# 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.
• 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

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.

• One of the few books that I actually read from cover to cover. :-) – Andrea Ferretti Aug 11 '10 at 12:17

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

• I'm still trying to find a way to get a decent copy of the original language version of this – Michael Hoffman Oct 30 '09 at 6:55
• I remember buying the first volume (Springer German 1954 edition) on ebay some time ago. – Pasha Zusmanovich Nov 3 '09 at 12:29

One of my all-time favorites for rigour and clarity:

Vistoli's Notes on Descent.

• It is incredible how Vistoli manages to make so clear a subject which is so abstract and notationally heavy. A masterpiece of mathematical writing. – Andrea Ferretti Aug 11 '10 at 12:20

Many of Atiyah's papers, especially those with Bott, are truly inspiring -- not only "The Yang-Mills Equations on Riemann Surfaces" but also "The Moment Map and Equivariant Cohomology" and "Convexity and Commuting Hamiltonians." He has a knack for writing in a style that, while not rigorous, allows the reader to fill in all rigorous details, and at the same time communicates high excitement.

His textbook Introduction to Commutative Algebra, written with Macdonald, is like a volume of poetry. I would guess (from internal stylistic evidence) that it was mostly written by Macdonald, but it's all great.

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.

Paolo Aluffi's Algebra: Chapter 0 presents the usual algebra material with special emphasis on category theory and homological algebra. It's written in a somewhat informal style that spurs on the reader and motivates constructions really well. I found it much better than Hungerford (which is often touted as an example of good writing itself); it is responsible for my interest in algebra and (the rudiments of) algebraic geometry.

• There is an impressive uniformity to Aluffi's book. He makes almost everything seem simple, almost guessable. – Arrow Dec 1 '15 at 9:48

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.

• Gian-Carlo Rota had a very nice review of one of Lam's books. Can anyone find it in free access? – darij grinberg May 15 '11 at 20:21
• I do recall Rota saying some nice things about Lam and A First Course in Noncommutative Rings in Indiscrete Thoughts, 242-244. I managed to find a snippet online by googling "professor neanderthal rota" (try it). – Todd Trimble Mar 11 '12 at 20:30
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.

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.

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

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.

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.

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."

• That forward is worthwhile just for this quote: "Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then zooming away and leaving us stranded in a blank haze." Yeah, I can relate to that. – Ben Webster Oct 14 '09 at 15:51

The American Mathematical Society awards the Leroy P. Steele Prize, in part for recognition of mathematical exposition:

The Leroy P. Steele Prize for Lifetime Achievement The Leroy P. Steele Prize for Mathematical Exposition The Leroy P. Steele Prize for Seminal Contribution to Research

These prizes were established in 1970 in honor of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein, and are endowed under the terms of a bequest from Leroy P. Steele. From 1970 to 1976 one or more prizes were awarded each year for outstanding published mathematical research; most favorable consideration was given to papers distinguished for their exposition and covering broad areas of mathematics. In 1977 the Council of the AMS modified the terms under which the prizes are awarded. Since then, up to three prizes have been awarded each year in the following categories: (1) for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students; (2) for a book or substantial survey or expository-research paper; (3) for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research. In 1993, the Council formalized the three categories of the prize by naming each of them: (1) The Leroy P. Steele Prize for Lifetime Achievement; (2) The Leroy P. Steele Prize for Mathematical Exposition; and (3) The Leroy P. Steele Prize for Seminal Contribution to Research. Each of these three US$5,000 prizes is awarded annually. The winners in the area of Mathematical Exposition, including prizes for specific books are listed here: http://www.ams.org/prizes/steele-prize.html Toric Varieties, about to be published by Cox, Little and Schenck is an unmeasurable amount of joy. It is impossible to get tired of it. Everything is well-bounded and it made me learn as much Algebraic Geometry as Toric Geometry itself. Its introductory sections to Algebraic Geometry before it develops the theory and shows you how to compute examples made me learn more than any dry full theoretic book in Algebraic Geometry. Definetely the best book I have read in two years. Available at Cox's website • Unfortunately, "Toric Varieties" is no longer available at Cox's website, as it has been published by AMS. – J W Apr 6 '12 at 19:38 • It was when I posted this :) – Jesus Martinez Garcia Apr 10 '12 at 15:10 From more introductory texts, some of the most well-written textbooks I came across are • Visual complex analysis, by Tristan Needham • Differential equations, by Blanchard, Devaney, and Hall • Please split this into two answers. The point of having a big list with one answer per post is that it makes it easy to vote things up and down. If you post more than one answer per post, this advantage of being able to bubble answers up and down is lost. – Anton Geraschenko Oct 18 '09 at 14:10 Probably the book(s) that most captivated me to study numerical analysis are the two by Forman Acton: "Numerical Methods That (usually) Work" and "Real Computing Made Real". The pithy and practical advice contained in both instilled in me the habits of trying to figure out just how structured the givens of a problem may be, among other things. 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. • Almost all of Donaldson's paper are a delight to read... – ARG Nov 13 '13 at 8:49 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. 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.

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.

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.

• Geometric invariant theory is also a perfect book. I think it isthe best of Mumford. – gauss Apr 6 '12 at 23:07
1. The book "Linear Algebra" by Greub; I've always thought his writing here was gorgeous, if a bit Spartan.

2. Most of John Stillwell's books.

"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.

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).

Category Theory by Steve Awodey (Oxford Logic Guides 49) is a very clear exposition of the subject.

And I know my students hate it, but I really like The Way of Analysis by Robert Strichartz.

• Personally, I think Mac Lane's Categories for the Working Mathematician is a much better introduction to category theory. – Daniel Miller Apr 5 '11 at 11:31
• Daniel, clearly the aims and audiences differ. Awodey was Mac Lane's last PhD advisee, and speaks well of Mac Lane's book (as so many of us who grew up on Mac Lane do), but he set out to write something suitable for different students. – Todd Trimble May 15 '11 at 12:24

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.

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

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.

• I second Fulton–Harris. – jmc Jan 14 '15 at 12:54

Anything and everything (including papers) by Marty Isaacs. I am more familiar with his "Finite Group Theory" book, but from the little I have seen, the same holds for his "Algebra" textbook. Of course, his book on Character Theory is the canonical text in that area.

The power of Isaacs' writing lies primarily in the clarity of his arguments. You are never left wondering where a certain bit came from, and he manages that without over-repeating himself, or over-explaining.

Then it's the content itself. In the first 70 pages of his FGT book, one already learns "exotic" stuff like the Chermak-Delgado measure, and the Theorems of Horosevskii, Lucchini, and Zenkov.

Another merit of his books is his choice of exercises. He really can't be commended enough on this. All are carefully chosen to supplement and strengthen the material presented in a flawless way.

A huge bonus is that nearly all arguments presented are his, and are as close to "from first principles" as possible.

• Isaacs' lovely books definitely deserve to be higher up on this list. – Elizabeth Henning Aug 5 at 0:55

This paper changed my life:

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