The Infinite

[If] a man had a positive idea of infinite…, he could add two infinites together: nay, make one infinite infinitely bigger than another, absurdities too gross to be confuted. (John Locke)

For over two thousand years the human intellect was baffled by the problem [of infinity]….

A long line of philosophers, from Zeno to M. Bergson, have based much of their metaphysics upon the supposed impossibility of infinite collections…. The definitive solution of the difficulties is due…to Georg Cantor. (Bertrand Russell)

Apart from an anti-Aristotelian backlash among the medievals, spear-headed by Gregory of Rimini and partly followed through by the rationalists (see above, Chapters 3 and 5), the time up until the early-mid nineteenth century saw nothing but hostility towards the actual mathematical infinite. Some of the hostility was towards the mathematical infinite per se. We saw something of this in Hegel. But some of it was emphatically not that. It was hostility specifically to the actual mathematical infinite. Here, of course, the key figure was Aristotle, for whom the infinite was certainly to be understood in mathematical terms; what had to be resisted was the idea that it could be given ‘all at once’. For over two thousand years this was the prevailing view.

As this view developed and achieved the status almost of orthodoxy, the notion of being given ‘all at once’ came to be understood in increasingly metaphorical terms. Often it just meant being a legitimate object of mathematical study in its own right. And, from that point of view, mathematics itself bore witness to the prevailing hostility. For the infinite never really was regarded as a legitimate object of mathematical study in its own right. True, it continually impinged on mathematical consciousness. But this was only because mathematicians, in their study of finite objects such as natural numbers and lines, would constantly step back and

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