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:
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