[mark-t]

To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does.

Now, on the other hand, linear algebra is almost universally important and is probably easier for a programmer to grasp. I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless, but they're fun and interesting, they'll give you a better idea of what mathematicians do, and you don't need much education to get into them.

My usual advice for building skills is to work on contest problems. See if you can find some AMC12 problems. If those are too easy, you can work your way up. AIME and Putnam would be good next steps (those can be found here: http://web.archive.org/web/20080205091131/http://www.kalva.d... ).

[aneesh]

Saying the Putnam is a next step from AMC12 problems is like saying the NBA is a next step from pickup basketball with friends in middle school! There are people who can do Putnam problems for fun, but those people generally know who they are already.

[albertni]

Solving problems 1-4 on each day of the Putnam with an "unlimited" amount of time is not a ridiculous expectation. Putnam's difficulty is partly due to its time format.

[mark-t]

Putnam problems are not considerably harder than AIME problems, and AIME is definitely the next step from AMC12. Anybody who can solve a few AIME problems can certainly solve A1 on the Putnam of almost every year.

[alxv]

> I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless

Useless? I dear to say that number theory is currently the most lucrative field of mathematics. Without number theory, modern day cryptography would not exist and thus everything that depends on secure communication of information would not exist. So forget about commerce over the Internet, bank wire transfers, credit cards, administrating computers remotely and, most importantly, hiding your huge porn collection from your wife.

And, combinatorics is useful for the study of algorithms. It is pretty much the foundation of computer science.

[mark-t]

I'm aware of their applications; by "practically", I meant "almost". There are certainly compelling uses for number theory and combinatorics (though I'm not convinced the study of algorithms is one of them), but they're nothing compared to calculus.

[jibiki]

> calculus isn't that important for mathematicians

That's a joke, right?

[mark-t]

No. Many pure math classes require no (or very little) calculus. Abstract algebra, number theory, combinatorics, and graph theory certainly fall into this category. Topology does, too, depending on which area you study and what you consider calculus. Sure, there are obviously fields that do rely heavily on calculus, as well as certain branches in the above fields, but my point was that it's nowhere near universally needed. I'm a graduate student at UCSD, and I can't remember the last time I used calculus in my research.

[mnemonicsloth]

This is terrible, wrongheaded advice. It's like Pablo Picasso, in the middle of his Blue Period, trying to convince younger painters that red isn't a useful color for serious artists.

If you want to study graph theory or combinatorics [1], then calculus will be pretty much useless to you, and you'll naturally go years without using it.

Calculus is also useless in some situations in abstract algebra (which are said to have combinatorial character). There are other parts of abstract algebra, e.g. Differential Galois Theory [2], in which calculus is pretty important.

Topology is similar. Elementary topology is part of the foundation supporting calculus, while algebraic topology is one of the tools that's useful when we try to do calculus (or solve differential equations) in non-Euclidean spaces.

Fields making heavy use of calculus include differential geometry, differential equations (ordinary or partial), dynamical systems or control theory. That subsumes most of physics. Fields underpinning (and largely inspired by) calculus include real and complex analysis, measure and integration theory (aka axiomatic probability theory). Also functional analysis, which is a generalization of linear algebra, which is the bookkeeping methodology of calculus in higher dimensions.

[1] The first sentence here says it all: http://en.wikipedia.org/wiki/Combinatorics

[2] http://en.wikipedia.org/wiki/Differential_Galois_theory

[mark-t]

That post does not contain any advice. It contains facts, none of which were contradicted by your post (a typical property of facts). I never told him not to study calculus. Given our current education system, that would be impossible anyway. I guided him more toward real analysis and suggested some other areas of math that might interest him. Since his only background is high school math, I felt it would be best to introduce him to something where proofs play a central role. If he can't stand that, then he probably shouldn't go into math.

[jibiki]

> calculus isn't that important for mathematicians

> Many pure math classes require no (or very little) calculus.

These are not the same thing (hence my confusion.) Your initial comment seemed to indicate that nobody does analysis anymore, which is just not true at all (look at the most recent fields medal.)

[mark-t]

Nope. I'm well aware that people still study analysis. I just meant that one doesn't necessarily need to learn calculus before taking the plunge into serious mathematics. Even in analysis, there's quite a bit you can do without knowing the stuff from a standard calculus class (though it certainly helps).

[TriinT]

I would claim that Calculus isn't that important for engineers / scientists / programmers either. Real Analysis is important if one needs to understand thing deeper. In the real world, problems can't be solved analytically... and many of the tools one learns in Calculus are kind of useless. I think Linear Algebra is much, much more important than Calculus. Linear Algebra is the arithmetic of higher mathematics, like Bellman said.

[jknupp]

>In the real world, problems can't be solved analytically...

Your other suggestions notwithstanding, you and I live in a very different "real world" my fried.

[slackenerny]

real world

Tell that to physicists “renormalizing”† it over and over all days long…

† I almost forgot not everyone on HN may know what that is, http://en.wikipedia.org/wiki/Renormalization

[TriinT]

If you can solve your problems analytically, then you're either working in a blessed field, or you're working with simplistic models...