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Being an engineer by training, I never got exposed to much algebra in my courses (beyond the usual high school stuff in high school). In fact did not miss it much either. Tried to learn some algebraic geometry then... oh the horror. For whatever reason, my intuition is very geometric and analytic (in the calculus sense). Even things like counting and combinatorics, they feel weird, like dry flavorless pretzels made of dried husk. Combinatorics is good only when I can use Calculus. Calculus, oh that's different, it's rich savoury umami buttery briskets. Yum.

That's not the interesting part. The interesting part is that I thought everyone is the same, like me.

It was a big and surprising revelation that people love counting or algebra in just the same way I feel about geometry (not the finite kind) and feel awkward in the kind of mathematics that I like.

It's part of the reason I don't at all get the hate that school Calculus gets. It's so intuitive and beautifully geometric, what's not to like. .. that's usually my first reaction. Usually followed by disappointment and sadness -- oh no they are contemplating about throwing such a beautiful part away.

by macromagnon7 hours ago|

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School calculus is hated because it's typically taught with epsilon delta proofs which is a formalism that happened later in the history of calculus. It's not that intuitive for beginners, especially students who haven't learn any logic to grok existential/universal quantifiers. Historically, mathematics is usually developed by people with little care for complete rigor, then they erase their tracks to make it look pristine. It's no wonder students are like "who the hell came up with all this". Mathematics definitely has an education problem.

by jjgreen6 hours ago|

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You can do it with infinitesimals if you like, but the required course in nonstandard analysis to justify it is a bastard.

by jonahx5 hours ago|

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Or you can hand wave a bit and trust intuition. Just like the titans who invented it all did!

The obsession with rigor that later developed -- while necessary -- is really an "advanced topic" that shouldn't displace learning the intuition and big picture concepts. I think math up through high school should concentrate on the latter, while still being honest about the hand-waving when it happens.

by zozbot2344 hours ago|

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You can do it with synthetic differential geometry, but that introduces some fiddliness in the underlying logic in order to cope with the fact that eps^2 really "equals" zero for small enough eps, and yet eps is not equal to zero.

by anthk3 hours ago|

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while (i > 0) { operate_over_time }

calculus works... because it was almost designed for Mechanics. If the machine it's getting input, you have output. When it finished getting input, all the output you get yields some value, yes, but limits are best understood not for the result, but for the process (what the functions do).

You are not sending 0 coins to a machine, do you? You sent X to 0 coins to a machine. The machine will work from 2 to 0, but 0 itself is not included because is not a part of a changing process, it's the end.

Limits are for ranges of quantities over something.

by cyberax6 hours ago|

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IMO, the calculus is taught incorrectly. It should start with functions and completely avoid sequences initially. Once you understand how calculus exploits continuity (and sometimes smoothness), it becomes almost intuitive. That's also how it was historically developed, until Weierstrass invented his monster function and forced a bit more rigor.

But instead calculus is taught from fundamentals, building up from sequences. And a lot of complexity and hate comes from all those "technical" theorems that you need to make that jump from sequences to functions. E.g. things like "you can pick a converging subsequence from any bounded sequence".

by srean6 hours ago|

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Interesting.

In Maths classes, we started with functions. Functions as list of pairs, functions defined by algebraic expressions, functions plotted on graph papers and after that limits. Sequences were peripherally treated, just so that limits made sense.

Simultaneously, in Physics classes we were being taught using infinitesimals, with the a call back that "you will see this done more formally in your maths classes, but for intuition, infinitesimals will do for now".

by Sharlin7 hours ago|

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"The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn's lemma?"

(attributed to Jerry Bona)

by yaks_hairbrush1 hours ago|

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It works if you don't care about magnitudes, distances, or angles of complex numbers. Those properties aren't algebraic.

by ajb6 hours ago|

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Hah. This perspective is how you get an embedding of booleans into the reals in which False is 1 and True is -1 :-)

(Yes, mathematicians really use it. It makes parity a simpler polynomial than the normal assignment).

by cperciva6 hours ago|

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The complex numbers are just elements of R[i]/(i^2+1). I don't even understand how people are able to get this wrong.

by FillMaths5 hours ago|

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Of course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.

by cperciva5 hours ago|

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I mean, yes of course i is an element in C, because it's a monic polynomial in i.

There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.

by FillMaths2 hours ago|

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It's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.

by 6 hours ago|

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deleted

by emil-lp8 hours ago|

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Obviously.