CONJECTURE

A friend of mine recently told me that the Dutch word for Mathematics is Wiskunde, a word coined in the 16th century by Simon Stevins. My friend suggested that it was derived from “Wis” meaning “sure” and “kunde” meaning “the study of”. Carrying this word back into English, we would come up with something like “Sure-ology”. But even if the subject, Mathematics, is named “Sure-ology”, its history would be poorly described by such a name. For like any Intellectual History, lots of the History of Mathematics is simply never captured: its major artifacts are ideas which spend much of their life in a volatile, unrecorded, state. Their eventual distillation as written record occurs long after their initial discovery. The presence of “Conjecture” as a pointedly formal element in the syntax of recent Mathematics may have the effect of counteracting this a bit, of bottling some vapours of mathematical ideas at an earlier stage than was previously done. Might this help in the historiography of Mathematics? When Aki Kanamori telephoned me asking whether I might be interested in giving a talk in the BU conference,1 I had been reading a book (Rapoport 1989) entitled Beilinson’s Conjectures on Special Values of Lfunctions. Telephone receiver in hand, I was suddenly struck by the thought that such a book, broadcasting “conjectures” in its very title, represents a large change in the way mathematical theories are worked on; this change has not been all that widely written about. At least I think it hasn’t. Here we have a book in Pure Mathematics, a whole book, fully devoted to the exposition of an extensive constellation of conjectures! It, and other mathematical works like it, signal that the art of conjecturing has achieved a formidable, and quite formal, prominence in the mathematical landscape. I believe that “formalization of conjecture” is a good thing for Mathematics, and an inevitable thing, as large mathematical theories grow larger. There is something perplexing and delicious when one finds a rarely discussed theme which one imagines to be interesting. The temptation to destroy its deliciousness by actually talking about it is then hard to resist!