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October 19, 2009

Halmos on Writing Mathematics

Posted by Simon Willerton

Over in a discussion at Math Overflow I was reminded about Halmos’ great article on writing mathematics, which I highly recommend to all graduate students (or anyone else, for that matter).

This caught my eye because I am in the middle of teaching an undergraduate course on how to present mathematics. As I flipped through Halmos’ article I realised that I discuss just about everything in it in my course. I had forgotten that I’d actually read the article, but I think that it had a lasting effect on me. Undoubtably, much of what is said is also said in most of the books on mathematical writing, however Halmos says it all rather well.

I won’t quote any of the article as I don’t want to focus on any one bit – it’s all great – instead I will give you the 2,000 year old quote that I start my course with. It’s by Marcus Fabius Quintilian, a Roman rhetorician, and seems as relevant now as it would have been then.

We should not write so that it is possible for our readers to understand us, but so that it is impossible for them to misunderstand us.

De Institutione Oratoria, Book VIII, 2, 24 (ca. 95AD).

Posted at October 19, 2009 7:30 PM UTC

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Re: Halmos on Writing Mathematics

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.

Posted by: Tom Leinster on October 20, 2009 3:06 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Right. I always stress with students that they should not assume that their work will all be read in one sitting nor that it will be read linearly. So I say that they should put plenty of ‘signposts’ in, by which I think I mean pointers to where they have been and to where they are heading, so the wandering reader will have a sense of where they are. Needless to say that such signposts are a helpful thing for the linear reader as well.

Posted by: Simon Willerton on October 20, 2009 3:47 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

In other words, they should do all their writing on a wiki or blog :)

I’m only half joking of course. I think the primary medium for all science writing in 10 years will be (wiki/blog)-like in nature.

Posted by: Eric Forgy on October 20, 2009 4:01 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Partly because of its marvelous hyperlinking, wiki or blog entries are rarely in publishable form. Maybe the younger generation will be less demanding.

Posted by: jim stasheff on October 24, 2009 2:19 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

That depends on your definition of “publishable form”. In a real sense, they are already published once they are on a public wiki or blog. In 10 years, I suspect the need to take some material you’ve developed on a wiki or blog and write it up into a pdf and submit to a journal for either electronic or paper publication will be diminished. In 20 years, that need will not exist. Or so I think :)

It would be interesting to see what Halmos would think about wikis and blogs :)

What if Einstein had a blog?

PS: Halmos is by far my favorite author for his clarity.

Posted by: Eric Forgy on October 24, 2009 4:48 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

In a real sense, they are already published once they are on a public wiki or blog.

I presume that you’re referring to the desires of hiring and tenure committees to see publications in “journals” as “fake,” which is fair enough as far as it goes—there’s no a priori reason that something needs to be in a “journal” to be “published.” But I don’t think we can yet assert that being on a public wiki or a blog has all the “real” or “important” properties of a journal publication, not at least until we figure out how to make peer review work. We’ve tossed around ideas but none have been implemented yet.

Regarding wikis, I also see a need in academics for writings to eventually exist in a stable form which can be referred to. When I cite theorem 3.82 of a paper [3], I want theorem 3.82 refer to the same theorem in that paper, now and forever after! This is only possible with a wiki if you refer to a particular revision of a page, which is clunky. And actually, with the ability to move pages we have at the nlab, even that isn’t reliable.

I do agree that eventually, the form of “publication” will probably change to be more wiki- or blog-like, but I don’t think we’re there yet, and not just because hiring and tenure committees are behind the times.

Posted by: Mike Shulman on October 24, 2009 5:01 PM | Permalink | PGP Sig | Reply to this

Re: Halmos on Writing Mathematics

I agree with what you say, but just point out that I was talking about 10 and 20 years from now and didn’t mean to indicate we are anywhere close to being there today. The writing is on the wall though so I am pretty confident with my time lines.

Posted by: Eric Forgy on October 25, 2009 2:53 AM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Regarding wikis, I also see a need in academics for writings to eventually exist in a stable form which can be referred to. When I cite theorem 3.82 of a paper [3], I want theorem 3.82 refer to the same theorem in that paper, now and forever after! This is only possible with a wiki if you refer to a particular revision of a page, which is clunky. And actually, with the ability to move pages we have at the nlab, even that isn’t reliable.

Thank you very much for this clue which strongly confirm an idea I had about how to organize info tidbits (roughly content based adressing).

Posted by: J-L Delatre on October 25, 2009 7:21 AM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

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.

Posted by: Stephen Harris on October 20, 2009 7:04 PM | Permalink | Reply to this

Math/Poetry; Re: Halmos on Writing Mathematics

I agree that, to first order, “the use of [connotation] is poetry” – as opposed to the use of denotation, which is Mathematics. I’ll not that Poetry and Mathematics are the two densest form of widely published language (meaning per character). And, for a complex combination of first order and higher-order effects, the two have much more in common.

The first-order duality is also expressed as Cognative/Conative.

Posted by: Jonathan Vos Post on October 24, 2009 4:17 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

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!

Posted by: Simon Willerton on October 20, 2009 7:29 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

The first sentence of Institutio Oratoria, Book VIII, Chap. 2, Sec. 24 is

quare non, ut intelligere possit, sed, ne omnino possit non intelligere, curandum.

From the Perseus Library.

Posted by: Tim Silverman on October 20, 2009 7:45 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Which is something like, “Therefore, it must be ensured not that he can understand, but that he cannot in any way fail to understand.”

Posted by: Tim Silverman on October 20, 2009 7:55 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Or, to be more explicit still, as I understand it with my somewhat meagre Latin,

quare		non,	ut	intelligere	possit,
therefore	not,	that	understand	he-can,
sed,	ne	omnino	possit	non	intelligere,	curandum.
but,	not	at-all	he-can	not	understand,	is-to-be-taken-care-of
Posted by: Tim Silverman on October 20, 2009 8:08 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Splendid. I think I owe you a coffee or a beer or something of that nature.

Posted by: Simon Willerton on October 20, 2009 8:17 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Thanks. Should we ever chance to be in the same place, I may take you up on that …

Posted by: Tim Silverman on October 20, 2009 8:50 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

quare non, ut intelligere possit, sed, ne omnino possit non intelligere, curandum.

If Quintilian thought being understandable was so darn important, why didn’t he say it in plain English?

Posted by: John Baez on October 20, 2009 8:26 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

de mortuis nil nisi bonum! :-)

Posted by: Tim Silverman on October 20, 2009 8:40 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Same to you too, buster!

Posted by: John Baez on October 20, 2009 8:52 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

There are some typos, but it looks like Harold Edgeworth Butler renders this a little more idiomatically as:

Therefore our aim must be not to put him in a position to understand our argument, but to force him to understand it.

Context:

For we must never forget that the attention of the judge is not always so keen that he will dispel obscurities without assistance, and bring the light of his intelligence to bear on the dark places of our speech. On the contrary, [he] will have many other thoughts to distract him unless what we say is so clear that our words will thrust themselves into his mind even when he is not giving us his attention, just as the sunlight forces itself upon the eyes. [24] Therefore our aim must be not to put him in a position to understand our argument, but to force him to understand it.

Posted by: Jon Awbrey on October 20, 2009 8:36 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

On some reflection, I can see how Dialecticians might prefer the double negation translation to the more dynamic idiom of Butler, relating as it does how watching one’s pp’s and qq’s can lead to their exchange, but I can also see how Intuitionistas might demur at the very idea that ¬¬understanding=understanding\not\not understanding = understanding.

Posted by: Jon Awbrey on October 22, 2009 1:08 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

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 ff, the element xx, the group GG, or the subgroup HH.” Most importantly, don’t expect that your reader has internalized the notation that you are using. If you introduced a symbol Θ i,j,k(x,y,z)\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,ki,j,k of the cocycle Θ\Theta indicate one thing while the arguments x,y,zx,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 GG that act on the space XX 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.

Posted by: Scott Carter on October 21, 2009 12:58 AM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Excellent advice, Scott. I think that learning to write mathematics (or other types of writing for that matter) is a life long process, much like learning mathematics itself is a life long process. You’re never done.

Posted by: Richard on October 21, 2009 3:57 AM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Scott wrote:

One trick that my advisor, Ronnie Lee, advocated was to use a descriptive term before using the symbolic name for the object.

For me:

One trick that my not official advisor, Jack Milnor, advocated was to
avoid if all possible ending a sentence or phrase with the symbolic name for an object then starting the next with another symbolic name.

Posted by: jim stasheff on October 22, 2009 2:23 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

Since “Characteristic Classes” by Milnor and Stasheff is among one of the greatest math books ever, that advice should be taken as gospel. Certainly, beginning a sentence with a symbol is bad-form in the extreme.

Posted by: Scott Carter on October 22, 2009 3:27 PM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

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

Posted by: Jason Starr on October 21, 2009 1:44 AM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

The Knuth notes look good, although I guess I wouldn’t get my students to read all the way through.

Posted by: Simon Willerton on October 21, 2009 8:32 AM | Permalink | Reply to this

Re: Halmos on Writing Mathematics

I don’t think the Knuth notes are really meant to be read all the way through. They’re basically the notes from the actual course (and I think they’re notes taken by members of the audience, not the notes that the people in the front of the room were using!)

Posted by: Michael Lugo on October 24, 2009 4:40 PM | Permalink | Reply to this

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