As far as I understand the notion of knowledge in Kantian philosophy, we cannot speak of knowing something unless there is a relation between its concept and some object of intuition in experience.

My question is, if my understanding is correct, and since, for example, we do not encounter any number or a perfect square in intuition as we do encounter an apple or a stone, how can we speak of mathematical or geometrical knowledge? I know that, for Kant, although the content of geometry and mathematics cannot be found in empirical experience, since they are universal and apodeictic, it still relates to experience in some sense. Does that mean that we do not need to have a corresponding object in intuition in order to have knowledge? Or should numbers and geometrical objects be taken as mathematical objects, in some Platonic sense, in Kantian terms?


We can speak of "mathematical or geometrical knowledge" because most of our knowledge is not about objects but about abstractions (classes of objects): I have the knowledge that to the object "my left hand" belongs a set of other objects that we call "finger". We map that to the mathematical concept of "set", we are able to define the cardinality of sets (pure mathematics) and map that back to counting our fingers. But we can also think, gain knowledge and communicate about even more anstract concepts like the set of subsets of a set of cardinality 5.

So: yes we have knowledge about abstract mathematical concepts and we use some of these to describe e.g. physical or sociological concepts.

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