Nature of pure mathematics research

Question

For a little background- I'm a 3rd year engineering student, planning to do masters in pure mathematics. I've been studying abstract group theory and real analysis, although I'm extremely interested about these kind of thinking and reasoning what worries me is that during my high school days there were a lot of problems where we didn't have to do a lot of reasoning rather we had to come up with clever tricks to reach a solution. Honestly I wasn't good at this and thought mathematics isn't much fun.

So my question stems from this worry that how much of pure mathematics research is about learning abstract topics and do reasoning with them? Or is it mostly coming up with clever tricks from scratch?

In a way I want to ask what kind of skills are absolutely necessary for professional pure mathematicians?

I know perhaps it has to do with the problem one is working on but if it's possible to answer generally, please do.

Answer 1

I'm a pure mathematician, and my first impression is that your question seems to be assuming a dichotomy in pure mathematics that isn't really there. The vast majority of pure mathematicians, it seems to me, at times have to deal with abstract structures and do the sort of "thinking and reasoning" with them that you've seen in your analysis and group theory courses. On the other hand, in a comment to another answer you defined tricks as "coming up with clever ways to prove certain things/attain certain results." My assertion is that according to this definition, virtually all research in pure mathematics involves the application of tricks. After all, there's only so much you can do that is of interest to the mathematical community that can be obtained purely by combining known results and definitions in straightforward ways.

Nevertheless, I should point out that pure mathematics does indeed lie on a spectrum when it comes to abstract reasoning versus clever problem solving. The former approach to mathematics is perhaps best epitomized by Alexander Grothendieck, who "avoided clever tricks that proved the theorem but did not develop insight. He likened his approach to softening a walnut in water so that, as he wrote, it can be peeled open 'like a perfectly ripened avocado.'" At the other extreme are mathematicians like Paul Erdos, who largely worked on concrete problems that one could solve using 'clever tricks'. (Although it should be pointed out that, as Federico Poloni noted in a comment, when one uses a trick more than once it becomes a method. Erdos' probabilistic method arose this way.) There is a nice quote about this approach to mathematics in Gowers' essay The Two Cultures of Mathematics:

At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied.

Having said all of this, I'll reiterate what I said at the beginning of my answer. This is largely a false dichotomy, and the vast majority of mathematicians have to use both clever tricks and abstract reasoning in their research.

Answer 2

It is not really clear to me where you draw the line between "reasoning" and "trick". I can assure you, though, that pure mathematics gets very abstract and very theoretical very quickly.

What skills are necessary: First of all, you will be frustrated very often and you need to "like" that (a mathematician is somebody wants to be frustrated, rather than bored). Many theories take years to understand (at least for most human beings), and you may spend months without really understanding anything.

Secondly, you need to really enjoy that "pure reasoning" because the reason you look at a particular question is usually just that it is "interesting", not that it has any connection to "reality".

Let me add two further pieces of advice:

• Pure mathematics is an area that is very hard to enter as an "outsider", and there are reasons why many trained mathematicians change to computer science or engineering, but very few go into the other direction.

• It is not likely that you find a job in pure mathematics later on. The area is really competitive. That does not mean that you will be unemployed, but your future job will probably not be doing research in pure mathematics.

Answer 3

Welcome to the forum! A mathematician, I forget who, once jokingly remarked that applied maths was like the dark side, because applied mathematicians had more money and dressed in cool clothes, whereas pure maths, well, you get the drift, I'm sure. (I think he was joking, at least).

Studying pure maths can be extremely satisfying, intellectually if not financially, but it can also be endlessly frustrating; pure maths is actually intensely practical in nature - there is always good, practical reasons for why a concept or a method has been developed, in my experience, but it can often be hard to see, if this isn't pointed out clearly, and good mathematicians can be amazingly poor at communication. The solution to this is to keep asking until you are completely satisfied.

As an engineer you will have experienced that you are not expected to actually understand maths at a deeper level - you need to know how to use the tools, not how to manufacture them; thus, you learn to rely on the results: the theorems and the formulas. In pure maths, what you need to learn is the methods - often the proofs are more important than the theorems. This is so you can go on and discover new theorems and/or develop new methods.

So, to sum up, slightly tongue-in-cheek, engineers/applied mathematicians are the tool-users, whereas pure mathematicians are the tool-makers.

Answer 4

Although this is not directly answering the OP’s question (I agree with a previous answer that the question presents a false dichotomy of mathematics), my answer aims to shed light on what mathematics is and is not.

Mathematics is an art-form and at the heart of this art-form is proof.

Whereas science relies on evidence and the scientific method, mathematics relies on argumentation to establish results.

The only requisite imposed on these arguments is that they logically follow from agreed-upon axioms. This is where the creative aspect of the art-form appears: many mathematicians place a premium on proofs that shed light on the problem and make connections to other areas of mathematics (Google the phrase ‘a short proof of’ or ‘an elementary proof of’) and it is quite often the case that there are many proofs of important results (for example, the infinitude of the prime numbers or the fundamental theorem of algebra).

These skills can only be acquired and honed in much the same way that a painter or musician acquired their skills: practice. Prove everything.

Answer 5

The best short answer I can give as to the nature of pure mathematics and what pure mathematicians actually do at the highest levels is exemplified, perhaps best, by David Hilbert in proposing a set (23 or 24, depending) of problems worth studying: Hilbert's Problems.

He looked at what was known to be true in math around 1900 and what was not known. From all of that he looked for things that were interesting and worth studying. He had no proofs of any of the things he was proposing. Of the problems, some are still unresolved 120 years later. Some have been proven impossible to resolve. Some have resulted in (proved) theorems. Some have partial results.

Look at what is known and where the holes in knowledge are. Think about what is interesting and think about what is worth the effort of further study. Then, get to work to determine the truth of what can be learned of those things. Some of this study results in proposed Theorems. Some of those can actually be proved.

Mathematics is a search of the unknown and an attempt to make it known. But there is also a filter of "meaningful or interesting" things that might be worth studying.

Tricks and methods and proofs and all come later. Sometimes much later and sometimes not at all. It is pure mind stuff.

Answer 6

What is your purpose with your pure math study?

If deepening your math knowledge, that's OK. Still, you can have struggles as your classmates will be having math bachelor degrees, they know a lot more about mathematics and see mathematics from a different angle than you. Proofs everywhere.

Pure math students live 24-hours for mathematics, it's their religion.

Pursuing a pure math career is a risky business. Especially if you are after a PhD. You need to absorb large chunks of mathematics and find some research topic that is interesting, doable and yet contains novelty.

But if you successfully attain a master degree in pure math, it's definitely a good starting point with your engineering degree to an applied math or computer science phd.