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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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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Beautiful analogy. I cried the first time I heard this years ago, and the ten times I had to repeat it now.
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I’m no mathematician, but that was a great analogy.
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