Singularities and Lunatic Fringes

I copied the following passage from Imre Lakatos’ unfinished dialog, “Proofs and Refutations,” which recasts the debate over Euler’s conjecture that: for all regular polyhedra, V – E + F = 2, where V stands for the number of vertices, E the number of edges, and F the number of faces on any polyhedra.

I think that if we want to learn about anything really deep, we have to study it not in its “normal,” regular, usual form, but in its critical state, in fever, in passion. If you want to know the normal healthy body, study it when it’s abnormal, when it’s ill. If you want to know functions, study singularities. If you want to know ordinary polyhedra, study their lunatic fringe. This is how one can carry mathematical analysis into the very heart of the subject … If you want to draw a borderline between counterexamples and monsters, you cannot do it in fits and starts.

Lakatos’ full dialog richly explores the central problem of mathematical (which is to say “logical”) proof /and/ does so by considering a particular geometric “monster” brought to life by Euler. There are so many wonderful ideas delineated by this imaginary dialog that it proves the wealth mathematics has to offer us all.

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