# 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
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 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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1. "Differential calculus on Normed Linear Banach Spaces" by Prof.Kalyan Mukherjee

This book gave me a very hands-on explorable window into the world of manifolds and Lie groups. Like it shows explicit calculations of derivative of matrix multiplication and determinant maps and also about computing tangents to curves inside Lie groups.

2. "Topology, Geometry and Gauge Fields" by Gregory Naber. (2 Volumes)

Its an exciting book which got me motivated into topology when it explained to me very simply the Heegard decomposition of S^3 and hence Hopf Fibration and how that relates to Dirac Monopoles! Before I read this book I had no clue that I would find mathematics exciting. Especially this revived my childhood interest in geometry.

Naber's are books that changed my career decision.

3. "Global Calculus" by S.Ramanan (in the AMS series)

This is a hard book to read initially but it excites the reader a lot and it was great to read alongside when Prof.Ramanan taught me topology and differential geometry. Anyway Prof.S.Ramanan is a great expositor. He could teach topics like modular forms and algebraic curves to a bunch of undergrads in their first complex analysis course in Chennai Mathematical Institute (CMI), India! He really pushes up the possible limits of exposition.

Prof.S.Ramanan's lectures in my alma mater CMI, affected my career choices almost as much as Naber's books did.

4. "Calculus on Manifolds" by Spivak

Its treatment of Fubini's theorem and related issues are great.

5. The writings on group theory by a college senior of mine called Vipul. His wiki "groupprops" is an amazing repository on finite group theory.

His extensive efforts into mathematical writing also inspired me into periodically LaTex-ing up interesting things in mathematics as I learn.

Can anyone here tell about nice expository writings on topics like Gromov-Witten theory or Reshetkhin-Turaev and Rozansky-Khovanov stuff and how these relate to QFT? Something which shows a lot of examples and may be also explicit calculations.

Most sources on Quantum Groups that I have tried looking at start off a bit harshly for the newcomer. I would be greatly interested to read of "great mathematical writing" in these areas.

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Hugo Steinhaus http://en.wikipedia.org/wiki/Hugo_Steinhaus book "Mathematical Kaleidoscope"

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I an algebraic geometer, so the book I'm going to propose is about as far from my subject as it can be. Still I think that Steele's book on stochastic calculus is one of the best written mathematical books I know. It really makes you enjoy probability, starting from the simplest examples of random walks and building a lot of theory, like martingales, Brownian motion and Ito's integral. I almost wanted to change my subject when I was reading it! :-)

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Mathematical logic has at least a couple of great writers. The canonical example is Bruno Poizat, especially the French originals; I would put Hodges in the same league. Both are emphatically not concise writers (at least in their most famous books). Their use of the full capabilities of language is very didactic, and often poetic. I greatly admire them both.

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

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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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Algebraic Functions and Projective Curves by David M. Goldschmidt; October 2000

Beautiful proofs; Chapter 6 (Zeta Functions), 6.3 Riemann Hypothesis, nice mathematics.

Anyone in David's field who has not at least sampled this book is leading an impoverished existence.

Falco

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I loved reading Zelevinsky's 'Representations of finite classical groups'. It gave a totally different perspective to everything that I knew before about representation theory.

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I wish someone had told me about this book when I was younger: A.I. Mal'cev, Algorithms and Recursive Functions.

The exposition is simple, thought provoking and rigorous. You are teased to delve into many strains when reading it.

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

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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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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 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

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

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