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Thoughts on the Riemann Hypothesis G. J. Chaitin
The Opinion column offers
mathematicians the opportunity to
write about any issue of interest to
the international mathematical
community. Disagreement and
controversy are welcome. The views
and opinions expressed here, however,
are exclusively those of the author,
and neither the publisher nor the
editor-in-chief endorses or accepts
responsibility for them. An Opinion
should be submitted to the editor-in
chief, Chandler Davis.
The simultaneous appearance in May 2003 of four books on the Rie
mann hypothesis (RH) provoked these reflections. I briefly discuss whether the RH should be added as a new axiom, and whether a proof of the RH might involve the notion of randomness.
New Pragmatically Justified
Mathematical Axioms that Are
Not at All Self-evident
A pragmatically justified principle is one that is justified by its many important consequences-which is precisely the opposite of normal mathematical practice.1 However, this is standard operating procedure in physics.
Are there mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified? And are these propositions fruitful? Do they yield many other significant results?
Yes, I think so. At present, the two best candidates2 for useful new axioms of the kind that GOdel and I propose [ 1) that are justified pragmatically as in physics are:
• the P =I= NP hypothesis in theoretical computer science that conjectures that many problems require an exponential amount of work to resolve, and
• the Riemann hypothesis concerning the location of the complex zeroes
of the Riemann zeta function
?cs) =I_!_= II -1 -. n ns p 1 - _l_
pS
(Here n ranges over positive integers and p ranges over the primes. )3 Knowing the zeroes of the zeta function, i.e., the values of s for which ?(s) = 0,
tells us a lot about the smoothness of the distribution of prime numbers, as is explained in these four books:
• Marcus du Sautoy, The Music of the Primes, Harper Collins, 2003.
• John Derbyshire, Prime Obsession, Joseph Henry Press, 2003.
• Karl Sabbagh, The Riemann Hypothesis, Farrar, Strauss and Giroux, 2003.
The Riemann zeta function is like my n number: it captures a lot of information about the primes in one tidy package. n is a single real number that contains a lot of information about the halting problem. 5 And the RH is useful because it contains a lot of number -theoretic information: many number-theoretic results follow from it.
Of the authors of the above four books on the RH, the one who takes Godel most seriously is du Sautoy, who has an entire chapter on Godel and Turing in his book In that chapter on p. 181, du Sautoy raises the issue of whether the RH might require new ax-
'However, new mathematical concepts such as v'=1 and Turing's definition of computability certainly are
judged by their fruitfu lness-Fran(:oise Chaitin-Chatelin, private communication.
2Yet another class of pragmatically justified axioms are the large cardinal axioms and the axiom of determi
nancy used in set theory, as discussed in Mary Tiles, The Philosophy of Set Theory, Chapters 8 and 9. For
the latest developments, see Hugh Woodin, "The continuum hypothesis," AMS Notices 48 (2001 ), 567-576,
681 -690.
3\fou start with this formula and then you get the full zeta function by analytic continuation.
4Supposedly Havil's book is on Euler's constant y, not the RH, but ignore that. Sections 1 5.6, 1 6.8, and 1 6. 1 3
of his book are particularly relevant t o this paper.
5!1 = 'lp ha�s 2-iol is the halting probability of a suitably chosen universal Turing machine. !1 is "incompress
ible" or "algorithmically random." Given the first N bits of the base-two expansion of !1, one can determine
whether each binary program p of size io1 ,; N halts. This information cannot be packaged more concisely. See
ioms. On p. 182 he quotes Godel,6 who specifically men
tions that this might be the case for the RH. And on p. 202
of that chapter, du Sautoy points out that if the RH is un
decidable this implies that it's true, because if the RH were
false it would be easy to confirm that a particular zero of
the zeta function is in the wrong place.
Later in his book, on pp. 256-257, du Sautoy again
touches on the issue of whether the RH might require a
new axiom. He relates how Hugh Montgomery sought re
assurance from Godel that a famous number-theoretic con
jecture-it was the twin prime conjecture, which asserts
that there are infinitely many pairs p, p + 2 that are both
prime-does not require new axioms. Godel, however, was
not sure. In du Sautoy's words, sometimes one needs "a
new foundation stone to extend the base of the edifice" of
mathematics, and this might conceivably be the case both
for the twin prime conjecture and for the RH.
On the other hand, on pp. 128-131 du Sautoy tells the
story of the Skewes number, an enormous number
wwlo34
that turned up in a proof that an important conjecture must
fail for extremely large cases. The conjecture in question
was Gauss's conjecture that the logarithmic integral
Li(x) = Jx du 2 ln u
is always greater than the number 1r (x) of primes less than
or equal to x. This was verified by direct computation for
all x up to very large values. It was then refuted by Little
wood without exhibiting a counter-example, and finally by
Skewes with his enormous upper bound on a counter
example. This raised the horrendous possibility that even
though Gauss's conjecture is wrong, we might never ever see a specific counter-example. In other words, we might
never ever know a specific value of x for which Li(x) is less
than 1r(x). This would seem to pull the rug out from under
all mathematical experimentation and computational evi
dence! However, I don't believe that it actually does.
The traditional view held by most mathematicians is that
these two assertions, P =I= NP and the RH, cannot be taken
as new axioms, and cannot require new axioms, we simply
must work much harder to prove them. According to the
received view, we're not clever enough, we haven't come
up with the right approach yet. This is very much the cur
rent consensus. However, this majority view completely ig-
no res 7 the incompleteness phenomenon discovered by
Godel and Turing, and extended by my own work [2] on in
formation-theoretic incompleteness. What if there is no
proof?
In fact, new axioms can never be proved; if they can,
they're theorems, not axioms. So they must either be justi
fied by direct, primordial mathematical intuition, or pragmatically, because of their rich and important consequences,
as is done in physics. And in line with du Sautoy's observa
tion, one cannot demand a proof that the RH is undecid
able before being willing to add it as a new axiom, because
such a proof would in fact yield the immediate corollary
that the RH is true. So proving that the RH is undecidable
is no easier than proving the RH, and the need to add the
RH as a new axiom must remain a matter of faith. The
mathematical community will never be convinced. 8 Someone recently asked me, "What's wrong with calling
the RH a hypothesis? Why does it have to be called an ax
iom? What do you gain by doing that?" Yes, but that's be
side the point; that's not the real issue. The real question
is, Where does new mathematical knowledge come from?
By "new knowledge" I mean something that cannot be
deduced from our previous knowledge-from what we al
ready know.
As I have been insinuating, I believe that the answer to
this fundamental question is that new mathematical knowl
edge comes from these three sources:
a. mathematical intuition and imagination ( \!=]\ b. conjectures based on computational evidence (explains
calculations), and
c. principles with pragmatic justification, i.e., rich in con-
sequences (explains other theorems). 9
And items (b) and (c) are much like physics, if you replace
"computational evidence" by "experimental evidence." In
other words, our computations are our experiments; the
empirical basis of science is in the lab, the empirical basis
of math is in the computer.
Yes, I agree, mathematics and physics are different, but
perhaps they are not as different as most people think, per
haps it's a continuum of possibilities. At one end, rigorous
proofs, at the other end, heuristic plausibility arguments,
with absolute certainty as an unattainable limit point.
I've been publishing papers defending this thesis for
more than a quarter of a century, 10 but few are convinced
by my arguments. So in a recent paper [ 1] I've tried a new
6Unfortunately du Sautoy does not identify the source of his Gbdel quote. I have been unable to find it in Gbdel's Collected Works.
7 As du Sautoy puts it, p. 1 8 1 , "mathematicians consoled themselves with the belief that any1hing that is really important should be provable, that it is only tortuous
statements with no valuable mathematical content that will end up being one of Gbdel's unprovable statements."
8The situation with respect to P * NP may be different. In a paper "Consequences of an exotic definition for P = NP," Applied Mathematics and Computation 145 (2003), pp. 655-665, N. C. A. da Costa and F. A. Doria show that if ZFC (Zermelo-Fraenkel set theory+ the axiom of choice) is consistent, then a version of P = NP is consistent with ZFC, so a version of P * NP cannot be demonstrated within ZFC. See also T. Okamoto, R. Kashima, "Resource bounded unprovability of compu
9A possible fourth source of mathematical knowledge is (d) probabilistic or statistical evidence: A mathematical assertion may be deemed to be true because the prob
ability that it's false is immensely small, say <1 0-99999
Here is a practical example of this: The fast primality testing algorithm currently used in Mathematica does not necessarily give the correct answer, but mistakes
are highly unlikely. Algorithms of this sort are called Monte Carlo algorithms.
10See, for example, the introductory remarks ·In my 1 974 J. ACM paper [3].
VOLUME 26, NUMBER 1 , 2004 5
tactic. I use quotes from Leibniz, Einstein, and Godel to
make my case, like a lawyer citing precedents in court ... .
Even though I am touting the Riemann hypothesis as an
excellent new-axiom candidate-whether Godel agrees or
merely thinks that a new axiom might be needed to prove
the RH, I'm not sure-let me briefly wax enthusiastic over
a possible approach to a proof of the RH. Disclaimer. I'm
not an expert on the RH. What I'm about to relate is defi
nitely an outsider's first impression, not an expert opinion.
A Possible Attack on the Riemann Hypothesis?
Here is a concrete approach to the RH, one that uses no
complex numbers. It's a probabilistic approach, and it in
volves the notion of randomness. It's originally due to Stielt
jes, who erroneously claimed to have proved the RH with
a variant of this approach.
The Mobius JL function is about as likely to be + 1 or - 1
(see Derbyshire, Prime Obsession, pp. 322-323).
{ 0 if k2 divides n, k > 1, f.L(n) = b rct·rr · ct· · r ( _ 1 )num er o 1 erent pnme tvtsors o n if n is square-free. The RH is equivalent to the assertion that as k goes from 1
to n, JL(k) is positive as often as negative. More precisely,
the RH is closely related to the assertion that the difference
between
• the number of k from 1 ton for which JL(k) = -1, and
• the number of k from 1 to n for which JL(k) = + 1
is O(Vn\ of the order of square root of n, i.e., is bounded
by a constant times the square root of n. This is roughly
the kind of behavior that one would expect if the sign of
the JL function were chosen at random using independent
tosses of a fair coin. u
This is usually formulated in terms of the Mertens func
tion M(n): 12
n M(n) = .2: p.,(k).
k�l
According to Derbyshire, pp. 249-251,
M(n) = O(Vn) implies the RH, but is actually stronger than the RH. The
RH is equivalent to the assertion that for any E > 0, I
M(n) = O(n2+E).
Could this formula be the door to the RH?!
This probabilistic approach caught my eye while I was
reading this May's crop of RH books.
I have always had an interest in probabilistic methods
in elementary number theory. This was one of the things
that inspired me to come up with my definition of algorithmic randomness and to find algorithmic randomness
in arithmetic [6] in connection with diophantine equations.
However, I doubt that this work on algorithmic random
ness is directly applicable to the RH.
In particular, these two publications greatly interested
me as a child:
• Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Mathematical
Monographs, vol. 12, Mathematical Association of Amer
ica, 1959.
• George P6lya, "Heuristic reasoning in the theory of num
bers," 1959, reprinted in Gerald W. Alexanderson, The Random Walks of George P6lya, Mathematical Associa
tion of America, 2000.
I think that anyone contemplating a probabilistic attack on
the RH via the JL function should read these two publica
tions. There is also some interesting work on random
sieves, which are probabilistic versions of the sieve of
Eratosthenes:
• D. Hawkins, "Mathematical sieves," Scientific American, December 1958, pp. 105-112.
As P6lya shows in the above paper-originally American Mathematical Monthly 66, pp. 375-384-probabilistic
heuristic reasoning can do rather well with the distribution
of twin primes. By the way, this involves Euler's y constant.
Can a refmement of P6lya's technique shed new light on JL and on the RH? I don't know, but I think that this is an in
teresting possibility.
By the way, P :1: NP also involves randomness, for as
Charles Bennett and John Gill showed in 1981-SJAM Journal on Computing 10, pp. 96-113-with respect (relative)
to a random oracle A, pA :1: NPA with probability one [7].
Further Reading-Four "Subversive" Books
• On experimental mathematics:
Borwein, Bailey, and Girgensohn, Mathematics by Ex
periment, Experimentation in Mathematics, A. K. Peters, 2003.
(See [8]. There is a chapter on zeta functions in volume
two.)
• On a quasi-empirical view of mathematics:
Tymoczko, New Directions in the Philosophy of Mathematics, Princeton University Press, 1998.
• On pragmatically justified new axioms and information
theoretic incompleteness:
Chaitin, From Philosophy to Program Size, Tallinn Cy
bernetics Institute, 2003.
(There is also an electronic version of this book [2].)
And regarding the adverse reaction of the mathematics
community to the ideas in the above books, I think that it
is interesting to recall Godel's difficulties at the Princeton
Institute for Advanced Study, as recounted in:
11 For a more precise idea of what to expect if the sign of the IL function were chosen at random, see the chapter on the law of the iterated logarithm in Feller, An In
troduction to Probability Theory and Its Applications, vol. 1 , Vlll.5 through VII I .?. 12See [4, 5].
6 THE MATHEMATICAL INTELLIGENCER
• John L. Casti, The One True Platonic Heaven, John
Henry Press, 2003.
According to Casti, one of the reasons that it took so long
for Godel's appointment at the lAS to be converted from
temporary to permanent is that some of Godel's colleagues
dismissed his incompleteness theorem. Now, of course,
Godel has become a cultural icon13 and mathematicians
take incompleteness more seriously-but perhaps not seri
ously enough.
Mathematicians shouldn't be cautious lawyers-! much
prefer the bold Eulerian way of doing mathematics. Instead
of endlessly polishing, how about some adventurous pioneer
spirit? Truth can be reached through successive approxi
mations; insistence on instant absolute rigor is sterile
that's what I've learned from incompleteness.14
WEB REFERENCES ( 1 ] Two philosophical applications of algorithmic information theory.
131n this connection, I should mention Incompleteness, a play and a theorem by Apostolos Doxiadis, which is a play about Gbdel. For more information, see [9]. 141n this connection, see da Costa and French, Science and Partial Truth, Oxford University Press, 2003.
Solution Kept Secret
VOLUME 26, NUMBER 1 , 2004 7
EUGENE GUTKIN
The Toeplitz-Hausdorff Theorem Revisited: Relating Linear Algebra and Geometry
Genesis
In the beautiful paper [24] 0. Toeplitz associated with any
complex n X n matrix a compact set in the complex plane.
As his title suggests, he was inspired by a theorem of L. Fe
jer [6] concerning a relationship between planar curves and
Fourier series. Apart from this, the paper [24] is self
contained. Let en be the standard vector space with the
scalar product <u, v>. I will not distinguish between the
n X n matrices and operators on en. Let C be one such. It is determined by its "bilinear" form <u, Cv>. The compact
set that Toeplitz introduces is the image, W = W(C) C e, of the unit sphere in en, under the quadratic map u �
<u, Cu>. He cof\jectures that W(C) is a convex set, and
proves that the outer boundary of W(C) is a convex curve.
A year later F. Hausdorff proved Toeplitz's col\iecture
[12]. The Toeplitz-Hausdorff theorem was born. For several
reasons, it continues to attract the attention of researchers.
Extensions of Toeplitz's setting came up in robust control;
hence the thriving engineering literature on the subject.
See [20, 21, 5]. My own preoccupation with the Toeplitz
Hausdorff theorem has its genesis in a joint project with
electrical engineers [15, 10]. Despite (or because of) the simplicity of the Toeplitz
Hausdorff framework, basic questions in the subject remain
open [14]. For instance, it is not known what domains are
realizable as W(C) for C on en. The present article aspires
to attract attention of the general mathematical readership
to the fascinating interplay of linear algebra, geometry, and
analysis that the papers [24, 12] initiated.
My plan is as follows. I analyze in some detail the original
papers of Toeplitz and Hausdorff. Then, following the view
point of [24], I associate with an arbitrary C a linear pencil
of hermitian operators H( · ) . This allows me to cast the analy
sis into the language of convex geometry: Support lines and
support functions come in. The crucial observation is that the
support function of W(C) is the highest eigenvalue, A(·), of
H( · ) . This brings in both the algebraic geometry and the con
vex geometry. R. Kippenhahn was the first to exploit this ob
servation. In his Dissertation [16] he introduces and develops
this point of view. To illustrate this approach, I immediately
derive rough bounds on the size of W( C) in terms of the spec
tral attributes of C. I also reproduce without proof the much
more sophisticated estimates of Kippenhahn [16]. Then I bring in the differential geometry by calculating
the curvature of the boundary curve aW(C). To show the
usefulness of this viewpoint, I apply it to obtain new bounds
on the size of W( C) in terms of the standard attributes of
C. These estimates, although still very crude, are sharper
than those I got out of the support function. The differen
tial geometry viewpoint turned out to be especially suitable
to study the multidimensional version of W( C), the joint nu
merical range [10].1 I conclude with a brief survey of the
literature and a personal remark
I thank the referees for helpful comments.
Historical Remarks
Toeplitz coined the name "Wertvorrat" for W(C). A literal
English translation is the value supply or the stock of values. Variations of "Wertvorrat" dominate the German liter
ature on the subject. For instance, A. Wintner, during the
Leipzig period of his prolific career, used the expressions
"Wertevorrat" (values supply) and "Wertbereich" (value domain) [26].2
The modern literature intermittently uses field of values
1There are many generalizations of the numerical range of an operator in the modern literature. It would take several pages just to give the relevant definitions. The con
cept of the joint numerical range and the awareness that it is the natural multi-dimensional extension of the numerical range is already in the founding papers [24, 1 2).
2Wintner emigrated to America shortly after the University of Leipzig refused to award hirn the Habilitation. The book [26) is apparently his Habilitationschrift.
and numerical range. s I don't like either expression. The
former adds one more item to the litany of mathematical "fields"; the latter is plain awkward. The original name is
better in every respect except one: It is German and therefore unacceptable in the English literature. 4 Some proposed alternatives (template, form range, contracted graph) did not fly. I fmd the expression numerical range the lesser of two evils, and I will use it in what follows. 5
Toeplitz proves several propositions relating W( C) and the spectrum of C. For instance, he shows that W(C) contains the spectrum, and if C is a normal operator, then W( C)
is the convex hull of the spectrum. But the centerpiece of [24] is "Satz 8," the convexity of the outer boundary.
The penultimate §5 of [24) offers several informal comments, and points out the difference between convexity of the outer boundary and convexity of the set. Then Toeplitz says: "I will now discuss a generalization of the entire setting, which . . . also shows the difficulties that stem in the general case from the possibility of holes." He goes on to introduce what is now called the joint numerical range of any number q of hermitian operators A1, .. . , Aq. The set in question, W.(A1, . .. , Aq) c !Rq, is the image of the unit sphere in en, under the map u f-i> ( <u, A1u>, . . . <u, Aqu> ). The decomposition C = A1 + iA2 implies W(C) = W.(A1,A2).6 Toeplitz demonstrates that W.(Al, . . . , Aq) is not convex, in general. He concludes: "Whether this can already happen for q = 2 remains possible, athough unlikely."
Toeplitz missed that he actually proved the desideratum! Indeed, to a modem reader, it seems that Toeplitz essentially settled the convexity conjecture. To us, it suffices to prove it for n = 2; for, if <u, Cu> and <v, Cv> belong to W(C), and the numerical range of the restriction of the form C to eu + ev is convex, then the claim holds. And in §5
Toeplitz shows that the numerical range of an operator on e2 is either an elliptic disc, or a segment, or a point-in each case, it is convex! In fact, this is how the ToeplitzHausdorff theorem is proved in modem textbooks [9, 11, 14].1 Amazingly, in the 80-some years since [24), nobody, including Hausdorff, noticed that the Toeplitz-Hausdorff theorem is implicitly proved in [24).
In the 3-page-long, focused, beautiful paper [ 12), Hausdorff proves Toeplitz's conjecture. On the one hand, he proves it from scratch, without using Satz 8 of [24). On the other, he goes just a step further than Toeplitz to show that the intersection of W(C) with any straight line is the image of a connected subset of the unit sphere under a continuous mapping, and hence is connected. 8 In a one-sentence remark Hausdorff points out that his results and the Toeplitz argument combine to prove the convexity of the
outer boundary of the surface W.(Al, A2, As) for any triple
of hermitian operators.
A natural generalization of the Toeplitz-Hausdorff theorem would have been the convexity of W.(A1, . . . , Aq) for all hermitian operators on any en. Although this claim is ''very false" [ 1 1), W.(A1, A2, As) for any triple A1. A2, As on en is convex if n 2:: 3. Remarkably, it was established 60 years after the papers [24, 12]! There are several proofs of this in the literature [10], and some are based on the Hausdorff connectedness idea [5] . The convexity claim for W.(Al, A2, As, A 4) for operators on en fails for any n [5]. Although this is unfortunate from the engineering viewpoint [21], there
are nontrivial interpretations of this "phase transition" [10].
But let us return to the subject. How could it be that neither Toeplitz nor Hausdorff realized that [24] contained a proof of the convexity of the numerical range? It is quite likely that Hausdorff overlooked the relevant part of [24] .
However, the Commentary by S. D. Chatterji in Hausdorffs Collected Works [ 13] reveals a curious fact in this respect. The Hausdorff Archives in Greifswald contain two handwritten notes for [ 12), dated September 19 and October 12
of 1918. In one of them Hausdorff works out the numerical range of any two-by-two matrix. He shows, as Toeplitz had already done, that it is a (possibly degenerate) ellipse.
Bringing in the Geometry
My interpretation of the approach of [24] is as follows. Let C be an n X n matrix, and let W(C) be the numerical range. Toeplitz associates with C a linear pencil of hermitian operators H( · ), parametrized by the circle of directions. The highest eigenvalue, A ( · ), of H( · ) is the support function of W(C). I will now explain this in detail.
Let <u, v> denote the standard scalar product on en, linear (resp. antilinear) in the second (resp. first) argument. As usual, I lull = V <u, u>. Let C be an operator on en with the adjoint C*, and let
C =A+ iB: A*= A, B* = B (1)
be the decomposition into hermitian operators. For 0 :S t < 27T set
The space of rays (i.e., oriented lines) in !R2 is parametrized by S1 X IR [22]. Namely, the ray r(t, p) has direction t, and the signed distance p from the origin. The notion of support lines is well known [1, 22]. I will associate with any compact set, XC IR2, the family, u(t), 0 :S t < 27T, of its support rays. For any 0 :S t < 27T the set of p E IR such that
3See [9] for historical comments on this terminology. The name "numerical range" is due to M . H. Stone [23].
4The German-English hybrids "eigenvalue, eigenvector" are the fortunate exceptions .. . . I don't know who coined them or how, but I am happy that I don't have to
use the awkward "proper value, proper vector, characteristic number," etc.
51t could have been worse. F. D. Murnaghan refers to W(C) as " ... the region of the complex plane covered by these values under the hypothesis that .. . " [1 8].
6Thus, the patent on the joint numerical range belongs to Toeplitz and not to Hausdorff [5].
7 A proof of the Toeplitz-Hausdorff theorem based on this idea is due to W. F. Donoghue [4). He explicitly calculates the ellipse in question. An elegant calculation of
aW(C) if n :s 3 is due to Murnaghan [1 8). Although he points out that aW(C) is an ellipse when n = 2, Murnaghan is not concerned with the region W(C) itself.
8Hausdorff's elegant argument is limited to finite dimensions, because he diagonalizes hermitian operators. The extension of the Toeplitz-Hausdorff theorem to infinite
dimensions is due to Stone [23). See [1 1 ) for a proof of N. P. Dekker [3) that combines Hausdorff's idea with the reduction to IC2
VOLUME 26, NUMBER 1, 2004 9
<3:>
Figure 1 . Support rays and the eigenvalues.
r( t, p) intersects X is compact; let p( t) be the maximal such p. Then a{t) = r(t,p(t)) is the support ray of X in ilirection t.
The following proposition is essentially Satz 8 of [24].
Proposition 1. Let C = A + iB be an operator on en and let H(t) = (cos t)A + (sin t)B, 0 :o; t < 27T, be the associated pencil of hermitian operators. Let
(3)
be the eigenvalues of H(t), and let Ei(O) c en be the eigenspace9 corresponding to Ai(O).
Let u(t), 0 ::5 t < 27T, be the support rays of W(C). Then the intersection point of a(t + 7T/2) with r(t,O) is A1(t)(cos t, sin t). Using this point as the origin in u(t + 7T/2), identify a{t + 7T/2) with R Then a{t + 7T/2) n W(C) C IRis the convex hull of the spectrum of the form H(t + 7T/2) restricted to E1(t).
Proof The unit circle acts on operators, C � e-iac, and on e, by rotations. The statement is equivariant with respect to these actions. Therefore, it suffices to verify the claims for the direction t = 0. We have H(O) =A, H(7T/2) = B, the ray r(O, 0) is the x-axis, and u( 7T/2) is the vertical ray supporting W from the right. See Figure 1. The points z = x + iy of the numerical range have the form z = <u, Cu>, llu ll = 1. By (1), x = <u, Au>, y = <u, Bu>. Therefore, the projection of Won the horizontal axis is the interval [An(A), A1(A)]. The right extremity of this interval is the intersection point with the ray lT( 7T/2). This proves one claim.
The intersection of lT( 7T/2) with W is given by
z = {<u, Au> + i <u, Bu>: llull = 1, <u, Au>= A1(A)}.
9Another fortunate hybrid!
10 THE MATHEMATICAL INTELLIGENCER
y
u(i)
1(A)
In view of the above, our subset of IR is formed by <u, Bu>, where u runs through the unit sphere in E1(A). The numerical range of an hermitian operator is the convex hull of its spectrum. This proves the other claim. •
Proposition 1 has several far-reaching consequences. First of all, it implies that the outer boundary aW(C) is convex [24]. Second, it describes the support rays of W(C) via the eigenvalues of the hermitian pencil H( · ) of (2). These support rays determine the convex hull of aW(C). Since W(C) is convex, as we now know, they determine the set W(C) itself. Thus, Proposition 1 yields a description of the numerical range of C in terms of the spectrum of the associated pencil H(·).
Since the publication of [24], many authors have developed this observation in several directions. One of these directions may be called algebra-geometric. Its starting point is the algebraic curve
det(xA + yB + zi) = 0. (4)
This paper exploits another direction, which may be called "proper geometric." It takes off with an immediate corollary of Proposition 1. To formulate it, I will recall the notions of the support function and the width function of a convex set [1, 22]. Let X c lh£2 be convex and compact, and let lT(t), 0 :o; t :o; 27T, be the support rays of X. The distance between the parallel lines lT(t), lT(t + 1r) is the width of X in direction t. The support function is the signed distance of lT( t) to the origin. Denote the support and the width functions by h(t) and w(t), respectively. Then w(t) = h(t) + h(t + 7T).
Corollary 1. Let C be an operator on en, let H(·) be the associated pencil of hermitian operators, and let An(·) :S
· · · ::::; A 1 ( ·) be the eigenvalues of H(-). Then the support and the width of the numerical range of C are
Proof. Proposition 1 yields the first claim. The second follows from the first and the identity H( t + 1r) = -H( t). •
Although the Toeplitz paper [24] is the precursor of both geometric directions, it was the work of R. Kippenhahn [ 16] that explicitly gave birth to them. 1 0 From now on I will concentrate on the proper geometric direction, referring the reader to the literature on the algebra-geometric direction. See, for instance [19].11
I will now use Corollary 1 to estimate the size of the numerical range of C in terms of the standard attributes of the operator C. The size of a planar convex compact set X is expressed via its area, diameter, breadth, and perimeter. Let w( ·) be the width of X. The breadth and the diameter of X are the minimum and the maximum of w, respectively. The perimeter and the area of X are also controlled by the width function [ 1]. If X = W( C), then w( ·) is determined by the spectrum of the hermitian pencil H( ·) which, in tum, is determined by the operator C. Among the standard attributes of C are its spectrum a( C) and the operator norm jcj. The number w(C) = maxAwCC)( IAi - A1} is the diameter of the spectrum.
For any a, b E I[ W(aC +b)= aW(C) + b.
Hence the size of the numerical range does not change under the transformations C � C + tl. Denote by Jtn the linear space of operators on en, and let .M� c Mn be the subspace of traceless operators. The function jCio = mintE<C jc + t� is a norm on the quotient space Jtn/{al}. The projection Co = C - tr(C)In I induces a linear isomorphism of Mn/{al) and Ml Note that jcj0::::; jC0j, and for generic C the inequality is strict. The following very rough estimates are essentially contained in [24].
Corollary 2. Let C be an operator on IC", and let W be 'its numerical range. Then
AreaCW) ::::; 4lcl6, Perimeter(W) ::::; 8jCjo;
w(C) :S Diameter(W) :S 2jC!o,
(6)
(7)
Proof For 0::::; t::::; 21r let R(t) be the rectangle formed by the four supporting rays. See Figure 2. Since w(H(t)) = A1(t) - An(t) and jH(t)j = max{ jA1(t)j, jAn(t)j }, we have
Using that jH(-)j ::::; lei, and the invariance of the preceding argument under C � C + al, we obtain (6). The upper bound in (7) follows from w(H(t)) ::::; 2jH(t)i :S 2jCj and the invariance principle. The obvious fact that W contains the spectrum of C implies the lower bound. •
The roughness of the estimates in Corollary 2 occurs for two reasons, one geometric and one analytic. The geometric reason is that W is much smaller than the circumscribed rectangles R(t). The analytic one is that the bounds A1(t) -An(t) :S 2jH(t)l ::::; 2jCj are very crude. Using convex geometry and subtle but elementary analysis, Kippenhahn obtained much better estimates [16]. Although the restriction tr C =
0 that [16] imposes, can be removed and the inequalities further improved, I will only state the relevant results of [16]. THEOREM 1. Let c be an operator on en such that tr c = 0, and let W = W(C) be its numerical range. Then
_!_ Vtr2(CC*) - itr C2j2::::; Area(W) ::::; n
(8) 2(n - 1)
Vtr2(CC*) - itr C2j2; n
4 Vn Vtr(CC*)
::::; Perimeter(W)::::; 21r � Vtr(CC*). (9)
The following corollary demonstrates the strength of Theorem 1. Corollary 3. The numerical range of a matrix C has empty interior iff C = a + bH, where H is hermitian.
I leave the proof to the reader, as an exercise. (Hint: Use (8) and the Cauchy-Schwarz inequality.) If C = a + bH, and H is hermitian, then C is normal. Hence, the numerical ranges of non-normal matrices have positive area.
Lest the reader think that the spectral properties of C matter only for the size of W(C), I hasten to add a few comments. Let X c IC be convex and compact. A point z E ax is a corner point12 if X has more than one support line at z. An eigenvalue A of C is normal if there exists an eigenvector v with the eigenvalue A such that Cl'v = Av. The following theorem [16, 4] gives an example of a completely different relation between the spectral characteristics of a matrix and the geometry of its numerical range. 1 3
THEOREM 2. Let c be an operator on en, and let w = W(C) be its numerical range. Then the corner points of W are among the normal eigenvalues of C.
Bringing in the Differential Geometry
An arbitrary convex compact W c I[ is determined by its support function. If the boundary a W is (piecewise) twice differentiable, then W is also determined by the curvature
1ilThe note [1 8] contains a few beautiful remarks about the algebraic geometry of iJW(C), but it does not pursue the matter.
11Complains about the scarcity of citations of Kippenhahn's work.
12Sharp point in the engineering literature. 1 3See [ 15] for a differential-geometric proof.
VOLUME 26, NUMBER 1 , 2004 1 1
y
u(t + 1r
Figure 2. Numerical range enclosed within a rectangle.
u(t
function, x(") 2: 0. As opposed to the support function, the
curvature is intrinsically defined by aw. The radius of curvature p(·) = x-1(·) is sometimes handier to use. Now let
W be the numerical range of a finite-dimensional operator,
C. Let H(·) be the corresponding pencil of hermitian oper
ators. By Corollary 1, the support function of W is the high
est eigenvalue A(·) of H(·). I will now express the radius of
curvature of aW in terms of A(·). A matrix is often called regular if its eigenvalues are
simple.
Definition 1. Let C be an operator on en, and let H(t), 0 :s t :s 27T, be the corresponding pencil of hermitian operators. Then C is Toeplitz regular if for all 0 :s t :s 27T the maximal eigenvalue of H(t) is simple.
THEOREM 3. Let W c 1Ri2 be the numerical range of an operator C on en. Let H(·) be the associated pencil of hermitian operators and let A(·) be the maximal eigenvalue of H( · ).
Suppose that C is Toeplitz regular. Then the junction A(·) is infinitely differentiable, and A + A" > 0. The set W is strictly convex, the boundary aw is twice differentiable, and its radius of curvature satisfies
p(t + 7T/2) = A(t) + A"(t). (10)
Proof Denote by E(t) c en the eigenspace of H(t) corre
sponding to the maximal eigenvalue. Let e E E(O) be a unit
vector. Then there is a unique vector function v(t), 0 :s t :s
27T, such that E(t) = Cv(t), llvCOII = 1, v(O) = e, and <v(t), v'(t)> = 0.14
141n general, v(21T) = {3v(O). The factor {3 has to do with Berry's phase.
1 2 THE MATHEMATICAL INTELLIGENCER
u(t + 37r/2)
By construction
H(t)v(t) = A(t)v(t). (11)
Differentiating this equation twice yields
and
(H' - A')v + (H - A)v' = 0 (12)
(H'- A")v + 2(H' - A')v' + (H - A)v" = 0. (13)
But H satisfies H' = -H. Substituting this into (13),
(A + A")v = 2(H' - A')v' + (H - A)v". (14)
Take the scalar product of (14) with v(t). Equation (12) im
plies
A + A"= 2 <v', (A - H)v'> .
But A is the top eigenvalue of H and v' is perpendicular to
its eigenspace E(A). Hence
A + A"> 0. (15)
Denote by ;£ the ray family (O"(t), 0 :s t < 27T}, where O"(t) has direction t + 7T/2 and intersects r(t, 0) at the point
A(t)(cos t, sin t). The positivity condition (15) implies that
the envelope, A(;£) C C, is a strictly convex curve, with the
parametric equations
x(t) = A(t) cost- A'(t) sin t, y(t) = A(t) sin t + A'(t) cost. (16)
Moreover, A(:£) is twice differentiable, and its radius of cur
vature is given by (10) [22, 1]. Since, by Proposition 1, :£ is
the family of support rays of W, we have A(:£) = aw. • Not every operator C on en is Toeplitz regular. If C is
normal, then W(C) is a polygon, hence it is not strictly con
vex. By Theorem 3, normal matrices are not Toeplitz reg
ular. In fact, by Theorem 2, the non-regularity of W(C) al
ways has to do with a partial normality of C. Fortunately,
there are plenty of Toeplitz regular operators.
Proposition 2. The complement to the set of Toeplitz regular operators in _Mn is contained in a closed hypersurface. Proof. Let �n denote the space of n X n hermitian opera
tors. By (1), _Mn =�nEB i�n. Replacing cost, sin t in (2) by
independent variables, we obtain an algebraic mapping, <p, from _Mn into the algebraic variety Gz(�n) of subspaces in
�n of dimension at most 2. The set of hermitian operators with multiple eigenval
ues is an algebraic variety, Xn C �n, of codimension 3. Therefore, the set of L E G2(�n), such that L n Xn i= 0 is a codimension one subvariety, Yn C G2(�n). Since
<p : _Mn >---7 G2C�n) is surjective, the preimage <p-1(Yn) C _Mn
is a hypersurface. But the complement of the set ofToeplitz
regular operators belongs to <p -1(Yn). • The following is immediate from Proposition 2.
Corollary 4. The set of Toeplitz regular operators on en is open and dense.
I will now use Theorem 3 to sharpen the bounds on the
size of the numerical range. Moreover, I will do it for
bounded operators on any Hilbert space �. Recall that if
dim � = oo and C is a bounded operator on it, the numeri
cal range W(C) c Cis bounded and convex [23], but not
necessarily closed. The operator norm C] and the reduced
operator norm ICio = mintEdC + til have the same basic
properties as in the case dim � < oo.
THEOREM 4. Let C be a bounded operator on any Hilbert space, and let W be its numerical range. Then
Perimeter(W) ::5 2'7TIC]o, Area(W) ::5 '7TIC]5. (17) Proof Let � be the Hilbert space where C acts. Assume
first that dim � < oo. Then � = en, and I will use the pre
ceding material. Let H( t), 0 < t < 2'7T, be the corresponding
pencil of hermitian operators and let A(·) be the highest
eigenvalue of H(·). Suppose first that C is Toeplitz regular.
From Theorem 3 and standard differential geometry [1, 22] we have
we conclude that (20) holds for C. Now use again the in
variance under C >---7 C + ti. •
Concluding Remarks
Although the bounds of Theorem 4 improve those of Corol
lary 2 by the factor of 411T, they are still very rough. The
same or better bounds on the size of the numerical range
W( C) can be obtained using elementary geometry. Let X C C be compact. Denote by r(X) the numerical radius of X, i.e., the radius of the smallest disc D(X), centered at (0,0) and containing X. Toeplitz proved in [24] that
19_::::; r(W(C)) ::::; lei. 2
(21)
Since W(C) c D(W(C)), (21) implies (20) and the inequal
ity Diameter(W) ::::; 2ICI. Invoking the invariance principle,
we obtain (17) and the upper bound of (7).
Set W1(C) = {zl- z2 : Z1, Zz E W(C)}. The set W1(C) C (: is symmetric about the origin and convex and satisfies [25]
W1(C) = { <u, Cv> + <v, Cu>: llull = llvll = 1,
This implies
Diameter(W(C)) =
<u, v> = 0}. (22)
max I <u, Cv> + <v, Cu> 1. (23) llull=llvJI= l,<u,v>=O
This in tum yields the bounds
Diameter(W(C))::::;
max 21 <u, Cv> I ::::; 2ICI. (24) llull=llvll= l,<u,v>=O
Invoking the same invariance principle, we obtain from (24) the upper bound of (7). There are other approaches to es
timating the size of W(C). For instance, [2] employs the
Gershgorin disc theorem to obtain quadratic bounds on the
area of W(C) for certain nilpotent matrices.
In view of these results and those of [16), of course, the
main justification of Theorem 3 is not in the bounds on the
size of the numerical range that it yields. The justification
is the elegant formula (10) for the curvature of the bound-
VOLUME 26, NUMBER 1, 2004 1 3
AUTHOR
&UOENE OUTKIN
ary of the numerical range. The estimates ( 17) follow from it by very crude estimates. The formula (10) seems to be novel. My only "precursor" M. Fiedler identified in spectral terms the boundary curvature of numerical range in special cases [7, 8]. There is no immediate relationship between his formulas and (10). I hope that (10) will find other applications to the remarkable subject that grew out of the Toeplitz-Hausdorff theorem.
It goes without saying that geometric considerations pervade the literature on numerical range. Several researchers have used the ideas above for purposes other than estimating the size of W(C). For instance, in [17] (16) helps to uncover new examples of domains satisfying the famous "porism of Poncelet."15
Before stopping, I will give unsolicited advice to the reader. There is a pervasive custom of concentrating on the latest literature while doing research. I am no exception to this rule. However, my experience with the study of numerical range brought me to the conclusion:
It is useful to read the work of "founding fathers"!
15A related way of using the numerical range to construct such examples is pre
sented in [27].
14 THE MATHEMATICAL INTELLIGENCER
REFERENCES [ 1 ] T. Bonnesen and W. Fenchel, Theorie der konvexen K6rper,
Springer-Verlag, Berlin, 1 97 4.
[2] M.-T. Chien, Y.-H. Lin, On the area of numerical range, Soochow
J. Math. 26 (2000), 255-269.
[3] N. P. Dekker, Joint numerical range and joint spectrum of Hilbert
space operators, Dissertation, Free University of Amsterdam , 1 969.
[4] W. F. Donoghue, Jr. , On the numerical range of a bounded oper
ator, Mich. Math. J. 4 (1 957), 261 -263.
[5] A. Feintuch and A. Markus, The Toeplitz-Hausdorff theorem and
This tale, like so many in mathematics, begins with a simple ques
tion, answers it, and ends with questions that have yet to be resolved.
Fred Richman in 1965 wondered whether it is possible to cut a square into an odd number of triangles of equal areas. The key word here is "odd," for a moment's reflection shows that a square can be cut into any even number of triangles of equal areas.
Before I go on to describe the research that grew out of that question over the last third of a century, I will stop to introduce a few terms for the sake of clarity.
A dissection of a polygon into triangles of equal areas I will call an equidissection. An equidissection into m triangles I call an m-equidissection. An m-equidissection with m odd will be called an odd equidissection, and with m even, an even equidissection. Richman was asking whether every
equidissection of a square is even.
Richman's colleague, John Thomas, became interested in the problem and proved that there is no odd equidissection of a unit square in standard position in the xy-plane if the coordinates of the vertices of the triangles are rational with odd denominators. When he submitted his work to Mathematics Magazine, "The referee thought the problem might be fairly easy (although he could not prove it) and possibly well-known (although he could find no reference to it)." The referee suggested that Thomas submit it as a Monthly problem and if no one solved it, the paper should be published. It appeared in 1968 [12].
Paul Monsky in 1970 [4], building on Thomas's proof, showed that the answer to Richman's question is, "No, there is no odd equidissection of a square." His argument uses two tools, Spemer's Lemma from combinatorial topology, and 2-adic valuations from algebra. I will describe both.
In 1928 Emanuel Spemer published a theorem which he used to prove sev-
eral topological theorems, including the fact that a ball of dimension n is not homeomorphic to a subset of a lower dimension space. It was soon applied by others to give a short proof of Brouwer's fixed-point theorem. He stated it for simplices in all dimensions, but I will present it just for polygons in the xy-plane.
Consider a polygon cut into triangles. For simplicity, assume that two triangles that touch each other intersect either in a complete edge of both or in a vertex of both. All the vertices are labeled A, B, or C. Figure 1 is an example.
l@ldii;IIM
An edge of a triangle whose ends are labeled A and B will be called complete. A triangle whose vertices are labeled A, B, and C will also be called complete. Spemer's reasoning shows that the number of complete edges on the boundary of the polygon has the same parity as the number of complete triangles. In Figure 1 the respective numbers are 3 and 9. In particular, if there are an odd number of complete edges on the boundary there must be at least one complete triangle. That implication is what Spemer used, and so will we.
The other tool is a 2-adic valuation, <P, which is a function defined on the real numbers. First, for a non-zero integer n, <P(n) is the number of 2's in the prime factorization of n. If a and b are non-zero integers, <P( alb) is defined as <P(a) - <P(b). At 0, <P is set equal to oo.
It follows from these properties that for <P(x) < <P(y), <P(x + y) = <P(x). As examples we have <P(1) = 0, <P( -1) = 0, <P( -x) = <P(x), <P(112) = -1, <P(V3/2) =
-1, and <P(\12) = 112. For each prime p there are p-adic
valuations defined in a similar manner. A more detailed treatment of Spemer's Lemma and p-adic valuations is to be found in Chapter 5 of [9].
With the aid of 2-adic valuations, we can divide the xy-plane into three sets, which I will call A, B, and C. The set A consists of the points (x,y) for which both <P(x) and <P(y) are positive. B consists of the points (x,y) for which <P(x) ::::; 0 and <P(x) ::::; <P(y). C consists of the remaining points, namely, those (x,y) for which <P(y)::::; 0 and <P(y) < <P(x). We label each point A, B, or C, depending on which of the three sets contains it. For instance (0,0) is labeled A, (1,0) is B, (1,1) is B, and (0,1) is C. The point (\13/2, \13/2) is labeled B. It is easy to check that translating a point by a point labeled A (viewing them for a moment as vectors) does not change the label: if P is a point, then P and P - A have the same labels.
One of the key tools, going back essentially to Thomas, is that if the three vertices of a triangle have all three labels, A, B, and C, then the valuation of the area of the triangle is less than or equal to -1. To show this, first translate the complete triangle by the vertex labeled A. Let us say that the three vertices are now (0,0), (a,b), and (e,d), with (a,b) labeled B and (e,d) labeled C. The area of the triangle is the absolute value of (ad - be )/2. Since <P(a) ::::; <P(b) and <P(d) < <P(e), <I>( ad) = <P(a) + <P(d) < <P(b) + <P(e) = <P(be), Thus <P(ad - be) = <P(ad)::::; 0, and it follows that <P((ad - be)/2) = ::::; -1.
Note that as a consequence a line in the xy-plane cannot meet all three sets A, B, and C.
Now consider an m-equidissection of a polygon of area A. Assume that at
18 THE MATHEMATICAL INTELLIGENCER
least one of its m triangles is complete. Since the area of the triangle is Aim, it follows that <P(Aim) ::::; -1.
Thus <P(A) - <P(m)::::; -1, from which we conclude that <P( m) ::::: <P(A) + 1.
So, if we knew that <P(A) is larger than -1, m must be even. In particular, if A is an integer, m is even.
To apply the information just obtained, we have to be sure that there is at least one complete triangle in the dissection. This is where Spemer's Lemma enters the picture.
After just one more observation, we will be ready to prove the RichmanThomas-Monsky theorem.
Consider a finite set of points in a complete line segment. Each of these points is labeled either A or B. The points divide the segment into shorter sections. The number of these sections that are complete must be odd. One way to show this is to drop pebbles in each section next to an end labeled A,
and then add them up in two ways, by points and by sections.
When a line segment that is not complete is divided into sections, the number of complete sections is always even. For instance, there are no complete sections when a segment with ends B and C (or A and C) is cut into sections, because no line meets all three sets, A, B, and C. A segment with ends A and A (or B and B) has an even number of complete sections.
With all the machinery in place, we are ready to prove that a square has only even equidissections.
Consider an equidissection of a square. It is no loss of generality to assume that the area of the square is 1 and that its vertices are (0,0), (1,0), (0,1), and (1,1). This is shown in Figure 2, with the labels of its four vertices.
lplijiJ;IfM
C (O, l )b
A (0, 0) B ( l , 0)
All the vertices, not just the comers of the square, are labeled A, B, or C. No matter how those vertices are labeled, there will be an odd number of complete sections along the bottom edge of the square, which is complete. The other three edges have no complete sections. Thus the total number of complete sections on the boundary of the square is odd. Hence there is at least one complete triangle in the dissection. It follows that the equidissection is even.
That is where the subject of equidissections remained until 1979, when David Mead [3] obtained a generalization from a square to a cube in any dimension. He proved that when an n
dimensional cube is divided into
simplices all of which have the
same volume, the number of the
simplices must be a multiple of n! .
In addition to Spemer's Lemma in higher dimensions, he used p-adic valuations for all primes p that divide n.
In 1985, when Elaine Kasimatis was presenting the result for a square in G.
Donald Chakerian's geometry seminar, I wondered, "What about the regular pentagon?" She found the answer and went on to prove that in any m-equidissection of a regular n-gon with at least
five sides, m must be a multiple of
n. In the proof she had to extend p-adic valuations to the complex numbers for the prime divisors of n. Her work appeared in 1989 [1]. In a sense it was another generalization of the theorem about equidissections of a square.
A year later she and I published [2] the results of an investigation of equidissections of trapezoids and other quadrilaterals. Among these was yet another generalization of a square, namely quadrilaterals whose four vertices are (O,Q), (1,0), (a, a), and (0,1), where a is any positive number, illustrated in Figure 3.
Mptdll;l¥4 (a, a)
(0, 1)
(0, 0) (1, 0)
The area of such a quadrilateral is a. If <P(a) is greater than 0, then the boundary of the quadrilateral has two complete edges, and the hypothesis of Spemer's Lemma doesn't hold. Incidentally, had it held, we would have been able to conclude that in any mequidissection of the quadrilateral m has to be a multiple of 4, since <P( m) would be greater than 1. Because m can be as low as 2, we could have predicted that there are an even number of complete edges on the boundary.
In this case we apply the linear mapping that takes (x,y) to (xla, y). The image of the original polygon has area 1 and vertices (0,0), (1/a,O), (1,a), and (0,1), as shown in Figure 4.
( 1 , a)
(0, 1)
(0, 0) ( l !a, 0)
Now there is only one complete edge on the boundary, and Spemer's Lemma applies. Hence m is even, as in the case of the square.
If -1 < <P(a) ::; 0, the labeling of the initial quadrilateral has one complete edge, and there is no need to introduce a linear mapping. Again, m must be even.
When <P(a) = -1, there may be odd equidissections as the case a = 3/2 shows. Figure 5 illustrates this case, where the quadrilateral is cut into three right triangles, each of area 1/2.
( 1 , 0)
More generally, if a = b/(2c), where b and c are odd integers, the corresponding quadrilateral has an odd equidissection. However, if <P(a) = -1 and a is irrational, I don't know what can be said. Even the case a = v3/2 is not settled. Does the quadrilateral with vertices (0,0), (1,0), ((v3/2, v'3!2), and (0, 1) have an odd equidissection?
In any case, this attempt to generalize the result for squares failed. The question remained: What is there about a square that forces all its equidissections to be even? In other words, What is the most general class ofpolygons that have no odd eq uidissection?
One simple generalization is that any parallelogram has no odd equidissection. This follows immediately from the result for a square, for any parallelogram is the image of a square by a linear mapping. Since a linear mapping magnifies all areas by a constant, it takes an equidissection into an equidissection.
A parallelogram being centrally symmetric suggests that perhaps any centrally symmetric polygon has no odd equidissection. Kasimatis's theorem about regular n-gons, when n is even, gave me enough extra evidence that I investigated centrally symmetric polygons, trying to produce a counter-example. Instead I proved in 1989 [8] that every centrally symmetric 6-gon or 8-gon has no odd equidissection. Monsky in 1990 [5] proved the theorem in general.
Even so, I did not feel that that was the last word. There was another class of polygons that I suspected would generalize the square. To construct this type of polygon, I start with the unit square in Figure 2 and then distort its boundary, changing opposite edges in the same way. The resulting polygon still tiles the plane by translates using all integer vee-
Mpt§ll;iiM
(0, 1 ) •�• ( 1 , 1 )
(0, 0) •�• (1 , 0)
tors. Figure 6 shows such a distorted square.
It seemed to me that complicating the boundary would lessen the chance that the resulting polygon would have an odd equidissection. I proved for a few simple families made this way, such as polygons formed by adding one dent, as in Figure 7, that my suspicion was valid.
·------· ( 1 , 0)
Some years later a surprising breakthrough occurred, which I described in a paper published in 1999 [10]. It concerns a unit square in the xy-plane whose comers have integer coordinates, such as the one in Figure 8.
+§lijii;i+:W (5, 7) • B
B A (5, 6) •'------• (6, 6)
Note that the square in Figure 8 has one complete edge. A moment's thought shows that exactly one vertex of any such square has both coordinates even, hence labeled A. Its two neighboring vertices are then labeled B and C. That implies that the square has exactly one complete edge.
It follows immediately that any
polygon in the xy-plane made up of
an odd number of such unit squares
has no odd equidissection. To see this, place a pebble inside each square in the polygon next to its complete edge. Because there is an odd number of pebbles, there must be an odd number of complete edges on the boundary. Moreover, since the area is an integer, it follows that the number of triangles must be even.
VOLUME 26, NUMBER 1 , 2004 1 9
It struck me as odd that by assum
ing that the polygon has an odd num
ber of squares I was able to deduce that
the number of triangles was even. I
checked a few cases where the poly
gon had an even number of squares,
enough to convince me that it was true
in general, but left it to someone else
to treat that case. Iwan Praton in 2002
[6] disposed of the even case. His proof
showed that if the number of squares
is of the form 2rb, where b is odd, then
there is a translate of the image of the
polygon by a linear mapping that takes
(x,y) to (x/2u, y/2v), where u + v ::5 r, to which a stronger version of Sperner's
lemma applies.
Consequently any polygon com
posed of the unit squares described has
no odd equidissection. There is an
other, more suggestive way to state this
result: Any polygon in the xy-plane
whose edges are parallel to the
axes and have rational lengths has
no odd equidissection. To show this,
first translate the polygon so that one
of its vertices is at the origin. Then
magnify this image by a mapping that
takes (x,y) to (qx, qy), where q is an in
teger divisible by all the denominators
of the lengths of the edges. The image
consists of congruent squares and has
no odd equidissection. Hence the orig
inal polygon has no odd equidissection.
The next conjecture is inevitable.
What if the assumption that the edges
have rational lengths is removed? I conjectured that any polygon whose edges are parallel to the axes has no
odd equidissection. I then faced three classes of poly
gons that I either knew or suspected
have no odd equidissections: centrally
symmetric, distorted square, edges
parallel to the axes. The first case was
already settled, and there was ample
evidence for the remaining two cases.
Figure 9 illustrates the three types. As
I stared at polygons like those, I no
ticed a property that they all shared. To
describe this property I orient the
boundary, turning each edge into a vec
tor whose direction is compatible with
the orientation. Then I call two vectors
on the boundary equivalent if they are
parallel. All three types have the prop
erty that the sum of the vectors in each equivalence class is the zero vector. I
20 THE MATHEMATICAL INTELLIGENCER
ii'riil;l¥+
O D Centrally symmetric Distorted square
Edges parallel to axes
called such a polygon special and con
jectured that each special polygon has
no odd equidissection.
I showed that the conjecture is true
when the special polygon has only a
few edges. The smallest possible num
ber of edges is four, and the polygon is
then a parallelogram, for which the
conjecture is true. There are no special
polygons with five sides, as may easily
be checked. There are three types of
special polygons with six sides, con
structed as follows.
The first step is to determine the
number of edges in an equivalence
class. There must be at least two in a
class and at most three, for if there
were four, two would be forced to be
adjacent. The partitions of six meeting
these conditions are 6 = 3 + 3 and 6 =
2 + 2 + 2. The second step is to see
how the equivalence classes could be arranged on the boundary. Take the
ljMiiijiitl
q
q � "' "' ... .. .. .. ... ... ..
P,,'' ....
\p
. . . . � : . . . .
q '" ...... _ ,' q - . · ... .. .. .. .. ..
p
p
(a)
p
q
(b)
p
3 + 3 case first. Denoting parallel vec
tors by the same letter, the only possi
bility is to alternate the vectors of the
two classes, as shown in Figure lOa.
That schema can be realized by a spe
cial polygon, as shown in Figure lOb.
Without loss of generality, we can as
sume its edges are parallel to the axes.
The 2 + 2 + 2 case leads to two es
sentially different schemas, as shown
in Figure l l and later in Figure 13.
I#Mil;li!i p
, .,. ... ... .. .. .. .. .... ..
q,/' "'\ r
. . . . ' I . . . .
r \.. ,/ q ... .. , ... .. .. .. .. .. ..
p
This schema can be realized by any
centrally symmetric polygon with six
sides, as shown in Figure 12.
l!'dli;Jifl p
p
The other possible schema is shown in Figure 13.
. .
q " ...... ..
p
p
It, too, can be realized by a special
polygon, shown in Figure 14.
Each case can be treated with the
aid of Sperner's Lemma, 2-adic valua
tions, and a variety of affine mappings,
that is, mappings that take (x,y) to
(ax + by + e, ex + dy + f), where a, b,
c, d, e, andf are constants and ad - be
is not 0.
p
To determine the special polygons
with seven sides, I first list the parti
tions of seven in which the summands
are at least two and at most three.
There is only one such partition,
namely 7 = 3 + 2 + 2. It can be real
ized in two different ways by schemas,
and each schema has a geometric re
alization, as shown in Figure 15.
+pMil;ii .. ii f.· · - < ' ' q/ \ p ' ' '
;- •• _ _ _ _ _ .·'q p
f!. · - -< ' ' r :
' \ P ' ' . : q\_ .. ·'q .. ... .. ...
p
p
q��r r[__fP
p
Again I managed to show that both of
these types of special polygons have no
odd equidissection [11) . Because the
proofs break into a couple of dozen
cases, I have hesitated to go on to the
eight-sided special polygons. In any
event, these 7-gons provide substantial
evidence for the general conjecture,
which I had wanted to call the "mother
of all conjectures," but was restrained
by the referee to name it simply a "gen
eralized conjecture."
As is customary in science, we are
left with more questions than we had
when we started. Perhaps we have
i
found the fundamental property of the
square that is the basis of the Richman
Thomas-Monsky theorem. Perhaps
not. That raises the first of several
questions:
Does a special polygon ever have an
odd equidissection?
The next four questions are suggested
by the special polygons.
How many partitions are there of a
positive integer n if the summands
are at least 2 and at most n/2?
Is each such partition representable
by a combinatorial schema?
If so, by how many?
Is each combinatorial schema rep
resentable by a special polygon?
Even if all these questions are an
swered, many questions about equidis
sections would remain. For instance,
does a trapezoid whose parallel edges
have lengths in the ratio of v'2 to 1 have any equidissections? I think that
the answer is no and make the follow
ing conjecture:
Consider a trapezoid whose parallel
edges have lengths in the ratio of r to 1, where r is algebraic. I conjec
ture that if r has at least one nega
tive conjugate, then the trapezoid
has no equidissection.
Little has been done about equidis
sections into simplices in higher dimen
sions aside from [3]. It was shown in [2)
that in any dissection of a regular octa
hedron into m simplices of equal vol
umes, m must be a multiple of 4. Is it
true that in any dissection of a centrally
symmetric polyhedron into m simplices
of equal volumes, m must be even?
That is where Richman's question
has led. The path that he discovered
seems to have no end.
i n t
- l i ha I I I .
Acknowledgment
I wish to thank Anthony Barcellos for
providing the illustrations, using Co
hort's software, Coplot.
REFERENCES 1 . E. A Kasimatis, Dissections of regular
and the American Economic Association (AEA) held their charter meetings in the mid-1880s. The American Political Science Association followed in 1903.3
The establishment of the professional social science disciplines, as well as the view among statisticians in the ASA that statistics could serve those disciplines, led to what became a regular practice in the ASA of holding its annual meetings in cooperation with such groups as the AEA and the AHA. Walter Willcox, a professor of economics and statistics at Cornell who served as president of both the ASA and the AEA, felt that statisticians benefited from association with other organizations. He particularly thought that the ASA's "best connections l[ay] with societies devoted to economics, political science, and law" [66, p. 288] .
These close ties between the ASA and the social science disciplines would eventually lead to tension between the main constituency of the ASA and the small but growing group of members who wanted to develop the mathematical aspects of statistical methods. Some of this mathematics began making its way into the work of statisticians in the ASA as early as the 1890s, but the relationship between collections of numerical data about society and the mathematical theory of probability had just begun to emerge, and several decades passed before a group of researchers would focus their inquiries on it specifically.4
In the 1920s, these researchers began to see the usefulness for research in economics, biology, and agriculture of the growing set of tools of statistical inference, but they found themselves struggling to find places to publish their results. Many of them were trained and employed as mathematicians and their interests overlapped
with a number of disciplines, but it seems that no single periodical or organization provided a comfortable professional home. 5 The mathematical statisticians felt no particularly warm welcome from either the ASA or the American Mathematical Society (AMS)-perhaps the most likely supporters of mathematical statistics. On the one hand, the members of the ASA were interested in data collection and the use of numerical information, but as Carver, founder of the Annals of Mathematical Statistics, put it, "most of their membership were economists, bankers and census people whose knowledge of mathematics was very limited."6 Most American mathematicians, on the other hand, focused their research on pure mathematics, and while papers in mathematical statistics included theorems and proofs, they often had a particular use of those theorems as a starting point. In a discussion leading up to the founding of the IMS, Henry Rietz, a mathematician at the University of Iowa, who would become the organization's first president, commented about the mathematical research community that "when it comes to practice accepting papers for publication it seems not much material is acceptable that is a bit tainted with possible applications to statistical data."7
Carver put out the first issue of the Annals in 1929 (initially with some financial backing from the ASA), and mathematical statisticians formally organized the IMS as an independent professional organization in 1935. In 1938, it assumed complete financial responsibility for the Annals.8
The IMS and the Annals provided a means for mathematical statisticians to establish formal citizenship in their emerging disciplinary community. They created a sort of boundary around that discipline, setting it off from its border-
3For discussions of the professionalization of American social science, see [41 , 1 1 ].
Figure 1. Abraham Wald. Illustration courtesy
of the Columbia University Archives-Colum
biana Library.
ing fields of inquiry. 9 But the boundary was not impenetrable. Although they tended to draw attention to the importance of theoretical statistics, mathematical statisticians still welcomed opportunities to support and collaborate with the users of statistics. As the mathematical statisticians were establishing their community, several groups of applied statisticians had organized themselves into societies supporting their particular interests. The Econometric Society was formed in 1930; the Psychometric Society, in 1935. By the late 1930s, biologists and medical scientists in the ASA had organized its first disciplinary section, the Biometric Section. The group began publishing a journal, Biometrics, in 1945. The work of these three organizations focused on applying statistics to studies based in their parent disciplines-economics, psychology, and biology.
The emerging community of mathematical statisticians in the United States had ties to these organizations.
4During the nineteenth century, the probabilistic tools of the normal distribution and the method of least squares had found most of their applications in the physical
sciences of astronomy and geodesy, occasionally appearing in actuarial work on mortality tables. For a discussion of these developments, see [35, 46].
5The variety of professional societies and journals in which the mathematical statisticians participated is discussed in [1 6].
6Harry C. Carver to Jerzy Neyman in [30, p. 1 72].
7Henry Rietz to E. B. Wilson, 27 July 1 935, in (1 3, p. 289].
8Until then, except for the ASA money, Carver had funded the journal out of his own pocket. See [30, p. 1 72].
9"fhe founding of the IMS marks perhaps only a middle chapter in the story of this process. The story continues with the impact of WWII on American science and
subsequent developments within American universities. While some consequences of the war will be explored below, the rest of the story is beyond the scope of this
discussion.
26 THE MATHEMATICAL INTELUGENCER
Of particular relevance to the place that Abraham W ald would find in the community when he emigrated from Europe were its connections to the Econometric Society. To understand the importance of those links for Wald's inclusion in the American mathematical statistics community, we must trace his path there back to his days in Vienna.
Mathematics and Economics in
Inter-war Vienna: A Network of
Communities
Wald arrived in Vienna in 1927. His hometown in Transylvania, known today by its Romanian name, Cluj, was part of Hungary when Wald was born on October 31, 1902. In 1920, Transylvania became part of Romania, but had a large Hungarian minority. W ald, a Jew, spoke Hungarian and "never developed any affinity for Romania," nor a knowledge of its language [29, p. 361] . Educated at home because the local school required attendance on the Sabbath, Wald had passed the gymnasium examination recognized by the University of Cluj, and after fmishing there enrolled in the University of Vienna at age 25 to study mathematics. He eventually took classes from and wrote a dissertation under Karl Menger, working on metric spaces and differential geometry. 10
Menger's father Carl Menger had made his mark on Austrian economics with his work on marginal utility. The younger Menger was well versed in his father's work and well connected to various "circles" of the Viennese intelligentsia, including the philosophical Vienna Circle and a number of overlapping groups of economists. 11 He led his own circle, the Mathematical Colloquium, which met and published its proceedings from 1928 to 1937. The Colloquium hosted an impressive array of local and international luminaries, including John von Neumann, Alfred Tarski, Karl Popper, and Kurt Godel
(whose incompleteness theorem was first presented to the Colloquium).l2
Wald joined the Colloquium in 1930. In some sense, this gathering formed for Wald the center of a network of communities that fundamentally shaped his research. He contributed twenty-one papers to the Colloquium's proceedings, Ergebnisse eines mathematischen Kolloquiums, between 1931 and 1937, co-editing the last two volumes with Menger, Gbdel, and Franz Alt. Many of these papers communicated his research in pure mathematics, but a few, discussed below, point to the connections among disciplines and communities that the Colloquium made for Wald.
Wald's connection to the world of Viennese economics would be among the most important for his passage to the United States. At Menger's invitation, the banker and economist Karl Schlesinger presented his work on equations of economic production to the Mathematical Colloquium, and W ald quickly became interested in the field, publishing [51, 52, 53] in 1935 and 1936.13 In addition to these "first publications in his long list of contributions to mathematical economics" [27, p. 18], Wald's contact with Schlesinger led to some work as the latter's private tutor in mathematics. Menger had encouraged this connection, knowing that as a Jew, W ald had no chance of employment at the university. For the same reason, he introduced Wald to Oskar Morgenstern, who hired him to work at the Austrian Institute for Business Cycle Research, which Morgenstern directed. According to Menger's recollection, Wald had not been disturbed by his lack of opportunity in the academic world: "W ald, with his characteristic modesty, told me that he would be perfectly satisfied with any small private position which would enable him to continue his work in our Mathematical Colloquium" [27, p. 18]. Not only did Wald's work in the Collo-
10Details of Wald's education and research can be found in [27, 29].
quium continue, but his private positions opened up new intellectual opportunities.
His relationships with Schlesinger and Morgenstern formed important threads in Wald's network of communities. These men were active in influential circles of Vienna's economists, both in and outside of the university. At the university, Schlesinger and Morgenstern participated in the seminar of Hans Mayer, appointed to a chair in economics in 1923.14 Apparently, Menger and Wald attended the seminar occasionally as well [5, p. 12] .
Morgenstern and Schlesinger also attended the private biweekly seminar of Ludwig von Mises. Passed over for a position at the university, Mises was nevertheless considered "the central figure in the Viennese economic community" at the time [5, p. 14]. He held his seminar in the 1920s and 1930s at the Vienna Chamber of Commerce where he was employed as the Secretary. Mises and his seminar participants formed the core of the National Economic Association, which Mises revived in the 1920s, becoming its vice-president, with Mayer as president. 15 The group met in a conference room of the National Banker's Association, thanks to its president, Karl Schlesinger. Papers they presented often appeared in the Zeitschrift fiir NationalOkonomie, a periodical edited by Mayer with Morgenstem's assistance [5, p. 18] . 16
Menger presented a paper at an Economic Association meeting on the Petersburg paradox, but he later recalled that Mayer discouraged its publication because of its strongly mathematical character [28, p. 259]. Morgenstern, on the other hand, encouraged the inquiry of mathematically minded researchers into economic questions. As the director of the Institute for Business Cycle Research, another Mises-promoted organization, Morgenstern employed not only Wald, but another student of
1 1 For discussions of Menger's many connections, particularly to the intellectual groups meeting outside the walls of the university, see [5, 1 0, 23, 44, 45]. 1 2The proceedings of the Colloquium have been reissued along with commentaries in [6].
1 3Schlesinger earned his Ph.D. in 1 91 4 under E. von Bohm-Bawerk at Vienna [2, p. 23].
14Mayer succeeded Friedrich von Wieser, famous for his work in opportunity cost theory. See [5].
15Mises seems to have wanted to insure that Mayer and his students were included in the professional society over against Mayer's rival at the university, Othmar
Spann. This motivation may explain why Mayer received the presidency [5, p. 1 7].
16The Zeitschrift was not, however, an official publication of the National Economic Association.
VOLUME 26, NUMBER 1 , 2004 27
Menger, Franz Alt, and the economist,
Gerhard Tintner, who later noted that
"at the Institute there was a much more
scientific attitude to economics than
elsewhere in Wien at this time. "17
As Wald's contact with economists
in Vienna fostered his interest in math
ematical economics, his work gradu
ally became known abroad. At a 1936
meeting of the Econometric Society in
Chicago, Tintner reported on some of
Wald's results [21, p. 188]. Schlesinger
and Wald himself attended a 1937
meeting of the Econometric Society in
France [36]. Wald's contacts in the
Econometric Society would form im
portant links for him to the statistics
community in the United States.
His economics research in Vienna
had touched upon issues related to sta
tistics, and he had published a paper in
the Ergebnisse on Richard von Mises's
notion of a collective, a concept that
played a role in the axiomatization of
probability.l8 But his immersion in the
ideas of mathematical statistics would
come in the United States. There, for
Wald, the web of communities woven
together for him in Vienna would con
verge with the statistics community
that had been forming in the United
States since the 1920s. The boundary
that American mathematical statisti
cians had drawn around their disci
pline as well as the connections cross
ing that boundary, particularly to the
econometrists, would further shape Wald's research, as well as his place in the scientific community.
Mathematical Statistics and
World War II: Communities
Converge
The Cowles Commission for Research
in Economics formed the thread link-
1 7From an interview reported in [5, p. 20].
ing Wald to the United States. The
Commission had been organized in
1932 by Alfred Cowles, the president of
an investment counseling firm in Col
orado Springs. He had come into con
tact with several members of the
Econometric Society in 1931, when he
began researching methods of fore
casting the stock market. At their sug
gestion, Cowles opened his research
institute and soon after began financ
ing the Society's new periodical Econometrica [ 4].
Both the Econometric Society and
the Cowles Commission had ties to the
Viennese economics and mathematics
communities, as well as to the Ameri
can statistics communities. Karl Menger
participated in the Society's organiza
tional meeting, held in Cleveland, Ohio,
in December 1930 [40]. Gerhard Tint
ner attended its meetings in the early
1930s and joined the Cowles Commis
sion staff in 1936. Other active mem
bers, also present at the organizational
meeting, included Harold Hotelling and
Walter A. Shewhart, both founding
members of the IMS. Hotelling became
one of the American mathematical sta
tistics community's most respected
spokesmen for the discipline [ 15, 16].
It may have been Tintner's connec
tions that resulted in Wald's invitation
to join the staff of the Cowles Com
mission in 1937, which he accepted,
though not without some delay and
hesitation about leaving Vienna. 19
Menger had departed in 1937 for the University of Notre Dame as the polit
ical climate in Austria was becoming
increasingly unbearable. When Hitler's
troops marched into Vienna in March
1938, Wald had not yet left. Morgen
stem was in the United States on a lec
ture tour and stayed, taking a position
in the economics department at Prince
ton when he heard that he had been
blacklisted by the Nazis [5, p. 29] . Wald,
having been dismissed by Morgen
stem's Nazi successor, fmally made his
way to Colorado Springs.20
His stay there was brief-within a
few months he left for Columbia Uni
versity to work with Harold Hotelling.
Hotelling had been teaching eco
nomics and building up a program in
mathematical statistics in Columbia's
economics department since 193!.21
Through the next decade, several early
members of the American mathemati
cal statistics community received
some training at Columbia under
Hotelling, including Samuel S. Wilks
and Joseph L. Doob [ 16].
Funded from 1938 to 1942 by a grant
Hotelling had obtained from the
Carnegie Corporation, Wald worked at
Columbia, first as a research assistant
and then teaching courses in mathe
matical statistics and economics.22 It
was during these first years at Colum
bia that Wald became immersed in the
ideas of mathematical statistics. Char
acteristically, according to his col
leagues, he "worked with prodigious
energy and endurance" [ 14, p. 18], with
"most of his waking moments during
this and the next several years . . .
given to work" [68, p. 2]. Wald became
an assistant professor of economics in
1942, and made his way through the
ranks to professor of mathematical statistics in 1945, finally becoming the chair of an independent department of
mathematical statistics at Columbia in
1946.23 Wald was a popular lecturer
from his first years of teaching at Co
lumbia. Students flocked to his lec
tures, which were "noted for their lu
cidity and mathematical rigor" [68, p.
18Wald 's interest in collectives followed a presentation on the subject at the Collocuium by Karl Popper. Richard von Mises was the brother of Ludwig. His collectives
briefly vied with Kolmorgorov's measure-theoretic formulation for a role in the foundations of statistics. See, for example, [47, 1 7] .
1 9Roy Weintraub makes this speculation about Tintner's role in [65].
20Except for one brother who eventually joined Wald in the U.S. , all of his immediate family perished in the Holocaust. Wald was just one of many scholars making their
way out of Europe in the wake of the Nazi takeover. For an account of the experiences of emigre mathematicians in the U.S., see [38]; on scholars more generally,
see [7]. 21As a graduate student, Harold Hotelling had applied unsuccessfully for an economics fellowship at Columbia. Hoping he could pursue his interests in probability and
economics elsewhere, he went to Princeton in 1 921 on a mathematics fellowship, but he found no one working in his areas of interest. Instead, he did his research in
topology and differential geometry with Oswald Veblen and Luther P. Eisenhart. This was only a temporary shift in Hotelling's focus, however. He later applied some
topological theory to his statistical research, but with the exception of the published version of his dissertation and one other research paper, the rest of his publica
tions dealt with statistical topics. See [1 6]. 22 See Series Ill .A., box 1 1 4, folder 5, Carnegie Corporation of New York Records, Columbia University. 23Hotelling had just left to chair a new department of mathematical statistics at the University of North Carolina. See [33].
28 THE MATHEMATICAL INTELLIGENCER
Figure 2. Harold Hotelling. Illustration cour
tesy of the Columbia University Archives
Columbiana library. Photo by Alman & Co.
3] . His colleagues described him as "a
gentle and kindly friend" [ 14, p. 19], re
porting that the students, who came
from all over the world, "loved and re
spected him" [29, p. 366].24
The recently organized American
mathematical statistics community
quickly became Wald's professional
home. By 1943, he was a fellow of the
IMS and was elected its president in
1948 while simultaneously serving as
vice-president of the ASA. But Wald
had also been a fellow of the Econo
metric Society since 1939, and his net
work still included economists, many
of them European emigres. One of
them, Jacob Marschak of the New
School for Social Research, had come
to New York in 1940 by way of the Uni
versity of Oxford's Institute of Statis
tics, which he had directed after being
dismissed from the University of Hei-
delberg in the wake of the Nazis' Jew
ish boycott [1 ] .25 He started a seminar
on econometric methods with others in
the New York area. Hotelling, Wald,
and several others with connections to
the mathematical statistics community
attended and contributed to the semi
nar. One of these, Henry Mann, had re
ceived his Ph.D. from Vienna in 1935
for a dissertation on algebraic number
theory and emigrated in 1938. He tu
tored in New York until obtaining fund
ing from the Carnegie Corporation to
study statistics at Columbia. He and
W ald collaborated on several papers,
including one that grew out of their
work in Marschak's seminar [24].
The local communities supporting
Wald's research in New York, like
those in Vienna, transcended univer
sity and disciplinary boundaries. In ad
dition to his colleagues in Marschak's
seminar and at Columbia, Wald worked
for more than two years with statisti
cians and economists on the staff of
the Statistical Research Group (SRG),
a branch of the National Defense Re
search Committee (NDRC). That orga
nization, the brainchild of Vannevar
Bush, president of the Carnegie Insti
tution of Washington, served to "cor
relate and support scientific research
on the mechanisms and devices of war
fare" [34]. Bush had organized the
NDRC in 1940 under an order from
President Roosevelt, and although it
initially had no division for research in
mathematics, Bush added the Applied
Mathematics Panel (AMP) in 1942,
which included the SRG. W. Allen
Wallis, Milton Friedman (both econo
mists), Hotelling, and Wald were
among the principal staff members of
the statistical group. Their work fo
cused on studies of damage to aircraft
from anti-aircraft guns, on methods of
most effectively bombing targets, and
on statistical methods of inspection in
production [37, 64].
As part of this team, Wald devel-
oped the sequential probability ratio
test, an idea that would later play im
portant roles in the theory and appli
cation of statistics. The details of
Wald's discovery have been recorded
by his colleague Allen Wallis [ 64]. Early
in 1943, Wallis had begun to work for
a Navy captain on some inference
problems involving ordnance testing.
Discussing the problems involved with
performing large numbers of tests, the
captain suggested that a "mechanical
rule which could be specified in ad
vance stating the conditions under
which the [testing] might be termi
nated earlier than planned" could serve
to eliminate waste in the testing
process [64, p. 325]. Wallis mentioned
the problems to Friedman, and the two
began discussing it informally, outside
of their regular work for the SRG. De
termining that the problem required
more statistical knowledge than they
possessed, Wallis and Friedman ex
plained the problem to Wald. Initially
unenthusiastic about the prospects of
solving it, Wald called two days later
with an outline of the basic ideas of the
SPRT, a test that uses data as they are
gathered to determine when to stop an
experiment or an inspection. Rather
than basing the experiment or test on
a fixed sample size, a sequential sam
pling plan provides a rule for deciding,
after each trial, whether to take a cer
tain action or to make another obser
vation.
Soon after formulating his ideas,
Wald began work on a monograph
treating the theoretical properties of
his test [57] , while Harold Freeman, a
professor of statistics in the depart
ment of economics and social science
at the Massachusetts Institute of Tech
nology, began a manual describing its
applications [8]. Wald also spoke about
some of the theory underlying his se
quential methods at the 1944 summer
meeting of the IMS held with the Amer
ican Mathematical Society [58] and
24 An historical analysis of the program begun by Hotelling and Wald at Columbia would provide 1nteresting information about the development of the American math
ematical statistics community in the second half of the twentieth century. Such an inquiry, which would go beyond an exploration of the connections between Wald's
communities and his research on sequential analysis, is outside the scope of this discussion. 25The New School for Social Research had opened in 1 91 9, founded by a group of progressive scholars that included John Dewey, Charles Beard, Thorstein Veblen,
and Franz Boas. In addition to providing opportunities for research for social scientists, the school offered an adult education program modeled on the German Volks
hochschulen. Reorganized in 1 922 under Alvin Johnson, an economist and editor of the New Republic, the New School focused more narrowly on adult education un
til 1 933 when Johnson saw an opportunity to rebuild the school 's research program. Over the next year, he brought a dozen social scientists dismissed from their po
sitions in Germany to New York and established what became the Graduate Faculty of Political and Social Science. See [20, 42).
VOLUME 26. NUMBER 1. 2004 29
published a long paper in the Annals of Mathematical Statistics in 1945 discussing the theory and applications of the SPRT [61]. That same year he contributed a non-technical, expository paper on the fundamental ideas and applications of the test to Journal of the ASA [60]. As the editor of the Journal explained in a footnote to the paper, W ald had specifically written it "to be accessible to statisticians with little mathematical background" [60, p. 277, note]. The appearance of Wald's ideas in these two periodicals-at different levels of mathematical sophistication-highlights the distinction still present in the mid-1940s between the communities of applied and mathematical statisticians.
Wald's ideas started a flurry of efforts on the part of other researchers to explore questions raised by his discoveries.26 Much of this research emerging from Wald's ideas found its way onto the pages of the Annals of Mathematical Statistics. That discussion about Wald's new ideas in sequential sampling occurred in what had become the official publication of the American mathematical statistics community suggests that the Annals had come to play a crucial role in advancing the community's discipline. No longer did its existence simply add to the distinctiveness of mathematical statistics by providing the discipline with an important professional accoutrement; mathematical statisticians like Wald seemed to regard it as having the credibility to record their continuing conversations about their theoretical work.
In addition to providing a subject for fruitful theoretical research of the sort that appeared in the Annals, sequential sampling offered a practical means of reducing the number of observations needed for testing and quality control. In his introduction to the Summary Technical Report of the Applied Mathematics Panel, Warren Weaver commented on the usefulness of the SPRT,
saying that the "Quartermaster Corps reported in October 1945 that at least 6,000 separate installations of sequential sampling plans had been made" [37, p. 614].
Sequential analysis provides an important example of an area of research that combined mathematical theory with statistical applications. Its problems and their solutions addressed the practical needs of manufacturers and scientists, and at the same time attracted the technical and theoretical interests of the mathematical statisticians. Abraham W ald himself seemed to be an ideal member of the community practicing this discipline situated between theory and application. He brought to his investigations, as his student and collaborator Jacob Wolfowitz wrote, "a high level of mathematical talent of the most abstract sort, and a true feeling for, and insight into, practical problems" [68, p. 4]. In this case, work on practical problems of defense promoted theoretical advances.
These theoretical advances extended beyond the field of inspection sampling in which the SPRT originated. In fact, the theory of sequential analysis became an important aspect of Wald's theory of decision functions. Decision theory generalized the questions addressed by statistical inference by determining a rule based on randomly selected observations for choosing the best course of action from a set of possibilities.
W ald had begun developing decision theory early in his time of formal study of modem statistics with Hotelling, several years before his World War II work on the SPRT. In 1939 he published a paper in the Annals of Mathematical Statistics introducing its central ideas. Here he articulated the idea of generalizing the problems of hypothesis testing and constructing confidence intervals, seeking to build a theory that would include them as special cases. Wald conceived of an approach that would pro-
vide a means of choosing among any number of hypotheses (in contrast to the Neyman-Pearson theory, which admitted only two) by specifying a system of acceptance regions according to criteria that would "depend on the relative importance of the different possible errors" [55, p. 301].
Wald lectured briefly on these ideas in a 1941 series of addresses organized by Menger at the University of Notre Dame, but only resumed research on them a few years later, after beginning his work in sequential analysis. 27 By then John von Neumann and Oskar Morgenstern had published their 1944 landmark work, Theory of Games and Economic Behavior [50]. Perhaps surprisingly, these two had not met before settling at Princeton. They had had a number of contacts in Vienna in common, including Wald and Menger. Both had also begun thinking about ideas related to game theory before meeting, and their earlier work had some connections to the economics and mathematics communities in Vienna. In particular, Menger had written a book taking a mathematical approach to social ethics that influenced some of Morgenstern's research in the 1930s.28
Since his days in Vienna Wald had been familiar with some of von Neumann's work in economics, having edited the latter's paper on equilibrium in a dynamic economy for the final volume of the Ergebnisse [49]. This paper had some connection to von Neumann's first work on game theory, published in 1928 [48], and while working with the SRG, W ald mentioned to a colleague that some of his ideas in decision theory were based on that 1928 paper.29 In a discussion of the work by von Neumann and Morgenstern in Mathematical Reviews, Wald pointed out in 1945 that "the theory of games has applications to statistics . . . , since the general problem of statistical inference may be treated as a zero-sum two-person game" [59].
His next paper on decision func-
26More than 1 8 papers related to sequential sampling were published between 1 945 and 1 950 [9, pp. 8-9]. A bibliography published in 1 960 lists 374 references deal
ing with sequential analysis that appeared through 1 959 [1 9] .
27Menger organized a Mathematical Colloquium at Notre Dame fashioned after the one i n Vienna. Wald's lectures at that Colloquium were published as [56].
28Menger's book and his motivation for writing it are discussed in [23]. The influence of Menger on Morgenstern's ideas is treated in [22].
29For an account of that conversation, see [64, p. 334]. The connection between von Neumann's two papers is described in [65].
30 THE MATHEMATICAL INTELUGENCER
tions, appearing the same year in the
Annals of Mathematics [62], elabo
rated on the connections between sta
tistical inference and the zero-sum
two-person game. The Annals of Mathematics was one of the key publication
venues for the (pure) mathematical re
search community in the United States,
and this paper was not Wald's only con
nection to that community. He was a
member of the American Mathematical
Society and had already published two
papers in the Society's Transactions. W ald would publish several more in the
Annals of Mathematics as well as in
the Bulletin of the AMS over the next
five years. These papers treated math
ematical issues raised by sequential
analysis, decision theory, and game
theory.
The results linking game theory
with decision function theory highlight
the interaction among the communi
ties in which Wald participated, both in
Vienna and in the United States. Wald's
success in raising and answering ques
tions of interest to a variety of re
searchers was perhaps due in part to
what one colleague described as "his
open-mindedness" to others' pursuits.
"He was ever ready to listen to the prob
lems other scholars encountered and he
was eager to speak about the work he
had in progress himself' [29, p. 366].
In the late 1940s, the publications of
Wald, von Neumann, and Morgenstern
influenced the research of economists
at the Cowles Commission. This work,
which was an "attempt to discover
what kind of behavior on the part of an
individual or group in specified cir
cumstances would most completely
achieve the goals pursued," drew on
the ideas of Wald and von Neumann
and Morgenstern, and led to research
in decision making under uncertainty
by Jacob Marschak and Leonid Hur
wicz [4, p. 48].30 So the convergence of
Wald's communities in the 1930s and
1940s, across national and disciplinary
lines, had fundamental connections to
his research in statistics, particularly to
his work on sequential analysis and de
cision function theory.
Figure 3. Abraham Wald in 1950. Illustration
courtesy of the Columbia University Archives
Columbiana Library.
Conclusion
Wald's early training with Menger in
geometry was far removed from his
work in decision theory-research that
colleagues at the time called his most
significant contribution to statistics. 31
Wald shifted his interests from pure
mathematics to statistics in less than a
decade, and from the perspective of
the disciplines themselves, this shift
has the appearance of a clean break, a
jump discontinuity. A wider historical
focus, however, that considers the con
text of the scientific communities to
which Wald belonged, brings some
continuity to light and helps explain
the connections between Wald's many
professional relationships and discipli
nary interests.
From his earliest years in Vienna,
although studying pure mathematics,
Wald found himself working with re
searchers engaged in a wide range of
intellectual pursuits. Karl Menger, in
particular, introduced him to the cir
cles of Viennese economists. Although
the use of mathematical methods in
economics research was not the dom
inant fashion in the discipline, Wald
met some economists whose mathe-
30Some of Marschak's work in this field appeared in [25]; some of Hurwicz's contributions can be found in [1 8] .
3 1See, for example, [68, p. 9 ] and [14 , p. 1 9] .
matical inclinations drew him into the
world of econometrics, a professional
community on the border of several
fields.
Because these economists were not
tied exclusively to the university, these
connections provided him immediate
employment in Vienna at a time when
the political and social climate barred
him from traditional academic em
ployment. The increasingly interna
tional character of this econometric
community had an even more far
reaching impact on Wald's opportuni
ties. Some of his colleagues, including
Menger, were active in an international
network of researchers with ties to the
Econometric Society, which had been
holding meetings in the United States
and Europe since its inception in 1930.
Wald's research and abilities caught
the attention of this network, resulting
first in a job at the Cowles Commis
sion-his ticket out of Nazi Europe
and then eventually in significant
influence on the direction of the econo
metric community's research.
In the meantime, however, after his
few months at the Cowles Commis
sion, Wald moved to Columbia to work
with Harold Hotelling. Here, his formal
contributions to the discipline of math
ematical statistics began. His connec
tions to the American statistics com
munity began to grow stronger as well.
Like the world of Viennese economics
that W ald had left behind, the statistics
community in the United States had
somewhat fluid disciplinary and insti
tutional boundaries, as well as impor
tant international connections. Its
members did research in the theory of
statistics as well as in statistical appli
cations to economics and biology,
among other fields. Universities, gov
ernment, and private organizations
supported the community's work.
Wald benefited from and con
tributed to the efforts of these patrons
of statistics-at the privately financed
Cowles Commission, as a researcher
and professor at Columbia with
Carnegie funding, and through his de
fense-related research in the SRG. His
VOLUME 26, NUMBER 1 , 2004 31
A U T H OR
PATTl WlLQIR HUNTER
international connections had helped
him move to the United States, and he
continued to interact with his fellow emigres as well as with researchers
abroad. In late 1950, Wald was on a lec
ture tour through London, Paris, and
Rome. En route to speak at the Indian
Statistical Institute, he died in a plane
crash on December 13.
As a reviewer of Wald's text on de
cision theory wrote in 1951, "Wald's
death [gave] Statistical Decision Functions [63] an altogether new sig
Mathematics East and West , Theory and Practice : The Example of Distributions
Science with the grievous glance It [mathematics} casts a grievous glance on mankind, and forces it to confront the solid reality, the real fact only, the fact which destroys alike the most magnificent and the most caustic fantasies.
-Robert Musil Notebooks. Excerpt from book #16, "The Spy"
(1923-1924), W I 1979-80
hat lessons can be drawn from the upheavals that characterized twentieth-
century scientific development? Did mathematics undergo the same up-
heaval? Was its status modified, or did it retain at the time of Hiroshima
the moral and aesthetic value which Plato praised? These
questions are too general, but they suggest a debate.
We will study here only a very precise situation and con
text-that of mathematical work conducted in Russia and
France from the 1930s, inspired among others by Jacques
Hadamard's seminal work, and which led to the worldwide
development of mathematical analysis and to the theory of
partial-differential equations. The relevant documents ex
ist, and after more than fifty years, a historical inquiry is
possible.
The death of Laurent Schwartz, a prominent French
mathematician, member of the Bourbaki group, and one of
the driving forces of the mathematical community for more
than twenty years, can be an occasion to think back about
the birth of the theory of distributions. The recent publi
cation of Soviet archives makes it possible to complement
the work of historians, in particular Adolph P. Yushkevich's
comments on the book [Lu] of Jesper Liitzen (whose rec
ognized competence in the history of mathematics and
whose conscientiousness are beyond question). In the Ap
pendix we provide a translation of Yushkevich's article,
where he examines very meticulously among others, the ar-
tides published in Russian (the references here comple
ment those given in his article). Indeed, while times have
changed, language barriers persist, slowing down the ex
change of ideas between the West and Russia, and hinder
ing a wider diffusion of Yushkevich's text, although it was
published as early as 1991 in the historical journal which
he founded. Naturally mathematics is not exempt from
chauvinistic behavior in the international competition (the
"Popov effect," both in the East and in the West), but
Yushkevich is aware of this and does not indulge in it. He
seems to have at heart to show that there was an intense
mathematical life in the East, in the USSR, isolated as it
was by the cold war and the "construction of socialism in
a single country." Let us take a look at the various sensi
bilities and styles revealed by this episode.
This is also an opportunity to take another look at the in
ternational scientific cooperation of the period, which has
hardly ever been studied. The Fields medal was awarded to
Laurent Schwartz at Harvard in 1950 during the Korean
War-and some called it "the Fields medal of the cold war,"
referring to the difficulties experienced by Hadamard and
his nephew Schwartz in getting a visa to the United States.
In any case this is a little-known episode in the relationship
between science and politics, as we shall see.
During the 1930s, the idea of generalized function or dis
tribution was "in the air": it was used by the great physi
cist Paul Adrien M. Dirac (1902-1984), and by Salomon
Bochner, whose work anticipated later work on distribu
tions, in particular with respect to the role of Fourier se
ries [Boc]: physicists were using distributions just as
Moliere's Monsieur Jourdain was producing prose-un
awares. The very birth of the theory of generalized func
tions/distributions can thus be rife with lessons for a time
when the relationships between mathematics and physics
are evolving (cf. [JQ]).
Above all, this study is an opportunity to bring into view
two different conceptions of the role of mathematics, in the
East and in the West (to simplify)-one, led by Schwartz
and Bourbaki, focusing on structures, and the other, cen
tered on Sobolev and the Saint Petersburg school, closely
linked to physical sciences. All these questions are of in
terest for the present, and we think that we owe it to the
memory of Laurent Schwartz and his keen sense of the
scholar's civic role to approach them-with honesty and
rigor-at last.
The Actors: Hadamard (1 865-1 963), Sobolev
(1 908-1 989), and Schwartz (1 91 5-2002)Two Different Worlds
Laurent Schwartz is a mathematician admired all over the
world, known indeed beyond specialists' circles for his role
as a "mathematicien dans le siecle" [S2]-a "secular" math
ematician, involved in the social and political world. One
of the active members of the Bourbaki group after World
War II, he was also a militant partisan of all the humani
tarian causes of the twentieth century, from militant Trot
skyism between 1936 and the Resistance to the "Comite
Au din" during the Algerian War, and the cause of the math
ematicians standing for human rights in Eastern Europe.
Schwartz's personality brings together the qualities of the
French intellectual, growing out of a family with a long tra
dition of social ascent, which supplied France with emi
nent intellectuals.
Nothing except mathematics is shared between a Lau
rent Schwartz and a Sergei Sobolev. The latter is also a fine
scholar, but gained much less fame in the West. Sergei
L'vovich Sobolev was born in Saint Petersburg in 1908, in
a family connected to the nobility; his father was a noted
lawyer from Saint Petersburg (later Leningrad). In the en
during rivalry between Moscow and Saint Petersburg, a city
created by Peter the Great in 1703, the mathematical
schools had a particular role: Saint Petersburg was the city
of Euler, who lived there for a large part of his life, as well
as Chebyshev (1821-1894), Markov (1856-1922), Lyapunov
(1857-1918). This already shows that mathematical life in
the city was very open to the sciences and technology. It
was also in Saint Petersburg where the managerial talents
of Steklov (1863-1926), an applied mathematician, led to
the creation of research institutes of the Academy which
later bore his name. A detailed account of political strug-
40 THE MATHEMATICAL INTELLIGENCER
gles in Moscow and Leningrad within mathematical soci
eties, and their dramatic consequences ("the Luzin affair"),
can be found in several recent publications like [De, Mar,
M-Sh, Viu, Y] , as well as in various issues of the history pe
riodical launched by A. P. Yushkevich.
Sobolev was a brilliant student at a particularly young
age, like a number of other 20th-century Russians. At the
University, which he entered in 1925, he followed the
courses of Grigori'i Mikhailovich Fikhtengolts (1888-1959)
and Nikolai Maksimovich Gunther (1871-1941) (the latter
in potential theory). He met Vladimir Ivanovich Smirnov
(1887-1974), who would be a professor then a co-worker
of Sobolev, a professor from 1925, and later dean of the
"Mat-Mekh" faculty for 25 years. (This did not spare him
the displeasure of criticism in 1957 on the occasion of a
tribute to Euler: after Smirnov praised the positive influ
ence of Frechet, who was attending the ceremony, on So
viet mathematics, Kolmogorov chided him publicly for his
"love of the foreign" ([Y] , page 31).
Sobolev's first publication was a counter-example to a
result announced by Saltykov and reused by Gunther in his
analysis course. In 1929, after obtaining his doctorate, he
joined the Institute of Seismology, where he collaborated
with Smirnov, before joining the Steklov Institute, becom
ing at 24 a corresponding member, then a full member
the youngest-of the USSR Academy of Sciences. Along
side his mathematical career, in which he was always open
to other sciences and other countries despite a difficult
context (he was fluent in French, which he had learned as
a child from his Belgian nanny), he conducted various pro
jects including the creation of the Siberian Centre of the
Academy of Sciences. He always displayed Russian pride
and a strong loyalty to the Soviet power, as a member of
the Party from the 1930s, yet this loyalty did not prevent
him from taking sometimes difficult and courageous posi
tions (for example in the Lysenko affair); but he sometimes
had a more orthodox stance, e.g., as one of the critics blam
ing Luzin in 1936 for his openness and his foreign publica
tions [De].
Somewhere between these two personalities is Jacques
Hadamard-"le petit pere Hadamard" (Daddy Hadamard), as he was familiarly called by his admirers, or "the living
legend of mathematics," an expression used by Hardy to
introduce him to the London Mathematical Society in 1944
[Ka]. After Poincare, Hadamard is without doubt the
Frenchman who most influenced twentieth-century math
ematics. He is also an illustration of the humanist and uni
versalist traditions in French culture at their best. For a
better understanding of the rest of this study, it must be
noted that Hadamard was Laurent Schwartz's great-uncle
by marriage, and followed his studies in secondary school,
then at the Ecole Normale Superieure (ENS). From his sem
inar grew the Bourbaki group (via the Julia Seminar). The
Hadamard seminar was formative for several generations
of students at ENS. Laurent Schwartz recognized (lac. cit.) the crucial part played by Hadamard in his education. We
know well the life of Hadamard [M-Sh]-the immensity of
his mathematical work, and also his left-wing radical com-
Figure 1 . S. L. Sobolev with his children, Moscow, 1940.
mitment, initially motivated by the Dreyfus affair, then by
the rise of Nazism, and his closeness to the French Com
munist Party along with Frederic Joliot-Curie. The archives
of the Academie des Sciences have copies of articles pub
lished during his stays in USSR, in which he praised the po
litical system and the merits of Soviet science [H1] .
The Facts
The 1 930s: Sobolev Functionals
Within the framework of his militant activities for friend
ship between peoples, Hadamard, an indefatigable traveler,
made numerous journeys in the East, in particular to China
and the USSR.
Visits to the USSR:
• 1930: he attends the Congress of Soviet mathematicians
in Kharkov, in July, then travels to Kiev. He meets
Sobolev in Kharkov and later they have discussions in
French in Leningrad. Hadamard asks Sobolev to keep
him informed of his work [M-Sh p. 217];
• May, 1934: Hadamard is a member of a delegation of nine
French academics travelling for the "week of French Sci
ence" in the USSR. In Leningrad he meets Sobolev but
does not participate in the second Congress of Soviet
Mathematicians (24-30 June, 1934), where Serge Sobolev
makes three presentations:
1. A new method for solving the Cauchy problem for hy
perbolic partial differential equations;
2. Generalized solutions of the wave equation;
3. On the diffraction problem for Riemann surfaces.
The contents of these talks were certainly discussed a fort
night earlier with Hadamard, who followed with interest
the works of his colleague: Sobolev himself acknowledged
the influence of the notion of finite part, discovered by
Hadamard in 1903 (!), in his discoveries of 1934-35 (see Ap
pendix).
As underlined in the obituary of Sobolev by Jean Leray
[L3] and the review of the [Lu] book by Yushkevich, the
discovery of generalized functions must be ascribed to
Sobolev in his articles of 1935 and 1936:
• The Cauchy problem in the space of functions, Proceedings (Doklady) of the USSR Academy of Sciences, 1935,
volume III (VIII), N 7 (67) (in French).
• New methods to solve the Cauchy problem for normal
hyperbolic linear equations, Mat. Sbornik, 1936, vol. 1
(43), 36-71 (in Russian).
In these two articles, Sobolev explicitly defines generalized
functionals as continuous forms on the space of differen
tiable functions of a given order m with support in a com
pact set K, for fixed m and K. He establishes the funda
mental properties of generalized functionals.
Why in French?
The year 1934, with the murder of Kirov, a popular Com
munist leader in Leningrad, was a turning point for the
USSR, which began to shut itself off, and where "ideologi
cal" struggles broke out, as illustrated by the campaign al
ready mentioned against Luzin. In this campaign the issue
of whether to publish in Russian or in a more widely ac
cessible language (as was done for most mathematical pub
lications until the war), played an important role. The pub
lication of the seminal article by Sobolev in Russian and in
French in the same volume of Doklady was purposeful.
Sobolev, who had criticized Luzin, was patriotically pub
lishing in Russian; the publication in French, though com
mon at that time, might be risky as a reminder of the so
cial background of the author. It is quite likely that this
double publication was perceived positively by Hadamard
at least, maybe even suggested by him.
In 1936 Hadamard was again in Moscow, returning from
China. In 1945 he made another journey to Moscow and
Leningrad as a member of the French delegation to the cel
ebrations of the 220th anniversary of the Russian Academy
of Sciences. He did not meet Sobolev (we shall see why).
However, as early as 1935, the reports he makes, back in
France, show Hadamard's awareness of the problems. He
evokes the tragic disappearance of a rising star, a clear al
lusion to the suicide of the young and brilliant mathemati-
Figure 2. S. L. Sobolev not reading mathematics, Novosibirsk, 1962.
VOLUME 26, NUMBER 1 , 2004 41
cian Schnirelman, a number theorist and topologist, in
1938. Hadamard praised the close relationships between
pure and applied science in the USSR, even in mathemat
ics [H2).
Sobolev's Discovery
Sobolev, inspired among other things by Hadamard's work,
first defmed generalized solutions of a wave equation, then,
in 1934-35, "generalized functions," without any mention
of a reference equation (contrary to the description in [Lu],
page 65), first under the name of "ideal" functions (as in
dicated by Mikhlin), probably in reference to the introduc
tion of ideal numbers by Kummer, then as "generalized
functions" in the seminal article of 1935. The older term
dangerously evoked idealist philosophy [M-Sh] at a time
when the Czech-born Marxist philosopher Kolman and the
other followers of "proletarian science" were stirring things
up in Leningrad. This hesitation over the naming, and the
double publication in Russian and in French, confirm that
Sobolev had a clear idea of the importance of his work and
its general character, contrary to the assertions made in
[Lu]. The reader can read in the Appendix a detailed analy
sis of the various articles written by Sobolev and his in
spirers and colleagues. There are hardly any clues to the
ongm of his discovery,
troduction) with a surprisingly partial and anti-chronolog
ical presentation of Sobolev's articles:
Soboleff Proceedings of the Soviet Academy of Sciences, 1, 1936, p. 279-282, Math. Sbornik, 4, 1938, 471---496, Friedrichs: (1939), . . . Kryloff (1947). . . . Some articles mentioned in previous notes were published later than the introduction of distributions, but the authors did not know of distributions, due to the slowness of the publishing process, the slowness of international communications, or delays in my publication. See also Soboleff's functionals (''New Methods" . . . ) .
The first two references have no critical interest. The last
one, "New Methods," is the article already quoted. On the
other hand, he "forgets" to mention the Doklady article of
1935 (received on 7/17/1935). Moreover, this Note remained
unchanged in later editions [S' 1 ] .
The Key to the Mystery
In his autobiography, Schwartz, after a minimal description
of the discovery made by Sobolev in 1935 as found in the
article not mentioned in Note 4 above, wonders ([S2), p.
236) why, after the war,
apart from Hadamard's
work. Sobolev had a clear idea of Sobolev did not continue
his work on generalized
Hadamard's curious and
enthusiastic mind could
not remain indifferent to
this ongoing work; he read
the 1936 article as soon as
the importance of h is work and its general character.
functions.
The answer is instruc
tive. Sobolev disappeared
from mathematical re
search circles and re
it got to the Ecole Normale Superieure. Moreover,
Hadamard always remained a subscriber to the main So
viet mathematical reviews [ManS] . Jean Leray was becom
ing a specialist in partial differential equations and also a
participant in the "prehistory of distributions" with his no
tion of weak solution [L1], the subject of his "Cours Pee
cot" in 1935 at the College de France. He told Sergei Sobolev
in the 1980s that he had discussed his 1936 article with Lau
rent Schwartz before the war (personal communication of
V. Chechkin, holder of the Chair of Partial Differential
Equations at Moscow University and grandson of S.
Sobolev).
It took Schwartz more than ten years, including several
years not dedicated to mathematics and some years of slow
maturation, to bring forth his work of 1945, which reuses
Sobolev's defmition. But meanwhile Sobolev had surrepti
tiously left the stage! Sobolev did not pursue his work in
this direction, though some work with Smirnov was not far
from it. He was awarded the Stalin Prize in 1941, and be
came a deputy of the Parliament of the Soviet Union and
Director (beginning in 1941) of the Steklov Institute. This
left Schwartz free to develop the theory. The missing parts
were mainly Fourier transforms and the topological struc
ture of the space of distributions (see below).
Moreover, the first publication in which Schwartz quotes
his sources [S1] contains a note (Note #4, page 5 of the In-
42 THE MATHEMATICAL INTELLIGENCER
frained from any foreign contacts from 1943 until 1953 be
cause he was busy with other activities in applied
mathematics-very applied indeed; he became the main
assistant of the director I. V. Kurchatov in "Laboratory 2,"
which was initially located within Moscow University, and
which became LIPAN, where the first Soviet atom bomb
was developed [Viz] .
It is not surprising that both in the West and in the East
great mathematicians played a critical role in the nuclear
projects [Go, p. 383). The complex physics of shock waves
involved in those projects entails the solution of nonlinear
equations, and Bethe (who told von Neumann about it) had
noticed the instability of the numerical approximation in the
solutions; the skills of top mathematicians were needed!
This work, essential to Soviet defense, led Sobolev to the
numerical solution of the equations for a spherical nuclear
reactor. He also studied the so-called gun effect and its vari
ation under neutron bombardment. This work is essential
in applications to assess water loss in reactors (Three Mile
Island and Chernobyl). In 1951 Sobolev received the most
prestigious civilian award, the Hero of Socialist Labour
medal. Naturally any foreign contact was totally forbidden
to him-even his wife did not know Sergei Sobolev's where
abouts when he left for periods of several months after
briefly visiting home. His publication list is much shorter
during this period-apart from his 1950 manual written in
Figure 3. Novosibirsk 1978: In the foreground 0. A. Oleinik, E. 5. Soboleva, 5. L. Sobolev, and French visitor Jean Leray.
a hospital where he was recovering with a broken leg-and
the main part of the work just mentioned is still unpub
lished.
The continuation is known in more detail; Sobolev re
sumed classic scientific activities in the 1960s. Mean
while, with Schwartz's book Distribution Theory and the
line of research pursued by him (tempered distributions
and Fourier transforms, applications of the theory of
topological vector spaces), he came to be considered the
father of the theory. The much-delayed acknowledgement
of Sobolev's paternity came only fifteen years later [L3,
L4] . Schwartz's main contribution, in the heritage of the
Bourbaki project (in a nutshell, the "algebraization of
analysis"), was to bring together Sobolev's definition and
the work begun by Dieudonne on topological vector
spaces in 1940 [Du] following Banach's famous Theorie des operations lineaires and Kothe's work. During the
period 1945-1950, Schwartz understood the importance
of applying TVS theory to the case of generalized func
tions.
This process of discovery by bringing together sepa
rate theories could be called "appropriation by bourbak
ization." It was frequently used-see, e.g., [Gr, Mi, S4]: a
beautiful idea by Minlos, which Gross had also had inde
pendently, was embodied ten years later in the theory of
"radonifying applications" without any acknowledgement
of Minlos. In the case of Sobolev, the author himself had
fostered the process! The term "bourbakization" of course
refers to the "Bourbaki project," which consisted in sin
gling out the deep structures of mathematics to reach the
degree of generality able to give a theory its extensive
power. This clear explanation of the topological vector
space structure paved the way to Schwartz's theorem of
kernels and to the theorem of Malgrange--a student of
Schwartz-on the existence of solutions to partial differ
ential equations in any open set in Euclidean space. Note
also that Schwartz's other student at that time, Jacques
Louis Lions (1928-2001), had been focusing since his dis
sertation on the use of Sobolev-style methods (Sobolev
spaces discovered in the 1930s ) , less elegant but more ef
fective than functional analysis, for example, for cubature
formulas. Lions later became the leader of French applied
mathematics.
"Percolation"
The "percolation" (or "illumination," as he also refers to it)
process discussed by Schwartz in his autobiography prob
ably consisted in the final linkage, made on the occasion
of a problem posed by Gustave Choquet, between Sobolev's
theory of functionals (defined as continuous linear forms)
and the work of Dieudonne and later Dieudonne and
Schwartz. Actually, contrary to the assertions in [Lu], by
January 1946 Schwartz had a good knowledge of Sobolev's
work: a participant recalls that during his "Cours Peccot"
at the College de France "he constantly had Sobolev's name
in his mouth."
Conclusions and Issues
Theory and Practice- East and West
In the era of triumphant socialism in Russia, science was
expected to be at the service of the people for the progress
of mankind. This notion was in fact the new face of an an
cient cultural tradition in Russia, still vital in Saint Pe
tersburg, even in the field of mathematics. One thinks for
example of Pafnutii L'vovich Chebyshev, whose concern
for linkages, ways to cut up fabric, and laws of chance,
were closely related to highly abstract concerns. Cheby
shev has very explicitly described [C] the mutual benefits
of mathematics and practical applications. In the case of
functionals, Smimov, in a profound analysis, shows how
central the experimental sciences remained among the
VOLUME 26. NUMBER 1 , 2004 43
concerns of Russian mathematicians (see Appendix). For the Russian school, in the period discussed here but also later, the value of mathematics is measured by its effectiveness. Even general topology, through Tychonov to Pontryagin, has applications to the study of control systems. More recently, this is also the case in the work of Arnold and his school. We can imagine the difficulties experienced by Lusitania, the famous school created by Luzin in Moscow around function theory, and largely inspired by (German) set theory or (French) function theory. In contrast, France, the country of Descartes, Galois, and Bourbaki, favored an interest in mathematical investigation "pour l'honneur de l 'esprit humain" (a phrase coined by Jacobi): the value of a theory is assessed by its degree of generality-a purifying quality of generality, synonym of efficiency, and evidenced in the connections between apparently remote domains for the production of new theories, and by the elegance of the concepts [B2]. (Similarly in Germany.) For Schwartz, for example, distribution theory develops as he associates the Sobolev definition to the theory of topological vector spaces, thus arriving at the properties of the topology of distribution spaces. This will make possible the work of his students Lions and Malgrange, after the presentation of the kernel theorem at the 1950 Congress in Cambridge, USA; this theorem was the cherry on the cake, and earned him the Fields Medal and the later paternity-in the West at least-of distributions. These two views of mathematics and their role were present simultaneously in both countries, and sometimes in the productions of the same mathematician, as in the cases of Gel'fand in the USSR or, earlier, Fourier in France. In the period we are interested in, the emphases were as we have mentioned above. Though this question goes beyond the scope of this article, we note that recent developments in physical and mathematical sciences show continuing ten-
Figure 4. The young Laurent Schwartz.
44 THE MATHEMATICAL INTELLIGENCER
sion between effectiveness and rigor (Feynmann integral, string theory; [JQ] for example gives an account of the debate). Should one rejoice that the political upheavals of the last decades threaten to standardize worldwide the practice of mathematical science and the answers to this "essential tension" [Ku]?
Theory and Practice- Probability and Measure
Measure theory and its relevance to probability deserve particular scrutiny: it was the first serious stumbling block in the development of the Bourbaki project [B2]. From the point of view that interests us, distribution theory obviously served as a weighty "ideological" argument at the time. As
an illustration, here is an excerpt from the introduction of [Bl] concerning measure theory: " . . . Thus, integration theory is connected, on the one hand to the general theory of duality in topological vector spaces, and on the other hand to distribution theory, which generalizes certain aspects of the notion of measure, and which we shall present in a later book" It can be seen how much this "structuralist" stancealso present in Schwartz's approach to distributions-concealed the real nature of the phenomena in question, for example the subtleties of random processes. (Another illustration of errors of judgment is found in Andre Weil [Wl], [W2]: " . . . The time has come to try, through closer analysis, to split up Lebesgue's discoveries into various elements in order to identify what is essential in the manipulation of an integral, and what is relevant to the specific operations over sets on which we are most frequently working.")
Rather than disregard for potential applications, it was the desire to give priority to the structure over the phenomenon, and to the architecture over the portrait, that caused a delay of fifteen years in French research on probability. Quite ironic, in the country of Laplace, Lebesgue, Borel, and most particularly Paul Levy, Fortet, Loeve, Ville,
Figure 5. Laurent Schwartz presented with a Vietnamese peasant hat.
and Doeblin, who, in the 1930s, were at the forefront of the
revival of probability theory by developing new trajectorial
aspects of processes, which were to have a wealth of appli
cations in the second half of the 20th century, including in
the solution of the great problems of classical analysis and
its renewal (PDE, Dirichlet's problem, potential theory, etc.).
We hope in subsequent work to revisit this question, on
which Schwartz himself was self-critical ([S2]).
Problems of Communication
From the Russian Revolution to the 1970s, interchanges be
tween mathematicians suffered from many difficulties be
cause of the lack of intellectual freedom in USSR, the cold
war, and internal conflict within the Soviet cultural and uni
versity system from the 1960s. Thus the Soviet delegation
as a group declined the invitation to the 1950 Congress at
Harvard, at the height of the Korean War. This was the Con
gress during which Laurent Schwartz was awarded the
Fields Medal. We suppose that Kolmogorov, even though
he was a member of the medal committee, did not even
mention the name of Sobolev, then assistant director of LI
PAN. In the 1960s, more problems appeared: we witnessed
difficulties of exchange and of publication of mathemati
cal articles in the USSR, which led for example to the creation of the review Funktional'niiAnaliz by Israel M. Gel'
fand in the 1970s.
Mathematics and Politics
At the end of the interview used as a working document by
Liitzen, Laurent Schwartz makes a surprising linkage be
tween distribution theory and political democracy, quoting
the eminent British Marxist historian Moses Finley, for
whom democracy was discovered by the Greeks: "It was
the Greeks, after all, who discovered not only democracy,
but also politics. I am not concerned to deny the possibil
ity that there were earlier examples of democracy . . . . What
ever the facts may be . . . their impact on history, on later
societies, was null. The Greeks, and only the Greeks, dis
covered democracy, precisely as Christopher Columbus,
not some Viking seaman-discovered America." [Fi]
In other words, Sobolev is cast as the Viking, Schwartz
as Columbus. Beyond the general debate on philosophic re
alism (was democracy discovered or invented? and distri
butions?), it is clear that neither mathematics nor political
concepts emerge ex nihilo, and that scientific work is a
process: Schwartz comes after Sobolev, Dirac, and even
Euler! ( cf. Appendix) In retrospect and based on the ex
amination above, this comparison appears to be not merely
excessive but unjustified. The same field of mathematical
analysis saw the emergence of the point of view of alge
braic analysis, whose importance seems much more
promising, if only-in Bourbaki's view-by the "bridges"
which it builds. Going farther, and taking into account the
frequent cases where Schwartz left some things unsaid (see
above), we can wonder whether there may be an allusion
to the ideological power gained by Bourbaki, sometimes
against the will of some members. For example, Claude
Chevalley remained a libertarian all his life. In a beautiful,
nostalgic interview [Che], he confesses that he thought he
was "enlightening the world of mathematics," in a common
desire for renewal. It is in fact in Chevalley's writings that
we find the most interesting remarks on the relationships
between Bourbaki and political thinking: he says that read
ing the political theorist Castoriadis made him understand
the wrongness of his view of mathematical logic!
Schwartz's aura personified that of Bourbaki: modem
mathematics and educational reform, the role of the
scholar in pronouncing what is right, and indirect power in
the life of society: the aura of the mathematician, which
Schwartz knew how to apply "for the good cause," is quite
evocative of Greece. Frequently, top mathematicians seem
to confound mathematical action, political struggle, and
moral principles.
The disappearance of such a strong personality evokes
the end (announced by some) of the era of "great narra
tives"-the disappearance of myth-creating romantic ac
tors (as Bourbaki, the dream of distributions). Is this hap
pening? Time will tell and History will judge.
Acknowledgments
The author thanks the Sobolev family for permission to use
the photographs of S. L. Sobolev; the photographs of
L. Schwartz are by kind permission of the editors of Pour la Science.
REFERENCES [Be] Beaulieu, Liliane. Bourbaki. Une histoire du groupe de mathemati
ciens franc;ais et de ses travaux (1934- 1944). Ph.D. Thesis, Univer
site de Montreal, 1 990, Paris, 1 992.
[Boc] Bochner, Salomon. Review of L. Schwartz's Theorie des distri
M Yushkevich, A P. Encounters with Mathematicians, Golden Years of
Moscow Mathematics ed. S. Zdravkovska, Peter L. Duren, History of
mathematics, vol. 6, American Mathematical Society, LMS, 1 991 .
(W1 ] Weil, Andre. Calcul des probabilites, methode axiomatique, inte
gration, in Revue Rose, vol. 1 of CEuvres completes, pp. 260-272.
Springer-Verlag, Berlin, New York, 1 979.
[W2] Weil, Andre. L 'integration dans les groupes topologiques et ses
applications, Hermann, Paris, 1 940; CEuvres, vol. 1 . See also the
commentaries on pages 551 -555.
Added in proof
Kutateladze, S. S. Sergei Sobolev and Laurent Schwartz: two fates and
two fames (in Russian), Novosibirsk, Sobolev Institute, preprint 1 2 1 ,
Oct. 2003.
Appendix
I.
Adolf P. Yushkevich: Some remarks on the history of the
theory of generalized solutions for partial differential
equations and generalized functions. Istoriko-matematicheskie issledovanie, 1991, 256-266 (Russian).1
Since 1968, I have been publishing appreciations of famous
French mathematicians on their Russian colleagues, on the
occasion of their applications as foreign member of the
Academie des Sciences de Paris (the election process has
remained unchanged since the middle of the 19th century).
Often these appreciations are interesting from the point of
view of the history of relationships between the scientists
of our two countries. Naturally these appreciations reflect
the personal point of view of the speakers, and frequently
the judgment on the candidates also depends on the inter
national situation. The appreciations published so far are
those of Chebyshev, Lyapunov, Bernstein, Vinogradov,
Lavrent'ev, and Kolmogorov. It was a great pleasure for me
to receive Paul Germain's authorization to publish Jean
Leray's appreciation on his colleague Sergei Sobolev.
II.
The best appreciation of the work of Sobolev is in the ref
erence [4], published for his 80th birthday: Sergei L'vovich
Sobolev (6. 10. 1908-3.01. 1989) completed his studies at
Leningrad University in 1928. His doctoral dissertation ad
visers were N. M. Gunther (1871-1941) and V. I. Smimov
(1887-1974), both students of V. A. Steklov (1863-1926), himself a student of A. M. Lyapunov (1857-1918). For most of their lives, these four professors worked on the theory of differential equations, the theory of partial differential equations, and their applications in mathematical physics and mechanics. They were eminent members of the mathematical school of Saint Petersburg, later Leningrad, headed by P. Chebychev (1821-1894), one of the professors of Lyapunov. As a student, Sobolev also followed lectures by Fikhtengolts (1888-1959), who was the first to develop in Leningrad the study of functions of a real variable, which prompted the extensive work of the school of Moscow with D. F. Egorov (1869-1931), N. N. Luzin (1883-1950), and their students.
Sobolev belongs to the fourth generation of Chebycheffs school, which systematically exploited the relationships between mathematics and the concrete problems of sciences and technology, without precluding a concern for the introduction of abstract questions-often over and above practical issues (even in number theory). It is necessary also to stress that Sobolev's professors themselves were already using the most recent developments in mathematics-topology, the theory of functions of a real variable, new areas of the theory of functions of a complex variable, integral equations, and the new area of functional analysis. Sobolev's research work began immediately after the end of his studies, in the department of seismology of the Academy of Sciences headed by V. I. Smirnov. While still a university student, he presented a Master's thesis on a topic suggested by Gunther. At the Institute of Seismology, Sobolev again conducted work closely related to this topic previously suggested by Gunther, viz. the analytical theory of partial differential equations and in particular the propagation of elastic waves. Some of his first publications were cosigned with Smirnov. On June 29, 1930, Sobolev presented a paper at the first Congress of Russian Mathematicians: "The Wave equation in a heterogeneous isotropic environment," an abstract of which appeared in the Notes aux Comptes-Rendus de l'Academie des Sciences de Paris. This work interested Jacques Hadamard (1865-1963), who attended the Congress and himself made a presentation on a topic close to Sobolev's: "Partial differential equations and the theory offunctions of a real variable" ([5], in French and in Russian). Sobolev's early work (summarized, after those of Gunther and Smimov, in section 8 of [6]) was already getting considerable attention from Soviet mathematicians, and Sobolev, not yet 25, was elected on 01 .02. 1933 a corresponding member of the Academy of Sciences. He was later elected a member on 29.01.1939.
Ill. In 1932 Sobolev enters the Physico-Mathematical Institute created by Steklov in 1921 . It is in this period that he develops his most important work, which establishes the beginning of the theory of generalized functions. He is the first to define them mathematically and to set about studying their fundamental properties. A summary of his ideas was written by Smirnov ([7] , p. 187-191). Sobolev started
articulating his ideas on distributions, which he calls functionals and were later called "generalized functions," from the late 1920s and the early 1930s-or possibly earlier. He presented them in his lecture "Generalized solutions of the wave equation" on June 29, 1934 at the second Congress of Soviet Mathematicians in Leningrad. Here is the laconic summary by the author: "The class of functions which we can consider as solutions to the wave equation from the classical point of view consists of twice-differentiable functions. But in various practical applications it seems convenient to consider functions with singularities of a welldefined type. We introduce a space of integrable functions in the sense of Lebesgue, in which it is possible to defme the generalized solutions of the wave equations as the limits of twice-differentiable solutions. Using a simple integrability criterion, we give a necessary and sufficient condition for a function to be a generalized solution, and we establish the link between the usual solutions and generalized solutions. Finally, this theory is applied to some concrete examples" ([8] , p. 259). Leray sees considerable importance in Sobolev's work in the theory of generalized functions, called distributions in western mathematical literature, but he dates them back to 1935 and 1936, not earlier. Smimov ([7], p. 187) refers to the article [8] of 1935 and to two other articles quoted in [9) and [ 10). In the bibliographical list [9] , the lecture of 1934 is not even mentioned.
In his two classic volumes on the history of mathematics in the last two centuries, Jean Dieudonne writes that Sobolev began the study of generalized functions in 1937 ( [11 ] , p. 2, [7], p. 171). In the 1982 article "Fonctions generalisees," Vladimirov quotes [9) along with the "generalized solutions" article ([12] , vol. 3, p. 1 102-1 1 10 and 1 116-11 17). It is only in the article written on the occasion of Sobolev's jubilee in 1989, that one of the authors, also named Vladimirov, indicates the article of 1934 "in which the theory of generalized functions appears for the first time." Notoriously, the establishing of chronological priority between several authors of a scientific discovery is not always a harmonious process, but nowadays it does not lead to such negative effects or violent quarrels as in the case, for example, of Newton and Leibniz, the creators of infinitesimal analysis.
IV.
The prehistory of the theory of Sobolev's generalized functions has not been investigated much. Gunther's work should probably be ascribed a role in laying down the core notions; in particular, his smoothing method for insufficiently differentiable functions, which is often quoted by Smimov ([7] p. 184). A path toward the theory of generalized functions is found earlier still in Hadamard's work, starting with his remark "on functional operations" and his "Le<;ons sur la propagation des ondes et les equations de l'hydrodynamique" (Lectures on wave propagation and the equations of hydrodynamics) published in 1903. The academician Steklov drew attention to these sources during the presentation of Hadamard's work when he was elected a corresponding member of our Academy on December 2, 1922.
VOLUME 26, NUMBER 1 , 2004 4 7
The article by Steklov is deep and definitely important.
He insists in particular on the significance of the first arti
cle, where Hadamard uses for the first time the term "func
tional," and he discusses in detail the results of the second
article. He insists in particular on the existence of "shock
waves" in compressible liquids and elastic bodies. One re
mark by Steklov is particularly interesting: issues of hydro
dynamics, translated into the language of mathematical
analysis, coincide with the theory of the characteristics of
independently from any physical origin." This remark shows
that Steklov understood perfectly the significance for later
applications of abstract basic research pursued in complete
independence from their use. Moreover he uses the classic
terminology he is familiar with (and uses the term "func
tional" only occasionally), and he could not foresee that a
few years later, it is essentially in his own institute that the
groundwork for the theory of generalized functions was go
ing to be laid. This speech by Steklov was not published un
til 1968 ([1] p. 1 10-115). As regards Hadamard's advances
toward the theory of generalized solutions for partial dif
ferential equations and generalized functions, let us quote
a statement made by G. Shilov (professor at Moscow Uni
versity from 1917 to 1975, and a recognized specialist on
this question), on February 10, 1964, during a memorial ses
sion of the Moscow Mathematical Society: "In solving hy
perbolic equations, Hadamard essentially introduces the de
vice of the theory of generalized functions of one or several
variables. This discovery remained dormant at the time
(Hadamard was many years ahead of the thinking of math
ematicians of his generation), and it was only in the mid
fifties that generalized functions spread worldwide in ques
tions of analysis" ( [ 13], p. 185). Shilov concludes by quoting
Szolem Mandelbrojt's words of 1922 about the famous
"Readings on Cauchy's problem" (translation to French in
1932, and to Russian in 1978): "The notions developed in
this work lead to general topology and to functional analy
sis, and the introduction of the notion of elementary solu
tion has a high degree of generality with respect to distrib
utions (generalized functions)" ([ 14], p. 4-5). Furthermore,
we owe to Hadamard the terms "functional" and "functional
analysis." Jean Leray also mentions this precursor work
What we say here does not by any means detract from
Sobolev's achievement; he is the first to give a rigorous def
inition-and in several ways-of the modem notion of gen
eralized function, and to lay the bases of later developments
in various domains of the theory of generalized solutions to
partial differential equations and generalized functions, as
an autonomous domain of analysis.
v. Almost all of Sobolev's work on the theory of solutions and
generalized functions was published in Russian, except the
article in French of 1936 ([9], 22). So it is not surprising that
in other countries [than the USSR] this work did not attract
immediately the interest it deserved. This remark also ap
plies to the book Some Applications of Functional Analysis in Mathematical Physics (Leningrad, 1950), which cor-
48 THE MATHEMATICAL INTELLIGENCER
responds to the course that Sobolev taught at the time at the
University. This book was not translated into English until
1963 (into German in 1964). It is quoted several times by
Leray, and, as noted by V. I. Smirnov, "this book played an
important part in the use of the modem ideas and methods
of function theory and functional analysis for the solution of
problems of the theory of partial differential equations" ([6]
p. 191). In Russia, Sobolev's new ideas, following those of
his masters Gunther and Smirnov, diffused fairly quickly, and
were extended and developed from the 1950s.
In the diffusion abroad of these new directions of math
ematical analysis, a major role must be ascribed to the book
Theorie des Distributions (in two volumes) by Laurent
Schwartz, a corresponding member (1973), then a full
member (1975), of the Academy of Sciences, and a profes
sor at the Ecole Polytechnique. Several articles published
by Schwartz between 1945 and 1948 already used the ex
pression "distributions." After the publication in 1950-51 of
Schwartz's book, distribution theory developed consider
ably and received numerous new applications.
The first historical study on research on distributions,
published by Jesper Liitzen in 1980, contains an accurate,
flawless mathematical analysis of the works of Sobolev,
Schwartz, and many earlier or contemporary mathemati
cians. In spite of all these achievements, Jesper Liitzen's
book has some gaps and, from my point of view, uncon
vincing evaluations, which can be explained by an imper
fect knowledge of work in Russian generally and of Sobolev
in particular (although his bibliography contains 1 1 refer
ences which were translated into English, as well as the
1950 book already mentioned and the thick course book in
its third version of 1954). Leray's note on Sobolev's work
is a substantial complement to Liitzen's study.
Without trying to write the history of the question, I shall
make here some remarks on Liitzen's book First, I cannot
agree with his evaluation of the results obtained by
Sobolev, then Schwartz, and their place in the development
of distribution theory. The essence of the differences be
tween their theories, according to Liitzen (p. 64), is that for
Sobolev distributions are a technique to resolve a specific
problem, while Schwartz developed distribution theory un
der multiple angles, and applied it to formulate and resolve
rigorously numerous problems. It is true that in 1934
Sobolev began with Cauchy's problem for the wave equa
tion (which is hyperbolic), but then he did not limit him
self to one of the applications which he had introduced,
and he considerably enriched them, as shown by Jean Leray
("work whose scope, variety, and power are admirable").
It is also true that these various contributions published in
successive articles were not collected into a monograph,
which would doubtless have had the seminal role of
Schwartz's book-which became the basic book for nu
merous researchers abroad and here. Liitzen briefly sum
marizes the fundamental difference between the works of
Sobolev and Schwartz: "So Sobolev invented distributions,
but distribution theory was created by Schwartz" (page 64).
Variants of this reflection occur in the book On page 67,
after quoting Lyusternik and Vishik's words in a speech pro-
nounced on the occasion of Sobolev's fiftieth birthday (1959), Liitzen supports what they say but immediately adds that "further development of the theory was not Sobolev's but Schwartz's achievement." Without intending in the least to detract from the essential importance of Laurent Schwartz's book of 1950, I find more balance in S. Vladimirov's judgment ([12], vol. 4, p. 1104): "The foundations of the mathematical theory of generalized functions were laid by Sobolev in 1936 with the aim of resolving Cauchy's problem for hyperbolic equations, but in the 1950s L. Schwartz gave a systematic statement of the theory and mentioned numerous applications." He could have added that the systematic account in modem terminology in Schwartz's work overshadowed Sobolev's. As regards Schwartz's possible knowledge of Sobolev's previous discoveries, according to the statements made by L. Schwartz in 1950-51 and in 1974, the latter did not know of them before 1945 (p. 67 of [16]). Elsewhere Liitzen writes that Schwartz's attention was called to Sobolev's work by Leray in 1946. Certainly Sobolev and Schwartz arrived at their discoveries of "generalized functions" and "distributions" by different paths-but certainly, too, there is no reason for assigning Sobolev's work to the "prehistory" of distribution theory, as Liitzen does three times (pages 64, 67, and 156).
More generally Liitzen devotes more attention in his book to Laurent Schwartz than to Serge! Sobolev. The statement of the results according to the bibliography is correct; but he could have gone into more detail. In this respect, Leray's note contains valuable complements, but even this note does not contain enough bibliographic data on Sobolev. These indications could have and should have been enriched by the inclusion of Lyustemik and Vishik's text (which Liitzen quotes and uses). There is no reference to Sobolev's teachers, in the text or in the reference index. L. Schwartz's biography is given a very contrasting treatment. In chapter 6, the reader is informed of all the stages in Schwartz's life, the names of the professors at the Ecole Normale (Leray, Leyy, Hadamard), Schwartz's membership in the Bourbaki group, his discovery in six months of distributions, his conversations with de Rham (also mentioned by Leray), etc. All this information is valuable, and it is regrettable that Sobolev's mature work is treated by Liitzen in merely half a page (p. 60).
To be sure, distinctions between the "prehistory" of a theory and its development are a matter of convention. The notion of "distribution" appeared in various authors of the beginning of the 20th century, and one could even go back to Euler (see below), whom Liitzen also mentions. However, we distinguish ideas belonging to prehistory-already born but not introduced yet into a well-defined framefrom ideas belonging to the history of a theory-where they have a precise definition and we focus on the study of their specific properties. Thus one reasoned with functions of one type or another in ancient Greece, in the Middle Ages, and at the beginning of the modem period, but functions as objects of mathematical investigation, in all their generality, appear only at the end of the 17th century. However, the title of Liitzen's book is "Prehistory of etc.", which
situates Schwartz-to whom the largest part of the book is devoted-as part of the prehistory of the theory.
If Liitzen had restricted his study of the prehistory of distributions to Western Europe, it would have been natural to insist on Schwartz's work But for a study of the development of mathematics as a worldwide process (which it has always been), the book's structure seems inadequate. This is shown in any case by the historical study of the facts in our country. Sobolev, following his teachers, played an important role by laying the bases of the numerous studies which began even before 1970, the publication date of Smimov's already mentioned article, which presents a summary of twenty years of work in the theory of partial differential equations-elliptic, hyperbolic, parabolic, or mixedas well as contributions to the general theory, and work by 0. A. Ladyzhenskaya, S. G. Mikhlin, N. N. Ural'tseva, and others. All this work was not isolated from foreign research. Collaborations took place between all the countries involved, albeit it was sometimes made difficult by problems of communication and the lack of personal contacts (which developed a lot in recent years, their earlier scarcity having been supplemented by numerous reference periodicals). The objective of my remarks on the history of the theory of generalized solutions and generalized functions is not only to clarify the conclusions of Liitzen's book, but also to introduce the presentation of Sobolev's candidacy by Leray, which is an essential complement to the Danish historian's account.
I must make a few additional remarks on the protohistory of the theme, which led to the solution of the equation of the vibrating string and to the dispute between d'Alembert and Euler, which went on for almost thirty years from 1750, and which somehow involved all the mathematicians of the 18th century. Briefly stated, d'Alembert completely excluded the case of discontinuity of a derivative, and even more stringently of the function itself. In my book on the history of Russian mathematics before 1917 (Nauka 1968), I showed that Euler, from physical considerations, deemed it necessary to admit, as solutions of problems of mathematical physics, what he called "broken" functions and curves; we would say that the initial position of the string and its initial velocity are functions of position which are continuous by segments, i.e., where discontinuities (in the modem sense) of the first two derivatives are allowed.
Not having the necessary mathematical means at his disposal, Euler gave a simplified geometrical description of the distribution of waves and their reflection for a string fixed at a single point. I take the liberty to quote my own book: "Threads are woven here between Euler's ideas and the new methods of the 20th century up to Sobolev and Schwartz's generalized functions" (p. 166, 169). During the more recent history of the notions of solutions of partial differential equations, the historian S. S Demidov relied as I did on a quote from d'Alembert (Opuscules, volume IX). "Euler essentially constructed a solution of the equation as a generalized solution-for which a correct defmition and, even more, construction, were beyond the capacities of the mathematicians of that time" (p. 179). I added in my book that because of its
VOLUME 26, NUMBER 1 , 2004 49
A U T HOR
IIQU8S bclis· cont Of
practical utility, Euler's construction had been the object of
the attention of numerous mathematicians over time. I used
Truesdell's well-known study of Euler's work in hydrody
namics and elasticity. References in Russian on this subject
are not known by Ltitzen (for example in volume IX of
d'Alembert's Opuscules, he only mentions Demidov's lecture
at the International Congress on the History of Mathematics
in 1977, which he also uses extensively on page 15). Ltitzen
also traces back to Euler the notion of generalized solution,
and he draws a parallel between Euler and Sobolev. Using
the definition of generalized solutions as limits of series of
classical functions, Ltitzen notes that this idea can be found
in Euler in 1765 and Laplace in 1772, and that the rigorous
definition was introduced in 1935 by Sobolev and later by
the other authors, in particular Schwartz in 1944.
In conclusion, we would say that Euler introduced func
tions which could seem strange to his contemporaries, for
example, (- 1 )x, x being an arbitrary real number . . . but
not the delta function!
REFERENCES [1 ] French-Russian Scientific Relationships, A Grigorian, A Yushke-
The Science of Conjecture: Evidence and Probabil ity before Pascal by James Franklin
BALTIMORE, THE JOHNS HOPKINS UNIVERSITY PRESS,
2001
600 pp. $22.50 PAPER ISBN 080-1 86569-7
REVIEWED BY NORMAN LEVITT
Franklin is nominally a mathematician and his book shows a mathe
matician's touch when it deals directly with mathematical matters, but this review must begin with a warning: this is not primarily a history of mathematics, nor is it a book that a strong mathe
matical background makes particularly accessible. Rather, The Science of Conjecture is a history of a lengthy philosophical investigation that has spanned a number of eras and civilizations, and which, of course, continues even now, with no discernible end. The central question is one of partial belief, belief that may be quite pronounced, but which stops short of demonstrative or "mathematical" certainty. Very simply, what kind of evidence and how much of it ought to be necessary to persuade a reasonable inquirer that it is more appropriate to accept a proposition than to reject it? How ought we to order degrees of belief that lie somewhere between absolute conviction and utter dismissal? What degree of belief is necessary to justify an action with grave consequences? This is not the sort of question that mathematicians are given to worrying about, at least not when going about their mathematical business. But it is the central concern of "practical reason," and in various forms it confronts us in many societal roles-as jurors, for instance, or as investors.
Franklin's view is that pre-modem and early modem thinking on this ques-
tion had a considerable influence on the origins of probability calculus, in the work, principally, of Fermat, Pascal, and Huygens. His history, which begins in classical times, closes at this juncture, the take-off point, we might think, of Western scientific rationalism. But by no means would he have us believe that formal, quantitative probability theory (or its evolution into statistics) made earlier modes of inquiry
obsolete or brought them to completion. Rather, the probabilist's quantification of likelihood and expectation, even in its most sophisticated development, covers only a narrow range of experience and informs our judgment at best in limited measure. Thus the meditations of ancient Aristotelians and cloistered Scholastics are far from being mere vestigial echoes of benight
edness. They touched on important epistemological points that have not become that much more transparent even a millennium or two down the line.
I am not an historian of ideas. My knowledge of the secondary, let alone the primary, literature relevant to The Science of Conjecture is too scant to be called even minimal. Yet I will risk the
opinion that Franklin's book is deeply researched and intensely learned. It is a throwback to the days when humanist scholarship meant thorough saturation in a vast ocean of sources, rather than picking out two or three texts and weaving elaborate postmodem curlicues around them. Franklin knows his authors, scores of them, thoroughly, and is scrupulously concerned to represent them with full fidelity to their ideas and their originality. He is not loath to judge those ideas, but his judgments do not come cheap; they are the fruit of care
ful reading and careful thought, not of a prefabricated agenda.
The book is roughly chronological in organization, so that the work of the inventors of mathematical probability culminates the inquiry. But the more important structural principle is to
iar with John Aubrey's Brief Lives, whose quirky biographical sketches of
noted contemporaries include dozens
of 17th-century British mathemati
cians, will appreciate this point.
In the end, however, Franklin does
not see quantitative probability theory
as the end point of the schools of
thought he reviews. Nor should he. The
word "probable" as now used, even by
scientists, rarely falls within the rubric
of quantitative probability calculus. It
is easy to see this even within the dis
course of mathematics itself. What do
we mean when we aver that conjecture
X is "probably" true or that a given strat
egy is "likely" to succeed in proving it?
Pretty clearly, there is no way to give a
quantitative significance to these asser
tions, nor, indeed, to translate them into
any suitable formalism. Worse, we re
ally have no idea of a systematic epis
temology that might justify them. Yet re
marks like these are really the working
discourse of research mathematicians;
we work on conjectures that seem
probable, using methods that seem
likely to get somewhere, but all this
"seeming" is tied up in unaccountable
subjective intuition, informed by anal
ogy and experience. The philosophical
status of all this is very unclear. The
same applies to science all down the
line; we think of string theory as prob
able (or not) and likewise for anthro
pogenic global warming or prions as the
cause of Alzheimer's. These judgments
are the stock-in-trade of everyday sci
ence. But there is no widely accepted
justificatory theory of judgment that
stands behind them. In that sense, we
are hardly further along than the cen
turies-dead heroes of Franklin's saga.
What Franklin makes of this is im
portant to note. Good ideas and suc
cesses, even partial ones, to hard prob-
lems are hard-won and tend to rest on
centuries of missed leads and blind al
lies. Thus they should be all the more
precious to us. We are obliged, then, to
reject the flighty notions now far too
popular that the "episteme" is tran
sient, arbitrary, and endlessly mutable,
that fundamental ideas are merely cul
tural fashion statements, and that sci
ence is bound, eventually, to recede as
surely as it once advanced. Science, for
Franklin (and, I hope, all of us), is a cu
mulative achievement as much rooted
in obscure toil as in famous triumph,
which should deepen, rather than di
lute, our esteem for it.
Department of Mathematics
Rutgers University
New Brunswick, NJ
Piscataway, NJ 08854
USA
Prime Obsession by John Derbyshire
WASHINGTON, DC, JOSEPH HENRY PRESS, 2003,
$27.95, ISBN 0-309-08549-7
The Music of the Primes by Marcus du Sautoy ��---------------NEW YORK, HARPER COLLINS, 2003, $24.95, ISBN 0-06-
621070-4
The Riemann Hypothesis by Karl Sabbagh NEW YORK, FARRAR, STRAUSS AND GIROUX, 2003,
$25.00, ISBN 0-374-25007-3
REVIEWED BY HAROLD M. EDWARDS
The nearly simultaneous publica
tion of three books for the general
public about the Riemann hypothesis
(hereinafter referred to as RH) can
probably be explained by the million
dollar prize offered by the Clay Mathe
matics Institute for the resolution of
RH (large sums of money evoke inter
est) and by the many books that were
sold to the general public about Fer
mat's last theorem in the wake of
Wiles's proof (selling books is the goal
of publishing). Whatever the reason for
this sudden flood of interest in one of
the frontiers of pure mathematics, it is
a welcome, if surprising, phenomenon.
Mathematicians are probably the
worst people to review such books. An
architect I once met pleased me by
telling me how he had become con
vinced of the power and beauty of
mathematics by reading a certain pop
ular book on mathematics that he
named. I was so gratified by this devi
ation from the usual "I was never any
good at math" that I rushed to the li
brary to see the book My disappoint
ment was great. To me, it was full of
dubious assertions, exaggerations, over
simplified history, and explanations of
mathematical ideas that could impart
no understanding other than false un
derstanding. But, as the architect
plainly demonstrated by his own ex
ample, the book had achieved its goal
brilliantly, at least for one reader.
Moreover, I have had the experi
ence-and most mathematicians I
have asked about it have had the same
experience-of rereading a book for
nonmathematicians that I had read in
my youth and that I remembered as
having inspired me, only to discover
that it had many explanations I now
found to be misleading at best and
statements I now found to be down
right wrong. Would I recommend the
book to a young reader today? My own
experience would say yes, but my
judgment as a mathematician would
say no.
These considerations have been on
my mind as I pondered these books on
RH. All three are quite well written, and
I can easily imagine any one of them
capturing the nonmathematical reader's
fancy. And, overall, I think each pre
sents a reasonably accurate picture of
the history of RH and the present-day
mathematicians who are working on it.
For that, the mathematical fraternity
can thank all three authors. But, after
all, I am a mathematician, and it is only
as a mathematician that I can evaluate
the books.
My lack of success over the years in
explaining the irrationality of Y2 to
reasonably able liberal arts students
has left me without much hope that ex-
VOLUME 26, NUMBER 1 , 2004 55
planations of RH intended for inter
ested non-mathematicians will suc
ceed. In other words, I am among the
"many people" who, according to the
first sentence of Karl Sabbagh's Pro
logue, "would say that the task I am
embarking on . . . is futile." He defends
his project by comparing it to anthro
pology and to "describing a remote
tribe whose customs and language are
unfamiliar to the reader, but whom I
understand enough to convey some
thing of their inner and outer lives."
Readers of the Mathematical Intelligencer, as members of that remote
tribe, will be interested to know
whether the descriptions he provides
are accurate and whether they illumi
nate our tribal culture. On both counts,
I am unenthusiastic.
The best parts of the Sabbagh book
are indeed the anthropological ones.
He tells who has worked on-or is
working on-the Riemann hypothesis,
how they became interested in mathe
matics and in this particular problem,
how they view their chances for suc
cess, and so forth. But what makes us
a tribe is our peculiar culture, and there
is no way to describe the interactions
of key members of the tribe without go
ing into the substance of our culture.
On this, Sabbagh is an unreliable guide.
For example, on p. 41 (page numbers
refer to the American edition-the orig
inal English edition is more compact,
so for example, this passage is on page 33 of that version) he says: "So, calcu
lating the value of the sum I lln8, which Riemann believed was possible
but couldn't say so for certain, would
result in a totally accurate number for
the number of primes less than n." Well, calculating the value of I 1/n8 is
certainly possible when the real part of
s is greater than 1, but Sabbagh does
not say which particular values of s will
be needed to produce his "totally ac
curate number." (Later in the book,
complex numbers are introduced, and
on the next-to-last page analytic con
tinuation is mentioned in passing, but
at this point I lln8 is far from being the
same thing as ?(s).) This "totally accu
rate number" must refer to Riemann's
explicit formula for 1r(x), which Sab
bagh seems to believe (see also the end
of Chapter 1) depends on RH; but in
56 THE MATHEMATICAL INTELLIGENCER
fact what is needed for this formula is
not the evaluation of ?(s) for one or
more values of s but a knowledge of
the zeros of ?C s) in the critical strip; the
formula is valid whether or not the ze
ros are on the critical line. This misap
prehension about the meaning of RH
probably underlies his answer, at the
end of his Prologue, to the question,
"Why is it [RH] so important? . . . A
proof . . . would . . . tell mathematicians
a huge amount about an important
class of numbers-the prime numbers,
which dominate the field of pure math
ematics." The notion that such a goal
accounts for the fascination of RH is a
profound misunderstanding of our
tribal culture, like believing moun
taineers want to climb Mount Everest
in order to get somewhere.
For another example, he often spec
ulates about who might or might not
prove the Riemann hypothesis. On p.
219, Martin Huxley is said to have "both
the desire and the ability to prove the
Riemann Hypothesis." On p. 240, we are
told that "many . . . feel that if anyone
is going to prove the Riemann Hypoth
esis, it will be [Alain Connes]." And not
only is there speculation that Louis de
Branges might be the one to prove RH,
the book includes an Appendix by de
Branges with the title "De Branges's
Proof." This isn't the way research in
mathematics goes. Perhaps in other
fields that require expensive equipment
one might, to a limited extent, predict where the next breakthrough might oc
cur, but in mathematics any attempt to
predict whether there will be a proof
any time soon, much less what shape it
might take or who might devise it, is
completely foolish.
In this connection, Sabbagh gives us
an interesting pair of speculations:
Henryk Iwaniec (p. 36) says, ''I'm only
worried that what may happen is that
a proof will be given by somebody and
I will be unable to understand it," while
Alain Connes (p. 263) worries about
something quite different: "It would be
a tragedy if it just needed a trick to
prove it." Different as these concerns
seem, I suspect that most mathemati
cians sympathize with both. Note that
neither has anything to do with "learn
ing a huge amount about an important
class of numbers."
Another of the authors, Marcus du
Sautoy, is a professional mathemati
cian, so we should expect his state
ments to be correct, but I am puzzled
by his statement (p. 11) that "a proof
of the Riemann Hypothesis would
mean that mathematicians could use a
very fast procedure guaranteed to lo
cate a prime number with, say, a hun
dred digits or any other number of dig
its you care to choose." He goes on to
relate this to RSA cryptography, clearly
implying that RH would have some
practical significance for cryptogra
phy, but I doubt that this is the case. I
suspect, rather, that he feels the gen
eral reader must be given some reason
for the significance of this million-dol
lar question in mathematics, but that
the real reason depends on aspects of
our tribal culture that are too difficult
to explain to the general reader. (Per
haps I am wrong; the supposed con
nection is again mentioned on page
243.) Similarly, on page 12 he says,
"The security of RSA depends on our
inability to answer basic questions
about prime numbers," but I thought it
depended on our inability to factor
large numbers. In fact, I thought the
practicality of RSA depended on the
disparity-in practice, primality test
ing is easy; factoring, hard.
His statement on page 5 that "Mas
tering these building blocks [primes]
offers the mathematician the hope of
discovering new ways of charting a
course through the vast complexities
of the mathematical world" puzzles me
in a different way. Whatever could it
mean? To my taste, this statement, and
much else in the du Sautoy book,
sounds too much like empty enthusi
asm, razzle-dazzle meant to impress
non-mathematicians not with sub
stance-because substantial mathe
matics is beyond their ken-but with
fanfares and flourishes.
Du Sautoy touches on a clash of cul
tures within mathematics that is sel
dom revealed to outsiders and that
might hold some interest for Sabbagh
and others interested in the anthropol
ogy of our tribe. In his last chapter he
sketches in very laudatory terms the
career of Alexander Grothendieck;
"Grothendieck's new language of geom
etry and algebra saw the creation of a
whole new dialectic which allowed
mathematicians to articulate ideas
which were previously inexpressible"
(p. 300). Then he goes on to say of the
brave new world of the Grothendieck
ists that "Even [Andre] Weil was rather
disconcerted by Grothendieck's new
abstract world," and, even more baldly,
quotes Carl Ludwig Siegel as saying, "I
was disgusted with the way in which
my own contribution to the subject had
been disfigured and made unintelligi
ble," and Atle Selberg as saying, "My
thought was that such lectures were
never given in earlier times. I said to
someone after the lecture a thought
which had come into my mind: if wishes
were horses, then beggars [could] ride."
Disagreements at the highest levels of
mathematics are extremely interesting,
and I applaud du Sautoy for bringing
them into the open, although he does
not pursue the subject.
I do not applaud, on the other hand,
his description of the relation between
RH and mental illness. In his last chap
ter, he says "Grothendieck is not the
only mathematician who has gone
crazy trying to prove the Riemann Hy
pothesis," as an introduction to a para
graph about John Nash. "Grothendieck
and Nash illustrate the dangers of math
ematical obsession," he concludes, but
mathematical obsession, whether it is
with RH or the continuum hypothesis,
is surely a symptom, not a cause. Our
tribe may have a stronger than average
association with madness that deserves
to be addressed, but, if so, it deserves
to be addressed with more seriousness
than to talk about going crazy trying to
prove the Riemann hypothesis.
As a sometime historian of mathe
matics, I am dismayed by du Sautoy's
failure to cite a single one of his his
torical sources. On page 104 he tells of
"several drafts" of a letter he says Rie
mann was writing to Chebyshev about
"his own progress" in the investigation
of the prime number theorem. In my
1974 book Riemann's Zeta Function I
published a jotting from Riemann's
helter-skelter notes showing that he
was aware of Chebyshev's existence; if
there is more evidence than this of a
Riemann-Chebyshev connection, I do
not know about it. The account of
Siegel's military history (p. 148) differs
substantially from the one given by
Benjamin Yandell in The Honors Class, and, since Yandell names his sources,
I believe his. I hope most readers will
realize that no sources could possibly
support such statements as "[Pythago
ras] filled an urn with water and
banged it with a hammer to produce a
note" and so forth on p. 77, but many
readers will not. Surely I am not the
only reader who wants to know what
lies behind the surprising reference (p.
128) to "Gauss and Einstein's belief
that space was indeed curved and non
Euclidean." That Gauss might have
considered the possibility of non
Euclidean physical space is plausible
enough, but that he believed it? The
propagation of unchecked and un
checkable anecdotes about the history
of mathematics is a form of pollution
to be combatted. An occasional tall
tale, with appropriate caveats, can cer
tainly be used to spice up the exposi
tion from time to time, but when no
sources are ever given for anything,
such tales become an unacceptable
norm.
Another feature of du Sautoy's writ
ing is his habit of introducing a private
phrase to describe something and for
ever calling it by his new name rather
than the one used by everyone else. For
example, he says on page 20 that "One
of Gauss's greatest early contributions
was the invention of the clock calcula
tor." He goes on to explain what he
means-modular arithmetic, of course,
the "clock" being a reference to arith
metic mod 12-but thereafter there is
no modular arithmetic, only "clock cal
culators" as in "That is because the cal
culations will be done not on a con
ventional calculator, but on one of
Gauss's clock calculators" (p. 234, deal
ing with RSA). Similarly, zeros of ((s) are first described as "points at sea level
in the zeta landscape" (p. 89) and are
called that for the remainder of the
book On page 79, rather than saying
that the harmonic series diverges, he
says it will "spiral off to infmity"-an
odd way to describe gradual increase
without bound-and thereafter series
never diverge but "spiral off to infmity."
The line where the real part of s is t is
not the critical line, it is "Riemann's
magic ley line" (p. 98) or "Riemann's ley
line." I have not found a definition of
"ley" in any American dictionary that
fits this use; it is apparently a term used
in British surveying.
But du Sautoy and Sabbagh were
not writing for mathematicians. It may
well be that the general readers they
have in mind will be intrigued and grat
ified by their descriptions of mathe
matics and mathematicians related to
RH. Certainly there is amusement to be
found in these books, and even math
ematicians will find many interesting
things in them if they are not too dis
tracted by questionable formulations
and implausible anecdotes. No harm is
done as long as cranks are not en
couraged and as long as genuinely in
quiring minds are not put off when
some of the purported explanations do
not seem to make sense.
The goal of the third book, the one
by John Derbyshire, appears to be dif
ferent from that of the other two. He
writes in his Prologue of "a general read
ership" (p. xii), but I think he is unduly
optimistic. He mentions, for example,
that he expects his readers to under
stand basic algebra, such as the fact that
S = 1 + xS becomes S = 1/(1 - x) when rearranged. Certainly anyone set
ting out to understand RH must be com
fortable with this rearrangement, but, in
the first place, I suspect that more edu
cated adults than we like to imagine
would not be comfortable with it, and,
in the second place, more mathematical
sophistication and ability is needed than
this example suggests. Perhaps Der
byshire set out to write a book for the
general reader, but as it developed I
think his goal had to change.
No matter! He has written a won
derful book He does not fudge the
mathematics, which will make parts of
it hard going for most non-mathemati
cians, but for the most important audi
ence of non-mathematicians-those
young ones who might consider becoming mathematicians-it will be a
great resource and inspiration. And for
mathematicians and readers with a fair
amount of mathematical sophistica
tion, it is a book that will inspire, in
form, and entertain. If you believe as I
do that RH for the general reader is a
futile project, you will agree that Der
byshire made the right choices. The
VOLUME 26, NUMBER 1 , 2004 57
copyright page states that the publisher, the Joseph Henry Press, "was created with the goal of making books on science, technology, and health more widely available to professionals and the public," a goal that is admirably served by this book
(Full disclosure: Derbyshire names me in his acknowledgments and mentions me a few times in the book I met him very briefly at the conference on RH held at NYU in 2002, and at that time I gave him a copy of my book, but I don't recall anything else requiring acknowledgment. Of course, I have made every effort to base my opinion of his book on the book alone.)
Even experts on RH will enjoy this book and learn from it, and I would encourage all readers of the Mathematical lntelligencer to try it. It is interestingly and skillfully written, and it approaches many aspects of the subject in imaginative and thought-provoking ways. For a quick probe, you might try reading pages 90-92. There you will find discussions of the contrast between measuring and counting (as he describes it, numbers legato and numbers staccato), Gauss's attitude toward Fermat's last theorem, Mallory's reason for wanting to climb Mount Everest, the rise of the Germans in 19th-century mathematics and how it may have been related to the Napoleonic wars and the Congress of Vienna-as well as a passing mention of Larry, Curly, and Moe.
If that rushed summary suggests that the writing is contrived or precious or pretentious, the fault is mine. To my taste it is always down-to-earth and treats its topics in natural and appropriate, but interesting, ways.
Naturally I have my disagreements and cavils with the book, but it is remarkable to me how few they are when I consider how dense with information and opinions the book is. The peasantpheasant story about Peter the Great on p. 56 should have been tossed out in the revision process. Derbyshire does an admirable job of keeping the calculus in the book to a minimum (he tells us in the Prologue that his original goal was to have no calculus at all, but that this goal "proved a tad overoptimistic"), but my alarm bells go off
58 THE MATHEMATICAL INTELLIGENCER
when, on p. 112 and again on p. 113, he describes a definite integral as an "area under a function." I gather that, competent as his mathematics is, he has never taught calculus and had to deal with students who persistently confuse functions with their graphs.
I am most disturbed by his statement about the formula em = - 1 that "Gauss is supposed to have said-and I wouldn't put it past him-that if this was not immediately apparent to you on being told it, you would never be a first-class mathematician," not only because I question the attribution of such a statement to Gauss and no source is given, but mainly because it strikes me as a terrible thing to say to a young student. One's reaction to em = -1 must be awe, not "oh, yes, of course!" If you tell me it was immediately apparent to you when you first saw it I will think you are a fool or a liar, or that your memory is faulty. Derbyshire is wrong to discourage his readers-who will need a good portion of ambition to allow them to penetrate his book-in any way, and particularly to do so on false grounds.
And he is indeed asking a lot of his readers. In his 21st chapter he walks the reader through Riemann's explicit formula
J(x) = Li(x) - I Li(xP) - log 2 + p
LX dt x t( t2 - 1) log t
where J(x) denotes 1T(x) + t'1T(Yx) + t1TCVx) + · · · (a terminating series because 1T(y), which is the number of primes less than y, is zero when y < 2) and where the complex numbers p are the zeros of the zeta function in the critical strip. And I don't mean that he simply explains the definitions of all the terms. He also explains how the series
Lr Li(xP) converges conditionally, so the order of the terms is of the essence and the convergence is very slow, and he actually provides numerical estimates of the various terms in the case x = 20. Once he has completed this, having shown in detail and with clarity how the formula yields the known value
7 J(20) = 9J2, he goes on to show the way in which Mobius inversion combines with this formula to give Riemann's ex-
plicit formula for 1T(x), taking for the sake of illustration the case x = 1,000,000 and carrying it through very clearly to show how it really does work out to give '1T(1,000,000) = 78498. (But he does not adequately explain how he evaluated the slowly converging "secondary terms." He gives them to five decimal places, but in his computation of J(20) he already confessed that 86,000 terms of the sum had to be computed to attain four-place accuracy for this term, and he certainly does not expect us to believe that he found the nine-place result he gives in that case by adding terms of the series!)
Can a beginner follow this chapter? Not unless the beginner is very talented. To tell the truth, I had to read it pretty attentively. But it is interesting. The talented beginner will learn from it, as I learned from it. And those who can't follow it are not being sold a bill of goods, not being encouraged to think they understand and appreciate something they don't understand at all, and not being condescended to. They can give it their best shot, and if they fail they can still admire it and still appreciate much of the rest of the book, and may someday come back to it when they are no longer beginners.
A parting thought. In my opinion, all three books grossly overstate the connection of RH to prime numbers. Derbyshire even chooses the title "Prime Obsession." True, an investigation of the distribution of primes and the Euler product formula led Riemann to RH, but Riemann himself quickly switched to another function he called g(t) (it is the value at s = t + it of f(�) ·
2 s(s; I) . '1T�s/2((s), which, as Riemann proves, is an even function of t that is real on the real axis) and his actual hypothesis was that the zeros of g( t) are real! To me, g( t) is a symmetrized version of ((s)-symmetrized to put the functional equation of ((s) in a simple form and to put the interesting part of the function on the real axis-that is an entire function of one complex variable. RH is simply the statement that its zeros are real. The connection with prime numbers may or may not play a role in explaining the amazing extent to which Riemann's hunch has been borne out by massive modern compu-
A Mathematician Grappling with His Century by Laurent Schwartz
BIRKHAUSER VERLAG. 2001
vi1i + 490 pp., ISBN 3764360526; US $49.95.
REVIEWED BY NORBERT SCHLOMIUK
Laurent Schwartz [1915-2002] was
one of the great mathematicians of
the twentieth century. His main con
tribution to mathematics is his work on
distribution theory.
In his "History of Functional Analy
sis" Jean Dieudonne wrote:
The role of Schwartz in the theory of distributions is very similar to the one played by Newton and Leibniz in the history of Calculus: Contrary to popular belief, they of course did not invent it, for derivation and integration were practised by men such as Cavalieri, Fermat and Roberval when Newton and Leibniz were mere schoolboys. But they were able to systematize the algorithms and notations of calculus in such a way that it became the versatile and powerful tool which we know.
The great importance of Laurent
Schwartz's contributions to mathe
matics was recognized by his being
awarded the Fields Medal in 1950, the
first French mathematician to receive it.
Laurent Schwartz was above all an
extraordinary human being: warm, gen
erous, wise, modest, deeply involved in
the struggle for the oppressed, for hu
man rights and the rights of people, a
great and noble figure of the twentieth
century. We are lucky that close to the
end of his life he decided to write an
autobiography.
A Mathematician Grappling with his Century is the translation by Leila
Schneps of the original French edition
published in 1997 by Editions Odile Ja
cob under the title Un mathematicien aux prises avec son siecle. For those
of us who had the good fortune to be
close friends of the author, reading the
book is listening to the beautiful voice
of Schwartz, a marvellous raconteur.
In the Foreword, the author pre
sents the content of the book:
I am a mathematician. Mathematics filled my life: a passion for research and teaching as a professor both in the University and at the Ecole Polytechnique. I have thought about the role of mathematics, research and teaching, in my life and in the lives of others. I have pondered on the mental processes of research and for decades I have devoted myself to urgently necessary reforms within the University and at Ecole Polytechnique. Some of my reflexions are contained in this book, as well as a description of the course of my life. However I do not discuss the University reforms since I have written many articles and books on the subject. Inevitably, mathematics appear in this book, one cannot conceive of an autobiography of a mathematician which contains no mathematics. I have written about them in a historical form which should be accessible to large non-specialist sections of the scientific public; readers impervious to their charm may simply skip them. They concern only about fifteen per cent of the volume.
The reader interested in the con
temporary problems facing universi
ties would find much to think about in
Schwartz's "Pour sauver l'universite"
(Seuil, 1983) and in "Pour la qualite de
l'Universite fran<;aise" by Pierre Merlin
and Laurent Schwartz (P.U.F. 1994).
The Introduction to his autobiogra
phy is titled "The Garden of Eden," a
reference to the property at Autouillet
purchased by Schwartz's parents in
1926. It was at Autouillet where the au
thor found the ideal conditions to work
Schwartz grew up in a very warm fam
ily. His father, a distinguished physi
cian, had a strong influence on his chil
dren. Here is a lesson given by the
father to his son:
If in a given circumstance you find that you are alone with your opinion against everybody else, try to listen to them, because maybe they are right and you are wrong. But if, after having thought it out, you still find yourself alone with your opinion, then you should say it and shout it and let everybody hear it.
His explanation, writes Schwartz,
remained engraved inside me and
guided me in all my political activities
in my adult life. Tolerance, inner free
dom, wisdom were qualities which im
pressed everyone who had the chance
to know him.
I remember one of the public lec
tures Schwartz gave in Montreal about
the life of a mathematician. Many in the
large audience were high school and
college students fascinated by the lec
ture and impressed by the sincerity of
his presentation when he spoke about
his self-doubt. I found his words in the
book:
In spite of my success [in school}, I was always deeply uncertain about my own intellectual capacity. I thought I was unintelligent. And it is true that I was and still am rather slow. I need time to seize things because I always need to understand them fully. Towards the end of the eleventh grade, I secretly thought of myself as stupid. Not only did I believe I was stupid, but I couldn't understand the contraindication between this stupidity and my own good grades. At the end of the eleventh grade I took the measure of the situation and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. Naturally it is help-
VOLUME 26. NUMBER 1 . 2004 59
ful to be quick, like it is to have a good memory. But it's neither necessary nor sufficient for intellectual success.
And here a last quotation from this
marvellous book:
Self-confidence is a condition of success; of course one must be modest, and every intellectual needs to recall this. I am perfectly conscious of the immensity of my ignorance compared with what I know. It's enough to meet other intellectuals to see that my knowledge is just a drop of water in an ocean. Every intellectual needs to be capable of considering himself relatively and measuring the immensity of his ignorance. But he must also have confidence in himself and in his possibilities of succeeding through the constant and tenacious search for truth.
The Mathematician Sophus Lie: It Was the Audacity of My Thinking by Arild Stubhaug, translated from the Norwegian by Richard H. Daly
BERLIN HEIDELBERG, SPRINGER-VERLAG 2002
556 pp. LIS $44.95, ISBN 3-540-421 37 ·8
REVIEWED BY JESPER LUTZEN
The year 2000 saw the publication of
two remarkable books about the
Norwegian mathematician Sophus Lie
and his work: Thomas Hawkins's
Emergence of the Theory of Lie Groups (Hawkins [2]) and Arild Stub
haug's Norwegian biography of Lie
(Stubhaug [3]). The present review
deals with the English translation of
the latter.
The two books complement each
other very well. Hawkins's book, which
represents the crowning result of many
years of research, gives a thorough
mathematical analysis of Lie's inven
tion of Lie groups and the further de
velopment of the subject for the next
half-century. It is aimed at an audience
of mathematicians who know the the
ory and can follow the technical details
of its development. Stubhaug, on the
other hand, draws a vivid picture of the
person Sophus Lie and the time he
lived in. His book is aimed at a general
audience, and does not go into mathe
matical detail. That does not mean that
mathematics is left out of Stubhaug's
biography; in fact, mathematics is con
stantly present in the book as Lie's
great passion, and it is clear that Lie's
greatness was due to his mathematical
creativity. Lie's mathematical cre
ations are described in general terms,
and in a chapter titled Into Mathemat
ical History Stubhaug does a good job
of setting them into historical and
philosophical perspective. A non-math
ematical reader will learn from Stub
haug that mathematics is fascinating
and important and will get a good
glimpse of the creative, dramatic, and
even existential aspects of mathemati
cal research. For this reason alone one
must hope that the book will be widely
read, especially at a time when scien
tific and mathematical research is un
der public scrutiny. The mathemati
cian-reader who finds the description
of Lie's mathematics superficial can
turn to Hawkins, or to Armand Borel's
new book Essays in the History of Lie Groups and Algebraic Groups [ 1 ] .
Both mathematicians and non-mathe
maticians will enjoy other aspects of
Stubhaug's book; like Stubhaug's ear
lier biography of Niels Henrik Abel, it
is both extremely well written and well
researched. Richard H. Daly, who also
translated the Abel biography, has
again done an excellent job. For the
non-Norwegian reader there have been
inserted a few explanations of locali
ties and personalities that are well
known to a Norwegian audience. In a
few cases this has been overdone, as
for example when the often-mentioned
Nordmarka is repeatedly explained to
be a forest north of Christiania. In one
respect the translation of the Lie book
is even more successful than the Abel
book: The mathematical terms have
with a few exceptions been translated
correctly.
The style is literary rather than
scholarly. For example, there are no
references and only a few footnotes in
the main text. And the style is narra
tive rather than analytic. The beginning
of the book may suggest otherwise.
Here Stubhaug explains that everyone
who met Lie would later tell stories and
anecdotes about him. "Nobody seems
to have been able to pass him by in si
lence. What, in detail, did the stories
recount, and what, on the other hand,
do we know with certainty? What was
imaginary and what was real?" This
may sound like an introduction to an
analysis of the sources pertaining to
Lie's life, but in fact the succeeding
chapter does not provide such an
analysis. The questions are rather a
stylistic trick to begin a 20-page over
view of Lie's life, that serves as a sort
of abstract of the book. This introduc
tion is helpful to the reader, who may
otherwise get lost in the wealth of ma
terial presented in the main part of the
book. For readers who do get lost,
however, there is a 5-page schematic
chronology at the end.
With its literary narrative style the
book reads like a novel. However, as is
revealed by the Bibliography, it is
based on a thorough study of a wealth
of sources. Many mathematicians and
historians of mathematics have writ
ten biographies of Lie-for example,
Friedrich Engel, Poul Heegaard, Elling
Holst, Max Noether, and Ludvig Sylow
(a list is provided in Stubhaug's book)
but this is the first book-length biogra
phy of the Norwegian mathematical ti
tan. One reason that Stubhaug's book
is longer than earlier biographies is
that he embeds the story of Lie's life in
a rich cultural, political, and institu
tional context. Another reason is that
new sources uncovered by Stubhaug's
research, in particular many collec
tions of letters to and from Lie, have al
lowed him to paint a very detailed pic
ture of Lie and his life. In this respect
Stubhaug's Lie book is more ground
breaking than his Abel book.
The scholarly aspects of Stubhaug's
work are also revealed by the 50 pages
of endnotes. Within them are refer
ences to the sources as well as addi
tional details that enhance the main
text. In many cases these enhance
ments seem just as relevant and inter
esting as the main text, and they may
have been relegated to endnotes only
because they would have hindered the
flow of the narrative.
The book is richly illustrated with
photographs of people and places of
significance to Lie, and with reproduc
tions of paintings representing the pre
vailing view of the grand Norwegian
Nature that give the reader an idea of
the existing zeitgeist. Unfortunately
the book does not contain a map of
Norway or Europe showing the places
mentioned in the text. That would have
been a great help, in particular for non
Norwegian readers.
As in most biographies, the main
part of Stubhaug's book is written
chronologically. After the summary
presented in the first part, the second
part deals with Lie's family background
and his upbringing. We learn that he
was born on December 17th, 1842, as
the penultimate child out of six of the
vicar Johan Herman Lie in Nordfjordeid
and Mette Maren nee Stabell, who ran
the vicarage as a model farm. When he
was 9 years old he moved with his fam
ily to Moss, where his father remained
a Parish Vicar for the rest of his life.
Sophus's mother died a year after the
move. In 1856 Lie completed the edu
cation at the local science school, and
after a year of private tutoring he fol
lowed his older brother to Nissens
School in Christiania (now Oslo). Such
are the bare essentials, but such a sum
mary does not do justice to the rich
ness of part 2 of this biography. The 27-
page section contains a host of details
imbedded in an extremely well-in
formed cultural-historic context. Stub
haug explains the family tree of the Lie
family; in particular he goes into some
detail about the life of Lie's father. We
learn that after having taken his theo
logical degree, father Lie began to
work as a teacher. Only when his ap
plication for the job as headmaster
of the school where he worked was
turned down did he tum to an ecclesi
astical career. However, he continued
to be very actively interested in the en
lightenment of his parishioners, not
only in religious but also in scientific
matters. In particular he initiated a se
ries of lectures on the natural sciences
for the workers in Moss, hoping that an
awareness of the laws of nature would
reveal the creative hand of God. Here
it is interesting to note that Sophus
Lie's later hero Abel had also been the
son of a vicar with rational scientific
leanings.
Stubhaug also tells about the village
of Nordfjordeid and in particular about
the vicarage and its buildings. We hear
about father Lie's versatility as a vio
linist, his two terms as mayor of the vil
lage, and many other things about the
rather happy everyday life in the vic
arage in Nordfjordeid. All this is seen
on the background of the general
trends in Norwegian society at the
time. After the move to Moss and the
death of Lie's mother, more stringent
and less happy conditions reigned in
the vicarage. But at school Lie did very
well and graduated as number one in
his class. In connection with his stud
ies at Moss Realschool, Stubhaug gives
a long explanation of the ongoing re
forms of the Norwegian school system.
This explanation continues in Part
3, which is devoted to Lie's time at Nis
sen's school in Christiania. Stubhaug
gives a 12-page account of how this
school was founded as a clear alterna
tive to the old Latin and Cathedral
schools, how it became an example for
many subsequent schools that valued
science and modem languages on a par
with the classical languages, and how
its creation was a reflection of and a
great influence on a broad debate in
Norway about the means and goals of
education, a debate in which Lie later
took an active part. We hear that the
founder of the school, Nissen, was a
former student of father Lie; we learn
about the school buildings, about the
teachers, both those that taught Lie
VOLUME 26, NUMBER 1 , 2004 61
and the ones that preceded them, and
about some of Lie's classmates and
their families.
As I present the content of this chap
ter it may sound far too detailed and
long, but in fact I found all of it inter
esting and relevant for an understand
ing of the context in which Lie grew
up. The only thing I missed was a more
detailed account of what happened
when Lie during his first year at Nis
sen's school had Ludvig Sylow as his
mathematics teacher. We learn only
that Lie was the best in his class in this
subject, and that Sylow later recounted
that he had not seen any special math
ematical genius in the young student.
Do the sources say nothing more about
the matter?
In the 3rd part of the book the char
acter Sophus Lie gradually becomes
visible. As a student at Nissen's school
he did very well in all his subjects, and
in the end he was number two in the
entrance exam to the University in
Christiania. He began to study science
at the University in 1861, and from then
on his whereabouts are rather well
documented. He continued to do bril
liantly in most subjects, in particular in
the mathematical sciences, physics, as
tronomy, and chemistry, and he was
very active in the Scientist Association.
He also began a habit, that lasted al
most to the end of his life, of taking
strenuous hikes of several weeks dur
ing the summer into the mountainous
regions of Norway. As a boy he had ex
celled in physical strength, and his long
and fast hikes became legendary.
What kind of a person was Lie? As
a short answer to this question and as
a good example of the beauty of Stub
haug's language, let me quote the open
ing words of the book:
According to most accounts of Sophus Lie, he was the embodiment of an archetypical character in a theatrical drama-with his forceful beard, his sparkling green eyes magnified by the stout lenses of his spectacles-the blond Nordic prototype, as it was called across Europe-the Germanic gigantic being-a primal force, a titan replete with the lust for life, with audacious goals and an indomitable
62 THE MATHEMATICAL INTELLIGENCER
will. These descriptions of his physical and mental strength also contained a subtext, an embryonic notion, not only about this brilliant man of science, the prophet, who intuitively conceived new mathematical truths, but also about the colossus who, in his constant zeal for new knowledge, might push others aside, and inadvertently trample them underfoot. He was described as highly committed, richly innovative, someone with unusual physical strength, and the stamina to overcome the majority of obstacles, but also, a man who afterwards had to pay for this with correspondingly great swings of mood and temperament.
One could add that he was a warm
person, often friendly, strong-willed,
direct in his speech, undiplomatic
"the blond Nord ic prototype . a t itan . . "
sometimes even raw in his manners,
and rather self-centered. He wrote
about himself that he "had little talent
for socialising with folk," and "what is
fatal is that I am so diametrically dif
ferent from [Felix] Klein with respect
to the ability to be able to get into the
thinking of others" (p. 257).
The mood swings that Stubhaug
mentions showed themselves for the
first time during Lie's last year as a uni
versity student. He had set himself the
high goal of graduating from University
with the highest grade overall. After
the first three years this still seemed an
attainable goal, but the last year's study
of the biological sciences did not go so
well, so he only got the second-highest
grade. He became depressed, was un
able to sleep, and even planned suicide.
This psychologically unstable state
lasted for the next few years, when he
was plagued by a sense of lacking a
calling for his life.
In fact it is remarkable that, unlike
Abel, who in high school had already
begun to study the masters and make
original contributions to mathematics,
Lie did not seriously begin to pursue a
career in mathematical research until
three years after his graduation from
university. Indeed, with his many phys
ical as well as intellectual talents he
might well have chosen a different pro
fession. In school he contemplated go
ing into philology, but at the entrance
exam to the university he got only the
second-highest mark in Greek. He was
so dissatisfied with his performance
that he opted for science instead. And
even as a science student he toyed with
the idea offollowing the example of his
older brother, who was an officer in the
army. In fact in 1864, when the Danish
borders were threatened by German
troops, Lie followed the general Scan
dinavism among the Norwegian stu
dents, and volunteered to defend the
brother country. However, before his
military training in Christiania was
complete, Denmark had surrendered.
Lie continued to serve for a few years
as a reserve lieutenant, but it turned
out that his eyes suffered from oblique
cornea, and therefore he could not pur
sue a military career.
Even after graduation from univer
sity, Lie did not have a clear idea about
his vocation. He began to work at the
observatory and to give popular lec
tures on astronomy in the Student So
ciety, and he planned to write a book
on the subject. However, probably as a
result of disagreements with the pro
fessor of astronomy, he did not obtain
the vacant job as assistant at the ob
servatory, and in 1867 he gradually
turned to mathematics and composed
a small textbook on trigonometry.
Only during the following year did
he become convinced that "there was
a mathematician in him." The turning
point was the meeting of the Scandi
navian natural scientists in Christiania
that year. Here Lie met Sylow again.
While a student at the university Lie
had followed a course on Galois theory
(the first lecture on that subject any
where in the world after Liouville's pri
vate lectures in the 1840s) that Sylow
had given while he was a substitute for
the mathematics professor Ole Jacob
Broch, who had been elected Member
of Parliament. Lie had in the meantime
lost his notes from those lectures, and
now asked Sylow permission to bor
row his notes. "I believe that group the
ory will become very important," Lie
prophetically told Sylow. Moreover,
Lie developed a friendship with the two
Danish mathematicians Adolph Steen
and Hieronymus Georg Zeuthen, who
attended the meeting. The former gave
a talk about integration of differential
equations, and Zeuthen talked about a
subject from the new geometry. These
three subjects, group theory, differen
tial equations, and geometry, became
the central elements in all of Lie's fu
ture mathematical work.
His depressive moods gone, Lie
threw himself into a study of various
recent works in geometry that Zeuthen
had referred him to, and he began to
do independent research. By Decem
ber he got the idea of his so-called
imaginary geometry. The following
year he lectured on it in the Science So
ciety and privately published a short
pamphlet about it, followed by two pa
pers in the Proceedings of the Christiania Academy of Sciences and a pa
per in Grelle's Journal fur die reine und angewandte Mathematik. This
closely paralleled the way the young
Abel had published his first original re
search. From the outset, Lie reached
out to an international audience. No
one in Norway could fully appreciate
the value of Lie's new contribution to
mathematics, and yet many of the best
Norwegian mathematicians and scien
tists were aware that it was important.
As early as 1869 he was given a travel
stipend that allowed him to stay for a
half-year in Berlin and for a half-year
in Paris, the two cities that Abel had
visited a half-century earlier. Unlike
Abel, however, Lie preferred the
French style in mathematics and was
not particularly fond of the rigor that
characterized the mathematicians in
Berlin. Still, his stay in Berlin became
very fruitful, and he had the triumph of
impressing Eduard Kummer by solving
a geometrical problem he could not
solve himself. This may be considered
Lie's breakthrough in international
mathematics.
But the most important aspect of his
stay in Berlin was his encounter with
the 20-year-old Felix Klein. The two
soon became friends, and they began
to collaborate on research. When Lie
continued on to Paris in the spring of
1870, Klein came along, and they both
enjoyed the regular meetings with
Camille Jordan and Gaston Darboux.
However, when the Prussian-French
war broke out in July, Klein immedi
ately returned to Germany. Lie also
left, for Italy. He had planned to walk
through France over the Alps to Milan,
where he wanted to meet with Luigi
Cremona. However, he only made it as
far as Fontainebleau before he was ar
rested as a German spy. When he was
released a month later, thanks to the
intervention of Darboux, he took the
next train to Switzerland.
In Berlin and Paris and in prison, Lie
pursued his geometrical research and
began to work on contact transforma
tions. His international success made
him hope for either a better stipend or
a permanent position in Norway, but
the applications he sent from abroad
did not bear fruit. When he came home
he composed a doctoral thesis about
the line-sphere transformation that he
had discovered in Paris. No one in Nor
way understood its content, but its im
portance was soon recognized interna
tionally, and when Lie in the fall of 1871
applied for a professorship in Lund,
Sweden, many Norwegian intellectuals
realized that Norway was on the verge
of repeating the mistake they had made
when they did not offer Abel a profes
sorship. The newspapers published let
ters of recommendation from Alfred
Clebsch, whom Lie had met in Gottin
gen, and from Cremona, whom Lie had
not succeeded in meeting in Milan. It
was mentioned that other letters of
support had come from Berlin, Copen
hagen, and Paris.
In 1872 Cabinet minister and math
ematician Broch convinced the Nor
wegian Parliament to appoint Lie as
extraordinary professor. Usually, pro
fessors were appointed by the Cabinet
of the Swedish king, who also ruled
Norway. Thus Lie's appointment was a
small part in a power struggle between
the freely elected Norwegian Parlia
ment and the Cabinet, and more gen
erally a strong statement in the Nor
wegian struggle for independence.
The same year Lie became engaged
to be married to Anna Birch, who was
a granddaughter of Abel's uncle. He
wanted to be married as soon as pos
sible, but she wanted to wait. In the
meantime they conducted an intense
correspondence, which Stubhaug de
tails in the beginning of Part 5. I must
admit that Lie's constant begging for an
early date of marriage became tire
some. It is the only part of the book
that in my opinion would have bene
fited from cutting. In 187 4 Lie finally
persuaded Anna to marry him, and
over the course of the next 10 years
they had three children: two daughters
and a son. Their marriage was happy.
The period 1872-1886, while Lie was
a Parliamentary professor in Christia
nia, was also his most productive pe
riod. He traveled often to Germany,
where he continued his friendship and
exchange of ideas with Felix Klein, and
to Paris, where he presented his new
work on differential equations and his
theory of transformation groups,
which he developed as a means to
solve them. He published a host of new
results, first in the Norwegian journal
Archiv for Mathematik og Naturvidenskab, which he founded together
with two colleagues, and later in inter
national journals, mainly in the Mathematische Annalen, which Klein edited
from 1877 together with Adolf Mayer.
In 1884 Klein arranged for Engel to go
to Christiania to help Lie present his
new ideas in a more polished book form.
Their intensive collaboration during
nine months in Christiania and later in
Leipzig resulted in Lie's main work
Theorie der Transformationsgruppen, published in three volumes in 1888--1893.
From 1873 Lie also worked with Sylow
on a new complete edition of Abel's
works. He also suggested to the
Swedish mathematician Gosta Mittag
Leffler that they found a new Scandi
navian research journal for mathemat
ics. This resulted in the creation of
Acta Mathematica. Lie sometimes lectured on aspects
of his new mathematics, but he did not
have any first-rate Norwegian students.
Therefore he immediately accepted
when, at the instigation of Klein, he
was offered the professorship vacated
VOLUME 26, NUMBER 1 , 2004 63
at Leipzig University when Klein moved to Gottingen. Lie's stay in Leipzig from 1886 to 1898 was a mixed experience. His lectures drew many good students. In particular Lie was proud that his Parisian colleagues sent some of their best students of the Ecole Normale, e.g., Arthur Tresse and Ernest Vessiot, to Leipzig to study with him. Moreover, he had several good assistants and Private Docents such as Engel (whose collaboration with Lie continued) and Eduard Study, Issai Schur, and Georg Scheffers, who published Lie's lectures in three large volumes: Differential Equations with Known Infinitesimal Transformations, Continuous Groups with Geometric and other Applications, and Geometry of Contact Transformations (all in German). Finally, he had a good relationship with some of the other professors such as Adolph Mayer and Wilhelm Ostwald, but his relationship with his mathematics colleague Carl Neumann was strained.
In 1889 Lie suffered a mental breakdown that confmed him to a psychiatric hospital for seven months. Even after he released himself from hospital and tried to walk the depression out of his system, he was strongly depressive for several years, and his personality seems to have changed for good. Even before the breakdown he had begun to assert his priorities vis-a-vis Klein, and after the breakdown there was a complete break between the two former friends. Lie also began to attack his other collaborators and supporters such as Engel, and he accused Wilhelm Killing of stealing his ideas. On the whole he turned away from his German colleagues and oriented himself more toward Paris, where his works won increasing acceptance.
Part 6 of Stubhaug's Lie biography, which deals with the Leipzig period, is in my opinion the weakest in the book. Stubhaug deviates more from a chronological presentation than in other parts, and that makes the part somewhat disconnected. In particular it is unfortunate that there is not a clear distinction between the periods before and after Lie's mental breakdown. Moreover, when writing about the cultural and institutional setting in Leipzig, Stubhaug
64 THE MATHEMATICAL INTELLIGENCER
seems to lack the mastery with which he deals with the Norwegian scene.
Yet Norwegian affairs are constantly mentioned also in this part. Indeed, Lie continued to keep abreast with the developments in his native country through regular reading of Norwegian newspapers, through correspondence, and through the young Norwegians who came to study with him in Leipzig. He even continued to participate in the public debate on educational affairs in his homeland, and he was active in the preparations for the centenary of Abel's birth in 1902. In particular, he tried to raise funds for an Abel Prize in Mathematics matching the newly founded Nobel Prizes in other disciplines. As is well known, this idea was not implemented for more than a century.
While his wife and children soon adapted to the new social situation and thrived in Leipzig, Lie had a hard time getting used to the tone at the University, to the much larger teaching load, to German militarism, and to the heat, and he found it difficult to lecture in a language that he had not mastered. He was happy to have left the provincial atmosphere in Christiania, but he missed his friends and in particular Norwegian nature. The first summers in Germany Lie rented a vacation home near Leipzig and went to the Alps for hiking tours, but from 1888 he began to spend parts of each summer in Norway.
His warm feelings for his home country and his hope for its freedom from Sweden became clear to everyone at the university during his inauguration as a Professor in Leipzig. During the ceremony the Rector mentioned that he had heard that the peasant representatives in the Norwegian Parliament had treated the noble King Oscar badly; therefore he could well understand that Lie wanted to leave Christiania. At this point Lie protested loudly and left the room.
In Norway the movement that resulted in independence from Sweden in 1905 was gaining momentum in the 1890s. One of the strategies among the cultural and academic elite was to display the great Norwegian talent. As a part of this strategy, the polar explorer Fridtjof Nansen, the mathematician Elling Holst, and the leading poet Bj0m-
stjeme Bj0mson, who had himself recently returned from a long stay in Paris, conceived a plan to bring Lie back to Norway. After negotiations with Lie in 1893 they succeeded in convincing the Norwegian Parliament to upgrade his professorship (from which he had obtained a leave when he moved to Leipzig) to a Professorship of Transformation Group Theory with a salary that matched his high German salary. Lie accepted, but it took him another 4 years before he finally resigned his position in Leipzig and returned for good to Christiania. During those four years he traveled back and forth from Norway several times, and even had an idea of trying to arrange a joint position at the two universities. When he returned to Christiania in September of 1898 he was not in good health, and soon it became clear that he suffered from pernicious anemia. He died from this disease on February 6 of the following year.
Many obituaries and later biographies have tried to capture this extraordinary mathematical genius, but none of them have been as complete, as well researched, or as well rooted in the cultural context as Stubhaug's wellwritten book. I can recommend it to all mathematicians as well as non-mathematicians, who have an interest in the human aspects of scientific creation.
REFERENCES [ 1 ] Armand Borel, Essays in the history of Lie
George Green, Mathematician and Physicist 1 793- 1 841 : The Background to H is Life and Work D. M. Cannell
SIAM, 2001 $75.00. ISBN 0-89871 -463-X XXXIX + 3 1 6 pp.
REVIEWED BY STEVEN G. KRANTZ
Every calculus student learns of Green's theorem as perhaps the
most benign multidimensional version of the fundamental theorem of calculus. Coupled with its delightful interpretations in terms of fluid flow and electrostatics, this result can be construed as one of the capstones of a freshman education in mathematics. But most calculus books contain almost no information about the life of George Green (1793-1841) himself, and most mathematicians have little knowledge of the man. More's the pity, for George Green was one of the more fascinating and important characters of early British science.
George Green was born in Nottingham. His father, an uneducated man, was a successful baker who set up his own mill in nearby Sneinton. He built a large family home on a substantial property near the mill. As was the custom of the time, George went to work for his father at an early age. Recognizing his son's talents, the elder Green sent his son to Robert Goodacre's academy at the age of 8. After four terms, young George had outstripped his teachers. He left the academy and returned to work at the mill.
George faithfully worked at his father's mill until 1829, when his father died. Then George was able to divest himself of the mill and devote himself to mathematics. In spite of his commitment to his father's business, George had been able to produce his fundamental (and first) paper, "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism." It was published by private
68 THE MATHEMATICAL INTELLIGENCER
subscription in 1828, and only about 50 people saw it. The response was polite indifference, and poor George returned despondently to milling. By good fortune, Sir Edward Bromhead (a member of the local intelligensia) became aware of Green's work in 1830, and he encouraged him to take up mathematics again.
With Bromhead's support, Green enrolled at Caius College, Cambridge in 1833. He was already 40 years old. He achieved 4th Wrangler on the dreaded Tripos in 1837 and was elected to the status of College Fellow in 1839. Unfortunately, ill health forced Green to leave his position after just four terms. He died in 1841.
George Green is considered by many to have been the father of British mathematical physics. He published just ten papers in the period 1828-1839. Of these, his first (referenced above) is thought to have been the most important, and the most influential. It contains or introduces
• the idea of potential function; • what we now call Green's theorem; • the idea of reciprocity; • the idea of singular value; • the idea of the Green's function.
Later papers include ( i) the first rendition of what we now call the Dirichlet principle, ( ii) an important asymptotic method for solving certain partial differential equations in divergence form, and (iii) a preliminary version of the idea of tensors (indeed a particular tensor is today named after George Green).
Mary Cannell's book is a remarkable and profound effort. Little is known of Green's early years, and (as a senior scientist) he left behind little correspondence, no diary, and no working papers. An especially arduous effort was required to piece together the story of his life. It should be stressed that this book is not a commentary on Green's scientific work (although some of the appendices do treat this aspect of Green's life). In fact, the avowed purpose of the book is to treat the personal aspects.
Cannell tells us that Green had seven children by Jane Smith, the daughter of his father's mill manager; yet they never married. It appears that
Smith remained in the background of Green's life-although it should be noted that all the children ultimately adopted the name Green. Cannell marvels over the fact that Green mastered many of the techniques of French analysis at a time when these ideas were virtually unknown in England. She goes to great lengths to trace his personal and intellectual heritage.
And she certainly mourns Green's lack of personal and scientific recognition during his lifetime. In fact it was Lord Kelvin who rediscovered Green's "Essay" (his first paper) in 1845-four years after the man's death. He ultimately arranged for the paper to be properly published in Grelle's Journal in the 1850s. Finally, on the 200th anniversary of George Green's birth, a plaque in his honor was placed in Westminster Abbey-in front of the statue of Isaac Newton.
One can only speculate, and Mary Cannell does so at length, about what sort of career George Green might have had if he had had benefit of a proper education at the appropriate time in his life, and if he had lived in a more nurturing environment (such as Paris), rather than the stultifying wreckage that was British science in the early nineteenth century. It seems inarguable that George Green had a profound influence on such leaders of nineteenth-century British mathematical physics as Maxwell, Stokes, and Rayleigh. Cannell makes a point of the great effect that Green's ideas had on twentieth-century mathematical physics. For example, the Nobel-Prizewinning work of Julian Schwinger on quantum electrodynamics makes considerable use of Green's function. On the occasion of the 200th anniversary of Green's birth, such luminaries as Freeman Dyson and Schwinger gathered to help pay tribute to him.
We owe a debt to D. Mary Cannell for penning this, the only full-length biography of George Green in existence. It is a shame that this new edition was published posthumously, but a tribute to her scholarship and dedication to an important cause. One of the more interesting and daring points that Cannell makes in her book is that a more formal educa-
Left: The windmill where George Green ground corn for a living. Be
low: Green's grave, St. Stephen's Courtyard, Sneinton, Nottingham.
(Photograph by Jan Crosbia.) Both figures reproduced from The Mathematical tntelligencer, vol. 1 1 (1989), no. 4, pp. 39, 40.
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VOLUME 26, NUMBER 1 , 2004 71
,.-jfi111.19.h.l§i Robin Wilson I
The Phi lamath' s Alphabet-(
Caratheodory: Constantin Cara
theodory ( 1873-1950) is the most
significant Greek mathematician of
recent times; but, encouraged by
Minkowski, Klein, and Hilbert, he spent
most of his life in Germany. He made
significant contributions to the calcu
lus of variations and its applications to
geometrical optics, the theory of func
tions (especially conformal represen
tation), and measure theory. In applied
mathematics, he wrote on thermody
namics and relativity theory.
Caratheodory
Cauchy
Chebyshev
Cauchy: Augustin-Louis Cauchy (1789-
1857) was the most important analyst
of the early nineteenth century. In the
1820s he transformed the whole area
of real analysis, providing a rigorous
treatment of the c:;alculus by formaliz
ing the concepts of limit, continuity,
derivative, and integral. In addition, he
almost single-handedly developed the
subject of complex analysis, and many
results in this area are named after him:
'Cauchy's integral formula' appears on
the stamp.
Chebyshev: Pafnuty Chebyshev (1821-
1894) is remembered mainly for his work on orthogonal functions ('Cheby