# How are mathematicians taught to write with such an expository style?

I wasn't sure if this question was appropriate for MSE.

One of the major complaints we see in industry is a person's ability to communicate which includes writing.

We see the same thing on questions that are posted on MSE by younger students.

However, the wonderful and skilled mathematicians on MSE both pose and answer questions with beautiful skill, directness, economy of words and all the rest. The grammar, flow and points are very clear and it is just beautiful expository writing.

In school, we were forced to take a class on technical writing, but this is not really the same thing. Even though I have a couple of different degrees, including mathematics, we were never required to take such a course.

My question is where and how do mathematicians learn this skill? Do some degree programs require a class on this or is this just learned by the school of hard knocks? I have even searched for books on the matter, but to no avail. Any insights would be appreciated.

-
My question is where and how do mathematicians learn this skill? Likely, it is similar to how great writers developed their writing ability: Constant practice, along with reading the work of great expositors. –  Eric Naslund Dec 28 '12 at 3:39
The standard way to learn how to write well is to read things that are well-written. –  KCd Dec 28 '12 at 3:47
Personally, I find a lot of mathematical writing to be pompous, pedantic, and opaque. The formalisms and abstractions are often poorly motivated, and are disconnected from the concrete examples from which they were (secretly) derived. The fixation with precision and possible exceptions often obscures the central ideas. But, I think this is a matter of taste. Some people think the abstractions and the nit-picky precision are "beautiful". Fair enough. –  bubba Feb 24 '13 at 11:06
Yes, indeed. And it is seldom "explicitly" taught, but usually "acquired"... –  amWhy Apr 18 '13 at 1:15
@JMCF125: That is an excellent observation and I totally agree! Regards –  Amzoti Jun 14 '13 at 15:28

## migrated from meta.math.stackexchange.comDec 28 '12 at 3:36

This question came from our discussion, support, and feature requests site for people studying math at any level and professionals in related fields.

Let me begin with the disclaimer that by answering this question I do not presume to behold my own writing as a model of the qualities that you admire. I'm answering because I've noticed my comfort with and talent for writing improve markedly as I've matured as a mathematician, and I've often wondered if there is a causal relationship. I believe that for a variety of reasons math and writing are co-productive skills.

First and foremost, learning and doing mathematics invariably requires one to read and write a great deal, and the reading and writing involved can be extraordinarily taxing. As with any talent there is no substitution for persistent, deliberate practice.

Second, good mathematicians and good writers must both develop the ability to use language in a very precise way. Every area of mathematics has its own vocabulary and grammar which one must master in order to become an expert, and indeed many of the great breakthroughs in mathematics are really linguistic revolutions (such as the advent of the formal definition of a limit in calculus). Because of this I've noticed that the writing mistakes which offend me the most are based on imprecise use of words; a common example is when students ask how to "prove a problem".

Third, every serious math student encounters at some point in life the stark contrast between really good and really bad exposition. The most beautiful ideas can feel like tedious nonsense in the hands of a bad writer, and even the most mundane details can come to life in the hands of a good one (of course, a huge part of good mathematical exposition is based on selection of detail). I know that this makes me self-conscious about my own writing, and I wonder if others feel the same way.

These three observations suggest ways in which mathematicians may naturally develop good writing skills, but it's worth noting that the mathematical community as a whole deliberately tries to nurture good writing as well. I never participated in any formal writing workshops, but several of my mentors over the years - particularly my PhD adviser - spent a great deal of time and energy helping me improve my writing.

-
I would add that in some cases it may be that cause and effect are reversed. By which I mean that it might be that some people learn how to write and express themselves effectively and that becomes a personal asset that leads them to success in mathematics. –  Todd Wilcox Dec 28 '12 at 23:03

I think this is a skill that is developed not only through wanting to show knowledge of maths but through wanting to explain things to the reader. Also I believe you have to have the right motivation for writing in maths. I come across many books which contain great material but written in a way I cannot cope with, whereas others think the books that I read are not very good for their style of learning. Either way most authors in maths communicate the same basic rules...be logical, rigorous and precise. The readers then pick this style up and not only copy it but learn to appreciate it.

To answer your question the more you understand something the better you will be at explaining it. This is the reason why you observe questions that are written "sloppily" but answers that are written extremely well. It is likely that the person asking does not understand the question fully and so the way the question is written reflects this (similarly with the answers, they are written by people that understand the question and the answer and so their replies will reflect this too).

-

A course on writing proofs? Yes, but on a really basic level. E.g. Daniel Solow, How to Read and Do Proofs. A student would see such a course when first starting to write proofs.

Nowadays, Ph.D. theses in many places are written in English, and advanced instruction is in English, even if that is not the usual language outside the classroom. Probably the professors work with students in these settings to improve their writing skills. Similarly, in the US there are Ph.D. students from China, India, Bulgaria, Zambia, etc., etc., and the American professors work with them to improve their English writing style for mathematics.

Advice for a student from China studying in the US ... get a professor who is NOT from China!

-
I like last one,it is true. :) –  chenbai May 8 '13 at 6:59

In my experience students go through a number of identifiable stages as they learn to write mathematics, and the end result is the excellence Amzoti refers to.

When students are first exposed to formal mathematical notation, there is a steep learning curve, and it takes a while to be able to read and write it fluently.

That barrier overcome, students -- even or perhaps especially the best ones -- fall in love with the power of that notation, and sprinkle $\forall$s and $\exists$s and $\epsilon$s and $\delta$s throughout their writing (I'm obviously thinking of first formal analysis classes here, which tend to occur at the stage I'm describing, but feel free to substitute your field of choice). And of course there are very real reasons for learning to manipulate formal logic in this way -- after all, modern mathematics is built on this sort of rigour.

But ultimately the point of doing mathematics is to explain what you have done to other mathematicians (narrow margins won't get most of us very far). And once you've been doing that for a while, you realise that natural language has features that make it a very good explanatory or expository tool. Terence Tao has a wonderful blog post on the many different ways in which a simple mathematical statement can be rendered into words.

Thus, just as learning a foreign language can increase your appreciation for the grammatical complexities of your own, so I believe good mathematicians write well because they have mastered formal mathematical notation, and as a result are better able to appreciate the nuances of natural language and how to wield it in efficient, effective, and expressive ways.

-