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Wikipedia has the following quote on John von Neumann:

Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: "Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods." He went on to explain that the three methods were:

  • A facility with the symbolic manipulation of linear operators;
  • An intuitive feeling for the logical structure of any new mathematical theory;
  • An intuitive feeling for the combinatorial superstructure of new theories.

And I am wondering what do 'logical structure' and 'combinatorial superstructure' mean in this context? Please explain these methods.

https://en.wikipedia.org/wiki/John_von_Neumann#Mastery_of_mathematics

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  • $\begingroup$ It seems to me that the three are not "methods" but "capabilities" : facility, intuition, ... It is hard to imagine a way to "learn" them. You can see : Giorgio Israel and Ana Millán Gasca, The World as a Mathematical Game: John von Neumann and Twentieth Century Science (2009) for a "book [that] provides the first comprehensive scientific and intellectual biography of John von Neumann, a man who perhaps more than any other is representative of twentieth century science." $\endgroup$ Oct 28, 2016 at 9:21

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The quote is from Adventures of a Mathematician (by Stanislaw Ulam):

"Von Neumann was different. He also had several quite independent techniques at his fingertips. (It is rare to have more than two or three.) These included a facility for symbolic manipulation of linear operators. He also had an undefinable "common sense" feeling for logical structure and for both the skeleton and the combinatorial superstructure in new mathematical theories. This stood him in good stead much later, when he became interested in the notion of a possible theory of automata, and when he undertook both the conception and the construction of electronic computing machines. He attempted to define and to pursue some of the formal analogies between the workings of the nervous system in general and of the human brain itself." (page 96)

In The World as a Mathematical Game (by Giorgio Israel and Ana Millán Gasca), we read that:

"In actual fact, during the Königsberg Congress, none of the eminent participants realized the full import and implications of the result announced by Gödel – with one exception: von Neumann. After the discussion the latter rushed up to Gödel and took him aside in order to get a better understanding of his demonstration. He then left the Congress in a state of extraordinary excitement and spent the next month working on the issue. Less than two months later he wrote to Gödel to announce he had demonstrated, as a consequence of the theorem of incompleteness, that the consistency of arithmetic cannot be proved. Gödel replied that he had in the meantime succeeded in obtaining this demonstration and sent him a copy of the article that had already been presented for publication." (Chapter 2, page 30, ISBN 978-3-7643-9895-8 Birkhäuser Verlag AG, Basel - Boston - Berlin)

And in this interview, Eugene Wigner tells a story where he asked Neumann if he could explain a theorem. Neumann asked Wigner whether he knows certain other theorems and a few more things. Subsequently, he provided an explanation (proof) on the spot using only theorems Wigner knew. Wigner concludes:

"He understood things, not only in one way, but also together/in combination [with other theorems] ['összefoghatóan']."

The Hilbertian view of mathematics was that of a gigantic "combinatorial game", and the above quotes demonstrate Neumann's ability to see how the elements combine together in a theory (skeleton) or how the theory (or theorem) connects (fit with) other theories (combinatorial superstructure).


Some further quotes from The World as a Mathematical Game:

"Moreover, his work on axiomatization and on proof theory led to a view of mathematics as «a combinatorial game played using primitive symbols in which it had to be determined in a finitely combinatorial way which combinations of primitive symbols the methods of construction or ‘proof’ led to», as he claimed at the Königsberg congress (Neumann (von) 1931). He never abandoned this view, and indeed built it up over the years, and this view helps to explain his interest in the scientific topics he concerned himself with in the 1940s and 1950s." (Chapter 2, page 30)

and

"Commenting on von Neumann’s scientific personality, Jean Dieudonné claimed that his genius lay in analysis and combinatorics, the latter being understood in a very wide sense, including the uncommon ability to organize and axiomatize complex situations that a priori do not seem amenable to mathematical treatment, as in quantum mechanics and the theory of games (Dieudonné 1976, 89)." (Chapter 2, page 48)

and

"For von Neumann, the world must be conceived of as a mathematical game, in the sense that in all cases it is useful and effective to seek axiomatic structures suitable for thinking of the phenomena in mathematical terms. The concept of strategic game is a kind of universal key for considering in terms of combinatorial structures all the interactions occurring in reality and to determine the conditions in which they allow an "acceptable" solution. But this in no way means that the world is actually a mathematical game. The conception of social interactions as a game, the combinatorial view on which the theory of automata will be founded, the analogy between brain and computer are tools of epistemological analysis, never ontological views." (Chapter 3.5, page 73)

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