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A Guide to Writing Mathematics [pdf]
- tkgally 6y agoThis is a sensible guide, by Kevin P. Lee, aimed at undergraduates taking math classes. A similar guide, aimed at people writing research papers, is “How to Write Mathematics” by Paul Halmos (1970) [1]. They both start from a similar assumption: Lee: “When you write a paper in a math class, your goal will be to communicate mathematical reasoning and ideas clearly to another person. The writing done in a math class is very similar to the writing done for other classes. You are probably already used to writing papers in other subjects like psychology, history, and literature. You can follow many of the same guidelines in a mathematics paper as you would in a paper written about these other subjects.” Halmos: “The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea.” [1] https://www.mathematik.uni-marburg.de/~agricola/material/halmos.pdf https://www.mathematik.uni-marburg.de/~agricola/material/hal...
- ZephyrBlu 6y agoI'm going to read through the guide linked in the post, but I don't have time (Or really the desire) to read through the guide aimed at people writing research papers. Do you have TL;DR for the difference between aimed at undergraduates vs research papers in terms of writing?
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- B-Con 6y agoAs a math major at UCD, I never saw Lee's essay, so I'm guessing it's from the past few years. I did read Halmos's, though. It was helpful for me because I was starting from a very programming centric frame of mind and I was surprised at how "conversational" math writing was. Reading it helped me start to learn how to express ideas precisely and clearly without a strict code-like structure.
- dnlggr 6y agoThanks for linking this! I like Halmos‘ clarifications on the editorial “we“. I don’t know if this is because I‘m living in a non English speaking country but at our university students often get told to use ”we” instead of ”I” in their papers. I always found this weird. If you‘re the only author you can‘t just refer to yourself as ”we” — just to avoid the use of ”I”. It sounds wrong — especially if the reader knows that you’re the only author — and eventually leads to absurd constructs such as the example in the essay: ”We thank our wife for her help”.
- tkgally 6y agoI understand the problem. I used to manage an English academic writing program at a university in Japan, and what guidance to give to students about pronoun usage was a frequent topic of discussion among the teachers. One problem was that the students had learned a moderately informal version of English in which first-person pronouns are common. Also, they were young and used to writing and speaking about themselves. That led to what some teachers perceived as excessive use of “I” for the research papers the students were being taught to write. Another issue was that the teachers themselves all had academic backgrounds, most with doctorates, and, we discovered through our discussions, pronoun usage varies a lot by field. Curious, I once looked through journals in a variety of fields—sociology, nursing, physics, gender studies, literature—and found that in some fields the authors never seemed to refer to themselves by “I” or “we” while in others it was common. The use of “we” in mathematical writing, especially proofs, may be a special case. The “we” in a sentence like “If we assume that M is a compact metric space, then we can prove that ...” doesn’t really refer to the author or authors; it seems to have a more abstract referent. Paul Halmos, by the way, was an excellent teacher as well as writer of mathematics. I was fortunate to take several classes from him when I was an undergraduate at the University of California, Santa Barbara, in the 1970s. Though I ended up not going into mathematics, I still have a very fond memories of learning with him.
- supernova87a 6y agoI don't know if anyone else experienced something similar in physics, but it was not until I read the textbooks of David Griffiths (his books on E&M and quantum especially), that either of these subjects became clear to me. I think the reason is that his writing was like a clear 1-on-1 tutorial session, with many of the "writing mathematics" practices described in the article here. It had a conversational style as if it was someone trying to explain but in written words. I recall phrases like "now look at the expression we have here, what does it tell us?" Or, "what follows is a somewhat long derivation, but you will find the effort of working through it pays off". Most other textbooks read like stilted reference manuals by comparison, with "exercises left to the reader".
- nicklecompte 6y agoIt has been a while since I read Griffiths. If I recall I liked his style for E&M, but the quantum mechanics book was far too discursive, full of unnecessary jokes and cutesy phrases that muddled the discussion instead of aiding understanding. I remember being frequently confused about when Griffiths was making a physical argument versus a mathematical argument when deriving results. My professor didn’t use Griffiths for the exercises, so I actually sold my copy back to the bookstore and bought a used copy of Shankar’s book: it is certainly drier, but I think it’s much more clear and precise.
- cionx 6y agoThere is also “Mathematical Writing” by Knuth: https://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematical_writing.pdf https://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematic...
- ptidhomme 6y agoWhat's not in the paper but I find quite annoying in scientific articles is excessively loaded notation. Some variables get to have like 4 or 5 subscripts and equations become unreadable. I really appreciate when papers get to say at some point "for the sake of clarity, let drop this or that notation" that are not that relevant to the key ideas. Usually it's in the good quality papers.
- supernova87a 6y agoYou mean you didn't find the triple-curly Greek xsi letter intuitive, even when subscripted with mu-nu, and superscripted double prime??
- davnn 6y agoMore of a rant, but since I began to implement algorithms from various papers for my PhD, my aversion to mathematical notation grows steadily. More often than not it's imprecise and hard to reproduce.
- wdkrnls 6y agoThis reminds me of: https://dspace.mit.edu/handle/1721.1/6707 https://dspace.mit.edu/handle/1721.1/6707 https://www.significancemagazine.com/2-uncategorised/593-cargo-cult-statistics-and-scientific-crisis https://www.significancemagazine.com/2-uncategorised/593-car...
- rahimiali 6y agoyour objection might be to the author of the notation, not the notation itself. or maybe to the review system that's supposed to catch this kind of imprecision. as with any language, it's possible to write nonsense or even self-conflicting statements in mathematical notation.
- davnn 6y agoWell, I believe the authors do their best to convey their ideas, but due to the fact that the notation is mostly informal it's easy to miss details. Also, different institutes might use their own special notation to describe the same ideas and even individual authors prefer some notation over another. All in all, imprecise probably was the wrong term. The statements are precise for the people working in this specific "bubble" of research, because they might know the implicit assumptions made and in the worst case, the only one who knows the implicit assumptions is the author.
- grayclhn 6y agoTBF, if key assumptions are left implicit, it’s not math.
- dorkwood 6y agoI also sometimes have to translate mathematical notation to algorithms. As a non-mathematician, math notation seems similar to naming all your variables 'a', 'b', 'c' etc. and then writing a key elsewhere that explains that 'b' stands for 'car deceleration rate'. This would be insane to do in code, why is it normal in mathematics? Note: I had this example in my head while writing https://www.traffic-simulation.de/info/IDMsstar.png https://www.traffic-simulation.de/info/IDMsstar.png