No one shall be able to drive us from the paradise that Cantor has created for us. (David Hilbert)
I would say, ‘I wouldn’t dream of trying to drive anyone from this paradise. ’ I would do something quite different: I would try to show you that it is not a paradise—so that you’ll leave of your own accord. I would say, ‘You’re welcome to this; just look about you. ’…
(For if one person can see it as a paradise…, why should not another see it as a joke?) (Ludwig Wittgenstein)
At one level, nobody could fail to be impressed by Cantor’s work. It showed mathematical craftsmanship of the very highest calibre. But there was great room for debate about its significance. The infinite seemed at last to have been subjected to precise mathematical scrutiny. But perhaps Cantor had been indulging in technical flights of fancy. Perhaps, as I suggested in the last chapter, he had actually confirmed some of the most deeply entrenched prejudices about the infinite with his talk of inconsistent totalities.
Certainly it was too much to expect that the infinite would now uncritically be accepted as an object of mathematical enquiry in just the way that Cantor had represented it. There was some immediate opposition and hostility to his work, as we saw in the last chapter. Later thinkers reacted against it at a somewhat deeper level, recognizing the importance and merit of what they were reacting against but using this to think through some of the most fundamental questions about the nature of the infinite (and of mathematics). It is to their work, and related work, that we turn in this chapter.
The Dutch mathematician L. E. J. Brouwer (1881-1966) founded what was to become one of the most influential schools in the philosophy of mathematics, intuitionism. One of his main contentions was that mathematical