Fitting Surprise in Mathematics

25 09 2012

In his faculty research colloquium this past Friday, my colleague, mathematician Andrew Shallue, gave a presentation titled “Constructing Large Numbers with Cheap Computers.”  As a part of this presentation, in which he discussed how he and his fellow researchers created the world’s largest Carmichael number, Andrew read a portion of the following, from chapter 18 of G.H. Hardy’s A Mathematician’s Apology:

”What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid’s and Pythagoras’s?  I will not risk more than a few disjointed remarks. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions.  There are no complications of detail – one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency.  We do not want many `variations’ in the proof of a mathematical theorem: `enumeration of cases,’ indeed, is one of the duller forms of mathematical argument.  A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”

Needless to say, I was intrigued. The whole talk (at least as much as I could understand) was excellent, but Hardy’s idea that the famous theorems of Euclid and Pythagoras are noteworthy because they seem both inevitable and unexpected struck a chord: Hardy was impressed by fitting surprise in mathematics, and fitting surprise is something—an aspect of poetry and many other arts (fiction, drama, painting)—that interests me greatly. Now, here it is in mathematics. Seems fitting. And surprising.