A History of the Infinite III: The Infinite in Mathematics

This lecture is concerned with the mathematics of the infinite, most of which has developed relatively recently. I outline the principal ideas of the brilliant German mathematician, Georg Cantor, who not only showed that we can distinguish between different infinite sizes, but who devised infinitely big numbers to measure them and who showed how to perform calculations with these numbers. Cantor’s work was both revolutionary and profound. But it greatly polarized opinion amongst his late nineteenth- and early twentieth-century contemporaries, contributing to a complete breakdown in Cantor’s mental health. His work also gave rise to several new paradoxes that contributed to another breakdown of sorts – a breakdown in work on the foundations of mathematics. I expound some of these paradoxes, including Bertrand Russell’s famous paradox of the set of all and only those sets that do not belong to themselves, and I discuss how mathematicians reacted to them. In particular, I talk about Gottlob Frege’s reaction, which was exacerbated by the fact that Russell’s paradox looked as though it had completely destroyed his life’s work, namely his attempt to provide mathematics with rigorous and secure foundations. I also discuss subsequent developments in mathematics that exploit some of these paradoxes, including some of Kurt Gödel’s work. I conclude that, by subjecting the infinite to formal scrutiny, mathematicians have ended up creating more problems for themselves than they have been able to solve and have found themselves having to reckon with some extraordinarily deep puzzles at the very heart of their discipline.