The Psychologistic Foundations of Hume's Critique of Mathematical Philosophy

Hume Studies Volume XXII, Number 1, April 1996, pp. 123-167 The Psychologistic Foundations of Hume's Critique of Mathematical Philosophy WAYNE WAXMAN Nearly every philosopher has encountered positions or arguments that seem fatally flawed yet been at a loss to diagnose the precise causes of the debility. There are cases, too, where a broad consensus exists that something is defective but, even after centuries of trying, no agreement as to why. So, when an author arrives claiming to establish a definitive diagnosis, as James Franklin does in "Achievements and Fallacies in Hume's Account of Infinite Divisibility,"1 one can only respond with hope and anticipation. Alas, once Franklin's case is subjected to scrutiny, most readers will, I believe, conclude that their hopes were misplaced. Nevertheless, its examination offers an excellent opportunity to improve our understanding of why Hume's reasons for denying infinite divisibility seem so uncharacteristically weak, while the source of the apparent infirmity remains so frustratingly elusive. In this paper, I shall advance a new diagnosis, locating the trouble not in Hume's argumentation but in the way readers tend to approach it. The error consists in treating the demonstrations of T I ii as self-contained exercises in philosophy of mathematics: so long as we remain wedded to the assumption that items established elsewhere in the Treatise may be ignored unless, and only insofar as, Hume expressly invoked them, his critique of mathematical philosophy cannot but seem shoddy and vainglorious. These demonstrations,2 together with the analysis of space and time of which they form a part, can be sustained only if buttressed by psychologistic supports drawn from T I i, most particularly the subjective, Wayne Waxman is at the Freie Universität, Berlin, Germany. 124 Wayne Waxman imagination-dependent character of relations and the reduction of universals to habitudes of comparison with an eye to resemblances. Once these and related tenets ("the elements of this philosophy," T 13)3 are factored in, and stand in readiness to ward off any challenge premised on the usual piecemeal approach, Hume's critique of mathematical philosophy gains immeasurably in force and coherence. Moreover, as their direct implication and application, an attack upon this critique necessarily strikes to the very heart of Hume's psychologism. Thus, if my diagnosis is correct, the proponent of mathematical philosophy can triumph over Hume only by redirecting his fire at the foundations of that psychologism. Unfortunately, it is impossible to undertake a systematic, detailed analysis of T I ii within the confines of a single paper. Instead, I propose to use Franklin's critique of Hume on indivisibles as a case study of what happens when the theory of space and time propounded in T I ii is detached from its foundations in T I i. What follows, then, is by design something of a hybrid, mixing elements of both critical discussion and philosophical interpretation. I. Franklin bases his belief that a final reckoning with the question of indivisibility is possible on advances in mathematics and set theory which establish once and for all the possibility that space and time are infinitely divisible. Since there are also consistent models of space and time as discrete or atomic (in one dimension the integers, and in higher dimensions, the lattice of points with integer coordinates), he concludes that it is now a merely empirical question, not to be settled either way by a priori reasoning, which model is correct (Franklin, 87-88). From this, Franklin draws two conclusions. First, Hume was warranted in rejecting the arguments of seventeenth and eighteenth century mathematicians who claimed to demonstrate infinite divisibility a priori, employing only the techniques then available (Franklin, 88-89); however, he was himself equally mistaken in believing that he was in possession of a priori demonstrative proofs for the indivisibility thesis. Second, and more controversially, Franklin holds that contemporary psychology and cognitive science lend support to Hume's imagistic approach to philosophical problems, including the philosophy of mathematics: Although mathematicians can claim that it is theoretically possible to do geometry purely by manipulating symbol strings ("Tis usual with mathematicians, to pretend, that those ideas, which are their objects, are of so refin'd...