# 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
You write "I wish someone had told me about this paper when I was younger", lucky you :-) When I was young(er) I was unable to read papers, just books and even that was not obvious. –  Patrick I-Z Nov 13 '13 at 8:51

I couldn't find Disquisitiones Arithmeticae listed as an answer, and find this strange. (It has been translated from the Latin). The book is a delight to read, and the proofs always seem to be exactly the right ones.

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So you feel that its proof of quadratic reciprocity is exactly the right one? –  KConrad Jan 22 at 4:35
@keith--I guess not, though it has its adherents on this site--I think Lemmermeyer speaks well of it. Isn't this the proof that essentially calculates K_2(Z)? –  paul Monsky Jan 23 at 0:09

Poincare's `Sur les Solutions Periodiques et le principe de moindre action'.

in Comptes rendus l'Academie de Sciences, t. 123, p 915-918, 1896

This paper humbled me. It me realize how much the founders of the subject(s) I work in really knew, and how far ahead they could see. I am in awe of how Poincare could give such a detailed trip report of his investigations without the formal language we use today being in place. (In many ways, formal language often gets in the way.) In this 2 and a half pages Poincare does most everything I did in a 13 page 102 years later. He does it more clearly and elegantly.

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Imre Lakatos: "Proofs and Refutations. The logic of mathematical discovery" is a fantastic read!

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Lam's Serre's Problem on Projective Modules. It contains everything: The big picture, the proof details, interesting techniques and the links between different methods. I wish more books were like this.

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I think I went through all questions without seeing Bourbaki. I think that his Algèbre Commutative is simply a dream to read; every time I open it, I found myself keeping on reading it about perfectly useless subjects (compared to what I was looking for) just for the pleasure, as I was used to do as a kid with some of my parents' books. The same happens to me with Topologie Générale

Besides this, I must also add two litte gems by John Tate, namely his Rigid Analytic Spaces and a small paper where he studies residues on curves in an adelic language, Residues of differentials on Curves, Ann. Sci. Ecole Norm. Sup. (1968).

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Weil's "Basic Number Theory" is my favorite book. Norbert Schappacher once said "If you learn number theory from this book, you will never forget it."

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I'm astonished that nobody has mentioned I. G. Macdonald's classic Symmetric Functions and Hall Polynomials. It is more than 50 years old but fun to read, compact, self-contained, elegant, deep.

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I nominate the name everyone knows: Walter Rudin.

I went through his PoMA and it was my first exposure to serious rigourous maths. It takes the minimal amount of words, and the proofs are ultra-clever. For instance, his treatment of upper and lower limits (3.17)are the best I've seen. When reading it you feel that after so many years' development this is the final reduced form of maths and it cannot be simplified anymore in the future. When going through it it's actually a lot of pain digesting the ideas and details, but the pain is definitely worth it.

Eventually (now) I'm working on his R&C and the feeling is so different: it's like the Louvre of mathematical theory: you just keep get surprised all the time. But his proofs follow the same principle: clever, minimal, most economical approach, but not necessarily to the maximal generality.(to simplify notation he even defines $dm$ in ch.9 as the Lebesgue measure devided by $\sqrt{2\pi}$!)But this time you feel like the proofs have the potential to be made easier in the future though.

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I believe that the papers on $A_\infty$-structures by Bernhard Keller are extremely well written: they provide the reader with an overview of the state of the art of research in the topic(s), applications and even understandable proofs. I suggest in particular

A-infinity algebras, modules and functor categories

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True story: When I was about to move to Stony Brook to start my PhD, one of my professors took me aside to tell me "You know, when I was a student Milnor was god, and Morse Theory was the bible." I found that nice and moved on, but a little later a younger professor took me aside to say "You know, when I was a student Milnor was god, and Introduction to Algebraic K-Theory was the bible." By then I knew that something was going on, but I was still taken by surprise when a more junior professor found me and said "You know, when I was a student Milnor was god, and Characteristic Classes was the bible."

Of course this was all planned. They succeeded in motivating me to take every opportunity to talk to and learn from the big names I met. But they made another point that I only recognized later, while writing my first paper: If you want to learn to write Mathematics well, read anything by Milnor.

When I was a student, Dynamics in One Complex Variable was the bible.

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Without a doubt, Arnold's Mathematical Methods of Classical Mechanics is the book most responsible for me deciding to be a geometer. Only some papers of Atiyah were able to replicate the feeling of awe I had reading Arnold's classic as an impressionable green undergrad. Very few authors are able to convey to me the feeling of completely unconstrained thinking as Arnold's writings do. They continue to be the go to place whenever if feel stuck or stale in my research. A few pages from him still do the trick: they remind me why I became a mathematician.

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I learned to read French because of Arnold's book. When I was an undergrad the library only had the Mir edition: Les méthodes mathématiques de la mécanique classique. I'm still in love with this book. –  alvarezpaiva Apr 28 '12 at 13:00

"A panoramic view of Riemannian Geometry" by M. Berger is an example of excellent mathematical writing to my taste. This book is great to learn what are the questions of interest in the field, and what are the main results. Although you will not find detailed proofs of the results, the main ideas are often explained in an intuitive way.

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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
@Andrea: probably you mean s/delusion/disappointment. As an Italian, I often make that mistake, too. :) –  Federico Poloni Apr 24 '12 at 15:33

Harris and fulton Representation theory is a nice book. Linear algebraic groups of Jim Humphreys is also very good one. On Riemann Surface the book of Forster's is really very good one.

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Riemann's paper, "On the number of primes less than a given magnitude," is the reason why I decided to study mathematics (at the graduate level and beyond). I read the paper as an undergraduate and I was very impressed by the techniques that Riemann used to study the properties of the prime counting function. In particular, I was blown away by Riemann's use of complex analysis, fourier analysis, and asymptotic analysis to study a problem in number theory, which I thought was a distant area of mathematics. This paper is truly a work of art and is less than 10 pages.

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Simon Donaldson - Riemann Surfaces Great writing, deep understanding. I believe that noone have mentioned it because this topic is older than the book.

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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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I second, third and double second this! –  Keerthi Madapusi Pera May 20 '11 at 21:53
I second Pera's comment. –  Joël Jun 11 '14 at 23:50

'Galois Cohomology' by Larry Washington in Cornell-Silverman-Stevens is my one stop reference for the eponymous topic. In about twenty pages (and with minimal prerequisites), he introduces Galois cohomology groups, explains Tate Local Duality Theorem and Euler Charateristic, shows the connection between extensions, deformation and cohomology groups, introduces generalized Selmer groups and proves a result that appears in Wiles' proof. Along the way, he also fully explains the Poitou-Tate nine-term exact sequence! Terrific stuff.

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I'm surprised I haven't seen Serge Lang. Some complain that he is too terse, but I really enjoy his style. Often times when i grab several books from the library on the same subject, it will be Lang's book I end up using the most. As for a single piece of writing, I think Lang's Algebra will do.

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I think Sipser's Introduction to the Theory of Computation deserves a mention. It is incredibly clear, full of valuable examples, and an absolute classic. I can't think of a better book for a mathematician who's interested in theoretical computer science. It also seems to serve computer scientists without a great deal of mathematical background by providing an introduction to proofs at the beginning. My favorite part: all theorems come with "Proof Idea" first and then proof after that. This helps the computer scientists who are not that familiar with proofs, but it's also great for a mathematician to get the main idea of the proof, fill in the blanks themselves, and then move on to the next result.

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How about Munkres Topology? This book certainly made me want to be a (point-set) topologist. Turns out I came along a bit late for that field, but I'm sure this book helped push me into algebraic topology. Anyway, Munkres is full of fantastic examples and pictures, it treats all the major aspects of the field, and it seems to be the most popular book for courses in point-set topology all over the US.

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Steven Strogatz's "Nonlinear Dynamics and Chaos" book is written in a manner that almost allows one to kick back in a recliner and enjoy. The style is one that draws one into the material on nonlinear ODEs... probably the best undergrad text book that I used.

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Donaldson and Kronheimer's "Geometry of 4-manifolds" - masters of the subject, they have a knack of explaining the crux of a difficult theorem in a concise and elegant way, and gauge theory has a lot of difficult theorems. After many years of reading, it still has new surprises.

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1. Milne's entire set of notes (algebraic number theory, class field theory, algebraic groups, complex multiplication, modular functions and modular forms, etc.), articles (abelian varieties, Jacobian varieties, Shimura Varieties, Tannakian Categories, etc.) and books (Elliptic Curves, Arithmetic Duality Theorems, Etale Cohomology etc.), available at www.jmilne.org/math/. They are indispensable for anyone who wishes to learn the fundamental concepts in arithmetic geometry. The Storrs lectures on Abelian Varieties and Jacobian Varieties are clear, succinct and give great references throughout. His notes on 'Class Field Theory' are superbly written. 'Etale Cohomology' is a standard reference for the subject, although I find his lecture notes on the same topic even more enjoyable. Finally, 'Arithmetic Duality Theorems' is quite possibly the only reference where one can find complete proofs of Tate's Duality Theorems as well as their generalizations using etale and flat cohomology.

2. Part 4 (particularly Chapter XX) of Lang's 'Algebra'. I may have learned (as little as I have) about homological algebra from Weibel or Gelfund-Manin as texts, but I always keep coming back to Lang's exposition. Not a lot of motivation, but it covers almost everything you need to know in a first course.

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Lou Kauffman's book "On Knots" inspired me to become a topologist. It conveys the feel of the way topologists think with copious hand-drawn pictures. It also gets into deep waters without losing a playful touch. It would actually be nice to have a similar book that covers the recent developments in knot theory as well.

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For anyone studying Stable Homotopy Theory, a fantastic text is Doug Ravenel's "green book" (although the second edition is red) Complex Cobordism and Stable Homotopy Theory

Not only is this book full of useful results for those in the field (making it an incredible reference for those starting out), it is also written in a very clear style and it's completely self-contained. I cannot think of a better book on how to do computation in homotopy theory. This definitely fits under "I wish someone had told me about this when I was younger" and "anyone in my field who hasn't read this is leading an impoverished existence"

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I'm shocked no one mentioned T.Y. Lam. He is perhaps the clearest expositor I've ever read, he motivates and gives history in every section, he fills his books with great examples and problems, and I have yet to find any errors. Indeed, his exercises are often finding counterexamples for errors in other published works, e.g. the following in Lectures on Modules and Rings:

In a ring theory text, the following statement appeared: "If $0\rightarrow C\rightarrow Q\rightarrow P\rightarrow 0$ is exact with $C$ and $Q$ finitely generated then $P$ is finitely presented" Give a counterexample.

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Harder's Algebraic Geometry 1 is a beautiful example of explaining why an abstract subject makes sense. The book has a conversational style without wasting words, and focuses on providing intuition for the subject.

When I get disheartened, this is the book I turn to for inspiration.

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I'll mention WL Burke's "Applied Differential Geometry." It's written for physicists, it will not be to the liking of the majority of mathematicians, but it changed this engineer's view of geometric methods forever.

The book changed the direction of my research because it presented a point of view that is not readily accessible if you follow the control and optimization literature. Becoming familiar with the differential geometry literature is an investment that a controls person is unlikely to make without a general idea of where the complete set of tools leads to. In this sense, the mathematics literature can present an obstacle. Burke's exposition is intuitive, though quite informal, and led me to read Spivak, Milnor, and other books, some mentioned here, which I would not have read if I had started with the math literature.

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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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## protected by Scott Morrison♦Nov 13 '13 at 12:11

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