The notion of knowledge for Kant and mathematical objects

Question

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?

Answer 1

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.

Answer 2

Math is, for Kant, synthetic, but apriori. So it is not clear and evident without external information, but it is not learned from external reality, either. (He addresses this as his first major topic in the "Prologomena to Any Future Metaphysics.")

That means it exists only in intuition, as that is the place where our nature and external reality meet. Only when some external reality is interpreted in intuition as having something to do with an overall pattern we are predisposed to seek out, do notions like the perfect square come to exist. The 'square' object is not a square, but neither is it irrelevant to our perception of it as a square. It is a form we shove reality into in the process of forming an intuition.

So geometry is built into people, and we impose it on our perceptions in order to make sense of them. Then we notice an underlying pattern among the perceptions so shaped. And we make that the subject of mathematics.

It is harder to see numbers that way. But Kantian-motivated mathematicians like Brouwer have interpreted the idea of numbers as a part of the intuition of time passing. As we count the three things in front of us, they are differentiated by the fact we focus on each of them at a different time in the counting process, we force them into separate events in memory. So the 'threeness' is not in the objects, it is in the counting process, which is built into our notion of time.

And time goes on within and around this succession of focusses. This is how we get the dual notion of real numbers flowing continuously and integers as fixed points. We feel time pass continuously, but we also clearly identify a specific succession of focal points as the events that happen to us. (Also, contrary to the observations of early anthropologists, everyone can count, even when their language does not have names for the numbers.)

(The labeling process that identifies these points in time is not as clearly connected, as we notice when we try to teach counting to children. Counting in any useful sense requires bringing together two separate kinds of intuition: succession and distinction (which are a priori) and a rule for naming (which is not).)

Answer 3

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.