Aristotle on Mathematical Existence

Do mathematical objects exist in some realm inaccessible to our senses? It may be tempting to deny this. For one thing, how we could come to know mathematical truths, if such knowledge must arise from causal interaction with non-empirical objects? However, denying that mathematical objects exist altogether has unsettling consequences. If you deny the existence of mathematical objects, then you must reject all claims that commit you to such objects, which means rejecting much of mathematics as it is standardly understood. For, as David Papineau (1990) vividly puts it, it is doublethink to deny that mathematical objects exist but to continue to believe, for example, that there are two prime numbers between ten and fifteen. Two current responses to this problem are literalism and fictionalism. Both literalists and fictionalists deny the existence of a world of mathematical objects distinct from the empirical world. But they differ markedly in this denial. Literalists argue that mathematical objects simply exist in the empirical world; on this account, mathematical assertions assert true beliefs about perceivable objects. Fictionalists, on the other hand, hold that, strictly speaking, mathematical objects do not exist at all, and so exist in neither the empirical world nor in some realm distinct from the empirical world. They argue that mathematical objects are not actual objects but rather harmless fictions; on this account, mathematical assertions do not assert true beliefs about the world but merely fictional attitudes. Although these two positions are apparently quite opposed to one another, they nonetheless have been both ascribed to Aristotle. Indeed, as I’ll argue, Aristotle’s philosophy of mathematics exhibits some of the features characteristic of literalism and some of the features characteristic of fictionalism. However, Aristotle’s position also exhibits features interestingly different from both literalism and fictionalism. The paper comes in three parts. In the first part, I’ll quickly survey the variety of descriptions which Aristotle uses to characterize the relation between mathematical objects and the perceivable world. This will help to explain how apparently opposed positions have been ascribed to Aristotle. In the second part, I’ll discuss literalism in contemporary philosophy of mathematics, the ascription of literalism to Aristotle and the points of agreement and disagreement between Aristotle and literalists. In the third and final part of the paper, I’ll discuss fictionalism in contemporary philosophy of mathematics, the ascription of fictionalism to Aristotle and the points of agreement and disagreement between Aristotle and fictionalists