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.
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
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.
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.
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:
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.
Notice the important words in parentheses: "caution: this backfilling is necessary".