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Involvement in Mathematics

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 John von Neumann left his mark on many fields of mathematics, from set theory, to operator algebra, unbounded operators and game theory.

His Ph.D. focused on the axiomatization of set theory. Further, he demonstrated two methods in which to avoid Russell's Paradox in set theory.

Von Neumann also highly respected Kurt Gödel, the mathematician known for his incompleteness theorems. Soon after Gödel announced his first theorem, von Neumann independently discovered the second. In correspondence between them, Gödel told him that he had already established that discovery but had yet to publish, so von Neumann vowed not to publish the proof (Rédei, 2005).

As you have established the theorem on the unprovability of consistency as a natural continuation and deepening of your earlier results, I clearly won't publish on this subject.
-John von Neumann, November 29, 1930

Von Neumann algebras were some of his contributions to operator algebra. These algebras were first referred to as rings of operators until mathematicians Jacques Dixmier and Jean Dieudonné suggested they be named "von Neumann algebras" (Rédei, 2005).

The field of unbounded operators received some clarification from von Neumann. He published several papers on the subject; however, the terminology he used differs from the terminology most commonly used today (Rédei, 2005). Nontheless, his contributions to the mathematics involved and led to developments in quantum mechanics.

Game theory is a particular field of mathematics that owes much of its creation to the Martian. His Minimax Theorem states that "every finite, sero-sum, two-person game has optimal strategies" (Weisstein). Regarding this, von Neumann once wrote "As far as I can see, there could be no theory of games on these bases without that theorem" (Rédei, 2005). He and economist Oskar Morgenstern went on to write Theory of Games and Economic Behavior in 1944, which is also considered a basis for the field of game theory.


 Sources:

Rédei, M. (Ed.) (2005). John von Neumann : selected letters. Providence, R.I.:American Mathematical Society/London Mathematical Society.

Weisstein, Eric W., MathWorld--A Wolfram Web Resource.  Minimax theorem. Retrieved from http://mathworld.wolfram.com/MinimaxTheorem.html



Comments

  1. Doug WardNovember 15, 2017 at 10:28 AM

    "two methods in which" I would say "by which" instead. Can you say a little bit about what the methods were?

    The Von Neumann result I know best is the minimax theorem. I hadn't heard about his work in set theory or connection with Godel before. Very interesting!

    ReplyDelete

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