Dissertation

http://academicworks.cuny.edu/gc_etds/703/

Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are grasped by intuition, or by conceptual analysis – being deducible from general principles and notions that are already understood. My answer to this question is both. In practice, the structural aspect of numbers is grasped by intuition. However, conceptual analysis reveals that intuition is not necessary, since the finite cardinal numbers are deducible from principles and notions that apply to every area of inquiry. This proposal is offered against the background of a rejection of the influential view that we grasp numbers, intuitively, via the innate sense of quantity known in the psychological literature as our “number sense.” Rather, I claim, we can grasp numbers, intuitively, due to our ability to visualize a geometric pattern that reveals the structural features of a progression. Further, I argue that while this ability to visualize a pattern is sufficient to grasp the aforementioned structural features, it does not completely reveal what numbers are. For this, conceptual analysis is required, which reveals the cardinal aspect of numbers – they are properties of sets, and, as such, are deducible from general principles and notions. I argue for this view by showing the extent to which it avoids the problems that plague other proposals in the literature, including the problems raised by Kripke against Frege. Special attention is paid to Kripke’s own proposal that numbers are finite sequences of objects that are ordered by length and then lexicographically, like the decimal numerals that stand for them, and to the relation between this proposal and his doctrine that there are contingent truths known a priori. I respond to Kripke that a good notation should reveal structural features of its subject matter visually, by helping one to see or visualize these features, and that decimal notation meets this requirement. Further, I argue that since there is a trade-off between the need to reveal the structural features of a subject matter tout court, and the need to reveal these features visually, to a creature with our limited parsing abilities, there are grounds for insisting that decimal notation has extra structure that is not shared by numbers.