One point on our Schopenhauer episode that we didn't take much time to get into was his attitude towards geometric demonstration, which was of course the model for all philosophy for thinkers like Descartes. Here's a short selection from section 39 of the Fourfold Root, which illustrates his idea that our knowledge of geometry is founded on our intuition of space ("knowledge from the reason of being), not deduction ("knowledge of the reason of knowing"):
When once the reason of being is found, we base our conviction of the truth of the theorem upon that reason alone, and no
longer upon the reason of knowing given us by the demonstration. Let us, for instance, take the sixth proposition of the first Book of Euclid :
"If two angles of a triangle are equal, the sides also which subtend, or are opposite to, the equal angles shall be equal to one another." Which Euclid demonstrates as follows:
"Let a b c be a triangle having the angle a b c equal to the angle a c b, then the side a c must be equal to the side a b also. For, if side a b be not equal to side a c, one of them is greater than the other. Let a b be greater than a c; and from b a cut off b d equal to c a, and draw d c. Then, in the triangles d b c, a b c, because d b is equal to a c, and b c is common to both triangles, the two sides d b and b c are equal to the two sides a c, a b, each to each; and the angle d b c is equal to the angle a c b, therefore the base d c is equal to the base a b, and the triangle d b c is equal to the triangle a b c, the less triangle equal to the greater, which is absurd. Therefore a b is not unequal to a c, that is, a b is equal to a c."
Now, in this demonstration we have a reason of knowing for the truth of the proposition. But who bases his conviction of that geometrical truth upon this proof? Do we not rather base our conviction upon the reason of being, which we know intuitively, and according to which (by a necessity which admits of no further demonstration, but only of evidence through intuition) two lines drawn from both extreme ends of another line, and inclining equally towards each other, can only meet at a point which is equally distant from both extremities; since the two arising angles are properly but one, to which the oppositeness of position gives the appearance of being two; wherefore there is no reason why the lines should meet at any point nearer to the one end than to the other.
It is the knowledge of the reason of being which shows us the necessary consequence of the conditioned from its condition in this instance, the lateral equality from the angular equality that is, it shows their connection; whereas the reason of knowing only shows their coexistence. Nay, we might even maintain that the usual method of proving merely convinces us of their coexistence in the actual figure given us as an example, but by no means that they are always coexistent; for, as the necessary connection is not shown, the conviction we acquire of this truth rests simply upon induction, and is based upon the fact, that we find it is so in every figure we make. The reason of being is certainly not as evident in all cases as it is in simple theorems like this 6th one of Euclid; still I am persuaded that it might be brought to evidence in every theorem, however complicated, and that the proposition can always be reduced to some such simple intuition.
What do folks with a deeper background in geometry than mine think? Schopenhauer is not arguing here that Euclid is wrong, but merely that what he's doing is maybe more what modern mathematicians think he's doing, i.e. establishing an axiomatic system based on a minimum number of assumptions which are supposed to be basic truths of reason, rather than in any sense explaining the epistemology of geometric truths. Elsewhere, Schopenhauer argues that only intuition can tell left from right, i.e. can capture indexicals, and the point about Euclid above is supposed to be another aspect of the same phenomenon: basic truths of reason (what Schopenhauer calls "paralogisms of reason"), such as the principle of non-contradiction, don't refer to specific points of view, whereas spatial relations are dependent upon these and claims about them ultimately (Schopenhauer says) derive their validity from intuitions expressing a specific spatial point of view. There seems room to me to venture either further into the "perspectivist" camp and argue that even the paralogisms in a more abstract sense rely on a point of view (though off hand I can't think of a coherent argument for this) or to hunker down and come up with a way to translate indexical claims into perspective-neutral claims; Schopenhauer would think that both of these attempts ignore the fundamental difference between these two versions of the Principle of Sufficient Reason.