(Here’s another tidbit which originally appeared on my physics blog.)
(Warning: Much of this was written in a cloud of decongestants. If there are any errors, well, you know why….)
Infinity can be a slippery concept, and it causes no end of woes to mathematicians. But, over the years, they have gotten a better and better handle on the concept. This was helped greatly by the work of Georg Cantor, who developed the basic mathematical tools used today for grappling with infinities. But he was by no means the first, nor the last. The legendary 18th century mathematician Leonhard Euler made great strides in devising methods for dealing with divergent infinite series, and the definitive work on that subject is the 1949 book by G. H. Hardy, Divergent Series.
Knowing how to deal with infinities and divergences can be crucial. The need to tame divergences in the blackbody radiation problem led Planck to take the first step in creating quantum mechanics (although, to be fair, Planck basically reformulated the problem in a form that didn’t result in infinities). Quantum electrodynamics calculations required the creation the “dippy procedure” of renormalization. And, on the bleeding edge of theoretical physics, the challenge of reconciling quantum theory and general relativity is fraught with seemingly intractable divergences. But such heady problems aren’t the only places where divergences crop up. They can arise in the most seemingly simple math problems. Read More