I'm waiting on some models to run, so here's another math thread about what it means when mathematicians prove a theorem and what an elegant proof looks like.
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Language in math is very precise. Definitions mean things. Definitions are everything! By asserting that V is a vector space, I know that V satisfies a specific set of properties. By asserting that X is a subspace of V, I know that X has properties with respect to V.
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When we look to prove something, we're not just juggling equations and throwing arcane symbols at the wall. We're looking to show that the consequences of certain properties and relationships imply some other truth necessarily.
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A theorem is essentially a consequence of various assertions and properties. A proof is the way to draw those connections in their logical order. Good proofs are all about organizing what you're given to get to where you want to be.
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You don't necessarily start from the top and go to the bottom. Good proofs will do some really clever things like set up a good one-two punch. The Galois proof I talked about earlier is like this.
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Galois proof is done in two major steps: first, we establish that a polynomial is solvable by radicals if and only if its splitting field has a solvable Galois group. This is not easy by any means.
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"If and only if" is a bidirectional implication. We have to prove both directions: 1. solvable by radicals implies the splitting field has a solvable Galois group; 2. the splitting field having a solvable Galois group implies solvability by radicals.
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But once we do that we set up the coup de grace: now we only need to compute the Galois group for a quintic polynomial (not so hard) and then show it is not solvable. This is probably not easy to follow if you don't know these terms. That's ok.
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What's elegant about this is it avoids a tremendous amount of tedium by doing a lot of overhead work first. We establish that the problem we care about looks like this other thing. Once we do that, the rest falls like dominoes.
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A good proof is like a good chef cooking a complex meal. It's chaos and complexity and drama! And then, all of a sudden, everything comes together at the end. It can be hard to understand what they're doing at first... but them BAM.
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Have you ever seen that clip of Susan Boyle at Britain's Got Talent? It's a trainwreck at first. She's dowdy, and old. She's out of her element. "I Dreamed a Dream?". Simon's making fun of her. She is mumbling her words... And then she sings.https://www.youtube.com/watch?v=RxPZh4AnWyk …
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Becoming good at math means being able to identify the setup work that needs to take place. This is not a special skill. Much like any professional, if you become good at something you know how to get started, whether it's starting a software project or a new patient record or
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But great mathematicians can see multiple steps into the future in this process, to intuit connections that are not obvious. Throughout my math undergrad, I viewed math as a mostly mechanical, procedural process. Which made sense for my discipline (scientific computing).
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But as I shifted more into formal analysis I began to finally understand proof. It took a mental maturity well-beyond my undergrad years to get there. It really took me until my late-20s. And I'm still learning that process.
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When I took Real Analysis at RPI I finished with a C, partly because it was my last semester and once I got a passing grade I stopped going. When I re-took the class at UVa, I had much more maturity and a finer grasp of proof-based math and got an A+.
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End of conversation
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