On Writing Math

There are a lot of examples for skills that despite being greatly needed, we never get any formal training for. At least nothing is built into our core educational programs. Few examples are: how to read well, how to listen well, or how to develop your can-do mental attitude. Writing well, in particular math-writing, is another such example. Here I share few pointers from my own experience of reading and writing math.

When I was a PhD candidate my promotor changed the notation I used in my research from $()_i$ to $()_h$ . I was working with hourly time series data so the subscript $()_h$ made more sense. Two things happened at that moment. The first is that the paper became much, much more readable. The second is that I realized the math should be treated as part of the text, rather than “here is the text” and “here is the associated equations”. To all you promotors\supervisors out there, in case you wonder about the impact you make on your protégés; such change-of-notation comments can make a massive difference in their career.

Here are few “dos and don’ts” which are worth generalizing in my opinion, in no particular order:

If 95% of the time the abbreviation RMSE refers to Root Mean Squared Error then don’t use it as Relative Mean Squared Error (as is done here by otherwise superb writers). Something like RelMSE makes more sense.
If you need a running subscript, the lower case $k$ is more similar to the upper case $K$ than $g$ is to $G,$ so the latter is better when you need to make a distinction between lower and upper case.
Prefer words. You can write $f(X,Y,\Theta)$ which reads ‘a function of the input, output and parameters’ but better to write simply residuals (or squared residuals) if that is what you aim for. Another recent example is taken from here where the authors use $a \vee b=\max \{a, b\}$ as a shorthand notation for the maximum of two numbers. I wonder about the efficiency vs readability tradeoff of replacing $\max$ with $\vee$ .
The notation $\vert x \vert$ is mainly used to denote the absolute value of $x$ , but it’s also used to denote the cardinality of a set (or determinant, or a norm). Same notation for more than one meaning in the same paper is energy-taxing for the reader. Solve it by using something like $card(x)$ to denote cardinality of x.
There are enough Greek letters to go around; lose the extravagance by choosing familiar/friendly letters when you can. All else equal, $\gamma$ is a better choice than $\zeta$ and $\delta$ is a better choice than $\psi$ . Research that uses pompous notations for no reason is self-degrading readability, which is a shame.
If you use the intuitive notation $r_t$ as a financial returns time series, you can use the letter $R$ as the column-collected data matrix. I often see $R$ stands for the correlation matrix, which the letter $C$ would make better sense. The point is that while $X$ is a common notation for the data, we can afford a change based on the context.
In the same vein, since there is no global math-writing standard just yet, I try to follow the paper Notation in econometrics: a proposal for a standard which offers internally consistent framework for notation and abbreviations (and an associated .sty file for latex).