'·'·"·"·'·'

Thoughts on the Riemann Hypothesis G. J. Chaitin

The Opinion column offers

mathematicians the opportunity to

write about any issue of interest to

the international mathematical

community. Disagreement and

controversy are welcome. The views

and opinions expressed here, however,

are exclusively those of the author,

and neither the publisher nor the

editor-in-chief endorses or accepts

responsibility for them. An Opinion

should be submitted to the editor-in­

chief, Chandler Davis.

The simultaneous appearance in May 2003 of four books on the Rie­

mann hypothesis (RH) provoked these reflections. I briefly discuss whether the RH should be added as a new ax­iom, and whether a proof of the RH might involve the notion of random­ness.

New Pragmatically Justified

Mathematical Axioms that Are

Not at All Self-evident

A pragmatically justified principle is one that is justified by its many important consequences-which is precisely the opposite of normal mathematical prac­tice.1 However, this is standard operat­ing procedure in physics.

Are there mathematical proposi­tions for which there is a considerable amount of computational evidence, ev­idence that is so persuasive that a physicist would regard them as exper­imentally verified? And are these propositions fruitful? Do they yield many other significant results?

Yes, I think so. At present, the two best candidates2 for useful new axioms of the kind that GOdel and I propose [ 1) that are justified pragmatically as in physics are:

• the P =I= NP hypothesis in theoretical computer science that conjectures that many problems require an ex­ponential amount of work to resolve, and

• the Riemann hypothesis concerning the location of the complex zeroes

of the Riemann zeta function

?cs) =I_!_= II -1 -. n ns p 1 - _l_

pS

(Here n ranges over positive inte­gers and p ranges over the primes. )3 Knowing the zeroes of the zeta function, i.e., the values of s for which ?(s) = 0,

tells us a lot about the smoothness of the distribution of prime numbers, as is ex­plained in these four books:

• Marcus du Sautoy, The Music of the Primes, Harper Collins, 2003.

• John Derbyshire, Prime Obsession, Joseph Henry Press, 2003.

• Karl Sabbagh, The Riemann Hy­pothesis, Farrar, Strauss and Giroux, 2003.

• Julian Havil, Gamma, Princeton Uni­versity Press, 2003.4

The Riemann zeta function is like my n number: it captures a lot of in­formation about the primes in one tidy package. n is a single real number that contains a lot of information about the halting problem. 5 And the RH is useful because it contains a lot of number -the­oretic information: many number-theo­retic results follow from it.

Of the authors of the above four books on the RH, the one who takes Godel most seriously is du Sautoy, who has an entire chapter on Godel and Turing in his book In that chapter on p. 181, du Sautoy raises the issue of whether the RH might require new ax-

'However, new mathematical concepts such as v'=1 and Turing's definition of computability certainly are

judged by their fruitfu lness-Fran(:oise Chaitin-Chatelin, private communication.

2Yet another class of pragmatically justified axioms are the large cardinal axioms and the axiom of determi­

nancy used in set theory, as discussed in Mary Tiles, The Philosophy of Set Theory, Chapters 8 and 9. For

the latest developments, see Hugh Woodin, "The continuum hypothesis," AMS Notices 48 (2001 ), 567-576,

681 -690.

3\fou start with this formula and then you get the full zeta function by analytic continuation.

4Supposedly Havil's book is on Euler's constant y, not the RH, but ignore that. Sections 1 5.6, 1 6.8, and 1 6. 1 3

of his book are particularly relevant t o this paper.

5!1 = 'lp ha�s 2-iol is the halting probability of a suitably chosen universal Turing machine. !1 is "incompress­

ible" or "algorithmically random." Given the first N bits of the base-two expansion of !1, one can determine

whether each binary program p of size io1 ,; N halts. This information cannot be packaged more concisely. See

[2], Sections 2.5 through 2.1 1 .

4 THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

ioms. On p. 182 he quotes Godel,6 who specifically men­

tions that this might be the case for the RH. And on p. 202

of that chapter, du Sautoy points out that if the RH is un­

decidable this implies that it's true, because if the RH were

false it would be easy to confirm that a particular zero of

the zeta function is in the wrong place.

Later in his book, on pp. 256-257, du Sautoy again

touches on the issue of whether the RH might require a

new axiom. He relates how Hugh Montgomery sought re­

assurance from Godel that a famous number-theoretic con­

jecture-it was the twin prime conjecture, which asserts

that there are infinitely many pairs p, p + 2 that are both

prime-does not require new axioms. Godel, however, was

not sure. In du Sautoy's words, sometimes one needs "a

new foundation stone to extend the base of the edifice" of

mathematics, and this might conceivably be the case both

for the twin prime conjecture and for the RH.

On the other hand, on pp. 128-131 du Sautoy tells the

story of the Skewes number, an enormous number

wwlo34

that turned up in a proof that an important conjecture must

fail for extremely large cases. The conjecture in question

was Gauss's conjecture that the logarithmic integral

Li(x) = Jx du 2 ln u

is always greater than the number 1r (x) of primes less than

or equal to x. This was verified by direct computation for

all x up to very large values. It was then refuted by Little­

wood without exhibiting a counter-example, and finally by

Skewes with his enormous upper bound on a counter­

example. This raised the horrendous possibility that even

though Gauss's conjecture is wrong, we might never ever see a specific counter-example. In other words, we might

never ever know a specific value of x for which Li(x) is less

than 1r(x). This would seem to pull the rug out from under

all mathematical experimentation and computational evi­

dence! However, I don't believe that it actually does.

The traditional view held by most mathematicians is that

these two assertions, P =I= NP and the RH, cannot be taken

as new axioms, and cannot require new axioms, we simply

must work much harder to prove them. According to the

received view, we're not clever enough, we haven't come

up with the right approach yet. This is very much the cur­

rent consensus. However, this majority view completely ig-

no res 7 the incompleteness phenomenon discovered by

Godel and Turing, and extended by my own work [2] on in­

formation-theoretic incompleteness. What if there is no

proof?

In fact, new axioms can never be proved; if they can,

they're theorems, not axioms. So they must either be justi­

fied by direct, primordial mathematical intuition, or prag­matically, because of their rich and important consequences,

as is done in physics. And in line with du Sautoy's observa­

tion, one cannot demand a proof that the RH is undecid­

able before being willing to add it as a new axiom, because

such a proof would in fact yield the immediate corollary

that the RH is true. So proving that the RH is undecidable

is no easier than proving the RH, and the need to add the

RH as a new axiom must remain a matter of faith. The

mathematical community will never be convinced. 8 Someone recently asked me, "What's wrong with calling

the RH a hypothesis? Why does it have to be called an ax­

iom? What do you gain by doing that?" Yes, but that's be­

side the point; that's not the real issue. The real question

is, Where does new mathematical knowledge come from?

By "new knowledge" I mean something that cannot be

deduced from our previous knowledge-from what we al­

ready know.

As I have been insinuating, I believe that the answer to

this fundamental question is that new mathematical knowl­

edge comes from these three sources:

a. mathematical intuition and imagination ( \!=]\ b. conjectures based on computational evidence (explains

calculations), and

c. principles with pragmatic justification, i.e., rich in con-

sequences (explains other theorems). 9

And items (b) and (c) are much like physics, if you replace

"computational evidence" by "experimental evidence." In

other words, our computations are our experiments; the

empirical basis of science is in the lab, the empirical basis

of math is in the computer.

Yes, I agree, mathematics and physics are different, but

perhaps they are not as different as most people think, per­

haps it's a continuum of possibilities. At one end, rigorous

proofs, at the other end, heuristic plausibility arguments,

with absolute certainty as an unattainable limit point.

I've been publishing papers defending this thesis for

more than a quarter of a century, 10 but few are convinced

by my arguments. So in a recent paper [ 1] I've tried a new

6Unfortunately du Sautoy does not identify the source of his Gbdel quote. I have been unable to find it in Gbdel's Collected Works.

7 As du Sautoy puts it, p. 1 8 1 , "mathematicians consoled themselves with the belief that any1hing that is really important should be provable, that it is only tortuous

statements with no valuable mathematical content that will end up being one of Gbdel's unprovable statements."

8The situation with respect to P * NP may be different. In a paper "Consequences of an exotic definition for P = NP," Applied Mathematics and Computation 145 (2003), pp. 655-665, N. C. A. da Costa and F. A. Doria show that if ZFC (Zermelo-Fraenkel set theory+ the axiom of choice) is consistent, then a version of P = NP is consistent with ZFC, so a version of P * NP cannot be demonstrated within ZFC. See also T. Okamoto, R. Kashima, "Resource bounded unprovability of compu­

tational lower bounds," http://eprint. iacr.org/2003/1 87 /.

9A possible fourth source of mathematical knowledge is (d) probabilistic or statistical evidence: A mathematical assertion may be deemed to be true because the prob­

ability that it's false is immensely small, say <1 0-99999

Here is a practical example of this: The fast primality testing algorithm currently used in Mathematica does not necessarily give the correct answer, but mistakes

are highly unlikely. Algorithms of this sort are called Monte Carlo algorithms.

10See, for example, the introductory remarks ·In my 1 974 J. ACM paper [3].

VOLUME 26, NUMBER 1 , 2004 5

tactic. I use quotes from Leibniz, Einstein, and Godel to

make my case, like a lawyer citing precedents in court ... .

Even though I am touting the Riemann hypothesis as an

excellent new-axiom candidate-whether Godel agrees or

merely thinks that a new axiom might be needed to prove

the RH, I'm not sure-let me briefly wax enthusiastic over

a possible approach to a proof of the RH. Disclaimer. I'm

not an expert on the RH. What I'm about to relate is defi­

nitely an outsider's first impression, not an expert opinion.

A Possible Attack on the Riemann Hypothesis?

Here is a concrete approach to the RH, one that uses no

complex numbers. It's a probabilistic approach, and it in­

volves the notion of randomness. It's originally due to Stielt­

jes, who erroneously claimed to have proved the RH with

a variant of this approach.

The Mobius JL function is about as likely to be + 1 or - 1

(see Derbyshire, Prime Obsession, pp. 322-323).

{ 0 if k2 divides n, k > 1, f.L(n) = b rct·rr · ct· · r ( _ 1 )num er o 1 erent pnme tvtsors o n if n is square-free. The RH is equivalent to the assertion that as k goes from 1

to n, JL(k) is positive as often as negative. More precisely,

the RH is closely related to the assertion that the difference

between

• the number of k from 1 ton for which JL(k) = -1, and

• the number of k from 1 to n for which JL(k) = + 1

is O(Vn\ of the order of square root of n, i.e., is bounded

by a constant times the square root of n. This is roughly

the kind of behavior that one would expect if the sign of

the JL function were chosen at random using independent

tosses of a fair coin. u

This is usually formulated in terms of the Mertens func­

tion M(n): 12

n M(n) = .2: p.,(k).

k�l

According to Derbyshire, pp. 249-251,

M(n) = O(Vn) implies the RH, but is actually stronger than the RH. The

RH is equivalent to the assertion that for any E > 0, I

M(n) = O(n2+E).

Could this formula be the door to the RH?!

This probabilistic approach caught my eye while I was

reading this May's crop of RH books.

I have always had an interest in probabilistic methods

in elementary number theory. This was one of the things

that inspired me to come up with my definition of algo­rithmic randomness and to find algorithmic randomness

in arithmetic [6] in connection with diophantine equations.

However, I doubt that this work on algorithmic random­

ness is directly applicable to the RH.

In particular, these two publications greatly interested

me as a child:

• Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Mathematical

Monographs, vol. 12, Mathematical Association of Amer­

ica, 1959.

• George P6lya, "Heuristic reasoning in the theory of num­

bers," 1959, reprinted in Gerald W. Alexanderson, The Random Walks of George P6lya, Mathematical Associa­

tion of America, 2000.

I think that anyone contemplating a probabilistic attack on

the RH via the JL function should read these two publica­

tions. There is also some interesting work on random

sieves, which are probabilistic versions of the sieve of

Eratosthenes:

• D. Hawkins, "Mathematical sieves," Scientific American, December 1958, pp. 105-112.

As P6lya shows in the above paper-originally Ameri­can Mathematical Monthly 66, pp. 375-384-probabilistic

heuristic reasoning can do rather well with the distribution

of twin primes. By the way, this involves Euler's y constant.

Can a refmement of P6lya's technique shed new light on JL and on the RH? I don't know, but I think that this is an in­

teresting possibility.

By the way, P :1: NP also involves randomness, for as

Charles Bennett and John Gill showed in 1981-SJAM Jour­nal on Computing 10, pp. 96-113-with respect (relative)

to a random oracle A, pA :1: NPA with probability one [7].

Further Reading-Four "Subversive" Books

• On experimental mathematics:

Borwein, Bailey, and Girgensohn, Mathematics by Ex­

periment, Experimentation in Mathematics, A. K. Pe­ters, 2003.

(See [8]. There is a chapter on zeta functions in volume

two.)

• On a quasi-empirical view of mathematics:

Tymoczko, New Directions in the Philosophy of Math­ematics, Princeton University Press, 1998.

• On pragmatically justified new axioms and information­

theoretic incompleteness:

Chaitin, From Philosophy to Program Size, Tallinn Cy­

bernetics Institute, 2003.

(There is also an electronic version of this book [2].)

And regarding the adverse reaction of the mathematics

community to the ideas in the above books, I think that it

is interesting to recall Godel's difficulties at the Princeton

Institute for Advanced Study, as recounted in:

11 For a more precise idea of what to expect if the sign of the IL function were chosen at random, see the chapter on the law of the iterated logarithm in Feller, An In­

troduction to Probability Theory and Its Applications, vol. 1 , Vlll.5 through VII I .?. 12See [4, 5].

6 THE MATHEMATICAL INTELLIGENCER

• John L. Casti, The One True Platonic Heaven, John

Henry Press, 2003.

According to Casti, one of the reasons that it took so long

for Godel's appointment at the lAS to be converted from

temporary to permanent is that some of Godel's colleagues

dismissed his incompleteness theorem. Now, of course,

Godel has become a cultural icon13 and mathematicians

take incompleteness more seriously-but perhaps not seri­

ously enough.

Mathematicians shouldn't be cautious lawyers-! much

prefer the bold Eulerian way of doing mathematics. Instead

of endlessly polishing, how about some adventurous pioneer

spirit? Truth can be reached through successive approxi­

mations; insistence on instant absolute rigor is sterile­

that's what I've learned from incompleteness.14

WEB REFERENCES ( 1 ] Two philosophical applications of algorithmic information theory.

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/dijon.html

[2] From philosophy to program size.

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ewscs.html

[3] Information-theoretic limitations of formal systems. http://www.

cs.auckland .ac. nz/CDMTCS/chaitin/acm 7 4. pdf

[4] Mertens function. http://mathworld.wolfram.com/

MertensFunction .html

[5] Mertens conjecture. http://mathworld .wolfram.com/

MertensConjecture.html

[6] Randomness in arithmetic. http://www.cs.auckland.ac.nz/

CDMTCS/chaitin/sciamer2. html

(7] Relative to a random oracle A, pA * NPA * co-NPA with proba­

bility 1 . http://www.research.ibm.com/people/b/bennetc/

bennettc1 981 497f3f4a.pdf

[8] Experimental mathematics website. http://www.expmath. info

[9] Apostolos Doxiadis home page. http://www.apostolosdoxiadis.com

IBM Research

Yorktown Heights, NY 1 0598

USA

e-mail: chaitin@watson.ibm.com

131n this connection, I should mention Incompleteness, a play and a theorem by Apostolos Doxiadis, which is a play about Gbdel. For more information, see [9]. 141n this connection, see da Costa and French, Science and Partial Truth, Oxford University Press, 2003.

Solution Kept Secret

VOLUME 26, NUMBER 1 , 2004 7

EUGENE GUTKIN

The Toeplitz-Hausdorff Theorem Revisited: Relating Linear Algebra and Geometry

Genesis

In the beautiful paper [24] 0. Toeplitz associated with any

complex n X n matrix a compact set in the complex plane.

As his title suggests, he was inspired by a theorem of L. Fe­

jer [6] concerning a relationship between planar curves and

Fourier series. Apart from this, the paper [24] is self­

contained. Let en be the standard vector space with the

scalar product <u, v>. I will not distinguish between the

n X n matrices and operators on en. Let C be one such. It is determined by its "bilinear" form <u, Cv>. The compact

set that Toeplitz introduces is the image, W = W(C) C e, of the unit sphere in en, under the quadratic map u �

<u, Cu>. He cof\jectures that W(C) is a convex set, and

proves that the outer boundary of W(C) is a convex curve.

A year later F. Hausdorff proved Toeplitz's col\iecture

[12]. The Toeplitz-Hausdorff theorem was born. For several

reasons, it continues to attract the attention of researchers.

Extensions of Toeplitz's setting came up in robust control;

hence the thriving engineering literature on the subject.

See [20, 21, 5]. My own preoccupation with the Toeplitz­

Hausdorff theorem has its genesis in a joint project with

electrical engineers [15, 10]. Despite (or because of) the simplicity of the Toeplitz­

Hausdorff framework, basic questions in the subject remain

open [14]. For instance, it is not known what domains are

realizable as W(C) for C on en. The present article aspires

to attract attention of the general mathematical readership

to the fascinating interplay of linear algebra, geometry, and

analysis that the papers [24, 12] initiated.

My plan is as follows. I analyze in some detail the original

papers of Toeplitz and Hausdorff. Then, following the view­

point of [24], I associate with an arbitrary C a linear pencil

of hermitian operators H( · ) . This allows me to cast the analy­

sis into the language of convex geometry: Support lines and

support functions come in. The crucial observation is that the

support function of W(C) is the highest eigenvalue, A(·), of

H( · ) . This brings in both the algebraic geometry and the con­

vex geometry. R. Kippenhahn was the first to exploit this ob­

servation. In his Dissertation [16] he introduces and develops

this point of view. To illustrate this approach, I immediately

derive rough bounds on the size of W( C) in terms of the spec­

tral attributes of C. I also reproduce without proof the much

more sophisticated estimates of Kippenhahn [16]. Then I bring in the differential geometry by calculating

the curvature of the boundary curve aW(C). To show the

usefulness of this viewpoint, I apply it to obtain new bounds

on the size of W( C) in terms of the standard attributes of

C. These estimates, although still very crude, are sharper

than those I got out of the support function. The differen­

tial geometry viewpoint turned out to be especially suitable

to study the multidimensional version of W( C), the joint nu­

merical range [10].1 I conclude with a brief survey of the

literature and a personal remark

I thank the referees for helpful comments.

Historical Remarks

Toeplitz coined the name "Wertvorrat" for W(C). A literal

English translation is the value supply or the stock of val­ues. Variations of "Wertvorrat" dominate the German liter­

ature on the subject. For instance, A. Wintner, during the

Leipzig period of his prolific career, used the expressions

"Wertevorrat" (values supply) and "Wertbereich" (value domain) [26].2

The modern literature intermittently uses field of values

1There are many generalizations of the numerical range of an operator in the modern literature. It would take several pages just to give the relevant definitions. The con­

cept of the joint numerical range and the awareness that it is the natural multi-dimensional extension of the numerical range is already in the founding papers [24, 1 2).

2Wintner emigrated to America shortly after the University of Leipzig refused to award hirn the Habilitation. The book [26) is apparently his Habilitationschrift.

8 THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

and numerical range. s I don't like either expression. The

former adds one more item to the litany of mathematical "fields"; the latter is plain awkward. The original name is

better in every respect except one: It is German and there­fore unacceptable in the English literature. 4 Some proposed alternatives (template, form range, contracted graph) did not fly. I fmd the expression numerical range the lesser of two evils, and I will use it in what follows. 5

Toeplitz proves several propositions relating W( C) and the spectrum of C. For instance, he shows that W(C) con­tains the spectrum, and if C is a normal operator, then W( C)

is the convex hull of the spectrum. But the centerpiece of [24] is "Satz 8," the convexity of the outer boundary.

The penultimate §5 of [24) offers several informal com­ments, and points out the difference between convexity of the outer boundary and convexity of the set. Then Toeplitz says: "I will now discuss a generalization of the entire set­ting, which . . . also shows the difficulties that stem in the general case from the possibility of holes." He goes on to introduce what is now called the joint numerical range of any number q of hermitian operators A1, .. . , Aq. The set in question, W.(A1, . .. , Aq) c !Rq, is the image of the unit sphere in en, under the map u f-i> ( <u, A1u>, . . . <u, Aqu> ). The decomposition C = A1 + iA2 implies W(C) = W.(A1,A2).6 Toeplitz demonstrates that W.(Al, . . . , Aq) is not convex, in general. He concludes: "Whether this can al­ready happen for q = 2 remains possible, athough unlikely."

Toeplitz missed that he actually proved the desideratum! Indeed, to a modem reader, it seems that Toeplitz essen­tially settled the convexity conjecture. To us, it suffices to prove it for n = 2; for, if <u, Cu> and <v, Cv> belong to W(C), and the numerical range of the restriction of the form C to eu + ev is convex, then the claim holds. And in §5

Toeplitz shows that the numerical range of an operator on e2 is either an elliptic disc, or a segment, or a point-in each case, it is convex! In fact, this is how the Toeplitz­Hausdorff theorem is proved in modem textbooks [9, 11, 14].1 Amazingly, in the 80-some years since [24), nobody, including Hausdorff, noticed that the Toeplitz-Hausdorff theorem is implicitly proved in [24).

In the 3-page-long, focused, beautiful paper [ 12), Haus­dorff proves Toeplitz's conjecture. On the one hand, he proves it from scratch, without using Satz 8 of [24). On the other, he goes just a step further than Toeplitz to show that the intersection of W(C) with any straight line is the image of a connected subset of the unit sphere under a continu­ous mapping, and hence is connected. 8 In a one-sentence remark Hausdorff points out that his results and the Toeplitz argument combine to prove the convexity of the

outer boundary of the surface W.(Al, A2, As) for any triple

of hermitian operators.

A natural generalization of the Toeplitz-Hausdorff theo­rem would have been the convexity of W.(A1, . . . , Aq) for all hermitian operators on any en. Although this claim is ''very false" [ 1 1), W.(A1, A2, As) for any triple A1. A2, As on en is convex if n 2:: 3. Remarkably, it was established 60 years after the papers [24, 12]! There are several proofs of this in the literature [10], and some are based on the Haus­dorff connectedness idea [5] . The convexity claim for W.(Al, A2, As, A 4) for operators on en fails for any n [5]. Although this is unfortunate from the engineering viewpoint [21], there

are nontrivial interpretations of this "phase transition" [10].

But let us return to the subject. How could it be that nei­ther Toeplitz nor Hausdorff realized that [24] contained a proof of the convexity of the numerical range? It is quite likely that Hausdorff overlooked the relevant part of [24] .

However, the Commentary by S. D. Chatterji in Hausdorffs Collected Works [ 13] reveals a curious fact in this respect. The Hausdorff Archives in Greifswald contain two hand­written notes for [ 12), dated September 19 and October 12

of 1918. In one of them Hausdorff works out the numeri­cal range of any two-by-two matrix. He shows, as Toeplitz had already done, that it is a (possibly degenerate) ellipse.

Bringing in the Geometry

My interpretation of the approach of [24] is as follows. Let C be an n X n matrix, and let W(C) be the numerical range. Toeplitz associates with C a linear pencil of hermitian op­erators H( · ), parametrized by the circle of directions. The highest eigenvalue, A ( · ), of H( · ) is the support function of W(C). I will now explain this in detail.

Let <u, v> denote the standard scalar product on en, linear (resp. antilinear) in the second (resp. first) argument. As usual, I lull = V <u, u>. Let C be an operator on en with the adjoint C*, and let

C =A+ iB: A*= A, B* = B (1)

be the decomposition into hermitian operators. For 0 :S t < 27T set

1 . . H(t) = 2 [e-'tC + e'tC*] = (cos t)A + (sin t) B. (2)

The space of rays (i.e., oriented lines) in !R2 is parame­trized by S1 X IR [22]. Namely, the ray r(t, p) has direction t, and the signed distance p from the origin. The notion of sup­port lines is well known [1, 22]. I will associate with any compact set, XC IR2, the family, u(t), 0 :S t < 27T, of its support rays. For any 0 :S t < 27T the set of p E IR such that

3See [9] for historical comments on this terminology. The name "numerical range" is due to M . H. Stone [23].

4The German-English hybrids "eigenvalue, eigenvector" are the fortunate exceptions .. . . I don't know who coined them or how, but I am happy that I don't have to

use the awkward "proper value, proper vector, characteristic number," etc.

51t could have been worse. F. D. Murnaghan refers to W(C) as " ... the region of the complex plane covered by these values under the hypothesis that .. . " [1 8].

6Thus, the patent on the joint numerical range belongs to Toeplitz and not to Hausdorff [5].

7 A proof of the Toeplitz-Hausdorff theorem based on this idea is due to W. F. Donoghue [4). He explicitly calculates the ellipse in question. An elegant calculation of

aW(C) if n :s 3 is due to Murnaghan [1 8). Although he points out that aW(C) is an ellipse when n = 2, Murnaghan is not concerned with the region W(C) itself.

8Hausdorff's elegant argument is limited to finite dimensions, because he diagonalizes hermitian operators. The extension of the Toeplitz-Hausdorff theorem to infinite

dimensions is due to Stone [23). See [1 1 ) for a proof of N. P. Dekker [3) that combines Hausdorff's idea with the reduction to IC2

VOLUME 26, NUMBER 1, 2004 9

<3:>

Figure 1 . Support rays and the eigenvalues.

r( t, p) intersects X is compact; let p( t) be the maximal such p. Then a{t) = r(t,p(t)) is the support ray of X in ilirection t.

The following proposition is essentially Satz 8 of [24].

Proposition 1. Let C = A + iB be an operator on en and let H(t) = (cos t)A + (sin t)B, 0 :o; t < 27T, be the associated pencil of hermitian operators. Let

(3)

be the eigenvalues of H(t), and let Ei(O) c en be the eigen­space9 corresponding to Ai(O).

Let u(t), 0 ::5 t < 27T, be the support rays of W(C). Then the intersection point of a(t + 7T/2) with r(t,O) is A1(t)(cos t, sin t). Using this point as the origin in u(t + 7T/2), iden­tify a{t + 7T/2) with R Then a{t + 7T/2) n W(C) C IRis the convex hull of the spectrum of the form H(t + 7T/2) re­stricted to E1(t).

Proof The unit circle acts on operators, C � e-iac, and on e, by rotations. The statement is equivariant with respect to these actions. Therefore, it suffices to verify the claims for the direction t = 0. We have H(O) =A, H(7T/2) = B, the ray r(O, 0) is the x-axis, and u( 7T/2) is the vertical ray sup­porting W from the right. See Figure 1. The points z = x + iy of the numerical range have the form z = <u, Cu>, llu ll = 1. By (1), x = <u, Au>, y = <u, Bu>. Therefore, the projection of Won the horizontal axis is the interval [An(A), A1(A)]. The right extremity of this interval is the intersec­tion point with the ray lT( 7T/2). This proves one claim.

The intersection of lT( 7T/2) with W is given by

z = {<u, Au> + i <u, Bu>: llull = 1, <u, Au>= A1(A)}.

9Another fortunate hybrid!

10 THE MATHEMATICAL INTELLIGENCER

y

u(i)

1(A)

In view of the above, our subset of IR is formed by <u, Bu>, where u runs through the unit sphere in E1(A). The numerical range of an hermitian operator is the con­vex hull of its spectrum. This proves the other claim. •

Proposition 1 has several far-reaching consequences. First of all, it implies that the outer boundary aW(C) is con­vex [24]. Second, it describes the support rays of W(C) via the eigenvalues of the hermitian pencil H( · ) of (2). These support rays determine the convex hull of aW(C). Since W(C) is convex, as we now know, they determine the set W(C) itself. Thus, Proposition 1 yields a description of the numerical range of C in terms of the spectrum of the as­sociated pencil H(·).

Since the publication of [24], many authors have devel­oped this observation in several directions. One of these directions may be called algebra-geometric. Its starting point is the algebraic curve

det(xA + yB + zi) = 0. (4)

This paper exploits another direction, which may be called "proper geometric." It takes off with an immediate corol­lary of Proposition 1. To formulate it, I will recall the no­tions of the support function and the width function of a convex set [1, 22]. Let X c lh£2 be convex and compact, and let lT(t), 0 :o; t :o; 27T, be the support rays of X. The distance between the parallel lines lT(t), lT(t + 1r) is the width of X in direction t. The support function is the signed distance of lT( t) to the origin. Denote the support and the width func­tions by h(t) and w(t), respectively. Then w(t) = h(t) + h(t + 7T).

Corollary 1. Let C be an operator on en, let H(·) be the associated pencil of hermitian operators, and let An(·) :S

· · · ::::; A 1 ( ·) be the eigenvalues of H(-). Then the support and the width of the numerical range of C are

h(t) = A1(t - 1r/2), w(t) = A1(t - 1r/2) - An(t - 1r/2). (5)

Proof. Proposition 1 yields the first claim. The second fol­lows from the first and the identity H( t + 1r) = -H( t). •

Although the Toeplitz paper [24] is the precursor of both geometric directions, it was the work of R. Kippenhahn [ 16] that explicitly gave birth to them. 1 0 From now on I will con­centrate on the proper geometric direction, referring the reader to the literature on the algebra-geometric direction. See, for instance [19].11

I will now use Corollary 1 to estimate the size of the nu­merical range of C in terms of the standard attributes of the operator C. The size of a planar convex compact set X is expressed via its area, diameter, breadth, and perimeter. Let w( ·) be the width of X. The breadth and the diameter of X are the minimum and the maximum of w, respectively. The perimeter and the area of X are also controlled by the width function [ 1]. If X = W( C), then w( ·) is determined by the spectrum of the hermitian pencil H( ·) which, in tum, is determined by the operator C. Among the standard at­tributes of C are its spectrum a( C) and the operator norm jcj. The number w(C) = maxAwCC)( IAi - A1} is the diameter of the spectrum.

For any a, b E I[ W(aC +b)= aW(C) + b.

Hence the size of the numerical range does not change un­der the transformations C � C + tl. Denote by Jtn the lin­ear space of operators on en, and let .M� c Mn be the sub­space of traceless operators. The function jCio = mintE<C jc + t� is a norm on the quotient space Jtn/{al}. The pro­jection Co = C - tr(C)In I induces a linear isomorphism of Mn/{al) and Ml Note that jcj0::::; jC0j, and for generic C the inequality is strict. The following very rough estimates are essentially contained in [24].

Corollary 2. Let C be an operator on IC", and let W be 'its numerical range. Then

AreaCW) ::::; 4lcl6, Perimeter(W) ::::; 8jCjo;

w(C) :S Diameter(W) :S 2jC!o,

(6)

(7)

Proof For 0::::; t::::; 21r let R(t) be the rectangle formed by the four supporting rays. See Figure 2. Since w(H(t)) = A1(t) - An(t) and jH(t)j = max{ jA1(t)j, jAn(t)j }, we have

Area(W)::::; 4jH(t)l·jH(t + 1r/2)j, Perimeter(W) ::::; 4IH(t)1 + 4jH(t + 7T/2)j.

Using that jH(-)j ::::; lei, and the invariance of the preceding argument under C � C + al, we obtain (6). The upper bound in (7) follows from w(H(t)) ::::; 2jH(t)i :S 2jCj and the invariance principle. The obvious fact that W contains the spectrum of C implies the lower bound. •

The roughness of the estimates in Corollary 2 occurs for two reasons, one geometric and one analytic. The geomet­ric reason is that W is much smaller than the circumscribed rectangles R(t). The analytic one is that the bounds A1(t) -An(t) :S 2jH(t)l ::::; 2jCj are very crude. Using convex geometry and subtle but elementary analysis, Kippenhahn obtained much better estimates [16]. Although the restriction tr C =

0 that [16] imposes, can be removed and the inequalities fur­ther improved, I will only state the relevant results of [16]. THEOREM 1. Let c be an operator on en such that tr c = 0, and let W = W(C) be its numerical range. Then

_!_ Vtr2(CC*) - itr C2j2::::; Area(W) ::::; n

(8) 2(n - 1)

Vtr2(CC*) - itr C2j2; n

4 Vn Vtr(CC*)

::::; Perimeter(W)::::; 21r � Vtr(CC*). (9)

The following corollary demonstrates the strength of Theorem 1. Corollary 3. The numerical range of a matrix C has empty interior iff C = a + bH, where H is hermitian.

I leave the proof to the reader, as an exercise. (Hint: Use (8) and the Cauchy-Schwarz inequality.) If C = a + bH, and H is hermitian, then C is normal. Hence, the numerical ranges of non-normal matrices have positive area.

Lest the reader think that the spectral properties of C matter only for the size of W(C), I hasten to add a few com­ments. Let X c IC be convex and compact. A point z E ax is a corner point12 if X has more than one support line at z. An eigenvalue A of C is normal if there exists an eigen­vector v with the eigenvalue A such that Cl'v = Av. The fol­lowing theorem [16, 4] gives an example of a completely different relation between the spectral characteristics of a matrix and the geometry of its numerical range. 1 3

THEOREM 2. Let c be an operator on en, and let w = W(C) be its numerical range. Then the corner points of W are among the normal eigenvalues of C.

Bringing in the Differential Geometry

An arbitrary convex compact W c I[ is determined by its support function. If the boundary a W is (piecewise) twice differentiable, then W is also determined by the curvature

1ilThe note [1 8] contains a few beautiful remarks about the algebraic geometry of iJW(C), but it does not pursue the matter.

11Complains about the scarcity of citations of Kippenhahn's work.

12Sharp point in the engineering literature. 1 3See [ 15] for a differential-geometric proof.

VOLUME 26, NUMBER 1 , 2004 1 1

y

u(t + 1r

Figure 2. Numerical range enclosed within a rectangle.

u(t

function, x(") 2: 0. As opposed to the support function, the

curvature is intrinsically defined by aw. The radius of cur­vature p(·) = x-1(·) is sometimes handier to use. Now let

W be the numerical range of a finite-dimensional operator,

C. Let H(·) be the corresponding pencil of hermitian oper­

ators. By Corollary 1, the support function of W is the high­

est eigenvalue A(·) of H(·). I will now express the radius of

curvature of aW in terms of A(·). A matrix is often called regular if its eigenvalues are

simple.

Definition 1. Let C be an operator on en, and let H(t), 0 :s t :s 27T, be the corresponding pencil of hermitian op­erators. Then C is Toeplitz regular if for all 0 :s t :s 27T the maximal eigenvalue of H(t) is simple.

THEOREM 3. Let W c 1Ri2 be the numerical range of an oper­ator C on en. Let H(·) be the associated pencil of hermit­ian operators and let A(·) be the maximal eigenvalue of H( · ).

Suppose that C is Toeplitz regular. Then the junction A(·) is infinitely differentiable, and A + A" > 0. The set W is strictly convex, the boundary aw is twice differentiable, and its radius of curvature satisfies

p(t + 7T/2) = A(t) + A"(t). (10)

Proof Denote by E(t) c en the eigenspace of H(t) corre­

sponding to the maximal eigenvalue. Let e E E(O) be a unit

vector. Then there is a unique vector function v(t), 0 :s t :s

27T, such that E(t) = Cv(t), llvCOII = 1, v(O) = e, and <v(t), v'(t)> = 0.14

141n general, v(21T) = {3v(O). The factor {3 has to do with Berry's phase.

1 2 THE MATHEMATICAL INTELLIGENCER

u(t + 37r/2)

By construction

H(t)v(t) = A(t)v(t). (11)

Differentiating this equation twice yields

and

(H' - A')v + (H - A)v' = 0 (12)

(H'- A")v + 2(H' - A')v' + (H - A)v" = 0. (13)

But H satisfies H' = -H. Substituting this into (13),

(A + A")v = 2(H' - A')v' + (H - A)v". (14)

Take the scalar product of (14) with v(t). Equation (12) im­

plies

A + A"= 2 <v', (A - H)v'> .

But A is the top eigenvalue of H and v' is perpendicular to

its eigenspace E(A). Hence

A + A"> 0. (15)

Denote by ;£ the ray family (O"(t), 0 :s t < 27T}, where O"(t) has direction t + 7T/2 and intersects r(t, 0) at the point

A(t)(cos t, sin t). The positivity condition (15) implies that

the envelope, A(;£) C C, is a strictly convex curve, with the

parametric equations

x(t) = A(t) cost- A'(t) sin t, y(t) = A(t) sin t + A'(t) cost. (16)

Moreover, A(:£) is twice differentiable, and its radius of cur­

vature is given by (10) [22, 1]. Since, by Proposition 1, :£ is

the family of support rays of W, we have A(:£) = aw. • Not every operator C on en is Toeplitz regular. If C is

normal, then W(C) is a polygon, hence it is not strictly con­

vex. By Theorem 3, normal matrices are not Toeplitz reg­

ular. In fact, by Theorem 2, the non-regularity of W(C) al­

ways has to do with a partial normality of C. Fortunately,

there are plenty of Toeplitz regular operators.

Proposition 2. The complement to the set of Toeplitz reg­ular operators in _Mn is contained in a closed hypersur­face. Proof. Let �n denote the space of n X n hermitian opera­

tors. By (1), _Mn =�nEB i�n. Replacing cost, sin t in (2) by

independent variables, we obtain an algebraic mapping, <p, from _Mn into the algebraic variety Gz(�n) of subspaces in

�n of dimension at most 2. The set of hermitian operators with multiple eigenval­

ues is an algebraic variety, Xn C �n, of codimension 3. Therefore, the set of L E G2(�n), such that L n Xn i= 0 is a codimension one subvariety, Yn C G2(�n). Since

<p : _Mn >---7 G2C�n) is surjective, the preimage <p-1(Yn) C _Mn

is a hypersurface. But the complement of the set ofToeplitz

regular operators belongs to <p -1(Yn). • The following is immediate from Proposition 2.

Corollary 4. The set of Toeplitz regular operators on en is open and dense.

I will now use Theorem 3 to sharpen the bounds on the

size of the numerical range. Moreover, I will do it for

bounded operators on any Hilbert space �. Recall that if

dim � = oo and C is a bounded operator on it, the numeri­

cal range W(C) c Cis bounded and convex [23], but not

necessarily closed. The operator norm C] and the reduced

operator norm ICio = mintEdC + til have the same basic

properties as in the case dim � < oo.

THEOREM 4. Let C be a bounded operator on any Hilbert space, and let W be its numerical range. Then

Perimeter(W) ::5 2'7TIC]o, Area(W) ::5 '7TIC]5. (17) Proof Let � be the Hilbert space where C acts. Assume

first that dim � < oo. Then � = en, and I will use the pre­

ceding material. Let H( t), 0 < t < 2'7T, be the corresponding

pencil of hermitian operators and let A(·) be the highest

eigenvalue of H(·). Suppose first that C is Toeplitz regular.

From Theorem 3 and standard differential geometry [1, 22] we have

127T Perimeter(W) = (A + A")(t)dt

and

0

= f7T A(t)dt::::; f7T IA(t)ldt (18) 0 0

Area(W) = _!_ J27T (A2- A'2)(t)dt::::; _!_ J2A2(t)dt. (19) 2 0 2 0

Bounding above lAO I as in the proof of Corollary 2, we ob­

tain the estimates

(20)

Both sides of these inequalities depend continuously on the

operator. Because the set of Toeplitz regular operators is

dense, (20) holds for arbitrary C on en. Using the invari­

ance under C >---7 C + tl, as in the proof of Corollary 2, we

replace the norm in (20) by the reduced norm. This proves

our claim in the finite-dimensional case.

Now let dim� = oo, and let C be a bounded operator on

�. Let ie C � be a finite-dimensional subspace, let C be

the restriction of the form C to ie, and let W be the nu­

merical range of C. Then W and C satisfy the bounds of

(20). Using that IC:I ::::; lei and that

Perimeter(W) = sup Perimeter(W), Area(W) = sup Area(W), �c� �c�

we conclude that (20) holds for C. Now use again the in­

variance under C >---7 C + ti. •

Concluding Remarks

Although the bounds of Theorem 4 improve those of Corol­

lary 2 by the factor of 411T, they are still very rough. The

same or better bounds on the size of the numerical range

W( C) can be obtained using elementary geometry. Let X C C be compact. Denote by r(X) the numerical radius of X, i.e., the radius of the smallest disc D(X), centered at (0,0) and containing X. Toeplitz proved in [24] that

19_::::; r(W(C)) ::::; lei. 2

(21)

Since W(C) c D(W(C)), (21) implies (20) and the inequal­

ity Diameter(W) ::::; 2ICI. Invoking the invariance principle,

we obtain (17) and the upper bound of (7).

Set W1(C) = {zl- z2 : Z1, Zz E W(C)}. The set W1(C) C (: is symmetric about the origin and convex and satisfies [25]

W1(C) = { <u, Cv> + <v, Cu>: llull = llvll = 1,

This implies

Diameter(W(C)) =

<u, v> = 0}. (22)

max I <u, Cv> + <v, Cu> 1. (23) llull=llvJI= l,<u,v>=O

This in tum yields the bounds

Diameter(W(C))::::;

max 21 <u, Cv> I ::::; 2ICI. (24) llull=llvll= l,<u,v>=O

Invoking the same invariance principle, we obtain from (24) the upper bound of (7). There are other approaches to es­

timating the size of W(C). For instance, [2] employs the

Gershgorin disc theorem to obtain quadratic bounds on the

area of W(C) for certain nilpotent matrices.

In view of these results and those of [16), of course, the

main justification of Theorem 3 is not in the bounds on the

size of the numerical range that it yields. The justification

is the elegant formula (10) for the curvature of the bound-

VOLUME 26, NUMBER 1, 2004 1 3

AUTHOR

&UOENE OUTKIN

ary of the numerical range. The estimates ( 17) follow from it by very crude estimates. The formula (10) seems to be novel. My only "precursor" M. Fiedler identified in spectral terms the boundary curvature of numerical range in spe­cial cases [7, 8]. There is no immediate relationship be­tween his formulas and (10). I hope that (10) will find other applications to the remarkable subject that grew out of the Toeplitz-Hausdorff theorem.

It goes without saying that geometric considerations pervade the literature on numerical range. Several re­searchers have used the ideas above for purposes other than estimating the size of W(C). For instance, in [17] (16) helps to uncover new examples of domains satisfying the famous "porism of Poncelet."15

Before stopping, I will give unsolicited advice to the reader. There is a pervasive custom of concentrating on the latest literature while doing research. I am no exception to this rule. However, my experience with the study of nu­merical range brought me to the conclusion:

It is useful to read the work of "founding fathers"!

15A related way of using the numerical range to construct such examples is pre­

sented in [27].

14 THE MATHEMATICAL INTELLIGENCER

REFERENCES [ 1 ] T. Bonnesen and W. Fenchel, Theorie der konvexen K6rper,

Springer-Verlag, Berlin, 1 97 4.

[2] M.-T. Chien, Y.-H. Lin, On the area of numerical range, Soochow

J. Math. 26 (2000), 255-269.

[3] N. P. Dekker, Joint numerical range and joint spectrum of Hilbert

space operators, Dissertation, Free University of Amsterdam , 1 969.

[4] W. F. Donoghue, Jr. , On the numerical range of a bounded oper­

ator, Mich. Math. J. 4 (1 957), 261 -263.

[5] A. Feintuch and A. Markus, The Toeplitz-Hausdorff theorem and

robust stability theory, Math. lntelligencer 21 ( 1 999), 33-36.

[6] L. Fejer, Ober gewisse durch die Fouriersche und Laplacesche

Reihe definierten Mittelkurven and Mittelflachen, Rend. Circ.

Matern. Palermo 38 (1 9 1 4) , 79-97.

[7] M . Fiedler, Geometry of the numerical range of matrices, Lin. Alg.

Appl. 37 (1 98 1 ), 8 1 -96.

[8] M. Fiedler, Numerical range of matrices and Levinger's theorem,

Lin. Alg. Appl. 220 (1 995), 1 71-180.

[9] K. E. Gustafson and D. K. M . Rao, Numerical Range, Springer­

Verlag, Berlin, 1 997.

[1 OJ E. Gutkin, E. Jonckheere, and M. Karow, Convexity of the joint nu­

merical range: Topological and differential geometric viewpoints,

preprint, 2002.

[1 1 ] P. Halmos, A Hilbert space problem book, Springer-Verlag, New

York, 1 982.

[1 2] F. Hausdorff, Der Wertvorrat einer 81/inearform, Math. Zeitschrift 3 (1 9 1 9), 31 4-3 1 6.

[1 3] F. Hausdorff, Gesammelte Werke, Band IV: Analysis, Algebra and

Zahlentheorie, Springer-Verlag, Berlin, 2001 .

[ 14] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge

University Press, Cambridge, 1 991 .

[1 5] E. Jonckheere, F. Ahmad, and E. Gutkin, Differential topology of

numerical range, Lin. Alg. Appl. 279 (1998), 227-254.

[1 6] R. Kippenhahn, Ober den Wertevorrat einer Matrix, Math. Nachr.

6 (1 951 ) , 1 93-228.

[ 1 7] B. Mirman, V. Borovikov, L. Ladyzhensky, and R. Vinograd, Nu­

merical ranges, Poncelet curves, invariant measures, Lin. Alg. Appl.

329 (2001 ) , 61 -75.

[1 8] F. D. Murnaghan, On the field of values of a square matrix, Proc.

Natl. Acad. Sci. USA 18 (1 932), 246-248.

( 1 9] H. Nakazato and P. Psarrakos, On the shape of numerical range

of matrix polynomials, Lin. Alg . Appl. 338 (2001 ), 1 05-1 23.

[20] D. H. Owens, The numerical range: a tool for robust stability stud­

ies?, Sys. Control Lett. 5 ( 1 984), 1 53-1 58.

[21 ] M . G. Safonov, Stability robustness of multivariable feedback sys­

tems, MIT Press, Cambridge, MA, 1 980.

[22] L. A. Santal6, Integral geometry and geometric probability, Addison­

Wesley, London, 1 976.

[23] M . H . Stone, Linear transformations in Hilbert space and their ap­

plications to analysis, A.M.S. , New York, 1 932.

[24] 0. Toeplitz, Das algebraische Analogon zu einem Satze von Fejer,

Math. Zeitschrift 2 (1 9 1 8) , 1 87-1 97.

[25] N . -K. Tsing, Diameter and minimal width of the numerical range,

Lin. Mult. Alg. 14 (1 983), 1 79-1 85.

[26] A. Wintner, Spektraltheorie der unendlichen Matrizen, Verlag S .

Hirzel, Leipzig, 1 931 .

[27] P. Y. Wu, Polygons and numerical ranges, Amer. Math. Monthly

107 (2000), 528-540.

Theories of Vision Enul · •I holz

I .

I I .

From Grosholz, Emily; Shorts and Headlands. Copyright © 1 988 by Princeton

University Press. Reprinted by permission of Princeton University Press.

1 6 THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

I l l .

IV.

v.

hildren

b · '< nd

aking

Department of Philosophy

Pennsylvania State University

University Park, PA 1 6802

USA

e-mail: erg2@psu.edu

1&ffij.i§j:@hl£113'fil§4fiijlj:i§:id M ichael Kleber and Ravi Vaki l , Ed itors

Cutting a Polygon into Triangles of Equal Areas Sherman Stein

This column is a place for those bits of

contagious mathematics that travel

from person to person in the

community, because they are so

elegant, suprising, or appealing that

one has an urge to pass them on.

Contributions are most welcome.

Please send all submissions to the

Mathematical Entertainments Editor,

Ravi Vakil, Stanford University,

Department of Mathematics. Bldg. 380,

Stanford, CA 94305-21 25 , USA

e-mail: vakil@math.stanford.edu

This tale, like so many in mathe­matics, begins with a simple ques­

tion, answers it, and ends with ques­tions that have yet to be resolved.

Fred Richman in 1965 wondered whether it is possible to cut a square into an odd number of triangles of equal areas. The key word here is "odd," for a moment's reflection shows that a square can be cut into any even number of triangles of equal areas.

Before I go on to describe the re­search that grew out of that question over the last third of a century, I will stop to introduce a few terms for the sake of clarity.

A dissection of a polygon into tri­angles of equal areas I will call an equidissection. An equidissection into m triangles I call an m-equidissection. An m-equidissection with m odd will be called an odd equidissection, and with m even, an even equidissection. Richman was asking whether every

equidissection of a square is even.

Richman's colleague, John Thomas, became interested in the problem and proved that there is no odd equidis­section of a unit square in standard po­sition in the xy-plane if the coordinates of the vertices of the triangles are ra­tional with odd denominators. When he submitted his work to Mathematics Magazine, "The referee thought the problem might be fairly easy (although he could not prove it) and possibly well-known (although he could find no reference to it)." The referee suggested that Thomas submit it as a Monthly problem and if no one solved it, the pa­per should be published. It appeared in 1968 [12].

Paul Monsky in 1970 [4], building on Thomas's proof, showed that the an­swer to Richman's question is, "No, there is no odd equidissection of a square." His argument uses two tools, Spemer's Lemma from combinatorial topology, and 2-adic valuations from al­gebra. I will describe both.

In 1928 Emanuel Spemer published a theorem which he used to prove sev-

eral topological theorems, including the fact that a ball of dimension n is not homeomorphic to a subset of a lower dimension space. It was soon ap­plied by others to give a short proof of Brouwer's fixed-point theorem. He stated it for simplices in all dimen­sions, but I will present it just for poly­gons in the xy-plane.

Consider a polygon cut into trian­gles. For simplicity, assume that two triangles that touch each other inter­sect either in a complete edge of both or in a vertex of both. All the vertices are labeled A, B, or C. Figure 1 is an example.

l@ldii;IIM

An edge of a triangle whose ends are labeled A and B will be called complete. A triangle whose vertices are labeled A, B, and C will also be called complete. Spemer's reasoning shows that the number of complete edges on the boundary of the polygon has the same parity as the number of complete tri­angles. In Figure 1 the respective num­bers are 3 and 9. In particular, if there are an odd number of complete edges on the boundary there must be at least one complete triangle. That implica­tion is what Spemer used, and so will we.

The other tool is a 2-adic valuation, <P, which is a function defined on the real numbers. First, for a non-zero in­teger n, <P(n) is the number of 2's in the prime factorization of n. If a and b are non-zero integers, <P( alb) is defined as <P(a) - <P(b). At 0, <P is set equal to oo.

It turns out that <P can be extended to

© 2004 SPRINGER-VERLAG NEW YORK, VOLUME 26, NUMBER 1 , 2004 17

all the reals in uncountably many ways, each having the following properties:

<P(xy) = <P(x) + <P(y) <P(x + y) ::::: minimum of <P(x)

and <P(y) <P(x) = oo only when x is 0.

It follows from these properties that for <P(x) < <P(y), <P(x + y) = <P(x). As ex­amples we have <P(1) = 0, <P( -1) = 0, <P( -x) = <P(x), <P(112) = -1, <P(V3/2) =

-1, and <P(\12) = 112. For each prime p there are p-adic

valuations defined in a similar manner. A more detailed treatment of Spemer's Lemma and p-adic valuations is to be found in Chapter 5 of [9].

With the aid of 2-adic valuations, we can divide the xy-plane into three sets, which I will call A, B, and C. The set A consists of the points (x,y) for which both <P(x) and <P(y) are posi­tive. B consists of the points (x,y) for which <P(x) ::::; 0 and <P(x) ::::; <P(y). C consists of the remaining points, namely, those (x,y) for which <P(y)::::; 0 and <P(y) < <P(x). We label each point A, B, or C, depending on which of the three sets contains it. For in­stance (0,0) is labeled A, (1,0) is B, (1,1) is B, and (0,1) is C. The point (\13/2, \13/2) is labeled B. It is easy to check that translating a point by a point labeled A (viewing them for a moment as vectors) does not change the label: if P is a point, then P and P - A have the same labels.

One of the key tools, going back es­sentially to Thomas, is that if the three vertices of a triangle have all three la­bels, A, B, and C, then the valuation of the area of the triangle is less than or equal to -1. To show this, first trans­late the complete triangle by the ver­tex labeled A. Let us say that the three vertices are now (0,0), (a,b), and (e,d), with (a,b) labeled B and (e,d) labeled C. The area of the triangle is the absolute value of (ad - be )/2. Since <P(a) ::::; <P(b) and <P(d) < <P(e), <I>( ad) = <P(a) + <P(d) < <P(b) + <P(e) = <P(be), Thus <P(ad - be) = <P(ad)::::; 0, and it follows that <P((ad - be)/2) = ::::; -1.

Note that as a consequence a line in the xy-plane cannot meet all three sets A, B, and C.

Now consider an m-equidissection of a polygon of area A. Assume that at

18 THE MATHEMATICAL INTELLIGENCER

least one of its m triangles is complete. Since the area of the triangle is Aim, it follows that <P(Aim) ::::; -1.

Thus <P(A) - <P(m)::::; -1, from which we conclude that <P( m) ::::: <P(A) + 1.

So, if we knew that <P(A) is larger than -1, m must be even. In particu­lar, if A is an integer, m is even.

To apply the information just ob­tained, we have to be sure that there is at least one complete triangle in the dissection. This is where Spemer's Lemma enters the picture.

After just one more observation, we will be ready to prove the Richman­Thomas-Monsky theorem.

Consider a finite set of points in a complete line segment. Each of these points is labeled either A or B. The points divide the segment into shorter sections. The number of these sections that are complete must be odd. One way to show this is to drop pebbles in each section next to an end labeled A,

and then add them up in two ways, by points and by sections.

When a line segment that is not com­plete is divided into sections, the num­ber of complete sections is always even. For instance, there are no com­plete sections when a segment with ends B and C (or A and C) is cut into sections, because no line meets all three sets, A, B, and C. A segment with ends A and A (or B and B) has an even number of complete sections.

With all the machinery in place, we are ready to prove that a square has only even equidissections.

Consider an equidissection of a square. It is no loss of generality to as­sume that the area of the square is 1 and that its vertices are (0,0), (1,0), (0,1), and (1,1). This is shown in Fig­ure 2, with the labels of its four ver­tices.

lplijiJ;IfM

C (O, l )b

A (0, 0) B ( l , 0)

All the vertices, not just the comers of the square, are labeled A, B, or C. No matter how those vertices are labeled, there will be an odd number of com­plete sections along the bottom edge of the square, which is complete. The other three edges have no complete sec­tions. Thus the total number of com­plete sections on the boundary of the square is odd. Hence there is at least one complete triangle in the dissection. It follows that the equidissection is even.

That is where the subject of equidis­sections remained until 1979, when David Mead [3] obtained a generaliza­tion from a square to a cube in any di­mension. He proved that when an n­

dimensional cube is divided into

simplices all of which have the

same volume, the number of the

simplices must be a multiple of n! .

In addition to Spemer's Lemma in higher dimensions, he used p-adic val­uations for all primes p that divide n.

In 1985, when Elaine Kasimatis was presenting the result for a square in G.

Donald Chakerian's geometry seminar, I wondered, "What about the regular pen­tagon?" She found the answer and went on to prove that in any m-equidissec­tion of a regular n-gon with at least

five sides, m must be a multiple of

n. In the proof she had to extend p-adic valuations to the complex numbers for the prime divisors of n. Her work ap­peared in 1989 [1]. In a sense it was an­other generalization of the theorem about equidissections of a square.

A year later she and I published [2] the results of an investigation of equidissec­tions of trapezoids and other quadri­laterals. Among these was yet another generalization of a square, namely quadri­laterals whose four vertices are (O,Q), (1,0), (a, a), and (0,1), where a is any pos­itive number, illustrated in Figure 3.

Mptdll;l¥4 (a, a)

(0, 1)

(0, 0) (1, 0)

The area of such a quadrilateral is a. If <P(a) is greater than 0, then the boundary of the quadrilateral has two complete edges, and the hypothesis of Spemer's Lemma doesn't hold. Inci­dentally, had it held, we would have been able to conclude that in any m­equidissection of the quadrilateral m has to be a multiple of 4, since <P( m) would be greater than 1. Because m can be as low as 2, we could have pre­dicted that there are an even number of complete edges on the boundary.

In this case we apply the linear map­ping that takes (x,y) to (xla, y). The image of the original polygon has area 1 and vertices (0,0), (1/a,O), (1,a), and (0,1), as shown in Figure 4.

( 1 , a)

(0, 1)

(0, 0) ( l !a, 0)

Now there is only one complete edge on the boundary, and Spemer's Lemma applies. Hence m is even, as in the case of the square.

If -1 < <P(a) ::; 0, the labeling of the initial quadrilateral has one complete edge, and there is no need to introduce a linear mapping. Again, m must be even.

When <P(a) = -1, there may be odd equidissections as the case a = 3/2 shows. Figure 5 illustrates this case, where the quadrilateral is cut into three right triangles, each of area 1/2.

( 1 , 0)

More generally, if a = b/(2c), where b and c are odd integers, the corre­sponding quadrilateral has an odd equidissection. However, if <P(a) = -1 and a is irrational, I don't know what can be said. Even the case a = v3/2 is not settled. Does the quadrilateral with vertices (0,0), (1,0), ((v3/2, v'3!2), and (0, 1) have an odd equidissection?

In any case, this attempt to general­ize the result for squares failed. The question remained: What is there about a square that forces all its equidissections to be even? In other words, What is the most general class ofpolygons that have no odd eq uidissection?

One simple generalization is that any parallelogram has no odd equidissec­tion. This follows immediately from the result for a square, for any parallelogram is the image of a square by a linear map­ping. Since a linear mapping magnifies all areas by a constant, it takes an equidissection into an equidissection.

A parallelogram being centrally sym­metric suggests that perhaps any cen­trally symmetric polygon has no odd equidissection. Kasimatis's theorem about regular n-gons, when n is even, gave me enough extra evidence that I in­vestigated centrally symmetric poly­gons, trying to produce a counter-ex­ample. Instead I proved in 1989 [8] that every centrally symmetric 6-gon or 8-gon has no odd equidissection. Monsky in 1990 [5] proved the theorem in general.

Even so, I did not feel that that was the last word. There was another class of polygons that I suspected would gen­eralize the square. To construct this type of polygon, I start with the unit square in Figure 2 and then distort its bound­ary, changing opposite edges in the same way. The resulting polygon still tiles the plane by translates using all integer vee-

Mpt§ll;iiM

(0, 1 ) •�• ( 1 , 1 )

(0, 0) •�• (1 , 0)

tors. Figure 6 shows such a distorted square.

It seemed to me that complicating the boundary would lessen the chance that the resulting polygon would have an odd equidissection. I proved for a few sim­ple families made this way, such as poly­gons formed by adding one dent, as in Figure 7, that my suspicion was valid.

·------· ( 1 , 0)

Some years later a surprising break­through occurred, which I described in a paper published in 1999 [10]. It con­cerns a unit square in the xy-plane whose comers have integer coordi­nates, such as the one in Figure 8.

+§lijii;i+:W (5, 7) • B

B A (5, 6) •'------• (6, 6)

Note that the square in Figure 8 has one complete edge. A moment's thought shows that exactly one vertex of any such square has both coordinates even, hence labeled A. Its two neighboring vertices are then labeled B and C. That implies that the square has exactly one complete edge.

It follows immediately that any

polygon in the xy-plane made up of

an odd number of such unit squares

has no odd equidissection. To see this, place a pebble inside each square in the polygon next to its complete edge. Because there is an odd number of pebbles, there must be an odd num­ber of complete edges on the bound­ary. Moreover, since the area is an in­teger, it follows that the number of triangles must be even.

VOLUME 26, NUMBER 1 , 2004 1 9

It struck me as odd that by assum­

ing that the polygon has an odd num­

ber of squares I was able to deduce that

the number of triangles was even. I

checked a few cases where the poly­

gon had an even number of squares,

enough to convince me that it was true

in general, but left it to someone else

to treat that case. Iwan Praton in 2002

[6] disposed of the even case. His proof

showed that if the number of squares

is of the form 2rb, where b is odd, then

there is a translate of the image of the

polygon by a linear mapping that takes

(x,y) to (x/2u, y/2v), where u + v ::5 r, to which a stronger version of Sperner's

lemma applies.

Consequently any polygon com­

posed of the unit squares described has

no odd equidissection. There is an­

other, more suggestive way to state this

result: Any polygon in the xy-plane

whose edges are parallel to the

axes and have rational lengths has

no odd equidissection. To show this,

first translate the polygon so that one

of its vertices is at the origin. Then

magnify this image by a mapping that

takes (x,y) to (qx, qy), where q is an in­

teger divisible by all the denominators

of the lengths of the edges. The image

consists of congruent squares and has

no odd equidissection. Hence the orig­

inal polygon has no odd equidissection.

The next conjecture is inevitable.

What if the assumption that the edges

have rational lengths is removed? I conjectured that any polygon whose edges are parallel to the axes has no

odd equidissection. I then faced three classes of poly­

gons that I either knew or suspected

have no odd equidissections: centrally

symmetric, distorted square, edges

parallel to the axes. The first case was

already settled, and there was ample

evidence for the remaining two cases.

Figure 9 illustrates the three types. As

I stared at polygons like those, I no­

ticed a property that they all shared. To

describe this property I orient the

boundary, turning each edge into a vec­

tor whose direction is compatible with

the orientation. Then I call two vectors

on the boundary equivalent if they are

parallel. All three types have the prop­

erty that the sum of the vectors in each equivalence class is the zero vector. I

20 THE MATHEMATICAL INTELLIGENCER

ii'riil;l¥+

O D Centrally symmetric Distorted square

Edges parallel to axes

called such a polygon special and con­

jectured that each special polygon has

no odd equidissection.

I showed that the conjecture is true

when the special polygon has only a

few edges. The smallest possible num­

ber of edges is four, and the polygon is

then a parallelogram, for which the

conjecture is true. There are no special

polygons with five sides, as may easily

be checked. There are three types of

special polygons with six sides, con­

structed as follows.

The first step is to determine the

number of edges in an equivalence

class. There must be at least two in a

class and at most three, for if there

were four, two would be forced to be

adjacent. The partitions of six meeting

these conditions are 6 = 3 + 3 and 6 =

2 + 2 + 2. The second step is to see

how the equivalence classes could be arranged on the boundary. Take the

ljMiiijiitl

q

q � "' "' ... .. .. .. ... ... ..

P,,'' ....

\p

. . . . � : . . . .

q '" ...... _ ,' q - . · ... .. .. .. .. ..

p

p

(a)

p

q

(b)

p

3 + 3 case first. Denoting parallel vec­

tors by the same letter, the only possi­

bility is to alternate the vectors of the

two classes, as shown in Figure lOa.

That schema can be realized by a spe­

cial polygon, as shown in Figure lOb.

Without loss of generality, we can as­

sume its edges are parallel to the axes.

The 2 + 2 + 2 case leads to two es­

sentially different schemas, as shown

in Figure l l and later in Figure 13.

I#Mil;li!i p

, .,. ... ... .. .. .. .. .... ..

q,/' "'\ r

. . . . ' I . . . .

r \.. ,/ q ... .. , ... .. .. .. .. .. ..

p

This schema can be realized by any

centrally symmetric polygon with six

sides, as shown in Figure 12.

l!'dli;Jifl p

p

The other possible schema is shown in Figure 13.

. .

q " ...... ..

p

p

It, too, can be realized by a special

polygon, shown in Figure 14.

Each case can be treated with the

aid of Sperner's Lemma, 2-adic valua­

tions, and a variety of affine mappings,

that is, mappings that take (x,y) to

(ax + by + e, ex + dy + f), where a, b,

c, d, e, andf are constants and ad - be

is not 0.

p

To determine the special polygons

with seven sides, I first list the parti­

tions of seven in which the summands

are at least two and at most three.

There is only one such partition,

namely 7 = 3 + 2 + 2. It can be real­

ized in two different ways by schemas,

and each schema has a geometric re­

alization, as shown in Figure 15.

+pMil;ii .. ii f.· · - < ' ' q/ \ p ' ' '

;- •• _ _ _ _ _ .·'q p

f!. · - -< ' ' r :

' \ P ' ' . : q\_ .. ·'q .. ... .. ...

p

p

q��r r[__fP

p

Again I managed to show that both of

these types of special polygons have no

odd equidissection [11) . Because the

proofs break into a couple of dozen

cases, I have hesitated to go on to the

eight-sided special polygons. In any

event, these 7-gons provide substantial

evidence for the general conjecture,

which I had wanted to call the "mother

of all conjectures," but was restrained

by the referee to name it simply a "gen­

eralized conjecture."

As is customary in science, we are

left with more questions than we had

when we started. Perhaps we have

i

found the fundamental property of the

square that is the basis of the Richman­

Thomas-Monsky theorem. Perhaps

not. That raises the first of several

questions:

Does a special polygon ever have an

odd equidissection?

The next four questions are suggested

by the special polygons.

How many partitions are there of a

positive integer n if the summands

are at least 2 and at most n/2?

Is each such partition representable

by a combinatorial schema?

If so, by how many?

Is each combinatorial schema rep­

resentable by a special polygon?

Even if all these questions are an­

swered, many questions about equidis­

sections would remain. For instance,

does a trapezoid whose parallel edges

have lengths in the ratio of v'2 to 1 have any equidissections? I think that

the answer is no and make the follow­

ing conjecture:

Consider a trapezoid whose parallel

edges have lengths in the ratio of r to 1, where r is algebraic. I conjec­

ture that if r has at least one nega­

tive conjugate, then the trapezoid

has no equidissection.

Little has been done about equidis­

sections into simplices in higher dimen­

sions aside from [3]. It was shown in [2)

that in any dissection of a regular octa­

hedron into m simplices of equal vol­

umes, m must be a multiple of 4. Is it

true that in any dissection of a centrally

symmetric polyhedron into m simplices

of equal volumes, m must be even?

That is where Richman's question

has led. The path that he discovered

seems to have no end.

i n t

- l i ha I I I .

Acknowledgment

I wish to thank Anthony Barcellos for

providing the illustrations, using Co­

hort's software, Coplot.

REFERENCES 1 . E. A Kasimatis, Dissections of regular

polygons into triangles of equal areas, Dis­

crete and Camp. Geometry 4 (1 989),

375-381 .

2 . -- and S . Stein , Equidissections of

polygons, Discrete Math. 85 (1 990),

281 -294

3. D. G. Mead, Dissection of hypercubes into

simplices, Proc. Amer. Math. Soc. 76

(1 979), 302-304.

4. P. Monsky, On dividing a square into tri­

angles, Amer. Math. Monthly 77 (1 970),

1 61 -1 64.

5. -- A conjecture of Stein on plane dis­

sections, Math. Zeit. 205 (1 990), 583-592.

6. I. Praton, Cutting Polyominos into Equal­

Area Triangles, Amer. Math. Monthly 1 09

(2002) 81 8-826.

7. F. Richman and J. Thomas, Problem 5471 ,

Amer. Math. Monthly 74 ( 1 967), 329.

8. S. Stein, Equidissections of centrally sym­

metric octagons, Aequationes Math. 37

(1 989), 3 1 3-31 8.

9. -- and S. Szabo, Algebra and Tiling,

Mathematical Association of America,

Washington, D . C. 1 994.

1 0 . -- Cutting a polyomino into triangles of

equal areas, Amer. Math. Monthly 1 06

(1 999), 255-257.

1 1 . -- A generalized conjecture abourt cut­

ting a polygon into triangles of equal areas,

Discrete and Camp. Geometry 24 (2000),

1 41 -1 45

1 2. J. Thomas , A dissection problem, Math.

Mag. 41 (1 968), 1 87-190.

Department of Mathematics

University of California Davis

1 Shields Avenue

Davis, CA 9561 6-8633

email: stein@math.ucdavis.edu

u 1

l lha r

VOLUME 26. NUMBER 1. 2004 21

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

The proof is in the pudding.

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 /?" Or even

"Who am !?" This sense of disorienta­

tion is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It's

mathematical, it's a humor column,

and it may even be harmless.

Column editor's address: Colin Adams,

Department of Mathematics, Bronfman

Science Center, Williams College,

Williamstown, MA 01 267 USA

e-mail: Colin.C.Adams@williams.edu

Col in Adam s , Editor

Rumpled Stiltskin Colin Adams

Once upon a time there was a topol­

ogist who lived with his daughter

in a tiny office in the Math Building at

the University of Chicago. One day the

Chairman of the department happened

to stop to talk to a colleague just out­

side the door of the topologist's office.

"The hiring season looks tough," said

the Chairman, a bit discouraged. "I hope

we can fmd someone extraordinary."

The topologist, who was barely

known to the Chair, stepped out of his

office. "Pardon me," he said timidly. "I

hate to interrupt. But I know of an

extraordinary mathematician. She can

turn coffee into theorems."

"Really?" said the Chair. "And who

is this mathematician?"

"She is my daughter," said the topol­

ogist.

"Then send her to my office this af­

ternoon," said the Chair.

That afternoon, the topologist's

daughter went to the Chair's office. She

was quite apprehensive, as she had no

idea how to turn coffee into theorems.

"Follow me," said the Chair as he led

her to the department lounge.

"Here you see a coffee maker, and

three pots of coffee. I want you to turn

the three pots of coffee into theorems

by morning. If you do not, then I will

see to it that the only job you ever get

is at a regional university with high

research expectations and a teaching

load of four courses per semester."

With that he left the lounge, locking

the door behind him.

The poor girl was disconsolate. She

fell sobbing on the couch. "Oh, what

ever will I do?" she cried. "My career

is over before it has even started."

Suddenly, as if by magic, the door to

22 THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

the lounge swung open, and in walked

a squat, disheveled creature, with long

matted beard and hair. He was a squat

man dressed in a dirty t-shirt, jeans, and

an even dirtier red sports coat. He

seemed surprised to see her.

"What are you doing here?" he

asked.

"Oh, the Chair has said that I must

turn this coffee into theorems, or he

will destroy my career."

"And you don't know how to tum

coffee into theorems?" asked the little

man, a smile playing across his lips.

"Oh, no. I have no idea how. My fa­

ther just said I could to impress the

Chair."

"And what will you give me if I turn

this coffee into theorems?" said the un­

sanitary fellow.

"Ummm, how about this mint con­

dition copy of Stewart's Calculus

book?" she suggested, pulling the book

from her briefcase.

"Let's see," he replied. "Is it the new

edition? I could get 50 bucks for that.

You got a deal."

And with that, the strange man

gulped down all three pots of coffee.

His bloodshot eyes began to glow. His

eyebrows started to twitch. Then he sat

down before a pad of paper and wrote

furiously for three hours. When he was

finished, the pages of three pads of pa­

per were covered with the most beau­

tiful lemmas the girl had ever seen.

"That ought to do the trick," said the

little man, and with that, he scooped

up the calculus book and was gone.

The next morning the Chair un­

locked the door, expecting to find the

girl crying or sleeping, with nothing to

show for her night. But his jaw dropped

when he saw the scribblings on the

pad. "This is really quite good," he said.

"Thank you," said the girl timidly.

"Can I go now?"

"What, are you kidding? This is the

beginning of some really good mathe­

matics. But you need to fill in the de­

tails. Flesh out the theory. Come back

this evening."

When the girl arrived that night, the

Chair pointed to six pots of coffee sit­

ting on the table. "If you don't turn this

coffee into theorems, I will make sure

the only work you get is as a recitation

instructor, teaching fifteen problem

sessions a week for large calculus lec­

tures." And again, he locked her in the

lounge.

The girl fell sobbing on the couch.

But she said to herself, if the little man

can do it, why can't I? With that she

went over and poured herself a cup of

the steaming brown liquid. She took

a sip and spat it out immediately.

"Ahhrgh," she said, "This tastes like it

has been sitting in the pot for the last

12 hours"-which in fact it had.

But then the door swung open again,

and in walked the little man. His jeans

were tom at the knee and his teeth ap­

peared never to have experienced the

friction of a toothbrush.

"Back again, are we?" he said.

"Yes, the Chair said I must tum

these six pots of coffee into theorems

by morning, or he will tum me into a

recitation instructor."

"And what, pray tell, will you give

me, if I do it for you?"

The girl thought for a moment and

then pointed to her computer brief­

case.

"How about my laptop?" she asked

hopefully.

She handed it to the man. "Hmmm,

looks like a Mac Titanium Power PC

G4, 800 MegaHertz, with one megabyte

L3 and 256K L2 cache. You got a deal."

So again, he gulped down the cof­

fee, and set to work Six hours later, he

had filled six pads of paper with the­

orems and proofs.

"This should do it," he said. And

grabbing up the laptop, he disappeared

out the door.

When the Chair arrived the next

morning, he was flabbergasted by the

beauty of the mathematics on the pads.

"This is really good stuff," he said

enthusiastically. "These are the germs

for a whole new theory. I am really im­

pressed."

"Good," said the girl nervously.

"Now can I go?"

"Yes, but you must come back

tonight," said the Chair. "You have

more work to do."

That evening, the Chair sat her

down before twelve pots of coffee.

"If you don't tum this coffee into

theorems," he threatened, "I will

make you into a permanent grader for

our remedial algebra course. But, if

you do succeed, I will give you a

tenure-track position on the faculty

here at Chicago." Then he turned and

left the lounge, locking the door be­

hind him.

The girl fell on the couch sobbing.

It was too much to hope the smelly

man would be back to help her once

more. And besides, she had nothing left

to give him.

Suddenly the doorknob turned and

in he walked.

"Still trying to turn coffee into the­

orems, are we? Haven't learned how to

do it yet?"

"Oh, no," she said. "I can't do it. And

the Chair is going to make me into a

permanent grader. Oh, woe is me."

"And what will you give me if I do

it for you?" asked the hair-encumbered

individual.

"I don't know," said the girl. "I don't

have anything left to give."

The little man grinned mischie­

vously. "Oh, I think you do," he said. "I

want you to give me your first-born

theorem."

"What do you mean?'' asked the girl.

"The first theorem that you prove

yourself, I want you to give it to me, to

claim as my own."

Now the topologist's daughter knew

that if she said no, she wouldn't ever

have the opportunity to create her own

theorems, anyway. So there wouldn't

be anything to lose. On the other hand,

if she did survive all this nonsense, and

had a career as a mathematician, what

was one theorem more or less? So she

agreed. The little man laughed delight­

edly.

"Oh, yes we have a bargain," he

laughed as he danced about the room.

Then he gulped down all twelve pots

of coffee, and worked through the en­

tire night, finishing just before day­

break.

"Remember our deal," he said as he

slipped out the door, leaving twelve

pads of paper filled with mathematics

on the table.

When the Chair arrived, he was

stunned by the level of work that he

saw.

"You have a job, a tenure-track job,"

said the chair, shaking her hand en­

thusiastically.

So the young woman began her

career at Chicago. She was an able

teacher, and enjoyed that aspect of her

job. But at first she found it difficult to

work on her research, as her other du­

ties were so numerous.

But one day, she attended a number

theory seminar. The speaker gave a dis­

cussion of Catalan's Conjecture, which

says that the only two consecutive

powers of whole numbers are the in­

tegers 8 and 9. She found the question

quite fascinating. Soon, she was spend­

ing all her time working on the prob­

lem. She would have worked even

more but sometimes exhaustion over­

came her. Finally, one evening, want­

ing to continue her work but unable to

keep her eyes open any longer, she

stumbled into the department lounge

and quickly swallowed a cup of coffee,

before she had a chance to gag.

Suddenly, she felt awake. Within

minutes, the caffeine was coursing

through her system, and her neurons

seemed to be firing every which way.

She worked all that night and by morn­

ing, she had proved Catalan's Conjec­

ture.

Although she was tired and in great

need of sleep, she decided to wait un­

til the Chair arrived at 8:00 to tell him

the good news. At 7:30, just as her eyes

were closing with exhaustion, the door

to her office swung open, and the little

man, whom she had not seen for the

last two years, bounded in.

"I am here to collect my debt," he

said.

"Oh no," pleaded the assistant pro­

fessor. "It's too good. You can have my

next one."

"I don't want your next one," said

the diminutive hairball. "I want this

one."

"Please, please, don't take it. It has

taken me all this time to learn how to

tum coffee into theorems. I can't give

it up."

'Til tell you what," said the little

man, an evil grin on his face. "If you

can guess my name, I will not take your

theorem. And I will give you three days

VOLUME 26, NUMBER 1 , 2004 23

to guess it." He laughed then and

scooted out of the office.

The young woman thought to her­

self that this couldn't be so hard. After

all, he had made no rules about the

guessing. She could guess as many

names as she wanted. Eventually she

would get it right.

The next morning, the door to her

office opened and in popped the minor

mutant.

"And what do you guess is my

name?" he asked.

"Is it Pythagoras?" she queried. "Is

it Zeno? Is it Euclid?"

"No, no, and no." He hopped de­

lightedly from one foot to the other.

"Is it Nicomachus? Is it Diophantus?

Is it Pappus?"

"Not even close."

"Is it Fibonacci? Is it Newton? Is it

Liebniz?"

"Ha."

"Is it Bernoulli, or Euler, or Lagrange?

"No, no, and no again. You will have

to do better than that." And with that

he was gone.

That day, the young woman

searched for every name she could

think of. She asked others around the

department for any other names they

might know.

When the little man arrived the next

morning, she asked, "Is it Gauss? Is it

Cauchy? Is it Mobius?"

"No, no, and no," he laughed, hardly

able to contain himself.

"Is it Lobachevsky? Dirichlet?

Liouville?"

"Be serious."

"How about Weierstrass? Cayley?

Hermite? Cantor? Dedekind? Bel­

trami?"

"No, no, no, no, no, no."

"What about Lie, or Poincare or

Peano or Hurwitz or Hilbert or Cartan?"

"Give me a break."

"Maybe Zermelo, Dickson, or

Lebesgue?"

"No, no, and no. Tomorrow is your

last chance." And with that, he disap­

peared out the door.

24 THE MATHEMATICAL INTELLIGENCER

The young professor was crushed.

She didn't know what to do.

"Oh, woe is me," she cried. All that

day, she wrung her hands. That evening,

as she went to get a tissue from the bath­

room to dab her tears, she heard a voice

singing from within the Men's Room.

"I am so happy. I could sing, as I

shower in the sink.

For she doesn't realize who I am,

And how with Chicago I link.

She doesn't know that I live in the

Lounge,

She doesn't know my game.

And she doesn't know the most

important part,

Rumpled Stiltskin is my name."

She immediately went to her fa­

ther's office.

"Pop," she said, "Have you ever heard

of someone named Rumpled Stiltskin?"

"Oh sure, everybody knows about

Rumpled Stiltskin. One of the most

brilliant minds ever to grace this

campus."

"Who is he?"

"Who was he is the more appropri­

ate question. Bob Stiltskin was a gradu­

ate student here thirty years ago. A real

star. But he got hooked on Catalan's

Conjecture. Spent all his time trying to

prove it. Couldn't bring himself to solve

an easier problem and get a Ph.D."

"So what happened?"

"After eight years, they cut his sup­

port, and threw him out of the program.

But he still hung around. Used to sleep

in the Math Lounge. Somehow he had

gotten hold of a key. About ten years

ago, he disappeared entirely. Nobody

knows where he went. But there are ru­

mors of a sighting every now and then."

"And why is he called Rumpled

Stiltskin?"

"Well, he always wore the same red

sports coat, and calling it rumpled is

being generous."

The next morning, the pungent per­

son sprang into her office, and said,

"Last chance. What's my name?"

The girl smiled and said, "Is it Ve­

blen, or Noether, or Sierpinski?"

"No, no, and no again."

"Is it Birkhoff, or Lefschetz, Little­

wood, or P6lya?"

"No, no, nope, and no."

"Maybe Ramanujan or Banach, Cech, or Bloch?"

"No, oh no, oh no, and no."

"Klein, Wiener, Nevanlinna, or

Urysohn?"

"Not even close."

"Artin or Zariski?"

"Double no."

"Church or Whitehead?"

"No and again a big no. Looks like

you are plumb out of luck." He was

grinning from ear to ear.

"I guess I don't know," she said

pausing for a second. "Unless of

course, perhaps, it is Rumpled Stilt­

skin."

The odoriferous oddball froze,

stunned for an instant. "How could

you? ... how did you?" he spluttered.

"I guess I keep the proof of Catala­

nis conjecture after all," she said.

"Arghhh," screamed the minute mis­

creant, his face turning as red as his

jacket. He stomped his feet and

gnashed his teeth, and pulled forcefully

on his matted hair. His eyes rolled up

in their sockets, and then he stormed

out of the office, never to be seen at

the University of Chicago again.

Since then, every once in a while

reports filter down from the Univer­

sity of Illinois at Chicago of coffee

pots found empty just minutes after

they had been full. And at Northwest­

em, department copies of Stewart's

Calculus disappear at an alarming

rate.

The young professor went on to a

very successful career at Chicago. She

and her father wrote some joint papers,

on the basis of which her father was

promoted to an office of reasonable

size. And although she did drink coffee

for the next five years, she switched to

herbal tea after receiving tenure. And

even then, the theorems kept coming.

1$Ffiili§i•£1h¥1MQ.Jj.i .. iii,ihi¥J Marjorie Senechal , Editor I

Connections, Context, and Community: Abraham Wald and the Sequential Probability Ratio Test Patti Wilger Hunter

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater

than one. What we say about the

communities is }ust as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 0 1 063 USA

e-mail: senechal@minkowski . smith. edu

In the early 1940s, Abraham Wald, a

Hungarian-born mathematician who

had emigrated to the United States from

Vienna in 1938, joined a group of scien­

tists doing defense-related research for

the U.S. government. Wald's work with

the Statistical Research Group (SRG)

resulted in the creation of a theory

which had immediate as well as long­

term implications for research in math­

ematical statistics-the theory of se­

quential analysis. W ald had developed

the idea that launched sequential analy­

sis, the sequential probability ratio test

(SPRT), in 1943, in response to some in­

spection sampling problems put to him

by two other members of the SRG-W.

Allen Wallis and Milton Friedman.

The story of the development of the

SPRT is inextricably bound to the im­

portance of community in the develop­

ment of scientific ideas. The events sur­

rounding Wald's discovery of the SPRT

are well documented and clearly reflect

the influence of his immediate col­

leagues on the problems he was work­

ing on in 1943 when he formulated the

test [64, 9]. What has not been explored

in detail is the broader historical context

in which W ald worked and the influence

of the wider intellectual community on

his contributions to sequential analysis

and to statistics more generally.

The mathematical statistics com­

munity in the United States, the com­

plex web of professional relationships

in which Wald was involved during his

years in Vienna, and World War II, with

its impact on the American scientific

and social science communities,

played important roles in Wald's work

in sequential analysis. Furthermore,

his ideas had an impact on the broader

historical context. This article exam­

ines these reciprocal connections.

American Statisticians:

Creating a Community

When Wald arrived in the United States

in 1938, the American community of

mathematical statisticians had been in

official existence for less than a decade.

Indeed, the principal professional sup­

port for mathematical statisticians in

the United States, the Institute of Math­

ematical Statistics (IMS), had been

formed only three years earlier, and the

first American periodical devoted to

the discipline, the Annals of Mathe­matical Statistics, had been in print

only since 1929. However, the origins

of the community go back nearly a cen­

tury, to 1839, when a small group of

physicians, lawyers, and ministers met

in Boston to organize the American

Statistical Association (ASA). 1 The

founders of the ASA, and indeed most

people who called themselves statisti­

cians in the nineteenth century, looked

upon statistics as "the collection and

comparison of facts which illustrate

the condition of mankind, and tend to

develop the principles by which the

progress of society is determined" [3,

p. v]. They emphasized the collection

and presentation of numerical data

about social and economic problems,

treating statistics as a tool that could

help explain social phenomena. 2 By the turn of the twentieth century,

that tool had come to be seen as a

means of making the solutions to so­

cial problems more scientific. In his

1908 presidential address to the ASA,

Simon N. D. North, director of the

United States Census, explicitly tied

the advancement of the disciplines of

sociology and economics to the use of

statistics. He charged that "if the claim

of these sciences to be exact sciences

is to be made good, it follows that the

economist or the sociologist must also

be a statistician" [31, p. 439]. The use

of statistics by social researchers to

certify their status as scientists came

as these researchers in the United

States were establishing the organiza­

tions and publication outlets of pro­

fessional academic disciplines. The

American Historical Association (AHA)

1 For some of the history of the founding of the ASA, see [67] and [1 5].

2See [1 5] for a brief discussion of this perspective.

© 2004 SPRINGER-VERLAG NEW YORK, VOLUME 26, NUMBER 1, 2004 25

and the American Economic Associa­tion (AEA) held their charter meetings in the mid-1880s. The American Politi­cal Science Association followed in 1903.3

The establishment of the profes­sional social science disciplines, as well as the view among statisticians in the ASA that statistics could serve those disciplines, led to what became a regular practice in the ASA of hold­ing its annual meetings in cooperation with such groups as the AEA and the AHA. Walter Willcox, a professor of economics and statistics at Cornell who served as president of both the ASA and the AEA, felt that statisticians benefited from association with other organizations. He particularly thought that the ASA's "best connections l[ay] with societies devoted to economics, political science, and law" [66, p. 288] .

These close ties between the ASA and the social science disciplines would eventually lead to tension between the main constituency of the ASA and the small but growing group of members who wanted to develop the mathemat­ical aspects of statistical methods. Some of this mathematics began mak­ing its way into the work of statisti­cians in the ASA as early as the 1890s, but the relationship between collec­tions of numerical data about society and the mathematical theory of proba­bility had just begun to emerge, and several decades passed before a group of researchers would focus their in­quiries on it specifically.4

In the 1920s, these researchers be­gan to see the usefulness for research in economics, biology, and agriculture of the growing set of tools of statisti­cal inference, but they found them­selves struggling to find places to pub­lish their results. Many of them were trained and employed as mathemati­cians and their interests overlapped

with a number of disciplines, but it seems that no single periodical or orga­nization provided a comfortable profes­sional home. 5 The mathematical statisti­cians felt no particularly warm welcome from either the ASA or the American Mathematical Society (AMS)-perhaps the most likely supporters of mathe­matical statistics. On the one hand, the members of the ASA were interested in data collection and the use of numeri­cal information, but as Carver, founder of the Annals of Mathematical Statis­tics, put it, "most of their membership were economists, bankers and census people whose knowledge of mathe­matics was very limited."6 Most Amer­ican mathematicians, on the other hand, focused their research on pure mathematics, and while papers in mathematical statistics included theo­rems and proofs, they often had a par­ticular use of those theorems as a start­ing point. In a discussion leading up to the founding of the IMS, Henry Rietz, a mathematician at the University of Iowa, who would become the organi­zation's first president, commented about the mathematical research com­munity that "when it comes to practice accepting papers for publication it seems not much material is acceptable that is a bit tainted with possible ap­plications to statistical data."7

Carver put out the first issue of the Annals in 1929 (initially with some fi­nancial backing from the ASA), and mathematical statisticians formally or­ganized the IMS as an independent pro­fessional organization in 1935. In 1938, it assumed complete financial respon­sibility for the Annals.8

The IMS and the Annals provided a means for mathematical statisticians to establish formal citizenship in their emerging disciplinary community. They created a sort of boundary around that discipline, setting it off from its border-

3For discussions of the professionalization of American social science, see [41 , 1 1 ].

Figure 1. Abraham Wald. Illustration courtesy

of the Columbia University Archives-Colum­

biana Library.

ing fields of inquiry. 9 But the boundary was not impenetrable. Although they tended to draw attention to the impor­tance of theoretical statistics, mathe­matical statisticians still welcomed op­portunities to support and collaborate with the users of statistics. As the mathematical statisticians were estab­lishing their community, several groups of applied statisticians had organized themselves into societies supporting their particular interests. The Econo­metric Society was formed in 1930; the Psychometric Society, in 1935. By the late 1930s, biologists and medical sci­entists in the ASA had organized its first disciplinary section, the Biometric Section. The group began publishing a journal, Biometrics, in 1945. The work of these three organizations focused on applying statistics to studies based in their parent disciplines-economics, psychology, and biology.

The emerging community of mathe­matical statisticians in the United States had ties to these organizations.

4During the nineteenth century, the probabilistic tools of the normal distribution and the method of least squares had found most of their applications in the physical

sciences of astronomy and geodesy, occasionally appearing in actuarial work on mortality tables. For a discussion of these developments, see [35, 46].

5The variety of professional societies and journals in which the mathematical statisticians participated is discussed in [1 6].

6Harry C. Carver to Jerzy Neyman in [30, p. 1 72].

7Henry Rietz to E. B. Wilson, 27 July 1 935, in (1 3, p. 289].

8Until then, except for the ASA money, Carver had funded the journal out of his own pocket. See [30, p. 1 72].

9"fhe founding of the IMS marks perhaps only a middle chapter in the story of this process. The story continues with the impact of WWII on American science and

subsequent developments within American universities. While some consequences of the war will be explored below, the rest of the story is beyond the scope of this

discussion.

26 THE MATHEMATICAL INTELUGENCER

Of particular relevance to the place that Abraham W ald would find in the community when he emigrated from Europe were its connections to the Econometric Society. To understand the importance of those links for Wald's inclusion in the American math­ematical statistics community, we must trace his path there back to his days in Vienna.

Mathematics and Economics in

Inter-war Vienna: A Network of

Communities

Wald arrived in Vienna in 1927. His hometown in Transylvania, known to­day by its Romanian name, Cluj, was part of Hungary when Wald was born on October 31, 1902. In 1920, Transyl­vania became part of Romania, but had a large Hungarian minority. W ald, a Jew, spoke Hungarian and "never developed any affinity for Romania," nor a knowl­edge of its language [29, p. 361] . Edu­cated at home because the local school required attendance on the Sabbath, Wald had passed the gymnasium exam­ination recognized by the University of Cluj, and after fmishing there enrolled in the University of Vienna at age 25 to study mathematics. He eventually took classes from and wrote a dissertation under Karl Menger, working on metric spaces and differential geometry. 10

Menger's father Carl Menger had made his mark on Austrian economics with his work on marginal utility. The younger Menger was well versed in his father's work and well connected to various "circles" of the Viennese intel­ligentsia, including the philosophical Vienna Circle and a number of over­lapping groups of economists. 11 He led his own circle, the Mathematical Col­loquium, which met and published its proceedings from 1928 to 1937. The Colloquium hosted an impressive array of local and international luminaries, including John von Neumann, Alfred Tarski, Karl Popper, and Kurt Godel

(whose incompleteness theorem was first presented to the Colloquium).l2

Wald joined the Colloquium in 1930. In some sense, this gathering formed for Wald the center of a network of communities that fundamentally shaped his research. He contributed twenty-one papers to the Colloquium's proceedings, Ergebnisse eines mathe­matischen Kolloquiums, between 1931 and 1937, co-editing the last two volumes with Menger, Gbdel, and Franz Alt. Many of these papers com­municated his research in pure mathe­matics, but a few, discussed below, point to the connections among disci­plines and communities that the Collo­quium made for Wald.

Wald's connection to the world of Viennese economics would be among the most important for his passage to the United States. At Menger's invita­tion, the banker and economist Karl Schlesinger presented his work on equa­tions of economic production to the Mathematical Colloquium, and W ald quickly became interested in the field, publishing [51, 52, 53] in 1935 and 1936.13 In addition to these "first publications in his long list of contributions to mathe­matical economics" [27, p. 18], Wald's contact with Schlesinger led to some work as the latter's private tutor in mathematics. Menger had encouraged this connection, knowing that as a Jew, W ald had no chance of employment at the university. For the same reason, he introduced Wald to Oskar Morgenstern, who hired him to work at the Austrian Institute for Business Cycle Research, which Morgenstern directed. According to Menger's recollection, Wald had not been disturbed by his lack of opportu­nity in the academic world: "W ald, with his characteristic modesty, told me that he would be perfectly satisfied with any small private position which would en­able him to continue his work in our Mathematical Colloquium" [27, p. 18]. Not only did Wald's work in the Collo-

10Details of Wald's education and research can be found in [27, 29].

quium continue, but his private posi­tions opened up new intellectual oppor­tunities.

His relationships with Schlesinger and Morgenstern formed important threads in Wald's network of commu­nities. These men were active in influ­ential circles of Vienna's economists, both in and outside of the university. At the university, Schlesinger and Mor­genstern participated in the seminar of Hans Mayer, appointed to a chair in economics in 1923.14 Apparently, Menger and Wald attended the seminar occasionally as well [5, p. 12] .

Morgenstern and Schlesinger also attended the private biweekly seminar of Ludwig von Mises. Passed over for a position at the university, Mises was nevertheless considered "the central figure in the Viennese economic com­munity" at the time [5, p. 14]. He held his seminar in the 1920s and 1930s at the Vienna Chamber of Commerce where he was employed as the Secre­tary. Mises and his seminar participants formed the core of the National Eco­nomic Association, which Mises revived in the 1920s, becoming its vice-presi­dent, with Mayer as president. 15 The group met in a conference room of the National Banker's Association, thanks to its president, Karl Schlesinger. Papers they presented often appeared in the Zeitschrift fiir NationalOkonomie, a pe­riodical edited by Mayer with Morgen­stem's assistance [5, p. 18] . 16

Menger presented a paper at an Eco­nomic Association meeting on the Pe­tersburg paradox, but he later recalled that Mayer discouraged its publication because of its strongly mathematical character [28, p. 259]. Morgenstern, on the other hand, encouraged the inquiry of mathematically minded researchers into economic questions. As the direc­tor of the Institute for Business Cycle Research, another Mises-promoted or­ganization, Morgenstern employed not only Wald, but another student of

1 1 For discussions of Menger's many connections, particularly to the intellectual groups meeting outside the walls of the university, see [5, 1 0, 23, 44, 45]. 1 2The proceedings of the Colloquium have been reissued along with commentaries in [6].

1 3Schlesinger earned his Ph.D. in 1 91 4 under E. von Bohm-Bawerk at Vienna [2, p. 23].

14Mayer succeeded Friedrich von Wieser, famous for his work in opportunity cost theory. See [5].

15Mises seems to have wanted to insure that Mayer and his students were included in the professional society over against Mayer's rival at the university, Othmar

Spann. This motivation may explain why Mayer received the presidency [5, p. 1 7].

16The Zeitschrift was not, however, an official publication of the National Economic Association.

VOLUME 26, NUMBER 1 , 2004 27

Menger, Franz Alt, and the economist,

Gerhard Tintner, who later noted that

"at the Institute there was a much more

scientific attitude to economics than

elsewhere in Wien at this time. "17

As Wald's contact with economists

in Vienna fostered his interest in math­

ematical economics, his work gradu­

ally became known abroad. At a 1936

meeting of the Econometric Society in

Chicago, Tintner reported on some of

Wald's results [21, p. 188]. Schlesinger

and Wald himself attended a 1937

meeting of the Econometric Society in

France [36]. Wald's contacts in the

Econometric Society would form im­

portant links for him to the statistics

community in the United States.

His economics research in Vienna

had touched upon issues related to sta­

tistics, and he had published a paper in

the Ergebnisse on Richard von Mises's

notion of a collective, a concept that

played a role in the axiomatization of

probability.l8 But his immersion in the

ideas of mathematical statistics would

come in the United States. There, for

Wald, the web of communities woven

together for him in Vienna would con­

verge with the statistics community

that had been forming in the United

States since the 1920s. The boundary

that American mathematical statisti­

cians had drawn around their disci­

pline as well as the connections cross­

ing that boundary, particularly to the

econometrists, would further shape Wald's research, as well as his place in the scientific community.

Mathematical Statistics and

World War II: Communities

Converge

The Cowles Commission for Research

in Economics formed the thread link-

1 7From an interview reported in [5, p. 20].

ing Wald to the United States. The

Commission had been organized in

1932 by Alfred Cowles, the president of

an investment counseling firm in Col­

orado Springs. He had come into con­

tact with several members of the

Econometric Society in 1931, when he

began researching methods of fore­

casting the stock market. At their sug­

gestion, Cowles opened his research

institute and soon after began financ­

ing the Society's new periodical Econo­metrica [ 4].

Both the Econometric Society and

the Cowles Commission had ties to the

Viennese economics and mathematics

communities, as well as to the Ameri­

can statistics communities. Karl Menger

participated in the Society's organiza­

tional meeting, held in Cleveland, Ohio,

in December 1930 [40]. Gerhard Tint­

ner attended its meetings in the early

1930s and joined the Cowles Commis­

sion staff in 1936. Other active mem­

bers, also present at the organizational

meeting, included Harold Hotelling and

Walter A. Shewhart, both founding

members of the IMS. Hotelling became

one of the American mathematical sta­

tistics community's most respected

spokesmen for the discipline [ 15, 16].

It may have been Tintner's connec­

tions that resulted in Wald's invitation

to join the staff of the Cowles Com­

mission in 1937, which he accepted,

though not without some delay and

hesitation about leaving Vienna. 19

Menger had departed in 1937 for the University of Notre Dame as the polit­

ical climate in Austria was becoming

increasingly unbearable. When Hitler's

troops marched into Vienna in March

1938, Wald had not yet left. Morgen­

stem was in the United States on a lec­

ture tour and stayed, taking a position

in the economics department at Prince­

ton when he heard that he had been

blacklisted by the Nazis [5, p. 29] . Wald,

having been dismissed by Morgen­

stem's Nazi successor, fmally made his

way to Colorado Springs.20

His stay there was brief-within a

few months he left for Columbia Uni­

versity to work with Harold Hotelling.

Hotelling had been teaching eco­

nomics and building up a program in

mathematical statistics in Columbia's

economics department since 193!.21

Through the next decade, several early

members of the American mathemati­

cal statistics community received

some training at Columbia under

Hotelling, including Samuel S. Wilks

and Joseph L. Doob [ 16].

Funded from 1938 to 1942 by a grant

Hotelling had obtained from the

Carnegie Corporation, Wald worked at

Columbia, first as a research assistant

and then teaching courses in mathe­

matical statistics and economics.22 It

was during these first years at Colum­

bia that Wald became immersed in the

ideas of mathematical statistics. Char­

acteristically, according to his col­

leagues, he "worked with prodigious

energy and endurance" [ 14, p. 18], with

"most of his waking moments during

this and the next several years . . .

given to work" [68, p. 2]. Wald became

an assistant professor of economics in

1942, and made his way through the

ranks to professor of mathematical sta­tistics in 1945, finally becoming the chair of an independent department of

mathematical statistics at Columbia in

1946.23 Wald was a popular lecturer

from his first years of teaching at Co­

lumbia. Students flocked to his lec­

tures, which were "noted for their lu­

cidity and mathematical rigor" [68, p.

18Wald 's interest in collectives followed a presentation on the subject at the Collocuium by Karl Popper. Richard von Mises was the brother of Ludwig. His collectives

briefly vied with Kolmorgorov's measure-theoretic formulation for a role in the foundations of statistics. See, for example, [47, 1 7] .

1 9Roy Weintraub makes this speculation about Tintner's role in [65].

20Except for one brother who eventually joined Wald in the U.S. , all of his immediate family perished in the Holocaust. Wald was just one of many scholars making their

way out of Europe in the wake of the Nazi takeover. For an account of the experiences of emigre mathematicians in the U.S., see [38]; on scholars more generally,

see [7]. 21As a graduate student, Harold Hotelling had applied unsuccessfully for an economics fellowship at Columbia. Hoping he could pursue his interests in probability and

economics elsewhere, he went to Princeton in 1 921 on a mathematics fellowship, but he found no one working in his areas of interest. Instead, he did his research in

topology and differential geometry with Oswald Veblen and Luther P. Eisenhart. This was only a temporary shift in Hotelling's focus, however. He later applied some

topological theory to his statistical research, but with the exception of the published version of his dissertation and one other research paper, the rest of his publica­

tions dealt with statistical topics. See [1 6]. 22 See Series Ill .A., box 1 1 4, folder 5, Carnegie Corporation of New York Records, Columbia University. 23Hotelling had just left to chair a new department of mathematical statistics at the University of North Carolina. See [33].

28 THE MATHEMATICAL INTELLIGENCER

Figure 2. Harold Hotelling. Illustration cour­

tesy of the Columbia University Archives­

Columbiana library. Photo by Alman & Co.

3] . His colleagues described him as "a

gentle and kindly friend" [ 14, p. 19], re­

porting that the students, who came

from all over the world, "loved and re­

spected him" [29, p. 366].24

The recently organized American

mathematical statistics community

quickly became Wald's professional

home. By 1943, he was a fellow of the

IMS and was elected its president in

1948 while simultaneously serving as

vice-president of the ASA. But Wald

had also been a fellow of the Econo­

metric Society since 1939, and his net­

work still included economists, many

of them European emigres. One of

them, Jacob Marschak of the New

School for Social Research, had come

to New York in 1940 by way of the Uni­

versity of Oxford's Institute of Statis­

tics, which he had directed after being

dismissed from the University of Hei-

delberg in the wake of the Nazis' Jew­

ish boycott [1 ] .25 He started a seminar

on econometric methods with others in

the New York area. Hotelling, Wald,

and several others with connections to

the mathematical statistics community

attended and contributed to the semi­

nar. One of these, Henry Mann, had re­

ceived his Ph.D. from Vienna in 1935

for a dissertation on algebraic number

theory and emigrated in 1938. He tu­

tored in New York until obtaining fund­

ing from the Carnegie Corporation to

study statistics at Columbia. He and

W ald collaborated on several papers,

including one that grew out of their

work in Marschak's seminar [24].

The local communities supporting

Wald's research in New York, like

those in Vienna, transcended univer­

sity and disciplinary boundaries. In ad­

dition to his colleagues in Marschak's

seminar and at Columbia, Wald worked

for more than two years with statisti­

cians and economists on the staff of

the Statistical Research Group (SRG),

a branch of the National Defense Re­

search Committee (NDRC). That orga­

nization, the brainchild of Vannevar

Bush, president of the Carnegie Insti­

tution of Washington, served to "cor­

relate and support scientific research

on the mechanisms and devices of war­

fare" [34]. Bush had organized the

NDRC in 1940 under an order from

President Roosevelt, and although it

initially had no division for research in

mathematics, Bush added the Applied

Mathematics Panel (AMP) in 1942,

which included the SRG. W. Allen

Wallis, Milton Friedman (both econo­

mists), Hotelling, and Wald were

among the principal staff members of

the statistical group. Their work fo­

cused on studies of damage to aircraft

from anti-aircraft guns, on methods of

most effectively bombing targets, and

on statistical methods of inspection in

production [37, 64].

As part of this team, Wald devel-

oped the sequential probability ratio

test, an idea that would later play im­

portant roles in the theory and appli­

cation of statistics. The details of

Wald's discovery have been recorded

by his colleague Allen Wallis [ 64]. Early

in 1943, Wallis had begun to work for

a Navy captain on some inference

problems involving ordnance testing.

Discussing the problems involved with

performing large numbers of tests, the

captain suggested that a "mechanical

rule which could be specified in ad­

vance stating the conditions under

which the [testing] might be termi­

nated earlier than planned" could serve

to eliminate waste in the testing

process [64, p. 325]. Wallis mentioned

the problems to Friedman, and the two

began discussing it informally, outside

of their regular work for the SRG. De­

termining that the problem required

more statistical knowledge than they

possessed, Wallis and Friedman ex­

plained the problem to Wald. Initially

unenthusiastic about the prospects of

solving it, Wald called two days later

with an outline of the basic ideas of the

SPRT, a test that uses data as they are

gathered to determine when to stop an

experiment or an inspection. Rather

than basing the experiment or test on

a fixed sample size, a sequential sam­

pling plan provides a rule for deciding,

after each trial, whether to take a cer­

tain action or to make another obser­

vation.

Soon after formulating his ideas,

Wald began work on a monograph

treating the theoretical properties of

his test [57] , while Harold Freeman, a

professor of statistics in the depart­

ment of economics and social science

at the Massachusetts Institute of Tech­

nology, began a manual describing its

applications [8]. Wald also spoke about

some of the theory underlying his se­

quential methods at the 1944 summer

meeting of the IMS held with the Amer­

ican Mathematical Society [58] and

24 An historical analysis of the program begun by Hotelling and Wald at Columbia would provide 1nteresting information about the development of the American math­

ematical statistics community in the second half of the twentieth century. Such an inquiry, which would go beyond an exploration of the connections between Wald's

communities and his research on sequential analysis, is outside the scope of this discussion. 25The New School for Social Research had opened in 1 91 9, founded by a group of progressive scholars that included John Dewey, Charles Beard, Thorstein Veblen,

and Franz Boas. In addition to providing opportunities for research for social scientists, the school offered an adult education program modeled on the German Volks­

hochschulen. Reorganized in 1 922 under Alvin Johnson, an economist and editor of the New Republic, the New School focused more narrowly on adult education un­

til 1 933 when Johnson saw an opportunity to rebuild the school 's research program. Over the next year, he brought a dozen social scientists dismissed from their po­

sitions in Germany to New York and established what became the Graduate Faculty of Political and Social Science. See [20, 42).

VOLUME 26. NUMBER 1. 2004 29

published a long paper in the Annals of Mathematical Statistics in 1945 dis­cussing the theory and applications of the SPRT [61]. That same year he con­tributed a non-technical, expository paper on the fundamental ideas and ap­plications of the test to Journal of the ASA [60]. As the editor of the Journal explained in a footnote to the paper, W ald had specifically written it "to be accessible to statisticians with little mathematical background" [60, p. 277, note]. The appearance of Wald's ideas in these two periodicals-at different levels of mathematical sophistica­tion-highlights the distinction still present in the mid-1940s between the communities of applied and mathe­matical statisticians.

Wald's ideas started a flurry of ef­forts on the part of other researchers to explore questions raised by his dis­coveries.26 Much of this research emerging from Wald's ideas found its way onto the pages of the Annals of Mathematical Statistics. That discus­sion about Wald's new ideas in se­quential sampling occurred in what had become the official publication of the American mathematical statistics community suggests that the Annals had come to play a crucial role in ad­vancing the community's discipline. No longer did its existence simply add to the distinctiveness of mathematical statistics by providing the discipline with an important professional ac­coutrement; mathematical statisticians like Wald seemed to regard it as hav­ing the credibility to record their con­tinuing conversations about their the­oretical work.

In addition to providing a subject for fruitful theoretical research of the sort that appeared in the Annals, sequential sampling offered a practical means of reducing the number of observations needed for testing and quality control. In his introduction to the Summary Technical Report of the Applied Math­ematics Panel, Warren Weaver com­mented on the usefulness of the SPRT,

saying that the "Quartermaster Corps reported in October 1945 that at least 6,000 separate installations of sequen­tial sampling plans had been made" [37, p. 614].

Sequential analysis provides an im­portant example of an area of research that combined mathematical theory with statistical applications. Its prob­lems and their solutions addressed the practical needs of manufacturers and scientists, and at the same time at­tracted the technical and theoretical in­terests of the mathematical statisti­cians. Abraham W ald himself seemed to be an ideal member of the commu­nity practicing this discipline situated between theory and application. He brought to his investigations, as his student and collaborator Jacob Wol­fowitz wrote, "a high level of mathe­matical talent of the most abstract sort, and a true feeling for, and insight into, practical problems" [68, p. 4]. In this case, work on practical problems of de­fense promoted theoretical advances.

These theoretical advances ex­tended beyond the field of inspection sampling in which the SPRT originated. In fact, the theory of sequential analy­sis became an important aspect of Wald's theory of decision functions. Decision theory generalized the ques­tions addressed by statistical inference by determining a rule based on ran­domly selected observations for choos­ing the best course of action from a set of possibilities.

W ald had begun developing deci­sion theory early in his time of formal study of modem statistics with Hotelling, several years before his World War II work on the SPRT. In 1939 he published a paper in the An­nals of Mathematical Statistics intro­ducing its central ideas. Here he artic­ulated the idea of generalizing the problems of hypothesis testing and constructing confidence intervals, seeking to build a theory that would in­clude them as special cases. Wald con­ceived of an approach that would pro-

vide a means of choosing among any number of hypotheses (in contrast to the Neyman-Pearson theory, which ad­mitted only two) by specifying a sys­tem of acceptance regions according to criteria that would "depend on the rel­ative importance of the different pos­sible errors" [55, p. 301].

Wald lectured briefly on these ideas in a 1941 series of addresses organized by Menger at the University of Notre Dame, but only resumed research on them a few years later, after beginning his work in sequential analysis. 27 By then John von Neumann and Oskar Morgenstern had published their 1944 landmark work, Theory of Games and Economic Behavior [50]. Perhaps sur­prisingly, these two had not met before settling at Princeton. They had had a number of contacts in Vienna in com­mon, including Wald and Menger. Both had also begun thinking about ideas re­lated to game theory before meeting, and their earlier work had some con­nections to the economics and mathe­matics communities in Vienna. In par­ticular, Menger had written a book taking a mathematical approach to so­cial ethics that influenced some of Mor­genstern's research in the 1930s.28

Since his days in Vienna Wald had been familiar with some of von Neu­mann's work in economics, having edited the latter's paper on equilibrium in a dynamic economy for the final vol­ume of the Ergebnisse [49]. This paper had some connection to von Neu­mann's first work on game theory, pub­lished in 1928 [48], and while working with the SRG, W ald mentioned to a col­league that some of his ideas in deci­sion theory were based on that 1928 pa­per.29 In a discussion of the work by von Neumann and Morgenstern in Mathematical Reviews, Wald pointed out in 1945 that "the theory of games has applications to statistics . . . , since the general problem of statistical inference may be treated as a zero-sum two-person game" [59].

His next paper on decision func-

26More than 1 8 papers related to sequential sampling were published between 1 945 and 1 950 [9, pp. 8-9]. A bibliography published in 1 960 lists 374 references deal­

ing with sequential analysis that appeared through 1 959 [1 9] .

27Menger organized a Mathematical Colloquium at Notre Dame fashioned after the one i n Vienna. Wald's lectures at that Colloquium were published as [56].

28Menger's book and his motivation for writing it are discussed in [23]. The influence of Menger on Morgenstern's ideas is treated in [22].

29For an account of that conversation, see [64, p. 334]. The connection between von Neumann's two papers is described in [65].

30 THE MATHEMATICAL INTELUGENCER

tions, appearing the same year in the

Annals of Mathematics [62], elabo­

rated on the connections between sta­

tistical inference and the zero-sum

two-person game. The Annals of Math­ematics was one of the key publication

venues for the (pure) mathematical re­

search community in the United States,

and this paper was not Wald's only con­

nection to that community. He was a

member of the American Mathematical

Society and had already published two

papers in the Society's Transactions. W ald would publish several more in the

Annals of Mathematics as well as in

the Bulletin of the AMS over the next

five years. These papers treated math­

ematical issues raised by sequential

analysis, decision theory, and game

theory.

The results linking game theory

with decision function theory highlight

the interaction among the communi­

ties in which Wald participated, both in

Vienna and in the United States. Wald's

success in raising and answering ques­

tions of interest to a variety of re­

searchers was perhaps due in part to

what one colleague described as "his

open-mindedness" to others' pursuits.

"He was ever ready to listen to the prob­

lems other scholars encountered and he

was eager to speak about the work he

had in progress himself' [29, p. 366].

In the late 1940s, the publications of

Wald, von Neumann, and Morgenstern

influenced the research of economists

at the Cowles Commission. This work,

which was an "attempt to discover

what kind of behavior on the part of an

individual or group in specified cir­

cumstances would most completely

achieve the goals pursued," drew on

the ideas of Wald and von Neumann

and Morgenstern, and led to research

in decision making under uncertainty

by Jacob Marschak and Leonid Hur­

wicz [4, p. 48].30 So the convergence of

Wald's communities in the 1930s and

1940s, across national and disciplinary

lines, had fundamental connections to

his research in statistics, particularly to

his work on sequential analysis and de­

cision function theory.

Figure 3. Abraham Wald in 1950. Illustration

courtesy of the Columbia University Archives­

Columbiana Library.

Conclusion

Wald's early training with Menger in

geometry was far removed from his

work in decision theory-research that

colleagues at the time called his most

significant contribution to statistics. 31

Wald shifted his interests from pure

mathematics to statistics in less than a

decade, and from the perspective of

the disciplines themselves, this shift

has the appearance of a clean break, a

jump discontinuity. A wider historical

focus, however, that considers the con­

text of the scientific communities to

which Wald belonged, brings some

continuity to light and helps explain

the connections between Wald's many

professional relationships and discipli­

nary interests.

From his earliest years in Vienna,

although studying pure mathematics,

Wald found himself working with re­

searchers engaged in a wide range of

intellectual pursuits. Karl Menger, in

particular, introduced him to the cir­

cles of Viennese economists. Although

the use of mathematical methods in

economics research was not the dom­

inant fashion in the discipline, Wald

met some economists whose mathe-

30Some of Marschak's work in this field appeared in [25]; some of Hurwicz's contributions can be found in [1 8] .

3 1See, for example, [68, p. 9 ] and [14 , p. 1 9] .

matical inclinations drew him into the

world of econometrics, a professional

community on the border of several

fields.

Because these economists were not

tied exclusively to the university, these

connections provided him immediate

employment in Vienna at a time when

the political and social climate barred

him from traditional academic em­

ployment. The increasingly interna­

tional character of this econometric

community had an even more far­

reaching impact on Wald's opportuni­

ties. Some of his colleagues, including

Menger, were active in an international

network of researchers with ties to the

Econometric Society, which had been

holding meetings in the United States

and Europe since its inception in 1930.

Wald's research and abilities caught

the attention of this network, resulting

first in a job at the Cowles Commis­

sion-his ticket out of Nazi Europe­

and then eventually in significant

influence on the direction of the econo­

metric community's research.

In the meantime, however, after his

few months at the Cowles Commis­

sion, Wald moved to Columbia to work

with Harold Hotelling. Here, his formal

contributions to the discipline of math­

ematical statistics began. His connec­

tions to the American statistics com­

munity began to grow stronger as well.

Like the world of Viennese economics

that W ald had left behind, the statistics

community in the United States had

somewhat fluid disciplinary and insti­

tutional boundaries, as well as impor­

tant international connections. Its

members did research in the theory of

statistics as well as in statistical appli­

cations to economics and biology,

among other fields. Universities, gov­

ernment, and private organizations

supported the community's work.

Wald benefited from and con­

tributed to the efforts of these patrons

of statistics-at the privately financed

Cowles Commission, as a researcher

and professor at Columbia with

Carnegie funding, and through his de­

fense-related research in the SRG. His

VOLUME 26, NUMBER 1 , 2004 31

A U T H OR

PATTl WlLQIR HUNTER

international connections had helped

him move to the United States, and he

continued to interact with his fellow emigres as well as with researchers

abroad. In late 1950, Wald was on a lec­

ture tour through London, Paris, and

Rome. En route to speak at the Indian

Statistical Institute, he died in a plane

crash on December 13.

As a reviewer of Wald's text on de­

cision theory wrote in 1951, "Wald's

death [gave] Statistical Decision Functions [63] an altogether new sig­

nificance. When it appeared, it was

threatened with rapid obsolescence by

the activity of Wald himself." All but

the fmal comments of this review had

been written before Wald's death,

when the reviewer "believed Wald's

great work on statistical decision to be

but a fraction of what he was about to

achieve" [43, p. 67] . How his later

achievements might have affected the

32 THE MATHEMATICAL INTELLIGENCER

research of the network of profes­

sional communities in which he par­

ticipated of course cannot be known.

The importance of the connections be­

tween that network and what Wald did

accomplish is clear.

REFERENCES [ 1 ] Kenneth J . Arrow. "Marschak, Jacob." In

International Encyclopedia of the Social

Sciences, ed. David L. Sills. The Free

Press, New York, 1 979.

[2] Kenneth J. Arrow, "Von Neumann and the

Existence Theorem for General Equilib­

rium. " In John von Neumann and Modern

Economics, ed. M . Dare et al. University

Press, Oxford, 1 989, 1 5-28.

[3] Bela B. Edwards. Preface. Collections of

the American Statistical Association 1 , part

1 (1 843). v-x.

[4] Carl F. Christ. "History of the Cowles Com­

mission, 1 932-1 952." In Economic Theory

and Measurement. Cowles Commission

for Research in Economics, Chicago,

1 952, 3-65.

[5] Earlene Craver. "The Emigration of the

Austrian Economists." History of Political

Economy 1 8 (1 986), 1 -32.

[6] Egbert Dierker and K. Sigmund, eds.

Ergebnisse eines Mathematischen Kof/o­

quiums. Springer-Verlag, Vienna, 1 998.

[7] Laura Fermi. Illustrious Immigrants, 2d ed. University of Chicago Press, Chicago,

1 97 1 .

[8] Harold Freeman. Sequential Analysis of

Statistical Data: Applications. Statistical

Research Group, Columbia University,

New York, 1 944.

[9] Bashkar K. Ghosh. "A Brief History of Se­

quential Analysis." In Handbook of Se­

quential Analysis, ed. Bashkar K. Ghosh

and Pranab K. Sen. Marcel Dekker, Inc. ,

New York, 1 991 , 1 -20.

[1 0] Louise Galland and Karl Sigmund. "Exact

Thought in a Demented Time: Karl Menger

and his Viennese Mathematical Collo­

quium." The Mathematical lntef/igencer 22

(2000), 34-45.

[1 1 ] Thomas L. Haskell. The Emergence of

Professional Social Science. University of

Il l inois Press, Urbana, 1 977.

[1 2] Frederick L. Hoffman. "Problems of Social

Statistics and Social Research." Publica­

tions of the American Statistical Associa­

tion 1 1 (1 908-1 909), 1 05-32.

[1 3] Robert V. Hogg. "On the Origins of the In­

stitute of Mathematical Statistics." Statis­

tical Science 1 (1 986), 285-291 .

[ 1 4] Harold Hotelling. "Abraham Wald." Amer­

ican Statistician 5 (1 951 ), 1 8-1 9.

(1 5] Patti W. Hunter. "Drawing the Boundaries:

Mathematical Statistics in 20th-Century

America." Historia Mathematica 23 (1 996).

7-30.

[1 6] Patti W. Hunter. "An Unofficial Community:

American Mathematical Statisticians be­

fore 1 935." Annals of Science 56 (1 999),

47-68.

[ 1 7] Patti W. Hunter. "Foundations of Statistics

in American Text books: Probability and

Pedagogy in Historical Context , " in Using

Recent History of Mathematics in Teach­

ing Mathemathics. Mathematical Associa­

tion of America, forthcoming.

[1 8] Leonid Hurwicz. "Theory of the Firm and

of Investment." Econometrica 1 4 (1 946),

1 09-1 36.

(1 9] J. Edward Jackson. "Bibliography on Se­

quential Analysis." Journal of the American

Statistical Association 55 (1 960), 561-580.

[20] Claus-Dieter Krohn. lntef/ectuals in Exile:

Refugee Scholars and the New School for

Social Research. Trans. Rita Kimber and

Robert Kimber. University of Massachu­

setts Press, Amherst, 1 993.

[21 ] Dickson H. Leavens. "Report of the

Chicago Meeting, December 28-30,

1 936." Econometrica 5 (1 937), 1 84-1 97.

[22] Robert J. Leonard. "From Parlor Games

to Social Science: von Neumann, Mor­

genstern, and the Creation of Game The­

ory: 1 928-1 944." Journal of Economic Lit­

erature 32 (1 995), 730-61 .

[23] Robert J. Leonard. "Ethics and the Ex­

cluded Middle: Karl Menger and Social

Science in Interwar Vienna." Isis 89 (1 998),

1 -26.

[24] Henry B. Mann and Abraham Wald. "On

the Statistical Treatment of Linear Sto­

chastic Difference Equations." Economet­

rica 1 1 (1 943), 1 73-220.

[25] Jacob Marschak. "The Role of Liquidity

under Complete and Incomplete Informa­

tion." The American Economic Review 39

(3) (1 949), Papers and Proceedings of the

Sixty-first Annual Meeting of the American

Economic Association, 1 82-1 95.

[26] Richmond Mayo-Smith. "Statistical Data

for the Study of the Assimilation of Races

and Nationalities in the United States."

Publications of the American Statistical

Association 3 (1 892-1 893), 429-449.

[27] Karl Menger. "The Formative Years of

Abraham Wald and His Work in Geome­

try." Annals of Mathematical Statistics 23

(1 952), 1 4-20.

[28] Karl Menger. Selected Papers in Logic,

Foundations, Didactics, and Economics.

D. Reidel Publishing Company, Dordrecht,

1 979.

[29] Oskar Morgenstern. "Abraham Wald,

1 902-1 950." Econometrica 1 9 (1 951 ) ,

361 -367.

[30] Jerzy Neyman. 'The Emergence of Math­

ematical Statistics. " In On the History of

Statistics and Probability, ed. D. B. Owen.

Marcel Dekker, New York, 1 976, 1 47-193.

[31 ] Simon N. D. North. "The Relation of Sta­

tistics to Economics and Sociology." Pub­

lications of the American Statistical Asso­

ciation 1 1 (1 908-09), 431 -446.

[32] Notes. Bulletin of the American Mathe­

matical Society 27 (1 920-1 921 ) , 336-337 .

[33] E. Shepley Nourse and Bernard G. Green­

berg. "Statistical Training and Research:

The University of North Carolina System. "

International Statistical Review 4 6 (1 978),

1 71 -207.

[34] Order Establishing the National Defense Re­

search Committee. In Scientists Against

Time, ed. James Phinney Baxter I l l . Little,

Brown and Company, Boston, 1 946, 451 .

[35] Theodore Porter. The Rise of Statistical

Thinking: 1820-1900. University Press,

Princeton, 1 986.

[36] Program of the Annecy Meeting, Septem­

ber 1 1 -15 , 1 937. Econometrica 6 (1 938),

9 1 .

[37] Mina Rees. "The Mathematical Sciences

and World War I I . " The American Mathe­

matical Monthly 87 (1 980), 607-621 .

[38] Nathan Reingold. " Refugee Mathemati­

cians in the United States of America,

1 933-1 941 : Reception and Reaction . "

Annals o f Science 38 (1 981 ) , 31 3-338, or

A Century of Mathematics in America ­

Part I, ed. Peter Duren et al . Providence:

American Mathematical Society, Provi­

dence, 1 988, 1 75-200.

[39] Henry L. Rietz et al. Handbook of Mathe­

matical Statistics. Houghton Mifflin Com­

pany, Boston, 1 924.

[40] Charles F. Roos. "The Organization of the

Econometric Society in Cleveland, Ohio,

December 1 930." Econometrica 1 (1 933),

7 1 -72.

[41 ] Dorothy Ross. The Origins of American

Social Science. University Press, Cam­

bridge, 1 991 .

[42] Peter M. Rutkoff and William B. Scott.

New School: A History of the New School

for Social Research. The Free Press, New

York, 1 986.

[43] L. J. Savage. "The Theory of Statistical

Decision." Journal of the American Statis­

tical Association 46 (1 951 ) , 55-67.

[44] Karl Sigmund. "Menger's Ergebnisse -A

Biographical Introduction." In Ergebnisse

eines Mathematischen Kolloquiums, ed.

E. Dierker and K. Sigmund. Springer-Ver­

lag, Vienna, 1 998, 5-31 .

[45] Karl Sigmund. "Karl Menger and Vienna's

Golden Autumn." In Karl Menger: Selecta

Mathematica, vol. 1 , ed. Bert Schweizer

et al. Springer-Verlag, Vienna, 2002,

7-21 .

[46] Stephen Stigler. The History of Statistics:

The Measurement of Uncertainty before

1900. Harvard University Press, Cam­

bridge, 1 986.

[47] R . von Mises and J. L. Doob. "Discussion

of Papers on Probability Theory." Annals

of Mathematical Statistics 1 2 ( 1 941 ) ,

2 1 5-2 1 7 .

[48] John von Neumann. "Zur Theorie der

Gesellschaftsspiele." Mathematische An­

nalen 1 00 (1 928), 295-320.

[49] John von Neumann. "Uber ein okonomi­

sches Gleichungssystem und eine Verall­

gemeinerung des Brouwerschen Fix­

punksatzes. " Ergebnisse eines mathema­

tischen Kolloquiums 8 (1 935-36), 73-83.

[50] John von Neumann and Oskar Morgen­

stern. Theory of Games and Economic Be­

havior. University Press, Princeton, 1 944.

[51 ] Abraham Wald. "Uber die eindeutige pos­

itive Losbarkeit der neuen Produktions­

gleichungen (Mitteilung 1) . " Ergebnisse

eines mathematischen Kol/oquiums 6

(1 935), 1 2-1 8.

[52] Abraham Wald. "Uber die Produktions­

gleichungen der okonomischen Wertlehre

(Mitteilung I I ) . " Ergebnisse eines mathe­

matischen Kol/oquiums 7 (1 936), 1 -6.

[53] Abraham Wald. "Uber einige Gleichungs­

systeme der mathematischen Okonomie."

Zeitschrift fur Nationalokonomie 7 (1 936),

637-670.

[54] Abraham Wald. "Die Widerspruchsfreiheit

des Kollektivbegriffes der Wahrschein­

lichkeitsrechnung." Ergebnisse eines math­

ematischen Kolloquiums 8 (1 937), 38-72.

[55] Abraham Wald. "Contributions to the The­

ory of Statistical Estimation and Testing

Hypotheses. " Annals of Mathematical Sta­

tistics 1 0 (1 939), 299-326.

[56] Abraham Wald. On the Pnnc1ples of Sta­

tistical Inference. Notre Dame Mathemat­

ical Lectures, No. 1 . University of Notre

Dame, 1 942.

[57] Abraham Wald. Sequential Analysis of

Statistical Data: Theory. Statistical Re­

search Group, Columbia University, New

York, 1 943.

[58] Abraham Wald. "On Cumulative Sums of

Random Variables ." Annals of Mathemat­

ICal Statist1cs 1 5 (1 944), 283-296.

[59] Abraham Wald. Review of Theory of

Games and Economic Behavior, by John

von Neumann and Oskar Morgenstern.

Mathematical Reviews 6 (1 945), 237.

[60] Abraham Wald. "Sequential Method of

Sampling for Deciding between Two

Courses of Action." Journal of the Ameri­

can Statistical Association 40 (1 945),

277-306.

[61 ] Abraham Wald. "Sequential Tests of Sta­

tistical Hypotheses." Annals of Mathemat­

ical Statistics 1 6 (1 945), 1 1 7-1 86.

[62] Abraham Wald. "Statistical Decision Func­

tions which Minimize the Maximum Risk."

Annals of Mathematics 46 (1 945), 265-

280.

[63] Abraham Wald. Statistical Decision Func­

tions. John Wiley and Sons, Inc. , New

York, 1 950.

[64] W. Allen Wallis. "The Statistical Research

Group, 1 942-1 945." Journal of the Amer­

ican Statistical Association 75 (1 980),

320-335.

[65] E. Roy Weintraub. "On the Existence of

a Competitive Equilibrium: 1 93Q-1 954."

Journal of Economic Literature 21 (1 983),

1 -39.

[66] Walter F. Willcox. "Cooperation between

Academic and Official Statisticians." Pub­

lications of the American Statistical Asso­

ciation 1 4 (1 9 1 4-1 9 1 5) , 281 -293.

[67] Walter F. Willcox. "Lemuel Shattuck, Sta­

tist, Founder of the American Statistical

Association . " Journal of the American Sta­

tistical Association 35 (1 940), 224-235.

[68] Jacob Wolfowitz. "Abraham Wald, 1 902-

1 950." Annals of Mathematical Statistics

23 (1 952), 1 -1 3 .

VOLUME 26, NUMBER 1 , 2004 33

Mathematics in the Library of El Escorial Pieter Maritz

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cafe

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

If so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

e-mail: dirk.huylebrouck@ping.be

The Real Monasterio de San Lorenzo

at El Escorial (henceforth referred

to just as El Escorial) near Madrid is

the monument which best represents

the ideological and cultural aspirations

of the Spanish Golden Age, that period

of nearly two hundred years that in­

cludes the reigns of Charles I, King of

Spain (better known as Emperor

Charles V), and his son Philip II.

El Escorial was originally designed

for a variety of purposes: (1) it is a

monastery for the monks of the order

of St. Jerome, whose church was the

pantheon of Emperor Charles V and his

wife, his son, Philip II, his relatives and

heirs; (ii) it is a palace to house the

King, patron of the foundation, and his

entourage; (iii) the college and semi­

nary complete the religious function of

the Monastery; (iv) and the Library

complements these three.

The victory of Philip II's army over

Henry II of France at Saint Quentin (a

town near Paris) coincided with the

feast of San Lorenzo (Saint Lawrence)

on August 10, 1557. The victory at Saint

Quentin led, in part, to the naming of

the Monastery at El Escorial. Philip II

had decided to build the Monastery for

two reasons: to give his father a digni­

fied burial after the latter had spent

his last years among the Jeronymite

monks of San Yuste, and to show his

gratitude for the victory over Henry II

of France. Therefore, Philip II decided

to build a temple in honor of San

Lorenzo, who was supposed to have

suffered martyrdom on August 10 of

the year 248. Philip II began his search

for the ideal site in 1558, which he fi­

nally located in 1562. The name given

to the place was El Sitio de San

Lorenzo el Real, which happened to be

a plateau on the foothills of the

Guadarrama. However, the Monastery

was better known from the nearby

hamlet-El Escurial, or El Escorial.

The Spaniard Juan Bautista de Toledo

was appointed architect to Philip II by

an ordinance dated 15 July 1559. In the

spring of 1562, Toledo measured the

34 THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

ground, plotted the foundations, and

organized the working teams for El Es­

corial [3]. The Monastery is situated on

the south side of the chain of moun­

tains dividing the central plain of

Castile. It took 21 years to build: April

23, 1563 to September 13, 1584.

El Escorial can by no means be con­

sidered the work of a single architect,

but rather the product of close collab­

oration between two men: Juan

Bautista de Toledo and Juan de Her­

rera, a young Asturian [1 , p. 10]. Juan

de Herrera first appeared at El Escor­

ial in 1563 when Philip II appointed him

Toledo's assistant. The main structure

of El Escorial was made of wood

beams and stone. The walls were hung

with tapestries from the Flemish

looms, and the Milanese, renowned at

that period for superb workmanship in

steel, gold, and precious stones, con­

tributed many exquisite specimens of

art [2, p. 164]. The roofs are of slate and

lead pieces. The architectural design is

Toscan and Doric order.

When Juan Bautista de Toledo died

in 1567, the entire south fac;ade had

been built, as well as the two-storied

grandiose Courtyard of the Evange­

lists. After Toledo's death, not Herrera

but Giovanni Battista Castello of Berg­

amo and Genoa was in charge at El

Escorial. It was not until 1572 that

Juan de Herrera was fully entrusted

with the immense labor. He was re­

sponsible for the completion of the

complex, including several parts of

the building which had not been

designed by Toledo. By 1571, the

Monastery area was almost complete;

work commenced on the House of the

King in 1572 and on the Basilica in

1574, which was consecrated in 1595,

the year in which most agree the

Monastery was completed. Nonethe­

less the last stone was placed in 1584,

and a few more years were spent on

its decoration.

The great monastic church, the

Basilica, is the "raison d'etre" of El Es­

corial. It consists of two churches,

Figure 1 . General view of the Library.

namely the people's church, or the So­

tocoro, and the Royal Chapel and

monastic church, which make up the

main body. On leaving the Basilica, one

crosses the Kings' Courtyard, ascend­

ing a staircase at the right of the

vestibule to reach the Library.

The Library, which lies at the very

center of the main side of the

Monastery, is one of the Monastery's

great rooms. That it is in such a promi­

nent position shows the importance

Philip II attributed to it in its total cre­

ation. The Library has over 40,000 edi­

tions, including an impressive number

of Latin, Greek, Arabic, and Hebrew

manuscripts.

Because of the number of windows,

the 55-by-10-meter Library is bright,

full of majesty and light [ 1 , p. 28]. The

frescoes on allegorical themes, painted

between 1586 and 1593 by the Italian

painters Pellegrino Tibaldi and Nicolas

Granello in the mannerist style on the

walls and ceilings, are clearly influ­

enced by Michelangelo. The complex

and extensive iconography, which

mostly represents or depicts the great

wise men of Antiquity, was the brain­

series of frescoes by Tibaldi begins at

the entrance to the Library with a rep­

resentation of Philosophy, and at the

child of Father Jose de Sigiienza. The Figure 2. Arithmetica.

far side, the south side or Convent end,

with a representation of Theology. Be­

tween the two extremes, the seven lib­

eral arts (Artes Liberales) are organized

under the medieval dictum of the Triv­

ium (Gramatica, Ret6rica, Dialectica)

and that of the Quadrivium (Aritmetica,

Musica, Geometria, Astrologia). There

is an allegorical representation of each

one of these arts situated in separate

sections of the ceiling with two learned

disciples of each science depicted in

the semicircular "windows" on either

side. Below the ceiling, the friezes are

adorned with yet more references to

the particular science depicted directly

above. Tibaldi's allegories are almost

fully Baroque in their realistic illusion

of space, their stress on volume, and

their contrasting illumination.

For the history of these arts, see [ 4],

and for a recent article on the seven

liberal arts in Munster's Hall of Peace,

see [5].

The floor of the Library is made of

gray marble and the walls are covered

with fmely elaborated Doric bookcases

(see figure 1) that were built by Jose

Flecha, Juan Senen, and Martin de

Gamboa according to designs by Juan

de Herrera.

VOLUME 26, NUMBER 1 , 2004 35

36 THE MATHEMATICAL INTELLIGENCER

On the five brown marble tables

distributed throughout the hall, a large

collection of terraqueous and celestial

globes, maps, astrolabes are on dis­

play, which suggests the scientific

cabinet status which the Library un­

doubtedly had. Also in the Library is

an armillary sphere, made by Antonio

Santucci around 1582, in accordance

with the Ptolemaic system, the earthly

and celestial spheres of Jean Blaeu

from around 1660, and the stone-mag­

net that was apparently found during

the Monastery foundational excava­

tions.

In 1573, Philip II began to assemble

the bodies of his dead relatives, and to

place them in temporary vaults pre­

pared to receive them. He commenced

with his father and mother, for he made

no effort to disturb the bodies of Fer­

dinand V and Isabella I at Granada; his

first and third wives were reburied at

El Escorial, but the remains of Mary

Tudor are in Westminster Abbey to this

day. The Mausoleum now holds all the

Spanish monarchs from Emperor

Charles V (that is, Charles I, King of

Spain) to Alfonso XIII (except the re­

mains of Philip V, Philip VI, and their

wives), and also of Don Juan de Bour­

bon and his wife, Dona Maria de las

Mercedes.

On the morning of September 13,

1598, in a little room off the Basilica in

El Escorial, just as the sun was rising

above the stony peaks of the Guadar­rarnas, Philip II died of (probably) dia­

betic gangrene at the age of seventy­

one [2, pp. 309, 310], [6, p. 726].

Figure 3. Detail of figure 2 (top left). Figure

4. Measuring (middle left). Figure 5. Detail of

figure 4 (bottom).

Figure 6. Calculating.

Figure 7. Detail of figure 6.

Acknowledgments

The author gratefully acknowledges

the assistance rendered by Joan Roux

and Soretha Swanepoel with some of

the photographs.

REFERENCES ( 1 ] Carmen Garcia-Frfas and Jose Luis Sancho

Gaspar. Real Monasterio de San Lorenzo

de el Escorial. Patrimonio Nacional, 1 999.

(2] Charles Petrie. Philip II of Spain. Eyre and

Spottiswoode, London, 1 963.

[3] George Kubler and Martin Soria. Art and Ar­

chitecture in Spain and Portugal and their

American dominions 1500 to 1 800. Pen­

guin books, Middlesex, 1 959.

(4] B. Artmann. The Liberal Arts. The

Mathematical ln telligencer 20(3), 40-4 1 ,

1 998.

[5] N. Schmitz. Mathematics in the Hall of

Peace. The Mathematical lntelligencer 24(4),

34-36, 2002.

[6] William Thomas Walsh. Philip II. Sheed and

Ward, London, 1 938.

Mathematics Department

University of Stellenbosch

Stellenbosch, South Africa

e-mail: pm@sun.ac.za

VOLUME 26. NUMBER 1 . 2004 37

JEAN-MICHEL KANTOR

Mathematics East and West , Theory and Practice : The Example of Distributions

Science with the grievous glance It [mathematics} casts a grievous glance on mankind, and forces it to confront the solid reality, the real fact only, the fact which destroys alike the most magnificent and the most caustic fantasies.

-Robert Musil Notebooks. Excerpt from book #16, "The Spy"

(1923-1924), W I 1979-80

hat lessons can be drawn from the upheavals that characterized twentieth-

century scientific development? Did mathematics undergo the same up-

heaval? Was its status modified, or did it retain at the time of Hiroshima

the moral and aesthetic value which Plato praised? These

questions are too general, but they suggest a debate.

We will study here only a very precise situation and con­

text-that of mathematical work conducted in Russia and

France from the 1930s, inspired among others by Jacques

Hadamard's seminal work, and which led to the worldwide

development of mathematical analysis and to the theory of

partial-differential equations. The relevant documents ex­

ist, and after more than fifty years, a historical inquiry is

possible.

The death of Laurent Schwartz, a prominent French

mathematician, member of the Bourbaki group, and one of

the driving forces of the mathematical community for more

than twenty years, can be an occasion to think back about

the birth of the theory of distributions. The recent publi­

cation of Soviet archives makes it possible to complement

the work of historians, in particular Adolph P. Yushkevich's

comments on the book [Lu] of Jesper Liitzen (whose rec­

ognized competence in the history of mathematics and

whose conscientiousness are beyond question). In the Ap­

pendix we provide a translation of Yushkevich's article,

where he examines very meticulously among others, the ar-

tides published in Russian (the references here comple­

ment those given in his article). Indeed, while times have

changed, language barriers persist, slowing down the ex­

change of ideas between the West and Russia, and hinder­

ing a wider diffusion of Yushkevich's text, although it was

published as early as 1991 in the historical journal which

he founded. Naturally mathematics is not exempt from

chauvinistic behavior in the international competition (the

"Popov effect," both in the East and in the West), but

Yushkevich is aware of this and does not indulge in it. He

seems to have at heart to show that there was an intense

mathematical life in the East, in the USSR, isolated as it

was by the cold war and the "construction of socialism in

a single country." Let us take a look at the various sensi­

bilities and styles revealed by this episode.

This is also an opportunity to take another look at the in­

ternational scientific cooperation of the period, which has

hardly ever been studied. The Fields medal was awarded to

Laurent Schwartz at Harvard in 1950 during the Korean

War-and some called it "the Fields medal of the cold war,"

referring to the difficulties experienced by Hadamard and

his nephew Schwartz in getting a visa to the United States.

© 2004 SPRINGER· VERLAG NEW YORK. VOLUME 26, NUMBER 1 , 2004 39

In any case this is a little-known episode in the relationship

between science and politics, as we shall see.

During the 1930s, the idea of generalized function or dis­

tribution was "in the air": it was used by the great physi­

cist Paul Adrien M. Dirac (1902-1984), and by Salomon

Bochner, whose work anticipated later work on distribu­

tions, in particular with respect to the role of Fourier se­

ries [Boc]: physicists were using distributions just as

Moliere's Monsieur Jourdain was producing prose-un­

awares. The very birth of the theory of generalized func­

tions/distributions can thus be rife with lessons for a time

when the relationships between mathematics and physics

are evolving (cf. [JQ]).

Above all, this study is an opportunity to bring into view

two different conceptions of the role of mathematics, in the

East and in the West (to simplify)-one, led by Schwartz

and Bourbaki, focusing on structures, and the other, cen­

tered on Sobolev and the Saint Petersburg school, closely

linked to physical sciences. All these questions are of in­

terest for the present, and we think that we owe it to the

memory of Laurent Schwartz and his keen sense of the

scholar's civic role to approach them-with honesty and

rigor-at last.

The Actors: Hadamard (1 865-1 963), Sobolev

(1 908-1 989), and Schwartz (1 91 5-2002)­Two Different Worlds

Laurent Schwartz is a mathematician admired all over the

world, known indeed beyond specialists' circles for his role

as a "mathematicien dans le siecle" [S2]-a "secular" math­

ematician, involved in the social and political world. One

of the active members of the Bourbaki group after World

War II, he was also a militant partisan of all the humani­

tarian causes of the twentieth century, from militant Trot­

skyism between 1936 and the Resistance to the "Comite

Au din" during the Algerian War, and the cause of the math­

ematicians standing for human rights in Eastern Europe.

Schwartz's personality brings together the qualities of the

French intellectual, growing out of a family with a long tra­

dition of social ascent, which supplied France with emi­

nent intellectuals.

Nothing except mathematics is shared between a Lau­

rent Schwartz and a Sergei Sobolev. The latter is also a fine

scholar, but gained much less fame in the West. Sergei

L'vovich Sobolev was born in Saint Petersburg in 1908, in

a family connected to the nobility; his father was a noted

lawyer from Saint Petersburg (later Leningrad). In the en­

during rivalry between Moscow and Saint Petersburg, a city

created by Peter the Great in 1703, the mathematical

schools had a particular role: Saint Petersburg was the city

of Euler, who lived there for a large part of his life, as well

as Chebyshev (1821-1894), Markov (1856-1922), Lyapunov

(1857-1918). This already shows that mathematical life in

the city was very open to the sciences and technology. It

was also in Saint Petersburg where the managerial talents

of Steklov (1863-1926), an applied mathematician, led to

the creation of research institutes of the Academy which

later bore his name. A detailed account of political strug-

40 THE MATHEMATICAL INTELLIGENCER

gles in Moscow and Leningrad within mathematical soci­

eties, and their dramatic consequences ("the Luzin affair"),

can be found in several recent publications like [De, Mar,

M-Sh, Viu, Y] , as well as in various issues of the history pe­

riodical launched by A. P. Yushkevich.

Sobolev was a brilliant student at a particularly young

age, like a number of other 20th-century Russians. At the

University, which he entered in 1925, he followed the

courses of Grigori'i Mikhailovich Fikhtengolts (1888-1959)

and Nikolai Maksimovich Gunther (1871-1941) (the latter

in potential theory). He met Vladimir Ivanovich Smirnov

(1887-1974), who would be a professor then a co-worker

of Sobolev, a professor from 1925, and later dean of the

"Mat-Mekh" faculty for 25 years. (This did not spare him

the displeasure of criticism in 1957 on the occasion of a

tribute to Euler: after Smirnov praised the positive influ­

ence of Frechet, who was attending the ceremony, on So­

viet mathematics, Kolmogorov chided him publicly for his

"love of the foreign" ([Y] , page 31).

Sobolev's first publication was a counter-example to a

result announced by Saltykov and reused by Gunther in his

analysis course. In 1929, after obtaining his doctorate, he

joined the Institute of Seismology, where he collaborated

with Smirnov, before joining the Steklov Institute, becom­

ing at 24 a corresponding member, then a full member­

the youngest-of the USSR Academy of Sciences. Along­

side his mathematical career, in which he was always open

to other sciences and other countries despite a difficult

context (he was fluent in French, which he had learned as

a child from his Belgian nanny), he conducted various pro­

jects including the creation of the Siberian Centre of the

Academy of Sciences. He always displayed Russian pride

and a strong loyalty to the Soviet power, as a member of

the Party from the 1930s, yet this loyalty did not prevent

him from taking sometimes difficult and courageous posi­

tions (for example in the Lysenko affair); but he sometimes

had a more orthodox stance, e.g., as one of the critics blam­

ing Luzin in 1936 for his openness and his foreign publica­

tions [De].

Somewhere between these two personalities is Jacques

Hadamard-"le petit pere Hadamard" (Daddy Hadamard), as he was familiarly called by his admirers, or "the living

legend of mathematics," an expression used by Hardy to

introduce him to the London Mathematical Society in 1944

[Ka]. After Poincare, Hadamard is without doubt the

Frenchman who most influenced twentieth-century math­

ematics. He is also an illustration of the humanist and uni­

versalist traditions in French culture at their best. For a

better understanding of the rest of this study, it must be

noted that Hadamard was Laurent Schwartz's great-uncle

by marriage, and followed his studies in secondary school,

then at the Ecole Normale Superieure (ENS). From his sem­

inar grew the Bourbaki group (via the Julia Seminar). The

Hadamard seminar was formative for several generations

of students at ENS. Laurent Schwartz recognized (lac. cit.) the crucial part played by Hadamard in his education. We

know well the life of Hadamard [M-Sh]-the immensity of

his mathematical work, and also his left-wing radical com-

Figure 1 . S. L. Sobolev with his children, Moscow, 1940.

mitment, initially motivated by the Dreyfus affair, then by

the rise of Nazism, and his closeness to the French Com­

munist Party along with Frederic Joliot-Curie. The archives

of the Academie des Sciences have copies of articles pub­

lished during his stays in USSR, in which he praised the po­

litical system and the merits of Soviet science [H1] .

The Facts

The 1 930s: Sobolev Functionals

Within the framework of his militant activities for friend­

ship between peoples, Hadamard, an indefatigable traveler,

made numerous journeys in the East, in particular to China

and the USSR.

Visits to the USSR:

• 1930: he attends the Congress of Soviet mathematicians

in Kharkov, in July, then travels to Kiev. He meets

Sobolev in Kharkov and later they have discussions in

French in Leningrad. Hadamard asks Sobolev to keep

him informed of his work [M-Sh p. 217];

• May, 1934: Hadamard is a member of a delegation of nine

French academics travelling for the "week of French Sci­

ence" in the USSR. In Leningrad he meets Sobolev but

does not participate in the second Congress of Soviet

Mathematicians (24-30 June, 1934), where Serge Sobolev

makes three presentations:

1. A new method for solving the Cauchy problem for hy­

perbolic partial differential equations;

2. Generalized solutions of the wave equation;

3. On the diffraction problem for Riemann surfaces.

The contents of these talks were certainly discussed a fort­

night earlier with Hadamard, who followed with interest

the works of his colleague: Sobolev himself acknowledged

the influence of the notion of finite part, discovered by

Hadamard in 1903 (!), in his discoveries of 1934-35 (see Ap­

pendix).

As underlined in the obituary of Sobolev by Jean Leray

[L3] and the review of the [Lu] book by Yushkevich, the

discovery of generalized functions must be ascribed to

Sobolev in his articles of 1935 and 1936:

• The Cauchy problem in the space of functions, Proceed­ings (Doklady) of the USSR Academy of Sciences, 1935,

volume III (VIII), N 7 (67) (in French).

• New methods to solve the Cauchy problem for normal

hyperbolic linear equations, Mat. Sbornik, 1936, vol. 1

(43), 36-71 (in Russian).

In these two articles, Sobolev explicitly defines generalized

functionals as continuous forms on the space of differen­

tiable functions of a given order m with support in a com­

pact set K, for fixed m and K. He establishes the funda­

mental properties of generalized functionals.

Why in French?

The year 1934, with the murder of Kirov, a popular Com­

munist leader in Leningrad, was a turning point for the

USSR, which began to shut itself off, and where "ideologi­

cal" struggles broke out, as illustrated by the campaign al­

ready mentioned against Luzin. In this campaign the issue

of whether to publish in Russian or in a more widely ac­

cessible language (as was done for most mathematical pub­

lications until the war), played an important role. The pub­

lication of the seminal article by Sobolev in Russian and in

French in the same volume of Doklady was purposeful.

Sobolev, who had criticized Luzin, was patriotically pub­

lishing in Russian; the publication in French, though com­

mon at that time, might be risky as a reminder of the so­

cial background of the author. It is quite likely that this

double publication was perceived positively by Hadamard

at least, maybe even suggested by him.

In 1936 Hadamard was again in Moscow, returning from

China. In 1945 he made another journey to Moscow and

Leningrad as a member of the French delegation to the cel­

ebrations of the 220th anniversary of the Russian Academy

of Sciences. He did not meet Sobolev (we shall see why).

However, as early as 1935, the reports he makes, back in

France, show Hadamard's awareness of the problems. He

evokes the tragic disappearance of a rising star, a clear al­

lusion to the suicide of the young and brilliant mathemati-

Figure 2. S. L. Sobolev not reading mathematics, Novosibirsk, 1962.

VOLUME 26, NUMBER 1 , 2004 41

cian Schnirelman, a number theorist and topologist, in

1938. Hadamard praised the close relationships between

pure and applied science in the USSR, even in mathemat­

ics [H2).

Sobolev's Discovery

Sobolev, inspired among other things by Hadamard's work,

first defmed generalized solutions of a wave equation, then,

in 1934-35, "generalized functions," without any mention

of a reference equation (contrary to the description in [Lu],

page 65), first under the name of "ideal" functions (as in­

dicated by Mikhlin), probably in reference to the introduc­

tion of ideal numbers by Kummer, then as "generalized

functions" in the seminal article of 1935. The older term

dangerously evoked idealist philosophy [M-Sh] at a time

when the Czech-born Marxist philosopher Kolman and the

other followers of "proletarian science" were stirring things

up in Leningrad. This hesitation over the naming, and the

double publication in Russian and in French, confirm that

Sobolev had a clear idea of the importance of his work and

its general character, contrary to the assertions made in

[Lu]. The reader can read in the Appendix a detailed analy­

sis of the various articles written by Sobolev and his in­

spirers and colleagues. There are hardly any clues to the

ongm of his discovery,

troduction) with a surprisingly partial and anti-chronolog­

ical presentation of Sobolev's articles:

Soboleff Proceedings of the Soviet Academy of Sci­ences, 1, 1936, p. 279-282, Math. Sbornik, 4, 1938, 471---496, Friedrichs: (1939), . . . Kryloff (1947). . . . Some articles mentioned in previous notes were published later than the introduction of distributions, but the authors did not know of distributions, due to the slowness of the publishing process, the slowness of international communications, or delays in my publication. See also Soboleff's functionals (''New Methods" . . . ) .

The first two references have no critical interest. The last

one, "New Methods," is the article already quoted. On the

other hand, he "forgets" to mention the Doklady article of

1935 (received on 7/17/1935). Moreover, this Note remained

unchanged in later editions [S' 1 ] .

The Key to the Mystery

In his autobiography, Schwartz, after a minimal description

of the discovery made by Sobolev in 1935 as found in the

article not mentioned in Note 4 above, wonders ([S2), p.

236) why, after the war,

apart from Hadamard's

work. Sobolev had a clear idea of Sobolev did not continue

his work on generalized

Hadamard's curious and

enthusiastic mind could

not remain indifferent to

this ongoing work; he read

the 1936 article as soon as

the importance of h is work and its general character.

functions.

The answer is instruc­

tive. Sobolev disappeared

from mathematical re­

search circles and re­

it got to the Ecole Normale Superieure. Moreover,

Hadamard always remained a subscriber to the main So­

viet mathematical reviews [ManS] . Jean Leray was becom­

ing a specialist in partial differential equations and also a

participant in the "prehistory of distributions" with his no­

tion of weak solution [L1], the subject of his "Cours Pee­

cot" in 1935 at the College de France. He told Sergei Sobolev

in the 1980s that he had discussed his 1936 article with Lau­

rent Schwartz before the war (personal communication of

V. Chechkin, holder of the Chair of Partial Differential

Equations at Moscow University and grandson of S.

Sobolev).

It took Schwartz more than ten years, including several

years not dedicated to mathematics and some years of slow

maturation, to bring forth his work of 1945, which reuses

Sobolev's defmition. But meanwhile Sobolev had surrepti­

tiously left the stage! Sobolev did not pursue his work in

this direction, though some work with Smirnov was not far

from it. He was awarded the Stalin Prize in 1941, and be­

came a deputy of the Parliament of the Soviet Union and

Director (beginning in 1941) of the Steklov Institute. This

left Schwartz free to develop the theory. The missing parts

were mainly Fourier transforms and the topological struc­

ture of the space of distributions (see below).

Moreover, the first publication in which Schwartz quotes

his sources [S1] contains a note (Note #4, page 5 of the In-

42 THE MATHEMATICAL INTELLIGENCER

frained from any foreign contacts from 1943 until 1953 be­

cause he was busy with other activities in applied

mathematics-very applied indeed; he became the main

assistant of the director I. V. Kurchatov in "Laboratory 2,"

which was initially located within Moscow University, and

which became LIPAN, where the first Soviet atom bomb

was developed [Viz] .

It is not surprising that both in the West and in the East

great mathematicians played a critical role in the nuclear

projects [Go, p. 383). The complex physics of shock waves

involved in those projects entails the solution of nonlinear

equations, and Bethe (who told von Neumann about it) had

noticed the instability of the numerical approximation in the

solutions; the skills of top mathematicians were needed!

This work, essential to Soviet defense, led Sobolev to the

numerical solution of the equations for a spherical nuclear

reactor. He also studied the so-called gun effect and its vari­

ation under neutron bombardment. This work is essential

in applications to assess water loss in reactors (Three Mile

Island and Chernobyl). In 1951 Sobolev received the most

prestigious civilian award, the Hero of Socialist Labour

medal. Naturally any foreign contact was totally forbidden

to him-even his wife did not know Sergei Sobolev's where­

abouts when he left for periods of several months after

briefly visiting home. His publication list is much shorter

during this period-apart from his 1950 manual written in

Figure 3. Novosibirsk 1978: In the foreground 0. A. Oleinik, E. 5. Soboleva, 5. L. Sobolev, and French visitor Jean Leray.

a hospital where he was recovering with a broken leg-and

the main part of the work just mentioned is still unpub­

lished.

The continuation is known in more detail; Sobolev re­

sumed classic scientific activities in the 1960s. Mean­

while, with Schwartz's book Distribution Theory and the

line of research pursued by him (tempered distributions

and Fourier transforms, applications of the theory of

topological vector spaces), he came to be considered the

father of the theory. The much-delayed acknowledgement

of Sobolev's paternity came only fifteen years later [L3,

L4] . Schwartz's main contribution, in the heritage of the

Bourbaki project (in a nutshell, the "algebraization of

analysis"), was to bring together Sobolev's definition and

the work begun by Dieudonne on topological vector

spaces in 1940 [Du] following Banach's famous Theorie des operations lineaires and Kothe's work. During the

period 1945-1950, Schwartz understood the importance

of applying TVS theory to the case of generalized func­

tions.

This process of discovery by bringing together sepa­

rate theories could be called "appropriation by bourbak­

ization." It was frequently used-see, e.g., [Gr, Mi, S4]: a

beautiful idea by Minlos, which Gross had also had inde­

pendently, was embodied ten years later in the theory of

"radonifying applications" without any acknowledgement

of Minlos. In the case of Sobolev, the author himself had

fostered the process! The term "bourbakization" of course

refers to the "Bourbaki project," which consisted in sin­

gling out the deep structures of mathematics to reach the

degree of generality able to give a theory its extensive

power. This clear explanation of the topological vector

space structure paved the way to Schwartz's theorem of

kernels and to the theorem of Malgrange--a student of

Schwartz-on the existence of solutions to partial differ­

ential equations in any open set in Euclidean space. Note

also that Schwartz's other student at that time, Jacques­

Louis Lions (1928-2001), had been focusing since his dis­

sertation on the use of Sobolev-style methods (Sobolev

spaces discovered in the 1930s ) , less elegant but more ef­

fective than functional analysis, for example, for cubature

formulas. Lions later became the leader of French applied

mathematics.

"Percolation"

The "percolation" (or "illumination," as he also refers to it)

process discussed by Schwartz in his autobiography prob­

ably consisted in the final linkage, made on the occasion

of a problem posed by Gustave Choquet, between Sobolev's

theory of functionals (defined as continuous linear forms)

and the work of Dieudonne and later Dieudonne and

Schwartz. Actually, contrary to the assertions in [Lu], by

January 1946 Schwartz had a good knowledge of Sobolev's

work: a participant recalls that during his "Cours Peccot"

at the College de France "he constantly had Sobolev's name

in his mouth."

Conclusions and Issues

Theory and Practice- East and West

In the era of triumphant socialism in Russia, science was

expected to be at the service of the people for the progress

of mankind. This notion was in fact the new face of an an­

cient cultural tradition in Russia, still vital in Saint Pe­

tersburg, even in the field of mathematics. One thinks for

example of Pafnutii L'vovich Chebyshev, whose concern

for linkages, ways to cut up fabric, and laws of chance,

were closely related to highly abstract concerns. Cheby­

shev has very explicitly described [C] the mutual benefits

of mathematics and practical applications. In the case of

functionals, Smimov, in a profound analysis, shows how

central the experimental sciences remained among the

VOLUME 26. NUMBER 1 , 2004 43

concerns of Russian mathematicians (see Appendix). For the Russian school, in the period discussed here but also later, the value of mathematics is measured by its effec­tiveness. Even general topology, through Tychonov to Pon­tryagin, has applications to the study of control systems. More recently, this is also the case in the work of Arnold and his school. We can imagine the difficulties experienced by Lusitania, the famous school created by Luzin in Moscow around function theory, and largely inspired by (German) set theory or (French) function theory. In con­trast, France, the country of Descartes, Galois, and Bour­baki, favored an interest in mathematical investigation "pour l'honneur de l 'esprit humain" (a phrase coined by Jacobi): the value of a theory is assessed by its degree of generality-a purifying quality of generality, synonym of efficiency, and evidenced in the connections between ap­parently remote domains for the production of new theo­ries, and by the elegance of the concepts [B2]. (Similarly in Germany.) For Schwartz, for example, distribution the­ory develops as he associates the Sobolev definition to the theory of topological vector spaces, thus arriving at the properties of the topology of distribution spaces. This will make possible the work of his students Lions and Mal­grange, after the presentation of the kernel theorem at the 1950 Congress in Cambridge, USA; this theorem was the cherry on the cake, and earned him the Fields Medal and the later paternity-in the West at least-of distributions. These two views of mathematics and their role were pres­ent simultaneously in both countries, and sometimes in the productions of the same mathematician, as in the cases of Gel'fand in the USSR or, earlier, Fourier in France. In the period we are interested in, the emphases were as we have mentioned above. Though this question goes beyond the scope of this article, we note that recent developments in physical and mathematical sciences show continuing ten-

Figure 4. The young Laurent Schwartz.

44 THE MATHEMATICAL INTELLIGENCER

sion between effectiveness and rigor (Feynmann integral, string theory; [JQ] for example gives an account of the de­bate). Should one rejoice that the political upheavals of the last decades threaten to standardize worldwide the practice of mathematical science and the answers to this "essential tension" [Ku]?

Theory and Practice- Probability and Measure

Measure theory and its relevance to probability deserve par­ticular scrutiny: it was the first serious stumbling block in the development of the Bourbaki project [B2]. From the point of view that interests us, distribution theory obviously served as a weighty "ideological" argument at the time. As

an illustration, here is an excerpt from the introduction of [Bl] concerning measure theory: " . . . Thus, integration the­ory is connected, on the one hand to the general theory of duality in topological vector spaces, and on the other hand to distribution theory, which generalizes certain aspects of the notion of measure, and which we shall present in a later book" It can be seen how much this "structuralist" stance­also present in Schwartz's approach to distributions-con­cealed the real nature of the phenomena in question, for example the subtleties of random processes. (Another il­lustration of errors of judgment is found in Andre Weil [Wl], [W2]: " . . . The time has come to try, through closer analy­sis, to split up Lebesgue's discoveries into various elements in order to identify what is essential in the manipulation of an integral, and what is relevant to the specific operations over sets on which we are most frequently working.")

Rather than disregard for potential applications, it was the desire to give priority to the structure over the phe­nomenon, and to the architecture over the portrait, that caused a delay of fifteen years in French research on prob­ability. Quite ironic, in the country of Laplace, Lebesgue, Borel, and most particularly Paul Levy, Fortet, Loeve, Ville,

Figure 5. Laurent Schwartz presented with a Vietnamese peasant hat.

and Doeblin, who, in the 1930s, were at the forefront of the

revival of probability theory by developing new trajectorial

aspects of processes, which were to have a wealth of appli­

cations in the second half of the 20th century, including in

the solution of the great problems of classical analysis and

its renewal (PDE, Dirichlet's problem, potential theory, etc.).

We hope in subsequent work to revisit this question, on

which Schwartz himself was self-critical ([S2]).

Problems of Communication

From the Russian Revolution to the 1970s, interchanges be­

tween mathematicians suffered from many difficulties be­

cause of the lack of intellectual freedom in USSR, the cold

war, and internal conflict within the Soviet cultural and uni­

versity system from the 1960s. Thus the Soviet delegation

as a group declined the invitation to the 1950 Congress at

Harvard, at the height of the Korean War. This was the Con­

gress during which Laurent Schwartz was awarded the

Fields Medal. We suppose that Kolmogorov, even though

he was a member of the medal committee, did not even

mention the name of Sobolev, then assistant director of LI­

PAN. In the 1960s, more problems appeared: we witnessed

difficulties of exchange and of publication of mathemati­

cal articles in the USSR, which led for example to the cre­ation of the review Funktional'niiAnaliz by Israel M. Gel'­

fand in the 1970s.

Mathematics and Politics

At the end of the interview used as a working document by

Liitzen, Laurent Schwartz makes a surprising linkage be­

tween distribution theory and political democracy, quoting

the eminent British Marxist historian Moses Finley, for

whom democracy was discovered by the Greeks: "It was

the Greeks, after all, who discovered not only democracy,

but also politics. I am not concerned to deny the possibil­

ity that there were earlier examples of democracy . . . . What­

ever the facts may be . . . their impact on history, on later

societies, was null. The Greeks, and only the Greeks, dis­

covered democracy, precisely as Christopher Columbus,

not some Viking seaman-discovered America." [Fi]

In other words, Sobolev is cast as the Viking, Schwartz

as Columbus. Beyond the general debate on philosophic re­

alism (was democracy discovered or invented? and distri­

butions?), it is clear that neither mathematics nor political

concepts emerge ex nihilo, and that scientific work is a

process: Schwartz comes after Sobolev, Dirac, and even

Euler! ( cf. Appendix) In retrospect and based on the ex­

amination above, this comparison appears to be not merely

excessive but unjustified. The same field of mathematical

analysis saw the emergence of the point of view of alge­

braic analysis, whose importance seems much more

promising, if only-in Bourbaki's view-by the "bridges"

which it builds. Going farther, and taking into account the

frequent cases where Schwartz left some things unsaid (see

above), we can wonder whether there may be an allusion

to the ideological power gained by Bourbaki, sometimes

against the will of some members. For example, Claude

Chevalley remained a libertarian all his life. In a beautiful,

nostalgic interview [Che], he confesses that he thought he

was "enlightening the world of mathematics," in a common

desire for renewal. It is in fact in Chevalley's writings that

we find the most interesting remarks on the relationships

between Bourbaki and political thinking: he says that read­

ing the political theorist Castoriadis made him understand

the wrongness of his view of mathematical logic!

Schwartz's aura personified that of Bourbaki: modem

mathematics and educational reform, the role of the

scholar in pronouncing what is right, and indirect power in

the life of society: the aura of the mathematician, which

Schwartz knew how to apply "for the good cause," is quite

evocative of Greece. Frequently, top mathematicians seem

to confound mathematical action, political struggle, and

moral principles.

The disappearance of such a strong personality evokes

the end (announced by some) of the era of "great narra­

tives"-the disappearance of myth-creating romantic ac­

tors (as Bourbaki, the dream of distributions). Is this hap­

pening? Time will tell and History will judge.

Acknowledgments

The author thanks the Sobolev family for permission to use

the photographs of S. L. Sobolev; the photographs of

L. Schwartz are by kind permission of the editors of Pour la Science.

REFERENCES [Be] Beaulieu, Liliane. Bourbaki. Une histoire du groupe de mathemati­

ciens franc;ais et de ses travaux (1934- 1944). Ph.D. Thesis, Univer­

site de Montreal, 1 990, Paris, 1 992.

[Boc] Bochner, Salomon. Review of L. Schwartz's Theorie des distri­

butions, Bull. Amer. Math. Soc. 58, ( 1 952), 78-85.

[B1 ] Bourbaki, N. Elements de mathematique, livre VI: Integration, Her­

mann, 1 952.

[B2] Bourbaki, N . L'Architecture des Mathematiques, p. 35-47, in Les

grands courants de Ia pensee mathematique, ed. F. Le Lionnais, Edi­

tions. Albert Blanchard, Paris, 1 962.

[Boul] Bouleau, Nicolas. Dialogues autour de Ia creation mathematique.

Association Laplace-Gauss, 1 997.

[C] Chebysheff, PafnutiT. Rapport du professeur extraordinaire de

VOLUME 26, NUMBER 1, 2004 45

I 'Universite de Saint-Petersbourg sur son voyage a I' stranger. In Oeu­

vres de P. L. Tchebychef, A Markoff and N . Sonin, editors, Chelsea,

New York, 1 961 .

[Che] Chevalley, Claude. Nicolas Bourbaki, Collective Mathematician.

The Mathematical lntelligencer, 7, no. 2, (1 985), 1 8-22.

[De] Demidov, Sergei S. The Moscow school of the theory of functions

in the 1 930s in: Golden years of Moscow Mathematics, ed. S.

Zravkovska, P. Duren, vol. 6, AMS, LMS, 1 993.

[Du] Dugac, Pierre. Jean Dieudonne mathematicien complet, Paris,

Jacques Gabay, 1 995.

[Fi] Finley, Moses I . Democracy Ancient and Modern, Rutgers Univer­

sity Press, 1 985.

[Ge1 ] Gel'fand Israel M. Some aspects of functional analysis and alge­

bra. Proc. Int. Cong. Math., 1 954, Amsterdam (1 957), 253-276.

[Ge2] Gel'fand, Israel M. and Shilov, GeorgiT E. Generalized Functions,

5 vols. , Academic Press (1 977) New York & London.

[Go] Godement, Roger. Analyse Mathematique, tome 2, Springer-Ver­

lag , 2000.

[Gr] Gross, Leonard. Harmonic Analysis on Hilbert Space, American

Mathematical Society, 1 963.

[H 1 ] Hadamard, Jacques. Le Mouvement Scientifique en URSS, Rap­

port presents en 1 935 a Paris aux journees d 'etude et d 'amitie franco­

sovietiques.

[H2] Hadamard, Jacques. Rapport paru en 1 945 apres le 2208 an­

niversaire de I 'Academie des Sciences de Russie.

[Ka] Kahane, Jean-Pierre. Jacques Hadamard, The Mathematical lntel­

ligencer, 13, no. 1 (1 991 ), 23-29.

[JQ] Jaffe, Arthur; Quinn, Frank. "Theoretical mathematics": toward a

cultural synthesis of mathematics and theoretical physics. Bull. Arner.

Math. Soc. (2) 29 (1 993), no. 1 , 1 -1 3.

[Ku] Kuhn, T, The essential tension, University of Chicago Press, 1 979.

[L 1] Leray, Jean. Sur le mouvement d'un liquide visqueux emplissant

l 'espace, Acta Math. 63 (1 934), 1 93-248.

[L2] Leray, Jean. Travaux de M. Laurent Schwartz, rapport annexe a Ia

candidature de Laurent Schwartz, 1 964, Academie des Sciences, Paris.

[L3] Leray, Jean. Rapport sur ! 'attribution du prix Cognac-Jay (Samar­

itaine) 1 972 a Laurent Schwartz, Jacques-Louis Lions et Bernard Mal­

grange.

[L4] Leray Jean. La vie et I'CEuvre de Serge Sobolev, La Vie des Sci­

ences, serie generale, vol. 7 , 1 990, No. 6, p. 467-471 .

[Lo] Lorentz, G. G. Mathematics and Politics in the Soviet Union, J. Ap­

proximation Theory 116 (2002), 1 69-223.

[Lu] Ultzen, Jesper, The Prehistory ofthe Theory of Distributions, Berlin,

Springer-Verlag, 1 982.

[ManS] Mandelbrojt, Szolem. Souvenirs a batons rompus, recueillis en

1 970 et prepares par Benoit Mandelbrot, Cahiers du Serninaire d'his­

toire des mathematiques, No 6, 1 985, p. 1 -46.

[ManB] Mandelbrot, Benoit: Chaos, Bourbaki and Poincare. The Math­

ematical lntelligencer 11, no. 3 (1 989), 1 0-1 2.

[Mar] Maritz, P. Around the graves of PetrovskiT and Pontryagin. The

Mathernatical lntelligencer 25, no. 2 (2003), 55-73.

(M-Sh] Maz'ya V.-Shaposhinikova T. Jacques Hadamard, AMS-LMS,

1 998.

[Mi] Minlos R. Continuation of a generalized random process to a com­

pletely additive measure, Dokl. Akad. Nauk SSSR (N. S.) 119 (1 958),

439-442.

1 From the French translation by Jean-Michel Kantor

46 THE MATHEMATICAL INTELLIGENCER

[S1 ] Schwartz Laurent. Theorie des Distributions, vol. 1 , Hermann, Paris,

1 950.

(S' 1 ] Schwartz, Laurent. Theorie des Distributions, Nouvelle edition en­

tierement corregee a fondue et augmentee. Hermann, Paris, 1 966.

[S2] Schwartz, Laurent. Un Mathematicien aux Prises avec le Siecle,

Editions Odile Jacob, 1 997.

[S3] Schwartz, Laurent, in "Les Mathematiciens," Pour Ia Science, Paris,

1 996.

[S4] Schwartz, Laurent. Seminaire "Applications radonifiantes, "

1 969-70, Ecole Polytechnique.

[Viu] Viucinich, A Soviet mathematics and dialectics in the Stalin Era.

Historia Mathematica 27 (2000), 54-76.

[Viz] Vizguin, V. lstoria sov. atornnogo proekta. lzdat. rousk. kirstian. gu­

manitarnogo instituta St. Petersburg.

M Yushkevich, A P. Encounters with Mathematicians, Golden Years of

Moscow Mathematics ed. S. Zdravkovska, Peter L. Duren, History of

mathematics, vol. 6, American Mathematical Society, LMS, 1 991 .

(W1 ] Weil, Andre. Calcul des probabilites, methode axiomatique, inte­

gration, in Revue Rose, vol. 1 of CEuvres completes, pp. 260-272.

Springer-Verlag, Berlin, New York, 1 979.

[W2] Weil, Andre. L 'integration dans les groupes topologiques et ses

applications, Hermann, Paris, 1 940; CEuvres, vol. 1 . See also the

commentaries on pages 551 -555.

Added in proof

Kutateladze, S. S. Sergei Sobolev and Laurent Schwartz: two fates and

two fames (in Russian), Novosibirsk, Sobolev Institute, preprint 1 2 1 ,

Oct. 2003.

Appendix

I.

Adolf P. Yushkevich: Some remarks on the history of the

theory of generalized solutions for partial differential

equations and generalized functions. Istoriko-mate­maticheskie issledovanie, 1991, 256-266 (Russian).1

Since 1968, I have been publishing appreciations of famous

French mathematicians on their Russian colleagues, on the

occasion of their applications as foreign member of the

Academie des Sciences de Paris (the election process has

remained unchanged since the middle of the 19th century).

Often these appreciations are interesting from the point of

view of the history of relationships between the scientists

of our two countries. Naturally these appreciations reflect

the personal point of view of the speakers, and frequently

the judgment on the candidates also depends on the inter­

national situation. The appreciations published so far are

those of Chebyshev, Lyapunov, Bernstein, Vinogradov,

Lavrent'ev, and Kolmogorov. It was a great pleasure for me

to receive Paul Germain's authorization to publish Jean

Leray's appreciation on his colleague Sergei Sobolev.

II.

The best appreciation of the work of Sobolev is in the ref­

erence [4], published for his 80th birthday: Sergei L'vovich

Sobolev (6. 10. 1908-3.01. 1989) completed his studies at

Leningrad University in 1928. His doctoral dissertation ad­

visers were N. M. Gunther (1871-1941) and V. I. Smimov

(1887-1974), both students of V. A. Steklov (1863-1926), himself a student of A. M. Lyapunov (1857-1918). For most of their lives, these four professors worked on the theory of differential equations, the theory of partial differential equations, and their applications in mathematical physics and mechanics. They were eminent members of the math­ematical school of Saint Petersburg, later Leningrad, headed by P. Chebychev (1821-1894), one of the professors of Lya­punov. As a student, Sobolev also followed lectures by Fikhtengolts (1888-1959), who was the first to develop in Leningrad the study of functions of a real variable, which prompted the extensive work of the school of Moscow with D. F. Egorov (1869-1931), N. N. Luzin (1883-1950), and their students.

Sobolev belongs to the fourth generation of Cheby­cheffs school, which systematically exploited the relation­ships between mathematics and the concrete problems of sciences and technology, without precluding a concern for the introduction of abstract questions-often over and above practical issues (even in number theory). It is necessary also to stress that Sobolev's professors them­selves were already using the most recent developments in mathematics-topology, the theory of functions of a real variable, new areas of the theory of functions of a complex variable, integral equations, and the new area of functional analysis. Sobolev's research work began immediately after the end of his studies, in the department of seismology of the Academy of Sciences headed by V. I. Smirnov. While still a university student, he presented a Master's thesis on a topic suggested by Gunther. At the Institute of Seismol­ogy, Sobolev again conducted work closely related to this topic previously suggested by Gunther, viz. the analytical theory of partial differential equations and in particular the propagation of elastic waves. Some of his first publications were cosigned with Smirnov. On June 29, 1930, Sobolev presented a paper at the first Congress of Russian Mathe­maticians: "The Wave equation in a heterogeneous isotropic environment," an abstract of which appeared in the Notes aux Comptes-Rendus de l'Academie des Sciences de Paris. This work interested Jacques Hadamard (1865-1963), who attended the Congress and himself made a presentation on a topic close to Sobolev's: "Partial differential equations and the theory offunctions of a real variable" ([5], in French and in Russian). Sobolev's early work (summarized, after those of Gunther and Smimov, in section 8 of [6]) was al­ready getting considerable attention from Soviet mathe­maticians, and Sobolev, not yet 25, was elected on 01 .02. 1933 a corresponding member of the Academy of Sci­ences. He was later elected a member on 29.01.1939.

Ill. In 1932 Sobolev enters the Physico-Mathematical Institute created by Steklov in 1921 . It is in this period that he de­velops his most important work, which establishes the be­ginning of the theory of generalized functions. He is the first to define them mathematically and to set about study­ing their fundamental properties. A summary of his ideas was written by Smirnov ([7] , p. 187-191). Sobolev started

articulating his ideas on distributions, which he calls func­tionals and were later called "generalized functions," from the late 1920s and the early 1930s-or possibly earlier. He presented them in his lecture "Generalized solutions of the wave equation" on June 29, 1934 at the second Congress of Soviet Mathematicians in Leningrad. Here is the laconic summary by the author: "The class of functions which we can consider as solutions to the wave equation from the classical point of view consists of twice-differentiable func­tions. But in various practical applications it seems conve­nient to consider functions with singularities of a well­defined type. We introduce a space of integrable functions in the sense of Lebesgue, in which it is possible to defme the generalized solutions of the wave equations as the lim­its of twice-differentiable solutions. Using a simple integra­bility criterion, we give a necessary and sufficient condition for a function to be a generalized solution, and we establish the link between the usual solutions and generalized solu­tions. Finally, this theory is applied to some concrete ex­amples" ([8] , p. 259). Leray sees considerable importance in Sobolev's work in the theory of generalized functions, called distributions in western mathematical literature, but he dates them back to 1935 and 1936, not earlier. Smimov ([7], p. 187) refers to the article [8] of 1935 and to two other ar­ticles quoted in [9) and [ 10). In the bibliographical list [9] , the lecture of 1934 is not even mentioned.

In his two classic volumes on the history of mathemat­ics in the last two centuries, Jean Dieudonne writes that Sobolev began the study of generalized functions in 1937 ( [11 ] , p. 2, [7], p. 171). In the 1982 article "Fonctions general­isees," Vladimirov quotes [9) along with the "generalized so­lutions" article ([12] , vol. 3, p. 1 102-1 1 10 and 1 116-11 17). It is only in the article written on the occasion of Sobolev's jubilee in 1989, that one of the authors, also named Vladimirov, indicates the article of 1934 "in which the the­ory of generalized functions appears for the first time." No­toriously, the establishing of chronological priority be­tween several authors of a scientific discovery is not always a harmonious process, but nowadays it does not lead to such negative effects or violent quarrels as in the case, for example, of Newton and Leibniz, the creators of infinites­imal analysis.

IV.

The prehistory of the theory of Sobolev's generalized func­tions has not been investigated much. Gunther's work should probably be ascribed a role in laying down the core notions; in particular, his smoothing method for insufficiently differ­entiable functions, which is often quoted by Smimov ([7] p. 184). A path toward the theory of generalized functions is found earlier still in Hadamard's work, starting with his re­mark "on functional operations" and his "Le<;ons sur la prop­agation des ondes et les equations de l'hydrodynamique" (Lectures on wave propagation and the equations of hydro­dynamics) published in 1903. The academician Steklov drew attention to these sources during the presentation of Hadamard's work when he was elected a corresponding member of our Academy on December 2, 1922.

VOLUME 26, NUMBER 1 , 2004 4 7

The article by Steklov is deep and definitely important.

He insists in particular on the significance of the first arti­

cle, where Hadamard uses for the first time the term "func­

tional," and he discusses in detail the results of the second

article. He insists in particular on the existence of "shock

waves" in compressible liquids and elastic bodies. One re­

mark by Steklov is particularly interesting: issues of hydro­

dynamics, translated into the language of mathematical

analysis, coincide with the theory of the characteristics of

partial differential equations, "which emerged completely

independently from any physical origin." This remark shows

that Steklov understood perfectly the significance for later

applications of abstract basic research pursued in complete

independence from their use. Moreover he uses the classic

terminology he is familiar with (and uses the term "func­

tional" only occasionally), and he could not foresee that a

few years later, it is essentially in his own institute that the

groundwork for the theory of generalized functions was go­

ing to be laid. This speech by Steklov was not published un­

til 1968 ([1] p. 1 10-115). As regards Hadamard's advances

toward the theory of generalized solutions for partial dif­

ferential equations and generalized functions, let us quote

a statement made by G. Shilov (professor at Moscow Uni­

versity from 1917 to 1975, and a recognized specialist on

this question), on February 10, 1964, during a memorial ses­

sion of the Moscow Mathematical Society: "In solving hy­

perbolic equations, Hadamard essentially introduces the de­

vice of the theory of generalized functions of one or several

variables. This discovery remained dormant at the time

(Hadamard was many years ahead of the thinking of math­

ematicians of his generation), and it was only in the mid­

fifties that generalized functions spread worldwide in ques­

tions of analysis" ( [ 13], p. 185). Shilov concludes by quoting

Szolem Mandelbrojt's words of 1922 about the famous

"Readings on Cauchy's problem" (translation to French in

1932, and to Russian in 1978): "The notions developed in

this work lead to general topology and to functional analy­

sis, and the introduction of the notion of elementary solu­

tion has a high degree of generality with respect to distrib­

utions (generalized functions)" ([ 14], p. 4-5). Furthermore,

we owe to Hadamard the terms "functional" and "functional

analysis." Jean Leray also mentions this precursor work

What we say here does not by any means detract from

Sobolev's achievement; he is the first to give a rigorous def­

inition-and in several ways-of the modem notion of gen­

eralized function, and to lay the bases of later developments

in various domains of the theory of generalized solutions to

partial differential equations and generalized functions, as

an autonomous domain of analysis.

v. Almost all of Sobolev's work on the theory of solutions and

generalized functions was published in Russian, except the

article in French of 1936 ([9], 22). So it is not surprising that

in other countries [than the USSR] this work did not attract

immediately the interest it deserved. This remark also ap­

plies to the book Some Applications of Functional Analy­sis in Mathematical Physics (Leningrad, 1950), which cor-

48 THE MATHEMATICAL INTELLIGENCER

responds to the course that Sobolev taught at the time at the

University. This book was not translated into English until

1963 (into German in 1964). It is quoted several times by

Leray, and, as noted by V. I. Smirnov, "this book played an

important part in the use of the modem ideas and methods

of function theory and functional analysis for the solution of

problems of the theory of partial differential equations" ([6]

p. 191). In Russia, Sobolev's new ideas, following those of

his masters Gunther and Smirnov, diffused fairly quickly, and

were extended and developed from the 1950s.

In the diffusion abroad of these new directions of math­

ematical analysis, a major role must be ascribed to the book

Theorie des Distributions (in two volumes) by Laurent

Schwartz, a corresponding member (1973), then a full

member (1975), of the Academy of Sciences, and a profes­

sor at the Ecole Polytechnique. Several articles published

by Schwartz between 1945 and 1948 already used the ex­

pression "distributions." After the publication in 1950-51 of

Schwartz's book, distribution theory developed consider­

ably and received numerous new applications.

The first historical study on research on distributions,

published by Jesper Liitzen in 1980, contains an accurate,

flawless mathematical analysis of the works of Sobolev,

Schwartz, and many earlier or contemporary mathemati­

cians. In spite of all these achievements, Jesper Liitzen's

book has some gaps and, from my point of view, uncon­

vincing evaluations, which can be explained by an imper­

fect knowledge of work in Russian generally and of Sobolev

in particular (although his bibliography contains 1 1 refer­

ences which were translated into English, as well as the

1950 book already mentioned and the thick course book in

its third version of 1954). Leray's note on Sobolev's work

is a substantial complement to Liitzen's study.

Without trying to write the history of the question, I shall

make here some remarks on Liitzen's book First, I cannot

agree with his evaluation of the results obtained by

Sobolev, then Schwartz, and their place in the development

of distribution theory. The essence of the differences be­

tween their theories, according to Liitzen (p. 64), is that for

Sobolev distributions are a technique to resolve a specific

problem, while Schwartz developed distribution theory un­

der multiple angles, and applied it to formulate and resolve

rigorously numerous problems. It is true that in 1934

Sobolev began with Cauchy's problem for the wave equa­

tion (which is hyperbolic), but then he did not limit him­

self to one of the applications which he had introduced,

and he considerably enriched them, as shown by Jean Leray

("work whose scope, variety, and power are admirable").

It is also true that these various contributions published in

successive articles were not collected into a monograph,

which would doubtless have had the seminal role of

Schwartz's book-which became the basic book for nu­

merous researchers abroad and here. Liitzen briefly sum­

marizes the fundamental difference between the works of

Sobolev and Schwartz: "So Sobolev invented distributions,

but distribution theory was created by Schwartz" (page 64).

Variants of this reflection occur in the book On page 67,

after quoting Lyusternik and Vishik's words in a speech pro-

nounced on the occasion of Sobolev's fiftieth birthday (1959), Liitzen supports what they say but immediately adds that "further development of the theory was not Sobolev's but Schwartz's achievement." Without intending in the least to detract from the essential importance of Lau­rent Schwartz's book of 1950, I find more balance in S. Vladimirov's judgment ([12], vol. 4, p. 1104): "The founda­tions of the mathematical theory of generalized functions were laid by Sobolev in 1936 with the aim of resolving Cauchy's problem for hyperbolic equations, but in the 1950s L. Schwartz gave a systematic statement of the theory and mentioned numerous applications." He could have added that the systematic account in modem terminology in Schwartz's work overshadowed Sobolev's. As regards Schwartz's possible knowledge of Sobolev's previous dis­coveries, according to the statements made by L. Schwartz in 1950-51 and in 1974, the latter did not know of them be­fore 1945 (p. 67 of [16]). Elsewhere Liitzen writes that Schwartz's attention was called to Sobolev's work by Leray in 1946. Certainly Sobolev and Schwartz arrived at their dis­coveries of "generalized functions" and "distributions" by different paths-but certainly, too, there is no reason for assigning Sobolev's work to the "prehistory" of distribution theory, as Liitzen does three times (pages 64, 67, and 156).

More generally Liitzen devotes more attention in his book to Laurent Schwartz than to Serge! Sobolev. The state­ment of the results according to the bibliography is correct; but he could have gone into more detail. In this respect, Leray's note contains valuable complements, but even this note does not contain enough bibliographic data on Sobolev. These indications could have and should have been enriched by the inclusion of Lyustemik and Vishik's text (which Liitzen quotes and uses). There is no reference to Sobolev's teachers, in the text or in the reference index. L. Schwartz's biography is given a very contrasting treat­ment. In chapter 6, the reader is informed of all the stages in Schwartz's life, the names of the professors at the Ecole Normale (Leray, Leyy, Hadamard), Schwartz's membership in the Bourbaki group, his discovery in six months of dis­tributions, his conversations with de Rham (also mentioned by Leray), etc. All this information is valuable, and it is re­grettable that Sobolev's mature work is treated by Liitzen in merely half a page (p. 60).

To be sure, distinctions between the "prehistory" of a theory and its development are a matter of convention. The notion of "distribution" appeared in various authors of the beginning of the 20th century, and one could even go back to Euler (see below), whom Liitzen also mentions. How­ever, we distinguish ideas belonging to prehistory-already born but not introduced yet into a well-defined frame­from ideas belonging to the history of a theory-where they have a precise definition and we focus on the study of their specific properties. Thus one reasoned with functions of one type or another in ancient Greece, in the Middle Ages, and at the beginning of the modem period, but functions as objects of mathematical investigation, in all their gen­erality, appear only at the end of the 17th century. How­ever, the title of Liitzen's book is "Prehistory of etc.", which

situates Schwartz-to whom the largest part of the book is devoted-as part of the prehistory of the theory.

If Liitzen had restricted his study of the prehistory of distributions to Western Europe, it would have been nat­ural to insist on Schwartz's work But for a study of the de­velopment of mathematics as a worldwide process (which it has always been), the book's structure seems inadequate. This is shown in any case by the historical study of the facts in our country. Sobolev, following his teachers, played an important role by laying the bases of the numerous studies which began even before 1970, the publication date of Smimov's already mentioned article, which presents a sum­mary of twenty years of work in the theory of partial differ­ential equations-elliptic, hyperbolic, parabolic, or mixed­as well as contributions to the general theory, and work by 0. A. Ladyzhenskaya, S. G. Mikhlin, N. N. Ural'tseva, and others. All this work was not isolated from foreign re­search. Collaborations took place between all the countries involved, albeit it was sometimes made difficult by prob­lems of communication and the lack of personal contacts (which developed a lot in recent years, their earlier scarcity having been supplemented by numerous reference period­icals). The objective of my remarks on the history of the theory of generalized solutions and generalized functions is not only to clarify the conclusions of Liitzen's book, but also to introduce the presentation of Sobolev's candidacy by Leray, which is an essential complement to the Danish historian's account.

I must make a few additional remarks on the proto­history of the theme, which led to the solution of the equa­tion of the vibrating string and to the dispute between d'Alembert and Euler, which went on for almost thirty years from 1750, and which somehow involved all the mathemati­cians of the 18th century. Briefly stated, d'Alembert com­pletely excluded the case of discontinuity of a derivative, and even more stringently of the function itself. In my book on the history of Russian mathematics before 1917 (Nauka 1968), I showed that Euler, from physical considerations, deemed it necessary to admit, as solutions of problems of mathematical physics, what he called "broken" functions and curves; we would say that the initial position of the string and its initial velocity are functions of position which are continuous by segments, i.e., where discontinuities (in the modem sense) of the first two derivatives are allowed.

Not having the necessary mathematical means at his dis­posal, Euler gave a simplified geometrical description of the distribution of waves and their reflection for a string fixed at a single point. I take the liberty to quote my own book: "Threads are woven here between Euler's ideas and the new methods of the 20th century up to Sobolev and Schwartz's generalized functions" (p. 166, 169). During the more recent history of the notions of solutions of partial differential equa­tions, the historian S. S Demidov relied as I did on a quote from d'Alembert (Opuscules, volume IX). "Euler essentially constructed a solution of the equation as a generalized so­lution-for which a correct defmition and, even more, con­struction, were beyond the capacities of the mathematicians of that time" (p. 179). I added in my book that because of its

VOLUME 26, NUMBER 1 , 2004 49

A U T HOR

IIQU8S bclis· cont Of

practical utility, Euler's construction had been the object of

the attention of numerous mathematicians over time. I used

Truesdell's well-known study of Euler's work in hydrody­

namics and elasticity. References in Russian on this subject

are not known by Ltitzen (for example in volume IX of

d'Alembert's Opuscules, he only mentions Demidov's lecture

at the International Congress on the History of Mathematics

in 1977, which he also uses extensively on page 15). Ltitzen

also traces back to Euler the notion of generalized solution,

and he draws a parallel between Euler and Sobolev. Using

the definition of generalized solutions as limits of series of

classical functions, Ltitzen notes that this idea can be found

in Euler in 1765 and Laplace in 1772, and that the rigorous

definition was introduced in 1935 by Sobolev and later by

the other authors, in particular Schwartz in 1944.

In conclusion, we would say that Euler introduced func­

tions which could seem strange to his contemporaries, for

example, (- 1 )x, x being an arbitrary real number . . . but

not the delta function!

REFERENCES [1 ] French-Russian Scientific Relationships, A Grigorian, A Yushke-

vich, Nauka, 1 968.

[2] lstori. Math. /ss/edovanie, 31 (1 989), Nauka.

[3] Ibidem, 90, vol. 32-33.

[4] Bakhalov-VIadimirov-Gonchar, Sergei L'vovich Sobolev, Uspekhi

Mat. Nauk, vol. 43, 5 (1 989), 3-1 3.

[5] Proceedings of the First Congress of Mathematicians of the USSR,

Kharkov 1 930, Gonti, 1 935.

[6] Sobolev SergeT L. Smirnov Seminar, Soviet Academy of Sciences,

1 949.

50 THE MATHEMATICAL INTELLIGENCER

[7] Smirnov S. (ed .) , Partial Differential Equations, Department of Math­

ematics of the University of Leningrad, 1 970.

[8] Proceedings of the Second Congress of Mathematicians of the

USSR, Leningrad, June 24-30, 1 935.

[9] Forty Years of Mathematics in USSR, 191 7-57, Fizmatgiz, 1 959,

vol. 2, Lyusternik-Vishik article.

[1 0] Mathematics in USSR, 1958-67, Bibliography, Nauka, 1 970, vol.

2.

[ 1 1 ] Dieudonne, J . (dir.), Abrege d'Histoire des MatMmatiques,

1 700-1900, Hermann, 1 996.

[1 2] Encyclopedie des MatMmatiques, vol. 5.

[1 3] Shilov, J . Hadamard and the Birth of Functional Analysis, Usp. Mat.

Nauk, 19, 3, (1 964), 1 83-186.

[1 4] Mandelbrojt, S. Article "Hadamard Jacques, " Dictionary of Scien­

tific Biography, vol. 6, Charles Scribner, 1 972.

[1 5] Schwartz, Laurent. Theorie des Distributions, Hermann, Paris,

1 950-1 951 '

[1 6] Li.ltzen . The Prehistory of Distribution Theory, Springer-Verlag,

1 980, 232 pages.

[1 7] Leray, Jean. La vie et I 'CEuvre de Serge Sobolev, La Vie des Sci­

ences, Comptes Rendus, serie generale, vol. 7 , 1 990, No 6,

467-471 '

A Prest ig ious Col lection from the London Mathematica l Society

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together th some more apphed areas of analys•s. ma hematiCal p ys . t eoretiCal computer sc ence. probab1hty and s a 1s 1cs

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VOLUME 26, NUMBER 1 , 2004 51

In

52 THE MATHEMATICAL INTELLIGENCER

The Reception of the Theory of Distributions Peter Lax

Courant Institute of Mathematical Sciences

New York University

New York, NY 1 00 1 2

USA

e-mai l : lax@cims.nyu.edu

ld§ijl§',ifj Osmo Pekon e n , Ed itor I

Feel like writing a review for The

Mathematical lntelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or, if

you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor: Osmo Pekonen, Agora

Center, University of Jyvaskyla, Jyvaskyla,

40351 Finland

e-mail: pekonen@mit.jyu.fi

The Science of Conjecture: Evidence and Probabil ity before Pascal by James Franklin

BALTIMORE, THE JOHNS HOPKINS UNIVERSITY PRESS,

2001

600 pp. $22.50 PAPER ISBN 080-1 86569-7

REVIEWED BY NORMAN LEVITT

Franklin is nominally a mathemati­cian and his book shows a mathe­

matician's touch when it deals directly with mathematical matters, but this re­view must begin with a warning: this is not primarily a history of mathematics, nor is it a book that a strong mathe­

matical background makes particularly accessible. Rather, The Science of Con­jecture is a history of a lengthy philo­sophical investigation that has spanned a number of eras and civilizations, and which, of course, continues even now, with no discernible end. The central question is one of partial belief, belief that may be quite pronounced, but which stops short of demonstrative or "mathematical" certainty. Very simply, what kind of evidence and how much of it ought to be necessary to persuade a reasonable inquirer that it is more appropriate to accept a proposition than to reject it? How ought we to or­der degrees of belief that lie some­where between absolute conviction and utter dismissal? What degree of be­lief is necessary to justify an action with grave consequences? This is not the sort of question that mathemati­cians are given to worrying about, at least not when going about their math­ematical business. But it is the central concern of "practical reason," and in various forms it confronts us in many societal roles-as jurors, for instance, or as investors.

Franklin's view is that pre-modem and early modem thinking on this ques-

tion had a considerable influence on the origins of probability calculus, in the work, principally, of Fermat, Pas­cal, and Huygens. His history, which begins in classical times, closes at this juncture, the take-off point, we might think, of Western scientific rational­ism. But by no means would he have us believe that formal, quantitative proba­bility theory (or its evolution into sta­tistics) made earlier modes of inquiry

obsolete or brought them to comple­tion. Rather, the probabilist's quantifi­cation of likelihood and expectation, even in its most sophisticated develop­ment, covers only a narrow range of ex­perience and informs our judgment at best in limited measure. Thus the med­itations of ancient Aristotelians and cloistered Scholastics are far from be­ing mere vestigial echoes of benight­

edness. They touched on important epistemological points that have not become that much more transparent even a millennium or two down the line.

I am not an historian of ideas. My knowledge of the secondary, let alone the primary, literature relevant to The Science of Conjecture is too scant to be called even minimal. Yet I will risk the

opinion that Franklin's book is deeply researched and intensely learned. It is a throwback to the days when humanist scholarship meant thorough saturation in a vast ocean of sources, rather than picking out two or three texts and weaving elaborate postmodem curli­cues around them. Franklin knows his authors, scores of them, thoroughly, and is scrupulously concerned to represent them with full fidelity to their ideas and their originality. He is not loath to judge those ideas, but his judgments do not come cheap; they are the fruit of care­

ful reading and careful thought, not of a prefabricated agenda.

The book is roughly chronological in organization, so that the work of the inventors of mathematical probability culminates the inquiry. But the more important structural principle is to

© 2004 SPRINGER-VERLAG NEW YORK, VOLUME 26, NUMBER 1 , 2004 53

study various notions of likelihood and

less-than-absolute certainty as they

carne to bear on other fields of thought.

The most important of these, overall,

is law, an area in which the same ques­

tions of credibility and weight of evi­

dence recur from era to era and civi­

lization to civilization. Franklin surveys

classical (that is, Hellenic and Roman)

ideas, along with Islamic, Judaic, me­

dieval, and Renaissance thinking on

the primary question: what degree of

direct and indirect evidence, what lines

of argument, will suffice to convict the

accused or to win redress for a plain­

tiff. Obviously, there is no consensus

to be found here. More discouraging,

perhaps, is that it is far from evident

that the principles of judgment now in­

stituted in our courts represent the dis­

tilled essence of hard-won wisdom.

The best to be said, I think, is that we

have abandoned divination, along with

the notion that evidence of guilt re­

quires confession, which in turn ne­

cessitates torture. But this is a long

way from our having a coherent and

systematic legal epistemology.

Because medieval Europe is a cen­

tral focus, canon law, as well as civil,

is an area where Franklin finds inter­

esting thinkers. This naturally con­

nects to the prolific literature of moral

philosophy, whose "practical" side in­

cluded the vast array of "manuals" for

the priesthood, guides, principally, to

the quasi-judicial ritual of Confession.

Here, "cases of conscience" come to

the fore: how much doubt must arise

in the mind of the faithful believer in

order to proscribe a possible course of

action as potentially sinful (or, put the

other way, how much doubt must be

effaced before the action may be un­

dertaken with a clear conscience)? The

lack of agreement among the school­

men, here, as well as the ferocity of

their disputes, is one of the most in­

triguing topics of the book. We must

adjust our minds to the ancient usage

of "probable." During the period in

question, this does not mean "more

likely than not" but rather, "sustainable

by at least some evidence, argument,

and authority," in other words, merely

plausible. Thus, "probabilism" is the

doctrine that a justification merely

"probable" in this weak sense suffices

54 THE MATHEMATICAL INTELLIGENCER

to render actions licit, even in the face

of weightier opposing arguments. This

doctrine carne to be identified with Je­

suit thought in particular, and con­

tributed to the Jesuits' reputation for

sly equivocation.

Franklin also considers epistemo­

logical and evidentiary standards in

pre-modem science. There are some

surprises here; some "modem" philos­

ophy turns out not to be so modem as

all that. Even Ockharn's famous Razor

is revealed not so much as a revolu­

tionary device heralding the end of me­

dieval obscurantism as a tool already

quite familiar and well worn with use

before Ockharn took it up. On the ba­

sis of Franklin's shrewd observations,

one learns new respect for the long-de­

spised Scholastics. He joins a growing

corps of intellectual historians newly

appreciative of the stunning, singular

mathematical genius of the 14th-cen­

tury cleric Nicole Oresme, as amazing,

in its way, as that of Rarnanujan.

At the same time, Franklin shows

that many of the developments we look

upon as practical solutions to thor­

oughly secular problems grew out of

attempts to create a calculus capable

of evaluating financial and commercial

dealings from the point of view of sanc­

tioned morality. Thus, notions of "ex­

pected value" that might apply to such

arrangements as annuities, lotteries,

risk insurance, and what we would

now call futures contracts were devel­

oped primarily to distinguish fair deal­

ing from usury. Of course, this applies

in large part to the formal treatises of

scholars. One assumes that, in parallel,

men of practical affairs developed their

own formularies for these things, but

here the record is scanty. What is most

interesting in this regard is that the ori­

gins of true probability calculus in the

mid-17th century seem closely con­

nected to the tradition of parsing

doubtful situations to decide what is

truly fair, equitable, or licit. We usually

think of the problems addressed by

Fermat and Pascal as gambling or gam­

ing problems, and thus we assume that

they grew out of serious interest in

these practices, with the mathemati­

cians seeking to justify, generalize, or

improve rules of thumb known to de­

voted gamblers. Franklin demon-

strates that for the most part this is

quite doubtful. The classic problems of

the early probabilists are usually

phrased in terms of finding an equi­

table division of stakes in an inter­

rupted game, which would be a curi­

ous way of looking at things if the

object were to improve the acumen of

dicers and card-players. But it makes

perfect sense when seen as a continu­

ation of the Scholastic tradition of de­

ciding what course must be followed

to avoid sinful error. (The sin, note, lies

not in gambling, but in a lack of even­

handedness in the setup of the game.)

This is not to say that real-life gamblers

were not on the scene, or that their in­

sights were never absorbed by the

unworldly mathematicians. But their

practical concerns are not central. And

we know, of course, that Pascal him­

self carne to prefer the high-stakes

game of winning souls through con­

version. (Franklin does discuss the fa­

mous Wager, but his account is fairly

conventional.)

This explains some curious omis­

sions in the underlying justification for

probabilistic formulae. The early mas­

ters never seem to refer to the idea that

the probability of an outcome (of, say,

a throw of the dice) is essentially iden­

tical with the ratio of the number of

times that outcome is achieved to the

total number of trials when the latter

is very large. Similarly, though notions

of expected value of a game occur, and

are correctly computed, this is not re­

lated in any direct way to the average

result over a large ensemble of actual

games. By the same token, the actuar­

ial approach, deriving parameters em­

pirically from observed frequencies

and averages, and then using these as

probabilities or expected values in fur­

ther calculation, is not part of the pro­

gram either. Basic probabilities-the

one-in-six that a given face will tum up

on a given throw of a die-are simply

assumed ad hoc, without any sense

that real life may depart from these

ideal frequencies. As well, notions that

we think of as quite elementary-the

rules for deriving probabilities of con­

junctions and disjunctions, for in­

stance-are not clearly formalized.

Thus, the invention of probability cal­

culus, though undeniably a brilliant

leap, is placed firmly in its historical

and cultural context; musty tradition of

all sorts clings to it as it emerges. Why

does it emerge at all, and why at this

time? Franklin's answer flatters math­

ematicians: The chief reason was the

existence of a large, active community

of mathematicians who inspired, in­

structed, and challenged each other,

and provided an admiring audience

when success was achieved by one of

their number. Readers who are famil­

iar with John Aubrey's Brief Lives, whose quirky biographical sketches of

noted contemporaries include dozens

of 17th-century British mathemati­

cians, will appreciate this point.

In the end, however, Franklin does

not see quantitative probability theory

as the end point of the schools of

thought he reviews. Nor should he. The

word "probable" as now used, even by

scientists, rarely falls within the rubric

of quantitative probability calculus. It

is easy to see this even within the dis­

course of mathematics itself. What do

we mean when we aver that conjecture

X is "probably" true or that a given strat­

egy is "likely" to succeed in proving it?

Pretty clearly, there is no way to give a

quantitative significance to these asser­

tions, nor, indeed, to translate them into

any suitable formalism. Worse, we re­

ally have no idea of a systematic epis­

temology that might justify them. Yet re­

marks like these are really the working

discourse of research mathematicians;

we work on conjectures that seem

probable, using methods that seem

likely to get somewhere, but all this

"seeming" is tied up in unaccountable

subjective intuition, informed by anal­

ogy and experience. The philosophical

status of all this is very unclear. The

same applies to science all down the

line; we think of string theory as prob­

able (or not) and likewise for anthro­

pogenic global warming or prions as the

cause of Alzheimer's. These judgments

are the stock-in-trade of everyday sci­

ence. But there is no widely accepted

justificatory theory of judgment that

stands behind them. In that sense, we

are hardly further along than the cen­

turies-dead heroes of Franklin's saga.

What Franklin makes of this is im­

portant to note. Good ideas and suc­

cesses, even partial ones, to hard prob-

lems are hard-won and tend to rest on

centuries of missed leads and blind al­

lies. Thus they should be all the more

precious to us. We are obliged, then, to

reject the flighty notions now far too

popular that the "episteme" is tran­

sient, arbitrary, and endlessly mutable,

that fundamental ideas are merely cul­

tural fashion statements, and that sci­

ence is bound, eventually, to recede as

surely as it once advanced. Science, for

Franklin (and, I hope, all of us), is a cu­

mulative achievement as much rooted

in obscure toil as in famous triumph,

which should deepen, rather than di­

lute, our esteem for it.

Department of Mathematics

Rutgers University

New Brunswick, NJ

Piscataway, NJ 08854

USA

Prime Obsession by John Derbyshire

WASHINGTON, DC, JOSEPH HENRY PRESS, 2003,

$27.95, ISBN 0-309-08549-7

The Music of the Primes by Marcus du Sautoy ��---------------NEW YORK, HARPER COLLINS, 2003, $24.95, ISBN 0-06-

621070-4

The Riemann Hypothesis by Karl Sabbagh NEW YORK, FARRAR, STRAUSS AND GIROUX, 2003,

$25.00, ISBN 0-374-25007-3

REVIEWED BY HAROLD M. EDWARDS

The nearly simultaneous publica­

tion of three books for the general

public about the Riemann hypothesis

(hereinafter referred to as RH) can

probably be explained by the million­

dollar prize offered by the Clay Mathe­

matics Institute for the resolution of

RH (large sums of money evoke inter­

est) and by the many books that were

sold to the general public about Fer­

mat's last theorem in the wake of

Wiles's proof (selling books is the goal

of publishing). Whatever the reason for

this sudden flood of interest in one of

the frontiers of pure mathematics, it is

a welcome, if surprising, phenomenon.

Mathematicians are probably the

worst people to review such books. An

architect I once met pleased me by

telling me how he had become con­

vinced of the power and beauty of

mathematics by reading a certain pop­

ular book on mathematics that he

named. I was so gratified by this devi­

ation from the usual "I was never any

good at math" that I rushed to the li­

brary to see the book My disappoint­

ment was great. To me, it was full of

dubious assertions, exaggerations, over­

simplified history, and explanations of

mathematical ideas that could impart

no understanding other than false un­

derstanding. But, as the architect

plainly demonstrated by his own ex­

ample, the book had achieved its goal

brilliantly, at least for one reader.

Moreover, I have had the experi­

ence-and most mathematicians I

have asked about it have had the same

experience-of rereading a book for

nonmathematicians that I had read in

my youth and that I remembered as

having inspired me, only to discover

that it had many explanations I now

found to be misleading at best and

statements I now found to be down­

right wrong. Would I recommend the

book to a young reader today? My own

experience would say yes, but my

judgment as a mathematician would

say no.

These considerations have been on

my mind as I pondered these books on

RH. All three are quite well written, and

I can easily imagine any one of them

capturing the nonmathematical reader's

fancy. And, overall, I think each pre­

sents a reasonably accurate picture of

the history of RH and the present-day

mathematicians who are working on it.

For that, the mathematical fraternity

can thank all three authors. But, after

all, I am a mathematician, and it is only

as a mathematician that I can evaluate

the books.

My lack of success over the years in

explaining the irrationality of Y2 to

reasonably able liberal arts students

has left me without much hope that ex-

VOLUME 26, NUMBER 1 , 2004 55

planations of RH intended for inter­

ested non-mathematicians will suc­

ceed. In other words, I am among the

"many people" who, according to the

first sentence of Karl Sabbagh's Pro­

logue, "would say that the task I am

embarking on . . . is futile." He defends

his project by comparing it to anthro­

pology and to "describing a remote

tribe whose customs and language are

unfamiliar to the reader, but whom I

understand enough to convey some­

thing of their inner and outer lives."

Readers of the Mathematical Intelli­gencer, as members of that remote

tribe, will be interested to know

whether the descriptions he provides

are accurate and whether they illumi­

nate our tribal culture. On both counts,

I am unenthusiastic.

The best parts of the Sabbagh book

are indeed the anthropological ones.

He tells who has worked on-or is

working on-the Riemann hypothesis,

how they became interested in mathe­

matics and in this particular problem,

how they view their chances for suc­

cess, and so forth. But what makes us

a tribe is our peculiar culture, and there

is no way to describe the interactions

of key members of the tribe without go­

ing into the substance of our culture.

On this, Sabbagh is an unreliable guide.

For example, on p. 41 (page numbers

refer to the American edition-the orig­

inal English edition is more compact,

so for example, this passage is on page 33 of that version) he says: "So, calcu­

lating the value of the sum I lln8, which Riemann believed was possible

but couldn't say so for certain, would

result in a totally accurate number for

the number of primes less than n." Well, calculating the value of I 1/n8 is

certainly possible when the real part of

s is greater than 1, but Sabbagh does

not say which particular values of s will

be needed to produce his "totally ac­

curate number." (Later in the book,

complex numbers are introduced, and

on the next-to-last page analytic con­

tinuation is mentioned in passing, but

at this point I lln8 is far from being the

same thing as ?(s).) This "totally accu­

rate number" must refer to Riemann's

explicit formula for 1r(x), which Sab­

bagh seems to believe (see also the end

of Chapter 1) depends on RH; but in

56 THE MATHEMATICAL INTELLIGENCER

fact what is needed for this formula is

not the evaluation of ?(s) for one or

more values of s but a knowledge of

the zeros of ?C s) in the critical strip; the

formula is valid whether or not the ze­

ros are on the critical line. This misap­

prehension about the meaning of RH

probably underlies his answer, at the

end of his Prologue, to the question,

"Why is it [RH] so important? . . . A

proof . . . would . . . tell mathematicians

a huge amount about an important

class of numbers-the prime numbers,

which dominate the field of pure math­

ematics." The notion that such a goal

accounts for the fascination of RH is a

profound misunderstanding of our

tribal culture, like believing moun­

taineers want to climb Mount Everest

in order to get somewhere.

For another example, he often spec­

ulates about who might or might not

prove the Riemann hypothesis. On p.

219, Martin Huxley is said to have "both

the desire and the ability to prove the

Riemann Hypothesis." On p. 240, we are

told that "many . . . feel that if anyone

is going to prove the Riemann Hypoth­

esis, it will be [Alain Connes]." And not

only is there speculation that Louis de

Branges might be the one to prove RH,

the book includes an Appendix by de

Branges with the title "De Branges's

Proof." This isn't the way research in

mathematics goes. Perhaps in other

fields that require expensive equipment

one might, to a limited extent, predict where the next breakthrough might oc­

cur, but in mathematics any attempt to

predict whether there will be a proof

any time soon, much less what shape it

might take or who might devise it, is

completely foolish.

In this connection, Sabbagh gives us

an interesting pair of speculations:

Henryk Iwaniec (p. 36) says, ''I'm only

worried that what may happen is that

a proof will be given by somebody and

I will be unable to understand it," while

Alain Connes (p. 263) worries about

something quite different: "It would be

a tragedy if it just needed a trick to

prove it." Different as these concerns

seem, I suspect that most mathemati­

cians sympathize with both. Note that

neither has anything to do with "learn­

ing a huge amount about an important

class of numbers."

Another of the authors, Marcus du

Sautoy, is a professional mathemati­

cian, so we should expect his state­

ments to be correct, but I am puzzled

by his statement (p. 11) that "a proof

of the Riemann Hypothesis would

mean that mathematicians could use a

very fast procedure guaranteed to lo­

cate a prime number with, say, a hun­

dred digits or any other number of dig­

its you care to choose." He goes on to

relate this to RSA cryptography, clearly

implying that RH would have some

practical significance for cryptogra­

phy, but I doubt that this is the case. I

suspect, rather, that he feels the gen­

eral reader must be given some reason

for the significance of this million-dol­

lar question in mathematics, but that

the real reason depends on aspects of

our tribal culture that are too difficult

to explain to the general reader. (Per­

haps I am wrong; the supposed con­

nection is again mentioned on page

243.) Similarly, on page 12 he says,

"The security of RSA depends on our

inability to answer basic questions

about prime numbers," but I thought it

depended on our inability to factor

large numbers. In fact, I thought the

practicality of RSA depended on the

disparity-in practice, primality test­

ing is easy; factoring, hard.

His statement on page 5 that "Mas­

tering these building blocks [primes]

offers the mathematician the hope of

discovering new ways of charting a

course through the vast complexities

of the mathematical world" puzzles me

in a different way. Whatever could it

mean? To my taste, this statement, and

much else in the du Sautoy book,

sounds too much like empty enthusi­

asm, razzle-dazzle meant to impress

non-mathematicians not with sub­

stance-because substantial mathe­

matics is beyond their ken-but with

fanfares and flourishes.

Du Sautoy touches on a clash of cul­

tures within mathematics that is sel­

dom revealed to outsiders and that

might hold some interest for Sabbagh

and others interested in the anthropol­

ogy of our tribe. In his last chapter he

sketches in very laudatory terms the

career of Alexander Grothendieck;

"Grothendieck's new language of geom­

etry and algebra saw the creation of a

whole new dialectic which allowed

mathematicians to articulate ideas

which were previously inexpressible"

(p. 300). Then he goes on to say of the

brave new world of the Grothendieck­

ists that "Even [Andre] Weil was rather

disconcerted by Grothendieck's new

abstract world," and, even more baldly,

quotes Carl Ludwig Siegel as saying, "I

was disgusted with the way in which

my own contribution to the subject had

been disfigured and made unintelligi­

ble," and Atle Selberg as saying, "My

thought was that such lectures were

never given in earlier times. I said to

someone after the lecture a thought

which had come into my mind: if wishes

were horses, then beggars [could] ride."

Disagreements at the highest levels of

mathematics are extremely interesting,

and I applaud du Sautoy for bringing

them into the open, although he does

not pursue the subject.

I do not applaud, on the other hand,

his description of the relation between

RH and mental illness. In his last chap­

ter, he says "Grothendieck is not the

only mathematician who has gone

crazy trying to prove the Riemann Hy­

pothesis," as an introduction to a para­

graph about John Nash. "Grothendieck

and Nash illustrate the dangers of math­

ematical obsession," he concludes, but

mathematical obsession, whether it is

with RH or the continuum hypothesis,

is surely a symptom, not a cause. Our

tribe may have a stronger than average

association with madness that deserves

to be addressed, but, if so, it deserves

to be addressed with more seriousness

than to talk about going crazy trying to

prove the Riemann hypothesis.

As a sometime historian of mathe­

matics, I am dismayed by du Sautoy's

failure to cite a single one of his his­

torical sources. On page 104 he tells of

"several drafts" of a letter he says Rie­

mann was writing to Chebyshev about

"his own progress" in the investigation

of the prime number theorem. In my

1974 book Riemann's Zeta Function I

published a jotting from Riemann's

helter-skelter notes showing that he

was aware of Chebyshev's existence; if

there is more evidence than this of a

Riemann-Chebyshev connection, I do

not know about it. The account of

Siegel's military history (p. 148) differs

substantially from the one given by

Benjamin Yandell in The Honors Class, and, since Yandell names his sources,

I believe his. I hope most readers will

realize that no sources could possibly

support such statements as "[Pythago­

ras] filled an urn with water and

banged it with a hammer to produce a

note" and so forth on p. 77, but many

readers will not. Surely I am not the

only reader who wants to know what

lies behind the surprising reference (p.

128) to "Gauss and Einstein's belief

that space was indeed curved and non­

Euclidean." That Gauss might have

considered the possibility of non­

Euclidean physical space is plausible

enough, but that he believed it? The

propagation of unchecked and un­

checkable anecdotes about the history

of mathematics is a form of pollution

to be combatted. An occasional tall

tale, with appropriate caveats, can cer­

tainly be used to spice up the exposi­

tion from time to time, but when no

sources are ever given for anything,

such tales become an unacceptable

norm.

Another feature of du Sautoy's writ­

ing is his habit of introducing a private

phrase to describe something and for­

ever calling it by his new name rather

than the one used by everyone else. For

example, he says on page 20 that "One

of Gauss's greatest early contributions

was the invention of the clock calcula­

tor." He goes on to explain what he

means-modular arithmetic, of course,

the "clock" being a reference to arith­

metic mod 12-but thereafter there is

no modular arithmetic, only "clock cal­

culators" as in "That is because the cal­

culations will be done not on a con­

ventional calculator, but on one of

Gauss's clock calculators" (p. 234, deal­

ing with RSA). Similarly, zeros of ((s) are first described as "points at sea level

in the zeta landscape" (p. 89) and are

called that for the remainder of the

book On page 79, rather than saying

that the harmonic series diverges, he

says it will "spiral off to infmity"-an

odd way to describe gradual increase

without bound-and thereafter series

never diverge but "spiral off to infmity."

The line where the real part of s is t is

not the critical line, it is "Riemann's

magic ley line" (p. 98) or "Riemann's ley

line." I have not found a definition of

"ley" in any American dictionary that

fits this use; it is apparently a term used

in British surveying.

But du Sautoy and Sabbagh were

not writing for mathematicians. It may

well be that the general readers they

have in mind will be intrigued and grat­

ified by their descriptions of mathe­

matics and mathematicians related to

RH. Certainly there is amusement to be

found in these books, and even math­

ematicians will find many interesting

things in them if they are not too dis­

tracted by questionable formulations

and implausible anecdotes. No harm is

done as long as cranks are not en­

couraged and as long as genuinely in­

quiring minds are not put off when

some of the purported explanations do

not seem to make sense.

The goal of the third book, the one

by John Derbyshire, appears to be dif­

ferent from that of the other two. He

writes in his Prologue of "a general read­

ership" (p. xii), but I think he is unduly

optimistic. He mentions, for example,

that he expects his readers to under­

stand basic algebra, such as the fact that

S = 1 + xS becomes S = 1/(1 - x) when rearranged. Certainly anyone set­

ting out to understand RH must be com­

fortable with this rearrangement, but, in

the first place, I suspect that more edu­

cated adults than we like to imagine

would not be comfortable with it, and,

in the second place, more mathematical

sophistication and ability is needed than

this example suggests. Perhaps Der­

byshire set out to write a book for the

general reader, but as it developed I

think his goal had to change.

No matter! He has written a won­

derful book He does not fudge the

mathematics, which will make parts of

it hard going for most non-mathemati­

cians, but for the most important audi­

ence of non-mathematicians-those

young ones who might consider be­coming mathematicians-it will be a

great resource and inspiration. And for

mathematicians and readers with a fair

amount of mathematical sophistica­

tion, it is a book that will inspire, in­

form, and entertain. If you believe as I

do that RH for the general reader is a

futile project, you will agree that Der­

byshire made the right choices. The

VOLUME 26, NUMBER 1 , 2004 57

copyright page states that the pub­lisher, the Joseph Henry Press, "was created with the goal of making books on science, technology, and health more widely available to professionals and the public," a goal that is admirably served by this book

(Full disclosure: Derbyshire names me in his acknowledgments and men­tions me a few times in the book I met him very briefly at the conference on RH held at NYU in 2002, and at that time I gave him a copy of my book, but I don't recall anything else requiring ac­knowledgment. Of course, I have made every effort to base my opinion of his book on the book alone.)

Even experts on RH will enjoy this book and learn from it, and I would en­courage all readers of the Mathemati­cal lntelligencer to try it. It is interest­ingly and skillfully written, and it approaches many aspects of the sub­ject in imaginative and thought-pro­voking ways. For a quick probe, you might try reading pages 90-92. There you will find discussions of the con­trast between measuring and counting (as he describes it, numbers legato and numbers staccato), Gauss's attitude to­ward Fermat's last theorem, Mallory's reason for wanting to climb Mount Ever­est, the rise of the Germans in 19th-cen­tury mathematics and how it may have been related to the Napoleonic wars and the Congress of Vienna-as well as a passing mention of Larry, Curly, and Moe.

If that rushed summary suggests that the writing is contrived or pre­cious or pretentious, the fault is mine. To my taste it is always down-to-earth and treats its topics in natural and ap­propriate, but interesting, ways.

Naturally I have my disagreements and cavils with the book, but it is re­markable to me how few they are when I consider how dense with information and opinions the book is. The peasant­pheasant story about Peter the Great on p. 56 should have been tossed out in the revision process. Derbyshire does an admirable job of keeping the calculus in the book to a minimum (he tells us in the Prologue that his origi­nal goal was to have no calculus at all, but that this goal "proved a tad over­optimistic"), but my alarm bells go off

58 THE MATHEMATICAL INTELLIGENCER

when, on p. 112 and again on p. 113, he describes a definite integral as an "area under a function." I gather that, com­petent as his mathematics is, he has never taught calculus and had to deal with students who persistently confuse functions with their graphs.

I am most disturbed by his state­ment about the formula em = - 1 that "Gauss is supposed to have said-and I wouldn't put it past him-that if this was not immediately apparent to you on being told it, you would never be a first-class mathematician," not only be­cause I question the attribution of such a statement to Gauss and no source is given, but mainly because it strikes me as a terrible thing to say to a young stu­dent. One's reaction to em = -1 must be awe, not "oh, yes, of course!" If you tell me it was immediately apparent to you when you first saw it I will think you are a fool or a liar, or that your memory is faulty. Derbyshire is wrong to discourage his readers-who will need a good portion of ambition to al­low them to penetrate his book-in any way, and particularly to do so on false grounds.

And he is indeed asking a lot of his readers. In his 21st chapter he walks the reader through Riemann's explicit formula

J(x) = Li(x) - I Li(xP) - log 2 + p

LX dt x t( t2 - 1) log t

where J(x) denotes 1T(x) + t'1T(Yx) + t1TCVx) + · · · (a terminating series be­cause 1T(y), which is the number of primes less than y, is zero when y < 2) and where the complex numbers p are the zeros of the zeta function in the crit­ical strip. And I don't mean that he sim­ply explains the definitions of all the terms. He also explains how the series

Lr Li(xP) converges conditionally, so the order of the terms is of the essence and the convergence is very slow, and he actually provides numerical esti­mates of the various terms in the case x = 20. Once he has completed this, hav­ing shown in detail and with clarity how the formula yields the known value

7 J(20) = 9J2, he goes on to show the way in which Mobius inversion combines with this formula to give Riemann's ex-

plicit formula for 1T(x), taking for the sake of illustration the case x = 1,000,000 and carrying it through very clearly to show how it really does work out to give '1T(1,000,000) = 78498. (But he does not adequately explain how he evaluated the slowly converging "sec­ondary terms." He gives them to five decimal places, but in his computation of J(20) he already confessed that 86,000 terms of the sum had to be com­puted to attain four-place accuracy for this term, and he certainly does not ex­pect us to believe that he found the nine-place result he gives in that case by adding terms of the series!)

Can a beginner follow this chapter? Not unless the beginner is very tal­ented. To tell the truth, I had to read it pretty attentively. But it is interesting. The talented beginner will learn from it, as I learned from it. And those who can't follow it are not being sold a bill of goods, not being encouraged to think they understand and appreciate something they don't understand at all, and not being condescended to. They can give it their best shot, and if they fail they can still admire it and still ap­preciate much of the rest of the book, and may someday come back to it when they are no longer beginners.

A parting thought. In my opinion, all three books grossly overstate the con­nection of RH to prime numbers. Der­byshire even chooses the title "Prime Obsession." True, an investigation of the distribution of primes and the Euler product formula led Riemann to RH, but Riemann himself quickly switched to another function he called g(t) (it is the value at s = t + it of f(�) ·

2 s(s; I) . '1T�s/2((s), which, as Riemann proves, is an even function of t that is real on the real axis) and his actual hy­pothesis was that the zeros of g( t) are real! To me, g( t) is a symmetrized ver­sion of ((s)-symmetrized to put the functional equation of ((s) in a simple form and to put the interesting part of the function on the real axis-that is an entire function of one complex vari­able. RH is simply the statement that its zeros are real. The connection with prime numbers may or may not play a role in explaining the amazing extent to which Riemann's hunch has been borne out by massive modern compu-

tations unimaginable in his day. De­

spite this modem evidence in its favor,

and despite its connection to a raft

of fascinating and theoretically impor­

tant "generalized Riemann hypotheses"

that also stand up to computational

scrutiny, no one seems to have any idea

why it should be true. Who wants to be

a millionaire?

Courant Institute of Mathematical Sciences

New York University

New York 1 001 2, USA

e-mail: edwards@cims.nyu.edu

A Mathematician Grappling with His Century by Laurent Schwartz

BIRKHAUSER VERLAG. 2001

vi1i + 490 pp., ISBN 3764360526; US $49.95.

REVIEWED BY NORBERT SCHLOMIUK

Laurent Schwartz [1915-2002] was

one of the great mathematicians of

the twentieth century. His main con­

tribution to mathematics is his work on

distribution theory.

In his "History of Functional Analy­

sis" Jean Dieudonne wrote:

The role of Schwartz in the theory of distributions is very similar to the one played by Newton and Leibniz in the history of Calculus: Contrary to popular belief, they of course did not invent it, for derivation and integra­tion were practised by men such as Cavalieri, Fermat and Roberval when Newton and Leibniz were mere schoolboys. But they were able to sys­tematize the algorithms and nota­tions of calculus in such a way that it became the versatile and powerful tool which we know.

The great importance of Laurent

Schwartz's contributions to mathe­

matics was recognized by his being

awarded the Fields Medal in 1950, the

first French mathematician to receive it.

Laurent Schwartz was above all an

extraordinary human being: warm, gen­

erous, wise, modest, deeply involved in

the struggle for the oppressed, for hu­

man rights and the rights of people, a

great and noble figure of the twentieth

century. We are lucky that close to the

end of his life he decided to write an

autobiography.

A Mathematician Grappling with his Century is the translation by Leila

Schneps of the original French edition

published in 1997 by Editions Odile Ja­

cob under the title Un mathematicien aux prises avec son siecle. For those

of us who had the good fortune to be

close friends of the author, reading the

book is listening to the beautiful voice

of Schwartz, a marvellous raconteur.

In the Foreword, the author pre­

sents the content of the book:

I am a mathematician. Mathematics filled my life: a passion for research and teaching as a professor both in the University and at the Ecole Poly­technique. I have thought about the role of mathematics, research and teaching, in my life and in the lives of others. I have pondered on the men­tal processes of research and for decades I have devoted myself to ur­gently necessary reforms within the University and at Ecole Polytech­nique. Some of my reflexions are con­tained in this book, as well as a de­scription of the course of my life. However I do not discuss the Univer­sity reforms since I have written many articles and books on the sub­ject. Inevitably, mathematics appear in this book, one cannot conceive of an autobiography of a mathematician which contains no mathematics. I have written about them in a histor­ical form which should be accessible to large non-specialist sections of the scientific public; readers impervious to their charm may simply skip them. They concern only about fifteen per cent of the volume.

The reader interested in the con­

temporary problems facing universi­

ties would find much to think about in

Schwartz's "Pour sauver l'universite"

(Seuil, 1983) and in "Pour la qualite de

l'Universite fran<;aise" by Pierre Merlin

and Laurent Schwartz (P.U.F. 1994).

The Introduction to his autobiogra­

phy is titled "The Garden of Eden," a

reference to the property at Autouillet

purchased by Schwartz's parents in

1926. It was at Autouillet where the au­

thor found the ideal conditions to work

Schwartz grew up in a very warm fam­

ily. His father, a distinguished physi­

cian, had a strong influence on his chil­

dren. Here is a lesson given by the

father to his son:

If in a given circumstance you find that you are alone with your opinion against everybody else, try to listen to them, because maybe they are right and you are wrong. But if, after hav­ing thought it out, you still find your­self alone with your opinion, then you should say it and shout it and let everybody hear it.

His explanation, writes Schwartz,

remained engraved inside me and

guided me in all my political activities

in my adult life. Tolerance, inner free­

dom, wisdom were qualities which im­

pressed everyone who had the chance

to know him.

I remember one of the public lec­

tures Schwartz gave in Montreal about

the life of a mathematician. Many in the

large audience were high school and

college students fascinated by the lec­

ture and impressed by the sincerity of

his presentation when he spoke about

his self-doubt. I found his words in the

book:

In spite of my success [in school}, I was always deeply uncertain about my own intellectual capacity. I thought I was unintelligent. And it is true that I was and still am rather slow. I need time to seize things be­cause I always need to understand them fully. Towards the end of the eleventh grade, I secretly thought of myself as stupid. Not only did I be­lieve I was stupid, but I couldn't un­derstand the contraindication be­tween this stupidity and my own good grades. At the end of the eleventh grade I took the measure of the situation and came to the conclusion that rapidity doesn't have a precise relation to in­telligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. Naturally it is help-

VOLUME 26. NUMBER 1 . 2004 59

ful to be quick, like it is to have a good memory. But it's neither necessary nor sufficient for intellectual success.

And here a last quotation from this

marvellous book:

Self-confidence is a condition of suc­cess; of course one must be modest, and every intellectual needs to recall this. I am perfectly conscious of the immensity of my ignorance com­pared with what I know. It's enough to meet other intellectuals to see that my knowledge is just a drop of water in an ocean. Every intellectual needs to be capable of considering himself relatively and measuring the immen­sity of his ignorance. But he must also have confidence in himself and in his possibilities of succeeding through the constant and tenacious search for truth.

Very interesting, sincere and illumi­

nating are the chapters about the po­

litical involvement of Schwartz. Very

early during the infamous Moscow tri­

als (1937) he realized the nature of

Stalin's dictatorship. He became a Trot­

skyite in 1944, and was an unsuccess­

ful candidate as a Trotskyite for the

French Legislative election in 1945.

Much too free a thinker to accept a

"party line," he broke with Trotskyism

in 1947.

The center of his life continued to

be in mathematics. After a one-year ap­

pointment in Grenoble, Schwartz joined

in 1945 the Faculty of Science in Nancy,

where he spent seven extremely fruit­

ful years both for his research and for

attracting brilliant young mathemati­

cians: Grothendieck, Malgrange, Lyons,

Treves, Bruhat. He moved to the Sor­

bonne in 1952 and to the Ecole Poly­

technique in 1958.

The torture and murder of the young

and gifted mathematician Maurice Au­

din by repressive French forces in Al­

geria led Schwartz to a deep involve­

ment in chairing the Audin Committee,

in protesting against torture and mur­

der committed by some of the official

French forces, and in signing the fa­

mous "Manifesto of the 121 ," which

proclaimed the right of French youth

to rebel against the Algerian war. For

60 THE MATHEMATICAL INTELLIGENCER

this he was dismissed from the Ecole

Polytechnique by the then Minister of

Defense Pierre Messmer.

Later, during the Vietnam war,

Schwartz again became politically in­

volved, and again that involvement was

not one-sided. During trips to Vietnam

after Ho Chi Minh's victory, he contin­

ued to intervene with the Vietnamese

authorities on behalf of dissidents.

Together with Henri Cartan and

Marcel Broue, Laurent Schwartz was a

founder of the Committee of Mathe­

maticians. The task of the Committee

was to fight against human rights vio­

lations of mathematicians, mainly in

the Soviet Union but also in Morocco,

in Czechoslovakia, in Uruguay-and

indeed, all over the world.

Each chapter of Schwartz's autobi­

ography is beautiful and interesting.

Mathematicians will be particularly in­

terested in chapter VI, "The invention

of distributions." Few mathematicians

have written about the act of creation

in mathematics, Poincare and Hadamard

for instance. The author's pages are, I

think, in the same league.

Laurent Schwartz is no longer among

us; we cannot listen to his beautiful

warm voice carrying messages of wis­

dom, humor, and understanding; but

we have this marvellous autobiography

of a splendid man.

Departement de Mathematiques et de

Statistique

Universite de Montreal

C. P. 61 28, Succursale Centre-Vil le

Montreal, H3C 3J7 , Canada.

e-mail: schlomiu@dms.umontreal .ca

The Mathematician Sophus Lie: It Was the Audacity of My Thinking by Arild Stubhaug, translated from the Norwegian by Richard H. Daly

BERLIN HEIDELBERG, SPRINGER-VERLAG 2002

556 pp. LIS $44.95, ISBN 3-540-421 37 ·8

REVIEWED BY JESPER LUTZEN

The year 2000 saw the publication of

two remarkable books about the

Norwegian mathematician Sophus Lie

and his work: Thomas Hawkins's

Emergence of the Theory of Lie Groups (Hawkins [2]) and Arild Stub­

haug's Norwegian biography of Lie

(Stubhaug [3]). The present review

deals with the English translation of

the latter.

The two books complement each

other very well. Hawkins's book, which

represents the crowning result of many

years of research, gives a thorough

mathematical analysis of Lie's inven­

tion of Lie groups and the further de­

velopment of the subject for the next

half-century. It is aimed at an audience

of mathematicians who know the the­

ory and can follow the technical details

of its development. Stubhaug, on the

other hand, draws a vivid picture of the

person Sophus Lie and the time he

lived in. His book is aimed at a general

audience, and does not go into mathe­

matical detail. That does not mean that

mathematics is left out of Stubhaug's

biography; in fact, mathematics is con­

stantly present in the book as Lie's

great passion, and it is clear that Lie's

greatness was due to his mathematical

creativity. Lie's mathematical cre­

ations are described in general terms,

and in a chapter titled Into Mathemat­

ical History Stubhaug does a good job

of setting them into historical and

philosophical perspective. A non-math­

ematical reader will learn from Stub­

haug that mathematics is fascinating

and important and will get a good

glimpse of the creative, dramatic, and

even existential aspects of mathemati­

cal research. For this reason alone one

must hope that the book will be widely

read, especially at a time when scien­

tific and mathematical research is un­

der public scrutiny. The mathemati­

cian-reader who finds the description

of Lie's mathematics superficial can

turn to Hawkins, or to Armand Borel's

new book Essays in the History of Lie Groups and Algebraic Groups [ 1 ] .

Both mathematicians and non-mathe­

maticians will enjoy other aspects of

Stubhaug's book; like Stubhaug's ear­

lier biography of Niels Henrik Abel, it

is both extremely well written and well

researched. Richard H. Daly, who also

translated the Abel biography, has

again done an excellent job. For the

non-Norwegian reader there have been

inserted a few explanations of locali­

ties and personalities that are well

known to a Norwegian audience. In a

few cases this has been overdone, as

for example when the often-mentioned

Nordmarka is repeatedly explained to

be a forest north of Christiania. In one

respect the translation of the Lie book

is even more successful than the Abel

book: The mathematical terms have

with a few exceptions been translated

correctly.

The style is literary rather than

scholarly. For example, there are no

references and only a few footnotes in

the main text. And the style is narra­

tive rather than analytic. The beginning

of the book may suggest otherwise.

Here Stubhaug explains that everyone

who met Lie would later tell stories and

anecdotes about him. "Nobody seems

to have been able to pass him by in si­

lence. What, in detail, did the stories

recount, and what, on the other hand,

do we know with certainty? What was

imaginary and what was real?" This

may sound like an introduction to an

analysis of the sources pertaining to

Lie's life, but in fact the succeeding

chapter does not provide such an

analysis. The questions are rather a

stylistic trick to begin a 20-page over­

view of Lie's life, that serves as a sort

of abstract of the book. This introduc­

tion is helpful to the reader, who may

otherwise get lost in the wealth of ma­

terial presented in the main part of the

book. For readers who do get lost,

however, there is a 5-page schematic

chronology at the end.

With its literary narrative style the

book reads like a novel. However, as is

revealed by the Bibliography, it is

based on a thorough study of a wealth

of sources. Many mathematicians and

historians of mathematics have writ­

ten biographies of Lie-for example,

Friedrich Engel, Poul Heegaard, Elling

Holst, Max Noether, and Ludvig Sylow

(a list is provided in Stubhaug's book)­

but this is the first book-length biogra­

phy of the Norwegian mathematical ti­

tan. One reason that Stubhaug's book

is longer than earlier biographies is

that he embeds the story of Lie's life in

a rich cultural, political, and institu­

tional context. Another reason is that

new sources uncovered by Stubhaug's

research, in particular many collec­

tions of letters to and from Lie, have al­

lowed him to paint a very detailed pic­

ture of Lie and his life. In this respect

Stubhaug's Lie book is more ground­

breaking than his Abel book.

The scholarly aspects of Stubhaug's

work are also revealed by the 50 pages

of endnotes. Within them are refer­

ences to the sources as well as addi­

tional details that enhance the main

text. In many cases these enhance­

ments seem just as relevant and inter­

esting as the main text, and they may

have been relegated to endnotes only

because they would have hindered the

flow of the narrative.

The book is richly illustrated with

photographs of people and places of

significance to Lie, and with reproduc­

tions of paintings representing the pre­

vailing view of the grand Norwegian

Nature that give the reader an idea of

the existing zeitgeist. Unfortunately

the book does not contain a map of

Norway or Europe showing the places

mentioned in the text. That would have

been a great help, in particular for non­

Norwegian readers.

As in most biographies, the main

part of Stubhaug's book is written

chronologically. After the summary

presented in the first part, the second

part deals with Lie's family background

and his upbringing. We learn that he

was born on December 17th, 1842, as

the penultimate child out of six of the

vicar Johan Herman Lie in Nordfjordeid

and Mette Maren nee Stabell, who ran

the vicarage as a model farm. When he

was 9 years old he moved with his fam­

ily to Moss, where his father remained

a Parish Vicar for the rest of his life.

Sophus's mother died a year after the

move. In 1856 Lie completed the edu­

cation at the local science school, and

after a year of private tutoring he fol­

lowed his older brother to Nissens

School in Christiania (now Oslo). Such

are the bare essentials, but such a sum­

mary does not do justice to the rich­

ness of part 2 of this biography. The 27-

page section contains a host of details

imbedded in an extremely well-in­

formed cultural-historic context. Stub­

haug explains the family tree of the Lie

family; in particular he goes into some

detail about the life of Lie's father. We

learn that after having taken his theo­

logical degree, father Lie began to

work as a teacher. Only when his ap­

plication for the job as headmaster

of the school where he worked was

turned down did he tum to an ecclesi­

astical career. However, he continued

to be very actively interested in the en­

lightenment of his parishioners, not

only in religious but also in scientific

matters. In particular he initiated a se­

ries of lectures on the natural sciences

for the workers in Moss, hoping that an

awareness of the laws of nature would

reveal the creative hand of God. Here

it is interesting to note that Sophus

Lie's later hero Abel had also been the

son of a vicar with rational scientific

leanings.

Stubhaug also tells about the village

of Nordfjordeid and in particular about

the vicarage and its buildings. We hear

about father Lie's versatility as a vio­

linist, his two terms as mayor of the vil­

lage, and many other things about the

rather happy everyday life in the vic­

arage in Nordfjordeid. All this is seen

on the background of the general

trends in Norwegian society at the

time. After the move to Moss and the

death of Lie's mother, more stringent

and less happy conditions reigned in

the vicarage. But at school Lie did very

well and graduated as number one in

his class. In connection with his stud­

ies at Moss Realschool, Stubhaug gives

a long explanation of the ongoing re­

forms of the Norwegian school system.

This explanation continues in Part

3, which is devoted to Lie's time at Nis­

sen's school in Christiania. Stubhaug

gives a 12-page account of how this

school was founded as a clear alterna­

tive to the old Latin and Cathedral

schools, how it became an example for

many subsequent schools that valued

science and modem languages on a par

with the classical languages, and how

its creation was a reflection of and a

great influence on a broad debate in

Norway about the means and goals of

education, a debate in which Lie later

took an active part. We hear that the

founder of the school, Nissen, was a

former student of father Lie; we learn

about the school buildings, about the

teachers, both those that taught Lie

VOLUME 26, NUMBER 1 , 2004 61

and the ones that preceded them, and

about some of Lie's classmates and

their families.

As I present the content of this chap­

ter it may sound far too detailed and

long, but in fact I found all of it inter­

esting and relevant for an understand­

ing of the context in which Lie grew

up. The only thing I missed was a more

detailed account of what happened

when Lie during his first year at Nis­

sen's school had Ludvig Sylow as his

mathematics teacher. We learn only

that Lie was the best in his class in this

subject, and that Sylow later recounted

that he had not seen any special math­

ematical genius in the young student.

Do the sources say nothing more about

the matter?

In the 3rd part of the book the char­

acter Sophus Lie gradually becomes

visible. As a student at Nissen's school

he did very well in all his subjects, and

in the end he was number two in the

entrance exam to the University in

Christiania. He began to study science

at the University in 1861, and from then

on his whereabouts are rather well

documented. He continued to do bril­

liantly in most subjects, in particular in

the mathematical sciences, physics, as­

tronomy, and chemistry, and he was

very active in the Scientist Association.

He also began a habit, that lasted al­

most to the end of his life, of taking

strenuous hikes of several weeks dur­

ing the summer into the mountainous

regions of Norway. As a boy he had ex­

celled in physical strength, and his long

and fast hikes became legendary.

What kind of a person was Lie? As

a short answer to this question and as

a good example of the beauty of Stub­

haug's language, let me quote the open­

ing words of the book:

According to most accounts of Sophus Lie, he was the embodiment of an ar­chetypical character in a theatrical drama-with his forceful beard, his sparkling green eyes magnified by the stout lenses of his spectacles-the blond Nordic prototype, as it was called across Europe-the Germanic gigantic being-a primal force, a ti­tan replete with the lust for life, with audacious goals and an indomitable

62 THE MATHEMATICAL INTELLIGENCER

will. These descriptions of his physi­cal and mental strength also con­tained a subtext, an embryonic no­tion, not only about this brilliant man of science, the prophet, who intu­itively conceived new mathematical truths, but also about the colossus who, in his constant zeal for new knowledge, might push others aside, and inadvertently trample them un­derfoot. He was described as highly committed, richly innovative, some­one with unusual physical strength, and the stamina to overcome the ma­jority of obstacles, but also, a man who afterwards had to pay for this with correspondingly great swings of mood and temperament.

One could add that he was a warm

person, often friendly, strong-willed,

direct in his speech, undiplomatic

"the blond Nord ic prototype . a t itan . . "

sometimes even raw in his manners,

and rather self-centered. He wrote

about himself that he "had little talent

for socialising with folk," and "what is

fatal is that I am so diametrically dif­

ferent from [Felix] Klein with respect

to the ability to be able to get into the

thinking of others" (p. 257).

The mood swings that Stubhaug

mentions showed themselves for the

first time during Lie's last year as a uni­

versity student. He had set himself the

high goal of graduating from University

with the highest grade overall. After

the first three years this still seemed an

attainable goal, but the last year's study

of the biological sciences did not go so

well, so he only got the second-highest

grade. He became depressed, was un­

able to sleep, and even planned suicide.

This psychologically unstable state

lasted for the next few years, when he

was plagued by a sense of lacking a

calling for his life.

In fact it is remarkable that, unlike

Abel, who in high school had already

begun to study the masters and make

original contributions to mathematics,

Lie did not seriously begin to pursue a

career in mathematical research until

three years after his graduation from

university. Indeed, with his many phys­

ical as well as intellectual talents he

might well have chosen a different pro­

fession. In school he contemplated go­

ing into philology, but at the entrance

exam to the university he got only the

second-highest mark in Greek. He was

so dissatisfied with his performance

that he opted for science instead. And

even as a science student he toyed with

the idea offollowing the example of his

older brother, who was an officer in the

army. In fact in 1864, when the Danish

borders were threatened by German

troops, Lie followed the general Scan­

dinavism among the Norwegian stu­

dents, and volunteered to defend the

brother country. However, before his

military training in Christiania was

complete, Denmark had surrendered.

Lie continued to serve for a few years

as a reserve lieutenant, but it turned

out that his eyes suffered from oblique

cornea, and therefore he could not pur­

sue a military career.

Even after graduation from univer­

sity, Lie did not have a clear idea about

his vocation. He began to work at the

observatory and to give popular lec­

tures on astronomy in the Student So­

ciety, and he planned to write a book

on the subject. However, probably as a

result of disagreements with the pro­

fessor of astronomy, he did not obtain

the vacant job as assistant at the ob­

servatory, and in 1867 he gradually

turned to mathematics and composed

a small textbook on trigonometry.

Only during the following year did

he become convinced that "there was

a mathematician in him." The turning

point was the meeting of the Scandi­

navian natural scientists in Christiania

that year. Here Lie met Sylow again.

While a student at the university Lie

had followed a course on Galois theory

(the first lecture on that subject any­

where in the world after Liouville's pri­

vate lectures in the 1840s) that Sylow

had given while he was a substitute for

the mathematics professor Ole Jacob

Broch, who had been elected Member

of Parliament. Lie had in the meantime

lost his notes from those lectures, and

now asked Sylow permission to bor­

row his notes. "I believe that group the­

ory will become very important," Lie

prophetically told Sylow. Moreover,

Lie developed a friendship with the two

Danish mathematicians Adolph Steen

and Hieronymus Georg Zeuthen, who

attended the meeting. The former gave

a talk about integration of differential

equations, and Zeuthen talked about a

subject from the new geometry. These

three subjects, group theory, differen­

tial equations, and geometry, became

the central elements in all of Lie's fu­

ture mathematical work.

His depressive moods gone, Lie

threw himself into a study of various

recent works in geometry that Zeuthen

had referred him to, and he began to

do independent research. By Decem­

ber he got the idea of his so-called

imaginary geometry. The following

year he lectured on it in the Science So­

ciety and privately published a short

pamphlet about it, followed by two pa­

pers in the Proceedings of the Chris­tiania Academy of Sciences and a pa­

per in Grelle's Journal fur die reine und angewandte Mathematik. This

closely paralleled the way the young

Abel had published his first original re­

search. From the outset, Lie reached

out to an international audience. No

one in Norway could fully appreciate

the value of Lie's new contribution to

mathematics, and yet many of the best

Norwegian mathematicians and scien­

tists were aware that it was important.

As early as 1869 he was given a travel

stipend that allowed him to stay for a

half-year in Berlin and for a half-year

in Paris, the two cities that Abel had

visited a half-century earlier. Unlike

Abel, however, Lie preferred the

French style in mathematics and was

not particularly fond of the rigor that

characterized the mathematicians in

Berlin. Still, his stay in Berlin became

very fruitful, and he had the triumph of

impressing Eduard Kummer by solving

a geometrical problem he could not

solve himself. This may be considered

Lie's breakthrough in international

mathematics.

But the most important aspect of his

stay in Berlin was his encounter with

the 20-year-old Felix Klein. The two

soon became friends, and they began

to collaborate on research. When Lie

continued on to Paris in the spring of

1870, Klein came along, and they both

enjoyed the regular meetings with

Camille Jordan and Gaston Darboux.

However, when the Prussian-French

war broke out in July, Klein immedi­

ately returned to Germany. Lie also

left, for Italy. He had planned to walk

through France over the Alps to Milan,

where he wanted to meet with Luigi

Cremona. However, he only made it as

far as Fontainebleau before he was ar­

rested as a German spy. When he was

released a month later, thanks to the

intervention of Darboux, he took the

next train to Switzerland.

In Berlin and Paris and in prison, Lie

pursued his geometrical research and

began to work on contact transforma­

tions. His international success made

him hope for either a better stipend or

a permanent position in Norway, but

the applications he sent from abroad

did not bear fruit. When he came home

he composed a doctoral thesis about

the line-sphere transformation that he

had discovered in Paris. No one in Nor­

way understood its content, but its im­

portance was soon recognized interna­

tionally, and when Lie in the fall of 1871

applied for a professorship in Lund,

Sweden, many Norwegian intellectuals

realized that Norway was on the verge

of repeating the mistake they had made

when they did not offer Abel a profes­

sorship. The newspapers published let­

ters of recommendation from Alfred

Clebsch, whom Lie had met in Gottin­

gen, and from Cremona, whom Lie had

not succeeded in meeting in Milan. It

was mentioned that other letters of

support had come from Berlin, Copen­

hagen, and Paris.

In 1872 Cabinet minister and math­

ematician Broch convinced the Nor­

wegian Parliament to appoint Lie as

extraordinary professor. Usually, pro­

fessors were appointed by the Cabinet

of the Swedish king, who also ruled

Norway. Thus Lie's appointment was a

small part in a power struggle between

the freely elected Norwegian Parlia­

ment and the Cabinet, and more gen­

erally a strong statement in the Nor­

wegian struggle for independence.

The same year Lie became engaged

to be married to Anna Birch, who was

a granddaughter of Abel's uncle. He

wanted to be married as soon as pos­

sible, but she wanted to wait. In the

meantime they conducted an intense

correspondence, which Stubhaug de­

tails in the beginning of Part 5. I must

admit that Lie's constant begging for an

early date of marriage became tire­

some. It is the only part of the book

that in my opinion would have bene­

fited from cutting. In 187 4 Lie finally

persuaded Anna to marry him, and

over the course of the next 10 years

they had three children: two daughters

and a son. Their marriage was happy.

The period 1872-1886, while Lie was

a Parliamentary professor in Christia­

nia, was also his most productive pe­

riod. He traveled often to Germany,

where he continued his friendship and

exchange of ideas with Felix Klein, and

to Paris, where he presented his new

work on differential equations and his

theory of transformation groups,

which he developed as a means to

solve them. He published a host of new

results, first in the Norwegian journal

Archiv for Mathematik og Naturvi­denskab, which he founded together

with two colleagues, and later in inter­

national journals, mainly in the Mathe­matische Annalen, which Klein edited

from 1877 together with Adolf Mayer.

In 1884 Klein arranged for Engel to go

to Christiania to help Lie present his

new ideas in a more polished book form.

Their intensive collaboration during

nine months in Christiania and later in

Leipzig resulted in Lie's main work

Theorie der Transformationsgruppen, published in three volumes in 1888--1893.

From 1873 Lie also worked with Sylow

on a new complete edition of Abel's

works. He also suggested to the

Swedish mathematician Gosta Mittag­

Leffler that they found a new Scandi­

navian research journal for mathemat­

ics. This resulted in the creation of

Acta Mathematica. Lie sometimes lectured on aspects

of his new mathematics, but he did not

have any first-rate Norwegian students.

Therefore he immediately accepted

when, at the instigation of Klein, he

was offered the professorship vacated

VOLUME 26, NUMBER 1 , 2004 63

at Leipzig University when Klein moved to Gottingen. Lie's stay in Leipzig from 1886 to 1898 was a mixed experience. His lectures drew many good students. In particular Lie was proud that his Parisian colleagues sent some of their best students of the Ecole Normale, e.g., Arthur Tresse and Ernest Vessiot, to Leipzig to study with him. More­over, he had several good assistants and Private Docents such as Engel (whose collaboration with Lie contin­ued) and Eduard Study, Issai Schur, and Georg Scheffers, who published Lie's lectures in three large volumes: Differential Equations with Known Infinitesimal Transformations, Con­tinuous Groups with Geometric and other Applications, and Geometry of Contact Transformations (all in Ger­man). Finally, he had a good relation­ship with some of the other professors such as Adolph Mayer and Wilhelm Ostwald, but his relationship with his mathematics colleague Carl Neumann was strained.

In 1889 Lie suffered a mental break­down that confmed him to a psychi­atric hospital for seven months. Even after he released himself from hospital and tried to walk the depression out of his system, he was strongly depressive for several years, and his personality seems to have changed for good. Even before the breakdown he had begun to assert his priorities vis-a-vis Klein, and after the breakdown there was a com­plete break between the two former friends. Lie also began to attack his other collaborators and supporters such as Engel, and he accused Wilhelm Killing of stealing his ideas. On the whole he turned away from his German colleagues and oriented himself more toward Paris, where his works won in­creasing acceptance.

Part 6 of Stubhaug's Lie biography, which deals with the Leipzig period, is in my opinion the weakest in the book. Stubhaug deviates more from a chrono­logical presentation than in other parts, and that makes the part somewhat dis­connected. In particular it is unfortu­nate that there is not a clear distinction between the periods before and after Lie's mental breakdown. Moreover, when writing about the cultural and in­stitutional setting in Leipzig, Stubhaug

64 THE MATHEMATICAL INTELLIGENCER

seems to lack the mastery with which he deals with the Norwegian scene.

Yet Norwegian affairs are con­stantly mentioned also in this part. In­deed, Lie continued to keep abreast with the developments in his native country through regular reading of Norwegian newspapers, through cor­respondence, and through the young Norwegians who came to study with him in Leipzig. He even continued to participate in the public debate on ed­ucational affairs in his homeland, and he was active in the preparations for the centenary of Abel's birth in 1902. In particular, he tried to raise funds for an Abel Prize in Mathematics matching the newly founded Nobel Prizes in other disciplines. As is well known, this idea was not implemented for more than a century.

While his wife and children soon adapted to the new social situation and thrived in Leipzig, Lie had a hard time getting used to the tone at the Univer­sity, to the much larger teaching load, to German militarism, and to the heat, and he found it difficult to lecture in a language that he had not mastered. He was happy to have left the provincial at­mosphere in Christiania, but he missed his friends and in particular Norwegian nature. The first summers in Germany Lie rented a vacation home near Leipzig and went to the Alps for hiking tours, but from 1888 he began to spend parts of each summer in Norway.

His warm feelings for his home coun­try and his hope for its freedom from Sweden became clear to everyone at the university during his inauguration as a Professor in Leipzig. During the cere­mony the Rector mentioned that he had heard that the peasant representatives in the Norwegian Parliament had treated the noble King Oscar badly; therefore he could well understand that Lie wanted to leave Christiania. At this point Lie protested loudly and left the room.

In Norway the movement that re­sulted in independence from Sweden in 1905 was gaining momentum in the 1890s. One of the strategies among the cultural and academic elite was to dis­play the great Norwegian talent. As a part of this strategy, the polar explorer Fridtjof Nansen, the mathematician Elling Holst, and the leading poet Bj0m-

stjeme Bj0mson, who had himself re­cently returned from a long stay in Paris, conceived a plan to bring Lie back to Norway. After negotiations with Lie in 1893 they succeeded in convincing the Norwegian Parliament to upgrade his professorship (from which he had obtained a leave when he moved to Leipzig) to a Professorship of Transfor­mation Group Theory with a salary that matched his high German salary. Lie ac­cepted, but it took him another 4 years before he finally resigned his position in Leipzig and returned for good to Chris­tiania. During those four years he trav­eled back and forth from Norway sev­eral times, and even had an idea of trying to arrange a joint position at the two universities. When he returned to Christiania in September of 1898 he was not in good health, and soon it became clear that he suffered from pernicious anemia. He died from this disease on February 6 of the following year.

Many obituaries and later biogra­phies have tried to capture this extra­ordinary mathematical genius, but none of them have been as complete, as well researched, or as well rooted in the cultural context as Stubhaug's well­written book. I can recommend it to all mathematicians as well as non-mathe­maticians, who have an interest in the human aspects of scientific creation.

REFERENCES [ 1 ] Armand Borel, Essays in the history of Lie

groups and algebraic groups. History of

Mathematics 21 , Providence, Rl: American

Mathematical Society; Cambridge: London

Mathematical Society, 2001 .

[2] Thomas Hawkins, Emergence of the The­

ory of Lie Groups. An Essay in the History

of Mathematics 1 869-1926, New York:

Springer-Verlag, 2000.

[3] Arild Stubhaug, Oat var mine tankers

djervhet-Matematikeren Sophus Lie. Oslo:

Aschehoug, 2000.

[4] Arild Stubhaug, Niels Henrik Abel and his

Times. Called too Soon by Flames Afar,

Berlin, Heidelberg: Springer-Verlag, 2000.

Department of Mathematics

Copenhagen University

Universitetsparken 5

DK-21 00 Copenhagen 0 Denmark

e-mail: lutzen@math.ku.dk

Fibonacci Numbers by Nicolai N. Vorobiev

BIRKHAUSER. 2002 1 76 pp. €31 ISBN 3-7643-61 35-2

paperback

REVIEWED BY FREDRIC T. HOWARD

The Fib�nacci �umbers ar� defined

by u1 - 1 , Uz - 1, and Un - Un�l + Un�2 for n 2: 3. They were first men­

tioned in 1202 in the Liber Abaci, a

book written by Leonardo of Pisa to in­

troduce the Hindu-Arabic numeral sys­

tem to western Europe. Leonardo, per­

haps the greatest mathematician of the

Middle Ages, wrote under the name of

Fibonacci, a contraction of "filius

Bonacci" (son of Bonacci). In Liber Abaci the numbers appeared in the fa­

mous rabbit problem, but they were

not called "Fibonacci numbers" until

much later.

The rich and interesting early his­

tory of the Fibonacci numbers can be

found in L. E. Dickson's History of the Theory of Numbers, volume 1. During

the last half of the twentieth century,

interest in the numbers and their ap­

plications increased dramatically, and

in 1961 N. Vorobiev published his ele­

mentary but influential book titled Fi­bonacci Numbers. Written for high

school students, this book introduced

the basic properties of the Fibonacci

numbers to a new generation of readers.

Vorobiev revised his book in 1989,

and it has now been re-issued by

Birkhauser Verlag. I will briefly sum­

marize the chapters and compare them

to the material in the 1961 edition.

Chapter 1 is titled "The Simplest

Properties of the Fibonacci Numbers."

Vorobiev first proves some simple

formulas, like u1 + u2 + · · · + Un = Un+2 - 1, and then works his way up to

more sophisticated formulas like the

one for ur + u� + . . . + u�. Along the

way, he discusses binomial coefficients

and proves the Binet formula. In the old

edition, he ends the chapter with the re­

sult that Un is the nearest integer to

an!v'5, where a = (1 + V5)/2. The new

edition goes further; Vorobiev continues

to discuss the last result, and he studies

Sn, the sum of the reciprocals of the first

n Fibonacci numbers, and the limit lim n�x

Sn. He shows how to write a given num­

ber as a sum of Fibonacci numbers, and

he makes a comparison of the Fibonacci

system and the decimal system. He gen­

eralizes to other bases.

Chapter 2 is titled "Number-Theo­

ertic Properties of Fibonacci Num­

bers." Vorobiev proves the well-known

fact that if m divides n, then Um divides

Un. He then discusses general properties

of the greatest common divisor, and he

proves that gcd(um, Un) = Ugcd(m, n} In

this chapter he has a statement and

proof of the Euclidean Algorithm. The

new edition extends the divisibility

properties of Un far beyond this. As the

chapter progresses, Vorobiev proves

and uses Fermat's Little Theorem and

The Fundamental Theorem of Arith­

metic; he defines and discusses The

Euler phi-function and congruence

modulo m. He even gives an example

of the Quadratic Reciprocity Law. This

chapter, with parts of chapters 1 and 3,

could be used as an introductory

course in number theory (with an em­

phasis on Fibonacci numbers).

Chapter 3 is called "Fibonacci Num­

bers and Continued Fractions." Voro­

biev proves some basic and well­

known properties of continued fractions

and their convergents. Naturally, he is

especially interested in representa­

tions of numbers like Un+ 11un. He dis­

cusses infinite continued fractions, and

he proves some of the standard in­

equalities for convergents. In the new

edition, he goes much further and

proves some nonstandard inequalities.

He proves a well-known inequality due

to Legendre, and then he proves lesser

known theorems of Vahlen, Borel, and

Hurwitz. He also discusses "equivalent

numbers" in some detail. Unfortu­

nately, the new material does not have

much to do with Fibonacci numbers.

Chapter 4 is "Fibonacci Numbers

and Geometry." Vorobiev defines the

"golden section," and he gives examples

in geometry. The pentagon and the

golden section rectangles are discussed,

and the familiar "proof' that 64 = 65 (by

cutting up a certain square of side Uzn and reassembling it) is given. The new

edition continues with more examples

of patterns that occur in nature, and it

discusses at length a game using di­

rected graphs and Fibonacci numbers.

Chapter 5 is a completely new chap­

ter titled "Fibonacci Numbers and

Search Theory." Evidently Vorobiev felt

this material should be added because,

in his opinion, number theory lost its

paramount position in mathematical re­

search and "the interest in optimization

problems gained a sudden weight." The

chapter is long and detailed, and the

use of the Fibonacci numbers seems

minimal. In the reviewer's opinion, this

material is the least appropriate of the

new additions, but that might be a mat­

ter of personal taste.

In general, this book provides an ex­

cellent introduction to the Fibonacci

numbers, and it also covers some of the

other basic topics in elementary num­

ber theory. On the whole, it is suitable

for a good high school student with a

minimal background in mathematics.

The changes from the old edition, with

the possible exceptions of chapter 5

and the game of chapter 4, strengthen

the book One thing that is missing is

a bibliography; at the very least, there

should be a list of articles and books

for recommended reading.

Mathematics Department

Wake Forest University

Box 7388 Reynolda Station

Winston-Salem, NC 271 09

USA

e-mail: howard@wfu.edu

New Visual Perspectives on Fibonacci Numbers by Krassimer Atanassov, Vassia

Atanassova, Anthony Shannon,

John Turner

WORLD SCIENTIFIC, SINGAPORE, 2002 332 pp. paperback U.S $36 ISBN 981 -238-734-1

REVIEWED BY FREDRIC T. HOWARD

Back in 1202, Leonardo of Pisa, bet­

ter known as "Fibonacci," intro­

duced the sequence of numbers 1, 1, 2, 3, 5, 8, 13, . . . as the solution to a prob­

lem involving reproducing rabbits.

Leonardo could never have predicted

that eight hundred years later his num­

bers would still be of intense interest

VOLUME 26, NUMBER 1 , 2004 65

to people all over the world, and he

could never have foreseen the amazing

number of books and articles that have

been published about the numbers. As

A. F. Horadam says in the introduction

to the book under review, "It has been

observed that three things in life are

certain: death, taxes and Fibonacci

numbers."

It is refreshing to see a new book

that is totally different from all the

other Fibonacci publications. This

book certainly contains nonstandard

and highly original material that is not

found anywhere else. The book has

weaknesses, but its great strength is its

originality and its suitability for stu­

dent research projects.

The book is organized into two

parts: part A (Number Theoretic Per­

spectives) and part B (Geometric Per­

spectives). Part A, written mainly by

K. Atanassov, V. Atanassova, and

Tony Shannon, constitutes about one

fourth of the book. Part B is primar­

ily the work of John Turner. All of

these authors are well known and

highly respected in the Fibonacci

community.

In part A, the authors discuss "2-Fi­

bonacci Sequences," which generalize

the usual sequence. Suppose {an} and

(bn} are sequences such that an+z = an+ 1 + bn and bn+2 = bn+ 1 + an. The

authors work out properties and for­

mulas for these numbers, and they con­

sider several variations, like bn+Z = an+ 1 + bn and an+2 = bn+ 1 + an. They

extend this idea by using three se­

quences and by using sequences with

four-term recurrences. They also look

at the recurrence an+2 = an+ 1bn. All of

these new sequences are interesting,

and they would make good research

topics for students. The proofs are

pretty routine and computational, with

a heavy use of induction. In this part of

the book there are no new visual per­

spectives, so the title does not really

fit. The second half of part A is con­

cerned with the number trees of John

Turner. The basic idea is a method for

constructing a sequence ( Tnl of tree

graphs such that Tn has Fn leaf nodes,

where Fn is the Fibonacci number.

Some generalizations and extensions

66 THE MATHEMATICAL INTELLIGENCER

of this idea are worked out. Here again,

as the reviewer can attest from his own

experience, the ideas are suitable for

student projects.

In part B, the first section is con­

cerned with "Fibonacci vector geome­

try." The basic idea is the Fibonacci

vector <Fn- 1, Fn, Fn+ 1) and its gener­

alizations. Among other things, the au­

thor works out some elementary geo­

metric results and proves some vector

identities. The material in this section

is highly original, and it contains many

novel definitions; for example, the au­

thor studies the "honeycomb plane"

x + y - z = 0, which contains all of

the Fibonacci vectors. He examines in

some detail the locations and various

configurations of the integer-points in

the plane.

The last half of part B is a presen­

tation of "Goldpoint Geometry," which

is defined as the "study of geometric

figures into which golden-mean points

(called goldpoints) have been con­

structed or introduced." The author

studies fractals in goldpoint geometry,

triangles and squares marked with

goldpoints, plane tessellations with

goldpoint squares, and games (such as

"goldpoint chess") with goldpoint

tiles. In this section there is a host of

new definitions which are probably

not found anywhere else in the litera­

ture.

As mentioned earlier, the strength

of this book is its originality and its

suitability as a basis for student re­

search. It is a refreshing and stimulat­

ing book, but it could benefit from

tighter editing. Here are some exam­

ples, most of which are admittedly

picky. In the first part of the book, the

period is used as the multiplication

symbol, and this is irritating. For ex­

ample, the number 3k + 2 is written

3.k + 2 (see page 10). On page 39 the

notationfn is used, when it should ob­

viously be Fn. The notation fn is used

later (page 151) in another way. In

chapter 2, for clarity the notation Xnm should always be written Xn,m· The no­

tation is inconsistent here. The same

sort of problem occurs on page 72,

where Tijk should always be written

TiJ,k, as it is in some places. On page

63 it is not clear why the notation gi. is

used instead of gi (without the period).

On pages 121-122, it is not clear if

Li(QUV) means the area of QUV or (as

on page 122) just the triangle QUV.

Perhaps some of the tables, like the

one on pages 261-263, are overdone

and not that useful. In a few places, the

exposition could be made clearer and

less idiosyncratic. For example, in the

definition of goldpoint on page 184, the

notation lAP : BPI is used without ex­

planation. As another example, the un­

defined term "pink tile" is used on page

246.

On the whole, however, this is an in­

teresting and unique book. As Profes­

sor Horadam says in the introduction,

"Indeed, there is something to be

gleaned from this book by most read­

ers." The reviewer agrees.

Mathematics Department

Wake Forest University

Box 7388 Reynolda Station

Winston-Salem, NC 27709

e-mail: howard@wfu.edu

Kepler's Conjecture: How Some of the Greatest M inds in History Helped Solve One of the Oldest Math Problems in the World by George G. Szpiro

HOBOKEN, NEW JERSEY: JOHN WILEY & SONS, INC. 2003

296 pp. US $24.95 1SBN 0-471 -0860 1 -0

REVIEWED BY KARL SIGMUND

If you pour unit spheres randomly

into a large container, you will fill

only some 55 to 60 percent of the

space. If you shake the box while you

are filling it, you will get a denser pack­

ing-something like 64 percent. What

is the densest packing possible? In a lit­

tle booklet which he published in 1611 ,

Johannes Kepler claimed that the

hexagonal close packing did the trick:

pack one horizontal layer so densely

that each sphere is surrounded by six

spheres, then add the next layer by

placing spheres into the dimples

formed by the first layer, etc. The den­

sity is now 74.05 percent. And so Kep­

ler's conjecture was born.

Kepler did not fail to point out that

street vendors stocked their apple

carts in exactly that way. (He must

have relished this kind of "market re­

search": his later booklet on volumes

started out with observing how the

content of wine-casks is gauged by in­

troducing a measuring rod).

In a remarkably candid piece of in­

trospection, the young Kepler once

wrote that his appearance, his ap­

petites, his habits, and his character

traits displayed "in every way a dog­

like nature. " We may add that Kepler's

conjecture still has a vicious bite. Who­

ever approaches it does so at a con­

siderable risk

George Szpiro, the author of "Kep­

ler's Conjecture," will surely agree.

Right after Thomas Hales announced,

in August 1998, that the problem was

at long last solved, and that the dens­

est way of packing unit spheres is the

obvious way, Szpiro sat down to do, for

Kepler's conjecture, what Singh had

done for Fermat's, and he wrote a

splendid book full of insights, anec­

dotes, and context to celebrate Hales's

achievement. Now, five years later, the

book has appeared but the proof still

hasn't. It is true that the paper by Hales

and Ferguson will be published even­

tually in the Annals of Mathematics, but with an introductory remark by the

editors, a disclaimer as it were, stating

that they had been unable to verify the

correctness of the 250-page manu­

script with absolute certainty. The

proof is so huge, and based to such an

extent on massive use of computers,

that the platoon of mathematicians

charged with the task of checking it ran

out of steam. Robert MacPherson, the

Annals editor in charge of the project,

stated that "the referees put a level of

energy into this that is, in my experi­

ence, unprecedented." But they ended

up being only "99 percent certain" that

the proof was correct.

Hales, too, was exhausted. Rather

than take up MacPherson's suggestion

to re-write the manuscript, he started

on a new project, the Formal Proof of

Kepler, FPK (felicitously dubbed Pro­

ject Flyspeck). After computer-based

theorem-proving, this is the next great

leap forward: computer-based proof­

checking. Pushed to the limit, this

would seem to entail a self-referential

loop. Maybe the purists who insist that

a proof is a proof if they can under­

stand it are right after all. On the other

hand, computer-based refereeing is

such a promising concept, for review­

ers, editors, and authors alike, that it

seems unthinkable that the community

will not succumb to the temptation.

It appears that Flyspeck will require

twenty man-years to verify every single

step of Hales's proof. If all goes well,

we then can be 100 percent certain.

You shouldn't wait twenty years before

reading about the tormented history of

Kepler's conjecture, though. In fact,

Szpiro's book, by appearing so patently

at the wrong moment, gains an extra

quality. It becomes what some of my

colleagues from the humanities call a

"meta-text." The whole book, which is

mostly about the treacherous nature of

the sphere-packing problem, has itself

become yet another victim of that

treachery.

This does not speak at all against

this well-crafted piece of popular sci­

ence writing. The topic leaves, of

course, nothing to be desired: many as­

pects are based on simple geometric

arguments which are easy to illustrate.

Crystals and wallpapers, Euler's theo­

rem and Voronoi's cells, packings and

coverings, nets and knots, all have an

immediate appeal to the brain and the

senses alike.

More importantly, the four hundred

years of persistent intellectual struggle

with the elusive conjecture offer a

wonderful mine of historical detail,

starting with Sir Walter Raleigh and his

concern for storing cannon-balls, and

including Buckminster Fuller with his

re-invention of the geodesic dome. The

"greatest minds in history" include, of

course, the usual suspects: Kepler,

Newton, Lagrange, Gauss, and Hilbert.

But Szpiro avoids the lure of touring

the summits only, and offers lively

vignettes of many figures less well­

endowed with biographers. Axel Thue's

muddled attempts at solving the circle-

packing problem, the polite collabora­

tion of Schutte and van der Waerden

on the "kissing number" (how many

unit spheres can touch a unit sphere),

and the almost hundred-year-long race

for an upper bound in sphere-packing

densities are carefully researched and

retold, as is the valiant struggle of Las­

zlo Fejes-T6th, who predicted in the six­

ties that computers would be needed to

crack the conjecture. George Szpiro, a

mathematician who became a reporter

(he currently works as the Israel cor­

respondent of the prestigious Neue Zurcher Zeitung), displays his inves­

tigative talents for digging up unlikely

stories, and a professional eye for the

telling detail. The review which Gauss

wrote about a book by Seeber (proving

in a few lines a lot more than Seeber

had achieved in two hundred pages) is

described with gusto and wit, as are the

recent debates surrounding Hsiang's

claims to having solved the problem.

A substantial part of Szpiro's book

is devoted to the diffidence with which

many mathematicians approach com­

puter-based proofs. Most mathemati­

cians agree that they are ugly, should

only be used as a last resort, and ought

to be replaced, as soon as possible, by

some of these short and elegant

"proofs from the book" in which God,

according to Erdos, keeps all true

mathematical insights. At the time

Szpiro wrote his text, the proof by

Hales and Ferguson was seen as such

a necessary evil (and shared the same

social status, among working mathe­

maticians) as the classification of finite

groups: no single human brain had ever

understood it in its entirety, but the

community seemed to have resigned it­

self to accept it. But now, this appears

to be no longer the case. Hence, the

story of Kepler's conjecture will have

to go on for a few more chapters. It is

to be hoped that it will eventually be

told with the same elan as in Szpiro's

book

Institute for Mathematics

University of Vienna

Strudlhofgasse 4

1 090 Vienna

Austria

e-mail: karl .sigmund@univie.ac.at

VOLUME 26, NUMBER 1, 2004 67

George Green, Mathematician and Physicist 1 793- 1 841 : The Background to H is Life and Work D. M. Cannell

SIAM, 2001 $75.00. ISBN 0-89871 -463-X XXXIX + 3 1 6 pp.

REVIEWED BY STEVEN G. KRANTZ

Every calculus student learns of Green's theorem as perhaps the

most benign multidimensional version of the fundamental theorem of calcu­lus. Coupled with its delightful inter­pretations in terms of fluid flow and electrostatics, this result can be con­strued as one of the capstones of a freshman education in mathematics. But most calculus books contain al­most no information about the life of George Green (1793-1841) himself, and most mathematicians have little knowledge of the man. More's the pity, for George Green was one of the more fascinating and important characters of early British science.

George Green was born in Notting­ham. His father, an uneducated man, was a successful baker who set up his own mill in nearby Sneinton. He built a large family home on a substantial property near the mill. As was the cus­tom of the time, George went to work for his father at an early age. Recog­nizing his son's talents, the elder Green sent his son to Robert Goodacre's academy at the age of 8. After four terms, young George had outstripped his teachers. He left the academy and returned to work at the mill.

George faithfully worked at his fa­ther's mill until 1829, when his father died. Then George was able to divest himself of the mill and devote himself to mathematics. In spite of his commit­ment to his father's business, George had been able to produce his funda­mental (and first) paper, "An Essay on the Application of Mathematical Analy­sis to the Theories of Electricity and Magnetism." It was published by private

68 THE MATHEMATICAL INTELLIGENCER

subscription in 1828, and only about 50 people saw it. The response was polite indifference, and poor George returned despondently to milling. By good for­tune, Sir Edward Bromhead (a member of the local intelligensia) became aware of Green's work in 1830, and he en­couraged him to take up mathematics again.

With Bromhead's support, Green enrolled at Caius College, Cambridge in 1833. He was already 40 years old. He achieved 4th Wrangler on the dreaded Tripos in 1837 and was elected to the status of College Fellow in 1839. Unfortunately, ill health forced Green to leave his position after just four terms. He died in 1841.

George Green is considered by many to have been the father of British mathematical physics. He published just ten papers in the period 1828-1839. Of these, his first (referenced above) is thought to have been the most impor­tant, and the most influential. It con­tains or introduces

• the idea of potential function; • what we now call Green's theorem; • the idea of reciprocity; • the idea of singular value; • the idea of the Green's function.

Later papers include ( i) the first rendi­tion of what we now call the Dirichlet principle, ( ii) an important asymptotic method for solving certain partial dif­ferential equations in divergence form, and (iii) a preliminary version of the idea of tensors (indeed a particular ten­sor is today named after George Green).

Mary Cannell's book is a remarkable and profound effort. Little is known of Green's early years, and (as a senior sci­entist) he left behind little correspon­dence, no diary, and no working papers. An especially arduous effort was re­quired to piece together the story of his life. It should be stressed that this book is not a commentary on Green's scien­tific work (although some of the ap­pendices do treat this aspect of Green's life). In fact, the avowed purpose of the book is to treat the personal aspects.

Cannell tells us that Green had seven children by Jane Smith, the daughter of his father's mill manager; yet they never married. It appears that

Smith remained in the background of Green's life-although it should be noted that all the children ultimately adopted the name Green. Cannell mar­vels over the fact that Green mastered many of the techniques of French analysis at a time when these ideas were virtually unknown in England. She goes to great lengths to trace his personal and intellectual heritage.

And she certainly mourns Green's lack of personal and scientific recog­nition during his lifetime. In fact it was Lord Kelvin who rediscovered Green's "Essay" (his first paper) in 1845-four years after the man's death. He ulti­mately arranged for the paper to be properly published in Grelle's Journal in the 1850s. Finally, on the 200th an­niversary of George Green's birth, a plaque in his honor was placed in West­minster Abbey-in front of the statue of Isaac Newton.

One can only speculate, and Mary Cannell does so at length, about what sort of career George Green might have had if he had had benefit of a proper education at the appropriate time in his life, and if he had lived in a more nur­turing environment (such as Paris), rather than the stultifying wreckage that was British science in the early nineteenth century. It seems inar­guable that George Green had a pro­found influence on such leaders of nineteenth-century British mathemati­cal physics as Maxwell, Stokes, and Rayleigh. Cannell makes a point of the great effect that Green's ideas had on twentieth-century mathematical physics. For example, the Nobel-Prize­winning work of Julian Schwinger on quantum electrodynamics makes con­siderable use of Green's function. On the occasion of the 200th anniversary of Green's birth, such luminaries as Freeman Dyson and Schwinger gath­ered to help pay tribute to him.

We owe a debt to D. Mary Cannell for penning this, the only full-length biogra­phy of George Green in existence. It is a shame that this new edition was pub­lished posthumously, but a tribute to her scholarship and dedication to an impor­tant cause. One of the more interesting and daring points that Cannell makes in her book is that a more formal educa-

Left: The windmill where George Green ground corn for a living. Be­

low: Green's grave, St. Stephen's Courtyard, Sneinton, Nottingham.

(Photograph by Jan Crosbia.) Both figures reproduced from The Math­ematical tntelligencer, vol. 1 1 (1989), no. 4, pp. 39, 40.

VOLUME 26, NUMBER 1 , 2004 69

tion�specially in the England of the

early nineteenth century-might have

actually suppressed George Green's

creativity. It does seem that Green was

the product of his environment-but in

a surprising and delightful way.

Department of Mathematics

Washington University in St. Louis

St. Louis, MO 631 30

USA

e-mail: sk@math.wustl .edu

Mathematical Apocrypha by Steven G. Krantz

MATHEMATICAL ASSOCIATION OF AMERICA, 2002, 205 pp $32.95 US, ISBN 0·88385-539-9

REVIEWED BY MARION D. COHEN

In Woody Allen's Midsummer Night's Sex Comedy, a stodgy old philosophy

prof, on the eve of his planned wed­

ding to especially-lovely-in-this-film

Mia Farrow, waxes quite enthusiastic

about introducing his new wife to his

faculty associates. "I can't WAIT to in­

troduce you to Prof. Eddy and his wife.

They're such an entertaining and amus­

ing couple. He specializes in Dr. John­

son, she teaches Boswell . . . . " Mia Far­

row's demure non-expression betrays

that her enthusiasm does not match

his. As the evening progresses, we see

the prof at the piano, blissfully singing

Schubert Lieder, oblivious to Mia Far­

row's boredom as she fans herself and

yawns, contemplating the prospect of

the rest of her life.

The academic life is possibly an ac­

quired taste. At any rate, it's rather spe­

cialized, an arena which, among other

things, has its particularities, its ru­

mors, its "in-jokes," "in-wisdom"-in

other words, apocrypha. And often

what academic people such as Woody

Allen's professor find "entertaining and

amusing" wouldn't be given the time of

day by people who are perhaps not so

hard up.

The anecdotes in Mathematical Apocrypha are not all meant to be

funny per se, but humor seems, to var­

ious degrees, to be their underlying

feeling and raison d'etre, if not theme.

70 THE MATHEMATICAL INTELLIGENCER

Indeed, the very contrast between the

mathematical life and the "ordinary"

life provides a solid basis for humor.

Some of these stories are funny-ha-ha,

but others are, by turns, funny-familiar,

funny-wise, funny-sad, and funny-dark

(Apocryphes noirs?). Working backwards, we have tales

of the "notorious anti-Semite" Ludwig

Bieberbach and of murderous mathe­

maticians, such as the recently infa­

mous "Unabomber" and the perhaps

less known Walter Petryshyn, who

hammered his wife to death.

As an example of funny-sad (pathos),

we have the aged and senile Heinz Hopf

walking up to someone in the math de­

partment hallways and remarking, "I

hear that the great Professor Hopf is

visiting us next week"

I wish there were more wise tales,

and perhaps the reason there aren't is

that smart people are often not wise. At

any rate, we do have Kakutani (Or, asks

the author, was it P6lya?) explaining

Brownian motion in three versus two

dimensions: "A drunken man will usu­

ally find his way home. A drunken bird

has no hope." And Hardy and Little­

wood's four "axioms" concerning their

collaboration shows considerable hu­

morous wisdom. Here are the first two:

"Axiom 1: . . . it was completely indif­

ferent whether what he wrote was right

or wrong . . . . Axiom 2: . . . When one

received a letter from the other, he was

under no obligation whatsoever to read

it, let alone to answer it. . . . " Here's my favorite funny-familiar

anecdote (otherwise known as an "in­

joke"): "One year at MIT they decided

that calculus would henceforth be

taught non-rigorously. The dreaded ep­

silons and deltas would no longer be

part of the course, and that was de­

partment policy . . . some professors

took umbrage . . . one professor walked

into his calculus class on the first day

and announced, 'In this class we will

make use of certain constants called

sigma and tau. We defme a limit as fol­

lows: For every tau greater than zero

there exists a sigma greater than zero

such that . . . '

As for funny ha-ha, the book offers

some downright slapstick Professor X,

in an effort to explain infinity as "like

a long line that never stops," walked

"through and out the window" and . . .

wound up "two floors below, spread­

eagled in the bushes (and unharmed)."

Also deserving mention are the

funny-interesting, such as the descrip­

tions of R. L. Moore's unusual way of

teaching. There are even the funny­

poignant, like the clash of egos be­

tween Atle Selberg and Paul Erdos.

"Paul, " began a third party, Kaplansky,

"you always say that mathematics is

part of the public trust. Nobody owns

the theorems. They are out there for all

to learn and to develop. So why do you

continue this feud with Selberg?" "Ah," replied Erdos, "but this is the prime

number theorem."

And then there are those which, like

the colleagues of the stodgy professor

in the Woody Allen movie, are perhaps

simply NOT funny, such as the mes­

sage wired by Dirichlet to his father-in­

law when his first child was born: "2 +

1 = 3". (Or perhaps this is funny­

subtle, or funny-ya-hadda-be-there.)

Are there apocrypha that specifi­

cally concern math itself? Happily, the

answer is yes-for example, the story

about von Neumann. "The lecturer ex­

hibited a slide with many pieces of ex­

perimental data and, although they

were badly scattered, he argued that

most of them lay on a curve. It is said

that von Neumann murmured, 'At least

they lie on a plane.' " Another example:

"Moore began a lecture by saying, 'Let

a be a point and b be a point.' Lefschetz

shouted, 'But why don't you just say,

"Let a and b be points"?' Moore replied,

'Because a may equal b.' " All told, this book gives a good pic­

ture of "the life" and, as the author

says in the Preface, "the people who

live it"-although the book is, un­

avoidably, more accessible to those

who themselves "live it." "I have stren­

uously," he continues, "avoided the

telling of stories that are mean-spirited

or critical or that depict people in a

bad light. I want these stories to make

people happy, not sour." However, in

the next paragraph he admits that they

"are not always flattering," and much

of his book bears him out. And while

the consensus of opinion might be that

it portrays mathematicians as "hu­

man," perhaps a little psychotherapy

would have made some of these same

mathematicians MORE human, and

less INhuman. Indeed, even in the

chapters on "Great Ideas" and "Great

People" we find some "not always flat­

tering" examples. To wit: "Wiener

woke up in the middle of a lecture. He

peered slowly at each of the black­

boards, evidently saw nothing of in­

terest, burst into a fit of coughing,

staggered from the room, and was

seen no more. The coughing ceased as

soon as he left the lecture room."

Krantz divides his book into six

chapters: "Great Foolishness", "Great

Affrontery", "Great Ideas", "Great Fail­

ures", "Great Pranks", and "Great Peo­

ple". Sometimes it didn't seem clear

whether a particular anecdote had

been placed in the correct chapter. For

example, certainly the one about the

mathematician who hammered his

wife to death has no place in "Great

Ideas".

Other ways of dividing up the sto­

ries might come to mind: Great Self­

Absorption, Great Psychos, Great Be­

fuddlement, Great Egos, Great Repres­

sion, Great Manipulations, Great Ex­

pectations, Great Power-trips, and Great

Bad-Jokes, perhaps all coming under the

heading Great Personality Defects, or

even Dysfunctional Analysis!

What does this book say about the

psychology of academic life? How

much do the stories betray a longing

for the ordinary (perhaps non-aca­

demic) life? Amidst the pages of this

book Bertrand Russell muses, "I've

made an odd discovery. Every time I

talk to a savant I feel quite sure that

happiness is no longer a possibility. Yet

when I talk with my gardener, I'm con­

vinced of the opposite."

What do the anecdotes have to say

about mathematicians and their brand

of yearnings, vulnerabilities, strengths?

Might academics have something to

learn about their lives and how to live

them, in particular how to communi­

cate? What price do some of these

mathematicians and their psyches pay

for the kind of super-human-ness which

they possess?

Department of Physics, Mathematics,

Statistics, and Computer Science

University of the Sciences in Philadelphia

Philadelphia, PA 1 91 04

USA

e-mail: mathwoman1 99436@aol.com

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VOLUME 26, NUMBER 1 , 2004 71

,.-jfi111.19.h.l§i Robin Wilson I

The Phi lamath' s Alphabet-(

Caratheodory: Constantin Cara­

theodory ( 1873-1950) is the most

significant Greek mathematician of

recent times; but, encouraged by

Minkowski, Klein, and Hilbert, he spent

most of his life in Germany. He made

significant contributions to the calcu­

lus of variations and its applications to

geometrical optics, the theory of func­

tions (especially conformal represen­

tation), and measure theory. In applied

mathematics, he wrote on thermody­

namics and relativity theory.

Caratheodory

Cauchy

Chebyshev

Cauchy: Augustin-Louis Cauchy (1789-

1857) was the most important analyst

of the early nineteenth century. In the

1820s he transformed the whole area

of real analysis, providing a rigorous

treatment of the c:;alculus by formaliz­

ing the concepts of limit, continuity,

derivative, and integral. In addition, he

almost single-handedly developed the

subject of complex analysis, and many

results in this area are named after him:

'Cauchy's integral formula' appears on

the stamp.

Chebyshev: Pafnuty Chebyshev (1821-

1894) is remembered mainly for his work on orthogonal functions ('Cheby­

shev polynomials') and probability

('Chebyshev's inequality') and for an im­

portant contribution to the proof of the

1 2 � 8

nederland

Please send all submissions to

the Stamp Corner Editor,

Computer graphics

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6AA, England

e-mail: r.j.wilson@open.ac.uk

72 THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

Condorcet

prime number theorem. He also worked

on quadratic forms and integrals and

studied theoretical mechanics and link­

ages. He taught at the University of St

Petersburg, and founded the St Peters­

burg School of Mathematics.

Computer graphics: Computer-aided

design has developed rapidly in recent

decades, and in 1970 the Netherlands

produced the first set of computer-gen­

erated stamp designs. The stamp below

shows an isometric projection in which

the circles at the centres of the faces

gradually expand and become trans­

formed into squares.

Condorcet: The Marquis de Condorcet

(1743-1 794) clarified the foundations

of probability theory, and studied the

solutions of ordinary and partial dif­

ferential equations. His most signifi­

cant contributions, however, were to

'social mathematics'-in particular, to

the analysis of models of voting pat­

terns. After the French Revolution he

was arrested while fleeing for his life,

and died in captivity.

Counting on the fingers: From ear­

liest times, people have needed to be

able to count and measure the objects

around them. Early methods of count­

ing included forming stones into piles,

cutting notches in sticks, and counting

on the fingers. It is undoubtedly due to such finger counting that our familiar

decimal system emerged.

Counting on the fingers