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Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.

Do you have such an example?

Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.

And as usual let's do one per post and vote freely, vote a lot.

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closed as off topic by Kevin H. Lin, Andrés Caicedo, Felipe Voloch, Hailong Dao, Bill Johnson Jul 9 '11 at 15:01

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Why are so many answers big-picture papers and philosophical tracts? I'm sure many of them are good papers, but is this really what the question was about? Am I right in suspecting that posters only read the title of the question and not the question itself? – Thierry Zell Sep 4 '10 at 0:23
Perhaps it's time to close this question. – S. Carnahan Oct 22 '10 at 17:40
Agreed, as Thierry and Tobias say, there are too many recommendations for punditry. – Robin Chapman Nov 17 '10 at 11:48

41 Answers 41

Perhaps not really a paper, but i think a "must-read" is A Mathematician's Lament by Paul Lockhart.

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It might not be a bad idea to read Scott Aaronson's thoughts on it afterwards, though: – Qiaochu Yuan Oct 25 '09 at 16:10
I wish all math teachers read that link. – Ilya Nikokoshev Oct 25 '09 at 18:40
I strongly disagree with the "must-read" label. Lockhart's article is mostly composed of half-truths and empty dramatic language. – S. Carnahan Nov 7 '09 at 5:30
IMHO, one can say "I prefer geometric proofs/arguments to algebraic ones" in shorter space than 25 pages. – Ketil Tveiten Nov 17 '10 at 11:26
Absolutely, the only way to reintroduce honesty into the prevailing math curriculum is to abolish the requirement that everyone must study math. – Michael Hardy Nov 17 '10 at 13:53

William Thurston's On Proof and Progress in Mathematics is a wonderful read, enlightening many aspects of the practice of mathematics.

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I am surprised to see that so many people suggest meta-mathematical articles, which try to explain how one should do good mathematics in one or the other form. Personally, I usually find it a waste of time to read these, and there a few statements to which I agree so wholeheartedly as the one of Borel:

"I feel that what mathematics needs least are pundits who issue prescriptions or guidelines for presumably less enlightened mortals."

The mere idea that you can learn how to do mathematics (or in fact anything useful) from reading a HowTo seems extremely weird to me. I would rather read any classical math article, and there are plenty of them. The subject does not really matter, you can learn good mathematical thinking from each of them, and in my opinion much easier than from any of the above guideline articles. Just to be constructive, take for example (in alphabetical order)

  • Atiyah&Bott, The Yang-Mills equations over Riemann surfaces.
  • Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts.
  • Furstenberg, A Poisson formula for semi-simple Lie groups.
  • Gromov,Groups of polynomial growth and expanding maps.
  • Tate, Fourier analysis in number fields and Hecke's zeta-functions.

I am not suggesting that any mathematician should read all of them, but any one of them will do. In fact, the actual content of these papers does not matter so much. It is rather, that they give an insight how a new idea is born. So, if you want to give birth to new ideas yourself, look at them, not at some guideline.

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I would argue for Shannon's "A Mathematical Theory of Communication". Its wonderfully written, started an entire field of research (or two), and struck a very nice balance between abstraction and transparency in the mathematics. The ideas first introduced in that paper are powerful tools even today!

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One of the wonderful things about that paper is that many of the actual proofs border on trivial -- the important things are the "big ideas." – Harrison Brown Nov 1 '09 at 14:44
yes, exactly! That is why it is such a wonderful role model of a paper. I think everyone should dream of writing something so transparent and so groundbreaking – Carter Tazio Schonwald Nov 1 '09 at 14:57
Well, it's better to have a copy of Cover and Thomas handy if you read that! (Shannon does not give rigorous proofs, and it took some years before it was all cleaned up.) – jon Nov 19 '09 at 4:42
This is my all-time favorite maths paper. – Amritanshu Prasad Oct 27 '10 at 4:24

Missed Opportunities, Freeman Dyson

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Really up to the point. After reading it, I also think any mathematician (or physicist) should do the same. Thanks! – Jose Brox Nov 19 '09 at 19:19
I wonder if there is a progress since that time in the directions that Dyson mentioned as what should be done. – Sergei Akbarov Jul 31 '13 at 12:17

I would recommend Gowers' The Two Cultures of Mathematics. It talks about the two types of mathematicians, the "theory builders" and the "problem solvers."

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Dyson's Birds and Frogs (mentioned below, from the Notices of the AMS, February 2009) is similar. This "two cultures" thread has been discovered quite a few times, as was pointed out in various letters to the editor published in the June 2009 Notices ( – Michael Lugo Oct 27 '09 at 15:40

I had recommended to me from several prominent faculty the paper:

The Yang-Mills Equations over Riemann Surfaces Author(s): M. F. Atiyah and R. Bott Source: Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 Published by: The Royal Society Stable URL:

One professor called it "the basis for truly 21st century mathematics." It is also reportedly accessible by beginning graduate students with some exposure to differential geometry and suitable for independent study or as a reading course. It is a 93 page paper and develops a lot of fundamental constructions and ideas from scratch. Here is Martin Guest's review on MathSciNet.

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For about 5 years I carried my copy with me everywhere I went, in an increasingly decrepit 3-ring binder weighed down by page after page of my own notes and explanations. One day, at a conference, a dispute arose over whether the main result of the paper held with integral coefficients or required one to work over the rationals. In the flash of an eye, four or five of us pulled out our copies and opened to the relevant page. Luckily, I was right: integral coefficients. The first time I left home without the paper, it felt like a rite of passage. Or at least that's the way I remember it. – Dan Ramras Sep 4 '10 at 4:49

Another suggestion: A beginner's guide to forcing by Tim Chow.

It really explains the continuum hypothesis, in a very accessible and captivating way. People often talk about the continuum hypothesis, but it's nice to know what's going on for real.

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This is a nice high level summary. One objection I'd have is that it suggests that standard models are necessary, rather than a convenience. An Introduction to Independence for Analysts by Dales and Woodin clears this up, as well as being a more thorough introduction to the subject designed for non-set-theorists. However, it is unfortunately out of print. – Richard Dore Oct 26 '09 at 20:15
@Richard: In the paper I do mention that infinitary set theory is not needed for forcing and that a purely finitary proof is possible; I don't think I say anywhere that standard models are necessary. However, since you got that erroneous impression, presumably other readers will too, so it's good to clear up that point explicitly. – Timothy Chow May 31 '10 at 17:20
what's going on for real I want to interpret that as a pun – David Corwin Jul 11 '12 at 16:47

I think "What is good mathematics?" by Terry Tao is a great paper because it argues that we do not need to all be pursuing the same ideal of good mathematics (and indeed, people should pursue disjoint ideals), and it provides an interesting case study of a nice result, Szemerédi's theorem.

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I am really tempted to accept an answer, though this would probably be unfair to other people who posted a bunch of interesting and diverse stuff to read... – Ilya Nikokoshev Oct 26 '09 at 23:29
A very nice case study indeed, but I found the general remarks surrounding it superfluous and besides the point (and even insulting in parts). Not to talk about the horrible title, which raises expectations that the paper cannot keep. Why not call it "On Szemeredi's theorem", skip Sections 1 and 3 and leave it to the reader which conclusions to draw? I was very disappointed to see that a great mathematician like Tao felt the need to write such a strange convolute of nice insights (in the case study) and complete trivialities (in Section 1). But then, my position seems to be an isolated one. – Tobias Hartnick Nov 17 '10 at 12:41
@Tobias: The "complete trivialities" in Section 1 are not obvious to all mathematicians. (I have certainly seen people whose personal definitions of "good mathematics" wouldn't include some qualities in the list—which is fine—and who seemed unaware that others could value those qualities.) Even these insights, "trivial" to you, may be encouraging and illuminating to a young person somewhere. There are people who find Section 1&3 even more valuable than the case study. If mathematicians kept all their trivial thoughts to themselves, everyone to whom those were not trivial would be much poorer. – shreevatsa Jan 7 '12 at 19:22

"On the Number of Primes Less Than a Given Magnitude", B. Riemann.

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If you are a geometer I would say it is worth to read the paper of Gromov, called "Spaces and Questions", this paper is not about one single result, it rather gives a point of view on geometry, which seems very inspiring, at least to me, here is the refference:

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Well, the main purpose of the question was to mention "papers everyone should read", and not just geometers; I started Gromov's paper not being a geometer myself and didn't find it very inspiring... – Jose Brox Nov 19 '09 at 17:50

One paper that I want to share with any of my colleagues, although it is not in my field, is Doyle and Conway, Division by Three, math/0605779v1.

To emphasize why this paper is so great, let me quote the entirety of the conclusion (saving you the trouble of reading the rest of the paper):

What’s wrong with the axiom of choice?

Part of our aversion to using the axiom of choice stems from our view that it is probably not ‘true’. A theorem of Cohen shows that the axiom of choice is independent of the other axioms of ZF, which means that neither it nor its negation can be proved from the other axioms, providing that these axioms are consistent. Thus as far as the rest of the standard axioms are concerned, there is no way to decide whether the axiom of choice is true or false. This leads us to think that we had better reject the axiom of choice on account of Murphy’s Law that ‘if anything can go wrong, it will’. This is really no more than a personal hunch about the world of sets. We simply don’t believe that there is a function that assigns to each non-empty set of real numbers one of its elements. While you can describe a selection function that will work for finite sets, closed sets, open sets, analytic sets, and so on, Cohen’s result implies that there is no hope of describing a definite choice function that will work for ‘all’ non-empty sets of real numbers, at least as long as you remain within the world of standard Zermelo-Fraenkel set theory. And if you can’t describe such a function, or even prove that it exists without using some relative of the axiom of choice, what makes you so sure there is such a thing?

Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat doubtful whether large natural numbers (like 805000, or even 2200) exist in any very real sense, and we’re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic—and hence also of set theory—are inconsistent. (See [E. Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.]) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today.

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Or the first phrase: "In this paper we show that it is possible to divide by three." – Ilya Nikokoshev Nov 8 '09 at 13:01

I would have to go with the "Five Worlds" paper by Impagliazzo. It is a beautiful overview of how many complexity/cryptographic results relate to each other and what they "mean" for the real world (as of 1995, at least). It is a great way to web all those buzz words from class and coffee discussions into a cohesive unit.


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Birds and Frogs by Freeman Dyson, which explains nicely that the world of mathematics is both , broad and deep.

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In recent years Manin has put out several philosophical writings on mathematics, physics, and other related topics:

Truth as value and duty: lessons of mathematics

Mathematical knowledge: internal, social and cultural aspects

The notion of dimension in geometry and algebra

Georg Cantor and his heritage

Von Zahlen und Figuren

There's also a book, Mathematics as Metaphor, that collects even more of Manin's philosophical material.

These are all very nice reads and I would recommend them to almost anyone, mathematician/physicist or not.

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Imre Lakatos "Proofs and Refutations". Great book about origin of mathematical reasoning and rise of formal theories.

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Stallings's How Not To Prove the Poincare Conjecture is the funniest paper I've ever read.

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If you ever - as in my case - quoted a textbook to your students claiming that pointwise convergence of Fourier series for piecewise continuous functions is difficult and subtle, you'll feel stupid after reading Paul Chernoff's two-page paper "Pointwise Convergence of Fourier Series."

I can't find a free online copy of it, but you should be able to read it here with university access: JSTOR (Actually, you can see the first page for free, which already proves the main result.)

(Or get it from the library: The American Mathematical Monthly, Vol. 87, No. 5 (May, 1980), pp. 399-400.)

EDIT: There is a free copy here:

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Pointwise convergence for continuous functions is indeed very difficult and subtle. Chernoff's paper is about the far easier differentiable case. – Richard Borcherds Sep 3 '10 at 16:31

Two additional papers in combinatorics (That I managed to find on line) each having a beautiful and simple result.

On the Shannon Capacity of a Graph by Laszlo Lovasz

The Upper Bound Conjecture and Cohen Macaulay Rings by Richard Stanley

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Not technically a paper but a lecture (in pdf form) full of pretty pictures and cool ideas:

The Mysteries of Counting: Euler Characteristic versus Homotopy Cardinality by John Baez.

We all know what it means for a set to have 6 elements, but what sort of thing has -1 elements, or 5/2? Believe it or not, these questions have nice answers. The Euler characteristic of a space is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive real values. These concepts shed new light on basic mathematics. For example, the space of finite sets turns out to have homotopy cardinality e, and this explains the key properties of the exponential function. Euler characteristic and homotopy cardinality share many properties, but it's hard to tell if they are the same, because there are very few spaces for which both are well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series---and the two then agree! The challenge of unifying them remains open.

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Proofs from the Book! (Ok it's a book rather than a paper, but just pick any chapter.) Every line is amazing.

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Two notes on notation by Knuth. This paper discusses "Iverson" notation, which is of use to almost all mathematicians, and good notation for Stirling numbers.

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I like Musical Actions of Dihedral Groups pretty much. It gives a nice view of harmony (the art of using chords in music), considering the set of chords as the dihedral group of order 24 (12 major + 12 minor).

Unfortunately, this is useful only for people into music and maths. I would also like to share it with my musician friends, but most of them will probably run away at the sight of the first mathematical term...

Please don't vote down if you're not a musician ;).

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"On the Electrodynamics of Moving Bodies", Albert Einstein

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I love Minkowski's rebuttal to that paper. – Ryan Budney Nov 19 '09 at 2:27

Andre Weil's "Two lectures on number theory. past and present." L'Enseignement Mathematique. Revue Internationale. fie Serie. 20: 87-110. 1974

available here

Great historical perspective on number theory up to the early 1970's. Easy to read too!

[I should remark that despite the article's great virtues, Weil is (apparently) unfair to Hardy and that many topics in number theory are left untouched.]

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An Elementary Theory of the Category of Sets

I always had a problem with ZFC because it makes too many arbitrary choices: why do we choose this countable set to be the natural numbers and not this other one? Why do we choose Kuratowski ordered pairs instead of some other version? This paper turned me on to the idea that all of mathematics could be done in a "nice" way, where things are only determined up to unique isomorphism by the properties you want them to satisfy. It was also my first exposure to category theory, and so holds a special place in my heart.

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I object to the idea that basic definitions in set theory are arbitrary. The goal is to minimize primitives (just containment). The other definitions have natural justifications in this setting. For defining the natural numbers, we want < to agree with containment. For the Kuratowski pairing, first it is tricky just to pick something that works. Then you want a definition which increases set rank as little as possible. To be clear, I have no objection to thinking about logic categorically. In set theory it hasn't done much IMO, but it has been very fruitful in other areas of logic. – Richard Dore Oct 26 '09 at 19:51
See my comment on FOM The idea is that if you encode mathematical objects as sets, you will get all of the theorems you want, but because your choice had to be somewhat arbitrary, you overspecify the problem,and end up with weird identifications like the number 3 being a function. That is cool if it is what you like, but I prefer a foundations which doesn't say anything more about natural numbers than what is shared by all isomorphic copies of them. ETCS does that. – Steven Gubkin Oct 26 '09 at 22:08

One paper that I've read a few times and always loved was Who Can Name the Bigger Number? (also available in Spanish and French, for those who prefer to read in those). It discusses how our concept of "big numbers" has evolved over time, and talks about Turing machines and the "busy beaver" numbers, which represent a non-computable function.

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