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by xanderlewis 558 days ago
> the definitions are actually the exact truth (by definition! tautologically so)

It’s easy to forget that non-math people find this — the idea that the definition is its own ‘model’ rather than an approximation of something more ‘real’ — somewhat hard to stomach. Outside of pure mathematics the idea is that mathematics is a tool for (usually lossy) modelling of reality, not a collection of already perfectly well-motivated objects to be studied in their own right.
1 comments
gowld 558 days ago
More generally and symmetrically:

When you are studying science and technology, and the math theorem doesn't match experiment, the theory is probably wrong (or incomplete, missing factors), so you can discard it or try to improve it.

When you are studying math, and the intuition doesn't match the theorem, the intuition is probably wrong (or incomplete).

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kmill 558 days ago
Things get a bit messier once you're doing research mathematics — definitions don't just come from nothing, and a good definition is one that serves its theorems. Definitions can be "wrong" (they might be generalizable, they might have unexpected pathological examples, etc.), and it's the result of lots of hard work by lots of mathematicians throughout history that we have the definitions we enjoy the use of today.

But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.

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xanderlewis 558 days ago
Yes. It’s slightly disingenuous of me to suggest that any definition is as valid and ‘real’ as any other. Obviously, mathematicians care about some ideas more than others.
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kmill 558 days ago
Yeah, during department teas you can hear mutters of "interesting" as ideas are exchanged and evaluated.

But, in my last comment I was just trying to temper my previous comment's claim about how important definitions are. At some point you get so used to a definition that even if you don't know a particular formulation word for word, you could still write a textbook on the subject because you know how the theory is supposed to go.

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