According to Steven Krantz's Mathematical Apocrypha (pg. 186):

As was custom, Weil often attended tea at [Princeton] University . Graduate student Steven Weintrab one day went about the room asking various famous mathematicians who was the greatest mathematician of the twentieth century. When he asked Weil, the answer (without hesitation) was "Carl Ludwig Siegel (1896-1981)."

As the title of Krantz's book suggests, the anecdote may be apocryphal. However, there are other better grounded accounts of great mathematicians expressing the highest admiration for Siegel:

(A) In The Map of My Life Shimura wrote:

I always thought that few people really understood my work. I knew that Chevalley, Eichler, Siegel, and Weil understood my work, and that was enough for me [...] Of course [Siegel] established himself as one of the giants in the history of mathematics long ago [...] Among his contemporaries, [Weil] thought highly of Siegel [...]

(B) In an published interview (pg. 30) Selberg said

[Siegel] was in some ways, perhaps, the most impressive mathematician I have met. I would say, in a way, devestatingly so. The things that Siegel tended to do were usually things that seemed impossible. Also after they were done, they seemed still almost impossible.

Why might Weil, Shimura and Selberg have been so impressed by Siegel? I should emphasize that I'm not trying to precipitate a debate about the relative standing of historical mathematicians - rather - I'm hoping to learn about aspects of Siegel's work that I might otherwise overlook. I'm also not looking for, e.g. quotations from the Wikipedia article on him, but rather, less familiar material.

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    with all respect, why don't you read some of his work? just reading his topics in complex analysis was enough to convince me of his depth and breadth. – roy smith Nov 5 '12 at 5:34
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    I like this question (and the answer it received). – Dan Petersen Nov 5 '12 at 8:30
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    I also like very much the question and the way it is adressed, even if my own personal interest in Siegel is rather modest. I must add that I do not quite understand the policy of closing posts that are perfectly fair, do comply with all rules of MO, and in general seem to have nothing offensive besides looking stupid/ugly/uninteresting to a handful of moderators. I really wish MO were more democratic, in this regard. I do not believe another style of moderation would necessarily lead the site any closer to – Delio Mugnolo Nov 5 '12 at 19:58
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    I reacted negatively to this question, but only because I have a conflict of interest... – Paul Siegel Nov 5 '12 at 22:13
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    @Mahdi: humor aside, Weil knew Grothendieck very well but pretended he wasn't interested in Grothendieck's schemes. Weil, not precisely a modest mathematician, was certainly not going to compliment the man whom he knew had condemned his Foundations of Algebraic Geometry to total oblivion. – Georges Elencwajg Nov 5 '12 at 22:43
up vote 92 down vote accepted

No one with any familiarity with his work can doubt that Siegel was one of the greatest mathematicians of the 20th century. Weil was a decisive, opinionated man -- just the type of person who would have an answer to this question ready at hand. And "Carl Ludwig Siegel" is a totally unsurprising answer from anyone. (Also "Andre Weil" would be a totally unsurprising answer from anyone: it might be my answer!)

But it is especially unsurprising coming from Weil. The list of contemporary mathematicians of the Siegel-Weil caliber is short enough, and among mathematicians on that list -- e.g. Wiener, von Neumann, Kolmogorov, Godel -- the research interests of Siegel and Weil were especially close: for instance, there is a Siegel-Weil formula. Both brought their prodigious knowledge and technique to bear on number theory, but with distinct, and distinctive, styles. To be very brief and crude, Weil had a fundamentally algebraic approach, whereas Siegel had a fundamentally analytic approach. My own approach to mathematics is rather close to Weil's (although in magnitude, microscopic compared to his): I very much appreciate that finding the right bit of "structure" can make the solution of your problems self-evident. A lot -- by no means all -- of Weil's work is like that: the finished product is so tidy and efficacious that you too easily forget to ask how he thought of any of it in the first place. To someone with this "algebraic" style, Siegel's work looks like a sequence of miracles. So it is unsurprising to me that someone like Weil would select someone like Siegel to give his top regards.

I think you can also gain some insight into why Weil named Siegel by considering their ages: Siegel (born in 1896) was ten years older than Weil (born in 1906). Ten years is long enough for Siegel always to have been ahead of Weil in his career and stature, but short enough for them to be true contemporaries and competitors. Most other great mathematicians that spring to mind when I think of Weil are actually quite a bit younger, e.g. Serre (born 1926), Tate (born 1925), Shimura (born 1930); it makes sense that Weil is not going to name any of these as the greatest mathematician of the 20th century. Indeed all three are alive well into the 21st century.

[Added: I just remembered that Chevalley (born 1909) was a contemporary of Weil of a similar stature. But Chevalley was very close to Weil, both personally and in mathematical styles and tastes. It is psychologically natural to esteem (and fear) most that which is most different from ourselves, not that which is most similar. Anyway, for Weil to name Chevalley would have sounded arrogant, as if not being able to name himself he picked the person standing right next to him.]

By the way, I think that Shimura and Siegel are quite similar in style as well as stature. I read Shimura's autobiography, and I think he is right to be profoundly disappointed that Siegel did not take more of an interest in his work. Shimura's work is closer to being a continuation of Siegel's (including a continuation of the brilliance, creativity and orginality!) than any other mathematician I can think of, so it is natural that Shimura holds Siegel in high regard.

There is also something "organic" in the work of both Siegel and Shimura which naturally bristles a bit at the "Bourbakistic" influence of the French school: it seems clear enough, for instance, that the modern theory of "Shimura varieties" is both an addition and a subtraction from what Shimura himself intended. I know several of Shimura's students, and though they work in what the rest of the mathematical world thinks of as parts of algebraic number theory and arithmetic geometry, in the way they actually think about mathematics they take a more analytic Siegel. I have even fewer credentials to speak for Selberg than I do for any of these others, but I imagine that he may have felt a similar kinship to Siegel, i.e., the use of an "analytic" approach to studying problems that others regard as being more algebraic.

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    Thanks Pete! I really appreciate your historical perspective. – Jonah Sinick Nov 5 '12 at 7:04
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    ...and very thorough answer. – Jonah Sinick Nov 5 '12 at 7:08
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    This is a very nice answer and, a bit paradoxically, illustrates how the question is not very strange. On the one hand, I do understand how some people may be wary of questions that somehow feel 'gossipy.' However, I understood Jonah's question to be something like this: 'Of course Siegel is great, but there are many great mathematicians. What special circumstance made Weil name Siegel in particular? I think a good answer to this question by someone knowledgeable in number theory and some history would be quite illuminating from a mathematical point of view.' – Minhyong Kim Nov 5 '12 at 7:31
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    Pete's answer demonstrates very well the correctness of this expectation, but I agreed even before I saw it. – Minhyong Kim Nov 5 '12 at 7:31
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    Note that Weyl \neq Weil. :) – Joel Dodge Nov 5 '12 at 20:30

In addition to the other insightful and informative answers, for veracity it probably should be noted that there is a significant chance that Weil's remark about Siegel was disingenuous or sarcastic, with some ulterior motivations, all the better that it is defensible, and perhaps out-of-the-blue to listeners at the time.

Being somewhat acquainted with many of the bigger-than-life persons mentioned in other answers, and while greatly respecting their work, I would not have much hope of getting a straightforward, sincere answer from any of them about any questions that touched their own accomplishment and potential place in the historical record, or even touched a question of their taste.

The defensibility on scientific grounds of nominating Siegel at mid-century for the greatest mathematician in that century is a little misleading, also, given the non-smoothness of mathematical activity (what with wars and such getting in the way, too).

In addition to complicated sarcasm, there is a large possibility that Weil chose that moment to attempt to invalidate other parvenues' claims to "kewlness", by referring to an ur-classical figure.

(Another point, referring to another answer: it is my impression that Weil would not have thought his "Foundations" (of alg geom) was a long-lasting edifice, but, rather, that it was a stop-gap measure. The point at the time was that the "geometric Italian school" had not provided proofs of a certain sort... and that that issue had produced false conclusions, not only that the proofs/heuristics were not clearly airtight. It was a different time. There was no definition of "Jacobian" in positive characteristic. Arguments "by continuity" that had plausible sense, if still unrigorous, in characteristic $0$ had dubious sense in positive characteristic. Indeed, in 1970, say, as I can personally aver, it was certainly not the case that all the world had capitulated to Grothendieck's idea of alg geom.)

But Pete Clark's essay is more constructive, less dragged down by issues of personality, ego, prejudice, arrogance! :)

Nevertheless, one should reserve endless discounting for the effects of personality, ego, prejudice, arrogance on questions of taste or judgement. :)

... exemplified in the silly-but-profoundly-explanatory "Why can't a woman ... be more like a man?"

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    Very interesting answer, Paul. As informative and interesting might be Pete's answer, it fails somehow to account for the big surprise that Weil, one of the cofounders of Bourbaki, considered Siegel, a mathematician with well-known hostility to Bourbaki's style, as the best mathematician of the (last) century. Your answer, as well as psteldr's (whoever it may be), has the merit of addressing this surprising element. – Joël Nov 7 '12 at 1:45
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    This is an interesting point of view. Yet, I would just like to add that I think one can assert that Weil at least really had a high opinion of Siegel, and not only very early on. I did not yet find the time to check each mention and do not have it handy now, but in the Cartan--Weil correspondence Siegel is mentioned a couple of times and where I checked Weil's tone was often positive; e.g. suggesting work of Siegel to Cartan for a seminar. And since in this correpondence Weil seems really not shy to be critical of people, I think this means something. – user9072 Nov 7 '12 at 10:58
  • Dear Paul, This is a very interesting answer, and I think I understand your caution with respect to the larger-than-life people. However, I would have to agree with quid in this case: Weil speaks highly of Siegel in a number of different places. John Coates points out to me that Siegel was the person who really thrust Weil's name into the mathematical landscape by using the Mordell-Weil theorem to prove the finiteness of points on affine hyperbolic curves. – Minhyong Kim Nov 8 '12 at 13:58
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    (That should have been integral points.) It's easy to forget now that the significance of proving the finite generation of points on something as exotic as a Jacobian, not corresponding in any obvious way to an actual equation, must not have been obvious in the early twentieth century. Siegel's concrete theorem was what made it clear that this was a useful thing. It's also easy to forget that Siegel's theorem was essentially the best thing Diophantine geometry had to offer for many decades preceding Faltings, and probably one of the best theorems in all of number theory. – Minhyong Kim Nov 8 '12 at 14:04

Apparently, the respect was not reciprocated. I heard a story about Siegel (near the end of his life) asking Paul Cohen (who was no fan of Weil, since they had intersected at Chicago in the fifties): So what happened to that promising young man Andre Weil? He wrote a nice thesis, but what happened since then?

  • Source? BTW, what's Weil's thesis about? Mordell-Weil? – 36min Nov 6 '12 at 4:15
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    Second hand from Paul Cohen, and yes. – Igor Rivin Nov 6 '12 at 4:19
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    @anon: I am quite sure CLS knew exactly what AW was up to in the fifty years since his thesis. What he thought of it was perhaps expressed by his question. – Igor Rivin Nov 7 '12 at 21:00
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    @Igor: I don't mean anything personal by this, but: "Igor Rivin says that Paul Cohen said that Carl Siegel said" is not my idea of an unimpeachable source. (You can change the names of the people involved to X,Y,Z; the problem is that this is double hearsay.) – Pete L. Clark Nov 8 '12 at 4:29
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    More to the point, all opportunity for context is lost. When I said that the statement was implausible, I meant that it was implausible that Siegel was sincerely inquiring what happened to Andre Weil in a period of at least 30 years since his thesis. If Siegel was making some kind of joke or ironic comment: okay, but I think to appreciate it we had to be there...and we weren't. Maybe the quip means that Siegel didn't respect Weil; maybe it means something else entirely. I am wary of drawing any serious conclusions from anecdotes like this one. – Pete L. Clark Nov 8 '12 at 4:39

On top of all the classical work that has been mentioned, Siegel made important contributions to other areas, and eventually became a major figure in Dynamical Systems. It is not just anyone who can switch fields and remain at the top. See my account of his proof of existence of Siegel disks.

I would just like to mention that in addition to the many shared research interests of Siegel and Weil, they were clearly well acquainted with one another, with both attending Max Dehn's seminar on the history of mathematics over a period of years. From pages 124-127 of Yandell's book "The Honors Class" and pages 51-53 of Weil's "The Apprenticeship of a Mathematician", it shows that Siegel attended the seminar regularly from 1921-1935 while Weil began attending in 1926 and during subsequent visits to Frankfurt says "as often as I could". So Weil's high esteem for Siegel is doubtless influenced by personal interaction at the seminar (and elsewhere) in addition to familiarity with his research.

For Weil, the Mordell-Weil theorem; for Siegel, the theorem on integral points on curves (genus at least 1). Think about the use of abelian varieties here. Mordell-Weil is sort of about making Mordell's descent work with a general abelian variety rather than just a one-dimensional one. Siegel's result has to go through Jacobians but brings a sharper version of Thue's work on diophantine approximation to bear. These are both "diophantine geometry" today but Siegel's work still looks like a major advance.

Edit: Maybe I left out the point. Didn't Hadamard tell Weil that he should prove the Mordell conjecture, to do a proper job? And the chronology: both those results came out in 1929.

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