Infinity

The idea of a quantity or process with no final limit, debated by philosophers (Zeno's paradoxes, Aristotle's distinction between a merely potential infinity and a completed, actual one) for two millennia before mathematics gave it rigorous, working content. The now-familiar lemniscate-shaped symbol, a sideways figure eight, is a separate matter from the concept itself: it was introduced by the English mathematician John Wallis in 1655 as a notation and carries no special mathematical meaning of its own beyond standing in for the idea, whichever way the idea is made precise (a limit growing without bound, in ordinary calculus, or a genuine completed infinite size, in Georg Cantor's set theory). Cantor's work from 1874 onward was the decisive mathematical achievement: he showed that actual, completed infinities exist as legitimate mathematical objects and that they come in different sizes, a result that met fierce resistance from some contemporaries precisely because it treated Aristotle's merely potential infinity as something considerably more real.