Abstract

This book is about universals. "Mathematics," we learn, "is the theory of universals". Natural numbers turn out to be universals, as do real numbers, complex numbers, and sets. It is natural, therefore, that the reader demand some evidence that universals abound sufficiently to supply models of canonical mathematical theories. The author devotes nearly a third of his book to one potential source of such evidence: the Truthmaker axiom. In the case of some propositions P, Truthmaker entails that if P is true, then there are things whose joint existence strictly implies P's truth. These things are truthmakers for P. Might Truthmaker, then, transform evidence for the truth of propositions into evidence for the existence of truthmaking universals? And might we thus be guaranteed a rich supply of numbers and sets? The author thinks not: "universals must not be construed as truthmakers". Something is a universal only if it can be instantiated by individuals x and y such that x's being an instance does not strictly imply y's being an instance. Since this characteristic of universals is incompatible with natural assumptions about truthmakers, the author's disheartening conclusion follows: "universals should not be expected to play any distinctive role in truthmaking". So Truthmaker is not a wellspring of universals. Why should we, then, believe that universals abound? One must understand, first, that the author embraces David Armstrong's a posteriori realism: "Everything there is is physical.... Hence universals, too, are physical. That is to say, the universals which exist are all real physical properties and relations among physical things". Universals are reasonably posited, then, only if their existence is entailed by our best accounts of the physical universe: "In cutting universals away from truthmakers, I am making their existence a matter, not for a priori proof from logic alone, but for total science". Sentences not interpretable in optimal physical theory will, on this view, not be modeled by universals and, hence, will be either bad mathematics or no part of mathematics at all.