Saturday, June 13, 2009

Visual Proofs

We've been enjoying more raucous debate at the Math Forum about whether we have "visual proofs" in mathematics. This is a question for investigation, ala Wittgenstein into language games, not a knee-jerk response.

For example, suppose we wanted to prove a tetrahedron is simpler than a cube. The skeptic says "prove it!" and we go into some counting procedure, pairing edges with ordinal numbers and showing the cube has twice as many, ergo is over-built for the purposes of enclosing volume. QED.

However, the skeptic sees a lot of skulduggery in the above proof and will want to go back and "define terms" e.g. "simpler". It's not fair if he loses, is the basic premise.

I'm certainly in favor of using cartoons for the purposes of marshaling one's faculties, giving intuitive insights "in a flash" where that's doable. Satisfying mathematics includes gestalt switches of this kind. But that's not to say I'm against plodding (pedantic) proofs where necessary. Sometimes you get a poky pony to ride, been there done that.

On behalf of the skeptic: oft times those swoopy flashy proofs bleep over key points, aren't truly open source in that you can't do them at home, more like legerdemain (sleight of hand), and really not truly proofs in that sense (more like clever hoaxes, spoof proofs).

But remember: once arrived at, by whatever means, a clever visual proof may be just what the doctor ordered and "swoopy flashy" is not necessarily a criticism in this case.

By way of analogy, consider some of Ramanujan's astounding "generator expressions" (Pythonic namespace) -- the proof is in the pudding as they say (you've got your decimal type, check it out), but does that mean we always know where he got 'em in the first place? Not really. A proof can be like that too (out of a hat).