What does the following quote on John von Neumann mean? 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
 A: 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)

