>>14121819> Any book from The Republic mogs a century's worth of modern thinking on the relevant subject.Indeed, but no one has expanded on the ideas contained in Book VII since, particularly those incredibly rich ideas regarding geometry. From Plato to the Grundlagenkrise no one to my knowledge explored the foundations explicitly, not even to reiterate the old ideas, and even then in the XIX no one did it from a platonic perspective, instead attempting to find a different base. Even Bernays in his "Sur le platonisme dans les mathématiques", which is the most important text on the foundations that dares name the man, doesn't take the time to explain Plato's position. The path to the answer was staring them right in the face and they chose the perpendicular route. Perhaps they didn't have the tools to go deeper --- plus none of them were really philosophers or looking to approach the foundations of mathematics from a philosophical position --- but the point stands: The Republic hits the spot, modern philosophy of mathematics doesn't.
> Is there's a list of crucial subjects which will cover my bases or is every path in math its own little rabbithole?The latter, although paths sometimes cross.
Start with the master --- Euclid. Reading any of the books in the Elements will teach you more about proofs than anything else ever could. If you're interested in mathematics it'll also put you in the right headspace: learning to "get" the objects is the real goal here and the high art of writing proofs is closely related to an intimate understanding of the properties/characteristics of said objects. After that, go crazy. Ultimately the only thing that'll keep you from any area of mathematics is familiarity with the language. Introductory books will help.
"Finite-Dimensional Vector Spaces" by P. Halmos is a great, great, great beginner-friendlyish book which has taught me a lot on linear algebra and how to write good demonstrations.
