Content-Type: text/shitpost

There are a number of anecdotes in circulation among mathematicians about how dumb philosophers are about mathematics. Not all of these are deserved.

The story that Hegel claimed (in 1801) that it was a priori impossible that there be more than seven planets is one of these. The reality is definitely rather more interesting, and I think that to get the full picture I would have to read and understand Hegel's argument, which I have not yet done. I have not yet found the full text, and the original was in Latin. (“Quadratum est lex naturae,” says Hegel, “triangulum, mentis.” That's “the square is the law of nature; the triangle, of the mind.” Ummmm, okay.)

Another story tells that Kant claimed (1781) that Euclidean geometry was a priori correct and that a non-Euclidean geometry was not merely impossible but inconceivable. The claim of course is false, but I wonder how much of Kant's claim is really recognizable in there, and also how many mathematicians might have made similar claims in 1781.

Philosophy can be really tricky. Philosopher X seems to be saying Y, but then it turns out that you have misunderstood Y and it really means Z. Here's what the Stanford Encyclopedia of Philosophy says:

Kant introduced the notion of a priori knowledge in contrast to a posteriori, and synthetic knowledge in contrast to analytical knowledge to allow for the existence of knowledge that did not rely on experience (and was thus a priori) but was not tautological in character (and therefore synthetic and not analytic). Analytic statements are a priori, the contentious class of a priori non-analytic statements contains those that could not be otherwise and so provide certain knowledge. Among them are the statements of Euclidean geometry; Kant ascribed synthetic a priori status to the knowledge of space. He also ascribed certainty to Euclidean geometry.

(Gray, Jeremy, "Epistemology of Geometry", The Stanford Encyclopedia of Philosophy (Fall 2017 Edition), Edward N. Zalta (ed.))

Okay, maybe? Gray continues:

But, wrote Kant, it is not the philosopher who knows that the angle sum of a triangle is two right angles, it is the mathematician, because the mathematician makes a particular construction that makes the truth of the claim demonstrable.

Now I wonder if, where the whole situation explained to him, Kant would want to classify this theorem not as synthetic and a priori but as tautological, since the mathematician can only prove it by assuming that the triangle is in a Euclidean space.

I am willing to give Kant and Hegel the benefit of the doubt here. But if anyone is looking for a really good example of an indefensible mathematical fuckup by a first-class philosoper, check out this article I wrote about Thomas Hobbes a while back.