I mean you can't learn math in a linear order. Can I just read a paper first on a subject I haven't studied and just work backwards? For example, I have never studied combinatorics but I sort of have a fuzzy idea about the paper on alternating permutations by Stanley.

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    $\begingroup$ I assume you are talking about Ravi Vakil's advice on this page: math.stanford.edu/~vakil/potentialstudents.html If not, could you be more specific? $\endgroup$ May 30, 2011 at 16:42
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    $\begingroup$ @Qiaochu Yuan: Yes...the tendrils of knowledge. $\endgroup$
    – Rick
    May 30, 2011 at 16:51
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    $\begingroup$ You could try. If I point you at the Wikipedia page for, say, de Rham cohomology, you could in principle just keep following the links until you reach something you can understand, and then work your way back up to the starting point. I learned a fair bit of differential geometry that way. $\endgroup$
    – Zhen Lin
    May 30, 2011 at 17:20
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    $\begingroup$ I'll second Zhen Lin. That's how I've learned most of what I know. $\endgroup$
    – Sam Lisi
    May 30, 2011 at 18:57
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    $\begingroup$ Different individuals learn best in different ways. Whether you can learn in this extreme backward way depends on you. Try it if you want, and see if it works for you. $\endgroup$
    – GEdgar
    Jun 1, 2011 at 13:28

5 Answers 5


This may in the end, for an absolute beginner, be quite unproductive. Normally introductory texts are written in such a way as to be helpful to a newcomer by first building a knowledge-base of basic facts about a field of mathematics, and then perhaps having a few chapters on selected topics of moderate difficulty towards the back.

I would recommend a compromise. First pick up an elementary text which is well-known and accepted among your peers, or accepted among the people who are writing the papers you wish to read. (This information is not very hard to find.) Then read and study the first few chapters with the basic material thoroughly. Then you may read the advanced paper, and be thoroughly confused, and work your way backwards as you describe.

But without the first step I do believe you will be not only creating more problems for yourself in the beginning, but also in the end, where you may indeed miss the entire point of a paper. Very often top research papers replace a standard part of a big elementary machine with a clever alternative to attack a difficult problem and find a new solution. You will not understand or appreciate this without a basic knowledge of how the solution works in the standard case first.

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    $\begingroup$ Glen: I was not expecting a reasonable answer to this question anymore, but you expressed exactly my feelings towards this question. $\endgroup$
    – t.b.
    Jun 1, 2011 at 14:51

I think it can be a very good idea to start with more advanced material and work backward to foundations. Certainly, in the case of combinatorics, it is quite reasonable to start out try to understand a particular problem and be brought backwards into the general theory, and Stanley is a very clear writer. To chose an example from my own life, I learned a good deal of real analysis by trying to understand how one proved that $\sum \cos (kx)/k^2$ converged to a periodic function, each period of which was a parabola.

What worries me about your question is the word "fuzzy". If you are going to skip ahead, I would advise you to have a very sharp understanding of those things you do not know yet, but expect to know when you have mastered the more basic material. Choose some particular theorem from the advanced paper and try to write out or talk through a complete proof of it. (Note that I had no trouble stating the above Fourier result and, indeed, watching it occur on my graphing calculator.) When you reach a point that you can't explain, this is when you need to dig back into the foundational material and figure out where that point is explained.

I think this sort of study requires you to be scrupulously honest with yourself about whether you actually know a complete proof, and not to be satisfied until you do. I've seen a lot of students whose "fuzzy" understanding proved, on examination, to be no understanding at all.

  • $\begingroup$ What do you mean by "each period of which was a parabola"? $\endgroup$ Jun 1, 2011 at 19:21
  • $\begingroup$ Ahhhhhh. I see :) $\endgroup$ Jun 1, 2011 at 19:22
  • $\begingroup$ @David Speyer I extremely disagree with the last paragraph. I would be interested in hearing your thoughts to my answer, which I have written in response to it. $\endgroup$ Oct 2, 2017 at 18:47

The "tendrils of knowledge" passage is quoted below. It is something that most graduate students experience. I don't think that the author advocates it as an overall way of learning but as:

  • a reassurance that mathematical intuition and vocabulary are often acquired more by osmosis and gestalt and less as a pyramid of carefully stacked bricks

  • encouragement to attend lectures (or read papers, or participate in conversations) that might not be fully comprehensible but nevertheless impart a feeling for the language and imagery that support detailed understanding later on.

The title of the question should be edited to avoid the implication that this is one professor's (or any person's) recommmendation of a complete learning method. It is advice to graduate students in a very theoretical discipline, algebraic geometry, where there is so much background to learn that it is rare for specialists in the field to read the proofs of everything before (or after) commencing research. The message is to not be intimidated by the formal complexity of the subject and not to bury oneself only in books while disengaging from talks and research papers.

Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

( from http://math.stanford.edu/~vakil/potentialstudents.html )

Notice the important words in parentheses: "caution: this backfilling is necessary".


Why not? You should try it and see if the approach suits you.


This is too long to be a comment. I address the last paragraph in David Speyer's answer, which has prompted quite a bit of discussion amongst friends in real life recently, mathematical and otherwise.

I think this sort of study requires you to be scrupulously honest with yourself about whether you actually know a complete proof, and not to be satisfied until you do. I've seen a lot of students whose "fuzzy" understanding proved, on examination, to be no understanding at all.

I strongly disagree. I have "faked it until I have made it" quite a bit in my undergraduate career, confused beyond all belief by trying to study mathematics way beyond me that struck me as more interesting than the mathematics I was supposed to learn at the time. It has only been in hindsight I realized I had learned quite a lot, even though at the time I had drawn the very severe criticism and ridicule of elders for:

  • not "going slow and steady",
  • not "knowing what I am talking about".

We do not need more students to be repulsed by nor fall in love with the Sylow theorems and very basic results from linear algebra and general topology. This is what has turned off so many of my friends from pursuing mathematics beyond the undergraduate level.

I end with two quotations from some writings about mathematics for a popular level, courtesy of Quanta.


Huh tried to use these lunches to ask Hironaka questions about himself, but the conversation kept coming back to math. When it did, Huh tried not to give away how little he knew. “Somehow I was very good at pretending to understand what he was saying,” Huh said.


At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which says that the equation xn + yn = zn has no nonzero whole-number solutions if n is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”

As Scholze burrowed into the proof, he became captivated by the mathematical objects involved — structures called modular forms and elliptic curves that mysteriously unify disparate areas of number theory, algebra, geometry and analysis. Reading about the kinds of objects involved was perhaps even more fascinating than the problem itself, he said.

  • $\begingroup$ I just saw this. I am trying to figure out how I can agree with what you wrote and not think I'm wrong :). I guess I'll just say that it is both true that I have benefited a lot by running way ahead of the curriculum and gone back to fill in things I skipped, and that it was very important to eventually go back and fill those things in. $\endgroup$ Oct 2, 2019 at 13:45

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