The Mathematical Experience

by Philip J Davis & Reuben Hersh

6.Comparative Aesthetics (p298)

WHAT ARE THE ELEMENTS that make for creativity? Is it a deep analytic ability deriving from ease of combinatorial or geometric visualization - a mind as restless as a swarm of honeybees in a garden, flitting from fact to fact, perception to perception and making connections, aided by a prodigious memory-a mystic intuition of how the universe speaks mathematics -a mind that operates logically like a computer, creating implications by the thousands until an appropriate configuration emerges?
Or is it some extralogical principle at work, a grasp and a use of metaphysical principles as a guide? Or, as Henri Poincaré thought, a deep appreciation of mathematical aesthetics?
There is hardly a science of mathematical aesthetics. But we can look at an instance and discuss it at length. We can get close to the reason for Poincaré's assessment. I shall take a famous mathematical theorem, one with a high aesthetic component, and present two different proofs of it.
The theorem is the famous result of Pythagoras that Ö2 is not a fraction. The first proof is the traditional proof.

Proof I. One supposes that Ö2 = p/q where p and q are integers. This equation is really an abbreviation of 2 =p²/q² . One supposes that p and q are in lowest terms, i.e. they have no common factor. (For if they do, strike it out.) Now 2 =p²/q² implies p² =2q² . Therefore p² is an even number. Therefore p is even (for if it were odd, p² would be odd since odd x odd = odd). If p is even it is of the form p = 2r, so that we have (2r)² = 2q² or 4r² = 2q² or q² = 2r² . Thus, as before, q² is even so that q must be even. Now we are in a logical bind, for we have proved that p and q are both even, having previously asserted that they have no common factor. Thus, the equation Ö2 = p/q must be rejected if p, q are integers.
The second proof is not traditional and is argued a bit more loosely.

Proof II. As before, suppose that p² =2q². Every integer can be factored into primes, and we suppose this has been done for p and q . Thus in p² there are a certain number of primes doubled up (because of p² = p . p). And in q² there are a certain number of doubled-up primes. But (aha!) in 2 . q² there is a 2 that has no partner. Contradiction.
I have no doubt that nine professional mathematician,out of ten would say that Proof II exhibits a higher level of aesthetic delight. Why? Because it is shorter? (Actually, we have elided some formal details.) Because, in comparison, Proof I with its emphasis on logical inexorableness seems heavy and plodding? I think the answer lies in the fact that Proof II seems to reveal the heart of the matter, while Proof I conceals it, starting with a false hypothesis and ending with a contradiction. Proof I seems to be a wiseguy argument; Proof II exposes the "real" reason. In this way,the aesthetic component is related to the purer vision.

Further Reading See Bibliography S.A.Papert, [1978]