Highly complex proofs and implications of such proofs

Description

USA.: California Institute of Technology, [Phil. Trans. R. Soc. A (2005) v.363, 2401–2406, eBook, English]Contents Conventional wisdom says the ideal proof should be short, simple, and elegant. However there are now examples of very long, complicated proofs, and as mathematics continues to mature, more examples are likely to appear. Such proofs raise various issues. For example it is impossible to write out a very long and complicated argument without error, so is such a ‘proof’ really a proof? What conditions make complex proofs necessary, possible, and of interest? Is the mathematics involved in dealing with information rich problems qualitatively different from more traditional mathematics. I have some experience with one such effort: the Classification of the finite simple groups. I’m going to use the Classification theorem and its proof as a basis for discussion, but I’m not going to state the theorem or go into details about the proof. Rather I’ll treat the Classification and its proof as a black box, in that I’ll begin by listing some features of the theorem and its proof, and later use them to help illustrate some of the points I hope to make.