We are floating in a medium of vast extent, always drifting uncertainly, blown to and fro; whenever we think we have a fixed point to which we can cling and make fast, it shifts and leaves us behind; if we follow it, it eludes our grasp, slips away, and flees eternally before us. Nothing stands still for us. This is our natural state and yet the state most contrary to our inclinations. We burn with desire to find a firm footing, an ultimate, lasting base on which to build a tower rising up to infinity, but our whole foundation cracks and the earth opens up into the depth of the abyss. (Blaise Pascal)
What I want to do in this chapter is to continue what I was doing in the last section of the last chapter, but on a larger scale: that is, to reassess the history in the light of the ideas that have begun to emerge and, at the same time, to develop the ideas.
I shall continue to assume that the truly mathematically infinite is something that resists mathematical scrutiny. It does not follow that, , and suchlike are really finite. This only follows given the further assumption that they are susceptible to mathematical scrutiny. As we saw in Part One, there has been a good deal of scepticism about whether they are. The mere consistency of set theory, for example, would not be enough to assuage this scepticism with respect to . There remains the question of what any set-theoretical symbolism has to do with the natural numbers. If the answer is, ‘Not enough to bring them under the control of the set theorist as a determinate, mathematically investigable totality, ’ then the natural numbers can continue to be regarded as providing a kind of paradigm of infinitude, in its truest sense—just as they have throughout the history of the topic. My assumption is not meant to prejudice any of these issues. It is just part of the model that I was beginning to develop in the last chapter.
One thing that the modern formalism does serve to emphasize is this: there is no single way of drawing together and subjugating everything in mathematical reality. For even if there is a determinate totality of natural