# 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.
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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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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

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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I've always really enjoyed the papers by Graeme Segal. They are short and I often feel like they have been distilled down into the essence of what is important. I keep going back to them and extracting new nuggets of beauty.

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Can you give a concrete example that you would say is quintessential Graeme Segal? –  Anton Geraschenko Oct 12 '09 at 23:28

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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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

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

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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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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

Another paper that I really like, because it makes a lot of things that are muddled in the literature very clear, is Sawin's Quantum Groups at Roots of Unity and Modularity. In particular the lesson to learn is that if it takes you a while to sort through which papers use which conventions or to find all the relevant constants attached to Lie algebras, then you really owe it to your readers to put that information in your paper where it's easy to find (preferably in a table).

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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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The survey paper Hilbert's Tenth Problem over rings of number-theoretic interest by Bjorn Poonen (my advisor) is one of my favorites. He has several survey papers on his web page. My first year of grad school I read many of these and decided I wanted to work with him, so in a sense this paper did change my life'.

In general Bjorn's writing is extremely clear and I have tried to model my own writing on his.

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John Hubbard's book on Teichmuller theory is clear, beautiful, inspiring, and (amazingly) essentially self-contained. He has a fantastic ability to take very technical and difficult results and make them seem clear and natural.

Bill Thurston wrote a preface for it which can be read here :

http://matrixeditions.com/Thurstonforeword.html

The money quote : "I only wish that I had had access to a source of this caliber much earlier in my career."

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Actually a book that ended up changing my life: Kaczynski, Mischaikow, Mrozek: Computational Homology

I read it while working on my master's thesis in computational homological algebra, in order to see what they had to say about efficient implementations of simplicial homology.

After reading it, I first realized that algebraic topology has applications far outside what I had seen thus far - and now, a doctorate later, I'm active in the field of Applied Algebraic Topology and Topological Data Analysis.

I'm not certain I'd peg it for great writing as such, but the criteria above did state "changed my life".

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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

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

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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I love Grothendieck's Tohoku paper.

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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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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

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

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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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

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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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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This paper changed my life:

Vaughan Jones, "Hecke algebra representations of braid groups and link polynomials"

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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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books I enjoyed very much: Mumford "Lectures on Curves on an Algebraic Surface"; Mumford "Curves and their Jacobians"; Mumford "Basics of Torus Embeddings; Examples of the Theory": Chapter 1 in "Smooth Compactification of Locally Symmetric Varieties"; Koblitz "Introduction to Elliptic Curves and Modular Forms"; Deligne SGA 4 1/2

In general I found Bourbaki-seminar texts often very helpfull and readable. Unfortunately many of the newer issues seem not to be free available on the web.

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Geometric invariant theory is also a perfect book. I think it isthe best of Mumford. –  gauss Apr 6 '12 at 23:07

Since you wanted to learn writing from examples: J. Kock and I. Vainsencher's book "An Invitation to Quantum Cohomology" is wonderful reading, simply because of its incredibly friendly style. It gives you the feeling that the authors take you by the hand and lead you through their garden of wonders (always uphill of course). The achievement of the book is to give you lots of intuition - for moduli stacks, strategies for proofs in enumerative geometry, the necessity of a virtual fundamental class, how generating functions work... This is something very difficult to do in mathematical writing - in this respect you could compare it to John Baez's blog, only that it is a longer, coherent book on a single subject.

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I have had this feeling a few times. For example: Gillman & Jerison, Rings of Continuous Functions.

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