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?

  • Kant's subjective essential points to an intuitive mathematical function which is a part of a priori knowledge. Let's just say it straight out, math is entirely human originated it is an instinctive reaction to the recognition of self as 'one', a numerical origin. Math can have no life outside of a human origin. There is simply no way to explain its origin other than in human intuition. See my 'To Discern Divinity; The story of Gift Star, p. 39. charlessaunders5@academia.edu. This 'story' depicts a mythical 'person', who first imagined and then conceived agricultural spatial relationships. CMS – Charles M Saunders Dec 9 '19 at 12:29
  • @CharlesMSaunders - I share your view. I'm interested but I had a look on academia-edu but could not see the essay you mention here. – PeterJ Dec 9 '19 at 13:28
  • @Peter J- Thanks Peter, the piece is in a book. It is titled ' To Discern Divinity- A Discussion and Interpolation of Spinoza's Ethics Part 1- Concerning God' , the piece is on page fifty something and mentions 'Gift Star' which is the name I gave to the protagonist. Cheers! – Charles M Saunders Dec 10 '19 at 13:33
  • @CharlesMSaunders - Thanks. Got it. – PeterJ Dec 11 '19 at 13:02

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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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.<

This is WRONG : knowledge requires the union of concept and intuition, but not necessarily "in experience". Kant denies that all intuition is empirical.

According to Kant, experience involves sensibility ( intuition) , but it is not the case that everything that involves sensibility is empirical.

In sensibility ( intuition) have to be distinguished 2 aspects :

  • matter, given by actual sensations

  • and form ( or structure) which is given a priori.

Kant claims that space and time are a priori forms of intuition ( hence, sensible, but not empirical).

Being given that the human mind is able to construct a priori mathematical concepts in the pure form of intuition ( in space for geometrical objects, in time for arithmetical objects) , this construction yields universal and necessary results; and this is what explains how mathematics is possible a priori.

Reference: Prolegomena to any future metaphysics ( for a first discovery of Kant); also : Critics of Pure reason.

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