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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That is what a perfect proof feels like.
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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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