The concept of a proof from the Book was devised by Paul Erdős. The Book contains the best possible proof of each theorem of mathematics. The idea has been popularised in a book of the same name by Martin Aigner and Gunter Ziegler, who have assembled a fine collection of short and beautiful proofs. All of mathematics is there, and I am sure that many people have found their book inspirational.
Yesterday in a lecture (I am writing this on Tuesday but it will be posted on Monday because of a time warp), Geoff Whittle suggested a related but slightly different concept, which I like. This is perhaps aimed more at professional mathematicians than at beginners. This is “Fifty proofs to read before you die”. Whereas the Book is something that only God (and perhaps now Erdős) can read, Whittle’s concept seems to me to be much more human. As with other lists of “Fifty Xs to Y before you die”, everyone can produce their own list, according to their own criteria, stated or implicit.
Geoff’s suggestion for such a proof is Nash-Williams’ proof of Kruskal’s Theorem asserting that finite trees are well-quasi-ordered by the minor relation. As far as I understood his thinking, this is a substantial proof, which is not only a work of art to be savoured, but has also provided tools and techniques of wide applicability. This last criterion makes Geoff’s concept a bit different, since a clumsy (or even ugly) proof may be more fruitful in the long run.
I don’t want to be dogmatic about the criteria, but I would be very interested in any suggestions you may have for this list. Either add your suggestion as a comment, or email it to me; if there is a response, I will produce a commented list at some point.