The n-Category Café

It’s sound stuff. The main thing I found to disagreee with was that he seems to work under the assumption that readers read from beginning to end. (He doesn’t say that he assumes this. I just got that impression.) This assumption is wonderfully simplifying when it comes to solving the problem of how to structure your book, paper, etc., but totally unrealistic.

SH: I’ve always admired Paul Halmos and have kept my copy of “Naive Set Theory” over the years. Most great writers give the advice of searching for the best and most precise word that maps to the context. I really liked your choice of topic. I was so surprised when I came across Halmos’ criticism of eigenvalue that I never forgot it.

Paul Halmos celebrating 50 years of mathematics By Paul Richard Halmos, John H. Ewing, Frederick W. Gehring

“Halmos has been involved in such [word] battles. The word _eigenvalue is an abomination made by lumping together an adjective from one language with a noun from another. Halmos fought against the word, using instead “proper value”, but in 1967 he finally acceded [2, p. x]”

For many years I have battled for proper values, and against the one and a half times translated German-English hybrid that is often used to refer to them. I have now become convinced that the war is over, and eigenvalues have won it … [and a comment on blending creativity with precision]

If the use of [connotation] is poetry, then I insist on being a poet when I write and I appreciate poetry when I read. A cleverly chosen word that means other things than the one it explicitly says – that suggests a whole aura, an ambiance, an atmosphere – that puts the reader in the right frame of mind to appreciate and understand the denotation – that’s a good thing. That’s style, that’s poetry if you like, that’s efficient communication.

SH: It is unfortunate that some emulate this advice in a tasteless manner, which is of course a subjective impression.

I realise that I should be more honest about the Qunitilian quote. I couldn’t find the actual Latin on the web, but could only find various English translations – the version above is my own interpretation of those translations. If anyone can find the original Latin and translate it then I would be very happy!

I wrote in response to a question about good mathematical writing the following:

One trick that my advisor, Ronnie Lee, advocated was to use a descriptive term before using the symbolic name for the object. Thus write, “the function $f$, the element $x$, the group $G$, or the subgroup $H$.” Most importantly, don’t expect that your reader has internalized the notation that you are using. If you introduced a symbol $\Theta_{i,j,k}(x,y,z)$ on page 2 and you don’t use it again until page 5, then remind them that the subscripts $i,j,k$ of the cocycle $\Theta$ indicate one thing while the arguments $x,y,z$ indicate another.

Another trick that is suggested by literature —- and can be deadly in technical writing —- is to try and find synonyms for the objects in question. A group might be a group for a while, or later it may be giving an action. In the latter case, the set of symmetries $G$ that act on the space $X$ is given by $\ldots$. Context is important.

Vary cadence. Long sentences that contain many ideas should have shorter declarative sentences interspersed. Read your papers out loud. Do they sound repetitive?

My last piece of advice is one I have been wanting to say for a long time. Don’t write your results up. Write your results down.

Let me expand upon the last paragraph. There should be a balance in mathematical writing between abstraction and exemplification. Sometimes a familiar example should be introduced before a definition (e.g. groups before categories). In this way, the reader can be convinced that she understands the material. Sometimes the concept is so broad (e.g. vector space) that it requires the abstract definition before a replete list of examples is given. The goal of mathematical writing is not to prove to the world how smart you are, but rather to tell the world about a clever idea that you have. You have to explain why it is clever, and if you do so well, your reader will be wondering why you are so excited about the idea. Your reader should believe that she could have thought of that, or she might believe that she did think of it.

One thing that I have noticed about many math texts —- Rudins and Jacobson BAI, and BAII come to mind especially —- is that when I first read them, they seemed opaque, but years later they seem crystal clear.

I don’t know if I have learned to write yet.

I am organizing a mathematical writing seminar this semester for grad students. I also enjoyed Halmos’s article. But I enjoyed even more the lecture notes from Knuth’s course on writing.

http://tex.loria.fr/typographie/mathwriting.pdf

I found my way back to this post thanks to the random-past-entries widget in the sidebar, and I noticed that the link has gone bad. This appears to be a working one now.