Zettelkasten Forum

Using a zettelkasten in mathematics research

@bradfordfournier wrote here:

Any shared experiences re mathematics research in your zettelkasten would be great.

I think this deserves a discussion of its own, with a more telling title than the one where it came up.
@bradfordfournier - I hope you are OK with this.


  • @thomasteepe said:
    @bradfordfournier wrote here:

    Any shared experiences re mathematics research in your zettelkasten would be great.

    I think this deserves a discussion of its own, with a more telling title than the one where it came up.
    @bradfordfournier - I hope you are OK with this.

    Of course. Thanks.

  • Side note on quickly getting some latex into your editor / zk :

    One external piece of software / app that I've found incredibly helpful is the iOS app "Snip" which takes a picture of some mathematics and converts via ocr, to latex in multiple formats. It's excellent with hand-written OCR as well.

    Attached is the image taken of a simple bit of math and the app interface which provides the result.

    The result provides multiple formats as well as the confidence in the OCR. It can handle much more complex images including category theory etc. Very helpful for quickly getting some math into your preferred editor / zk software.

    • Disclosure: I've finished my dissertation in applied mathematics in 2001 with a topic in discrete Markov chain theory / first hitting times of Genetic Algorithms, and I haven't been active in academic math research since then.
    • I knew next to nothing about ZKs when writing it (apart from Arno Schmidt's hypernovel "Zettel's Traum", which gave me no hints about using a ZK for research).
    • My entire thinking about general problem solving in a ZK environment is deeply rooted in my experiences with math problem solving.
    • My current ideas on paper-ZK-based problem solving is outlined here, and the crucial question to me seems that of problem solving tools: Which tools are the most useful, given one's mathematical domain of work, knowledge, skills and expertise?
      (In a way, I suspect the tool collection could be a kind of plug-in - you could have tool collections for maths, for physics, for philosophy etc., with a certain overlap in general methods and any amount of diversity in specific methods.)
      Having a tool collection that can be adapted with ease and that co-evolves with its user seems of paramount importance.

    • The math tool collection from George Polya's "How to Solve It" is perhaps the most famous in mathematics, but personally, I never found it very helpful with its four stages of
      1) Understanding
      2) Making a Plan
      3) Carrying Out the Plan
      4) Looking Back,
      and its heavy reliance on using "related problems" one had solved earlier.
      I found the methods developed by Mason / Burton / Stacey in "Thinking Mathematically" and by Spyros Kalomitsines in "How to Solve Problems: New Methods and Ideas" much more accessible. Kalomitsines in particular gives a number of examples that show the interplay between thinking tools and "writing for insights" in the process.

    • In my years of study and at my university, there was practically no instruction about the methodology of problem solving, and although I had a number of friends that were really good at maths, there was little informal talk about methods.
      Even in retrospect, I cannot describe the reasons for this.

    • I think that even a very small community of ZK users communicating about their experiences could profit considerably. Given an explicit framework of concepts on how to write work notes, how to organize notes, how to use tools would enable them to progress decently - by discussing questions like
      how do you deal with confusion,
      what do you try when you are stuck,
      how do you generate new approaches,
      how do you organize your tree of solution trials" etc.

  • Here's a passage from a 1984 interview with Michael Atiyah:

    [Question:] How do you select a problem to study?
    ATIYAH: I think that presupposes an answer. I don’t think that’s the way I work at all. Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.
    (Source: The Two Cultures of Mathematics by W. T. Gowers)

    The essay examines differences between "problem solvers" and "theory builders".

    I'm mentioning it here because Atiyah's approach seems to correspond to straightforward zettelkasten work.

    About his own work habits, Atiyah said:

    I'm not the sort of person who does my mathematics writing on paper. I do that at the last stage of the game. I do my mathematics in my head. I sit down for a hard day's work and I write nothing all day. I just think. And I walk up and down because that helps keep me awake, it keeps the blood circulating, and I think and think.
    (Source: https://www.theguardian.com/education/2004/apr/21/highereducation.uk)

    Not exactly the principal witness for my "write for insights" mantra, as it seems.

  • @thomasteepe

    Great quotes. And while he may not be the poster boy for "write for insights" he is the poster boy for "walk for insights". I still think putting the two together is a dynamite combination.

  • Here's advice by Terence Tao on "Write down what you've done".

    On his page "What's new" he has subsections on "Career Advice" and "On Writing".

    There is no explicit connection to zettelkasten work, but tons of good advice on doing and on writing maths that can be used in any form of organizing knowledge.

  • One of the most relevant discussion on mathoverflow.net (a question and answer site for professional research mathematicians) seems to be "How do you keep your research notes organised?"

    The discussion was closed 9 years ago, with the following howler of an argument:

    This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet.

    There are 16 answers, mentioning paper notebooks, TeX files, wikis, photos of handwritten notes in Evernote, etc. I'm pretty sure that many of the commenters back then would agree today that a zettelkasten, digital or perhaps paper-based or hybrid, would provide a compelling basis for their work.

  • @GeoEng51
    At this very moment, I'm looking for information about how mathematicians organise their work.
    Here's something from an interview with Andrew Wiles.

    [Question:] Usually people work in groups and use each other for support. What did you do when you hit a brick wall?
    [Andrew Wiles:] When I got stuck and I didn't know what to do next, I would go out for a walk. I'd often walk down by the lake. Walking has a very good effect in that you're in this state of relaxation, but at the same time you're allowing the sub-conscious to work on you. And often if you have one particular thing buzzing in your mind then you don't need anything to write with or any desk. I'd always have a pencil and paper ready and, if I really had an idea, I'd sit down at a bench and I'd start scribbling away.
    (Source: https://www.pbs.org/wgbh/nova/article/andrew-wiles-fermat/)

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