43
$\begingroup$

This is quite a philosophical, soft question which can be moved if necessary.

So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an enjoyable and rewarding experience. However, I have come to realise how much further I still have to go to reach a point where I could even think about publishing original contributions in the literature given how intensively everything has already been studied and the discoveries already made.

For example, I have just finished a 600 page textbook on graduate level mathematics. Although it took me a while to understand everything in it, I learned from this and enjoyed doing the exercises, but realised by the end that I still basically know nothing and that it is really intended as a springboard to slightly more advanced texts. I picked up another book which starts to delve more into one of the specific aspects in the book and again, it is 500 pages long.

Do I have to read another 500 page book to get a sense of something more specific which I can contribute? At this rate, it will be years and years before I am ever able to publish anything. I only ask as I was a few years older than normal when I started studying mathematics so I feel like I am already playing catch-up slightly and that at this rate I will be an old man by the time I know enough to actually contribute anything.

Edit: Many thanks for the comments and I have taken them on board. I don't regret reading the entire book which I have already read as it is a classic but for now on I will focus on reading papers and trying to solve problems, then work backwards and use books as references if necessary.

$\endgroup$
  • 25
    $\begingroup$ This varies from area to area, but ultimately it is your Ph.D. advisor's job to chart for you a narrowish path throw the literature to a point where you can make an original contribution. I have this model that in your undergraduate you are learning the basics in a ball around 0, but in your Ph.D. you have to start drilling a fairly narrow path through the vast body of mathematics to an accessible point on the frontier, before making that tunnel wider over time. Your Ph.D. advisor should have some idea of which points on the frontier are accessible to you and how to get there. $\endgroup$ – Alex B. Jul 3 at 12:28
  • 26
    $\begingroup$ You don't need to know everything before you can do anything. Also, the nature of Mathematics is such that even people who work at all their lives feel that they know nothing of what is there to be known, so the feeling you mention is not unique to you. It's good to keep learning, and there may well naturally (hopefully soon) come a point when you have an insight which no-one else seems to have had, or you can answer a question that was previously unanswered, or ask an interesting question previously unasked. $\endgroup$ – Geoff Robinson Jul 3 at 12:29
  • 10
    $\begingroup$ I think it is good if you start with an actual problem, and start working backwards. Read related papers, and then read the definitions needed to make sense of the papers. You do not need to understand papers in detail, but make notes of the main ideas, so that you can return and study the techniques in detail once you believe you need them. $\endgroup$ – Per Alexandersson Jul 3 at 12:36
  • 24
    $\begingroup$ My experience is that reading entire textbooks is a very bad way to start doing research. Read papers instead $\endgroup$ – Stanley Yao Xiao Jul 3 at 14:16
  • 21
    $\begingroup$ @AlexB. your model (which I agree with) is nearly identical to this one: matt.might.net/articles/phd-school-in-pictures $\endgroup$ – Terry Tao Jul 3 at 23:08
43
$\begingroup$

It is something of a myth that everything has already been studied and that you have to master thousands of pages of prior work before you can contribute something new.

To be sure, there are some subfields of mathematics that are highly technical, and you're unlikely to be able to contribute something new to them unless you've studied a lot of background material. However, there are also areas of mathematics that don't require that much background knowledge. For example, Aubrey de Grey recently made spectacular progress on a longstanding open problem in combinatorics, and almost no background knowledge was needed for that problem. Even in supposedly highly technical areas of mathematics, people sometimes come up with breakthroughs that employ very little advanced machinery.

As others have mentioned, more crucial than "knowing everything" are (1) finding a good problem to work on, and (2) having problem-solving ability. If you have both of these, then you can typically learn what you need as you go along. When you're at an early stage in your career, finding a good problem generally requires an advisor, unless you have the rare ability to smell out good problems yourself just by reading the literature and listening to talks. Problem-solving ability is probably innate to some extent, but a lot of it comes down to experience and persistence. Of course you will be a more powerful problem solver if you have a lot of tools in your toolbox, but generally speaking, you get better at solving problems by spending your time directly attempting to solve problems, and only reading the 500-page books when it becomes clear that they are needed to solve the problem you have in mind.

$\endgroup$
  • 6
    $\begingroup$ I recently wrote a paper on quantum mechanics essentially just by realising that something I had written in my notebook from reading papers on QM was a good problem to study, then when I realised that I required certain pieces of mathematics during the writing of the paper, I learned what I needed listening to talks and reading a few more papers, so I suppose I could just carry on like that. $\endgroup$ – Tom Jul 3 at 21:38
  • 1
    $\begingroup$ Wow, did Aubrey de Grey really make such progress on that problem? I knew him for something completely different, that's very impressive. $\endgroup$ – Tom Jul 3 at 21:40
  • 2
    $\begingroup$ Indeed, most people know Aubrey de Grey for something completely different, and his work on that problem was very impressive! $\endgroup$ – Timothy Chow Jul 4 at 1:57
35
$\begingroup$

Mathematics is not learned by reading books. One becomes a research mathematician by solving problems. Most people need an adviser to recommend a good problem. Then you start thinking and reading what is relevant to your specific problem. General education by reading books with hundreds of pages can be done as a parallel process, but the main emphasis should be on a specific problem. It is a duty of the adviser to find a problem which does not require too much reading.

There are many examples that demonstrate these principles. Many good mathematicians obtained their first original results before the age of 18 or even much earlier, at the time when they learned very little.

Myself, I published my first paper at the age of 18, when I was a second year undergraduate student. I did not know much of mathematics at that time. I do not say that this paper is among my best, and at present I would not publish such a result, but this is irrelevant. The main point I am trying to make is that one has to solve problems, not to read books. It is not necessary that problems you solve in the beginning are new/publishable. But eventually you will obtain new results. Finding a good adviser is a crucial matter, for most people.

$\endgroup$
  • 5
    $\begingroup$ I read the book to get an idea and then solve the problems in the book. I disagree with you in part: mathematics is partly solving problems but it's also about the ideas to a certain extent. Thanks for your advice, I will maybe start focussing on trying to solve problems as I feel like my knowledge is sufficient to tackle some problems I have in mind which are quite basic. $\endgroup$ – Tom Jul 3 at 14:29
  • 6
    $\begingroup$ @Tom: My main recommendation is to find a good adviser. In my case, the adviser proposed a research problem (unsolved) and pointed 3 or 4 relevant papers to read. $\endgroup$ – Alexandre Eremenko Jul 3 at 14:36
  • 2
    $\begingroup$ OK, that's a good idea. I know a research mathematician who has corresponded with me before on a problem they are trying to solve, so I am probably going to email them now with my thoughts on it and asking how we could progress or if they have more recommendations for papers to read. $\endgroup$ – Tom Jul 3 at 14:39
  • 28
    $\begingroup$ “Mathematics is not learned by reading books.” Well, mathematics as a body of knowledge can be learned with reasonable effectiveness by reading books, and that’s done all the time. Perhaps you meant that the skill of doing research in mathematics can not be learned by reading books? I would happily agree with that statement, but I feel that it’s important to draw a distinction (which may seem artificial to you as an accomplished research mathematician) between “mathematics” and “research in mathematics”, since for most users of mathematics those are two very different things. $\endgroup$ – Dan Romik Jul 3 at 19:32
  • 8
    $\begingroup$ Also when I took analysis courses in uni, I found that I often automatically knew the answers to many of the questions the instructor was asking because I had read Hardy's book on analysis, I don't think it's as simple as 'never read mathematics'. $\endgroup$ – Tom Jul 3 at 21:14
11
$\begingroup$

As a partial answer, learn to trust peer review. When you are starting out in graduate-level mathematics you focus on proofs and seldom go beyond statements which can be exhaustively reduced to basic axioms. In some ways that is the ideal of mathematics. But, when you get to the research frontier, you might discover that you need to use a statement which is contained in paper A, which at a crucial step in its proof invokes a result from paper B, which in turn invokes papers C, D and E, ... It could perhaps take months of work to see how that single statement ultimately follows from what you currently know. If you plunged down every such rabbit hole you encountered, it is unlikely that you would ever make progress. Exhaustive background knowledge is not a prerequisite for the creation of new knowledge. You can explore what follows from what is currently known, without first reducing what is currently known to first principles.

$\endgroup$
  • $\begingroup$ Thanks for your advice. $\endgroup$ – Tom Jul 5 at 15:59
7
$\begingroup$

I put myself in a position where I could contribute to mathematics without needing to publish. I don't suggest spending half your time on MathOverflow as a career substitute or career booster. However, thinking about a variety of problems allows you to make connections to things you have studied and to define and modify your areas of interest. Good participation on this forum can be a contribution to mathematics.

Alain Valette was kind enough in a paper to mention an example https://mathoverflow.net/a/64754 of mine to a question of his. It is a contribution on a small scale, but one of a growing collection of mine based on my activity here. I encourage you to share some of your contributions here, through questions, answers and comments. It will prepare you for the kind of contribution off MathOverflow that you may wish to make.

Gerhard "Come And Join The Party" Paseman, 2019.08.24.

$\endgroup$
2
$\begingroup$

I believe one is ready to make an original contribution when one understands the problem and also understands why their solution solves it.

Mathematics is meant to talk about ideas, abstractions, and truths over prestige or authority.

This topic often comes up when we stop thinking for ourselves and place mathematics in a sociological mindset.

Instead of thinking "Why are we defining it this way? Why are the existing tools insufficient? Why does this proof work? Can I revise this theory to be more concise?", we begin to think "Man, this person wrote 1000 pages of mathematics, there's no way I could possibly understand it all."

There is irony here: the less of some theory one understands, the more difficult it is to challenge. In many instances, one is indeed lacking key insights to fully understand the theory. But there will ALSO inevitably be instances where every member of the population assumes that some other member knows it better than they do, thus no single member attempts to challenge the theory. Each chapter or argument may be optimized locally, but opportunities nonetheless exist in the global scope for huge simplifications.

It's important to be able to challenge the arguments. If a theory is so large and complex that amendments cannot be understood within the overall context of the theory, it is unlikely that such an amendment will be simplified or assimilated. Mathematical theories in such cases will usually undergo a process of explosion

The point being that having very large theories taking years of work to understand is all the more reason to believe that opportunities exist for contributions.

$\endgroup$
  • 2
    $\begingroup$ "Mathematical theories in such cases will usually undergo a process of explosion" Can you give an example of a mathematical theory exploding? $\endgroup$ – Gerry Myerson Aug 25 at 0:08
  • $\begingroup$ @Gerry One good example is the AGDA programming language (agda.readthedocs.io/en/v2.6.0.1/index.html), a dependently-typed programming language designed to encode mathematical proofs. Over time, the number of built-in language features has expanded, but mostly to resolve special cases or counterexamples. Such cases have struggled to assimilate into the main Curry-Howard/functional programming theory. $\endgroup$ – user1318416 Aug 25 at 17:54
  • $\begingroup$ Very interesting, but exactly which mathematical theory was it that exploded in that case, and what leads you to liken it to an explosion? $\endgroup$ – Gerry Myerson Aug 25 at 22:26
  • $\begingroup$ It would be the theory relating to functional programming and the Curry-Howard Isomorphism as a medium for mathematical statements and their associated proofs. Comparatively simple representations exist independently for representing logical statements (syntax of terms, logical connectives, and quantifiers) and functional programming (think Lisp or lambda calculus), but not their union. $\endgroup$ – user1318416 Aug 26 at 0:26
  • $\begingroup$ I liken the process to an explosion once the theory fails to be self-encapsulating. It explodes because subsequent work is more likely to increase the complexity of the given subject matter rather than decrease it. In some fields, even coming to understand the definitions and prerequisite vocabulary is quite the process. $\endgroup$ – user1318416 Aug 26 at 0:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.