Mathematics Is the Subfield of Philosophy That Humans Are Good At

By philosophy I understand the discipline of discovering truths about reality by means of thinking very carefully. Contrast to science, where we try to come up with theories that predict our observations. Philosophers of number have observed that the first ten trillion nontrivial zeros of the Riemann zeta function are on the critical line, but people don't speak of the Riemann hypothesis as being almost certainly true, not necessarily because they anticipate a counterexample lurking somewhere above ½ + 1026i (although "large" counterexamples are not unheard-of in the philosophy of numbers), but rather because while empirical examination is certainly helpful, it's not really what we do. Mere empiricism is usually sufficient for knowing (with high probability) what is true, but as philosophers, we want to explain why, and moreover, why it could not have been otherwise.

When we try this on topics like numbers or shapes, it works really, really well: our philosophers quickly reach ironclad consensuses about matters far removed from human intuition. When we try it on topics like justice or existence ... it doesn't work so well. I think it's sad.