What are the essential skills and abilities for pure mathematics?

Question

I am interested in learning pure mathematics and I want to be better at it so I want to know what skills and abilities are most important for this field so I can improve them. I have asked two related questions here before, this one about improving 3D visualisation or visualisation in general, and this one about arithmetic calculating or calculating in general like calculating many more steps in my head faster and more accurate so I can know if certain approach in proof or a problem like integrals will lead to nothing . However, the answers I received suggested that these skills are not very useful or relevant for pure mathematics (although I think they are important). So, what are the skills that I should focus on developing and practising? I have heard that pure mathematics requires a lot of creativity, abstraction, and rigour, but I am not sure how to measure or improve these aspects.

I would appreciate any advice or guidance from experienced pure mathematicians or anyone who has studied or worked in this field. What are the skills that you use most often ? How do you practice or improve these skills?

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Edit: I should also mention that I am a self-learner of pure math and I don't have anyone to help me or to talk to about math I work pretty much alone and currently I am studying the undergraduate topics, but I study math to be able to do research. So I guess I am interested in both the skills and abilities of undergraduates and researchers, but for now I need to know the skills and abilities of undergraduates.

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Edit: I also want to know what it means to be good at pure math in general. What does it even mean to say someone is better or smarter than someone in math? I mean, if I compare myself to someone like Terence Tao he is much better than I am or I will ever be even in 100 lifetimes. And the question is, why is he better? The short answer is, he is smarter than me, but what does that really mean in mathematics? What does “smart” mean in pure math? Being smart in math means having many abilities that he has and I don’t. But what are those abilities exactly and how can I improve them? But at least if I can break down what exactly being smarter means, I can identify each skill and try to develop it and become generally good at mathematics. That is why I asked this question, hoping to get some insight from other people who have more experience and knowledge than me. I know I will not be a brilliant mathematician or maybe not even qualified to be one in the first place. Also, I am a self-learner. I don’t have someone to talk to about math and everybody around me is not even interested in math. I study alone and I don’t have any help except online.

Answer 1

One of the most important—and underrated—skills in pure mathematics is the ability to make (or at least identify) good questions. Certainly there are well-known problems that can consume a career (Riemann hypothesis, Collatz conjecture, Twin Prime conjecture, ...), but far more relevant is that someone working in the field be able to generate good questions... ones that are neither impossibly difficult nor trivially simple, whose answers lead to new (good) questions or solution techniques, reveal hidden connections among disparate mathematical concepts, and more. This requires knowledge, dedication, and creativity that is just a bit different from the skills in solving problems. There are many examples of celebrated mathematicians who attained fame by posing a good problem or conjecture.

I used to teach at one of America's most elite liberal arts colleges and my prize math student could solve nearly every problem I posed for him. (We published two or three peer-reviewed papers while he was an undergraduate.) He then went off to graduate school in one of the nation's most elite pure math departments where... alas... he foundered. He struggled for years to find a problem appropriate for a dissertation. I was so disappointed to learn when he dropped out of school as a result.

My recommendation: practice making problems. Analyze outstanding problems and understand why they are important and worthy of effort. Conversely, identify problems that are "weak" and not particularly worthy of your (or anyone else's) time.

And talk to your professors about how they identify good problems.

A famous professor at my undergraduate school, AI pioneer Marvin Minsky, said it perfectly: Don't just learn what your professors know... learn how they think.

Answer 2

You mention creativity, abstraction, and rigor, which are all good answers. Creativity is hard to really nail down, but the other two are quite concrete skills that can be measured and practiced.

Abstraction is the process of taking some interesting piece of something and dropping the rest. The goal is to get "the" heart of what makes the original thing so useful or interesting in the first place ("the" in quotes because there can often be many).

An example would be to look at the real number and ask "what's so great about these?", which might lead you to a number of different answer. You could notice that there's a very strict concept of order, where any two objects are either the same exact object or one is bigger than the other. You could notice that arithmetic is possible. You could notice that, between any two real numbers, there is another real number. If you explore any of those ideas, you end up with a different abstraction of the real numbers.

Rigor, on the other hand, has more to do with how deeply you think (and talk/write) about the objects you encounter and the relationships between them. It's kind of like that thing kids do where they just keep asking "why" until you get frustrated or go in circles.

Taking the example of the real numbers as above, you might first ask what they really are. An answer like "I know one when I see one" is not rigorous, because it can't be used to make any precise statements about the real numbers. We could say that any set which satisfies all the interest properties we care about with real numbers is effectively a set of real numbers -- even better if we can prove that there is, in some sense, only one possibility.

In the beginning, I think it's valuable to take these approaches to their extremes. Keep abstracting a bunch of stuff away until you have nothing left. Keep adding rigor until you've described everything as deep as it goes. The more you get experience with these concepts, the better you get at knowing which directions to go and when to stop before it stops being useful.

Answer 3

There's a great blog post by Terence Tao on the subject of mathematical development in humans (and presumably most other species of mathematician). He distinguishes three phases of maturation:

1. The "pre-rigorous" stage.
2. The "rigorous" stage.
3. The "post-rigorous" stage.

At the undergrad level, your main goal is to get to stage 2. This means, first and foremost, learning to write proofs. There are many ways to achieve this. One of the most popular is to go carefully through Baby Rudin, or something similar, and do many, many problems. Other useful approaches include math competitions, like the Putnam, a course on discrete math/combinatorics, or a dedicated proofs course. It is critically important to do many, many problems, and write out the proofs carefully.

Eventually there are other skills you will need to acquire, but by the time you've gotten to the "rigorous" stage, you will have a much better appreciation for those things.