INTRODUCTION

Providing truthmakers for the truths of mathematics and set theory (mathematics for short) is an important task for metaphysicians. The demand I place on such truthmakers is that they be compatible with naturalism, defined as the hypothesis that the world of space-time is all that there is. I very much doubt that naturalism is a necessary truth – after all, is not the existence of space-time a contingent matter? But I do think it is a true proposition. The truthmaker for this truth <space-time is all that there is> will be the space-time system itself together with the ‘closure’ state of affa-irs that this system is all there is. The deep structure, and the extent, of space-time are of course to be sought by science. The ‘manifest image’ of space-time is extremely unlikely to be the last scientific word. But I do have a metaphysical hypothesis of a very abstract sort: that the ultimate nature of space-time is a structure of states of affairs (argued for in my 1997).

If the truths of mathematics do not require truthmakers then my task will be eased. I have, however, committed myself to Truthmaker Maximalism: every truth has a truthmaker. So I must find truthmakers for the truths of mathematics in the space-time system. One thought has special influence here. We do not know whether space-time is finite or infinite. It may involve ‘structures all the way down’ or there may be atoms.