Friday 3 December 2021

The Threefold Root of the How-Question About Mathematical Knowledge

Platonism is the default, almost obviously correct view about mathematical objects. One of the major things that puts pressure on Platonism is the question 'How do we know about mathematical objects, then?'. What gives this question its power? I think three things conspire and that the third might be under-appreciated:

1. Real justificatory demands internal to mathematical discourse. For particular mathematical claims, there are very real 'How do we know?' questions, and they have substantive mathematical answers. The impulse to ask the question then gets generalized to mathematical knowledge in general, except that then there's no substantial answer.

2. A feeling of impossibility engendered by a causal theory of knowledge. If you only think about certain kinds of knowledge, it can seem plausible that, in general, the way we get to know about things is via their causal impacts on us. This then makes mathematical knowledge seem impossible.

3. Our deeply-ingrained habit of giving reasons. The social impulse to justify one's claims to another is hacked by a monster: the philosophical question at the heart of the epistemology of mathematics.

If it were just 1 and 2 getting tangled up with each other, the how-question would not be so persistent. With existing philosophical understanding we'd be able to see our way past it. But 3 hasn't been excavated yet and that keeps the whole thing going.