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I learn and produce mathematics. In that process, I had to read quite a number of research articles.

Question :

What are some efficient ways to keep a note of results of a research article in mathematics?

I keep a note of definitions (in detail) and results (with out proofs) for each paper I read.

Are there any other efficient methods to keep track of what results does a paper contain?

Reading introduction or abstract of a paper does not count as they are written in cryptic manner.

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  • This is not restricted to mathematics.. please feel free to add any general advice.. Apr 8 '20 at 12:53
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I'm a big believer in using simple index cards for this sort of thing. But there are modern alternatives also.

For the traditional approach, make a card for each paper. It contains title and author as well as publisher/source. It also contains a one or two sentence summary of the paper. What is the reason that this paper is important/interesting?

Cards are kept in a file box, perhaps in author order.

The advantage of this approach is that your card deck will still be available to you in fifty years after multiple operating systems and software systems have become extinct. Paper lasts. Another advantage of index cards is that they are small so you can only write essential things.

As a supplement, you can keep a notebook with a page or two for each paper. The index cards contain a page reference into your notebook system.

But the disadvantage is that such a paper-only system is hard to search. So you might consider an alternative. There is software for "digital index cards" that let you do the above on a computer and provide the opportunity for both searching and for updating the formats and organization of your notes. This might be a better solution for you in general, but don't overlook the permanence of paper. If you are really serious, use such a software system but also use a printer that conveniently and efficiently prints real index cards. Now you have your bomb-proof backup that you will be able to refer to in your 70's in the post-computer era.


Personal note: I have, in fact saved such things as my college course notes from deep into the previous century and have, very occasionally, gone back to refer to them. Later, I kept things on floppy disks that I can no longer access, even though I have the equipment to do so. They degrade as paper does not. Provided you use fairly good paper, of course. I found my first "paper" done back in secondary school (on paper, obviously, created when computers were made of vacuum tubes) and found it "interesting". Since it was typed (on a typewriter), I can scan it for more modern use.

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  • Thanks. I did not knew about digital index cards... Apr 8 '20 at 13:17
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    Some software is "explicitly" digital index cards. Others are similar but can be used in that way. The only one of the first kind that I know of is for Windows 10, which I don't use, but there are also useful things for other systems.
    – Buffy
    Apr 8 '20 at 13:19
  • I have come across papers which develop several 'incremental' results (usually methods / theorems), all combined to create that one interesting piece that goes into the abstract, is the highlight of the paper, and obviously goes into my 'index card' for that paper. I wonder those 'incremental' results/methods/theorems can come in use to me sometime in future as standalones separate from the interesting result they created in combination with each other. Do you face such a concern? If you do, how do you work around this? Apr 9 '20 at 12:35
  • Re index cards: If you have access to ISO A-formatted paper (A0 with surface of 1 square metre, side length ratio 1:square root of 2) than you may fold and cut / tear them quickly from a A4 paper used for printers which may be less carry weight while on a travel. Or you cut an 8-page minibook (quickly folded, e.g., youtube.com/watch?v=21qi9ZcQVto) after returning home into index cards.
    – Buttonwood
    Apr 9 '20 at 22:32
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How inefficient for each reader of a given paper to do this work privately and independently! Ideally, this should be done on a common, open platform. There is no standard dedicated platform for this activity at the moment (and arXiv does not accept reader comments), here are nevertheless two suggestions:

  • Wikipedia: In principle this is for non-technical content, but mathematicians get away with a lot of technical stuff. In principle you should avoid primary sources (i.e. research articles) and cite secondary sources (review articles and books), but again this is not a strict rule. So long what you write is not excessively technical, subjective or personal, you may probably use Wikipedia, and benefit from its collaborative editing, wikilinks, version control, etc.

  • Wikiversity: a sister project of Wikipedia, with the same infrastructure, but no rules against original research, primary sources or technical content.

My own (experimental) practice is to write in Wikipedia about relatively old and well-established results, and in Wikiversity about more recent results.

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