# Life after Differential Equations

I am currently taking calculus, and will be done with differential equations by the time I graduate high school, maybe by junior year. I was wondering what comes after differential equations, because up to differential equations there was a series of maths that you were supposed to take:

algebra 1, geometry, algebra 2, trig, precalc, calc 1,2,3, linear algebra, then differential equations. But I haven't really heard anything about what comes after.

I find it laughable that you'll be done with differential equations before you even start college. I've never even heard of a high school offering calc III, and of course, it's difficult to find a high school that'll treat any part of calculus with any rigor.

Anyway, math kinda branches off with algebra. Differential equations comes in several varieties, the two most notable of which are ordinary and partial differential equations; to say you've 'covered differential equations' is to say that you've probably glanced at a few ordinary differential equations. There's modern algebra, which is (AFAIK) unrelated to calculus. There's analysis, which AFAIK is 'calculus done right', which is to say, you prove a bunch of theorems rather than taking definite integrals and such. There's topology, which AFAIK is a generalization of geometry. I think those are the biggies.

I find it laughable that you'll be done with differential equations before you even start college. I've never even heard of a high school offering calc III, and of course, it's difficult to find a high school that'll treat any part of calculus with any rigor.
I mainly lurk around here, but I've noticed that you like to say a lot (often dismissively) about things you don't seem to have much experience with, and also that you like to throw in your (often dismissive) opinion when it hasn't been asked for. Calc III, Linear algebra and differential equations are taught in most community colleges, so it's entirely possible that an advanced high school student could take them before graduating.

OP: Angry Citizen's cursory discussion of higher math, while not entirely accurate, is mostly correct. Algebra and Analysis are the main branches of "real" mathematics, although they are not entirely unrelated. Topology is sort of neat, because it takes elements of analysis, algebra, and geometry and is really hard to pin down into any disciple. Then again, deep study will reveal this about any branch of mathematics. There's also complex analysis, which is calculus using complex numbers, notorious for being both beautiful and inscrutible.

I would not worry about what comes next right now, though. There's just too much math to even get a good idea about the basics before you reach college. And in fact in college it might be a good idea to take honors versions of the courses you've already taken, because being super-comfortable with the fundamentals will help you a whole lot more than just taking a lot of advanced classes. If you're not already taking physics and computer programming courses, those will give you a good idea of how mathematical ideas are applied. If you are interested in pure math and want an accessible look at contemporary (and highly abstract) mathematics, I highly recommend https://www.amazon.com/dp/052171916X/?tag=pfamazon01-20&tag=pfamazon01-20.

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Also, I completely neglected to mention probability, which is a more applied but extremely applicable branch of math, and logic, which is sort of the interface where math meet philosophy (not to be confused with philosophy of mathematics, which doesn't actually have much to do with math, though it is interesting if somewhat pointless). And there is the whole field of discrete mathematics, which in a way is an extension of calculus and algebra, though they are used in completely different ways.

So there isn't really any set path to higher mathematics, if that's what you were wondering. It's all very diverse and interconnected.

I find it funny that you criticize another member for their dismissiveness and then turn right around and call a major branch of analytic philosophy 'somewhat pointless'. I imagine the philosophers of mathematics and the pure mathematicians pursue their respective disciplines for largely similar reasons, and I would wager that philosophy's lack of 'applicability' would hardly be seen as a fault in the eyes of a pure mathematician, who is likely indifferent to the applicability of his or her own discipline.

I was wondering what comes after differential equations, because up to differential equations there was a series of maths that you were supposed to take:
Well, exactly. You've listed the core courses for most science majors, and after those, you take whatever specialty courses are recommended for your major.

If you are that far advanced in high school, you should surely be thinking about getting into a good university. I suggest you look at the websites of the colleges you are considering, and see what they recommend for math and science majors. Most have a sample curriculum, and if not, go ahead and email the department and explain your situation. They might be impressed enough to offer you a scholarship.

You may be able to extend your head start even more by taking some of the free online courses offered by MIT and other schools.

Good luck to you.

I find it funny that you criticize another member for their dismissiveness and then turn right around and call a major branch of analytic philosophy 'somewhat pointless'. I imagine the philosophers of mathematics and the pure mathematicians pursue their respective disciplines for largely similar reasons, and I would wager that philosophy's lack of 'applicability' would hardly be seen as a fault in the eyes of a pure mathematician, who is likely indifferent to the applicability of his or her own discipline.
Fair enough :) It would indeed be very hippocritical for someone interested in pure math to dismiss philosophy as pointless! I only meant that philosophy of mathematics have very little to do with actually doing mathematics, so it can be safely disregarded by a mathematician. I do find philosophy of science very interesting and didn't mean to imply that the subject itself is pointless.

Well pointless and non-applicable aren't really synonymous. Besides, philosophy of mathematics, like applied math, doesn't seem all that well-defined. Wikipedia hints strongly that mathematical logic counts as philosophy of mathematics, and well some mathematicians and certainly many computer scientists would consider the subject to be important. Unfortunately I don't know anything about mathematical logic, but hey Paul Cohen won the Fields Medal.

Also I think Angry Citizen meant to say that calculus, as opposed to analysis, focuses on the mechanical aspects. To imply that analysis doesn't employ the full range of tools available from elementary calculus is kind of silly.

I was in geometry my freshmen year, it was really boring and with a terrible teacher so I took from algebra 2 to calc 1 which I'm in right now at my local community college

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Well pointless and non-applicable aren't really synonymous. Besides, philosophy of mathematics, like applied math, doesn't seem all that well-defined. Wikipedia hints strongly that mathematical logic counts as philosophy of mathematics, and well some mathematicians and certainly many computer scientists would consider the subject to be important. Unfortunately I don't know anything about mathematical logic, but hey Paul Cohen won the Fields Medal.
This is sort of what I meant by distinguishing logic from philosophy, though. Obviously there's not a strict divide--for example, Godel's ideas were influenced by Betrand Russel, who was a philosopher. But since Russel's time, philosophy of mathematics and mathematical logic have mostly diverged into the separate fields of philosophy and mathematics (as far as you would call those separate). For examples, one of the central problems of mathematical philosophy is whether math inherent in the universe, or if it is strictly a human invention. I find this question very interesting, but I can't see why someone actually doing mathematics would need to worry about it.

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I find it laughable that you'll be done with differential equations before you even start college. I've never even heard of a high school offering calc III, and of course, it's difficult to find a high school that'll treat any part of calculus with any rigor.
Laugh away, although I am not so sure what is so funny.

I finished differential equations before I got to college, which was decades ago. MIT gave me credit for 18.03, and let me register for a graduate course with it as a prereq, which I did just fine in.

I mainly lurk around here, but I've noticed that you like to say a lot (often dismissively) about things you don't seem to have much experience with, and also that you like to throw in your (often dismissive) opinion when it hasn't been asked for.
I'm an opinionated individual. If people ask for advice or for academic guidance, then they would (or should, rather) be appreciative of outside the box thinking, and of healthy skepticism. I've never heard of anyone completing such an advanced curriculum of mathematics at such a young age; certainly not to any rigor. So naturally I doubt the claim in the hopes that he would explain it further.

Also I think Angry Citizen meant to say that calculus, as opposed to analysis, focuses on the mechanical aspects.
That is correct.

I'm an opinionated individual. If people ask for advice or for academic guidance, then they would (or should, rather) be appreciative of outside the box thinking, and of healthy skepticism. I've never heard of anyone completing such an advanced curriculum of mathematics at such a young age; certainly not to any rigor. So naturally I doubt the claim in the hopes that he would explain it further.

That is correct.
I'm not certain if it's you or Ryker who did A-Levels, but I'll ask anyway. Are most of these courses not covered in A2 Maths and Further Maths? I do realise the OP is at an American high school but I just figured that was worth pointing out. (in the event that it is, actually correct)

Laugh away, although I am not so sure what is so funny.

I finished differential equations before I got to college, which was decades ago. MIT gave me credit for 18.03, and let me register for a graduate course with it as a prereq, which I did just fine in.
Further; applicants having other high school qualifications (like the IB; A-Levels; French Bac') can skip 18.03 (pretty certain that's the one) if they have A/B (at A2-level) or 6/7 (IB HL) in Mathematics.

I'm not certain if it's you or Ryker who did A-Levels, but I'll ask anyway. Are most of these courses not covered in A2 Maths and Further Maths? I do realise the OP is at an American high school but I just figured that was worth pointing out. (in the event that it is, actually correct)
It wasn't me

Nor me, I'm an American and an engineering major.

I must've dreamed that.

Well yeah, lots of these topics are covered in my A2 syllabus. (complex numbers as well) What I am uncertain though, is how rigorous it is compared to college courses.

Differential Equations are just a tiny part of the calculus we study, though.