Connections between Abstract Algebra and High School Algebra (2015)

Re:Connections between Abstract Algebra and High School Alge... by qubex on Thursday December 07, 2017 @02:52AM (#15868437)

Yeah, I’m aware of the numerics... but I’m also aware that if you do the same process backwards (naive numerical integration), for example for simulating planetary motions, you get into ridiculous situations where the collective momentum of a closed system rises exponentially after a ”close flyby”. This is precisely the kind of situation where ”false intuition” created by these shallow teachings cause the most harm.

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Re:Connections between Abstract Algebra and High School Alge... by romwell on Thursday December 07, 2017 @05:46PM (#15875105)

I don't think it's harmful - without stumbling onto an example like that, it's hard to justify why we need solid foundations and exact solutions.

On the other hand, very brute numerics work for an awful lot of scenarios - that's why epsilon-delta came centuries after Calculus was invented.

For instance, "first-order optics" and "third-order optics" rise from chopping off the Taylor series after the 1st and 3rd term, resp. And it works! In many places, 1st order approximations are just good enough. A lot of scenarios are inherently stable.

So I don't think the intuition you build up is wrong - it just has a scope. There's nearly always a place for counter-examples where "things work the way you think they should" wouldn't apply, however you think about things :)

On a philosophical note, the continuity is a human construct - down there, things seem to be discrete, just with a very small step size. Continuity models these pretty well, until it doesn't - but that doesn't mean the intuition you build up is wrong. Just limited in scope.

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Re:Connections between Abstract Algebra and High School Alge... by qubex on Friday December 08, 2017 @12:40AM (#15876911)

I see what you are saying, but to each his own: I’m one of those kids (there's one in every mechanics class) that grow irate at the sin(ϴ︎)≈︎ϴ︎ approximation in the pendulum solution and then waste weeks using the full series expansion, only to find out the results differ only in the third or fourth decimal place. The thing is, I emerged from the experience thinking to myself “that was a useful learning moment”.

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