The beauty of a good mathematical proof

Throughout his life, John von Neumann, the Hungarian-American polymath, leaned towards pure mathematics, or pure mathematics with recognised applications. In his 1947 essay entitled The Mathematician, he described his personal concept of mathematics, showing him to be thoughtful and original concerning the philosophical underpinnings of the discipline. One word von Neumann repeatedly uses is aesthetical; he defends mathematics for mathematics’ sake, consciously posting analogies to the visual arts. For example, in listing the qualities of a good mathematical proof:

One also expects “elegance” in its “architectural,” structural make-up. Ease in stating the problem, great difficulty in getting hold of it and in all attempts at approaching it, then again some surprising twist by which the approach, or some part of the approach, becomes easy, etc. Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which ”explains” the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc. These criteria are clearly those of any creative art, and the existence of some underlying empirical, worldly motif in the background — overgrown by aestheticizing developments and followed by a multitude of labyrinthine variants — all this is much more akin to the atmosphere of art pure and simple than to that of the empirical sciences.

Nevertheless, von Neumann insisted that the best mathematics was usually inspired by practical problems, perhaps in partial defence of game theory from fellow mathematicians who at the time deprecated it as an applied field.

(also see: Feynman, Bethe and the beauty of mathematics)

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