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