How to study mathematics

Question

I've been self studying math for 6 months, i'm a computer engineering student and I just know a little bit of calculus, linear algebra, abstract algebra, logic, set theory, but not much at all, my problem is that i'm never satisfied, even if maybe "I know a lot" (for example I read 4 logic books cover to cover) in my head I always feel like I still dont get it. I would like understand when someone reaches this famous "mathematical maturity" and really understand a topic. I was used to studying calculus which was pretty much all about computation and I could consider myself ready to move on with the chapter when I could solve most of the exercises, right now i'm studying set theory and most of the exercises are only proofs (with no solutions unfortunately) and I can't really test myself to see if I really get the topic or not, unless it's all about proving things but I doubt it. So my question is, how do you study a math book? Usually I can test myself with other subjects solving problems etc (physics, chemistry etc), but what about math? When should I consider myself ready to move on to the next topic?

Answer 1

The first question you seem to be asking is what constitutes a "mathematically mature" understanding of a field of mathematics. Understanding math fundamentally requires understanding proofs. To truly comprehend a field of mathematics, you should be able to understand the proofs of the most fundamental theorems in the field and be able to apply these theorems and variations of their proofs to other problems in the field. This makes understanding mathematics at a high level both challenging and rewarding.

Thus, the real question is how to master proofs. For beginners, this really amounts to recognising that their own proofs are either correct or incorrect. There are two answers to this question which, depending on how you look at the situation, are either competing or complementary.

The first way is to master the notion of "formal proof". A formal proof is basically a sequence of statements, each of which is expressed in formal logic, and each of which follows from the previous ones by an application of a single formal logic rule (or is an axiom). Obviously, nobody actually writes proofs of any nontrivial complexity this way. However, if you understand formal proofs, you'll be able to see how to take a proof and (at least in principle) translate it into formal logic. Thus, the test of whether a proof is correct is simply whether it can be translated into formal logic. According to this school of thought, the key when writing an informal proof is that at every step of the way, you are able to understand from the informal proof how that portion of the formal proof would be written.

The other way is by writing proofs to be read by other people. A well-written proof under this definition is one that convinces a competent mathematician that the statement is proved. Mastering proofs using this method requires that you have someone to check your work and point out where things should be made clearer.

Once you understand how to recognise a correct proof and how in principle to write one, mastering a particular field will require you to learn certain theorems and techniques which come up over and over again in the field. For example, when learning real analysis, there are quite a few proofs involving taking an interval and repeatedly splitting it in half (for example, the intermediate value theorem). At a higher level, proofs in real analysis will involve more general concepts like connectedness and compactness.

At the highest level of abstraction, category theory provides techniques which prove extremely helpful in most (if not all) branches of math; many theorems across disparate fields of math often prove to be special cases of a theorem from category theory. For example, Stone-Cech compactification and free groups are examples of the same categorical phenomenon; a left adjoint to a forgetful functor. The existence of both of these follows from the Adjoint Functor Theorem.

At the end of the day, mathematicians spend years attaining this "mathematical maturity". Don't be discouraged if it takes longer than six months. If you really want to master mathematics, focus on proofs and take your time. Don't go on to the next chapter in the book until you can prove every theorem in the chapter and can work every exercise. If you had somehow mastered linear algebra, set theory, logic, abstract algebra, and calculus in just six months while also studying computer engineering, you would be a prodigy. Don't burn yourself out; learning is a marathon, not a sprint.