Well-written math

February 28, 2014 7:03 AM

Which mathematicians write exceptionally well or exceptionally clearly? (Individual books or articles would also be acceptable.)

As I contemplate assembling the bits of my dissertation, I realise that my writing leaves something to be desired. It's like if Kafka wrote math with no exposition between propositions. Kafka's great and all, but that many clauses doesn't go so well for me. But I find myself not knowing who to look to for clear exposition. I've heard Atiyah's name mentioned for this once or twice, but who else? Or are there exceptionally well-written books or articles I could look at?

Note: Simon Singh is not an answer to this question. 'Popular math' is not what I'm looking for.

posted by hoyland to Science & Nature (19 answers total)

A similar question has been asked on MathOverflow.
posted by madcaptenor at 7:12 AM on February 28, 2014 [1 favorite]

The prerequisites are your own command of the material and organization. And, I guess, motivation. What is the example that caused your line of research to be interesting?
posted by SemiSalt at 7:34 AM on February 28, 2014

Martin Gardner is legendary for being a superb writer about math.
posted by Chocolate Pickle at 7:43 AM on February 28, 2014

Barry Mazur?
posted by HoraceH at 8:00 AM on February 28, 2014

Terrence Tao.

Both his actual papers and his blog posts are pretty readable.
posted by empath at 8:02 AM on February 28, 2014

I'm constantly surprised by how clear Gerhard Gentzen manages to be.

And finally, you said not 'popular math', but what about purely expository work? I found Hans Hahn's "Infinity" in The World of Mathematics Vol. 3 to be incredibly lucid compared to other treatments of the same subject. In fact, it was this article that made me realize Wikipedia's treatment of topics might forever be inferior to the sole works of talented individuals.

I'd probably be a lot more helpful in over a year. I'm working through a number of works now and would have a larger sample to pull from later. Had I the liberty, I'd look at your field's predecessors: Hilbert, Dedekind, Weyl, Poincare, Weierstrass, etc. and see if anything stands out as particularly worthy of emulation.
posted by SollosQ at 8:02 AM on February 28, 2014

I always vote for J-P. Serre as a model of good mathematical writing. His papers and books are very readable and extremely well-structured. He also gave a hilarious and informative talk on How to Write Mathematics Badly, which is a very good lecture on what not to do when writing math.
posted by Frobenius Twist at 8:04 AM on February 28, 2014 [1 favorite]

Oh, and as examples of how not to write mathematics in a clear and easy-to-follow way (since such things are also useful), I nominate anything by Bourbaki or EGA/SGA. That is not to say they aren't worth reading - I've read large chunks of Bourbaki, EGA, and SGA, and have found them immensely useful; they are the standard references for some very important results - but they are quite difficult to read since they're organized as huge info dumps with no clear overarching structure. (Ironically, Serre, a master expositor, was a member of Bourbaki.)
posted by Frobenius Twist at 8:14 AM on February 28, 2014

I like Sheldon Axler's Linear Algebra Done Right a lot. Right now I can't remember if the actual prose is exceptionally good, but the whole structure and pedagogy of it is very careful and well-crafted.

Donald Knuth's writing is exceptional, too.
posted by mbrock at 8:15 AM on February 28, 2014 [1 favorite]

Atiyah is a good choice. I came in to say Serre but Frobenius Twist beat me. So yeah, Serre. Now Serre's style is very crystalline and terse, yet very very clear -- not a word wasted. His books A Course in Arithmetic and Linear Representations of Finite Groups are terrific models, or the Duke paper if you want something more along the lines of a research paper. (Looking it up, I see I'd forgotten it's in French, which just goes to show you how clear it is!) From what you say, though, I get the impression that actually you are looking for a somewhat more explanatory style? I also agree with the suggestions of Terry Tao's blog -- sometimes the blog is "popular," but more often than not it's written at the level ranging between "graduate textbook" and "research monograph." He really excels speaking directly about the motion of ideas through an argument without resorting to vague handwaving.

But I should mention someone who doesn't appear elsewhere, so: I think the graduate textbooks by Joe Silverman on elliptic curves, by David Eisenbud on commutative algebra, and by Benson Farb and Dan Margalit on the mapping class group are excellent examples of mathematical writing that conveys deep ideas swiftly and precisely but which are a pleasure to read.
posted by escabeche at 8:45 AM on February 28, 2014 [1 favorite]

Absolutely - seconding Eisenbud's Commutative Algebra book, which in fact I'm currently reading because I want a more solid foundation in the subject. It says something about the readability of this book that you can just sit down and read it straight through without difficulty. It's a model of how to balance heuristic big-picture points of view with details and clearly written proofs. It even has lots of nice pictures!
posted by Frobenius Twist at 9:09 AM on February 28, 2014

Response by poster: From what you say, though, I get the impression that actually you are looking for a somewhat more explanatory style?

I think so. Not as a matter of principle, but because I don't think I can pull terse off successfully. My terse is just horribly confusing.

When I was thinking about this earlier, I was debating whether Eisenbud's Commutative Algebra was an answer. So maybe I should go read a bit of it.
posted by hoyland at 9:24 AM on February 28, 2014

Korners Fourier Analysis book is a nice example of clear mathematical exposition although he's maybe a bit too discursive for what you want.
posted by crocomancer at 10:14 AM on February 28, 2014

Prime Obsession and Unknown Quantity by John Derbyshire
posted by IndigoJones at 10:32 AM on February 28, 2014

I, too, like Sheldon Axler's linear algebra book. I've been reading the late Bill Thurston's notes on the geometry and topology of 3-manifolds, and they're very good explanations written with a lot of verve; I think the first few chapters are expanded in his book which may have the feel you're going for. Which field are you writing in?
posted by bluefly at 12:01 PM on February 28, 2014

I was blown away by Eric Temple Bell's "Mathematics: Queen and Servant of the Sciences."
posted by RichardS at 12:51 PM on February 28, 2014

Response by poster: bluefly: combinatorics, with a heavy dose of algebra
posted by hoyland at 3:38 PM on February 28, 2014

One of the best-written and clearest mathematics texts I've seen is Tristan Needham's "Visual Complex Analysis".
posted by logopetria at 9:57 AM on March 13, 2014

Also, John Baez has been writing and blogging on a huge variety of mathematical topics for a long time, and has a very friendly easy-going style that's a good model of how to write colloquially while maintaining the right amount of rigour. For example, here's one of his posts on combinatorics and generating functions, and the "Seven Trees in One" isomorphism.
posted by logopetria at 10:07 AM on March 13, 2014