I listened to a talk this afternoon, with the title “Short theorems with long proofs”, given by Professor Colva Roney-Dougal. It has long been known in mathematics that there are short concise statements of fact, that take inordinate lengths of time to establish. Take Fermat’s Last Theorem as an example. The four colour theorem is another example. There is seemingly no limit to how long a proof of a simple statement needs to be. Colva explained this in a remarkably simple way – a proof does not have a length, it has an area. What matters when you are trying to prove a theorem, therefore, is not how long the proof is, but how wide.
If you’ve ever seriously tried to prove the four colour theorem, or Fermat’s Last Theorem, or anything else for that matter, you will have noticed how awkward cases keep running off to the side, and avoiding your direct assault on the problem. Army Generals will tell you the same thing – it is not so much a matter of how far you can advance, it is much more important to measure how wide the front is. You do not advance by yards, you advance by acres.
Mathematicians understand this aspect of progress. I’m not convinced that physicists always do. It’s all very well sending String Theory off advancing towards Moscow in the middle of winter, but when they have lost all contact with the reality of Paris, what really is the point?
Colva explained one more thing about the geometry of a proof – and that is its curvature. The idea comes from the four colour theorem, again. If you’ve got a map you want to colour with four colours, then the most important thing to realise is that it is a map of (part of) (the surface of) the Earth. And therefore it is curved. Euler in the 18th century calculated exactly how much curvature a sphere has got. That curvature can be re-distributed over the map – that, after all, is the point of a map, to sweep all the curvature off to the edges. The proof of the four colour theorem boils down to a very careful inventory of exactly how much curvature you have managed to sweep where. And, ultimately, the fact that you cannot sweep it under the carpet, because it will always pop up somewhere.
The same is true of mathematical proofs. Ultimately you have to tie everything together into one big coherent sphere of argument, with no holes in it. When you blow up the balloon, it must not leak. It can be made of the most delicate silk, but it must be painstakingly sewn together, and it must not leak. But you cannot make a balloon out of string – it will not fly, however hard you try.