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We were discussing this question at dinner this evening: Who is the last mathematician who had an understanding of a large proportion of mathematics (at the time they were alive)?

I think it is safe to say that the mathematician lived and worked before the secong world war. After that period it has become impossible to aquire knowledge of such a large and rapidly growing subject.

Picard (1856-1941) was the best suggestion we have come up with because he published research papers and wrote text books in a vast range of different mathematical subjects.

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closed as not constructive by Kevin H. Lin, José Figueroa-O'Farrill, Harry Gindi, Steve Huntsman, S. Carnahan Jun 12 '10 at 0:56

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This question would make for an interesting blog post, with an interesting thread of comments. For a MO question, probably not so much. –  Mariano Suárez-Alvarez Jun 11 '10 at 23:15
I agree with Mariano. –  Kevin H. Lin Jun 11 '10 at 23:20
This calls for a meta thread - tea.mathoverflow.net/discussion/439/… –  François G. Dorais Jun 11 '10 at 23:43
@alext87: I think this should be a community wiki question. –  François G. Dorais Jun 11 '10 at 23:47
You mean a ploymath? May be Poincare. –  Unknown Jun 12 '10 at 7:19

5 Answers 5

John von Neumann.

From Wikipedia:

John von Neumann (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician who made major contributions to a vast range of fields,[1] including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history.[2] The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians",[3] while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century.[4] Even in Budapest, in the time that produced geniuses like von Kármán (b. 1881), Szilárd (b. 1898), Wigner (b. 1902), and Edward Teller (b. 1908), his brilliance stood out.[5]

Von Neumann was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory[1][6] and the concepts of cellular automata[1] and the universal constructor. Along with Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.

My own personal favorite of his ideas are the von Neumann ordinals: every ordinal number is precisely the set of smaller ordinals.

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Yes. That's a good one. Thanks! –  alext87 Jun 11 '10 at 23:17
I doubt that Von Neumann had an understanding of all of mathematics. For example, algebraic geometry was already being developed in Italy by this time, and it doesn't seem like Von Neumann ever studied it. I think we probably have to travel a lot further back than Von Neumann or Hilbert to actually find someone who understood all of the mathematics of his time. I would say Euler, Gauss, or Riemann are good bets. –  Harry Gindi Jun 12 '10 at 0:10
The text of the question mentions "large proportion" rather than "all" mathematics. –  Joel David Hamkins Jun 12 '10 at 0:18
Somewhere I read an anecdote about von Neumann talking to some grad students who were stunned to realize that he didn't know anything about some rather simple concepts of topology. The point of the anecdote I think was in fact to illustrate that you don't have to know everything. –  Mike Benfield Jun 12 '10 at 1:42
One factor seems to be practical application and modeling of abstractions. This is opposed to purely abstract thinking or specialized thinking, the domains where most of us function. Von Neumann's famous accomplishments include introducing rigorous mathematical models to areas that previously had none. The trend of Von Neumann's accomplishments is almost unsurprisingly predictable given his particular skills. While impressive, it does not encompass "all mathematics" or even the larger portion of it. Although, he probably knew more than most of us about the nature of math itself. –  Alexander Pritchard 2 days ago

Poincaré is often mentioned in that light.

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I can only attest that it is common folklore that Hilbert is the last mathematician to have understood all of mathematics (I can't recall where I've read this; but I know I've seen or heard this in more than one place).

But this is folklore; I of course can't judge whether this is true, or if other mathematicians (you mention Picard, and Joel David Hamkins even says von Neumann) truly fit this criteria. I feel this is very subjective and the only people who may be allowed to make such claims should be experts very knowledgeable in the history of mathematics.

EDITED: Picard was born 6 years before Hilbert, and they both died in the early 40's. So chronologically it would make sense if they both represent the last generation that could have had a full understanding of all of mathematics.

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Hilbert (1862-1943) is a good suggestion. This is slightly earlier than John von Neumann but definely a true contender. –  alext87 Jun 11 '10 at 23:24

By the way, facetiously one could add "Bourbaki" to that list.

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Bourbaki doesn't know any hard analysis, only the soft stuff which requires no difficult estimates. –  BCnrd Jun 12 '10 at 0:37
Maybe Bourbaki thinks he knows everything? –  Michael Hardy Jun 12 '10 at 0:40
Dear Michael, according to Dieudonné, Bourbaki was created to write down only the most elementary and foundational results (cf. The Work of Nicholas [sic] Bourbaki, which is an interview with Dieudonné). Clearly, if we go just by the work published under the name "Nicolas Bourbaki", then Bourbaki certainly didn't think he knew everything. If instead we ascribe to Bourbaki the sum of the knowledge of all of his collaborators, then Bourbaki is really a serious contender. I realize that it's fashionable nowadays to dismiss Bourbaki's accomplishments, but I personally think it's in poor taste. –  Harry Gindi Jun 12 '10 at 9:28

Many people have mentioned Jean Dieudonné (1906-92) in this regard.

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Maybe, as his name suggest, he really was godsend... –  François G. Dorais Jun 12 '10 at 22:32
Dieudonné presumably had not a very deep knowledge of the theory of probability of his time. He was also not fond of logic and deep foundational questions. That said, his knowledge was truly prodigious. –  Joël Jul 18 '12 at 1:05

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