In what order should mathematical fields be learned?

Question

This could be considered a broader version of this question, with all fields.

I know that when high-level maths are reached, the fields being to split quickly (i.e. specializing in this type of algebra, or this type of calculus). What I'm interested in is the more "basic" order of what should be learned to understand mathematics as much as possible.

So what would this order be for someone who wants to pursue mathematics as a hobby?

For the sake of analogy, in case my wording is a bit ambiguous, let me use English as an analogy - one recommended pattern of learning would be:

1. The alphabet
2. Constructing words
3. Differentiation between nouns, verbs, adjectives and adverbs
4. Constructing basic sentences

And so on. Just a general guide for what to learn in math.

Answer 1

The answer to this question is very difficult actually. At least in my opinion. The best piece of advice I have been given is to follow your interests and desires while learning math. That has kept my interest in mathematics growing since I was exposed to the parts that actually got my attention.

One of the major difficulties I feel in answering this question comes from the fact that in my own studies I made two big mistakes:

1. I learned material that was more advanced before fully understanding the basics that applied to it.

2. I foolishly decided there were areas of math that would never be important to what I wanted to do and then ignored them. It turned out that some of those areas became important and I had to start learning things I should have learned already.

If you go by the standard model of universities you have the progression of college algebra/trigonometry, calculus 1 - 3, and usually thrown in for math majors is a discrete math course and linear algebra. Then you get into advanced calculus, real analysis/complex analysis, and abstract algebra getting progressively more abstract. One key point here is that as you get more advanced you will see overlap in areas that I once thought wouldn't happen. For example in parts of functional analysis you end up using techniques from integration and measure theory combined with techniques from abstract algebra. So at the beginning it is very good to learn as much of the basics of analysis and algebra as possible.

As a hobbyist I would start with a site like mit opencourseware and start with the first class single variable calculus. If that doesn't make sense then move to a college algebra level book. Then I would go down the list they have and see what gets my attention as you progress in skills and abilities. You don't necessarily need to know calculus to do a discrete math book but in my experience if you really want to get good at math doing both would be more helpful than singling one out at the exclusion of another until you get a firm idea of where you're going.

Hope that helps.

Answer 2

I wholeheartedly agree with every word quantumzorn has said. MIT's Opencourseware classes have proven to be an incredible help in learning mathematics to me as well ( I apply it all to Physics accordingly.) I've made the same mistake he has, but another approach that is available--and I won't recommend it. It's just the approach I take--is to start out with what you're wanting to learn, and look at an introductory text for what you're wanting to learn. Pretty quickly you'll find out whether or not you have the mathematical backing to do it or not. Nevertheless, once you get to a point where you have no clue what's going on, figure out what branch of mathematics is being applied there, and look at an introductory text for that. From there, it's just rinse and repeat until you get to a point where you can build upon knowledge you have, or gain new knowledge.

This method, however, can be very expensive and time consuming. For example:

I've been wanting to learn the mathematical foundations of Einstein's Relativity. This is arguably higher level mathematics ( Mostly Tensor Calculus ). Tensor Calculus was my top level math branch. Coming down from there, I learned that tensor Calculus is derived from Differential Geometry. My Second level down from Relativity. From there, I figured out that Differential Geometry is laden with Vector Analysis ( Vectors are merely a type of tensor. Level 3. Fortunately, I was already familiar with Vector Analysis, but vector Analysis picks up at the end of most college calculus courses ( Calc III ), which happens to be multivariable calculus. Level 4. Which then at the point, it's just a step down to single variable calculus, algebra, then your high school level mathematics.

One of the benefits of this approach though is that you'll quickly be able to figure what branch of mathematics something is, and sub branches required to learn what you want rather quickly. The way I've described my Relativity experience appears to be just a linear progression of doctrines of math. Most of the time though it isn't that straight forward though. I find it very helpful to make tree diagrams--probably pretty obvious that I was alluding to that. I also came to find that tensor not only include vectors, but matrices. What is the mathematics of matrices in general? Linear Algebra. That would go on level 2 alongside Differential Geometry. I'd go so far as to say that those two course can be taken concurrently, but I digress.

But once you've learned enough to ( in my case ) get past a certain part of a Relativity text, continue on until you hit a point in which something doesn't make sense, and then rinse and repeat. This process, again, can be very time consuming, but the more often you do it, the quicker you'll get at it. I'm only 19, and started with just an introductory level calculus text with just a high school knowledge of mathematics--and I wasn't even good at math to start lol--when I was 18, and utilizing MIT's resources, as well as any book I could get my hands on, I have no problem with figuring out what I need to learn for what I want to do. I won't say when doing this I was doing this religiously, it was more of--just like your case--kind of a hobby, and even now that I don't have much time, this method is still very practical.

Good luck!