Abstract

To the Pythagoreans, to apeiron, that which has no bounds, the infinite, “was something abhorrent[aut]Moore, Adrian” (Moore, The infinite, Routledge, p 19, 2001). Nevertheless the infinite thrust itself upon them once they became aware of the fact that there was no common length for which the sum of finitely many instances of it would be equal to both the length of the side and the length of the diagonal of a square. Again and again mathematicians have engaged warily in infinitary reasoning, and most often the methods were found to be useful and robust even when rigorous justification was only to be obtained at a later time. In this essay I will survey some of these encounters with the infinite, beginning with Torricelli’s[aut]Torricelli, Evangelista horn that was so shocking in the seventeenth century, and ending with the very enormous infinite sets that contemporary set theorists study without any hope of a proof that their existence is consistent with the axioms of ordinary mathematics.