Maths and Logic workshop: Emily Carson

Emily Carson gave a paper entitled ‘Arithmetic and the Possibility of Experience’. In it she sought to work out Kant’s position on how arithmetic is grounded in the transcendental structures of experience. This questions asks, very roughly, how we can explain that every ‘discursive’ thinker - as Kant puts it - possesses some arithmetical knowledge, and how arithmetic applies absolutely generally to every empirical object (every such object is quantifiable). In the case of geometry Kant’s position is fairly clear: geometry is the pure science of space and our pure intuition of space is the source of our a priori geometrical knowledge. But there is no comparable science of number for Kant, the source of which would be our pure intuition of time.

Emily argued that the source of arithmetical knowledge are the categories of quantity, categories that are involved in every synthesis of any manifold of intuition. She explained this by arguing that the threefold synthesis cited in the A-edition Deduction, a synthesis apparently necessary for any determinate thought content whatsoever, is a synthesis that minimally involves the categories of quantity. Thus the formal element of possible experience that arithmetic is grounded in is this synthesis, and this explains Kant’s cryptic statement that the formal element in experience that grounds arithmetical knowledge is‘the universal in the synthesis of one and the same thing in time and space, and the magnitude of an intuition in general (number) that arises from that.’

I for one found this a very convincing account of Kant’s position. But one question that arose was that if all that grounds arithmetic is the categorical synthesis of quantity, doesn’t that give us an unKantian answer to how arithmetic and the possibility of experience relate? That is, arithmetical capacities now start to look wholly conceptual. It came up in discussion that it seems necessary for there to be some intuitive structure to index the categories to, so that arithmetical abilities didn’t depend on contingencies like the amount of objects in the world (if there were only four objects, the sentence 7+5+12 couldn’t be formulated unless there were an infinite iterative structure like time that has enough places to index the constituent numbers of the sentence to). What was then asked was, why should this indexing structure be specifically time or space? To my mind, against Kant, this question is wrongheaded: he is not trying to show how any discursive thinker would be capable of arithmetic, but how we are.