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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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    $\begingroup$ 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. $\endgroup$
    – Anton Geraschenko
    Commented Oct 19, 2009 at 6:39
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    $\begingroup$ 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. $\endgroup$
    – Patrick I-Z
    Commented Nov 13, 2013 at 8:51

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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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    $\begingroup$ "Characteristic Classes" is particularly great indeed, I do love most of Milnor's books. $\endgroup$
    – Sam Derbyshire
    Commented Oct 18, 2009 at 4:46
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    $\begingroup$ I read Milnor's "Dynamics in One Complex Variable" as a graduate student -- it was wonderful. $\endgroup$
    – Sam Nead
    Commented Nov 15, 2009 at 1:59
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    $\begingroup$ 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. $\endgroup$
    – Łukasz Grabowski
    Commented Jan 9, 2011 at 1:19
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    $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Commented May 15, 2011 at 11:31
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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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Canonical submission: Anything by J.-P. 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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    $\begingroup$ 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. $\endgroup$
    – Alon Amit
    Commented Oct 12, 2009 at 18:45
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    $\begingroup$ 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. $\endgroup$
    – Sean Tilson
    Commented Mar 29, 2010 at 5:46
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    $\begingroup$ When I first spotted this question, my mind immediately went to Serre's books. The 1st part of "Cours d'Arithmetique" is one of my main references for my introductory undergraduate course in Number Theory. I also used his book "Linear representation of finite groups" as textbook once. "Corps Locaux" is another great book. I also like to mention his fundamental papers "FAC" and "GAGA" which I greatly enjoyed reading back in my grad student days. P.S.: I never read "Trees", but after reading Scott's answer, I feel a urge to do so! :-) $\endgroup$
    – Andrea Mori
    Commented Aug 23, 2010 at 16:15
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    $\begingroup$ 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. $\endgroup$
    – Łukasz Grabowski
    Commented Jan 9, 2011 at 1:17
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    $\begingroup$ My personal story: in my early years I always wanted too much. I wanted to read books which I would be only able to read after learning years' worth of further mathematics. In every single case I failed and had to give up after few first pages. The only exception I remember was "Groupes algébriques et corps de classes". It is so incredibly well written that I was able to at least understand what's going on. I vividly remember feeling of ave. How on earth can this man achieve to make even guys like I was then to understand?! $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 21, 2018 at 19:07
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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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    $\begingroup$ I completely agree with you! In my opinion that is one of the best books ever! $\endgroup$
    – Vincenzo Zaccaro
    Commented Nov 21, 2018 at 21:03
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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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    $\begingroup$ Agreed. These books are very clearly written and motivate the subject well. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 15, 2009 at 18:59
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    $\begingroup$ Thirded. This series immediately came to mind when I saw this question. $\endgroup$
    – Darsh Ranjan
    Commented Oct 21, 2009 at 2:08
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    $\begingroup$ Agreed further; in my opinion all three books of the series are "great mathematical writing". $\endgroup$
    – Pietro
    Commented Aug 23, 2010 at 15:30
  • $\begingroup$ Agree, this is what I wished for (and got) last christmas. $\endgroup$
    – Per Alexandersson
    Commented Dec 15, 2012 at 18:58
  • $\begingroup$ There are four books in the series. $\endgroup$
    – Per Alexandersson
    Commented Nov 13, 2013 at 9:24
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Michael Spivak's Calculus made me want to study analysis.

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    $\begingroup$ I learned calculus out of that book. It convinced me to become a math major ... $\endgroup$
    – Micah Milinovich
    Commented Aug 9, 2010 at 15:16
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    $\begingroup$ I'm reading Spivak now as a high school junior - it has inspired me to become a math major. $\endgroup$
    – CaptainAmerica16
    Commented Nov 24, 2018 at 5:28
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    $\begingroup$ can you give an example of Spivak's book that shows why it is amazing? (I know he is great in all aspects). $\endgroup$
    – C.F.G
    Commented Sep 20, 2019 at 13:26
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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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    $\begingroup$ His notes on three-manifolds are very pretty. $\endgroup$
    – Sam Nead
    Commented Nov 15, 2009 at 2:00
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    $\begingroup$ Everybody who wants to learn about the Serre spectral sequence should also not waste his time with McCleary or something like this but turn to Hatcher's notes. $\endgroup$
    – Lennart Meier
    Commented Aug 23, 2010 at 20:23
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    $\begingroup$ I've personally always found this text way too verbose. Not to the point. It's very hard to read quickly or to find things. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jul 11, 2017 at 21:53
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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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    $\begingroup$ Not just undergraduates ... and not just mathematicians ... can enjoy reading Lee's fine text. $\endgroup$
    – John Sidles
    Commented May 20, 2011 at 19:12
  • $\begingroup$ I second this, absolute classic. $\endgroup$
    – Hollis Williams
    Commented Jul 4, 2019 at 4:50
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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 language 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 language/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 étale 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 literature.

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

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    $\begingroup$ I second, third and double second this! $\endgroup$
    – Keerthi Madapusi
    Commented May 20, 2011 at 21:53
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    $\begingroup$ I second Pera's comment. $\endgroup$
    – Joël
    Commented Jun 11, 2014 at 23:50
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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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    $\begingroup$ 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. $\endgroup$
    – Jose Capco
    Commented Nov 10, 2009 at 22:30
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    $\begingroup$ 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. $\endgroup$
    – Erik Davis
    Commented May 3, 2010 at 21:56
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    $\begingroup$ What about people who don't like his first book to start with? :-) $\endgroup$
    – Andrea Ferretti
    Commented Aug 11, 2010 at 12:13
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    $\begingroup$ I strongly disagrees with this recommendation. $\endgroup$
    – Kerry
    Commented Aug 24, 2010 at 3:42
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    $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Commented May 15, 2011 at 11:39
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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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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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    $\begingroup$ 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. $\endgroup$
    – alvarezpaiva
    Commented Apr 28, 2012 at 13:00
  • $\begingroup$ My favourite too. $\endgroup$
    – Michael
    Commented Sep 12, 2014 at 21:33
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    $\begingroup$ In the same vein one could add the terrific book "Ergodic Problems of Classical Mechanics" by Arnold and Avez. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Nov 22, 2018 at 15:57
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I love Grothendieck's Tohoku paper.

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    $\begingroup$ Me too. A brilliant evergreen. $\endgroup$
    – Ady
    Commented Dec 23, 2009 at 4:16
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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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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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    $\begingroup$ Great maths: yes. Great writing: no. If you want to know why a particular theorem is true, you have to go back to all the theorems that it builds upon. And then farther back. And farther back. $\endgroup$
    – Helene Sigloch
    Commented Jan 14, 2015 at 12:44
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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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    $\begingroup$ I strongly disagree. I've never found the style of proofs in it particularly beautiful--they're all just little combinatorial tricks. $\endgroup$
    – Eric Wofsey
    Commented Oct 12, 2009 at 22:25
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    $\begingroup$ It was a big delusion for me as well. If there is a Book, I seriously doubt it contains any of these proofs. $\endgroup$
    – Andrea Ferretti
    Commented Aug 11, 2010 at 12:14
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    $\begingroup$ Well, I think Joyal's proof of Cayley's theorem does belong in "The Book". That's a proof one could never forget. On the other hand, I've never been keen on Zagier's one-sentence proof of Fermat's two-squares theorem -- it's just not that memorable, even after one has worked through the details. $\endgroup$
    – Todd Trimble
    Commented May 15, 2011 at 11:50
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    $\begingroup$ @Andrea: probably you mean s/delusion/disappointment. As an Italian, I often make that mistake, too. :) $\endgroup$
    – Federico Poloni
    Commented Apr 24, 2012 at 15:33
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    $\begingroup$ "Disillusion" might be the sought-after word. $\endgroup$
    – Todd Trimble
    Commented Jun 27, 2017 at 13:41
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Silverman and Tate's "Rational Points on Eliptic Curves."

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    $\begingroup$ My guess is you want this to go under "I wish someone had told me about this when I was younger." I'd have to agree, at least for the first few chapters. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 15, 2009 at 18:58
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    $\begingroup$ Actually someone did tell me about this when I was younger. I read it after senior year of highschool and greatly enjoyed it. $\endgroup$
    – Noah Snyder
    Commented Oct 16, 2009 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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  • $\begingroup$ Did you mean "Rational points on elliptic curves" as Noah mentioned, or Silverman's book with the title you gave? $\endgroup$
    – S. Carnahan ♦
    Commented Oct 16, 2009 at 9:05
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    $\begingroup$ No, I meant The Arithmetic of Elliptic Curves. I've also looked at the book Noah mentioned, but I found this one more enjoyable. $\endgroup$
    – Akhil Mathew
    Commented Oct 16, 2009 at 11:03
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    $\begingroup$ 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. $\endgroup$
    – Sam Derbyshire
    Commented Oct 18, 2009 at 4:45
  • $\begingroup$ I find the exposition confusing and unnecessarily long, due to the fact that he assumes that the reader has no previous knowledge of either algebraic geometry or group cohomology. $\endgroup$
    – Andrea Ferretti
    Commented Aug 11, 2010 at 12:16
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    $\begingroup$ Also very nice: he keeps up a list of errata on his website, so this book really is a great book to learn from: math.brown.edu/~jhs/AEC/AECErrata.pdf $\endgroup$
    – David White
    Commented May 15, 2011 at 19:55
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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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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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  • $\begingroup$ I totally agree. +1 $\endgroup$
    – Jose Brox
    Commented Aug 11, 2010 at 12:02
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    $\begingroup$ Also, perhaps, "Mathematics: A Very Long Introduction". (This is what I call The Princeton Companion to Mathematics, of which Gowers was editor.) $\endgroup$
    – Michael Lugo
    Commented Aug 23, 2010 at 17:19
  • $\begingroup$ I like this book, but despite several attempts, I've yet to find a non-mathematician who 'gets it'. I worry that it's not suitable for its target audience. $\endgroup$
    – HJRW
    Commented Dec 15, 2012 at 8:36
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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 for example at http://i11www.iti.kit.edu/~awolff/lehre/scientific_writing/h-hwm-70.pdf.

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  • $\begingroup$ Personally, I don't like this article. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 14, 2015 at 22:12
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    $\begingroup$ @WłodzimierzHolsztyński I like it. You might not agree with everything he says (Halmos knows you won't, and even says so), but there's a lot of good advice in there and he not only talks the talk, but walks the walk. Care to say why you don't like it? $\endgroup$
    – Todd Trimble
    Commented Jun 27, 2017 at 13:36
  • $\begingroup$ I like Halmos comment that there should be one topic in a writing. Sometimes that's unfeasible, but I feel that it is a sensible point to keep in mind. $\endgroup$
    – ACL
    Commented Dec 12, 2017 at 20:21
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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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    $\begingroup$ It's certainly written with a lot of pedagogical care and consideration and, I'd have to say, human warmth. I continue to consult it regularly. $\endgroup$
    – Todd Trimble
    Commented Jun 24, 2014 at 13:01
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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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  • $\begingroup$ Yes. This is a wonderful paper! $\endgroup$
    – Richard Montgomery
    Commented Jan 21, 2015 at 15:56
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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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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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    $\begingroup$ Surely you mean "but not less", not "but not more"? $\endgroup$
    – LSpice
    Commented Jan 14, 2015 at 1:44
  • $\begingroup$ +1 for Artin. On the other hand, I had "Pure Mathematics" by Hardy years ago, but each of the three authors (three separate works, textbooks on Mathematical Analysis): Banach, Sierpiński, Kuratowski--were so much better(!). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 14, 2015 at 22:18
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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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  • $\begingroup$ The other volumes are useful as well, even if you don't read all of the material (a likely situation, the one issue with the volumes is that they are long). For example volume 5 gives a useful geometric perspective on partial differential equations which is not always presented in PDE texts. $\endgroup$
    – Pait
    Commented May 15, 2011 at 15:19
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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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  • $\begingroup$ +1: nice to see such strong +ve statements. $\endgroup$
    – Suvrit
    Commented May 15, 2011 at 8:41
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    $\begingroup$ Qiaochu: you are a combinatorialist! :-) $\endgroup$
    – Todd Trimble
    Commented May 15, 2011 at 11:52
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    $\begingroup$ @Todd: I'm flattered. It feels wrong for me to label myself anything until I at least start graduate school. $\endgroup$
    – Qiaochu Yuan
    Commented May 15, 2011 at 12:25
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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.

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    $\begingroup$ There is an impressive uniformity to Aluffi's book. He makes almost everything seem simple, almost guessable. $\endgroup$
    – Arrow
    Commented Dec 1, 2015 at 9:48
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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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  • $\begingroup$ One of the few books that I actually read from cover to cover. :-) $\endgroup$
    – Andrea Ferretti
    Commented Aug 11, 2010 at 12:17
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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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    $\begingroup$ Bar-Natan's papers in general tend to be well-written. $\endgroup$
    – Jim Conant
    Commented May 15, 2011 at 20:14
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