No title

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 = mintE d C + til have the same basic properties as in the case dim � < oo.

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

Bounding above l AO 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), �c�

Proof

::5

2'7TIC]o,

Area(W) ::5

(17)

'7TIC]5.

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 Perimeter(W)

2 = 7T (A + A")(t)dt 10

f7T A(t)dt::::; f7T IA(t)ldt

(18)

J27T (A2- A'2)(t)dt::::; _!_ J2A 2(t)dt

(19)

=

0

0

and Area(W) =

_!_ 2

0

2

0

.

�c�

we conclude that (20) holds for C. Now use again the in­ • variance under C >---7 C + ti. Concluding Remarks

Although the bounds ofTheorem 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 Cbe 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 (1 7) 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) =

{ + : llull

THEOREM

Perimeter(W)

Area(W) = sup Area(W),

=

llvll = 1, = 0}. (22)

This implies Diameter(W(C)) = max llull=llvJI= l,=O

I + 1. (23)

This in tum yields the bounds Diameter(W(C))::::; max llull=llvll= l,=O

21 I ::::; 2ICI. (24)

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

13

REFERENCES

AUTHOR

[ 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

&UOENE OUTKIN

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 - 1 80. [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 . [ 1 4] 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.

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 spe­ cial cases [7, 8]. There is no immediate relationship be­ tween 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 re­ searchers 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 nu­ merical range brought me to the conclusion:

6 (1 951 ), 1 93-228.

[ 1 7] B. Mirman, V. Borovikov, L. Ladyzhensky, and R. Vinograd, Nu­ merical ranges, Poncelet curves, invaria nt measures, Lin. Alg. Appl. 329 (200 1 ) , 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.

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

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

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

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

Hirzel, Leipzig, 1 931 . sented in [27].

14

THE MATHEMATICAL INTELLIGENCER

107 (2000), 528-540.

Theories of Vision Enul

·

•I

holz

I.

hildren

Ill.

IV.

b·

II.

'<

nd

v.

aking

Department of Philosophy Pennsylvania State University University Park, PA 1 6802 From Grosholz, Emily; Shorts and Headlands. Copyright © 1 988 b y Princeton

USA

University Press. Reprinted by permission of Princeton University Press.

e-mail: [email protected]

16

THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

1&ffij.i§j:@hl£113'fil§4fiijlj:i§:id

Cutting a Polygon into Triangles of Equal Areas

M i chael Kleber a n d Ravi Vaki l , Ed itors

his tale, like so many in mathe­

Tmatics, begins with a simple ques­

elegant, suprising, or appealing that

tion, answers it, and ends with ques­ tions 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 re­ search 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 tri­ angles 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

one has an urge to pass them on.

equidissection of a square is even.

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

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: [email protected]

Richman's colleague, John Thomas, became interested in the problem and proved that there is no odd equidis­ section of a unit square in standard po­ sition in the xy-plane if the coordinates of the vertices of the triangles are ra­ tional 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 pa­ per should be published. It appeared in 1968 [12]. Paul Monsky in 1970 [4], building on Thomas's proof, showed that the an­ swer 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 al­ gebra. 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 ap­ plied by others to give a short proof of Brouwer's fixed-point theorem. He stated it for simplices in all dimen­ sions, but I will present it just for poly­ gons in the xy-plane. Consider a polygon cut into trian­ gles. For simplicity, assume that two triangles that touch each other inter­ sect 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 tri­ angles. In Figure 1 the respective num­ bers 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 implica­ tion is what Spemer used, and so will we. The other tool is a 2-adic valuation,
© 2004 SPRINGER-VERLAG NEW YORK, VOLUME 26, NUMBER 1 , 2004

17

all the reals in uncountably many ways, each having the following properties: (ad) =
18

THE MATHEMATICAL INTELLIGENCER

least one of its m triangles is complete. Since the area of the triangle is Aim, it follows that
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 com­ plete sections along the bottom edge of the square, which is complete. The other three edges have no complete sec­ tions. Thus the total number of com­ plete 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 equidis­ sections remained until 1979, when David Mead [3] obtained a generaliza­ tion from a square to a cube in any di­ mension. He proved that when an n­

lplijiJ;IfM

Mptdll;l¥4

C (O, l )

A

(0, 0)

b

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 val­ uations 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 pen­ tagon?" She found the answer and went on to prove that in any m -equidissec­

tion 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 ap­ peared in 1989 [1]. In a sense it was an­ other generalization of the theorem about equidissections of a square. A year later she and I published [2] the results of an investigation of equidissec­ tions of trapezoids and other quadri­ laterals. Among these was yet another generalization of a square, namely quadri­ laterals whose four vertices are (O,Q), (1,0), (a, a), and (0,1), where a is any pos­ itive number, illustrated in Figure 3.

(a, a)

(0 , 1 )

B

( l , 0)

(0 , 0 )

(1 , 0 )

The area of such a quadrilateral is a. If
( 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 <
More generally, if a = b/(2c), where b and c are odd integers, the corre­ sponding quadrilateral has an odd equidissection. However, if
parallelogram has no odd equidissec­ tion. This follows immediately from the result for a square, for any parallelogram is the image of a square by a linear map­ ping. Since a linear mapping magnifies all areas by a constant, it takes an equidissection into an equidissection. A parallelogram being centrally sym­ metric suggests that perhaps any cen­ trally symmetric polygon has no odd equidissection. Kasimatis's theorem about regular n-gons, when n is even, gave me enough extra evidence that I in­ vestigated centrally symmetric poly­ gons, trying to produce a counter-ex­ ample. 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 gen­ eralize the square. To construct this type of polygon, I start with the unit square in Figure 2 and then distort its bound­ ary, changing opposite edges in the same way. The resulting polygon still tiles the plane by translates using all integer vee-

Mpt§ll;iiM

�• ( 1 , 1 )

(0 , 1 ) •

( 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 sim­ ple families made this way, such as poly­ gons formed by adding one dent, as in Figure 7, that my suspicion was valid.

�• ( 1 , 0)

( 0, 0) •

·------ · ( 1 , 0) Some years later a surprising break­ through occurred, which I described in a paper published in 1999 [10]. It con­ cerns a unit square in the xy-plane whose comers have integer coordi­ nates, 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 num­ ber of complete edges on the bound­ ary. Moreover, since the area is an in­ teger, it follows that the number of triangles must be even.

VOLUME 26, NUMBER 1 , 2004

19

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

ii'riil;l¥+

3 + 3 case first. Denoting parallel vec­ tors by the same letter, the only possi­

OD Centrally symmetric

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.

Distorted square

Without loss of generality, we can as­

in general, but left it to someone else

sume its edges are parallel to the axes.

to treat that case. Iwan Praton in 2002

The 2 + 2 + 2 case leads to two es­

[6] disposed of the even case. His proof

sentially different schemas, as shown

showed that if the number of squares

in Figure l l and later in Figure 13.

is of the form 2rb, where b is odd, then there is a translate of the image of the (x,y) to (x/2u, y/2v), where

u + v ::5 r,

to which a stronger version of Sperner's

called such a polygon special and con­

any

polygon

com­

I showed that the conjecture is true

no odd equidissection. There is an­

when the special polygon has only a

other, more suggestive way to state this

few edges. The smallest possible num­

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,

ber of edges is four, and the polygon is

first translate the polygon so that one

be checked. There are three types of

of its vertices is at the origin. Then

special polygons with six sides, con­

magnify this image by a mapping that

structed as follows.

(qx, qy), where q is an in­

. .

r \..

I

... ..

.. .. .. .. .. ..

, ...

,/ q

p

l!'dli;Jifl

p

The first step is to determine the

class and at most three, for if there were four, two would be forced to be

inal polygon has no odd equidissection.

adjacent. The partitions of six meeting these conditions are 6

=

3 + 3 and 6

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

have rational lengths is removed? I

how the equivalence classes could be

conjectured that any polygon whose

arranged on the boundary. Take the

ljMiiijiitl

gons that I either knew or suspected

,'' P,

have no odd equidissections: centrally

q

� "' "' ...

.. ..

.. ... ...

..

..

..

. .

edges

�

q

already settled, and there was ample evidence for the remaining two cases.

'"..

The other possible schema is shown in Figure 13.

p

p \..

:

. .

parallel to the axes. The first case was

p

=

What if the assumption that the edges

square,

. .

sides, as shown in Figure 12.

no odd equidissection. Hence the orig­

distorted

. .

'

polygons with five sides, as may easily

class. There must be at least two in a

I then faced three classes of poly­

.

centrally symmetric polygon with six

of the lengths of the edges. The image

odd equidissection.

"'\ r

conjecture is true. There are no special

consists of congruent squares and has

edges are parallel to the axes has no

..

This schema can be realized by any

number of edges in an equivalence

The next conjecture is inevitable.

'

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

then a parallelogram, for which the

teger divisible by all the denominators

symmetric,

q/ , .

no odd equidissection.

posed of the unit squares described has

takes (x,y) to

p

jectured that each special polygon has

lemma applies. Consequently

I#Mil;li!i

Edges parallel to axes

polygon by a linear mapping that takes

. .

,

.. .. _ -

... .. .. .. .. ..

.·

. .

' q

" q .. .. ..

p

Figure 9 illustrates the three types. As I stared at polygons like those, I no­

(a )

ticed a property that they all shared. To

p

describe this property I orient the

..

p

It, too, can be realized by a special polygon, shown in Figure 14.

boundary, turning each edge into a vec­

Each case can be treated with the

tor whose direction is compatible with

aid of Sperner's Lemma, 2-adic valua­

the orientation. Then I call two vectors

p

q q

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

p

(b)

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.

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? 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

q/

'

f. · · - <

' '

;

- •• _ _ _ _ _

p

f!. · - - < .. ... .. ... . p

q��r r[__fP p

'

\' P :

' .

q\_

\' p

.·'q

' '

r:

p

'

.

·'q

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

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

The next four questions are suggested by the special polygons.

crete and Camp.

Geometry 4 (1 989),

375-381 . 2 . -- and S . Stein , Equidissections of

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

polygons,

Math.

85

(1 990),

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 6 1 -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

A ssociation

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. 11.

--

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. Thom as , 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: [email protected]

u

- l i ha I I I .

Discrete

281 -294

l lha

1

r

VOLUME 26. NUMBER 1. 2004

21

Mathematically Bent

Col i n Adam s , Editor

Rumpled Stiltskin Colin Adams 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 0 1 267 USA e-mail: [email protected]

22

nce 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

O

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."

That evening, the Chair sat her When the girl arrived that night, the Chair pointed to six pots of coffee sit­ down before twelve pots of coffee. "If you don't tum this coffee into ting on the table."If you don't turn this coffee into theorems, I will make sure theorems," he threatened, "I will the only work you get is as a recitation make you into a permanent grader for instructor, teaching fifteen problem our remedial algebra course. But, if sessions a week for large calculus lec­ you do succeed, I will give you a tures." And again, he locked her in the tenure-track position on the faculty here at Chicago. " Then he turned and lounge. The girl fell sobbing on the couch. left the lounge, locking the door be­ But she said to herself, if the little man hind him. The girl fell on the couch sobbing. can do it, why can't I? With that she went over and poured herself a cup of It was too much to hope the smelly the steaming brown liquid. She took man would be back to help her once a sip and spat it out immediately. more. And besides, she had nothing left "Ahhrgh," she said, "This tastes like it to give him. Suddenly the doorknob turned and has been sitting in the pot for the last 12 hours"-which in fact it had. in he walked. "Still trying to turn coffee into the­ But then the door swung open again, and in walked the little man.His jeans orems, are we? Haven't learned how to were tom at the knee and his teeth ap­ do it yet?" peared never to have experienced the "Oh, no," she said. "I can't do it.And friction of a toothbrush. the Chair is going to make me into a permanent grader. Oh, woe is me." "Back again, are we?" he said. "And what will you give me if I do "Yes, the Chair said I must tum these six pots of coffee into theorems it for you?" asked the hair-encumbered by morning, or he will tum me into a individual. "I don't know," said the girl. "I don't recitation instructor." "And what, pray tell, will you give have anything left to give." me, if I do it for you?" The little man grinned mischie­ The girl thought for a moment and vously. "Oh, I think you do," he said."I then pointed to her computer brief­ want you to give me your first-born theorem." case. "What do you mean?'' asked the girl. "How about my laptop?" she asked "The first theorem that you prove hopefully. She handed it to the man. "Hmmm, yourself, I want you to give it to me, to looks like a Mac Titanium Power PC claim as my own." G4, 800 MegaHertz, with one megabyte Now the topologist's daughter knew L3 and 256K L2 cache. You got a deal." that if she said no, she wouldn't ever So again, he gulped down the cof­ have the opportunity to create her own fee, and set to work Six hours later, he theorems, anyway. So there wouldn't had filled six pads of paper with the­ be anything to lose. On the other hand, orems and proofs. if she did survive all this nonsense, and "This should do it," he said. And had a career as a mathematician, what grabbing up the laptop, he disappeared was one theorem more or less? So she agreed. The little man laughed delight­ out the door. When the Chair arrived the next edly. "Oh, yes we have a bargain," he morning, he was flabbergasted by the beauty of the mathematics on the pads. laughed as he danced about the room. "This is really good stuff," he said Then he gulped down all twelve pots enthusiastically. "These are the germs of coffee, and worked through the en­ for a whole new theory. I am really im­ tire night, finishing just before day­ break. pressed." "Remember our deal," he said as he "Good," said the girl nervously. slipped out the door, leaving twelve "Now can I go?" "Yes, but you must come back pads of paper filled with mathematics tonight," said the Chair. "You have on the table. When the Chair arrived, he was more work to do."

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 ..

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,

Marjorie Senechal , E d itor

I

n 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. Wald 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 Wald 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.

historians, anthropologists, and others. American Statisticians:

Please send all submissions to the

Creating a Community

Mathematical Communities Editor,

When Wald arrived in the United States in 1938, the American community of

Marjorie Senechal, Department

I

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 Mathe­ matical 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)

of Mathematics, Smith College, Northampton, MA 0 1 063 USA

1 For some of the history of the founding of the ASA,

e-mail: senechal@minkowski. smith. edu

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

see [67] and [1 5].

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

25

and the American Economic Associa­ tion (AEA) held their charter meetings in the mid-1880s. The American Politi­ cal Science Association followed in 1903.3 The establishment of the profes­ sional 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 hold­ ing 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 mathemat­ ical aspects of statistical methods. Some of this mathematics began mak­ ing its way into the work of statisti­ cians in the ASA as early as the 1890s, but the relationship between collec­ tions of numerical data about society and the mathematical theory of proba­ bility had just begun to emerge, and several decades passed before a group

with a number of disciplines, but it seems that no single periodical or orga­ nization provided a comfortable profes­ sional home.5 The mathematical statisti­ cians felt no particularly warm welcome from either the ASA or the American Mathematical Society (AMS)-perhaps the most likely supporters of mathe­ matical statistics. On the one hand, the members of the ASA were interested in data collection and the use of numeri­ cal information, but as Carver, founder of the Annals of Mathematical Statis­ tics, put it, "most of their membership were economists, bankers and census people whose knowledge of mathe­ matics was very limited."6 Most Amer­ ican mathematicians, on the other hand, focused their research on pure mathematics, and while papers in mathematical statistics included theo­ rems and proofs, they often had a par­ ticular use of those theorems as a start­ ing 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 organi­ zation's first president, commented about the mathematical research com­ munity that "when it comes to practice accepting papers for publication it seems not much material is acceptable that is a bit tainted with possible ap­ plications to statistical data."7 Carver put out the first issue of the Annals in 1929 (initially with some fi­ nancial backing from the ASA), and

of researchers would focus their in­

mathematical statisticians formally or­

quiries on it specifically.4 In the 1920s, these researchers be­ gan to see the usefulness for research in economics, biology, and agriculture of the growing set of tools of statisti­ cal inference, but they found them­ selves struggling to find places to pub­ lish their results. Many of them were trained and employed as mathemati­ cians and their interests overlapped

ganized the IMS as an independent pro­ fessional organization in 1935. In 1938, it assumed complete financial respon­ sibility 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-

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 impor­ tance of theoretical statistics, mathe­ matical statisticians still welcomed op­ portunities to support and collaborate with the users of statistics. As the mathematical statisticians were estab­ lishing their community, several groups of applied statisticians had organized themselves into societies supporting their particular interests. The Econo­ metric Society was formed in 1930; the Psychometric Society, in 1935. By the late 1930s, biologists and medical sci­ entists 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 mathe­ matical statisticians in the United States had ties to these organizations.

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

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 Wald 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 math­ ematical 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 to­ day by its Romanian name, Cluj, was part of Hungary when Wald was born on October 31, 1902. In 1920, Transyl­ vania became part of Romania, but had a large Hungarian minority. Wald, a Jew, spoke Hungarian and "never developed any affinity for Romania," nor a knowl­ edge of its language [29, p. 361]. Edu­ cated at home because the local school required attendance on the Sabbath, Wald had passed the gymnasium exam­ ination 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 intel­ ligentsia, including the philosophical Vienna Circle and a number of over­ lapping groups of economists. 11 He led his own circle, the Mathematical Col­ loquium, 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 mathe­ matischen Kolloquiums, between 1931 and 1937, co-editing the last two volumes with Menger, Gbdel, and Franz Alt. Many of these papers com­ municated his research in pure mathe­ matics, but a few, discussed below, point to the connections among disci­ plines and communities that the Collo­ quium 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 invita­ tion, the banker and economist Karl Schlesinger presented his work on equa­ tions of economic production to the Mathematical Colloquium, and Wald 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 mathe­ matical 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, Wald 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 opportu­ nity in the academic world: "Wald, with his characteristic modesty, told me that he would be perfectly satisfied with any small private position which would en­ able him to continue his work in our Mathematical Colloquium" [27, p. 18]. Not only did Wald's work in the Collo-

quium continue, but his private posi­ tions opened up new intellectual oppor­ tunities. His relationships with Schlesinger and Morgenstern formed important threads in Wald's network of commu­ nities. These men were active in influ­ ential circles of Vienna's economists, both in and outside of the university. At the university, Schlesinger and Mor­ genstern 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 com­ munity" 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 Secre­ tary. Mises and his seminar participants formed the core of the National Eco­ nomic Association, which Mises revived in the 1920s, becoming its vice-presi­ dent, 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 pe­ riodical edited by Mayer with Morgen­ stem's assistance [5, p. 18].16 Menger presented a paper at an Eco­ nomic Association meeting on the Pe­ tersburg 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 direc­ tor of the Institute for Business Cycle Research, another Mises-promoted or­ ganization, Morgenstern employed not only Wald, but another student of

1 0Details of Wald's education and research can be found in [27, 29]. 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]. 1 4Mayer 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]. 1 6The Zeitschrift was not, however, an official publication of the National Economic Association.

VOLUME 26, NUMBER 1 , 2004

27

Menger, Franz Alt, and the economist,

ing Wald to the United States. The

in the economics department at Prince­

Gerhard Tintner, who later noted that

Commission had been organized in

ton when he heard that he had been

"at the Institute there was a much more

1932 by Alfred Cowles, the president of

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

scientific attitude to economics than elsewhere in Wien at this time. " 17

an investment counseling firm in Col­

having been dismissed by Morgen­

As Wald's contact with economists

orado Springs. He had come into con­ tact with several members

of the

stem's Nazi successor, fmally made his

way to Colorado Springs. 20

in Vienna fostered his interest in math­

Econometric Society in 1931, when he

His stay there was brief-within a

ematical economics, his work gradu­

began researching methods of fore­

few months he left for Columbia Uni­

ally became known abroad. At a 1936

casting the stock market. At their sug­

versity to work with Harold Hotelling.

meeting of the Econometric Society in

gestion, Cowles opened his research

Hotelling

Chicago, Tintner reported on some of

institute and soon after began financ­

nomics and building up a program in

metrica

mathematical statistics in Columbia's economics department since 193 ! . 2 1

Wald's results [21, p. 188]. Schlesinger and Wald himself attended a 1937

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

had

been

teaching

eco­

Both the Econometric Society and

Through the next decade, several early

France [36]. Wald's contacts in the

the Cowles Commission had ties to the

members of the American mathemati­

meeting of the Econometric Society in Econometric Society would form im­

Viennese economics and mathematics

cal

portant links for him to the statistics

communities, as well as to the Ameri­

some

statistics

community

community in the United States.

can statistics communities. Karl Menger

Hotelling, including Samuel S. Wilks and Joseph L. Doob [ 16].

training

at

received

Columbia

under

His economics research in Vienna

participated in the Society's organiza­

had touched upon issues related to sta­

tional meeting, held in Cleveland, Ohio,

tistics, and he had published a paper in

in December 1930 [40]. Gerhard Tint­

Hotelling

ner attended its meetings in the early

Carnegie Corporation, Wald worked at

the Ergebnisse on Richard von Mises's

Funded from 1938 to 1942 by a grant had

obtained

from

the

notion of a collective, a concept that

1930s and joined the Cowles Commis­

Columbia, first as a research assistant

played a role in the axiomatization of probability. l8 But his immersion in the

sion staff in 1936. Other active mem­ bers, also present at the organizational

and then teaching courses in mathe­ matical statistics and economics. 22 It

ideas of mathematical statistics would

meeting, included Harold Hotelling and

was during these first years at Colum­

come in the United States. There, for

Walter A.

bia that Wald became immersed in the

Wald, the web of communities woven

members of the IMS. Hotelling became

ideas of mathematical statistics. Char­

together for him in Vienna would con­

one of the American mathematical sta­

acteristically,

verge with the statistics community

tistics

leagues, he "worked with prodigious

that had been forming in the United

spokesmen for the discipline [ 15, 16].

Shewhart, both founding

community's most respected

according to his col­

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

States since the 1920s. The boundary

It may have been Tintner's connec­

"most of his waking moments during

that American mathematical statisti­

tions that resulted in Wald's invitation

this and the next several years . . .

cians had drawn around their disci­

to join the staff of the Cowles Com­

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

pline as well as the connections cross­

mission in 1937, which he accepted,

an assistant professor of economics in

ing that boundary, particularly to the

though not without some delay and hesitation about leaving Vienna. 19

ranks to professor of mathematical sta­

econometrists, would further shape

Wald's research,

as

well as his place in

Menger had departed

in 1937

for the

1942, and made his way through the

tistics in

1945,

finally becoming the

the scientific community.

University of Notre Dame as the polit­

chair of an independent department of

ical climate in Austria was becoming Mathematical Statistics and

increasingly unbearable. When Hitler's

mathematical statistics at Columbia in 1946. 23 Wald was a popular lecturer

World War II: Communities

troops marched into Vienna in March

from his first years of teaching at Co­

Converge

1938, Wald had not yet left. Morgen­

lumbia. Students flocked to his lec­

The Cowles Commission for Research

stem was in the United States on a lec­

tures, which were "noted for their lu­

in Economics formed the thread link-

ture tour and stayed, taking a position

cidity and mathematical rigor" [68, p.

1 7 From an interview reported in [5, p. 20]. 1 8Wald '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 9 Roy 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]. 2 1As 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 Hei24

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 Wald 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

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 dis­

cussing the theory and applications of the SPRT [61]. That same year he con­ tributed a non-technical, expository paper on the fundamental ideas and ap­ plications of the test to Journal of the ASA [60]. As the editor of the Journal explained in a footnote to the paper, Wald 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 sophistica­ tion-highlights the distinction still present in the mid-1940s between the communities of applied and mathe­ matical statisticians. Wald's ideas started a flurry of ef­ forts on the part of other researchers to explore questions raised by his dis­ coveries.26 Much of this research emerging from Wald's ideas found its way onto the pages of the Annals of Mathematical Statistics. That discus­ sion about Wald's new ideas in se­ quential 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 ad­ vancing 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 ac­ coutrement; mathematical statisticians like Wald seemed to regard it as hav­ ing the credibility to record their con­ tinuing conversations about their the­ oretical 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 Math­ ematics Panel, Warren Weaver com­

mented on the usefulness of the SPRT,

saying that the "Quartermaster Corps reported in October 1945 that at least 6,000 separate installations of sequen­ tial sampling plans had been made" [37, p. 614]. Sequential analysis provides an im­ portant example of an area of research that combined mathematical theory with statistical applications. Its prob­ lems and their solutions addressed the practical needs of manufacturers and scientists, and at the same time at­ tracted the technical and theoretical in­ terests of the mathematical statisti­ cians. Abraham Wald himself seemed to be an ideal member of the commu­ nity practicing this discipline situated between theory and application. He brought to his investigations, as his student and collaborator Jacob Wol­ fowitz wrote, "a high level of mathe­ matical 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 de­ fense promoted theoretical advances. These theoretical advances ex­ tended beyond the field of inspection sampling in which the SPRT originated. In fact, the theory of sequential analy­ sis became an important aspect of Wald's theory of decision functions. Decision theory generalized the ques­ tions addressed by statistical inference by determining a rule based on ran­ domly selected observations for choos­ ing the best course of action from a set of possibilities. Wald had begun developing deci­ sion 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 An­ nals of Mathematical Statistics intro­ ducing its central ideas. Here he artic­ ulated the idea of generalizing the problems of hypothesis testing and constructing confidence intervals, seeking to build a theory that would in­ clude them as special cases. Wald con­ ceived of an approach that would pro-

vide a means of choosing among any number of hypotheses (in contrast to the Neyman-Pearson theory, which ad­ mitted only two) by specifying a sys­ tem of acceptance regions according to criteria that would "depend on the rel­ ative importance of the different pos­ sible 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 sur­ prisingly, these two had not met before settling at Princeton. They had had a number of contacts in Vienna in com­ mon, including Wald and Menger. Both had also begun thinking about ideas re­ lated to game theory before meeting, and their earlier work had some con­ nections to the economics and mathe­ matics communities in Vienna. In par­ ticular, Menger had written a book taking a mathematical approach to so­ cial ethics that influenced some of Mor­ genstern's research in the 1930s.28 Since his days in Vienna Wald had been familiar with some of von Neu­ mann's work in economics, having edited the latter's paper on equilibrium in a dynamic economy for the final vol­ ume of the Ergebnisse [49]. This paper had some connection to von Neu­ mann's first work on game theory, pub­ lished in 1928 [48], and while working with the SRG, Wald mentioned to a col­ league that some of his ideas in deci­ sion theory were based on that 1928 pa­ per.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 [62], elabo­ rated on the connections between sta­ tistical inference and the zero-sum two-person game. The Annals of Math­ ematics 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. Wald 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.

Annals of Mathematics

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-

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 Wald 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

30Some of Marschak's work in this field appeared in [25]; some of Hurwicz's contributions can be found in [1 8] . 31See, for example, [68, p. 9 ] and [14, p. 1 9] .

VOLUME 26, NUMBER 1 , 2004

31

research of the network of profes­

A U T H OR

sional communities in which he par­ ticipated of course cannot be known.

[1 4] Harold Hotelling. "Abraham Wald." Amer­ ican Statistician 5 (1 951 ), 1 8-1 9.

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

The importance of the connections be­

Mathematical Statistics in 20th-Century

tween that network and what Wald did

America." Historia Mathema tica 23 (1 996).

accomplish is clear.

7-30. [1 6] Patti W. Hunter. "An Unofficial Community:

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6

[52] Abraham Wald. " U ber die Produktions­ gleichungen der okonomischen Wertlehre

tistical Hypotheses." Annals of Ma themat­ 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. Sta tistical 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

(Mitteilung II)." Ergebnisse eines mathe­

Academic and Official Statisticians." Pub­

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

lications of the American Statistical Asso­

[53] Abraham Wald. " U ber einige Gleichungs­

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

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

systeme der mathematischen O konomie."

Econometric Society in Cleveland, Ohio,

Zeitschrift fur Nationalokonomie 7 (1 936),

tist, Founder of the American Statistical

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

637-670.

Association. " Journal of the American Sta­

7 1 -72.

[54] Abraham Wald. "Die Widerspruchsfreiheit

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

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

[41 ] Dorothy Ross. The Origins of American

des Kollektivbegriffes der Wahrschein­

Social Science. University Press, Cam­

lichkeitsrechnung." Ergebnisse eines math­

1 950." Annals of Ma thema tical Statistics

bridge, 1 99 1 .

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

23 (1 952), 1 -1 3.

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

VOLUME 26, NUMBER 1 , 2004

33

Mathematics in T 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: [email protected]

34

he 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

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 2 1 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 child of Father Jose de Sigiienza. The

series of frescoes by Tibaldi begins at the entrance to the Library with a rep­ resentation of Philosophy, and at the

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.

Figure 2. Arithmetica.

VOLUME 26, NUMBER 1 , 2004

35

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 Guadar­ rarnas, 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).

36

THE MATHEMATICAL INTELLIGENCER

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.

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

Peace. The Mathematical lntelligencer 24(4), 34-36, 2002.

Spottiswoode, London, 1 963. [3] George Kubler and Martin Soria. Art and Ar­ chitecture in Spain and Portugal and their

[6] William Thomas Walsh. Philip II. Sheed and Ward, London, 1 938.

American dominions 1 500 to 1 800. Pen­

guin books, Middlesex, 1 959. REFERENCES

( 1 ] Carmen Garcia-Frfas and Jose Luis Sancho Gaspar. Real Monasterio de San Lorenzo de el Escoria l. Patrimonio Nacional, 1 999.

(4] B.

Artmann.

The

Liberal Arts.

The

Mathematics Department

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

University of Stellenbosch

1 998.

Stellenbosch,

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

South

Africa

e-mail: [email protected]

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 twentiethcentury scientific development? Did mathematics undergo the same upheaval? 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. (1 865-1 963), Sobolev (1 908-1 989), and Schwartz (1 91 5-2002) ­

The Actors: Hadamard 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 Audin" 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 (182 1-1894), 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-

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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-

•

•

The Cauchy problem in the space of functions, Proceed­ ings (Doklady) of the USSR Academy ofSciences, 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? 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:

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-

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:

Figure 2. S. L. Sobolev not reading mathematics, Novosibirsk, 1 962.

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, apart from Hadamard's work. Hadamard's curious and enthusiastic mind could not remain indifferent to this ongoing work; he read the 1936 article as soon as 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-

troduction) with a surprisingly partial and anti-chronolog­ ical presentation of Sobolev's articles:

Soboleff Proceedings of the Soviet Academy of Sci­ ences, 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, Sobolev did not continue his work on generalized functions. The answer is instruc­ tive. Sobolev disappeared from mathematical re­ search circles and re­ 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

Sobolev had a clear idea of the i m portance of his work and its general character.

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THE MATHEMATICAL INTELLIGENCER

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

sion between effectiveness and rigor (Feynmann integral,

the Russian school, in the period discussed here but also

string theory; [JQ] for example gives an account of the de­

later, the value of mathematics is measured by its effec­

bate). Should one rejoice that the political upheavals of

tiveness. Even general topology, through Tychonov to Pon­

the last decades threaten to standardize worldwide the

tryagin, has applications to the study of control systems.

practice of mathematical science and the answers to this

More recently, this is also the case in the work of Arnold

"essential tension" [Ku]?

and his school. We can imagine the difficulties experienced by Lusitania, the famous school created by Luzin in

Theory and Practice- Probability and Measure

Moscow around function theory, and largely inspired by

Measure theory and its relevance to probability deserve par­

(German) set theory or (French) function theory. In con­

ticular scrutiny: it was the first serious stumbling block in

trast, France, the country of Descartes, Galois, and Bour­

the development of the Bourbaki project [B2]. From the

baki, favored an interest in mathematical investigation

point of view that interests us, distribution theory obviously

"pour l'honneur de l'esprit humain" (a phrase coined by

served as a weighty "ideological" argument at the time. As

Jacobi): the value of a theory is assessed by its degree of

an illustration, here is an excerpt from the introduction of

generality-a purifying quality of generality, synonym of

[Bl] concerning measure theory: " . . . Thus, integration the­

efficiency, and evidenced in the connections between ap­

ory is connected, on the one hand to the general theory of

parently remote domains for the production of new theo­

duality in topological vector spaces, and on the other hand

ries, and by the elegance of the concepts [B2]. (Similarly

to distribution theory, which generalizes certain aspects of

in Germany.) For Schwartz, for example, distribution the­

the notion of measure, and which we shall present in a later

ory develops as he associates the Sobolev definition to the

book" It can be seen how much this "structuralist" stance­

theory of topological vector spaces, thus arriving at the

also present in Schwartz's approach to distributions-con­

properties of the topology of distribution spaces. This will

cealed the real nature of the phenomena in question, for

make possible the work of his students Lions and Mal­

example the subtleties of random processes. (Another

grange, after the presentation of the kernel theorem at the

lustration of errors of judgment is found in Andre Weil [Wl],

il­

1950 Congress in Cambridge, USA; this theorem was the

[W2]: " . . . The time has come to try, through closer analy­

cherry on the cake, and earned him the Fields Medal and

sis, to split up Lebesgue's discoveries into various elements

the later paternity-in the West at least-of distributions.

in order to identify what is essential in the manipulation of

These two views of mathematics and their role were pres­

an integral, and what is relevant to the specific operations

ent simultaneously in both countries, and sometimes in the

over sets on which we are most frequently working.")

productions of the same mathematician, as in the cases of

Rather than disregard for potential applications, it was

Gel'fand in the USSR or, earlier, Fourier in France. In the

the desire to give priority to the structure over the phe­

period we are interested in, the emphases were as we have

nomenon, and to the architecture over the portrait, that

mentioned above. Though this question goes beyond the

caused a delay of fifteen years in French research on prob­

scope of this article, we note that recent developments in

ability. Quite ironic, in the country of Laplace, Lebesgue,

physical and mathematical sciences show continuing ten-

Borel, and most particularly Paul Levy, Fortet, Loeve, Ville,

Figure 4. The young Laurent Schwartz.

44

THE MATHEMATICAL INTELLIGENCER

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 cre­ ation 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

[S1 ] Schwartz Laurent. Theorie des Distributions, vol. 1 , Hermann, Paris,

New York, 1 961 .

(S' 1 ] Schwartz, Laurent. Theorie des Distributions, Nouvelle edition en­

vres de P. L. Tchebychef, A Markoff and N . Sonin, editors, Chelsea,

1 950. tierement corregee a fondue et augmentee. Hermann, Paris, 1 966.

[Che] Chevalley, Claude. Nicolas Bourbaki, Collective Mathematician. The Mathematical lntelligencer, 7, no. 2, (1 985), 1 8-22.

[S2] Schwartz, Laurent. Un Mathematicien aux Prises avec le Siecle,

[De] Demidov, Sergei S. The Moscow school of the theory of functions in the 1 930s in: Golden years of Moscow Mathematics, ed. S.

Editions Odile Jacob, 1 997. [S3] Schwartz, Laurent, in "Les Mathematiciens," Pour Ia Science, Paris, 1 996.

Zravkovska, P. Duren, vol. 6, AMS, LMS, 1 993. [Du] Dugac, Pierre. Jean Dieudonne mathematicien complet, Paris,

[S4]

Jacques Gabay, 1 995.

Schwartz,

Laurent.

Seminaire

"Applications

radonifiantes, "

1 969-70, Ecole Polytechnique.

[Fi] Finley, Moses I. Democracy Ancient and Modern , Rutgers Univer­

[Viu] Viucinich, A Soviet mathematics and dialectics in the Stalin Era. Historia Mathematica 27 (2000), 54-76.

sity Press, 1 985. [Ge 1 ] Gel'fand Israel M. Some aspects of functional analysis and alge­

[Viz] Vizguin, V. lstoria sov. atornnogo proekta. lzdat. rousk. kirstian. gu­

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.

manitarnogo instituta St. Petersburg. M Yushkevich, A P. Encounters with Mathematicians, Golden Years of Moscow Mathematics ed. S. Zdravkovska, Peter L. Duren, History of

[Go] Godement, Roger. Analyse Mathematique, tome 2, Springer-Ver­ lag , 2000.

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.

[Gr] Gross, Leonard. Harmonic Analysis on Hilbert Space, American

Springer-Verlag, Berlin, New York, 1 979.

Mathematical Society, 1 963. [H 1 ] Hadamard, Jacques. Le Mouvement Scientifique en URSS, Rap­

[W2] Weil, Andre. L 'integration dans les groupes topologiques et ses

port presents en 1 935 a Paris aux journees d'etude et d 'amitie franco­

applications, Hermann, Paris, 1 940; CEuvres, vol. 1 . See also the

sovietiques.

commentaries on pages 551 -555.

[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

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 ,

ligencer, 13, no. 1 (1 991 ), 23-29.

Oct. 2003.

[JQ] Jaffe, Arthur; Quinn, Frank. "Theoretical mathematics": toward a cultural synthesis of mathematics and theoretical physics. Bull. Arner.

Appendix

Adolf P. Yushkevich: Some remarks on the history of the theory of generalized solutions for partial differential equations and generalized functions. Istoriko-mate­ maticheskie issledovanie, 1991, 256-266 (Russian). 1

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

46

by Jean-Michel

THE MATHEMATICAL INTELLIGENCER

Kantor

I.

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 math­ ematical school of Saint Petersburg, later Leningrad, headed by P. Chebychev (1821-1894), one of the professors of Lya­ punov. 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 Cheby­ cheffs school, which systematically exploited the relation­ ships 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 them­ selves 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 Seismol­ ogy, 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 Mathe­ maticians: "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 al­ ready getting considerable attention from Soviet mathe­ maticians, and Sobolev, not yet 25, was elected on 01.02. 1933 a corresponding member of the Academy of Sci­ ences. 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 de­ velops his most important work, which establishes the be­ ginning of the theory of generalized functions. He is the first to define them mathematically and to set about study­ ing 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 func­ tionals 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 func­ tions. But in various practical applications it seems conve­ nient to consider functions with singularities of a well­ defined 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 lim­ its of twice-differentiable solutions. Using a simple integra­ bility 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 solu­ tions. Finally, this theory is applied to some concrete ex­ amples" ([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 ar­ ticles 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 mathemat­ ics 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 general­ isees," Vladimirov quotes [9) along with the "generalized so­ lutions" article ([12], vol. 3, p. 1 102-1 1 10 and 1 1 16-11 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 the­ ory of generalized functions appears for the first time." No­ toriously, the establishing of chronological priority be­ tween 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 infinites­ imal analysis. IV.

The prehistory of the theory of Sobolev's generalized func­ tions 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 differ­ entiable 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 re­ mark "on functional operations" and his "Le<;ons sur la prop­ agation des ondes et les equations de l'hydrodynamique" (Lectures on wave propagation and the equations of hydro­ dynamics) 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

47

The article by Steklov is deep and definitely important.

responds to the course that Sobolev taught at the time at the

He insists in particular on the significance of the first arti­

University. This book was not translated into English until

cle, where Hadamard uses for the first time the term "func­ tional," and he discusses in detail the results of the second

1963 (into German in 1964). It is quoted several times by

Leray, and, as noted by V. I. Smirnov, "this book played an

article. He insists in particular on the existence of "shock

important part in the use of the modem ideas and methods

waves" in compressible liquids and elastic bodies. One re­

of function theory and functional analysis for the solution of

mark by Steklov is particularly interesting: issues of hydro­ dynamics, translated into the language of mathematical

problems of the theory of partial differential equations" ([6]

p. 191). In Russia, Sobolev's new ideas, following those of

analysis, coincide with the theory of the characteristics of

his masters Gunther and Smirnov, diffused fairly quickly, and

partial differential equations, "which emerged completely

were extended and developed from the 1950s.

independently from any physical origin." This remark shows

In the diffusion abroad of these new directions of math­

that Steklov understood perfectly the significance for later

ematical analysis, a major role must be ascribed to the book

applications of abstract basic research pursued in complete

Theorie des Distributions

independence from their use. Moreover he uses the classic

Schwartz, a corresponding member (1973), then a full

terminology he is familiar with (and uses the term "func­

member ( 1975), of the Academy of Sciences, and a profes­

(in two volumes) by Laurent

tional" only occasionally), and he could not foresee that a

sor at the Ecole Polytechnique. Several articles published

few years later, it is essentially in his own institute that the

by Schwartz between 1945 and 1948 already used the ex­

groundwork for the theory of generalized functions was go­

pression "distributions." After the publication in 1950-51 of

ing to be laid. This speech by Steklov was not published un­

Schwartz's book, distribution theory developed consider­

til 1968 ( [ 1 ] p. 1 10-115). As regards Hadamard's advances

ably and received numerous new applications.

toward the theory of generalized solutions for partial dif­

The first historical study on research on distributions,

ferential equations and generalized functions, let us quote

published by Jesper Liitzen in 1980, contains an accurate,

a statement made by G. Shilov (professor at Moscow Uni­

flawless mathematical analysis of the works of Sobolev,

versity from 1917 to 1975, and a recognized specialist on

Schwartz, and many earlier or contemporary mathemati­

this question), on February 10, 1964, during a memorial ses­

cians. In spite of all these achievements, Jesper Liitzen's

sion of the Moscow Mathematical Society: "In solving hy­

book has some gaps and, from my point of view, uncon­

perbolic equations, Hadamard essentially introduces the de­

vincing evaluations, which can be explained by an imper­

vice of the theory of generalized functions of one or several

fect knowledge of work in Russian generally and of Sobolev

variables. This discovery remained dormant at the time

in particular (although his bibliography contains 1 1 refer­

(Hadamard was many years ahead of the thinking of math­

ences which were translated into English, as well as the

ematicians of his generation), and it was only in the mid­

1950 book already mentioned and the thick course book in

fifties that generalized functions spread worldwide in ques­

its third version of 1954). Leray's note on Sobolev's work

tions of analysis" ( [ 13], p. 185). Shilov concludes by quoting

is a substantial complement to Liitzen's study.

Szolem Mandelbrojt's words of 1922 about the famous

Without trying to write the history of the question, I shall

"Readings on Cauchy's problem" (translation to French in

make here some remarks on Liitzen's book First, I cannot

1932, and to Russian in 1978): "The notions developed in

agree with his evaluation of the results obtained by

this work lead to general topology and to functional analy­

Sobolev, then Schwartz, and their place in the development

sis, and the introduction of the notion of elementary solu­

of distribution theory. The essence of the differences be­

tion has a high degree of generality with respect to distrib­

tween their theories, according to Liitzen (p. 64), is that for

utions (generalized functions)" ([ 14], p. 4-5). Furthermore,

Sobolev distributions are a technique to resolve a specific

we owe to Hadamard the terms "functional" and "functional

problem, while Schwartz developed distribution theory un­

analysis." Jean Leray also mentions this precursor work

der multiple angles, and applied it to formulate and resolve

What we say here does not by any means detract from

rigorously numerous problems. It is true that in 1934

Sobolev's achievement; he is the first to give a rigorous def­

Sobolev began with Cauchy's problem for the wave equa­

inition-and in several ways-of the modem notion of gen­

tion (which is hyperbolic), but then he did not limit him­

eralized function, and to lay the bases of later developments

self to one of the applications which he had introduced,

in various domains of the theory of generalized solutions to

and he considerably enriched them, as shown by Jean Leray

partial differential equations and generalized functions, as

("work whose scope, variety, and power are admirable").

an autonomous domain of analysis.

It is also true that these various contributions published in

successive articles were not collected into a monograph, v.

which would doubtless have had the seminal role of

Almost all of Sobolev's work on the theory of solutions and

Schwartz's book-which became the basic book for nu­

generalized functions was published in Russian, except the

merous researchers abroad and here. Liitzen briefly sum­

article in French of 1936 ([9], 22). So it is not surprising that

marizes the fundamental difference between the works of

in other countries [than the USSR] this work did not attract

Sobolev and Schwartz: "So Sobolev invented distributions,

immediately the interest it deserved. This remark also ap­

but distribution theory was created by Schwartz" (page 64).

Some Applications of Functional Analy­ sis in Mathematical Physics (Leningrad, 1950), which cor-

Variants of this reflection occur in the book On page 67,

plies to the book

48

THE MATHEMATICAL INTELLIGENCER

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 Lau­ rent Schwartz's book of 1950, I find more balance in S. Vladimirov's judgment ([12], vol. 4, p. 1104): "The founda­ tions 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 dis­ coveries, according to the statements made by L. Schwartz in 1950-51 and in 1974, the latter did not know of them be­ fore 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 dis­ coveries 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 state­ ment 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 treat­ ment. 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 dis­ tributions, his conversations with de Rham (also mentioned by Leray), etc. All this information is valuable, and it is re­ grettable 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. How­ ever, we distinguish ideas belonging to prehistory-already born but not introduced yet into a well-defined frame­ from 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 gen­ erality, appear only at the end of the 17th century. How­ ever, 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 nat­ ural to insist on Schwartz's work But for a study of the de­ velopment 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 sum­ mary of twenty years of work in the theory of partial differ­ ential equations-elliptic, hyperbolic, parabolic, or mixed­ as 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 re­ search. Collaborations took place between all the countries involved, albeit it was sometimes made difficult by prob­ lems of communication and the lack of personal contacts (which developed a lot in recent years, their earlier scarcity having been supplemented by numerous reference period­ icals). 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 proto­ history of the theme, which led to the solution of the equa­ tion 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 mathemati­ cians of the 18th century. Briefly stated, d'Alembert com­ pletely 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 dis­ posal, 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 equa­ tions, 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 so­ lution-for which a correct defmition and, even more, con­ struction, 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

[7] Smirnov S. (ed.), Partial Differential Equations, Department of Math­

A U T HOR

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, 1 9 1 7-57, Fizmatgiz, 1 959, vol. 2, Lyusternik-Vishik article. [1 0] Mathematics in USSR, 1 958-67, Bibliography, Nauka, 1 970, vol. 2. [1 1 ] Dieudonne, J. (dir.), Abrege d'Histoire des MatMmatiques, 1 700- 1 900, 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

IIQU8S cont

bclis·

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!

ences, Comptes Rendus , serie generale, vol. 7 , 1 990, No 6,

467-471 '

A Prestigious C ol lection from the London M athematica l Society The Proceedmgs. Journal.

athematiCs of the London

Mathematical Society are among h

ds lead•ng mathematical

research penod •cals . The subJect covera e ranges across a broad

spectrum of ma hemat1cs, covenng he who e of pure ma hema 1cs together

th some more apphed

areas of analys•s. ma hematiCal p

REFERENCES

ys . t eoretiCal computer sc ence.

probab1hty and s a 1s 1cs

[1 ] French-Russian Scientific Relationships, A Grigorian, A Yushkevich, Nauka, 1 968.

[2] lstori. Math. /ss/edovanie, 31 (1 989), Nauka. [3] Ibidem, 90, vol. 32-33.

To subscnbe pl ase cont

[4] Bakhalov-VIadimirov-Gonchar, Sergei L'vovich Sobolev, Uspekhi

journ Is cambrldg .org

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

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VOLUME 26, NUMBER 1 , 2004

51

The Recepti on of the Theory of Distributions Peter Lax

In

Courant Institute of Mathematical Sciences New York University New York, NY 1 00 1 2 USA e-mai l : [email protected]

52

THE MATHEMATICAL INTELLIGENCER

ld§ijl§',ifj

Osmo Peko n e n , Ed itor

I

The Science of Conjecture: Evidence and Probability 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

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: [email protected]

F

ranklin is nominally a mathemati­ cian and his book shows a mathe­ matician's touch when it deals directly with mathematical matters, but this re­ view 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 Con­ jecture is a history of a lengthy philo­ sophical 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 or­ der degrees of belief that lie some­ where between absolute conviction and utter dismissal? What degree of be­ lief is necessary to justify an action with grave consequences? This is not the sort of question that mathemati­ cians are given to worrying about, at least not when going about their math­ ematical 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, Pas­ cal, and Huygens. His history, which begins in classical times, closes at this juncture, the take-off point, we might think, of Western scientific rational­ ism. But by no means would he have us believe that formal, quantitative proba­ bility theory (or its evolution into sta­ tistics) made earlier modes of inquiry obsolete or brought them to comple­ tion. Rather, the probabilist's quantifi­ cation of likelihood and expectation, even in its most sophisticated develop­ ment, covers only a narrow range of ex­ perience and informs our judgment at best in limited measure. Thus the med­ itations of ancient Aristotelians and cloistered Scholastics are far from be­ ing 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 curli­ cues 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

to render actions licit, even in the face

strates that for the most part this is

less-than-absolute

certainty as they

of weightier opposing arguments. This

quite doubtful. The classic problems of

carne to bear on other fields of thought.

doctrine carne to be identified with Je­

the

The most important of these, overall,

suit thought in particular, and con­

phrased in terms of finding an equi­

is law, an area in which the same ques­

tributed to the Jesuits' reputation for

table division of stakes in an inter­

tions of credibility and weight of evi­

sly equivocation.

dence recur from era to era and civi­

early

probabilists

are

usually

rupted game, which would be a curi­

Franklin also considers epistemo­

ous way of looking at things if the

lization to civilization. Franklin surveys

logical and evidentiary standards in

object were to improve the acumen of

classical (that is, Hellenic and Roman)

pre-modem science. There are some

dicers and card-players. But it makes

ideas, along with Islamic, Judaic, me­

surprises here; some "modem" philos­

perfect sense when seen as a continu­

dieval, and Renaissance thinking on

ophy turns out not to be so modem as

ation of the Scholastic tradition of de­

the primary question: what degree of

all that. Even Ockharn's famous Razor

ciding what course must be followed

direct and indirect evidence, what lines

is revealed not so much as a revolu­

to avoid sinful error. (The sin, note, lies

of argument, will suffice to convict the

tionary device heralding the end of me­

not in gambling, but in a lack of even­

accused or to win redress for a plain­

dieval obscurantism as a tool already

handedness in the setup of the game.)

tiff. Obviously, there is no consensus

quite familiar and well worn with use

This is not to say that real-life gamblers

to be found here. More discouraging,

before Ockharn took it up. On the ba­

were not on the scene, or that their in­

perhaps, is that it is far from evident

sis of Franklin's shrewd observations,

sights were never absorbed by the

that the principles of judgment now in­

one learns new respect for the long-de­

unworldly mathematicians. But their

stituted in our courts represent the dis­

spised Scholastics. He joins a growing

practical concerns are not central. And

tilled essence of hard-won wisdom.

corps of intellectual historians newly

we know, of course, that Pascal him­

The best to be said, I think, is that we

appreciative of the stunning, singular

self carne to prefer the high-stakes

have abandoned divination, along with

mathematical genius of the 14th-cen­

game of winning souls through con­

the notion that evidence of guilt re­

tury cleric Nicole Oresme, as amazing,

version. (Franklin does discuss the fa­

quires confession, which in turn ne­

in its way, as that of Rarnanujan.

mous Wager, but his account is fairly

cessitates torture. But this is a long

At the same time, Franklin shows

way from our having a coherent and

that many of the developments we look

This explains some curious omis­

systematic legal epistemology.

upon as practical solutions to thor­

sions in the underlying justification for

conventional.)

Because medieval Europe is a cen­

oughly secular problems grew out of

probabilistic formulae. The early mas­

tral focus, canon law, as well as civil,

attempts to create a calculus capable

ters never seem to refer to the idea that

is an area where Franklin finds inter­

of evaluating financial and commercial

the probability of an outcome (of, say,

esting thinkers. This naturally con­

dealings from the point of view of sanc­

a throw of the dice) is essentially iden­

nects to the prolific literature of moral

tioned morality. Thus, notions of "ex­

tical with the ratio of the number of

philosophy, whose "practical" side in­

pected value" that might apply to such

times that outcome is achieved to the

cluded the vast array of "manuals" for

arrangements as annuities, lotteries,

total number of trials when the latter

the priesthood, guides, principally, to

risk insurance, and what we would

is very large. Similarly, though notions

the quasi-judicial ritual of Confession.

now call futures contracts were devel­

of expected value of a game occur, and

Here, "cases of conscience" come to

oped primarily to distinguish fair deal­

are correctly computed, this is not re­ lated in any direct way to the average

the fore: how much doubt must arise

ing from usury. Of course, this applies

in the mind of the faithful believer in

in large part to the formal treatises of

result over a large ensemble of actual

order to proscribe a possible course of

scholars. One assumes that, in parallel,

games. By the same token, the actuar­

action as potentially sinful (or, put the

men of practical affairs developed their

ial approach, deriving parameters em­

other way, how much doubt must be

own formularies for these things, but

pirically from observed frequencies

effaced before the action may be un­

here the record is scanty. What is most

and averages, and then using these as

dertaken with a clear conscience)? The

interesting in this regard is that the ori­

probabilities or expected values in fur­

lack of agreement among the school­

gins of true probability calculus in the

ther calculation, is not part of the pro­ gram either. Basic probabilities-the

men, here, as well as the ferocity of

mid-1 7th century seem closely con­

their disputes, is one of the most in­

nected to the

of parsing

one-in-six that a given face will tum up

triguing topics of the book. We must

doubtful situations to decide what is

on a given throw of a die-are simply

tradition

ad hoc,

adjust our minds to the ancient usage

truly fair, equitable, or licit. We usually

assumed

of "probable." During the period in

think of the problems addressed by

that real life may depart from these

question, this does not mean "more

Fermat and Pascal as gambling or gam­

ideal frequencies.

likely than not" but rather, "sustainable

ing problems, and thus we assume that

we think of as quite elementary-the

by at least some evidence, argument,

they grew out of serious interest in

rules for deriving probabilities of con­

and authority," in other words, merely

these practices, with the mathemati­

junctions

plausible. Thus, "probabilism" is the

cians seeking to justify, generalize, or

stance-are

doctrine that a justification merely

improve rules of thumb known to de­

Thus, the invention of probability cal­

"probable" in this weak sense suffices

voted

culus, though undeniably a brilliant

54

THE MATHEMATICAL INTELLIGENCER

gamblers.

Franklin

demon-

and

without any sense

As well, notions that

disjunctions,

not

clearly

for in­

formalized.

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 0-06621070-4

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

T

he 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

fact what is needed for this formula is

ested

not the evaluation of

non-mathematicians

will

suc­

s

?(s)

for one or

Another of the authors, Marcus du Sautoy, is a professional mathemati­

ceed. In other words, I am among the

more values of

but a knowledge of

cian, so we should expect his state­

"many people" who, according to the

the zeros of ?Cs) in the critical strip; the

ments to be correct, but I am puzzled

first sentence of Karl Sabbagh's Pro­

formula is valid whether or not the ze­

by his statement (p.

logue, "would say that the task I am

ros are on the critical line. This misap­

of the

embarking on . . . is futile." He defends

prehension about the meaning of RH

mean that mathematicians could use a

his project by comparing it to anthro­

probably underlies his answer, at the

very fast procedure guaranteed to lo­

pology and to "describing a remote

end of his Prologue, to the question,

cate a prime number with, say, a hun­

tribe whose customs and language are

"Why is it [RH] so important? . . . A

dred digits or any other number of dig­

unfamiliar to the reader, but whom I

proof . . . would . . . tell mathematicians

its you care to choose." He goes on to

understand enough to convey some­

Riemann

11) that "a proof

Hypothesis

would

a huge amount about an important

relate this to RSA cryptography, clearly

thing of their inner and outer lives."

Mathematical Intelli­

class of numbers-the prime numbers,

implying that RH would have some

Readers of the

which dominate the field of pure math­

as members of that remote

practical

ematics." The notion that such a goal

phy, but I doubt that this is the case. I

gencer, tribe,

will

be

interested

to

know

whether the descriptions he provides

significance for cryptogra­

accounts for the fascination of RH is a

suspect, rather, that he feels the gen­

profound

eral reader must be given some reason

misunderstanding

of

our

are accurate and whether they illumi­

tribal culture, like believing moun­

for the significance of this million-dol­

nate our tribal culture. On both counts,

taineers want to climb Mount Everest

lar question in mathematics, but that

I am unenthusiastic.

in order to get somewhere.

the real reason depends on aspects of

The best parts of the Sabbagh book

For another example, he often spec­

our tribal culture that are too difficult

are indeed the anthropological ones.

ulates about who might or might not

to explain to the general reader. (Per­

He tells who has worked on-or is

prove the Riemann hypothesis. On p.

working on-the Riemann hypothesis,

219, Martin Huxley is said to have "both

how they became interested in mathe­

the desire and the ability to prove the

matics and in this particular problem,

Riemann Hypothesis." On p.

240, we are

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

how they view their chances for suc­

told that "many . . . feel that if anyone

inability to answer basic questions

cess, and so forth. But what makes us

is going to prove the Riemann Hypoth­

about prime numbers," but

a tribe is our peculiar culture, and there

esis, it will be [Alain Connes]." And not

depended on our inability to factor

I thought it

is no way to describe the interactions

only is there speculation that Louis de

large numbers. In fact, I thought the

of key members of the tribe without go­

Branges might be the one to prove RH,

practicality of RSA depended on the

ing into the substance of our culture.

the book includes an Appendix by de

disparity-in practice, primality test­

On this, Sabbagh is an unreliable guide.

41 (page numbers

Branges with the title "De Branges's

ing is easy; factoring, hard.

refer to the American edition-the orig­

mathematics goes. Perhaps in other

tering these building blocks [primes]

inal English edition is more compact,

fields that require expensive equipment

offers the mathematician the hope of

so for example, this passage is on page

one might, to a limited extent, predict

discovering new ways of charting a

33 of that version) he says: "So, calcu­

course through the vast complexities

lating the value of the sum

I lln8,

where the next breakthrough might oc­ cur, but in mathematics any attempt to

of the mathematical world" puzzles me in a different way. Whatever could it

but couldn't say so for certain, would

predict whether there will be a proof

any time soon, much less what shape it

For example, on p.

which Riemann believed was possible result

in

a totally accurate number for

Proof."

This isn't the way research in

His statement on page

5 that "Mas­

mean? To my taste, this statement, and

might take or who might devise it, is

much else in the du Sautoy book,

the number of primes less than

completely foolish.

sounds too much like empty enthusi­

certainly possible when the real part of

an interesting pair of speculations:

non-mathematicians

s

Henryk Iwaniec (p.

stance-because

n." Well, calculating the value of I 1 /n8 is is greater than

1,

but Sabbagh does

In this connection, Sabbagh gives us

36) says,

''I'm only

asm, razzle-dazzle meant to impress not

with

substantial

sub­

mathe­

not say which particular values of s will

worried that what may happen is that

matics is beyond their ken-but with

be needed to produce his "totally ac­

a proof will be given by somebody and

fanfares and flourishes.

curate number." (Later in the book,

I will be unable to understand it," while

complex numbers are introduced, and

Alain Connes (p.

263)

worries about

Du Sautoy touches on a clash of cul­ tures within mathematics that is sel­

on the next-to-last page analytic con­

something quite different: "It would be

dom revealed to outsiders and that

tinuation is mentioned in passing, but

a tragedy if it just needed a trick to

might hold some interest for Sabbagh

at this point

I lln8 is far from being the

prove it." Different as these concerns

and others interested in the anthropol­

?(s).) This "totally accu­

seem, I suspect that most mathemati­

ogy of our tribe. In his last chapter he

rate number" must refer to Riemann's

cians sympathize with both. Note that

sketches in very laudatory terms the

same thing as

1r(x),

which Sab­

neither has anything to do with "learn­

career

bagh seems to believe (see also the end

ing a huge amount about an important

"Grothendieck's new language of geom­

of Chapter

class of numbers."

etry and algebra saw the creation of a

explicit formula for

56

1)

depends on RH; but in

THE MATHEMATICAL INTELLIGENCER

of Alexander

Grothendieck;

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 be­ coming 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 pub­ lisher, 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 men­ tions 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 ac­ knowledgment. 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 en­ courage all readers of the Mathemati­ cal lntelligencer to try it. It is interest­ ingly and skillfully written, and it approaches many aspects of the sub­ ject in imaginative and thought-pro­ voking ways. For a quick probe, you might try reading pages 90-92. There you will find discussions of the con­ trast between measuring and counting (as he describes it, numbers legato and numbers staccato), Gauss's attitude to­ ward Fermat's last theorem, Mallory's reason for wanting to climb Mount Ever­ est, the rise of the Germans in 19th-cen­ tury 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 pre­ cious or pretentious, the fault is mine. To my taste it is always down-to-earth and treats its topics in natural and ap­ propriate, but interesting, ways. Naturally I have my disagreements and cavils with the book, but it is re­ markable to me how few they are when I consider how dense with information and opinions the book is. The peasant­ pheasant 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 origi­ nal goal was to have no calculus at all, but that this goal "proved a tad over­ optimistic"), 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, com­ petent 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 state­ ment 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 be­ cause 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 stu­ dent. 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 al­ low 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 +

X

p

L

x

dt

t(t2 - 1) log t

where J(x) denotes 1T(x) + t'1T(Yx) + t1TCVx) + · · · (a terminating series be­ cause 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 crit­ ical strip. And I don't mean that he sim­ ply 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 esti­ mates of the various terms in the case x = 20. Once he has completed this, hav­ ing 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 "sec­ ondary 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 com­ puted to attain four-place accuracy for this term, and he certainly does not ex­ pect 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 tal­ ented. 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 ap­ preciate 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 con­ nection of RH to prime numbers. Der­ byshire 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 hy­ pothesis was that the zeros of g(t) are real! To me, g(t) is a symmetrized ver­ sion 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 vari­ able. 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: [email protected]

A Mathematician Grappling with His Century by Laurent Schwartz BIRKHAUSER VERLAG. 2001 vi1i + 490 pp., ISBN 3764360526; US $49.95.

REVIEWED BY NORBERT SCHLOMIUK

L

aurent 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 integra­ tion were practised by men such as Cavalieri, Fermat and Roberval when Newton and Leibniz were mere schoolboys. But they were able to sys­ tematize the algorithms and nota­ tions 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 Poly­ technique. 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 men­ tal processes of research and for decades I have devoted myself to ur­ gently necessary reforms within the University and at Ecole Polytech­ nique. Some of my reflexions are con­ tained in this book, as well as a de­ scription of the course of my life. However I do not discuss the Univer­ sity reforms since I have written many articles and books on the sub­ ject. 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 histor­ ical 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 hav­ ing thought it out, you still find your­ selfalone 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 be­ cause 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 be­ lieve I was stupid, but I couldn't un­ derstand the contraindication be­ tween 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 in­ telligence. 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.

this he was dismissed from the E cole

Norwegian mathematician Sophus Lie

Polytechnique by the then Minister of

and

Defense Pierre Messmer. war,

Emergence of the Theory of Lie Groups (Hawkins [2]) and Arild Stub­

And here a last quotation from this

Schwartz again became politically in­

haug's Norwegian biography of Lie

Later, marvellous book:

Self-confidence is a condition of suc­ cess; of course one must be modest, and every intellectual needs to recall this. I am perfectly conscious of the immensity of my ignorance com­ pared 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 immen­ sity ofhis ignorance. But he must also have confidence in himself and in his possibilities ofsucceeding through the constant and tenacious search for truth.

during

the

Vietnam

his

work:

Thomas

Hawkins's

volved, and again that involvement was

(Stubhaug

not one-sided. During trips to Vietnam

deals with the English translation of

after Ho Chi Minh's victory, he contin­

the latter.

ued to intervene with the Vietnamese authorities on behalf of dissidents. Together with Henri Cartan and

[3]). The present review

The two books complement each other very well. Hawkins's book, which represents the crowning result of many

Marcel Broue, Laurent Schwartz was a

years of research, gives a thorough

founder of the Committee of Mathe­

mathematical analysis of Lie's inven­

maticians. The task of the Committee

tion of Lie groups and the further de­

was to fight against human rights vio­

velopment of the subject for the next

lations of mathematicians, mainly in

half-century. It is aimed at an audience

the Soviet Union but also in Morocco,

of mathematicians who know the the­

in Czechoslovakia, in Uruguay-and

ory and can follow the technical details

indeed, all over the world.

of its development. Stubhaug, on the

Each chapter of Schwartz's autobi­

other hand, draws a vivid picture of the

ography is beautiful and interesting.

person Sophus Lie and the time he

Mathematicians will be particularly in­

lived in. His book is aimed at a general

terested in chapter VI, "The invention

audience, and does not go into mathe­

of distributions." Few mathematicians

matical detail. That does not mean that

Very interesting, sincere and illumi­

have written about the act of creation

mathematics is left out of Stubhaug's

nating are the chapters about the po­

in mathematics, Poincare and Hadamard

biography; in fact, mathematics is con­

litical involvement of Schwartz. Very

for instance. The author's pages are, I

stantly present in the book as Lie's

early during the infamous Moscow tri­

think, in the same league.

great passion, and it is clear that Lie's

(1937) he realized the nature of

Laurent Schwartz is no longer among

Stalin's dictatorship. He became a Trot­

us; we cannot listen to his beautiful

creativity.

skyite in

warm voice carrying messages of wis­

ations are described in general terms,

als

1944, and was an unsuccess­

greatness was due to his mathematical Lie's

mathematical

cre­

ful candidate as a Trotskyite for the

dom, humor, and understanding; but

and in a chapter titled Into Mathemat­

French Legislative election in

we have this marvellous autobiography

ical History Stubhaug does a good job

1945.

Much too free a thinker to accept a

of a splendid man.

"party line," he broke with Trotskyism in

1947.

Departement de Mathematiques et de

The center of his life continued to

of setting them into historical and philosophical perspective. A non-math­

Statistique

ematical reader will learn from Stub­ haug that mathematics is fascinating

be in mathematics. After a one-year ap­

Universite de Montreal

and important and will get a good

pointment in Grenoble, Schwartz joined

C. P. 61 28, Succursale Centre-Ville

glimpse of the creative, dramatic, and

in

1945 the Faculty of Science in Nancy,

where he spent seven extremely fruit­

Montreal, H3C 3J7 , Canada.

even existential aspects of mathemati­

e-mail: [email protected]

cal research. For this reason alone one

ful years both for his research and for

must hope that the book will be widely

attracting brilliant young mathemati­

read, especially at a time when scien­

cians: Grothendieck, Malgrange, Lyons,

tific and mathematical research is un­

Treves, Bruhat. He moved to the Sor­ bonne in 1952 and to the E cole 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

The Mathematician Sophus Lie: It Was the Audacity of My Thinking by Arild Stubhaug, translated from the Norwegian by Richard H. Daly

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

to rebel against the Algerian war. For

THE MATHEMATICAL INTELLIGENCER

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 ofLie 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

REVIEWED BY JESPER LUTZEN

is both extremely well written and well

T

he year 2000 saw the publication of

translated the Abel biography,

two remarkable books about the

again done an excellent job. For the

121," which

proclaimed the right of French youth

60

BERLIN HEIDELBERG, SPRINGER-VERLAG 2002 556 pp. LIS $44.95, ISBN 3-540-42 1 37 ·8

der public scrutiny. The mathemati­

researched. Richard H. Daly, who also has

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 27page 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 ar­ chetypical 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 ti­ tan replete with the lust for life, with audacious goals and an indomitable

62

THE MATHEMATICAL INTELLIGENCER

will. These descriptions of his physi­ cal and mental strength also con­ tained a subtext, an embryonic no­ tion, not only about this brilliant man of science, the prophet, who intu­ itively 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 un­ derfoot. He was described as highly committed, richly innovative, some­ one with unusual physical strength, and the stamina to overcome the ma­ jority 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 Nordic prototype . " a titan . . 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 of following 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 Chris­ tiania 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 1874 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 Naturvi­ denskab, which he founded together with two colleagues, and later in inter­ national journals, mainly in the Mathe­ matische 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. More­ over, he had several good assistants and Private Docents such as Engel (whose collaboration with Lie contin­ ued) and Eduard Study, Issai Schur, and Georg Scheffers, who published Lie's lectures in three large volumes:

Differential Equations with Known Infinitesimal Transformations, Con­ tinuous Groups with Geometric and other Applications, and Geometry of Contact Transformations (all in Ger­ man). Finally, he had a good relation­ ship 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 break­ down that confmed him to a psychi­ atric 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 com­ plete 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 in­ creasing 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 chrono­ logical presentation than in other parts, and that makes the part somewhat dis­ connected. In particular it is unfortu­ nate that there is not a clear distinction between the periods before and after Lie's mental breakdown. Moreover, when writing about the cultural and in­ stitutional 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 con­ stantly mentioned also in this part. In­ deed, Lie continued to keep abreast with the developments in his native country through regular reading of Norwegian newspapers, through cor­ respondence, and through the young Norwegians who came to study with him in Leipzig. He even continued to participate in the public debate on ed­ ucational 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 Univer­ sity, 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 at­ mosphere 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 coun­ try 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 cere­ mony 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 re­ sulted in independence from Sweden in 1905 was gaining momentum in the 1890s. One of the strategies among the cultural and academic elite was to dis­ play 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 re­ cently 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 Transfor­ mation Group Theory with a salary that matched his high German salary. Lie ac­ cepted, but it took him another 4 years before he finally resigned his position in Leipzig and returned for good to Chris­ tiania. During those four years he trav­ eled back and forth from Norway sev­ eral 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 biogra­ phies have tried to capture this extra­ ordinary mathematical genius, but none of them have been as complete, as well researched, or as well rooted in the cultural context as Stubhaug's well­ written book. I can recommend it to all mathematicians as well as non-mathe­ maticians, 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 2 1 , 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- 1 926, 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: [email protected]

Sn. He shows how to write a given num­

Fibonacci Numbers by Nicolai N. Vorobiev BIRKHAUSER. 2002 1 76 pp. €31 ISBN 3-7643-6 1 35-2 paperback

REVIEWED BY FREDRIC T. HOWARD

he Fib�nacci �umbers ar� defined by u 1 - 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 Fi­ bonacci 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 u 1 + 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 n a !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

T

n�x

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 27 1 09

USA e-mail: [email protected]

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

ack 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

B

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 ( Tn l 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
used instead of gi (without the period). On pages 12 1-122, 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: [email protected]

Kepler's Conjecture: How Some of the Greatest Minds 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

I

f 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 161 1 , 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

Kepler, FPK (felicitously dubbed Pro­

packing problem, the polite collabora­ tion of Schutte and van der Waerden

formed by the first layer, etc. The den­

ject Flyspeck). After computer-based

sity is now 74.05 percent. And so Kep­

theorem-proving, this is the next great

on the "kissing number" (how many

ler's conjecture was born.

leap forward: computer-based proof­

unit spheres can touch a unit sphere),

checking. Pushed to the limit, this

and the almost hundred-year-long race

apple

would seem to entail a self-referential

for an upper bound in sphere-packing

carts in exactly that way. (He must

loop. Maybe the purists who insist that

densities are carefully researched and

have relished this kind of "market re­

a proof is a proof if they can under­

retold, as

search": his later booklet on volumes

stand it are right after all. On the other

zlo Fejes-T6th, who predicted in the six­

started out with observing how the

hand,

ties that computers would be needed to

content of wine-casks is gauged by in­

such a promising concept, for review­

crack the conjecture. George Szpiro, a

troducing a measuring rod).

ers, editors, and authors alike, that it

mathematician who became a reporter

Kepler did not fail to point out that street vendors stocked

their

In a remarkably candid piece of in­ trospection, the young Kepler once

computer-based refereeing is

is the valiant struggle of Las­

seems unthinkable that the community

(he currently works as the Israel cor­

will not succumb to the temptation.

respondent of the prestigious

wrote that his appearance, his ap­

It appears that Flyspeck will require

Zurcher Zeitung),

Neue

displays his inves­

tigative talents for digging up unlikely

petites, his habits, and his character

twenty man-years to verify every single

traits displayed "in every way a dog­

step of Hales's proof. If all goes well,

stories, and a professional eye for the

like nature. " We may add that Kepler's

we then can be 100 percent certain.

telling detail. The review which Gauss

conjecture still has a vicious bite. Who­

You shouldn't wait twenty years before

wrote about a book by Seeber (proving

ever approaches it does so at a con­

reading about the tormented history of

in a few lines a lot more than Seeber

Kepler's conjecture, though. In fact,

had achieved in two hundred pages) is

siderable risk George Szpiro, the author of "Kep­

Szpiro's book, by appearing so patently

described with gusto and wit, as are the

ler's Conjecture, " will surely agree.

at the wrong moment, gains an extra

recent debates surrounding Hsiang's

Right after Thomas Hales announced,

quality. It becomes what some of my

claims to having solved the problem.

in August

1998, that the problem was

colleagues from the humanities call a

A substantial part of Szpiro's book

at long last solved, and that the dens­

"meta-text." The whole book, which is

is devoted to the diffidence with which

est way of packing unit spheres is the

mostly about the treacherous nature of

many mathematicians approach com­

obvious way, Szpiro sat down to do, for

the sphere-packing problem, has itself

puter-based proofs. Most mathemati­

Kepler's conjecture, what Singh had

become yet another victim of that

cians agree that they are ugly, should

treachery.

only be used as a last resort, and ought

done for Fermat's, and he wrote a

to be replaced, as soon as possible, by

splendid book full of insights, anec­

This does not speak at all against

dotes, and context to celebrate Hales's

this well-crafted piece of popular sci­

some

achievement. Now, five years later, the

ence

of

"proofs from the book" in which God,

writing.

The topic leaves,

of these

short

and

elegant

book has appeared but the proof still

course, nothing to be desired: many as­

according to Erdos, keeps all true

hasn't. It is true that the paper by Hales

pects are based on simple geometric

mathematical insights.

and Ferguson will be published even­

arguments which are easy to illustrate.

Szpiro wrote his text, the proof by

At the time

Annals of Mathematics,

Crystals and wallpapers, Euler's theo­

Hales and Ferguson was seen as such

but with an introductory remark by the

rem and Voronoi's cells, packings and

a necessary evil (and shared the same

editors, a disclaimer as it were, stating

coverings, nets and knots, all have an

social status, among working mathe­

that they had been unable to verify the

immediate appeal to the brain and the

maticians) as the classification of finite

correctness of the

senses alike.

groups: no single human brain had ever

tually in the

250-page manu­ The

More importantly, the four hundred

understood it in its entirety, but the

proof is so huge, and based to such an

years of persistent intellectual struggle

community seemed to have resigned it­

extent on massive use of computers,

with the elusive conjecture offer a

self to accept it. But now, this appears

that the platoon of mathematicians

wonderful mine of historical detail,

to be no longer the case. Hence, the story of Kepler's conjecture will have

script with absolute

certainty.

charged with the task of checking it ran

starting with Sir Walter Raleigh and his

out of steam. Robert MacPherson, the

concern for storing cannon-balls, and

to go on for a few more chapters. It is

Annals editor in charge of the project,

including Buckminster Fuller with his

to be hoped that it will eventually be

stated that "the referees put a level of

re-invention of the geodesic dome. The

told with the same elan as in Szpiro's

energy into this that is, in my experi­

"greatest minds in history" include, of

book

ence, unprecedented." But they ended

course,

up being only "99 percent certain" that

Newton, Lagrange, Gauss, and Hilbert.

the usual suspects:

Kepler,

the proof was correct.

But Szpiro avoids the lure of touring

University of Vienna Strudlhofgasse 4

Institute for Mathematics

Hales, too, was exhausted. Rather

the summits only, and offers lively

than take up MacPherson's suggestion

vignettes of many figures less well­

1 090 Vienna

to re-write the manuscript, he started

endowed with biographers. Axel Thue's

Austria

on a new project, the Formal Proof of

muddled attempts at solving the circle-

e-mail: [email protected]

VOLUME 26, NUMBER 1, 2004

67

George Green, Mathematician and Physicist 1 793- 1 841 : The Background to His Life and Work D. M. Cannell SIAM, 2001 $75.00. ISBN 0-8987 1 -463-X XXXIX + 3 1 6 pp.

REVIEWED BY STEVEN G. KRANTZ

very calculus student learns of Green's theorem as perhaps the most benign multidimensional version of the fundamental theorem of calcu­ lus. Coupled with its delightful inter­ pretations in terms of fluid flow and electrostatics, this result can be con­ strued as one of the capstones of a freshman education in mathematics. But most calculus books contain al­ most 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 Notting­ ham. 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 cus­ tom of the time, George went to work for his father at an early age. Recog­ nizing 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 fa­ ther'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 commit­ ment to his father's business, George had been able to produce his funda­ mental (and first) paper, "An Essay on the Application of Mathematical Analy­ sis to the Theories of Electricity and Magnetism." It was published by private

E

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 for­ tune, Sir Edward Bromhead (a member of the local intelligensia) became aware of Green's work in 1830, and he en­ couraged 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 impor­ tant, and the most influential. It con­ tains 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 rendi­ tion of what we now call the Dirichlet principle, (ii) an important asymptotic method for solving certain partial dif­ ferential equations in divergence form, and (iii) a preliminary version of the idea of tensors (indeed a particular ten­ sor 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 sci­ entist) he left behind little correspon­ dence, no diary, and no working papers. An especially arduous effort was re­ quired to piece together the story of his life. It should be stressed that this book is not a commentary on Green's scien­ tific work (although some of the ap­ pendices 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 mar­ vels 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 recog­ nition 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 ulti­ mately arranged for the paper to be properly published in Grelle's Journal in the 1850s. Finally, on the 200th an­ niversary of George Green's birth, a plaque in his honor was placed in West­ minster 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 nur­ turing environment (such as Paris), rather than the stultifying wreckage that was British science in the early nineteenth century. It seems inar­ guable that George Green had a pro­ found influence on such leaders of nineteenth-century British mathemati­ cal 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-Prize­ winning work of Julian Schwinger on quantum electrodynamics makes con­ siderable use of Green's function. On the occasion of the 200th anniversary of Green's birth, such luminaries as Freeman Dyson and Schwinger gath­ ered to help pay tribute to him. We owe a debt to D. Mary Cannell for penning this, the only full-length biogra­ phy of George Green in existence. It is a shame that this new edition was pub­ lished posthumously, but a tribute to her scholarship and dedication to an impor­ tant 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 Math­

ematical tntelligencer, vol. 1 1 (1 989), 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: [email protected]

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

n Woody Allen's Midsummer Night's 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.

ISex Comedy,

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 [email protected]

STATEMENT OF OWNERSHIP, MANAGEMENT, AND CIRCULATION (Required by 39 U.S. C. 3685). (1) Publication title: Mathematical Intelligencer. (2) Publication No. 001-656. (3) Filing Date: 10/03. (4) Issue Frequency: Quarterly. (5) No. of Issues Published Annually: 4. (6) Annual Subscription Price: $66.00. (7) Complete Mailing Address of Known Office of Publication: 175 Fifth Avenue, New York, NY 10010-7858. Contact Person: Joe Kozak, Telephone 2 12-460-1500 EXT 303. (8) Complete Mailing Address of Headquarters or General Business Office of Publisher: 175 Fifth Avenue, New York, NY 100107858. (9) Full Names and Complete Mailing Addresses of Publisher, Editor, and Managing Editor: Publisher: Springer-Verlag New York, Inc., 175 Fifth Av­ enue, New York, NY 10010-7858. Editor: Chandler Davis, Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 1Al. Managing Editor: Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010-7858. (10) Owner: Springer-Velag Export GmbH, Tiergartenstrasse 17, D69121 Heidelberg, Germany and Springer-Verlag GmbH

& Co. KG, Heidelberger Platz 3, D-14197 Berlin, Germany. (11) Known Bondholders, Mortgagees,

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VOLUME 26, NUMBER 1 , 2004

71

,.-jfi111.19.h.l§i R o b i n W i l s o n I

The Philamath's C

aratheodory: Constantin Cara­

prime number theorem. He also worked

theodory ( 1873-1950) is the most

on quadratic forms and integrals and

significant Greek recent

Alphabet-(

Caratheodory

but,

encouraged

of

studied theoretical mechanics and link­

by

ages. He taught at the University of St

Minkowski, Klein, and Hilbert, he spent

Petersburg, and founded the St Peters­

most of his life in Germany. He made

burg School of Mathematics.

significant contributions to the calcu­

Computer graphics: Computer-aided

lus of variations and its applications to

design has developed rapidly in recent

geometrical optics, the theory of func­

decades, and in 1970 the Netherlands

tions (especially conformal represen­

produced the first set of computer-gen­

tation), and measure theory. In applied

erated stamp designs. The stamp below

mathematics, he wrote on thermody­

shows an isometric projection in which

namics and relativity theory.

the circles at the centres of the faces

Cauchy: Augustin-Louis Cauchy ( 1 789-

gradually expand and become trans­

1857) was the most important analyst

formed into squares.

of the early nineteenth century. In the

Condorcet: The Marquis de Condorcet

1820s he transformed the whole area

(1 743-1 794) clarified the foundations

of real analysis, providing a rigorous

of probability theory, and studied the

treatment of the c:;alculus by formaliz­

solutions of ordinary and partial dif­

ing the concepts of limit, continuity,

ferential equations. His most signifi­

derivative, and integral. In addition, he

cant contributions, however, were to

almost single-handedly developed the

'social mathematics'-in particular, to

subject of complex analysis, and many

the analysis of models of voting pat­

results in this area are named after him:

terns. After the French Revolution he

'Cauchy's integral formula' appears on

was arrested while fleeing for his life,

the stamp.

and died in captivity.

Chebyshev: Pafnuty Chebyshev (18211894) is remembered mainly for his

Counting on the fingers: From ear­

work on orthogonal functions ('Cheby­

able to count and measure the objects

liest times, people have needed to be

probability

around them. Early methods of count­

('Chebyshev's inequality') and for an im­

ing included forming stones into piles,

portant contribution to the proof of the

cutting notches in sticks, and counting

shev

Cauchy

times;

mathematician

polynomials')

and

on the fingers. It is undoubtedly due to such finger counting that our familiar decimal system emerged.

12�8

nederland

Chebyshev

Please send all submissions to

Computer graphics

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

72

THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK

Condorcet

Counting on the fingers