Version: v2, Published online: 2011
Retrieved April 24, 2019, from https://www.rep.routledge.com/articles/thematic/infinity/v-2
The infinite is standardly conceived as that which is endless, unlimited, immeasurable. It also has theological connotations of absoluteness and perfection. From the dawn of civilization, it has held a special fascination: people have been captivated by the boundlessness of space and time, by the mystery of numbers going on forever, by the paradoxes of endless divisibility, and by the riddles of divine perfection.
The infinite is of profound importance to mathematics. Nevertheless, the relationship between the two has been a curiously ambivalent one. It is clear that mathematics in some sense presupposes the infinite, for instance in the fact that there is no largest integer. But the idea that the infinite should itself be an object of mathematical study has time and again been subjected to ridicule. The mathematical orthodoxy has been that there can be no formal theory of the infinite. In the nineteenth century this orthodoxy was challenged, with the advent of ‘transfinite arithmetic’. Many, however, have remained sceptical, believing that the infinite is inherently beyond our grasp.
Perhaps their scepticism should be trained on the infinite itself: perhaps the concept is ultimately incoherent. It is certainly riddled with paradoxes. Yet we cannot simply jettison it. This is why the paradoxes are so acute. The roots of these paradoxes lie in our own finitude: it is self-conscious awareness of that finitude which gives us our initial sense of a contrasting infinite, and, at the same time, makes us despair of knowing anything about it, or having any kind of grasp of it. This creates a tension. We feel pressure to acknowledge the infinite, and we feel pressure not to. In trying to come to terms with the infinite, we are trying to come to terms with a basic conflict in ourselves.
Moore, A.W.. Infinity, 2011, doi:10.4324/9780415249126-N075-2. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/infinity/v-2.
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