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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.

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    Step 0: are you sure you can get admitted in a graduate program to do pure math with a background in engineering? – ZeroTheHero Jan 19 at 21:41
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    "I've been studying abstract group theory and real analysis" ... taking courses in them? (good) or just reading about them? (perhaps not enough to get you admitted to a math Ph.D. program). – GEdgar Jan 19 at 22:10
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    As @GEdgar asks, if you cannot give some sort of evidence (for example, by confirmation by letter-of-reference writers) that you have self-studied usefully, simply claiming self-study of this-or-that is much weaker (in terms of persuading admissions committees) that having corroboration. – paul garrett Jan 19 at 22:45
  • No, just not reading about them, studying them. I have to appear for entrances before getting admitted to a masters program. – S_Mitter Jan 20 at 4:31
  • "what kind of skills are absolutely necessary for professional pure mathematicians? " -> being way above average in maths is a must-have. Being patient is somehow useful compared to applied maths. – m.raynal Jan 20 at 9:58
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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.

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    Yes, indeed. Well put. Too many simplistic mythologies mislead not only the general public, but also novice mathematicians, and even many professionals. Somehow over-simplification is often perversely attractive in coping (psychologically?) with profoundly complicated phenomena. – paul garrett Jan 19 at 22:55
  • Before I push myself towards pure mathematics research what are things I should consider first? What are the struggles of this profession that I should be comfortable with? Also, how can I better equip myself if I want to take up pure mathematics professionally? – S_Mitter Jan 20 at 4:27
  • I did research in pure and very applied mathematics. My personal experience: In pure mathematics, it is not uncommon to struggle with a question for months without making any visible process. In applied maths, you often have more "examples" which you can work out by hand or with your computer, and it is easier to see progress. In pure maths, it is not unusual to have nothing publishable two or three years into your PhD. – J Fabian Meier Jan 20 at 8:18
  • @S_Mitter The most obvious thing to consider is that it takes most people several years of studying maths full time to be ready for graduate study. I would recommend you find a maths prof you can talk to one-to-one, who can help you work out if this really is what you want, whether it's realistic that you will succeed, and if so what path you should take. – Jessica B Jan 21 at 11:30
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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.

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  • Tricks: Coming up with clever ways to prove certain things/attain certain results. Common example is high school combinatorics. It's not built upon rigorous structures. Different problems using same principles often require clever "strategies" which are difficult to think of from scratch, the leaps here from problem to the trick that solves it is not exactly explained, always. Whereas take for example Isomorphism theorems is built upon definitions and smaller results, there's nothing mysterious or unexplained about it. – S_Mitter Jan 19 at 20:23
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    S_Mitter: Math research has no grunt work. There are no experiments to do. Once someone figures it out, it's been figured out, and there is no more research to do on that question. So topics that have nothing mysterious or unexplained about them tend to be solved, and are not research questions anymore. Now it does frequently happen that a subject that originally looked like a bunch of tricks gets organized and becomes something that doesn't look like a bunch of tricks anymore, and that is part of the work of research in mathematics. – Alexander Woo Jan 19 at 21:08
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    "An idea that can be used only once is a trick. If one can use it more than once, it becomes a method.” --George Pólya and Gabor Szegö, Problems and Theorems in Analysis, 1972. – Federico Poloni Jan 19 at 21:54
  • @Alexander Woo: "Math research has no grunt work." --- Well, at the beginning there's the sometimes lengthy "library research" on a certain problem/topic (now mostly done outside the doors of an actual university library) that might need to be undertaken, and at the end there are the many successive theoretical refinements that one considers once the main method/idea is established/proved in the most important special case(s) (although given present-day publication quotas, these might be left for follow-up papers). – Dave L Renfro Jan 20 at 18:31
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    @AlexanderWoo Math research has no grunt work. - Really? I feel like a lot of PhD theses, and about half of the work I do in my own research, are grunt work. – Kimball Jan 21 at 2:23
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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.

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  • Great answer. What would be your advice if I tell you I aspire to be a pure mathematician? – S_Mitter Jan 20 at 16:35
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    @S_Mitter Well, I'd say "Go for it - but with a whole heart". Also, don't spread youself too thin - pick a favourite subject and get good at it; later, you can pick up more subjects, but you can't become an expert in all of algebra, topology, formal logic, category theory, .... And finally, keep bothering your professors, tutors and everybody else with questions. I think most mathematicians enjoy talking about and explaining their favourite subject. – j4nd3r53n Jan 20 at 16:52
  • Thanks for your encouraging words. – S_Mitter Jan 20 at 17:04
  • Here’s how to determine if you’re an applied or pure math person. What is 7/5? if you’re applied math, you answer 1.4. If you’re pure math you answer either a) over what field? or b) it’s equivalent to 14/10. – ZeroTheHero Jan 22 at 3:13
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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.

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  • Can you please elaborate on how the creative aspect mathematics emerge? Is it during judicious creation of new axioms/structures? – S_Mitter Jan 21 at 19:10
  • @S_Mitter: as an art, it is creative by definition; there are many proofs of the same result and deriving these proofs often requires inspired thought and effort. – Pietro Paparella Jan 21 at 19:13
  • Actually, this gets it a bit wrong. There are steps that come before proof and that are, possibly, even more important. Mathematics existed before Peano. The OP asks the critical question in the comment just above. The important issue is: What do you prove? But it would take a book to elaborate that. Was Hilbert not doing mathematics in his famous questions? – Buffy Jan 21 at 19:32
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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.

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  • What in your opinion beginners should strive to achieve during their coursework? – S_Mitter Jan 22 at 2:57
  • The correct answer is insight, but that only comes from hard work. Solve more than the minimum number of exercises. Maybe a lot more. Even variations on the same exercise. But it is the same for professionals also, not just beginners. – Buffy Jan 22 at 11:17
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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.

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    My first purpose is creation, it just took me some time to realize what kind of creative process will satisfy my hunger. After that I delved deep into pure mathematics, I was in a way roaming around it, all my life but this was the first time I actually took it up very seriously. Since then I figured out I had to learn a lot of tools in order to finally do some serious mathematics at some point of time in future. Besides I find the subject very enjoyable, getting comfortable with abstractions is perhaps the most fun and satisfying experience I have ever had. – S_Mitter Jan 23 at 11:18
  • This is a good motivation. If you are into abstractions, I suggest you to touch interactive theorem proving. My personal taste is the Isabelle theorem prover with lots of abstract mathematics formalised at isa-afp.org – Gergely Jan 23 at 11:44
  • Also of interest: cl.cam.ac.uk/~lp15/Grants/Alexandria – Gergely Jan 24 at 9:35

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