Mathematics for Humans: Kant's Philosophy of Arithmetic Revisited

According to Kant,3 mathematics is the pure formal science of quantity or magnitude. In turn, quantities or magnitudes are of two fundamentally different kinds: numerical and spatial. Arithmetic is the pure science of numbers, and geometry is the pure science of space. Whether arithmetic or geometry, however, mathematics for Kant is synthetic a priori, not analytic a priori – which is to say that it is a substantive or world-dependent science, not a purely logical science. But how can mathematics be at once a priori (i.e., experience-independent and necessary) and also substantive or world-dependent? As Brouwer correctly observes, for Kant mathematics is possible because it presupposes the innate human cognitive capacity for pure temporal and spatial representation, the innate human cognitive capacity for pure intuition (CPR A38–39/B55–56; P Ak. iv. 280–283). In turn, as the Transcendental Aesthetic shows, our pure intuitions of time and space are the non-empirical necessary subjective forms of inner and outer human sensibility. In this essay I revisit Kant’s much-criticized views on arithmetic. In so doing I make a case for the claim that his theory of arithmetic is not in fact subject to the most familiar and forceful objection against it, namely that his doctrine of the dependence of arithmetic on time is plainly false, or even worse, simply unintelligible; on the contrary, Kant’s doctrine about time and arithmetic is highly original, fully intelligible, and with qualifications due to the inherent limitations of his conceptions of arithmetic and logic, defensible to an important extent.