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I am studying Mathematics/Statistics at university, and realizing how important formal mathematical writing is—not just for assignments and papers, but as a general means of communicating ideas. Instead, I often have difficulty converting mathematical thoughts into polished, professional verbiage that is precise but accessible.

Professional mathematicians say a lot in very few words, but their proofs and explanations are clear. When I write proofs or try to explain concepts, they tend to sound long-winded, very informal, or, worse, sometimes clunky. I often use such expressions as "and then this happens" or "so it's obvious that," all of which aren't appropriate for formal writing.

These are my major inquiries:

  1. What are some formal elements of mathematical writing? (e.g., tone, structure, choice of words)
  2. Are there any resources (books, articles, guides) that may help improve mathematical writing skills?
  3. Which are the most common mistakes or informal habits that students make while writing mathematics?
  4. How can I make my writing clear but concise without oversimplifying or losing important detail?
  5. Are there any strategies for learning to think more formally while solving a problem? As I feel this might also add to my writing.

It's very underrated in university courses. Professors go a long way in showing you the quickest path to solving a problem, but rarely insist on showing you how to write in a polished, professional manner.

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    $\begingroup$ I recommend Velleman's book How to Prove It which helps you think about proofs in terms of standard outline templates for each type of proof (and also helps you think about what a standard looking template for a new type of proof would be). $\endgroup$
    – Alexander Woo
    Commented Dec 1 at 0:09
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    $\begingroup$ I don't have time or energy for a complete answer, but Steven Krantz's Primer of Mathematical Writing might be useful. $\endgroup$
    – Xander Henderson ♦
    Commented Dec 2 at 14:55

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Following on from my comment on @Sue VanHattum’s post, to provide more specific references to Solow’s points on reading condensed proofs.

I have the fifth edition of “How to read and do proofs” ISBN : 978-0-470-39216-4.

His key points in his book are:

  1. A proof is an exercise in communication between the writer and the reader(s).

  2. An important part of learning how to do proofs is learning how to read proofs.

  3. Often proofs presented are in "condensed” form where some detail of the working, and almost always the proof techniques, are omitted – the reader is supposed to fill in the details.

  4. When starting to learn to write proofs often more detail, rather than less, help the writer and the reader of the proof.

There are also very useful recordings of lectures given by Solow of the material in the books. I found the lectures on Youtube – best to search for them in case the URLs change. In the lectures Solow points out that proofs in books are often condensed – presumably to save on printing costs. In particular in my edition of his book on page 15 (may be different in other editions, of course) there are two relevant sub-sections (his capitals)

• Reasons Why Reading a Condensed Proof Is Challenging

• Steps for Reading a Condensed Proof

These may address the issues you are experiencing and reassure you that you are not alone in this.

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    $\begingroup$ +1 for this answer, but want to point out that proofs are not condensed to save on printing costs: they are condensed to convey information as efficiently as possible to the target (expert) audience. $\endgroup$
    – Steven Gubkin
    Commented Dec 2 at 17:14
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    $\begingroup$ @StevenGubkin - yep, agree. I was thinking of the (simple) example - not a proof but hopefully makes the point - say there is a quadratic equation with real coefficients and real roots, the explanation could go through step-by-step applying the formula to arrive at the answers or the author could just state the roots assuming the "expert reader" (for this level of problem) would know to, and know how to, apply the formula (or other technique). Similar idea for proofs, the author has to make some assumptions about the knowledge and skills of the intended reader. $\endgroup$
    – Clive Long
    Commented Dec 2 at 17:51
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I recommend proving simple things as practice, so that the mathematics is easy for you. For example, write your own proof that the square root of 2 is irrational, or prove that no Pythagorean triple will be odd-odd-even.

Find proofs that you like, and try to analyze what you like about them.

You could also include in your question a proof that you've written, so that folks here could help you improve one particular proof. (Professors don't teach "how to write in a polished, professional manner what you do" because the amount of one-on-one help needed would be too much.)

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    $\begingroup$ I think above is sound advice. In addition, the Solow book on "How to read and do proofs" makes the distinction between "long-winded" and concise proofs. I am working my way through the book and some videos accessible on youtube. My position on this is learn how to write correct proofs first so you understand what is a valid proof (it seems you are doing this) . Then as Sue recommends , study concise proofs and try to work out how they were constructed from 'longer" proofs. Solow provides advice and examples of this. $\endgroup$
    – Clive Long
    Commented Dec 1 at 14:23
  • $\begingroup$ @CliveLong, It might be a short answer, but this comment deserves to be in an answer. If you have the book with you, I wonder if you might give an example from the book. $\endgroup$
    – Sue VanHattum ♦
    Commented Dec 1 at 23:44
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Many of your issues here are general problems with technical writing, not math specific. Sure, there are some nuances of writing a math text versus a chemistry text. But this is a second order issue. The first order issue is general technical writing skill.

  1. Quickest practical thing would be to take an elective course in technical writing at your college. (It does not need to be math specific.)

  2. There are many texts on technical writing in general, and a few on math writing in particular. Buy a couple and at least skim them, expose yourself to them. You can surf Amazon and use the preview feature. Also, go to the library (the brick building) and peruse the shelves.

  3. For a quick initial/free section of advice, I am a huge fan of Katzoff's Clarity in Technical Reporting, which was like an underground bible at NASA Langley:

https://archive.blogs.harvard.edu/dlarochelle/2010/11/10/clarity-in-technical-reporting/

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    $\begingroup$ I don't want to steal Sue's point, but for sure "imitation and practice" is excellent advice. Aristotle said this a long time ago. And modern cognitive theory is showing more and more how fundamental this is to human learning. $\endgroup$
    – guest troll
    Commented Nov 30 at 22:46

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