# Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow.

How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to name as many examples of "great writing" as possible. Asking for "the best article you've read" isn't reasonable or helpful. Instead, ask yourself the question "what is a great article?", and implicitly, "what makes it great?"

If you think of a piece of mathematical writing you think is "great", check if it's already on the list. If it is, vote it up. If not, add it, with an explanation of why you think it's great. This question is "Community Wiki", which means that the question (and all answers) generate no reputation for the person who posted it. It also means that once you have 100 reputation, you can edit the posts (e.g. add a blurb that doesn't fit in a comment about why a piece of writing is great). Remember that each answer should be about a single piece of "great writing", and please restrict yourself to posting one answer per day.

I refuse to give criteria for greatness; that's your job. But please don't propose writing that has a major flaw unless it is outweighed by some other truly outstanding qualities. In particular, "great writing" is not the same as "proof of a great theorem". You are not allowed to recommend anything by yourself, because you're such a great writer that it just wouldn't be fair.

Not acceptable reasons:

• This paper is really very good.
• This book is the only book covering this material in a reasonable way.
• This is the best article on this subject.

Acceptable reasons:

• This paper changed my life.
• This book inspired me to become a topologist. (Ideally in this case it should be a book in topology, not in real analysis...)
• Anyone in my field who hasn't read this paper has led an impoverished existence.
• I wish someone had told me about this paper when I was younger.
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A useful feature of Math Overflow on a post like this one is the ability to sort answers chronologically as well as by number of votes. Just click the "newest" tab above the answers to see the most recent additions. –  Anton Geraschenko Oct 19 '09 at 6:39
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Anything by John Milnor fits the bill. In particular, "Topology from the differential viewpoint" made me feel that I understand what differential topology is about, and the "h-cobordism theorem" made me feel that it's beautiful. Many other books and papers by him are wonderful; the first that come to mind are "Characteristic Classes", "Morse Theory", lots of things in Volume 3 of his collected papers.

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"Characteristic Classes" is particularly great indeed, I do love most of Milnor's books. –  Sam Derbyshire Oct 18 '09 at 4:46
I read Milnor's "Dynamics in One Complex Variable" as a graduate student -- it was wonderful. –  Sam Nead Nov 15 '09 at 1:59
This should be so much higher than Serre :-). One of the differences between the two is that writings of Milnor can be appreciated almost by a layman, whereas apparently Serre needs a reasonably educated mathematician to appreciate. –  Łukasz Grabowski Jan 9 '11 at 1:19
Milnor's writing is masterful. Morse Theory for example is a fantastic book -- the writing is smooth and clear, and the proofs are remarkably detailed and complete. A most satisfying read. –  Todd Trimble May 15 '11 at 11:31

Canonical submission: Anything by Serre (e.g., Local Fields, Trees, Algebraic Groups and Class Fields,...). Reasons:

• I can't get enough of Trees, chapter 2. I spent a year working on automorphic forms on function fields in part because of this book (it didn't work out well, but that's another story).
• Peer pressure: several people (including my Ph.D. advisor) have told me that if I were to choose a role model for writing style, I should choose him.
• Mundane reasons: His writing is incredibly clear and concise, but not so brief as to be confusing. He has a keen eye for what is important in a theory or construction. He doesn't waste words having a conversation with the reader or expounding on his philosophy of mathematical practice.
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Although my French is fairly weak, I was able to read and immensely enjoy Serre's "Cours d'Arithmetique". Clear, motivated, engaging and simply memorable. "Trees" is incredible. –  Alon Amit Oct 12 '09 at 18:45
There is a video of a talk he gave entitled something like "How to write bad mathematics" or rather write mathematics poorly, it is a fun talk to watch. –  Sean Tilson Mar 29 '10 at 5:46
I haven't read Serre in years, but I remember that as an undergraduate his writing style was much too difficult, more precisely in my eyes your point 3 was not true. I specifically mean Trees, and more specifically definition of a graph. –  Łukasz Grabowski Jan 9 '11 at 1:17
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The book of Bott-Tu "Differential Forms in Algebraic Topology" was my door to enter the magic world of cohomology, Chern classes and similar topics. Moreover, it contains a wonderful (and in my opinion the best) exposition of spectral sequences with applications to the computation of some higher homotopy groups of the sphere. All that is presented in a self-contained way and in a magnificent style. A masterpiece!

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One of the math books I enjoy reading in most is Neukirch's book "Algebraic Number Theory". In my opinion, he presents the material beautifully and with a good degree of generality for a text book. Also, he manages to use language beautifully without losing mathematical rigor and without compromising clarity (this holds for the German version as well as for the English translation). When I have to look up some fact from algebraic number theory, Neukirch is usually the first book I try.

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Michael Spivak's Calculus made me want to study analysis.

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Walter Rudin's Real and Complex Analysis has long been one of my favorites. Like Serre, Rudin seems to strike a nice balance for detail, and his proofs are always slick and fun to read; I became heavily interested in analysis after reading that one.

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Rudin's writing may be good and clear, but I'm not sure of his approach. Rudin's Book on Functional analysis is not what I would recommend a beginner. When I was an undergraduate, I self-thought myself functional analysis and this was the first book that I read. I must admit it was a torture for me back then, I wouldn't teach functional analysis starting with the abstract topological vector spaces. –  Jose Capco Nov 10 '09 at 22:30
Although these comments are months old, I'd like to chime in anyways and say that a lot of the criticisms people have about Rudin's choice of topics sort of disappear when you take his books together as a 3 volume course. With that in mind, his choice for chapter 2 of his R&C Analysis, which presents the Riesz theorem as the way to construct measures, makes sense, because in his undergrad book he already spent a chapter building lebesgue measure on R^1. Likewise, his choices in Functional Analysis are justified by the chapters on Banach, Hilbert space, & Banach algebras in his R&C Analysis. –  Erik Davis May 3 '10 at 21:56
What about people who don't like his first book to start with? :-) –  Andrea Ferretti Aug 11 '10 at 12:13
I strongly disagrees with this recommendation. –  Kerry Aug 24 '10 at 3:42
Ever since Lakatos's Proofs and Refutations, Rudin's expository style has been something of a whipping boy for certain people. I'm not sure how much of it is deserved. You gotta give him some credit: his proofs are often elegant. –  Todd Trimble May 15 '11 at 11:39
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I love Grothendieck's Tohoku paper.

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Real Analysis by Elias Stein and Rami Shakarchi

I absolutely hated analysis until I read the Stein/Shakarchi analysis series (Fourier, Complex, Real Analysis). Now I find the subject to be very beautiful and full of deep ideas, and it is these books that really convinced me.

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Agreed. These books are very clearly written and motivate the subject well. –  Qiaochu Yuan Oct 15 '09 at 18:59
Thirded. This series immediately came to mind when I saw this question. –  Darsh Ranjan Oct 21 '09 at 2:08
Agreed further; in my opinion all three books of the series are "great mathematical writing". –  Pietro KC Aug 23 '10 at 15:30
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I think Algebraic Topology by Hatcher is one of my early favourites. It starts off being very basic but it manages to mention so much fascinating stuff, and I think the exposition is great. Definitely inspired me and got me interested in algebraic topology.

His book in progress "Vector Bundles and Characteristic Classes" is also very nice.

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Éléments de géométrie algébrique (EGA) (full text available from numdam) continues to be an inspirational text for me. I wish I'd started reading it earlier.

It's come up in a few other places here on MO. To quote Jonathan Wise's answer to another question,

Virtually every page I've read of EGA/SGA has been useful to me, and almost every page I've skimmed I've later wished I'd read in more detail. The reputation for difficulty is, I think, unfounded. They are certainly abstract, but virtually every detail is present; in many ways, that makes EGA/SGA easier to read than other sources. Opening a volume and reading a sub-paragraph from the middle can be difficult because of all the back-references, but reading linearly can be very pleasant and rewarding. The French language may be a barrier for some, but one doesn't have to "learn French" to learn enough to understand EGA.

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I'm not sure whether "expository" writing counts, but I'll go with it anyway...

I don't think it would necessarily change the life of anybody who was already into mathematics enough to pay for it, but I very much wish the Princeton Companion had come out when I was younger. You don't get the chance to get your hands dirty with the details of any of the topics the PCM covers, but sometimes you're not looking to get your hands dirty, and there's not much else of any quality that can compare in terms of breadth.

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John Lee's Introduction to Smooth Manifolds. This book reminded me of all the mathematics I kinda learned in undergrad, prepared me for graduate school, and taught me differential topology. I feel like every undergrad should have this book and work through it on their own.

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Not just undergraduates ... and not just mathematicians ... can enjoy reading Lee's fine text. –  John Sidles May 20 '11 at 19:12

Silverman and Tate's "Rational Points on Eliptic Curves."

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Actually someone did tell me about this when I was younger. I read it after senior year of highschool and greatly enjoyed it. –  Noah Snyder Oct 16 '09 at 2:22
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Silverman's The Arithmetic of Elliptic Curves got me interested in that area for some time, too. The exposition is fun to read, with both motivation and rigorous proofs.

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No, I meant The Arithmetic of Elliptic Curves. I've also looked at the book Noah mentioned, but I found this one more enjoyable. –  Akhil Mathew Oct 16 '09 at 11:03
Yes, this book is great. I like "Advanced Topics In The Arithmetic Of Elliptic Curves" even more, I love the description of complex multiplication it has, especially with regards to Kronecker's Jugendtraum and class field theory. –  Sam Derbyshire Oct 18 '09 at 4:45
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Atiyah & Bott's paper "The Yang-Mills Equations on Riemann Surfaces" is probably the most satisfying thing I've read. The writing is great, and the ideas are all cool.

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Just like in this thread, I am amazed that no one mentions Deligne. I think it was Illusie who said Grothendieck had a gift to build new theories and new langage while Serre's talent was to find new things to do with old tools. Deligne got the generality, abstraction and theory building from Grothendieck and the clarity of exposition and the constant reference to older langage/simple ideas from Serre. I think that's why he is sometimes overshadowed by his elders when someone asks this kind of question.

Here's a few examples. His "Théorie de Hodge I" explains the "yoga of weights" in just a few pages. The first sections of "La conjecture de Weil I" provide a great survey of both the theory of etale cohomology and Lefschetz theory for algebraic varieties almost from scratch. Another masterpiece is his "Le groupe fondamental de la droite moins trois points" where he builds a whole theory unifying several aspects of arithmetics, topology and differential equations but always comes back to very down to earth examples. Not to mention, his Bourbaki lectures or the uncountable number of private communications of his cited in the litterature.

If you are looking for great examples of mathematical writing, you should definitly read some articles by Deligne.

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"Algebraic curves and Riemann surfaces" by Rick Miranda is one of my favorite books. It is full of concrete examples and is full of very clear explanations for everything from the basics of Riemann surfaces and their projective embeddings though sheaf cohomology. Also, it assumes little more than elementary complex analysis.

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Proofs from the Book, Martin Aigner, Günter M. Ziegler, 2000.

Anyone in [mathematics] who hasn't read this [book] has led an impoverished existence.

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I strongly disagree. I've never found the style of proofs in it particularly beautiful--they're all just little combinatorial tricks. –  Eric Wofsey Oct 12 '09 at 22:25
It was a big delusion for me as well. If there is a Book, I seriously doubt it contains any of these proofs. –  Andrea Ferretti Aug 11 '10 at 12:14
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Dror's paper Khovanov's homology for tangles and cobordisms is one of the papers I loved back when I hated all math papers. In particular it's a paper that has a really good use of diagrams, a lot of papers use too few diagrams and suffer a lot for it.

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Bar-Natan's papers in general tend to be well-written. –  Jim Conant May 15 '11 at 20:14

This is not an example of great mathematical writing, but definitely something to know and probably even more useful: Paul Halmos' brilliant essay "How to write mathematics", which you'll find at http://retro.seals.ch/digbib/view?rid=ensmat-001:1970:16::59&id=browse&id2=browse2.

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Mumford's "The Red Book of Schemes and Varieties" was the first book trying to explain Grothendieck's new theory of schemes to the large public. It does this with a lot of examples from the 'real life' and even with drawings! It is far from complete, but it remains the best for communicating the love for the subject and for the clearness of the exposition. Another masterpiece!

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I'm surprised that nobody's mentioned almost anything by Emil Artin. His little monographs on Galois Theory and the Gamma Function are thrilling to read. They are so clear, and use the minimum necessary (but not more -- to paraphrase Einstein) I found them inspiring. Also his "Algebraic Numbers and Algebraic Functions" and "Geometric Algebra".

Another book, is G. H. Hardy's "Pure Mathematics". That's the book that I really learned analysis from (when I told that to Pat Gallagher he exclaimed that I was really lucky) when I was in high school. Reading that cemented my feeling that I wanted to be a mathematician.

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Gian-Carlo Rota's On the foundations of combinatorial theory I: Theory of Möbius Functions is an eye-opening gem. The same is true of practically every paper in Gian-Carlo Rota on Combinatorics, so consider this post a vote for the entire book. (If I become a combinatorialist, it will be because of this book.)

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In my opinion Atiyah's paper "vector bundles over an elliptic curve" is a gem that everyone interested in algebraic geometry should read.

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Every differential geometer should read at least the first two volumes of Spivak's A Comprehensive Introduction to Differential Geometry. In particular, volume 2 is an absolute gem. Not only does it reprint (translations of) original papers by Gauss and Riemann, complete with very enlightening notes and commentary, but (if I remember correctly) Spivak presents about 5 or 6 different proofs that a Riemannian manifold is flat if and only if it is locally isometric to Euclidean space. This gives the reader the best, most intuitive grasp of the concept of curvature that I have seen anywhere. (I am a firm believer in learning by repetition...)

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Well, for me Hartshorne was really the window into the brave new world — and yes, it fits several items from 'acceptable reasons'.

Though this prize should be shared with everyone else who was creating abstract algebraic geometry and scheme theory in the past century or so (I spare you the history, you already know it :) )

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Van der Waerden's Algebra. I became a mathematician because of this book.

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Some of my favourite authors are Serre, Mumford, Milnor, Fulton and Neukirch. There names are rather mainstream, I guess. But the most beautiful analysis textbook I ever read must be "Analysis now", by Gert Pedersen. Alas, the book is not as well known as it should be.

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Gowers' Mathematics: A Very Short Introduction. This is in Oxford's series of "very short introductions" on a variety of topics (hieroglyphics, film, Rousseau,...), each of which I think is a tremendous challenge to the (invariably eminent) writer. Gowers dispenses with "anecdotes, cartoons, exclamation marks, jokey chapter titles," and instead plunges right into details without apology. The chapter on proofs is especially important for nonmathematicians to understand. The explanations of concepts (e.g., "dimension") are lucid, achieving clarity without compromising on technical accuracy. I read it in one sitting, which may dismay the author who must have labored over these small 160 pages, but which is a testimony of how smoothly he conveys his insights.

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Also, perhaps, "Mathematics: A Very Long Introduction". (This is what I call The Princeton Companion to Mathematics, of which Gowers was editor.) –  Michael Lugo Aug 23 '10 at 17:19
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Lazarsfeld's book Positivity in Algebraic Geometry seems to fit the category Anyone in my field who hasn't read this paper has led an impoverished existence.''

I agree with Scott on `anything Serre.' His FAC and GAGA are gems; they will change your life.

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