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The whole approach of Grothendieck (the “rising sea” analogy) is to conquer deep theorems by first conjuring obscure (but simple) objects and prove simple prope
by joppy 5y ago
The whole approach of Grothendieck (the “rising sea” analogy) is to conquer deep theorems by first conjuring obscure (but simple) objects and prove simple properties of them, again and again, until finally the deep theorem is a result of many simple ones. In this project (and Grothendieck’s program) the deep result is the Weil conjectures.
- techdragon 5y agoI once tried to learn some math from first principles in order to solve a symbolic dice problem. I had a question deleted from mathoverflow (probably because I did a lousy job wording the open ended question of what the heck the mathematical terminology even was for the problem I was solving) but it lead down the rabbit hole of abstract algebra and into universal algebra. I had discovered a simple dice problem where the only mathematical representation was a a “commutative magma” and I found reading the surrounding material fascinating. The ideas behind universal algebra, the notion boiling mathematics down to a set of arbitrary objects and abstract and arbitrary operation(s) and classification and rules build up based on what you define/allow the rules to be. It’s surprisingly simple at a conceptual level but I rapidly stopped researching as I found the entire field seems to assume a phd level of math knowledge and terminology. For what I probably could have grasped in high school. Math needs more people to try and build a chain of understanding “up from the ground level” instead of arbitrary starting points based on assumptions regarding prior learning and educational pipelines/universities.