Writing mathematics literature for pedantics (and students)

Who this is for

This is meant as a collection of suggestions for anyone wanting to write a book on advanced mathematics, and wants to minimize the confusion of their readers and make their learning experience less stressful, less confusing and hence more effective from the perspective of the student.

Make new terms easy to find

A student needs to learn a lot of definitions and their terms. And a new term will not make any sense it is highly confusing to encounter a new term unless it is precisely defined, so please define a term before it is used the first time. Most will not remember each definition right away and will need to reread it, possibly several times, before it is internalized. So, make definitions easy to find on the page (i.e. use a definition box) and every time a term is defined, add it to the index. And there absolutely should be an index.

An example of a book that uses a new term (e.g. additive functor) before it is defined is

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Rotman, J. J. (2009). An Introduction to Homological Algebra. Springer New York. https://doi.org/10.1007/b98977

Enable navigation by numbering

Speaking of definitions, when numbering definitions, theorems, lemmas, examples etc, they should have the same counter. If one is looking for a specific box, it it much easier to orient oneself with respect to where what one is looking for is, if you can turn to the approximate location and if theorem/lemma/example 1.8 is on that page, you know that definition 1.3 is on an earlier page.

An example of a book that has separate counters is

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Fulton, W. (1998). Intersection Theory. Springer New York. https://doi.org/10.1007/978-1-4612-1700-8

Do not hide important stuff

Definitions and and even simple results of a certain generality and importance should not be be marked as an example or, for that matter, a remark. A student has a to prioritize their time, and the author should make this easier rather than penalizing a reader for skipping material that (appears as if it) is not needed for a full understanding.

An example is Gathmann’s otherwise very good lecture notes on algebraic geometry https://agag-gathmann.math.rptu.de/class/alggeom-2021/alggeom-2021.pdf. His (also, otherwise very good) notes on Commutative Algebra is also an example of this phenomena.

Don’t give unhelpful hints

It has happened more than once that the hint(s) given for an exercise has made the problem harder to solve, which (I assume) is not the intention, and is supremely frustrating.

An example of a book where this sometimes occurs is

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Bosch, S. (2022). Algebraic Geometry and Commutative Algebra. Springer London. https://doi.org/10.1007/978-1-4471-7523-0

Use cases for Git at university

Even if you are not developing developing software, the Free and Open Source Software (FOSS) version control system (VCS) Git, can be of much use for students. Features like branches and tags can make your life easier and save you time and grief.

For coursework

For each course you take, create a folder and initialize a Git repository. Afterwards, you can add all the files you use and keep histories and remote backups of all your files and homework. This works especially well for those of us who use the \LaTeX typesetting system or other markup based document systems. No more need to tell your professor that the dog ate your homework because your disk or computer crashed, or a coffee spilled on your keyboard.

Theses

Each year, I see posts by fellow students writing on message boards, asking how to recover files from thumb drives, desperate to recover their bachelor or master theses. Weeks or even days before the due date. No backup. Don’t be one of them! Use Git! Use one of the numerous free Git hosting services to back up your work af often as you want, for example after each writing session and write a short description of what you did in the commit message. Is your advisor asking for a major revision you may (or may not) agree with? Create a branch, and keep your original work while you work on your revision, and and work on both at the same time. Use tags to mark each draft version, and be able to restore an exact copy of each one.

But, but… a command line tool?

Well, yes. But firstly command line tools are not as scary as you might think. You might not agree, and there are plenty of free GUI tools to manage Git repositories, both open and closed source. I sometimes use SmartGit (https://www.syntevo.com/smartgit/) which is free for students. All major platforms are supported, Linux, MacOS and Windows, easily installed. For Windows and MacOS go to https://git-scm.com/. Linux users can either download the latest version from the mentioned website, or from their package manager of choice.