Predicativism as a Philosophical Position1

Predicativity requirements of explicit presentability of objects and predicatively acceptable proof are distinguished from predicativist theses of a philosophical character. Familiar among these are expressions of skepticism about the objectivity of full power sets of infinite sets. Articulation of strong, limitative theses, however, turns out to be problematic: impredicative commitments creep into the very formulations, e.g. that “predicative definability'' marks a limit of “intelligibility''. A thought experiment is proposed to undermine the predicativist idea that arbitrary parts of an infinite whole of atoms are “mindor language-dependent''. On the other hand, weaker claims, e.g., that predicative mathematics is “more secure'' than impredicative, are nearly platitudinous. The interesting philosophical force of predicativism seems to be negative, in its challenge to indispensability arguments, à la Gödel-Friedman, for the transfinite in pure mathematics, and à la Quine-Putnam, for abstract mathematics in the sciences. Evidence is mounting in favor of Gödel-Friedman, e.g. impredicativity of free-variable formulations of theorems such as Kruskal's and Graph Minor, and more far-reaching, recent work in Boolean Relation Theory. This may lead to a realization of Gödel's idea of justifying strong axioms of infinity through their unifying, explanatory role, in analogy with theoretical physics.