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It's a straw man to argue that math has brought new philosophical challenges, no one has posited against that. I've only stated that math is readily understood
by ethn 6y ago
It's a straw man to argue that math has brought new philosophical challenges, no one has posited against that. I've only stated that math is readily understood by transcendental idealism, disproving the false dilemma presented by the author of the article.
A more mature comment would be to point out an instance where there is some philosophical challenge to transcendental idealism that came from 'new mathematics'. Though I find the possibility of this to be impossible. I would expect at least the German-Idealist Godel to write about it.
If you're talking about Gauss's non-Euclidean geometry, that's completely compatible with transcendental idealism, in fact Gauss took his non-euclidean geometry from the second chapter of The Critique of Pure Reason. He read Kant's passage on the triangle and the line in respect to the sum of angles obsessively.
- btilly 6y agoI'm talking the collapse of infinitesmals as a basis for Calculus, the development of a more rigorous foundation, the rise of axiomatization, debates around the validity of pure existence proofs, the failure of naive set theory, and so on. On pure existence proofs, mathematicians have generally agreed that something can be proven to exist by proof by contradiction on an infinite set. The result is that we say that it exists even though there is no way to find it, and no way to verify that you have found it if you are presented with it. Nothing in Kant's philosophy had anything useful to say about the debate about whether to accept this kind of proof.
- ethn 6y agoThanks for your reply. The first part, although mostly true (axiomatization isn't new and Kant was well aware of Euclid), is wholly irrelevant so I'll comment on the second part. None of this contradicts the synthetic apriori of mathematics, in fact Kant's critique strictly tells you that analysis IS possible and guarantees its legitimacy. I would look to Wittgenstein and Godel for that expounded application of the transcendental idealism.