Sunday, January 19, 2014

Deep Waters

Almost 300 years ago Leonard Euler, who has to be in the competition for greatest mathematician of them all, discovered a "proof" that the sum of all the natural numbers (1 + 2 + 3 ...) is "equal to" -1/12. This result is not exactly intuitive, and it seems that PZ Myers recently learned about it and objected, ostensibly in the name of skepticism. Lumo heard about Myers objection, and objected to it. In his inimitable, or at least better unimitated, fashion.

The point is that Euler's mathematical curiosity turns out to be generalizable to whole classes of conventionally divergent series, and even more curiously, may well have some relevance to physics. In fact, Euler's series pops up in string theory, as Lumo explains - I also like the explanation in Zwiebach's A First Course in String Theory.

Is it coincidence that certain areas of physics are plagued with divergent series and that mathematicians have found ways to make sense of generalized sums of some such series? Beats the heck out of me, but it could be one of God's little hints.