83

In "How To Solve It", George Pólya writes:

"There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann." [2nd ed. (1957), p. xv]

Could someone tell me please what was that theorem which von Neumann proved?

Salech Alhasov
  • 6,946
  • 2
  • 30
  • 47
  • 2
    The fly infinite sum? – CAGT Feb 01 '14 at 03:31
  • 15
    @CAGT That's not a "theorem" that "is not proved" and "may be difficult". Polya is probably referring to an open problem here, since he says in other places that when he mentioned open problems in class that Von Neumman would often present him with the solution after class. – Bill Dubuque Feb 01 '14 at 03:52
  • 3
    Is thi srelated to the anecdote about a student coming late to class, copyiing what he thinks is a problem from the blackboard, handing in a solution next time saying "todays homework was quite tricky" - not having noticed that the "homework problem" was in fact an open problem? – Hagen von Eitzen Apr 12 '14 at 20:13
  • 5
    In the book Famous Puzzles of Great Mathematicians it says that George Pólya said

    "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."

     – Euna Apr 17 '14 at 05:44
  • 1
    I was not able to find the quote you included in the post in my copy of Polya's book (ISBN 9780691023564, it seems to be a reprint of the 2nd edition). I also failed to find it using Google Books. But maybe the editions of the book differ somehow. – Martin Sleziak Jun 09 '14 at 11:33
  • 13
    @HagenvonEitzen You probably mean the story about Dantzig: Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue. – Martin Sleziak Jun 09 '14 at 12:19
  • It was about a bird traveling between two opposite directed trains . – Fardad Pouran Jun 10 '14 at 05:52
  • 4
    @Fardad: No it wasn't. Thats a different von Neumann story. – TonyK Jun 10 '14 at 18:52
  • The version of the story outlined by @Hagen that I heard had Mary Ellen Rudin in the star role. – Jyrki Lahtonen Jul 12 '14 at 10:01
  • 1
    @HagenvonEitzen I've heard a similar story about John Milnor coming to class and proving hard, open topology problems resulting in his being offered tenure immediately upon termination of his graduate education. – Adam Hughes Jul 30 '14 at 03:12
  • 1
    @MartinSleziak This quote is indeed not in How to solve it, but it is in The Polya Picture Album. Encounters of a Mathematician, which Polya published in 1987. The variant that Bill Dubuque mentions above seems to be due to Paul Halmos, who wrote about von Neumann in the Monthly in 1973. – Per Erik Manne Jul 31 '14 at 18:57
  • Can someone who invented the von Neumann ordinals be a genius? – Han de Bruijn Sep 04 '14 at 20:05

1 Answers1

12

John von Neumann studied at Zürich during in Ph.D. and got Ph.D at the age of 22, in 1925. Before that year, Neumann published four papers (see here). The first one, published in 1922, is about roots of polynomials. The paper does not mention Pólya, but Pólya is known to have worked on ths subject.

The second paper is about the ordinals, the third about set theory which seems less likely. The fourth (in Hungarian) is about sequences and could contain the theorem, but I have not been able to access this article.

Of course, there is the possibility that von Neumann never wrote a paper about this theorem.

domotorp
  • 970
Tom-Tom
  • 7,017
  • 3
    In the Hungarian paper it is written that he heard this question from Fejér, so this can be also ruled out. – domotorp Jul 30 '21 at 07:14