members of the University of California who were dismissed in 1950 for refusing to sign a loyalty oath; he was reinstated only after the oath was declared unconstitutional. 33 Lehmer helped raise funds for the defense of colleagues prosecuted on charges of anti-American activities (most notably Lee Lorch in 1957).34 This was at the height of Vandiver's collaboration with the Lehmers on the use of electronic computers for increasingly high values of exponents for FLT.35 Political issues related to the Supreme Court's deliber­ ations frequently appear in letters from Emma Lehmer to Vandiver (though I could not find letters in which Vandiver explicitly addressed those issues and stated his own opinions).

Clearly there was no love lost be­ tween Vandiver and Moore as the UT crisis reached its climax, and these po­ larizing events surely ended whatever chance they might have had for sal­ vaging a civil relationship. Vandiver sided with most on the UT faculty, who felt that the Board of Regents had seri­ ously damaged academic freedom at the university. Moore, on the other hand, was among the minority who supported the Regents' policies and who actively opposed their critics. In a letter to the secretary of AAUP he de­ clared that its recent decision to cen­ sure UT only served to discredit the AAUP. "I do not know-he adduced­ a single instance in the last twenty years in which any board of regents of this University has violated what I consider to be sound principles, either of acad­ emic freedom or of tenure. "36 In a rare appearance at the General Faculty Meeting on May 12, 1945, Moore presented in great detail his views on the issue of tenure, a main source of contention between the UT faculty and the Board of Regents. At stake was a new scheme suggested by the faculty whereby any instructor would, after four years of service, either be offered a commitment for promotion or else would receive one year's notice to find alternative employment. Moore stated, axiomatically, two principles that in his view defined a first-class university: "(1) a very substantial amount of really fun­ damental research of a high order is car­ ried on by members of its faculty, and (2) there are some members of the fac­ ulty who are intensely on the alert to discover and develop outstanding re­ search ability on the part of their stu­ dents and who are both capable of rec­ ognizing such ability in the early stages of its manifestation and of developing it when it is discovered." He followed this with a detailed argument leading to the conclusion that UT "will never be of the first class . . . if it is dominated

by the ideals of those who are more concerned with unifonnity of standards and 'fair' treatment of the mediocre than they are with the establishment and maintenance of high standards and the discovery and fostering of outstanding ability. "37 In this highly self-serving perfor­ mance, Moore obviously preferred to ig­ nore the potential abuses of a weaker tenure system, which could be ex­ ploited as a political weapon by the Board of Regents. And while it seems likely that Vandiver would have agreed with Moore on the need to avoid tenure schemes that might lead to low acade­ mic standards, he clearly opposed the intrusions of politicians in the univer­ sity's academic affairs. By this time a deep chasm divided the Austin faculty into two clearly defined camps. Van­ diver and Moore found themselves in an additional and now very significant confrontation. In both mathematics departments, tensions only heightened as Moore be­ came more powerful than ever. Faculty members had been long openly com­ plaining that financial support was eas­ ily available to students of Moore, Et­ tlinger, and Wall,38 but not those working with Vandiver or other pro­ fessors in the department (Dodd, Lubben, Betchelder, and Beckenbach). The fistfight between Moore an Beck­ enbach took place at this time. Despite the sudden availability of funds for graduate students at the end of WWII, this situation did not change. In 1945 Vandiver submitted his res­ ignation. He gave no explicit reasons and many factors may have played a role, but surely the unbearable con­ frontation with Moore and the highly politicized atmosphere at UT were high among them. At that time, Vandiver was also deeply involved in his own re­ search and was overworked almost to the point of exhaustion. But the uni­ versity authorities, under increased pub-

32Vandiver to Ankeny: March 27, 1 951 (HSV). 33An interesting website containing information on this topic is http://sunsite. berkeley.edu/uchistory/archives_exhibits/loyaltyoath/symposium/timeline/short.html 34Several letters related to the Lehmers' support are found in the Emma & Dick Lehmer Archive, UC Berkeley. 35[Corry 2007a]. 36Quoted in [Parker 2005, 205]. 37Quoted in [Parker 2005, 203]. Emphasis in the original. 38Hubert Stanley Wall (1 902-1 97 1 ) joined the faculty at Austin in 1 946, at a late stage in his career, and became a devoted follower of Moore's method in teaching. See [Wall 1 963}. See also, "In Memoriam. Hubert Stanley Wall," Memorial Resolution, Documents and Minutes of the General Faculty, The University of Texas at Austin, 1 971 , 1 0433-1 0438 http://www.utexas.edu/faculty/councii/200Q-2001 /memorials/SCANNED/wall. pdf.

68

THE MATHEMATICAL INTELLIGENCER

lie scrutiny, worried that losing a math­ ematical star might expose them to fur­ ther criticism, external and internal. They suggested instead that Vandiver be transferred to the department of ap­ plied mathematics. Initially Vandiver saw this as a possible solution, but then for some reason the administration de­ cided to leave him in pure mathemat­ ics after all and appointed a committee to work with Vandiver to find a com­ promise. Typically, Vandiver conducted a good part of his negotiations with the university in writing, and from a safe distance; this time he did so in the calm surroundings of Princeton, where he was spending one of his frequent leaves of absence. The correspondence, in the summer of 1945, between Vandiver and President Theophilus S. Painter (1889-1969), a well-known geneticist and Rainey's successor, and Vice-presi­ dent]ames Clay Dolley suggests the dif­ ficulties the authorities had in dealing with Vandiver; he raised many different topics simultaneously, discussed with Dolley recent baseball games he had watched, and continually changed his positions vis-a-vis the administration's proposals. Finally, on December 1 2 , 1945,39 h e asked t o b e transferred to Applied Mathematics, and there he went. The Pure Mathematics Depart­ ment, surely under the initiative of Moore, insisted that Vandiver could not take his courses with him. Vandiver's old course "Theory of Numbers" thus became, in his new department, "The­ ory of Integers. " In 1952 the two departments were united, but this administrative act did not immediately translate into full col­ laboration. Indeed, according to Robert Greenwood ( 1 9 1 1-1993), who spent his 55-year mathematical career at Austin, a "spirit of antagonism developed in the minds of the young graduate students in the old Pure Mathematics Depart­ ment, and R. L. Moore was unrelenting in keeping pressure on former Applied Mathematics members. " Indeed, Moore

told UT administrators "that there wasn't a single person in the Applied Mathematics Department who was a real mathematician."40 He obviously in­ cluded Vandiver in his assessment.

A Mini-Cold War at Austin (1952-1969) In 1952 Moore turned seventy, the age at which, by university rules, a profes­ sor became a "modified service" mem­ ber of the faculty. He continued to work full-time for half the pay, and his pres­ ence was felt in all aspects of depart­ mental life as it always had been. In many ways, his influence became more visible than ever before. Thus, for in­ stance, between 1952 and 1969 Moore supervised twenty-eight doctoral stu­ dents, and six of his former students be­ came presidents of the Mathematical As­ sociation of America (MAA) after 1950. From a more general perspective, vari­ ous versions of the Moore Method of teaching became increasingly common in American universities. even though Moore himself never made any specific effort to foster such a development. 4 1 Vandiver, too, became a "modified service'' member of the faculty at this time. His earlier transfer to the Depart­ ment of Applied Mathematics and his new formal status only strengthened his natural tendency to estrange himself from departmental life. Moreover, the contrasts between the two mathemati­ cians became even more pronounced in their last years at UT, as Vandiver continued to be rather active in re­ search, actively collaborating with other researchers in his fields of expertise, whereas Moore had long before with­ drawn. The conflict between the two entered a phase of "cold war" that even­ tually became a source of embarrass­ ment for everyone at Austin. Yet, strangely, at the twilight of his career Vandiver began to discuss pub­ licly his ideas about school-level math­ ematical education and the proper train­ ing of teachers . This turn may have had some connections with contemporane-

ous debates on reforms in US mathe­ matics education, and the nomination of Ed Begle 0914-1978) as director of the School Mathematics Study Group (SMSG), from which the New Math later arose.42 Moore's ideas can be seen in­ directly in the background of these de­ bates, as Begle was a student of Ray­ mond L. Wilder. Likewise, Edwin Moise (1919-1998), another well-known Moore student, wrote influential high­ school textbooks. Interestingly, Moise emphasized that Moore himself never expressed any opinions on SMSG or about the New Math and made it clear that he did not want to be regarded as a pedagogue 43 Vandiver's ideas on teaching at this time appeared in a two-part research ar­ ticle published in 1952-53 in the Math­ ematics Magazine. "A Development of Associative Algebra and an Algebraic Theory of Numbers." Perhaps it is not a mere coincidence that this is one of the few places where Vandiver spent some effort in a technical discussion about a new system of postulates. This was a system for defining associative al­ gebras "in a bit unusual way," and he remarked that "many secondary school students are alienated from arithmetic and algebra because the only way they learn these topics . . . is by following a set of rules which are never stated ex­ plicitly by the teacher. " Vandiver said he learned this from his own experi­ ence as a high-school student. The mathematically gifted students, he thought, deserved a clear presentation of "a few explicit postulates in arith­ metic and algebra. " The ideas discussed i n the articles are of limited mathematical interest, but they are clearly related to Moore's method. Vandiver stated that he devel­ oped these ideas in his courses and seminars over twenty years, and espe­ cially in a recent seminar in which three of his five doctoral students partici­ pated.44 He also "discussed these top­ ics with sophomores with apparently some success," and attributed this to the

39Dolley to Vandiver: Aug 1 3, 1 945 (HSV). 4°[Greenwood 1 983, 47]. 41 [Parker 2005, 232-234]. 42[Usiskin 1 999]; [Raimi 2005].

43[Anderson & Fitzpatrick 2000]. The possible influence of Moore's ideas on New Math is a topic that deserves some further thought, but it cannot be pursued here for lack of space. 44 [Vandiver 1 953, 4].

© 2007 Spnnger Science+ Business Media, Inc., Volume 29. Number 4, 2007

69

Figures 2 and 3. Robert Lee Moore in his youth (left), and in October 1930 (right). Photos are published by permission of the

Center of American History, the University of Texas, Austin; they are part of the R. L. Moore Legacy Collection in the Archive of American Mathematics. fact that he "did not do anything except try to set up some rules to justify the operations they were already used to in algebra. " While in his advanced courses, he "suggested to the students that they forget everything they know about mathematics, since we would try to start from scratch"; he doubted this would be a "good suggestion to make to a sophomore."4 5 Somewhat later, in an unpublished manuscript, Vandiver also addressed the question of the proper training of teachers of mathematics. Besides other possible motivations, one gets the im­ pression that Vandiver at least wanted to stress what he saw as his own last­ ing contribution to the teaching of mathematics. A research scientist, Van­ diver wrote, is actually a good teacher by virtue of his very research activity, "and in some instances he fulfills the

45[Vandiver 1 953, 1 6]. 46[Vandiver, unpublished 1 ] . 47[Vandiver, unpublished 2].

70

THE MATHEMATICAL INTELLIGENCER

qualifications of a teacher even better than a professor who delivers lectures in a university. "46 Euler, for example, had no students but, by virtue of his enormous original work, "was the greatest teacher of mathematics who has lived in the last 200 years . " Gauss too would rank among the greatest teachers of mathematics according to Vandiver's definition, an individual A, who "communicates in any way what­ soever to an individual B some idea which is new to B and which B retains in his mind . " He concluded, At present I think the practice may be pretty widespread at universities to give their teachers time off and funds to travel to various places in order to consult the various research men in their own line of study; how­ ever, some scientist may be so situ­ ated that he cannot leave his uni-

versity for a long period. In this case I think it would be excellent if the university would pay for as many long distance telephone calls as he deemed necessary in order to keep in touch with other scientists with the idea of going forward in his work. (p. 9) Vandiver was also concerned with problems faced by mathematics teach­ ers in elementary schools.47 Elementary school teachers did not receive proper training. For example, an examination of many textbooks showed that they were not taught the essence and mean­ ing of-of all things-the decimal sys­ tem. Vandiver claimed that if his advice was followed, in five years time foun­ dations would be able to save huge amounts of money currently devoted to coaching teachers. Vandiver sent the editors of the American Mathematical

f',1onthzl' a manuscript of 49 pages: a

the communications gap.

Dickson

significant for Vandiver: little wonder

shorter version of five pages was not

once said that "every mathematician

that McShane's article struck a sympa­ thetic chord with him.

published either. though in several let­

owed a debt to mathematics that he

ters he mentioned that it \Yould soon

should repay by one hard job of

Inspired hy these ideas. Vandiver de­

1 959 volume of the TE'xas

scholarly \\'fiting . " His History ol thE'

cided to set forth his own views. Aside

W'hatever dre\v Vandiver into a dis­

son's own way of paying that debt.

he was convinced of the '·desirability of

cussion on pedagogical topics. \Ye find

McShane was a\Yare that few would

publishing complete bibliographies of

here a rather ironical situation. On the

consider an

undertaking of such

the literature on various branches of

one

so

magnitude: nonetheless, he insisted

much of his professional energies to his

that "each of us owes the debt, and

mathematics, with reviews when possi­ .. ble . I'J The editors of the Bulletin of the

appear in the

Quarterlv.

hand

Moore,

who

devoted

Theory (!l Su mhers had been Dick­

from the publication of research papers,

AMS may

have

had

d ifferent

ideas.

university teaching. distanced himself

should not repudiate it if he is math­

from the debate about mathematics in

ematicallv solvent . " (p. 3 1 3) Expos­

though more likely they rejected Van­

secondary and primary schools. On the

itory articles were needed for the

diver's article on this topic as inappro­

other hand, Vandiver, for whom teach­

continued renewal of the teacher's

priate for their journal. The editors of

ing was essentially a burden to his uni­

activity. On the research side. he he­

the .lfonthzv were also initially unen­

versity activities, contributed his own

moaned the low quality of writing

thusiastic, although they finally acqui­

ideas and tried to influence mathemat­

and the failure to make research pa­

esced and the paper was published in

ical

States

pers accessible beyond the limited

1960.

through improved training of teachers.

circle of specialists \Vith whom au­

education

in

the

United

thors were already in direct contact.

Vandiver based

his

A Role for Mathematical Scholarship

pository articles i n his own field of

the time of publication.

As

expertise.

own

already

suggested,

As the author of accomplished ex-

mathematical

Vandiver

read

McShane's

argument

on

Dickson's hook. which he considered as important in

1 960

as it had been at Quoting his

1 924 review of Volumes I

and I I , : "0

scholarship was of major importance for

speech with pleasure. I n fact. McShane

Vandiver but played a lesser role i n

made a flattering a llusion to Vandiver's

mathematics that a mathematician

�loore ·s

work:

becomes a specialist i n a particular

overall

conceptions.

This

It often happens in the history of

emerges in certain initiatives Vandiver

I am not recommending the writing

topic, and. a fter years of experience

undertook late in his career. when he

of expository papers as a sort of pas­

with it. he publishes a treatise giv­

attempted to inf1uence additional as­

time for gentlemen ( young. old, or

ing a harmonious and comprehen­

pects of mathematical life in the United

middle-aged) who have determined

sive development of the subject, the

1957 Vandiver read with great

by careful self-examination that they

material being arranged and pre­

interest the Retiring Presidential Address

haven't a research paper left in their

sented according to his own partic­

States. In delivered

by

0909-1989)

Edward

].

McShane

at the Annual Meeting of

the YIAA in December

1 956.

entitled .

,.1"

systems. A man of thirty may have

ular point of view. This treatise may

attained position

recognition

become a classic, and its readers are

and broad knowledge: a man past

likely to get in the habit of ignoring,

and

seventy may he active i n research.

to a considerable extent. the litera­

McShane worried that modern research

as the current volume of the Pro­

ture that preceded its publication. In

was running out of control and that

ceedings C
this way, the points of view of the

mathematics had grown wild and un­ structured during the last decades: this

On the other hand. McShane's case for

as these treatises rarely. if ever. re­

could lead to a breakdown into sub-dis­

the importance of expository writing in

produce all the older material on a

simply " Maintaining Communication .

older \'.Titers are often lost sight of,

ciplines in which only specialists could

teaching ran contrary to the essence of

particular topic . It would seem that

u nderstand each other. McShane \Yas

Moore's method. as one of Moore's

there is too great a preponderance

especially alarmed about the lack of

maxims was that students not read

of books of this sort i n the literature

general communication among modern

other people's work ( even though, in

and too few histories of reports of

researchers. He pointed to three main

thesis vvork. Moore definitely expected

the type of Dickson's work.

spheres of mathematical activity which,

novelty vis<'i-vis the existing literature.

Many works cited by Dickson con­

in his view. should complement one an­

which the students were expected to

tained results that hac! been published

other: teaching, research, and scholar­

know in detail ) . McShane called for in­

earlier by someone else. As Vandiver

ship. Yet this third sphere of activity " is

creased breadth of mathematical schol­

wrote to one of his correspondents, he

all too often left unmentione d . "

arship in teaching from the very early

himself had ·'been haranguing mathe­

McShane lamented. i n particular. the

stages of a student's training. The fact

maticians to do something about the

dearth of good expository articles,

that

situation , " with no visible resul t . " 1

\vhich \Yere badly needed to bridge

work in this context was also certainly

McShane

mentioned

Dickson's

Vandiver w a s aware of t h e immense

48[McShane 1 95 7].

49Vandiver to R . D. James (Editor of the Monthly): August 6, 1 958. 50[Vandiver 1 924].

5 1Vandiver to Leo F. Epstein: May 1 1 , 1 960 (HSV).

© 2007 Spnnger Sctence+Business Medta, Inc .. Volume 29, Number 4. 2007

71

effort that would be required to con­ tinue Dickson's work. The number of relevant references between 1920 and 1 956 were about 8,500. If five mature, top-rate number-theorists would collab­ orate, with five years to complete the job, each would need to write almost one review per day. Dickson had re­ viewed 8,382 works-leaving out major issues like algebraic number theory, Bernoulli numbers, and the law of qua­ dratic reciprocity. And these papers, Vandiver stressed, were far less difficult than more recent work, whose volume was rapidly growing. It would be diffi­ cult to find mathematicians willing to do the job. And this is just in number theory. In a field like differential equa­ tions, the most one could hope for would be the "preparation of a nearly complete list of references," while di­ viding the literature into sub-topics "so that a research man does not have to look up too many listed papers to de­ cide that the results he has arrived at are new. " Vandiver knew that "the work of a bibliophile on first glance is not attrac­ tive," and indeed many mathematicians reacted with "deep disgust at the idea . " Yet following McShane's lead, Vandiver argued that such bibliographies were crucial for the advancement of mathe­ matical research and teaching. Mathe­ maticians should undertake the task not only as a duty to the discipline but also for its personal benefits. He himself had found, preparing such lists for a num­ ber of topics in number theory (part of which are kept in his archive), "that his own knowledge of each topic increased greatly thereupon, and the publication of a number of his papers was due to this. " Vandiver received several letters in response to his article, most of them positive. He decided to transform his basic message into a plan for action along two fronts. First, he wanted to re­ form the existing reviewing system (which is essentially the one still in use today) to facilitate later compilations of complete bibliographies. Vandiver"s

main source of dissatisfaction with the current system was that "persons re­ porting on papers failed to remember that they are supposed to be reporters and not critics." 52 To correct this situa­ tion, authors should begin their articles with an abstract that could later be pub­ lished in the Mathematical Reviews or in a future bibliography of the subject. The referee for the journal would also approve the abstract for the Reviews. That such abstracts might merely reflect the opinions of the author was a minor problem, Vandiver felt, compared with the advantages gained in speed and ef­ ficiency. The situation was different for mathematical books; here criticism was welcome and necessary, "since there might be a question as to whether it would be advisable to have money spent to add such books to mathemat­ ics libraries. " 53 All of these ideas were inspired, Vandiver said, by what he had learned from his collaboration with Dickson, especially on the History. Vandiver wrote to various mathe­ maticians, especially editors of known journals, who he believed would sup­ port this project, among them Leonard Carlitz, ]. Barkley Rosser, Max Shiffer, Peter D. Lax, Joseph Walsh, Marshall Stone, Richard Bellman, and Gordon Whyburn. Some reacted with useful comments. Walsh suggested that au­ thors should be instructed to choose meaningful names for their papers (rather than, say, ''Proof of a Lemma due to Wye Zed") . Stone wrote that, al­ though he very much favored some of the suggestions, he would not like to have his name included as an uncon­ ditional backer. "> 4 One correspondent objected that reviewers sometimes make valuable suggestions for exten­ sions of results and this important in­ put would be lost under Vandiver's suggestion. Bellman fully supported Vandiver's initiative as he had a very low opinion of the current state of the refereeing system: I think that the only intelligent and efficient technique is one based upon a board of associate editors

52Vandiver to Bellman: May 1 2 . 1 960 (HSV). 53Vandiver to Bellman: May 1 2 , 1 960 (HSV). Emphasis in the original. 54Stone to Vandiver: January 2 1 , 1 962 (HSV). ssvandiver to Montgomery: January 1 3, 1 962 (HSV). 56Vandiver to Grad: February 27. 1 962 (HSV). 570n this matter, see [Fenster 1 999].

72

THE MATHEMATICAL INTELLIGENCER

empowered to present any paper which they think fit. The system of anonymous refereeing which we use now in most journals has so many defects and so many abuses that I think any unprejudiced observer would say that it had failed almost completely. Oddly enough, it serves the purpose of passing the mediocre paper along with no difficulty, and almost completely hindering the novel paper with original and un­ conventional results and ideas. Vandiver summarized the reactions and his own responses to them in a de­ tailed, formal letter to the President of the AMS, Deane Montgomery (19091992), and expected Montgomery to raise this matter in a forthcoming meet­ ing of the AMS Councii.55 It seems that his initiative did not reach any further, and his ideas on reviewing were never adopted in the Reviews, although Zen­ tralblatt often uses "Autor-referats". Vandiver was involved in a second undertaking that shows how he tried to turn his views on mathematical schol­ arship into a concrete plan of action. In 1961 William ]. LeVeque submitted a proposal to the National Science Foun­ dation calling for the publication of "A General Survey of the Theory of Num­ bers Leading to the Compilation of a Topical History and Critical Review of the Theory of Numbers, 1 9 1 5-1960." Not surprisingly, Vandiver was enthusi­ astic about this project and wrote a highly positive report.56 LeVeque mentioned three main top­ ics not originally covered by Dickson that should be included: Analytic The­ ory of Prime Numbers, Diophantine Ap­ proximations, and Algebraic Numbers. Vandiver suggested that a chapter on Bernoulli and Allied Numbers should also be included, as well as the very important topic of Higher Reciprocity that Dickson had left for a fourth vol­ ume but never published.57 He insisted that only abstracts of articles should be included, with somewhat longer ones when the original paper had appeared in an out-of-the-way journal. If Dickson

had included criticisms in his book "such material would now be worth­ less." Finally, he referred to the inten­ tion to rely on the Mathematical Re­

views:

Since experienced reviewers are hard to obtain in order to write re­ views for the Math. Reviews, I regard most of the reviews appearing in that journal as quite inadequate. And

from what I have seen of the other review journals, I do not think they are much, if any, better. 58 The NSF decided not to fund the pro­ ject, and it was postponed and even­ tually abandoned.59 Grad explained to Vandiver that although most reviews were favorable ''the bibliography was considered to be of second importance as compared with research of the usual type. "60 Vandiver replied, "if the NSF continues to support 'research of the usual type' to the exclusion of support of bibliography projects, then as time goes on it will be supporting the pub­ lication of the results of research which already are described in the litera­ 1 ture. ''6

Parting Company in Silence Both Moore and Vandiver remained ac­ tive until a very advanced age. For decades, Austin ' s two leading mathe­ maticians hardly exchanged a word so­ cially, if at all , and their careers ended quite differently. Not everyone in Austin welcomed Moore's "volunteering spirit" when he continued to work at the de­ partment after 1952 under a "modified service" contract. On becoming Dean of Arts and Sciences in 1967, John R. Sil­ ber "made no attempt to conceal his view that Moore's very presence and reputation hindered the recruitment of new faculty."62 Silber, formerly chair of the Philosophy Department, thought mathematics should be taught by ex­ perienced teachers in fewer sections with more students in them. This, of course, ran counter to Moore's peda­ gogical philosophy.

Silber brought in visiting scholars to evaluate the performance of various de­ partments and attempted to introduce mandatory retirement at the age of sev­ enty-five. This only raised tensions be­ tween the administration and Moore's still large and influential group of sup­ porters. A lengthy and rather nasty process ensued that finally led to Moore ·s forced retirement in September 1 969 at age eighty-seven. Almost seventy-one years after he arrived as a freshman and less than three years before his death, R. L. Moore walked off the University of Texas campus for the last time, refusing to attend any events to "honor" him. When, in 1973, the ne\v mathematics building was named the Robert Lee Moore Hall, he was noticeably absent at the dedication ceremony 63 Moore's long-time rival, Henry S. Van­ diver, voluntarily took emeritus status in 1966. Despite poor health in his later years, he continued to do research and even received a research grant at the age of seventy-six. Yet the only public hon­ ors conferred on him at the end of his career were quiet affairs that largely es­ caped notice. In 1961 he was invited to deliver the keynote address at the Texas Section of the Mathematical Association of America.61 Five years later, a few friends and collaborators put together in his honor a special issue of the Journal

time-was finally proved in 1994, it un­ leashed a t1urry of publicity inside and outside the mathematical community, hut Vandiver's noteworthy achievements were completely overlooked. Some fifty years before their passing, Moore and Vandiver had begun their mathematical careers at the University of Texas together. Each went on to be­ come distinguished in his own particu­ lar way, but their paths parted quickly and never again crossed. Vandiver died on January 4, 1 973, aged 9 1 ; Moore was dose to 92 when he passed away on October 4, 1974 . But both are buried in Austin's Memorial Park Cemetery.

of Mathematical Analysis and Applica­ tions, a publication otherwise devoted to

material and more. and the writing of an

topics unrelated to his own research.65 No buildings were named after Vandiver, nor did he leave a mark as a teacher at the University of Texas. None of his five doctoral students went on to become a leader within the American mathemati­ cal community. Of his many interesting contributions to mathematical research, only the conjecture of 1 934 bears his name, and this remains barely known, except to specialists. But most ironic of all, when Fermat's Last Theorem-the

without the basic work done at the

problem to which he devoted so much of his energy and on which he became the world's leading expert during his life-

ACKNOWLEDGMENTS

Albert C. Lewis and David Rowe read earlier versions of this article. I thank them for their critical remarks, which led to

significant

improvement.

Editorial

comments by Marjorie Senechal were also highly useful in preparing the final version. Primary sources used here were taken from the Archives of American Mathematics, Center for American His­ tory, the University of Texas at Austin, and cited with permission. Also some of the important secondary sources cited re­ lied on documents taken from the same archive. Albert. C. Lewis played an im­ portant role in putting together all of this article like this one would be impossible archives. REFERENCES

Anderson,

Richard D. and Ben Fitzpatrick

(2000), "An interview with Edwin Moise," Topological Commentary 5 (http ://at.yorku.

ca/t/o/p/c/88.htm). Corry, Leo (2004), Modern Algebra and the Rise of Mathematical Structures, Boston and

Basel, Birkhauser. --

(2007), " Fermat comes to America: Harry

Schultz Vandiver and FLT (1 9 1 4-1 963) , " Mathematical lntelligencer 29 no. 3 30-40.

58Vandiver had considered not including this latter comment so as not to jeopardize the prospects of the project's being approved, but he obviously changed his mind. See Vandiver to Grad: February 7, 1 962 (HSV). 59LeVeque to Vandiver: March 9, 1 962 (HSV). 60Grad to Vandiver: March 5, 1 962 (HSV). 51 Vandiver

to Grad: March 1 3, 1 962 (HSV).

62[Parker 2005 , 322]. 63[Parker 2005, 332].

64[Greenwood, et a/. 1 973, 1 0939]. See [Vandiver 1 961]. 65[Corry 2007].

© 2007 Spnnger Science+Business Media, Inc., Volume 29, Number 4, 2007

73

-- (2007a), "FLT Meets SWAC: Vandiver, the Lehmers, Computers and Number Theory," IEEE Annals of History of Computing (Forthcoming). Fenster, Della D. (1 999), "Why Dickson Left Quadratic Reciprocity out of His History of the Theory of Numbers," Am. Math. Monthly 1 06 (7), 61 8-629. Greenwood, Robert E. (1 983), "History of the Various Departments of Math­ ematics at the University of Texas at Austin: 1 883-1 983. " unpublished manuscript in the Greenwood archive, Archives of American Mathe­ matics, Center for American History, The University of Texas at Austin. -- (1 988), "The Benedict and Porter Years, 1 903-1 937 , " unpub­

Periodicals postage paid at New York, NY and additional mailing offices.

POSTMASTER:

Send address changes to The Mathematical

Springer Journals Dept.,

COPYRIGHT:

233 Spring St.,

New York, NY

10013.

Submission of a manuscript implies: that the work described

has not been published before (except in the forn1 of an abstract or as part of a published lecture, review or thesis); that it is not under consideration for publication elsewhere; that its publication has been approved by all coau­ thors, if any, as well astacitly or explicitly6by the responsiblE' authorities at the institution where the work was carried out. Transfer of copyright to Springer becomes effective if and when the article is accepted for publica­ tion. The copyright covers the exclusive right (for U.S. government employ­ ees: to the extent transferable) to reproduce and distribute the article, in­

lished oral interview (March 9, 1 988) (Oral History Project , The Legacy

cluding

of R . L. Moore, Archives of American Mathematics, Center for Amer­

electronic form (offline, online) or other reproductions of similar nature.

ican History, The University of Texas at Austin). Greenwood, Robert E., et a/. (1 973), "In Memoriam. Harry Schultz Van­ diver, 1 882-1 973," Memorial Resolution, Documents and Minutes of the General Faculty, The University of Texas at Austin, 1 974, 1 0926-1 0940. Lewis, Albert C. (1 989), "The Building of the University of Texas Math­ ematics Faculty, 1 883-1 938 , " in Peter Duren (ed.) A Century of Math­ ematics in America - Part Ill, Providence, R l , AMS, pp. 205-239.

Moore, Robert Lee (1 932), Foundations of Point Set Theory, Provi­ dence, Rl, AMS (revised edition, 1 962). McShane, Edward J. ( 1 957), "Maintaining Communication, " Am. Math. Monthly 64 (5), 309-31 7 .

Parker, John (2005), R. L. Moore. Mathematician & Teacher, Wash­ ington, DC, Mathematical Association of America. Raimi, Ralph (2005) , "Annotated Chronology of the New Math , " unpub­ lished manuscript available at the site Work in Progress, Concerning the History of the so-called New Math, of the Period 1 952-1975 (http:// www.rnath .rochester.edu/people/faculty/rarrn/the_new_math.htrnl).

Reid, Constance (1 993), The Search for E. T. Bell, Also Known as John Taine, Washington, DC, Mathematical Association of America.

Traylor, D. R . , Creative Teaching: The Heritage of R. L. Moore, Univer­ sity of Houston, 1 972. Usiskin, Zalman (1 999), "The Stages of Change" (1 999 LSC PI Meet­ ing Keynote Address), downloaded from http://lsc-net.terc.edu/do.cfm/ conference_material/6857/show/use_set -oth_pres. Vandiver, Harry Schultz (1 924), " Review of Volumes II and I l l of Dick­ son ' s History of the Theory of Numbers ," Bull. AMS 39, 65-70. -- (1 960), "On the desirability of publishing classified bibliographies of the mathematics literature," Am. Math. Monthly 67(1 ), 47-50. -- (1 961 ) "On developments in an arithmetic theory of the Bernoulli and allied numbers , " Scripta Mathematica 25, 273-303. -- (1 963), "Some aspects of the Fermat problem (fourth paper) , " Proc. NAS 49, 601 -608. --

(unpublished 1 ) , "On the relation between academic teaching and

academic research and the most effective working conditions for eac h , " unpublished manuscript (HSV), 1 2 pp. --

(unpublished 2), "On the training in universities which is desirable

reprints,

translations,

photographic

reproductions,

All articles published in this journal are protected by copyright, which cov­ reprints), as well as

all translation rights.

No material published in this jour­

nal may be reproduced photographically or stored on microfilm, in electronic data bases, video disks, etc., without first obtaining written pern1ission from Springer. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if not specifically identified, does not imply that thesE' names are not protected by the relevant laws and regulations.

An author may self-archive

an author-created version of his/her article on

his/her own website and his/her institution's repository, including his/her fi­ nal version; however he/she may not use the publisher's PDF version which is posted on www.springerlink.com. Furthermore, the author may only post his/her version provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's web­ site. The link must be accompanied by the following text: The original pub­ lication is available at www.springerlink.com. Please use the appropriate URL and/or DO! for the article (go to Linking Options in the article, then to OpenURL and use the link with the

DOl-).

Articles disseminated via

www.springerlink.com are indexed, abstracted and referenced by many ab­ stracting and infornmtion services, bibliographic networks, subscription agencies, library networks, and consortia. While the advice and information in this journal is believed to be tme and accurate at the date of its publication, neitlle r the authors, the editors, nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Springer publishes advertise­ ments in this journal in reliance upon the responsibility of the advertiser to comply with all legal requirements relating to tile marketing and sale of prod­ ucts or services advertised. Springer and the editors are not responsible for claims made in the advertisements published in the journal. The appearance of advertisements in Springer publications does not constitute endorsement, implied or intended, of the product advertised or the claims made for it by the advertiser. Specia,l regulations for photocopies in the USA. Photocopies may be made for personal or in-house use beyond the limitations stipulated under Section

107

or

108

of U.S. Copyright Law, provided a fee is paid. All fees should be

Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA (WEB: http://www.copyright.com; FAX: 978-646-8600; TEL: 978-750-8400), stating the ISSN 0343-6993, the volume, and the first and last

paid to the Copyright Clearance

page numbers of each article copied. The copyright owner's consent does not include copying for general distribution, promotion, new works or re­ sale. In these cases, specific written permission must first be obtained from the publisher. The Cancula Institute for Scientific and Technical Inj(mnation (CISTI) provides a comprehensive, world-wide document delivery service for all Springer journals. For more information, or to place an order for a copyright­ cleared Springer document, please contact Client Assistant, Document De­ livery, CISTI, Ottawa KIA OS2, Canada. TEL:

613-993-9251; FAX: 613-952-8243;

E-r>WL: [email protected]. You may also contact INFOTRIEVE (WEB: http://www.infotrieve.com/; E-MAIL: orders @infotrieve.com; fax:

published manuscript (HSV), 5 pp.

TEL:

Wall, Hubert (1 963), Creative Mathematics , Austin, UT Press.

Wilder, R. L. (1 959), "Axiomatics and the development of creative tal­ ent , " in L. Henkin, P. Suppes, and A. Tarski (eds), The Axiomatic Method with Special Reference to Geometry and Physics, Amster­

dam, North-Holland, pp. 474-488. Zitarelli, David E. and Marion D. Cohen (2004), "The Origin and Early I mpact of the Moore Method, " Am. Math. Monthly 1 1 1 (6) , 465-487 .

THE MATHEMATICAL INTELLIGENCER

microform,

ers the exclusive rights to reproduce and distribute the article (e.g., as

for potential teachers of mathematics in elementary schools," un­

74

Intel/igencer,

800-422-4633),

310-208-5971;

UnCover: http://www.ingenta.com; UMI: http://www.

umi.com/; lSI: http://www.isinet.com

RESPONSIBLE FOR ADVERTISEMENTS: M.J. Mn,ica Associates, Inc., 2 West Taunton Avenue; Berlin, NJ 08009, Tel. (856) 768-9360, Fax: (856) 7530064 Printed in the United States on acid-free paper.

© 2007

Springer Science+ Business Media, Inc.

I ;J§Ih§i,'tJ

O s rn o Pekon en ,

Editor

f

R. L. M oore: Mathematician & Teacher by john Parker WASHINGTON, DC, MATHEMATICAL ASSOCIATION

OF AMERICA (MAA), 2005 HARDCOVER,

XIV + 387 PP. US $52.50 (MAA M EMBERS

$41.95) ISBN 978-0-88385-550-8

REVIEWED BY PETER ROSS

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.

Col umn Editor: Osmo Pekonen, Agora Centre, 40014 Un iversity of Jyvaskyla , Finland e-mai l : [email protected]

has been written about Robert Lee Moore, the Texas topologist who is best known for the "Moore Method" of teaching. An early biography, D . R. Traylor's Creative Teaching: Heritage qf R. L. Moore [1], was published in 1972, two years be­ fore Moore's death. Its author was not unbiased; Traylor is an academic grand­ son of Moore-R H Bing was his the­ sis advisor-who attempted to get Moore hired at the University of Hous­ ton when Moore was forced into re­ tirement from the University of Texas at age 85. This hagiography is also a bit outdated; over half of the book consists of a list of publications by Moore and his academic descendants as of the early 1970s. A more current resource for Mooreabilia is The Legacy of R. L. Moore Project at http://www.discov­ ery.utexas.edu/rlm/. Its mission, as stated there, is "To disseminate the In­ quiry-Based Learning methodology of R. L. Moore and to further its imple­ mentation throughout the educational systems in the United States and abroad. " For readers who are uncon­ vinced of Moore's importance to math­ ematics, one can click there in the pho­ tos section to see all six MAA Presidents who were Moore students. How many can you name? MAA Presidents R. D . Anderson, R H Bing, Edwin Moise, R. L. Wilder, and Gail Young were doc­ toral students of Moore's, and Lida Bar­ rett was a master's student who received her Ph. D. under John R. Kline, Moore's

very first doctoral student. And of these six, Bing and Wilder were also Ameri­ can Mathematical Society (AMS) Presi­ dents, as was Moore student G. T. Why­ burn; Anderson and Moise were also AMS Vice-presidents. John Parker, the author of the new Moore biography, is a British journalist and eclectic writer who has authored 1 4 military o r investigative books, among them, Commandos, Tbe Gurkhas, and Inside the Foreign Legion. and 16 criti­ cal biographies including Prince Philip, Tbe Trial qfRock Hudson, and King of

Fools: Tbe Duke of Windsor and his Fas­ cist Friends. Given the author's back­

ground one wouldn' t expect to see much of Moore's mathematics in the book, and in fact there is little detail about his specific mathematical contri­ butions. For a survey of Moore's math­ ematical work, see R. L. Wilder's trib­ ute "Robert Lee Moore, 1 882- 1974'' [2] . Wilder classifies Moore's 68 research pa­ pers into three categories, namely Geometry, Foundations of Analysis, and Point Set Theory. This last category, Wilder points out, was so-named in def­ erence to Moore's own preferences, as many would call it set-theoretic topol­ ogy. In the Preface to the Moore biogra­ phy, Parker tells of an incident when Moore was being photographed for his Presidency of the AMS. When the pho­ tographer offered to airbrush a wart from his face, Moore replied, "Warts and all. " The author goes on to say, "And thus, in this account, I have followed the same guidance. This then is the ex­ traordinary story of R. L. Moore and how he developed the Moore Method, which was bigger than the man (with all his faults and idiosyncrasies), how it equipped its beneficiaries to excel in fields of excellence other than mathe­ matics, and how it has been modified to meet the educational requirements of today. " For the convenience o f the reader, in the remainder of this review I'll try to separate the Moore Method from Moore's life, although the book doesn't

© 2007 Spnnger Sc1ence +Business Media, Inc., Volume 29, Number 4, 2007

75

do this and in real life it wasn't easy to do. In both parts I'll include some in­ formation and anecdotes that may be new even to the cognoscenti. I'll con­ clude with a few particulars about the book itself, and a bottom line on whether you might want to read it.

Moore-the Method In the 1 966 MAA film Challenge in the Classroom, which is still available as a Classic Video from the MAA , Moore himself singled out a defining moment in the genesis of his method, when he taught himself calculus at age 1 5 . He would read the statement of a theorem and try to prove it, keeping the text's proof covered. If he needed to uncover too many lines of the proof to finish it, he felt that he had failed. Later, as a graduate student at the University of Chicago, Moore was intrigued by the laboratory system of instruction in mathematics that was advocated there by E . H. Moore (no relation). In his 1902 presidential retirement address to the AMS, E. H. Moore described how, in education, "The teacher should lead up to an important theorem gradually in such a way that the precise meaning of the statement in question, and further, the practical truth of the theorem is fully appreciated and the importance of the theorem is understood and indeed the desire for the formal proof of the propo­ sition is awakened before the formal proof itself is developed. Indeed, in most cases, much of the proof should be secured by the research work of the students themselves. " What then i s the Moore method? Parker summarizes it nicely: "Its prin­ cipal edicts virtually prohibit students from using textbooks during the learn­ ing process, call for only the briefest of lectures in class and demand no col­ laboration or conferring between class­ mates. It is in essence a Socratic method that encourages students to solve prob­ lems using their own skills of critical analysis and creativity. Moore summed it up in just eleven words: 'That stu­ dent is taught the best who is told the least. ' " The biography is permeated with discussions, often from old interviews with or articles by famous Moore stu­ dents, of the details in applying the method. For example, in 1959 Raymond Wilder, an early Moore student (Ph.D.

76

THE MATHEMATICAL INTELLIGENCER

1 923), gave six essentials of the Moore method as it was used in the mid-1930s [3]. (Parker lists just the first five.) 1 . Selection of students capable (as much as one can tell from personal contacts or history) of coping with the type of material to be studied. 2. Control of the size of the group par­ ticipating, from four to eight stu­ dents probably the best number. 3. Injection of the proper amount of in­ tuitive material, as an aid in the con­ struction of proofs. 4. Insistence on rigorous proof, by the students themselves, in accordance with the ideal type of axiomatic de­ velopment. 5. Encouragement of a good-natured competition. 6. Emphasis on method, not on subject matter. Wilder, later in his article, gave a sev­ enth feature, "Selection of material best suited to the method," but he down­ played it because of the vagueness of its terms and its ambiguity. Readers interested in a more recent appraisal of the method, including ob­ servations on its difficulties and draw­ backs, as well as who can use it and where, can consult the Monthly article [4] by another Moore alumnus, F. Bur­ ton ]ones (Ph.D. 1935). Jones notes that Moore wanted his topology classes to be as "homogeneously ignorant (topo­ logically) as possible," thus insuring a level playing-field where "competition was one of the driving forces." "Moore sternly prevented heckling," but it was rarely a problem as "the whole atmos­ phere was one of a serious community effort to understand the argument. " In contrast, Wilder, in talking about earlier Moore classes [3] , said that "good-na­ tured 'heckling' was encouraged. " Many Moore alumni single out his selection process, the filtering step, and his special ability to spot mathematical talent as key factors in his success. Mary Ellen Rudin met Moore at the registra­ tion table on her first day at the Uni­ versity of Texas. At this meeting, as de­ scribed in [5], Moore discovered in questioning her that she used "if'' and "then" , and "and" and "or", correctly, and decided to make her into a math­ ematician. In fact, according to [6], "It appears now that he [Moore] rearranged his schedule to teach her a mathemat­ ics course every semester she was in

college. " Parker's book describes how chemistry students F. Burton Jones and Gordon T. Whyburn, along with Ray­ mond Wilder, who intended to become an actuary, were all "hijacked into math­ ematics by Moore. " Moore was a master a t social engi­ neering and had, according to Rudin, a "terrific ability as a psychologist. " One of many techniques he employed was to add a student's name to a theorem, even when the student might only have slightly generalized it or had just found a better proof. Gail Young described Moore as an "incredible father substi­ tute," and pointed out that "We [Moore's students] were all fighting for approval." Young also singled out another impor­ tant Moore classroom technique, his us­ ing an "Inverse Order of Likelihood" cri­ terion for calling on students in class. An advantage in calling on students in reverse order of ability, F. Burton Jones remarked, was that this gave "the more unsuccessful students first chance when they did get a proof. " However, some­ times psychological bruising occurred. In [5] Mary Ellen Rudin described a class in 1945 with R. D. Anderson, R H Bing, Ed Moise, Ed Burgess, and "a sixth, whom we killed off right away. He was a very smart guy-1 think he went into computer science eventually-but he wasn't able to compete with the rest of us. " She added, "It builds your ego to be able to do a problem when some­ one else can't, but it destroys that per­ son's ego. I never liked that feature of Moore's classes. Yet I participated in it. " Another female Moore student, Mary Elizabeth Hamstrom, described her as­ tonishment at receiving a very long let­ ter from Moore even before her arrival at Texas for graduate school. In it Moore regrets her having already taken a course in real variable theory, and ad­ vises her not to read any point-set the­ ory in her summer reading, since for his course this "would make you too much of a spectator" and "an onlooker watch­ ing others work on theorems which are new and interesting to them but which you have already heard or read about." When students did make it into Moore's point-set topology course on the Foun­ dations of Mathematics they were vir­ tually forbidden from taking courses containing related material. Moore even once wrote a publisher, "I have even gone so far as to remove, or have re-

moved, from our university library a copy of my own book Foundations of Point Set Theory [7). On one occasion this university copy was lost or mis­ placed and I was inclined to be sorry afterwards that it was found and again shelved in our library." Many years after experiencing the Moore method, Hamstrom noted that it fostered the power and ability to do re­ search, as well as good research atti­ tudes. But there were limitations: "You do not get any training in learning the kind of mathematics other people do. It's something that you need; you need to learn to read the literature. " Mary Ellen Rudin observed in [5] that the mathematical language Moore used was his own, "completely different from the language of the mathematical litera­ ture. " For example, he used region for open set, and Rudin didn't learn the standard definitions of compact and limit point from his courses. When she received her Ph.D. she "didn't know any algebra; literally none. I didn't know any topology. I didn't know any analy­ sis-! didn't even know what an ana­ lytic function was . . . I never took an examination in mathematics in my life." Moreover, when she took her first fac­ ulty position she had never read a math­ ematical research paper, since in Moore's opinion reading other mathe­ maticians' work could destroy a stu­ dent's confidence. In his Monthly arti­ cle [4] F. Burton Jones discusses the problem of reading, and how difficult it is to get students to read mathemat­ ics after they have been working out their own proofs for a few months. He comments that even Moore himself complained that "when he wanted (finally) the students to read, they couldn't.'' Jones offered one effective technique he had found useful in his beginning graduate courses in general topology: to have students not read un­ til Christmas, then buy Kelley's General Topology for bedtime reading, and to prepare for the final examination. According to Gail Young, Moore be­ lieved algebraic topology was "the work of the devil. " Some Moore students who went on to direct their own doctoral students made certain that they learned some algebra. For example, Norman Steenrod wrote his first paper on point­ set topology after taking topology from Raymond Wilder at Michigan. But

Wilder then guided Steenrod towards algebraic topology and to Princeton, where Steenrod became Solomon Lef­ schetz's star Ph. D. student, with a the­ sis on "Universal Homology Groups," and later introduced what is now known as the Steenrod algebra. One Moore alumnus noted, "The Moore method is not for the faint hearted." Other students singled out Moore's ability to ask just the right ques­ tion at the right time, and also the pa­ tience required in doing this. In [3] Wilder compared the Moore method with a Socratic approach but with ca­ sualties, since only the fittest students survived. Wilder reported on his own use of a modified Moore method in classes of 30 or more students, although ''inevitably, a few, sometimes only two or three, students, would star in the pro­ duction. " Chapter 1 6 o f the biography de­ scribes Moore's calculus courses after World War II, when more and more en­ gineering and physics students took them. Moore made few concessions for

. I never took an examination in mathematics in my life . " these students, lecturing little and, for example, focusing on existence theo­ rems for extrema but avoiding word problems involving them and tech­ niques for finding the extrema. Mary Ellen Rudin observed that Moore's stu­ dents didn't end up learning much cal­ culus, which was disappointing for technical students who needed it as a tool. In addition, most poorer students dreaded and hated the classes and ended up with a feeling of defeat. They even became antagomsttc towards Moore himself, in spite of his being "a brilliant psychologist, " according to one student. John Green, a successful Moore student who went on to get two Ph.D.s, asserted, "He told us early on that he had no use for the university guidelines stating that we should expect three hours of outside class work for each hour in the classroom. He said he wanted us to think about his class all day, every day, to go to bed thinking about it, to wake up in the night think­ ing about it, to get up the next morn­ ing thinking about it, to think about it

walking to class, to think about it while we were eating. If we weren't prepared to do that, he didn't want us in his class. '' Moore used his calculus classes to discover mathematical talent. One such student was John Worrell, who went on to a very successful career in medicine and applied mathematics. Worrell recalled that in Moore's calcu­ lus courses, "he would give you no re­ ward for rote memorization. And in fact, he gave the veiled threat that if you try to memorize from the calculus I will de­

tect it, and I will excommunicate you . You will not be allowed in my class. "

[italics Worrell's] According to former Moore student Edwin Moise [8], the Moore Method re­ quired the instructor to dominate the environment that his students lived in. At places like Harvard, Princeton, and Chicago, said Moise (then teaching at Harvard), no one, not even Moore him­ self, could prevent students from work­ ing together or getting help from other professors, as Moore had done at Texas. Many of Moore's colleagues and con­ temporaries tried the method, but with mixed success. H. S. Vandiver, a dis­ tinguished number-theorist also at Texas, tried to teach as Moore taught but without using his very carefully or­ ganized structured sequence of ques­ tions. Vandiver eventually gave up, and went back to using texts in his classes. At Princeton both Steenrod and Lef­ schetz, an old Moore rival, used the method in topology seminars; Lefschetz even tried using it in an algebraic geom­ etry seminar. In his 1 977 Monthly arti­ cle [4] F. Burton Jones discusses areas other than topology where the Moore method works very well. These in­ cluded areas that start with an axiomatic development, such as group theory, foundations of geometry, and even an introduction to Hilbert Space. Paul Halmos in his "Automathogra­ phy" [9] discussed experimenting with a modified Moore Method and being converted after solving tactical prob­ lems. The method is not just for pro­ ducing research mathematicians: "The Moore Method is, I am convinced, the right way to teach anything and every­ thing-it produces students who can understand and use what they have learned. " Halmos used it at the Univer­ sity of Michigan in a beginning calcu­ lus course where, he found, 20 students

© 2007 Spnnger Science+Business Media, Inc., Volume 29, Number 4 , 2007

77

is the absolute maximum for which the method works, and in an honors linear algebra class in which he gave students on the first day a nineteen-page hand­ out with a bare-bones statement of fifty theorems for the course. For the latter, he noted that it took him several months of hard work to prepare for it, and com­ mented that teaching with the Moore Method "takes a lot out of you. " Hal­ mas also argued that the criticism that the method covers less material is a "red herring, " and that "with a little elastic­ ity you can even adapt the Moore Method to 'covering' a previously fixed amount of material. You'll be covering a lot of it a lot better than your lectur­ ing colleague (namely, the part that your students were led to discover and prove by themselves) . . . . " Elitism is a common criticism of both Moore and his method. Moore himself, being primarily interested in producing research mathematicians, spent more time and effort on the AMS than the MAA ; he once described the MAA as the Salvation Army of Mathematics. R. D . Anderson reported, "I was told that he [Moore] had joined the MAA three times, which meant that he must have resigned at least twice. " In spite of the elitism of the Moore Method itself, the biography points out its tremendous ef­ fect on both collegiate education and K-1 2 education in mathematics. The lat­ ter is due primarily to first-generation Moore students like Moise and second­ generation students like E. G. Begle, a Wilder student who was a founder of the School Mathematics Study Group (SMSG) in 1958. For collegiate educa­ tion, Anderson observed, while Moore did not believe in cooperative learning, in his classes "the overall learning ex­ perience was very similar to that now thought of as cooperative learning. " A footnote in the book credits the coop­ erative (or small-group) learning move­ ment in mathematics to Neil Davidson, a student of Bing's at Wisconsin, and cites the article "The Texas Method and the Small-Group Discovery Method" by Jerome Dancis and Neil Davidson at www. discovery.utexas.edu/rlm/refer­ ence/dancis_davidson.html. Davidson "started with the idea established by R. L. Moore that bright students can de­ velop mathematics, " but he changed the social environment. Instead of having students work individually within a

78

THE MATHEMATICAL INTELLIGENCER

competitive system, students worked to­ gether in a small group within a coop­ erative system. The article presents a nice overview of both methods, along with offering practical suggestions for using them.

Moore-the Man and the Mathematician Robert Lee Moore was born in Texas in 1882 and, as his name suggests, was raised in the southern tradition, in Texas. After an unexceptional boyhood he en­ tered the University of Texas at age 16, where he studied mathematics and was mentored by George Bruce Halsted, who had been a research assistant for]. ]. Sylvester at Johns Hopkins. Moore blossomed under Halsted's guidance, as had L. E. Dickson, an earlier student at Texas, and both proceeded to the Uni­ versity of Chicago for doctorates in mathematics. Moore's Ph.D . , which was received when he was just 20, was on "Sets of Metrical Hypotheses for Geom­ etry" and was supervised by E. H. Moore and Oswald Veblen. An indicator of Moore's single-mindedness is that his personal diary, written in this hotbed of mathematics, had, according to Parker, "virtually no mention of any social ac­ tivity, sports, girlfriends or other inter­ ests that might otherwise amuse and en­ gage someone of his age." The decade after Moore left Chicago was a sterile research period for him, but he did experiment with and refine his teaching method at Princeton, North­ western, and the University of Pennsyl­ vania. According to a Texas historian and future colleague of Moore's, at Penn, Moore developed a reputation "as a cur­ mudgeon that would last until his death. " But Moore's complaints about the office space there were clearly justified: he shared a basement room with eight other instructors or assistant professors, right next door to an odoriferous stable! While at Penn, Moore shifted his in­ terest from pure geometry to point-set topology. In 1 920 he returned to the University of Texas, where he stayed until his forced retirement in 1969. In 1931 he was elected to the National Academy of Sciences, and the follow­ ing year the AMS published his opus, Foundations of Point Set Theory [7]. Af­ ter that Moore gradually turned his at­ tention away from his own research to training future research mathematicians,

with remarkable success. From 1 940 un­ til his retirement, in what the biography calls his "golden age" for discovering mathematical talent, Moore produced 35 of his 50 Ph.D.s, 24 of them between 1953 and 1 969, when he was-not by choice-in modified service (half-time work with half-time pay). Moore ig­ nored the conditions and the spirit of this modified service, teaching 15 hours a week and 5 or 6 classes a semester. His doing so spurred a bizarre Faculty Council proposal in 1954 to restrict modified service to no more than half the regular duty of a full-time employee. In his eloquent argument against this proposal and its "union shop attitude," Moore amusingly referred to himself as "Professor X, " asserting, "I have full au­ thority to speak for him. " Moore won the battle against this proposal to limit his time in the class­ room but eventually lost the fifteen­ year-long war to forcibly retire him. (One of several reasons that the uni­ versity wanted Moore retired was that his presence was considered to be a hindrance to hiring new mathematics faculty.) Along with the messy details of this sad ending to a remarkable ca­ reer, the book includes some lighter moments, such as the strong rumor, when Moore started his mandatory modified service at age 70, that he sub­ scribed to an annuity that would not be­ gin paying out until he was 1 20! Parker does not shy away from dis­ cussing all aspects of Moore's person­ ality. In Chapter 1 5 he makes it clear that Moore was not a misogynist, al­ though Mary Ellen Rudin did observe in [5] , "He always pointed out that his women students were inferior." And Moore was a bigot, both in and out of class. He would not allow blacks in his class, and he once walked out of a topology lecture by a student of Bing's when he discovered that the student was black. Scott Williams, a black math­ ematician at the State University of New York at Buffalo, maintains a webpage documenting Moore's racism at www . math.buffalo.edu/mad/special/RLMoore­ racist-math. html. Moore's anti-Semitism was less pronounced than his racism but, according to Rudin, he claimed that Jews were inferior and enjoyed baiting and arguing in class with Edwin Moise, who was Jewish. Moore was also out­ spoken, both in class and as AMS Pres-

ident in the late 1930s, about the " in­

R. L. Moore-the Book

vasion" ofJewish emigre mathematicians

I particularly enjoyed the seventy pho­

and his personal loathing of President

tos Parker included in his biography­

Generalizing from this admittedly small

Roosevelt

pictures of Moore, his students, and his

sample, can we conclude that investiga­

cies. Anderson and Rudin, members of

contemporaries

of

tive journalists are perhaps best suited

Moore's renowned Class of '45,

and

his

New

Deal

poli­

at

the

Sylvia Nasar for John Nash (A Beaut[fu.l

University

/'vfind) and now Parker's book for Moore.

Chicago and the University of Texas.

for authoring critical biographies of out­

served that students \vould try to come

Many of these photos are informal, al­

standing mathematicians?

up with mathematical results for class

though Moore is never seen smiling, as

ob­

to prevent him from digressing to so­

he thought he looked silly when he

REFERENCES

cial or political issues.

smiled. Forty-four pages of Appendices

[ 1 ] Traylor, D.

I n these days of enforced collegial­ ity in academia,

in the book include descriptions of four

I am fascinated by

courses commonly taught by Moore at

Moore's feuds with several mathemati­

Texas, as well as short biographies of

cal colleagues at Texas. In 1 944, when

each of his fifty doctoral students, along

Edwin F. Beckenbach and Moore were

with their thesis titles. Many of these

competing

a

theses were in subject areas that are no

heated confrontation ended in fisticuffs!

for

graduate

students,

longer, if they ever were, part of main­

The next year Beckenbach left Texas

stream mathematics. ( R H Bing once

for UCLA.

remarked . regarding his thesis "Con­

A longer and more significant feud with

number-theorist

Harry

cerning simple plane webs , " that if any­

Schultz

one \\anted a journal article based on

Vandiver is discussed in the chapter

it he still had forty-eight of the fifty

"Clash of the Titans ( 1 944-1950). " 'If Van­

copies he had been give n . ) The book

diver and Moore had been colleagues

has a useful nine-page index, and for a

in the same department for 2 4 years

hardcover book with a quality binding

when their personal differences came

it is very reasonably priced.

Reginald Creative Teaching:

Heritage of R. L. Moore, University of Hous­

ton, 1 972. [2] Wilder, R . L. Robert Lee Moore 1 882-1 97 4 , Bulletin

o f the

American

Mathematical

Society 82 (1 976), 4 1 7-427 Available on­

line at http://www.discovery.utexas.edu/rlm/ reference/wilder2 . html [3] Wilder, R. L. Axiomatics and the develop­ ment of creative talent. In The Axiomatic Method with Special Reference to Geome­ try and Physics, L. Henkin, P. Suppes, and

A Tarski, Eds. North-Holland, 1 959, pp.

474-488. Available on-line at http://www. discovery. utexas.edu/rlm/reference/wilder1 . html [4] Jones, F.

Burton. The Moore Method.

to a head in 1 946. Vandiver threatened

We have reached the bottom line­

to resign but instead transferred from

should you read this book? Probably

the Department of Pure Mathematics to

not, especially if you've already read

that of Applied Mathematics, not the ob­

some of its high points in this review

vious place for a renowned expert on

and aren't just peeking at this conclu­

Mary Ellen Rudin in More Mathematical

Fermat's Last Theorem. Vandiver and

sion. The biography is readable and

People, D. Albers, G. Alexanderson , and C .

Moore never exchanged a single word

well-documented, but it has too much

Reid, Eds.

again.

minutiae for many if not most readers.

Boston, 1 990.

American Mathematical Monthly 84 (1 977),

273-278. [5] Albers, D., and Reid, C. An interview with

Harcourt Brace Jovanovich,

B y 1969 the two departments of Pure

In addition. as important as Moore was,

Mathematics and Applied Mathematics

he is not the most pleasant person to

of Women and Minorities in Mathematics,

had combined into one. Its members

read about. However. if you would like

voted 27 to 1 , with 9 abstaining, in fa­

to read more about the Moore Method

American

vor of .\1oore·s retirement. That summer

I suggest chapters: 6, 8, 9 . 1 1 ( "Moore

Moore taught his last class. He refused

the Teacher'' ) . 1 3 ( "Class of '45" ), 1 5

any symposium or other celebrations to

( "His Female Students " ) , 1 6 ( "Moore's

"honor'' his retirement. He also-at age

Calculus"), and 1 8 . Alternatively, you

85 !-turned down a professorship at the

might ignore this book altogether and

University of Houston. But he accepted

just read the descriptions of the method,

the title Professor Emeritus and spent

including its modifications and limita­

the remaining five years

tions, hy Halmos [9], Jones [4], Wilder

of his life

within walking distance of the Texas

[3], and Moise [8] .

campus.

I have a final comment. Many years

The book concludes with a nice sum­

ago when I read Constance Reid's en­

maiy of Moore's legacy to mathematics,

grossing

including not only the Moore Method in

Courant I knew that she had much math­

biographies

of

Hilbert

and

[6] Kenschaft, P. Change is Possible: Stories Mathematical

ory, American Mathematical Society Collo­

quium Publications 1 3, 1 932, revised ed. 1 962. [8] Moise, E. E. Activity and Motivation in Math­ ematics. American Mathematical Monthly 72 (1 965), 407-4 1 2 . [9] Halmos, P. I Want To Be a Mathemati­ cian: An Automathography in Three Parts ,

Springer-Verlag, 1 985 . pp. 255-265. [re­ printed as a Mathematical Association of America paperback, 1 988]

teaching but also the concept of Moore

ematical expertise

such as that of her sister Julia Robinson

numerous academic descendants. Ac­

and other distinguished mathematicians

cording to The Mathematics Genealogy

at Berkeley and elsewhere. More re­

Project at

cently, there have been several well-writ­

Santa Clara, CA 95053

nodak.edu/, there were 2 , 1 1 0 such de­

ten and mathematically accurate biogra­

USA

scendants as of July 2007.

phies of mathematicians by outsiders like

e-mail: [email protected]

�Al so see the article by Leo Corry in this issue, pp. 62-7 4.

- The

Provi­

[7] Moore, R . L. Foundations of Point Set The­

spaces in topology. It also mentions his

http:/ Igenealogy. math.ndsu.

Society,

dence, R . I . , 2005, p. 62.

available for help,

Department of Mathematics and Computer Science Santa Clara University

Editors

© 2007 Spnnger Science-.- Bus1ness Med1a, Inc., Volume 29, Number 4. 2007

79

The Poincare Conjecture: In Search of the Shape of the U niverse by Donal O 'Shea NEW YORK, WALKER

ix +

&

COMPANY, 2007,

293 PP, U S $26.95, ISBN-10: 0-8027-1532-X,

ISBN-13: 978-0-8027-1532-6

Poincare1s Prize: The H undred-Year Quest to Solve One of Math1s G reatest Puzzles by George G. Szpiro NEW YORK, DUTION, 2007,

ix + 320

PP

US $24.95, ISBN-10: 0525950249, ISBN-13: 978-0525950240

REVIEWED BY JOHN J. WATKINS

t is not often when something im­ portant in mathematics makes head­ lines around the world. But that did happen just this past year with the proof-after almost exactly one hun­ dred years-of the famous Poincare Conjecture by a reclusive Russian math­ ematician Grigory Perelman. Newspa­ pers everywhere ran articles trying to explain to their readers the significance of this brilliant achievement. Science magazine officially designated it the "Breakthrough of the Year" for 2006. 1be New Yorker magazine published a highly controversial article by Syvia Nasar (author of A Beautiful Mind ) and David Gruber, in which they portrayed a large cast of mathematicians behav­ ing like characters in a melodrama; the magazine has been threatened with a lawsuit for its trouble. Perhaps one of the clearest indica­ tions that the Poincare Conjecture had indeed captured the attention of the general public came during an episode of an American television series Studio 60 on the Sunset Strip when the fictional head comedy writer, wonderfully played

I

80

by Matthew Perry, had a running gag throughout the show as he struggled to write a comedy sketch about turning a bunny into a sphere. He knew this had something to do with the Poincare Con­ jecture, and that the conjecture itself is truly important, and he sensed that he should be able to make it funny. He just couldn't write the sketch. The script writers for Studio 60 are very smart people and they certainly de­ serve a lot of credit for inserting such a topical mathematical reference into one of their shows, so they can be excused if they don't have some of the details of the Poincare Conjecture itself exactly cor­ rect. They know it has something to do with topology and that topology in turn has something to do with deforming ob­ jects. For example, a topologist views a solid chocolate bunny as essentially the same as a solid sphere of chocolate be­ cause one of these objects could be grad­ ually deformed into the other. But they­ like almost everyone else in the world--don't have the vaguest idea what the Poincare Conjecture itself is really all about, much less why it might be im­ portant to anyone. For example, they clearly don't real­ ize that the Poincare Conjecture is about the three-sphere-that is, the three­ dimensional sphere-which, confus­ ingly, doesn't exist in three-dimensional space at all, but in four-dimensional space. So, if Matthew Perry or anyone else in the general public wants to know what it was that Grigory Perel­ man achieved when he proved the Poincare Conjecture and why anyone cares, they are clearly going to need some serious help. Fortunately, that help is at hand. In a truly remarkable book, The Poincare

Conjecture: In Search ofthe Shape ofthe Universe, Donal O'Shea tells the com­ plex story of the Poincare Conjecture in a way that provides just the help that any interested reader needs in order to gain an understanding of how Perelman did what he did, why it took so long, and why it is so important. O'Shea begins his story quite dra­ matically in a crowded lecture hall in April 2003 at MIT with Perelman at a blackboard explaining how he had used a very clever idea called Ricci flows that treat curvature in space as if it were heat. This allowed Perelman to recast the Poincare Conjecture in terms of a type

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media. Inc.

of equation known as partial-differen­ tial equations, an extremely well-studied branch of the area of mathematics that makes up most of mathematical physics. All of Perelman's training had prepared him for this particular method of attack on the Poincare Conjecture, from his high-school days at Leningrad Secondary School in Saint Petersburg, a school that specialized in mathematics and physics, to his later joining a group at the Saint Petersburg branch of the Steklov Insti­ tute that did legendary work in the field of partial-differential equations. As O'Shea lets his leisurely story un­ fold, we learn that there are many math­ ematicians whose work ultimately went into the solution of the Poincare Con­ jecture. It was Richard Hamilton who developed the method of Ricci flows in the early 1980s-and even the term itself-and did much of the real work behind solving the Poincare Conjecture. Within a decade Hamilton and others had achieved sensational results for two­ dimensional surfaces, and for dimen­ sions higher than three, but Hamilton had also showed that in three dimen­ sions there were genuine difficulties still to be overcome when using Ricci flows. It was by overcoming these seemingly insurmountable difficulties that Perel­ man was eventually able to crack the century-old Poincare Conjecture. Actually, and this may seem strange, Perelman and Hamilton were not really working on the Poincare Conjecture at all, they were working on a much more ambitious and much harder problem known as the Geometrization Conjec­ ture. In two dimensions we all know that there are exactly three different geometries: spherical, flat, and hyper­ bolic. But in three dimensions it turns out that there are eight different geome­ tries. In the 1 970s Bill Thurston made the extraordinary conjecture that in a natural way each individual piece of any three-manifold has to have one of these eight geometries! This became known as the Geometrization Conjecture. In turn, then, the Poincare Conjecture has long been known to be a rather simple consequence of Thurston's truly bold new vision of the way geometry works in three dimensions. While Perelman appears to have knocked off something much bigger than the Poincare Conjecture-namely the Geometrization Conjecture-it is of

course the Poincare Conjecture that is getting all the attention. It is the Poin­ care Conjecture after all that was listed by the Clay Institute in 2000 as one of its seven millennium problems and for which they offered one million dollars for a solution. It is for solving the Poin­ care Conjecture that the International Mathematical Union, in the summer of 2006 in Madrid, awarded Perelman the Fields Medal, the equivalent in mathe­ matics of a Nobel Prize. (Perelman, however, declined to accept it.) The Clay Institute will soon announce its de­ cision on whether they will award the full million dollars to Perelman or split it in some manner. In any event, it seems likely that, remarkably, Perelman will also decline to accept this award. The Poincare Conjecture is the fol­ lowing deceptively simple assertion: the only compact three-manifold on which any closed loop can be shrunk to a point is the obvious one, namely, the three­ sphere. O'Shea does a beautiful job of bringing the reader to the point of fully understanding this conjecture by talking about how we came to learn the shape of our own earth long before we could ever view it in its entirety from space and how we use maps-think about how Google Earth overlays a series of rectangular pictures to represent the en­ tire two-manifold that is the surface of our planet-and then he effortlessly takes the reader up one dimension to explain how the Poincare Conjecture is in fact one of the simplest and most fun­ damental questions we could ask about the shape of our own universe. We live in a three-manifold, we just don't know what it looks like. If we could view it in its entirety from some vantage point in four-dimensional space, we would know instantly what it looks like, but we can't. But, as O'Shea patiently ex­ plains, we can still represent our uni­ verse-just as was done with maps in the day of Columbus for the earth-with a series of pictures, each picture a three­ dimensional rectangular box, and the entire universe can be represented by overlaying this collection of pictures to form a map of the universe. Now, because the Poincare Conjec­ ture is at last a theorem, we know that if our universe is finite, which seems likely, and if it is also true that any closed loop in our u niverse can be shrunk to a point, then our universe must look like

a three-sphere. That's why the Poincare Conjecture is so important! Of course, it might not be true that any closed loop in our universe can be shrunk to a point. Just like a loop on the surface of a torus (a doughnut) that passes through the hole of the torus can't be shrunk to a point, it might be the case that some loops on the three­ dimensional surface in which we live might similarly get stuck if we try to shrink them. We just don't know-yet. Donal O'Shea takes the reader on a fascinating journey from the ancient world more than 2500 years ago when Pythagorus taught that the earth was a sphere to the present day, and provides the reader a solid intuitive understand­ ing of the complex details of what Grig­ ory Perelman and others have accom­ plished. While O'Shea takes great pains to explain carefully and skillfully all mathematical ideas along the way-in­ deed, were the fictional Matthew Perry to read The Poincare Conjecture: In Search of the Shape of the Universe, he would have not the slightest bit of trou­ ble finally being able to write that com­ edy sketch about turning a bunny into a sphere-this book is equally good reading for even expert mathemati­ cians. Anyone's understanding of the mathematics involved here will be en­ hanced by O'Shea's thoughtful discus­ sions and many readers will also ap­ preciate the many additional themes he manages to weave into his basic plot (although some may find these "tan­ gents" distracting) . I, for one, gained a far better feel for three-manifolds than I had before read­ ing this book, even though strictly speaking I have seen it all before, sim­ ply because O'Shea writes so well and so vividly. And I also found myself with a refreshing new awareness of Poin­ care's true greatness as a mathematician who, in O'Shea's words, "shaped the form of twentieth-century mathematics." I enjoyed his discussion of Euclid's Elements; his biographies of Gauss, Bolyai, and Lobachevsky; his placement of Riemann at the very heart of this story; his retelling of one of the most famous moments in intellectual his­ tory-when Poincare had an instanta­ neous flash of mathematical inspiration completely out of the blue just as he was stepping on a bus to go on an ex­ cursion, and his description of the key

role in this drama played by the stun­ ning example of the three-manifold pro­ duced by Poincare that we now know as the Poincare dodecahedral space. I especially enjoyed a joke that French schoolchildren recited gleefully about the most famous mathematician in France: "Qu'est-ce un cercle? Ce n'est point carre . " What is a circle? It is not a square. Which makes little sense in English, but becomes very clever in French where "point carre" is pro­ nounced the same as "Poincare . "

I d o have a few minor complaints. In such a fine book it is unfortunate that some of the computer-generated figures are of a rather poor quality by today's standard. For the most part this causes no difficulty whatsoever, it is merely unattractive. I might add that, by con­ trast, some of the hand-drawn figures are quite lovely. Half of one figure, however, is utterly incomprehensible, which seems strange since a portion of a subsequent figure could serve ad­ mirably if suitably adapted. There are also quite a few typos and several ref­ erences to the wrong figures, but none of these is too serious and each is rel­ atively easy to spot. It is not surprising that an event as earthshaking as a proof of the Poincare Conjecture would prompt the appear­ ance of several books. One book which takes a decidedly more sensational tack on this event is Poincare 's Prize: 1be

Hundred- Year Quest to Solve One of Math 's Greatest Puzzles by George G . Szpiro. The back cover carries a blaz­ ing headline worthy of a tabloid an­ nouncing in large red type: "AMAZ­ INGLY, THE STORY IS TRUE . " Well, much of Szpiro's story in the book is true, but not quite all. More troubling is that the front cover for this book, which after all is about

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

81

one of the great conjectures in topol­ ogy, contains a glaring topological blun­ der. The front cover features a sphere and a horse (it could have been a bunny, but a horse is much more graceful) and each of these objects has been given a very attractive checkerboard coloring. This coloring and a dashed line between the horse and the sphere form a visu­ ally compelling way to show that the horse and the sphere are equivalent ob­ jects from a topological point of view. So far, so good. The only problem is that it has been known since roughly the time of Euler that you can't color a sphere with a checkerboard pattern. The basic idea-see the book Across the Board for the details-is that in 1752 Euler found a formula for polyhedra that relates the number of vertices, edges, and faces, and this easily gener­ alizes to figures drawn on surfaces (where instead of counting faces you count regions). Remarkably then, it turns out that surfaces all have an "Euler characteristic, " and you can only draw a checkerboard figure on a surface whose Euler characteristic is 0. So, for example, you could draw a checker­ board figure on a torus (or a Klein bot­ tle if you know what that is) , but you can't ever draw such a figure on a sphere or a horse (or a bunny), simply because the Euler characteristic for a sphere is 1 . Ironically, Szpiro himself talks at some length in Poincare 's Prize about Euler's formula, so he could have no­ ticed this blunder on his own cover. He even makes matters worse by crediting Euler with proving his own formula in 1 7 5 2 . It is true that Euler did publish what he believed to be a proof in that year, but (see Graph Theory 1 7361936) Euler used a slicing procedure that does not work in all cases. Legendre did prove Euler's formula in 1 794. A somewhat similar historical inaccuracy occurs when, while not explicitly saying so, Szpiro leaves the reader with the im­ pression that the mathematical field of graph theory began with Euler and his solution to the famous problem of the seven bridges of Konigsberg, but graphs such as Szpiro has in mind did not ap­ pear until the second half of the nine­ teenth century. In another instance Szpiro misleads the reader mathematically as he tries to illustrate an early disagreement about

82

THE MATHEMATICAL INTELLIGENCER

convergence between Poincare and an astonomer by discussing infinite series and, in particular, the harmonic series , which 1 + 1/2 + 1/3 + 1/4 + 1/5 + of course diverges to infinity. Szpiro wants to emphasize how slowly this se­ ries diverges so he says "no less than 1 78 million entries must be added to ar­ rive at the sum of 20 . " But this is com­ pletely false. One of the most beautiful properties of the harmonic series is that while it is lazily strolling its way to in­ finity it somehow manages to miss ab­ solutely every integer except 1 , the in­ teger where it started! In order to be truthful Szpiro would need to say some­ thing like " 1 78 million entries must be added just to get beyond 20. " (Paul Hoffman claims in The Man Who Loved Only Numbers that you need about 300 million terms in order to exceed 20.) Elsewhere Szpiro talks about "the Eu­ clidean three-dimensional space that we live in" and while this phrase makes its appearance merely as part of a state­ ment whose proof we later learn is wrong, an unwary reader is again left with an incorrect impression, namely that the universe is Euclidean. In spite of the breezy style of the au­ thor, Poincare 's Prize is probably not the best choice for a reader who is seeking to understand the mathematics behind the Poincare Conjecture. Szpiro clearly made the decision not to include any di­ agrams in this book, but he pays a high price in terms of what he is able to com­ municate to a novice reader. Often he is effective, but not always. How many of his young readers will have seen an au­ tomobile tire that has an inner tube? Or know that a pretzel-when made in the traditional old way from a single long roll of dough with a quick twist and the two ends joined-has three holes? And even though his teaching instincts are well-meaning, the resulting metaphoric images can sometimes be amusingly dis­ concerting, as when he has us imagine ·

·

·

a fly taking a step back to lift off and look at a bagel at arm 's length. One very promising approach to con­ structing a counterexample to the Poin­ care Conjecture did involve pretzels, or at least it involved knots that can look a lot like pretzels. The idea is to start with a cube with a knot inside and then remove the knot so that you have a cube with a knotted hole. Then you attempt to put things back together again so that

loops shrink to a point and yet you man­ age to come up with something other than the three-sphere. Needless to say, this never quite worked.

Szpiro does have a flair for imagi­ native comparisons that can be in­ structive as well as entertaining, as when he uses LEGO bricks to help ex­ plain the Geometrization Conjecture or when he uses BOTOX injections to il­ lustrate the fundamentally important idea of how Ricci flows work. But he can also show a lamentable tendency to wander off into gratuitous and irrel­ evant comments, such as complaining about FDA approval of BOTOX, or when, in the middle of an already overly long biography of the man be­ hind the Clay Institute, he gets further sidetracked to lecture the reader about the right of certain museums to keep ancient artifacts of dubious provenance simply because they display them in a "more seemly manner." There are a good many things· I learned and liked in Poincare 's Prize. Szpiro traces in great detail a story he nicely frames from 9 October 1492, with Columbus standing on the bridge of the Santa Maria, to 22 August 2006, with Juan Carlos, the King of Spain, awarding the Fields Medal to Grigory Perelman who of course wasn't present in Madrid for this grand-opening cere-

mony for the Twenty-Fifth International Congress of Mathematicians. I'm sure each individual reader would find many things of interest and many new ideas in Poincare 's Prize; I will just mention a few that came my way, in order to suggest that there is indeed a genuine richness that lies be­ tween the covers of this book. Much to my amazement I learned that the Poincare dodecahedral space, which he discovered in 1904 as a coun­ terexample to the "theorem" he had claimed in 1 900, to this day remains the only known counterexample to this false theorem. I learned that RH were not the initials of that giant of topol­ ogy, R H Bing, but were in fact his ac­ tual first name. More importantly, I also learned that Bing, who himself made prolonged serious attempts on the Poincare Conjecture, in the end may have concluded that the conjecture was false. This , for me, was one of the very best moments in Szpiro's book because it showed so dramatically how hard it is in mathematics to know what is true and what is false if someone of R H Bing's stature can be so completely wrong about such a fundamental ques­ tion as the Poincare Conjecture. From a more personal point of view, I was delighted to learn that james Alexander, who is well known to most mathematicians as the discoverer of the famous and fabulous "Alexander horned sphere, " has a classic technical climbing route up 14,255 foot-high Longs Peak in Colorado, named Alexander's Chimney, after him. Simi­ larly, I found a rather lengthy history of the Ecole Polytechnique-which Poincare attended from 1 873 to 1875, graduating second in his class-ab­ solutely fascinating because my wife di­ rected a program at Colorado College for several summers during the mid1980s. That program was designed for students from the Ecole Polytechnique to guide them in the study of English and to acquaint them with American culture. Here is one final nugget in the same vein that takes on special irony now that Perelman has rejected the Fields Medal: at his death in 1 9 1 2 , Poincare had received the largest number of nominations for a Nobel Prize of any non-winner. Recall that there is not a mathematics category for Nobel Prizes

and that it is for this reason that the Fields Medal is considered to be the mathematical equivalent of a Nobel Prize. Poincare's nominations were all in physics. I must admit that, in the end, Szpiro does deliver on the tabloid promise from the back cover of his book. The most gripping, hard-to-put-down read­ ing are Szpiro's last two chapters, when he finally gets down to discussing the very messy controversy surrounding the solution of Poincare's famous con­ jecture. It is hard not to be intrigued by this controversy. There are some very serious issues here : What share of the credit for the solution does Hamil­ ton deserve' That the Clay Institute may well award him a large share of the mil­ lion-dollar prize is just one measure of the fact that many people believe that he and Perelman share equally in ar­ riving at the final solution. Another of the serious issues is the way in which Perelman bypassed traditionally ac­ cepted methods for publishing mathe­ matical proofs by placing his unrefer­ eed proofs on the Internet. It has taken three years and several teams of heav­ ily financed experts to conclude that Perelman's work is correct. Meanwhile, the really messy part of the controversy arose from a claim made by a team of Chinese mathematicians that they had published the first complete proof of the Poincare Conjecture. The article in Tbe New Yorker greatly inflamed this controversy by including a full-page drawing with Shing-Tung Yau looking as if he is about to rip the Fields Medal from the neck of Perelman (who looks a bit like Vincent Van Gogh in this drawing). Szpiro sorts through the details and complexities-including the ethical ones-of this controversy quite thor­ oughly and with what seems to be con­ siderable fairness and a great deal of wisdom, both concerning human na­ ture and with respect to maintaining al­ ways the highest regard for the well­ being of mathematics. Thus, he is able to bring us beyond the controversy to the point where we can celebrate the solution of the Poincare Conjecture, perhaps dream of solutions to one of the six remaining millennium prob­ lems, and find other ways-in the words of the mission statement of the Clay Institute-"to further the beauty,

power, and universality of mathemati­ cal thinking. " REFERENCES

1 . N. L. Biggs, E. K. Lloyd, and R. J. Wilson, Graph

Theory

1 736- 1936,

Clarendon

Press, 1 976. 2 . P. Hoffman, The Man Who Loved Only Num­ bers: The Story of Paul Erdos and the Search for Mathematical Truth , Hyperion, 1 998.

3. D . Mackenzie, The Poincare Conjecture­ Proved, Science 3 1 4 (22 December 2006), 1 848-1 849. 4 . S. Nasar and D . Gruber, Manifold Destiny, The New Yorker (August 28, 2006), 44-57 .

5. J . J . Watkins, Across the Board: The Math­ ematics of Chessboard Problems, Prince­

ton University Press, 2004.

More math comics by Courtney Gibbons are available online at: brownsharpie. courtneygibbons.org Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail : jwatkins@coloradocollege .edu

Fixing Frege john Burgess PRINCETON AND OXFORD: PRINCETON UNIVERSITY PRESS, 2005. PP. xii + 257. ISBN 0-691·12231-8, US$ 39.95

Reason's Proper Study: Essays Towards a NeoF regean Phi losophy of M athematics Bob Hale and Crispin Wright OXFORD, CLARENDON PRESS, 2001. PP. xiv + 455. US$ 45.00 ISBN 0-19-823639-5

REVIEWED BY 0YSTEIN LINNEBO

e know that there are infi­ nitely many prime numbers and that every natural number

© 2007 Spnnger Sc!ence +Buslness Med1a, Inc., Volume 29, Number 4, 2007

83

has a unique prime factorization. What sort of knowledge is this? Unlike our knowledge that Mogadishu is the capi­ tal of Somalia or that electrons have negative charge, arithmetical knowl­ edge does not seem to be empirical; that is, it does not seem to be based on observation or experiment. The German mathematician, logician, and philoso­ pher Gottlob Frege ( 1 848-1925) devel­ oped a bold new account of the nature of arithmetical knowledge: He argued that pure logic provides a source of such knowledge, and that arithmetic there­ fore is a priori rather than empirical. This view is now known as logicism and is one of the main philosophical accounts of mathematics (alongside for­ malism, intuitionism, conventionalism, and structuralism). Frege's defense of his logicist view of arithmetic proceeds in two steps. The first step consists in an account of how numbers are applied and of their iden­ tity conditions. Frege argues that count­ ing involves the ascription of numbers to concepts. For instance, when we say that there are eight planets, we ascribe the number eight to the concept " . . . is a planet". Let '#' abbreviate the op­ erator 'the number of'. Frege's claim is then that '#' applies to any concept F to form the expression '#F ', meaning "the number of Fs". Next Frege argues that the number of Fs is identical to the number of Gs if and only if the Fs and the Gs can be put in a one-to-one cor­ respondence. This principle (which is typically associated with Georg Cantor) is known in the philosophical literature as Hume's Principle (since it may have been anticipated by the philosopher David Hume). In order to formalize this principle, Frege makes essential use of the fact that his logic is second-order; that is, in addition to the ordinary first­ order quantifiers Vx and 3x, which range over some domain D, Frege's logic also has second-order quantifiers VR and 3R, which range over relations on D (of some particular adicity). Let 'F= G ' abbreviate the pure second­ order statement that there is a relation R that one-to-one correlates the Fs and the Gs. Hume's Principle can then be expressed as: (HP)

#F = #G � F = G

This makes sense because equivalence relation.

84

THE MATHEMATICAL INTELLIGENCER

=

is an

The second step of Frege's defense of logicism provides an explicit defini­ tion of terms of the form '#F'. Frege does this in a theory that consists of second-order logic and his "Basic Law V," which states that the extension of a concept F is identical to that of a con­ cept G if and only if the Fs and the Gs are co-extensional; or, in contemporary notation (V)

lxiFx) lxi Gx) � Vx(Fx � Gx). =

In this theory, Frege defines #F as the extension of the concept "x is an ex­ tension of some concept equinumerous with F." That is, he defines

#F = lxi3 G(x = lyiGy) l\ F= G)}. This definition is easily seen to sat­ isfy (HP). More interestingly, Frege proves in meticulous technical detail how this definition and his theory of ex­ tensions entail all of ordinary arithmetic. However, just as the second volume of his magnum opus was going to press in 1902, Frege received a letter from the English logician and philosopher Bertrand Russell, who reported that he had "encountered a difficulty" with Frege's theory of extensions. The diffi­ culty Russell had encountered is the paradox now bearing his name. Frege's theory of extensions is in effect a naive theory of sets. We may thus consider the set of all sets that are not members of themselves. In Frege's theory we can then prove that this set both is and is not an element of itself. Frege's re­ sponse to Russell's letter is remarkable. Sixty years later Russell described it as follows. As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege's dedication to truth. His en­ tire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his funda­ mental assumption was in error, he responded with intellectual plea­ sure, clearly submerging any feel­ ings of personal disappointment. Russell's paradox eventually led Frege to give up on logicism. Until the 1 980s both logicians and philosophers regarded Fregean logicism as a dead

end, and people attracted to the idea of logicism pursued other versions of it, such as Russell's very complicated "type theory." However, over the past two decades there has been a resurgence of interest in Fregean logicism. A variety of con­ sistent fragments of Frege's theory have been identified and explored, and their possible philosophical significance has been vigorously debated. The two books under review are without doubt among the most important products of this resurgence. Reason 's Proper Study is the most extensive philosophical ar­ ticulation and defense to date of a spe­ cific neo-Fregean programme, whereas Fixing Frege offers the deepest and most comprehensive technical investi­ gation of a variety of different neo­ Fregean approaches. Neo-Fregeanism began with Crispin Wright (Frege's Conception of Numbers as Objects, 1983) who suggested that the problem posed by Russell's paradox be evaded by making do with the first step of Frege's approach, abandoning alto­ gether the second step and its incon­ sistent theory of extensions. This ap­ proach is made possible by two relatively recent technical discoveries. The first discovery is that (HP), unlike (V), is consistent. More precisely, let Frege Arithmetic be the second-order theory, with (HP) as its sole non­ logical axiom. Frege Arithmetic can then be shown to be consistent if and only if second-order Peano Arithmetic is. The second discovery is that Frege Arith­ metic and some very natural definitions suffice to derive all the axioms of sec­ ond-order Peano Arithmetic. This result is known as Frege's Theorem. It is an amazing result. For more than a century now, informal arithmetic has almost without exception been given some Peano-Dedekind style axiomatization, where the natural numbers are regarded as finite ordinals, defined by their po­ sition in an omega-sequence. Frege's Theorem shows that an alternative and conceptually completely different ax­ iomatization of arithmetic is possible, based on the idea that the natural num­ bers are finite cardinals, defined by the cardinalities of the concepts whose numbers they are. Technically speaking, the neo­ Fregean foundation of arithmetic is thus a success: it is consistent and strong

enough to prove all of ordinary arith­ metic. But what about its philosophical significance? Reason 's Proper Study, which brings together 1 5 essays by the two foremost neo-Fregeans, is an extended argument that neo-Fregeanism is a philosophical success as well. It is argued that this ap­ proach enjoys most of the philosophical benefits promised by Frege's original but sadly inconsistent form of logicism. I can here mention only three of the questions that Hale and Wright grapple with in their defense of this claim. The first question is whether the first of the two steps of Frege's approach (which I described above) can stand on its own. Frege himself thought it could not because (HP) fails to settle all ques­ tions about the identity of numbers. For instance, (HP) fails to settle whether the number of planets is identical to the Ro­ man emperor julius Caesar! In order to settle such pesky questions, Frege thought it necessary to proceed to the second step and give an explicit defin­ ition of the natural numbers. Hale and Wright disagree, arguing in­ stead that (HP) gives the criterion of identity for numbers; that non-numbers have different such criteria; and that this implies that Caesar cannot be a number. The second question concerns the epistemological status of Hume's Princi­ ple. As Hale and Wright admit, (HP) does not particularly "look like" a logi­ cal principle. They defend instead the slightly weaker claim that (HP) can serve as an explanation of the meaning of the #-operator and thus be known a priori. If correct, this claim would be very sig­ nificant, as it would establish that arith­ metic-with its infinite ontology of num­ bers-can be known a priori. This would be almost as good as what was promised by Frege's original logicism. The third question concerns the de­ mand for a deeper and more general understanding of the kind of explana­ tion that (HP) is supposed to provide. The demand is made particularly acute by the structural similarity between (HP) and the inconsistent principle (V). What if it is just a happy accident that (HP) is consistent? If so, the neo-logi­ cists can hardly claim that merely lay­ ing (HP) down as an explanation of the meaning of the #-operator yields a pri­ ori knowledge that (HP) is true. For surely a belief cannot count as know!-

edge if it is just a happy accident that it is true! Hale and Wright respond by arguing that knowledge does not re­ quire any kind of antecedent guarantee against error. It seems to this reviewer that this can at most postpone, not elim­ inate, the need for a deeper explana­ tion. After all, it is part of the very na­ ture of both mathematics and philosophy to seek general explana­ tions whenever such are possible. Whereas the agenda of Reason s Proper Study is predominantly philo­ sophical, that of Fixing Frege is pre­ dominantly mathematical. For instance, Fixing Frege has little to say about the first two questions mentioned above but a lot to say about the third. The opening chapter provides a very readable introduction to the mathemat­ ical aspects of neo-Fregeanism. Burgess first provides a useful summary of Frege's own constructions, of Russell's paradox and Frege's response to it, and of Russell's competing form of logicism. He then develops a sophisticated frame­ work in which various neo-Fregean the­ ories can be analyzed and their strengths compared. Particularly useful is his explanation of a hierarchy of mathematical theories, ranging from very weak subsystems of arithmetic up to very strong systems of set theory. This hierarchy provides a unified sys­ tem of targets for neo-Fregean recon­ struction, which enables us to measure the strength of a neo-Fregean theory in terms of how much of this hierarchy the theory allows us to reconstruct. The remaining two chapters explore the two main ways of ensuring the con­ sistency of Frege-inspired theories. The standard way-already encountered above and the topic of Chapter 3-is to abandon Frege's Basic Law V in favor of related but weaker principles such as (HP). An alternative way-which forms the topic of Chapter 2-is by placing re­ strictions on which open formulas can define relations. In order to explain this alternative way we need some defini­ tions. A comprehension axiom is an ax­ iom which states that an open formula ¢, with free variables x1 , . . . , Xn, suc­ ceeds in defining an n-adic relation R, under which n objects fall if and only if they satisfy the formula ¢, or in symbols:

3RVxl . . . Vxn[Rxl, . . . , Xn � ¢Cx1 , . . . , x�]

A comprehension axiom is said to be predicative if ¢ contains no second­ order quantifiers and impredicative otherwise. If regarded as definitions, predicative comprehension axioms have a nice property of non-circularity, namely that the relation R is defined without quantifying over a totality that includes R itself. The philosopher Michael Dummett has proposed an exciting but contro­ versial analysis of "the cause" of Rus­ sell's paradox: He blames the contra­ diction not on Basic Law V but rather on the presence of impredicative com­ prehension axioms in the background theory. To substantiate this analysis, it must be shown that restricting oneself to predicative comprehension restores consistency. Chapter 2 gives a nice pre­ sentation of some earlier theorems which show that this is indeed the case. But for the analysis to be plausible, it must also be shown that this restriction leaves the character of the relevant the­ ories intact; otherwise consistency will be restored not by excising a precisely circumscribable "cause of paradox" but more bluntly by rendering the theories impotent. But some new results from Chapter 2 show that the resulting the­ ories are very weak. So this bodes ill for Dummett's claim that impredicative comprehension is "the serpent that en­ tered Frege's paradise. " The final chapter examines t h e stan­ dard way of restoring consistency. Call a principle of the logical form

() *

§F = § G � F - G

an abstraction principle. Some abstrac­ tion principles-such as (HP)-appear to be acceptable, whereas others-such as (V)-clearly are not. Can a sharp and well-motivated line be drawn between abstraction principles that are accept­ able and those that are not? A natural thought is that (V) is made unaccept­ able by the fact that it requires a one­ to-one map from the concepts on a do­ main into the domain itself, which we know by Cantor's theorem to be im­ possible. Say that an abstraction prin­ ciple ( ) is non - inflationary on a do­ main D if the equivalence relation - is such that there are no more --equiva­ lence classes of concepts than there are objects in D. One easily sees that every non-inflationary abstraction principle has a model. *

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

85

The question of when a system of abstraction principles is acceptable is harder. For even if the principles that make up the system are individually non-inflationary, the system as a whole need not be. This question receives a penetrating analysis in Kit Fine's Tbe Limits ofAbstraction (2002), where it is shown that a system of abstraction prin­ ciples is acceptable provided that each principle is individually non-inflationary and based on an equivalence relation that satisfies a certain "logicality con­ straint'' (involving invariance under per­ mutations of the domain D) . In Fixing Frege, Burgess gives a nice exposition of this analysis and substantially ad­ vances the discussion by pinpointing the strength of the resulting system: that of "third-order arithmetic . " Although the strength o f this sys­ tem is thus substantial, it falls far short of the stronger target theories in Burgess's hierarchy. In order to reach higher, Burgess proposes a new way to motivate the axioms of ordinary set theory. He first uses "limitation of size" considerations to motivate a so­ called reflection principle, from which he derives (drawing on earlier work by Paul Bernays) most of the axioms of ZFC set theory, as well as some large cardinal hypotheses. Although this is an impressive feat, Burgess ad­ mits that the motivation and the re­ sulting theory are no longer particu­ larly Fregean. Summing up, it is now clear, in a way it was not two decades ago, that a wealth of philosophical and techni­ cal insights can be rescued from the ruins of Frege's logicism. Whether these insights add up to a coherent and attractive philosophy of mathe­ matics is still (in my opinion) an open question. But Reason 's Proper Study and Fixing Frege are warmly recom­ mended as the best places to start for an examination of, respectively, the philosophical and technical insights to be learnt from Frege and the prospects for a neo-Fregean philosophy of math­ ematics. Department of Philosophy University of Bristol 9 Woodland Rd Bristol BS8 1 TB UK e-mail: oystein . [email protected]

86

THE MATHEMATICAL INTELLIGENCER

Letters to a Young Mathematician by Ian Stewart NEW YORK, BASIC BOOKS, 2006. HARDCOVER US $22.95 ISBN: 9780465082315

REVIEWED BY REUBEN HERSH

Dear grandchildren David, Jessica, and Ze'ev, As is customary, I list you in chrono­ logical order, by your year of birth. As all of you probably know, your grand­ father in New Mexico is a retired math professor. If it should some day hap­ pen, by some good book or some good teacher, that one or two of you get turned on to math, I'll be more than ea­ ger to help you, with my information and my advice. In all fairness then, I am letting you know that the new book by my friend Ian Stewart already offers such information and advice. Therefore, for you three, and also for any other potential readers of this prestigious pe­ riodical, I am going to tell you about Stewart's book. First I'll tell you about his informa­ tion. Then I'll tell you about his advice. And finally, I'll tell you my own advice. The book consists of 21 letters to "Meg . " The quotation marks inform us, I suppose, that "Meg" is imaginary or fictitious. In the first letter "Meg" is "at school." Since Ian doesn't hesitate to speak seriously and deeply to "Meg", I guess she's already in what we here in the States call "high-school." Meg is wondering what mathematicians do, and how her Uncle (??) Ian became a mathematician. As time passes, Meg chooses to study math(s) at University. By the last letter, she's concerned with what a tenure-track Assistant Professor is concerned with: the mad struggle to attain tenure. ("Tenure" is the college professors' word for "job security.) I am sure you , or anyone interested in mathematical life today, will find the "Letters" interesting and enjoyable. Stewart freely confides to Meg some of his own personal story, of how he was drawn to mathematics, and of some of his pleasures and successes as a math­ ematician. There is a really wonderful account of how an investigation into the abstract theory of groups turned out to

be of great use in analyzing the gait of four-footed creatures, like dogs! Who knew that there was even such an aca­ demic specialty as "Gait Studies"? Several of the chapter titles tell enough to make clear their messages: "The Breadth of Mathematics," "Hasn't it All Been Done?" "How Mathemati­ cians Think," "How to Learn Math," "Fear of Proofs," "Can't Computers Solve Everything?" "Impossible Prob­ lems. " Every sentence is clear and com­ prehensible. The love of mathematics that impels Stewart is always there; if the reader is susceptible at all, she or he may well become infected. Starting with Chapter 14, and going on to Chapter 20, the next to last, there is a noticeable change of tone and focus. "The Career Ladder," "Pure or Applied," "Where Do You Get Those Crazy Ideas?" "How to Teach Math," "The Mathemati­ cal Community," "Perils and Pleasures of Collaboration. " Ian is no longer talking to a child, sharing his enthusiasm and en­ joyment. He is talking to someone who is committed to mathematics, and is wor­ ried about how to make a living at it. Of course, this is a very realistic kind of conversation to imagine. In fact, many senior mathematicians, responsi­ ble for guiding advanced undergradu­ ates, graduate students, postdocs, and faculty just starting to teach, do have such conversations many times over their teaching careers. I imagine that while the "Meg" of the first half of the book is partly based on real acquain­ tance with school children, and partly a creation of Stewart's imagination, the "Meg" of the second part may well be an amalgam of many young mathe­ maticians Stewart has counseled. So what kind of advice does he give? I would say, sound and sober advice. Realist advice, how to get on in the math­ ematical world as it really is. (Meaning, of course, not necessarily as we would most, in our heart of hearts, desire it to be.) Stewart knows what's what, and he most kindly and sincerely wants Meg to make it, to get that job and that tenure. That means, knowing what hiring com­ mittees look for, and what promo­ tion and tenure committees look for,

etc.etc.etc. "THE REAL WORLD. " So, very good, what could be wrong with that? Nothing at all. And yet I can't help remembering a

young English mathematician I met years ago who did some crazy things. For instance, he wrote and illustrated COMIC BOOKS about fractals and chaos! And since, I guess, he couldn't get them published in English, he pub­ lished them in France, IN FRENCH! He even gave me copies of those two won­ derful works of his. That wasn't all that he did which some would have considered ill-ad­ vised. A "hot" new specialty in mathe­ matics appeared with a wonderful name: "Catastrophe Theory." My English friend became an active worker in this new field, and an active public advo­ cate of it, even though everyone knew that it was controversial. Many influen­ tial senior mathematicians disliked it, considered it a shallow fad, vastly over­ publicized because of its exciting name. Many of his older friends must have had doubts whether this was really the wis­ est career move he could make. Of course, he also did fine work in other noncontroversial specialties. But he also did something much more un­ wise. He knew, of course, that many re­ search mathematicians don't have the highest admiration or respect for jour­ nalists. And that to many people high up in mathematics, popular books that can be understood by anybody are hardly above journalism. And yet, what did he do but write lots of popular books! Over 20 are listed in the front matter of "Let­ ters to a Young Mathematician." I would say that the career of Ian Stewart is the career of a supremely suc­ cessful mathematician whose first con­ cern does not seem to have been "play­ ing it safe . " S o , now I a m ready t o offer m y own advice. I have tried to tell you about Stewart's advice. You may know, from experience or from hearsay, about a parent who admits that he isn't in all ways an ideal role model for his child. He may admit to some weaknesses or even vices. But, "Child," says he, "do as I say, not as I do . " I am enough of an optimist to look at things the opposite way. Read Stewart's book, enjoy it, hut do as he Does, not as he Says. 1 000 Camino Rancheros Santa Fe New Mexico 87505 USA e-mail: [email protected]

Funf M inuten Mathematik by Ehrhard Behrends WIESBADEN. FRIEDR. VIEWEG 2006, 256 PP,

&

SOHN VERLAG,

22.90, ISBN-10: 3834800821,

ISBN-13: 978-3834800824

REVIEWED BY VAGN LUNDSGAARD HANSEN

inner is served: One hundred well assorted German mathe­ matical "tapas·· online. Each of them takes five minutes to consume, and what an enjoyable five minutes! The recipes for the mathematical tapas are collected in a book by Ehrhard Behrends based on the first one hun­ dred of his columns ''Fiinf Minuten Mathematik," written for the German daily newspaper Die Welt in the years 2003 and 2004. The tapas are easily di­ gestible and well prepared, indeed. Behrends is an experienced popu­ lariser of mathematics. His columns in Die Welt have attracted much attention, and by reading his book you can see why. There is something for everybody with the slightest interest in mathemat­ ics in the book. And Behrends is not compromising with his presentation of mathematical results and methods. One gets real and valuable information about mathematics in small doses. This is, I think, a fruitful way of reaching out to the general public. Having read the col­ umn one week, I am sure that many readers have looked forward to seeing what fascinating mathematics the next column would bring the following week. It is a pleasure to notice that Behrends covers almost all subjects of mathemat­ ics: number theory, algebra, geometry, analysis, probability theory, stochastics, and pure as well as applied topics, like coding theory. The hook demonstrates that, with the right care, almost anything from mathematics can be presented so that a lay person can get some feeling for it. Space prevents me from describing the 100 columns in detail; I can only present a few of my own personal favourites. In a nice column on mathe­ matics and music, Behrends explains the Pythagorean and the chromatic tone scales, revealing his insights and strong

interest in music. His mathematical re­ search field is probability theory, and there are several interesting columns on topics from combinatorics and the the­ ory of games-among others, a column on mathematics and chess. I also liked columns on the old classics from geom­ etry about constructions with ruler and compass. In a column titled 'How much mathematics do human beings need?' Behrends shows that he is also prepared to take up this kind of discussion, which I suppose all mathematicians engage in from time to time. The book is attractively prepared by Vieweg, and several illustrations are in colour. The book would make a good gift for a lay person with an interest in mathematics. The hook is also valuable for the mathematician who needs ideas for the occasional, unprepared mathe­ matical conversations at dinner parties. Given the opportunity, why not offer your guests some mathematical tapas from Behrends' book "Fiinf Minuten Mathematik"? They will probably be sur­ prised hut they will enjoy it. And those five minutes spent with mathematics will truly be remembered! Department of Mathematics Matematiktorvet, Building 303 Technical University of Denmark DK-2800 Kgs. Lyngby Denmark e-mail: [email protected]

Images of a Complex World The Art and Poetry of Chaos by Robin Chapman and julien Clinton Sprott SINGAPORE, WORLD SCI ENTIFIC PUBLISHING CO. PTE. LTD., 2005, 175 PP. PLUS CD, $34.00 ISBN-13: 978-9812564016

REVIEWED BY JOHN PASTOR

he goal of both mathematician and poet is to seek clarity and beauty of expression about the world around us through elegant use of their respective languages. While nature

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 4, 2007

87

is a common source of inspiration for both mathematician and poet, the poet examines the human response to nature while the mathematician explores the logical order of nature. Despite these similarities, creative use of both mathematics and poetry together is uncommon. Several mathematicians have written poetry-]. ]. Sylvester and James Clerk Maxwell sometimes incor­ porated poems into their papers, al­ though these are now forgotten (perhaps with good reason) while their mathe­ matics continues to inform new research. Many of us have enjoyed the light verse of Ralph Boas, Jr. [1] and poetry has of­ ten graced the pages of 7be Mathemat­ ical lntelligencer. But few poets have dared to incorporate mathematical themes in their exploration of the hu­ man condition, although Anne Michaels has captured Kepler's life and thought superbly in her long first-person poem, A Lesson from the Earth [2), which be­ gins "I begged scraps from the Rudol­ phine Table-the rinds of orbits, stars scattering like pips spat from Tycho's chewing mouth . . . " and continues with " . . . We must learn this lesson from the earth, that the greater must make room for the small, just as the earth attracts the smallest stone . . . " and " . . . I used to think that we escape time by disap­ pearing into beauty. Now I see the op­ posite. Beauty reveals time." In Images of a Complex World, a poet and research psychologist (Chapman) and a physicist and dynamicist (Sprott), both at the University of Wisconsin, col­ laborate on exploring the beautiful world of dynamical systems and nature through poems, illustrations, and thumbnail es­ says. Although billed as an addition to your coffee table, this book really be­ longs in your classroom instead. By adding depth and dimension to many dynamical ideas and concepts, Chapman's poems enrich our and our students' understanding of them. The accompanying poem, Fixed Point, is from a set of poems titled Stillness (in the second column). The clarity and depth of her poem is not, of course, the pithy clarity and depth of :x! = f (:x!). But just as we teach our students how to unpack an equation to discover its hidden meaning, so does Chapman unpack the concept of a fixed point to uncover its hidden poetic beauty. The poems in this book also explore

88

THE MATHEMATICAL INTELLIGENCER

The Rxed Point The dot: how it stops everything. Finishes the thought. Ends t.h sentence. \\7h r everything vanishes in the end. P riod. l3u r it is not the all of it, though all come to it. It is only the idea of no di m ens i on over which \Ve exclaim, the van ishing point that lends the observer perspective.

a fiction of the eye too far away to see­ speck, mote. egg.

the human condition at the same time they illustrate mathematical ideas. How many of us have had difficulty ex­ plaining to students the meaning behind the property of nonlinear systems that f(x + y) =I= f(x) + j(y)? Here's Chap­ man's illustration of nonlinear systems, poignant not only for capturing the essence of this property but also for de­ scribing an all-too common condition of contemporary human existence:

Def 2: One in which does not equal f (x)

f

(x

+

+ f (y)

y)

This is easily enough understood By an} child of divorce­ .\l om's house And Dad's house are not the same

As the house with both Dad and

'\t om bdo rc . Or think off as h a ppine ss , And kn
Of course, the whole thrust of mathe­ matical teaching for the past 1 50 years or so has been to disembody mathe­ matics, to get away from this "human dimension, " but when used with dis­ cretion and taste as in the hands of skilled practitioners and teachers such as Chapman and Sprott, I see no harm and even much good in exploring the poetic dimension. Julien Sprott, an accomplished au­ thor [3) and scientific educator (see his website at http://sprott.physics.wisc. edu/), provides clear expositions of the mathematical concepts that are the springboards of the poems, serving as prose counterpoints to Chapman's verse, including entropy, state space, basins of attraction, and the three-body problem, among many others. Here is his explanation of hysteresis, in a sec­ tion called Time 's Arrow. "Hysteresis is a form of memory . . . . Hysteresis re­ quires a nonlinearity in the governing equations, and the nonlinearity is often in the form of a threshold where the behavior switches abruptly from one form to another as a parameter is changed (a bifurcation). " This neatly balances the first two lines of the next poem, called Hysteresis: "Whatever we expected would happen next I it wasn't this . . . " One could spend a mathe­ matical or poetic lifetime exploring the deep meaning behind these descrip­ tions. Sprott's other contribution to the book are the stunning illustrations of at­ tractors (strange and otherwise), iter­ ated function systems, and Julia sets. At this point, the reader might wonder why she or he needs to see yet more ex­ amples of these, but the cleanliness, clarity, and visual beauty of these illus­ trations are truly marvelous. All of the attractors are in color, with the color de­ picting the value in the z-dimension, and most are on a white background, which gives them a crispness lacking in many other books. Because many non­ mathematicians find the unstructured appearance of strange attractors difficult to appreciate, Sprott introduces sym­ metry into some of these illustrations by an interesting transformation of the x and y coordinates to polar coordinates, a technique which he has previously published in [4). This gives the attractor image the shape of a wedge, and sev­ eral wedges are then assembled into a

radially symmetric image, like the arms of snowflakes or bursts of fireworks. I suspect that underlying these polar transformations there may be some in­ teresting ideas in the mathematics of symmetry which may make a nice stu­ dent project or undergraduate thesis. There is an Appendix for the Mathe­ matically Inclined that discusses these and other mathematics behind the im­ ages. This collection of Chapman's poems with Sprott's visual art and mathemati­ cal expositions is a welcome compila­ tion and nicely shows the breadth of their work, both separately and in col­ laboration. The book comes with a CD with these and many other images,

� Springer

the language of science

readings of some of the poems by Chapman herself, weblinks, and a vari­ ety of other classroom resources. Im­ ages of a Complex World should set a standard for collaborations between mathematicians or scientists and artists: indeed, the line separating which au­ thor is a scientist from which is an artist is often blurred in this book. Buy it or have your library purchase it, share it with your students, and dip into it when your creative pump needs priming.

The Mathematical Association of America, Dolciani Mathematical Expositions, Volume 1 5, Washington, DC, 1 995. [2] Anne Michaels. Poems. Alfred A. Knopf, New York, 200 1 . (3) Julien Clinton Sprott. Chaos and Time-Se­ ries Analysis . Oxford University Press, Ox­

ford, 2003. [4] Julien

Clinton Sprott. Strange attractor

symmetric icons. Computers & Graphics 20: 325-332, 1 996.

REFERENCES

Department of Biology

[1 ] Gerald Alexanderson and Dale Mugler, ed­

University of Minnesota Duluth

itors. Lion Hunting & Other Mathematical

Pursuits.

A Collection of Mathematics,

Verse, and Stories by Ralph P. Boas, Jr.

Duluth, MN 558 1 2 USA e-mail: jpastor@d . umn.edu

springer.com

Quick, what's the best way to keep up o date with new developments ur field? The answer i s SpringerAlerts! As an alerts subscriber you'll get ..,. Reliable "first-word" on upcoming books and journal content ..,. Free e-book downloads ..,. Excl usive trial access to journals

SpringerAlerts - it's fast, it's free, and best of all it's flexible springer.com/springeralerts 011759x

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 4, 2007

89

41fl.,j.M9.h.J§I

Robin Wi l s o n

M athe matics i n I nd ian P h i l ate l y ANIL NAWLAKHE

I

stamp depicts the logo of the Games, featuring Jantar Mantar, permanent as­ tronomical instruments of huge dimen­ sion, constructed by the Indian Ma­ harajah]ai Singh ( 1 686-1743). The logo also incorporates the sun, the symbol of the Asian Games.

The Great Trigonometrical Survey: Starting in 1802 Colonel William Lamb­ ton carried out a trigonometrical survey of the Indian peninsula in the vicinity of Madras. Initially planned for a short arc, it later grew in scale, marking it as one of the most ambitious and adven­ turous of scientific undertakings. Accu­ rate measurement of the Himalayas ad­ vanced our knowledge of the shape of our planet by producing new values for the curvature of the earth's surface. Rad­ hanath Sikdar joined the survey at Mus­ sourie at the age of 2 1 . He calculated the height of Mount Everest and pre­ pared the first edition of auxiliary tables. Nain Singh, an intrepid schoolteacher, made an invaluable contribution, sur­ veying Tibet; the 2000-km trade route took 3 1 latitude measurements, and he determined elevations of 33 places. He followed the course of the great Tibetan river, the Tsangpo, for 800 km and even­ tually proved that the Tsangpo and Brahmaputra rivers are the same.

HURl: In Islam, there are historical reasons for adopting AD 622 as the commencement of the Hijri calen­ dar. In Arabic, Hijra means migration. The Holy Prophet of Islam, Hazrat Muhammad migrated to Medina in AD 622, and the migration is a sig­ nificant event in the spread of Islam; the Hijri calendar, based on twelve lu­ nar months, commences from this date. The stamp was issued to com­ memorate the 1 4th century of the Hi­ jri era.

Department of Physics, J. M. Patel Co llege , Bhandara -441 904,

jantar Mantar: In 1982 India commemorated the IX Asian Games. The

M.S. India e-mail: [email protected]

'h
The Great Trigonometrical Survey

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

The Open Un iversity, M i lton Keynes, MK7 6AA, England e-ma i l : [email protected] k

92

Jantar Mantar

THE MATHEMATICAL INTEWGENCEA Ct:l 2007 Springer Science +Business Media. Inc.

HIJRI

utvey