6

In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-themselves, saying that, for example, geometry was the spatial lens by which we viewed phenomena whilst number arithmetic was the temporal lens.

If so, if one were to classify this position into a contemporary position on the ontology of mathematics, would it be fair to say that he was an intuitionist (i.e. he believed mathematical objects to be human constructs)? Or was he still a platonic realist in this regard?

9

You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle differences.

First of all, it would be anachronistic to call Kant an intuitionist, since Brouwer didn't develop intuitionist mathematics until well after Kant's death. As a result, intuitionism as we know it was not an available position in Kant's time. So one would have to make the case that the anachronistic label was really warranted. This by itself, of course, does not mean that it would be inappropriate to retroactively sort Kant into the intuitionist paradigm.

And there is some initial reason for finding this categorization attractive. Brouwer and Kant certainly agree on one really fundamental issue: namely, that mathematical truths are grounded in the nature of the mind. Certainly Brouwer and other intuitionists drew inspiration from Kant, e.g., from Kant's notion of intuition and the role that time plays in structuring intuition. In particular, Brouwer agrees with Kant that arithmetic truths are synthetic a priori. Brouwer departs strongly from Kant, though, with respect to the view of space. Brouwer denies that geometry is synthetic a priori, therefore denying the possibility of an a priori intuition of space.

Furthermore, it is not so clear whether Kant was a constructivist about mathematics. By contrast, constructivism is at the heart of intuitionism (i.e., the position that mathematical existence can only be proven by constructing the alleged mathematical object, and not simply showing that its existence is consistent). Intuitionism is a species of constructivism, so that if Kant was not a constructivist, then you could not call him an intuitionist in any sense. It's true that Kant uses the word "construction" and claims that mathematics is in the business of constructing its concepts. But this is a potentially very misleading connection. Kant's definition of construction is:

to construct a concept means to exhibit a priori the intuition corresponding to it (A713/B741).

It's not at all clear that this is the notion of construction that intuitionists work with. For example, in constructing proofs the intuitionist will frequently construct an object by demonstrating an algorithm that is sufficient to construct it (this, in fact, is part of the appeal of intuitoinist logic for mathematicians interested in computability). In turn, the logic underlying intuitionism was certainly not Kant's logic; it requires a revisionary definitions of terms such as function and real number that would have certainly been alien to Kant. Let's put it this way: if Kant found himself revived from the dead and walked into a classroom where intuitionist mathematics was being taught, there would be much that he would not recognize.

Would our revived, 21st-century Kant adopt intuitionism now that he finds it an available option? This is an intriguing question. What you'd have to do is to understand some of the subtle differences (just a couple of which I've highlighted above) and determine which of these doctrines are essential to Kant's philosophy of mathematics and which he might give up while retaining the integrity of his kernel of truth. It's also worth pointing out that several other mathematical philosophies take inspiration from the spirit, if not the letter, of Kant's own philosophy of mathematics (e.g., structuralism). This is important to note because it brings out the sense in which intuitionism certainly has no unique claim to the legacy of Kantian philosophy of mathematics, which points to the sense in which intuitionism is just one option among many that one could find implicit in the spirit of Kant's writings.

Finally, though it is of course entirely possible to disconnect intuitionist mathematics from the idiosyncracies of its founder (Brouwer), it's also important to note that Brouwer thought of mathematics in a very different way than Kant. Brouwer held a mystical view of the mind's construction of mathematical objects that would be, to say the least, in violation of Kant's transcendental idealism.

In short: No, Kant was not an intuitionist in any straightforward sense. However, the central resonances of Kant's view with certain general, fundamental intuitonist ideas would make an investigations of the details of the points of departure well worth studying.

  • Your answer is a godsend! Thank you so much. – J.M.W Turner Apr 22 at 8:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.