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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

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

PDE as a Unified Subject by Sergiu Klainerman.
An essay on partial differential equations written by a leading expert in the field, I strongly recommend to anyone who aspires to know more on the subject as well as to those who are not interested strictly in PDE's, but would like to get a grasp of interactions between Mathematics and Physics. There are also many interesting references.

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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

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

"An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" by Jonathan Shewchuk at UC Berkeley

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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

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

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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"Rigor and Proof in Mathematics: A Historical Perspective" by Israel Kleiner. Mathematics Magazine December 1991, 64:291-314.

This paper gives a very nice overview of how the understanding of rigor in mathematics has evolved from the early ages to the 20th century.

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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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I highly recommend this lucid, little book (with the length of a paper):
Mathematics: A very short introduction, by Fields Medalist Timothy Gowers

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The Unreasonable Effectiveness of Mathematics in the Natural Sciences

by Eugene Wigner

Although Wigner is physicist, I consider this article about mathematical physics very important both for physicists and mathematicians. It's a wonderful feeling to realize to what extent our world can be mathematical.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. E.Wigner

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This is not only a historically important paper,it focuses on an aspect of mathematics that Western culture has hesitated to return to after the wholesale rejection of it during the Bourbaki era. –  The Mathemagician Oct 22 '10 at 19:04
Even if bashing Bourbaki seems to have become hip again (as it was actually during the Bourbaki era as well), and thus the myth of the "wholesale rejection" of applications of mathematics "during the Bourbaki era" has become generally accepted in certain circles, it still remains a myth, which like all myths contains a germ of truth surrounded by a lot of prejudices, misunderstandings and plainly wrong statements. –  Tobias Hartnick Nov 17 '10 at 13:44

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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On the theorem of Pythagoras by by E.W. Dijkstra. (Did you know that in every plane triangle sgn$(\alpha + \beta - \gamma)$ = sgn$(a^2 + b^2 - c^2)$, a "theorem, say, 4 times as rich [as the original]"?)

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I don't think this is any news to most mathematicians. This is even in some good German schoolbooks from the 1960's-70's. Often when there is a statement like "if $a=b$ then $c=d$" one could check whether $a\leq b$ implies $c\leq d$, and lots of geometric inequalities have been created this way from identities. –  darij grinberg Nov 17 '10 at 14:08

That's easy just off the top of my head,Illya: Nets And Filters In Topology by the late Robert G. Bartle;appearing in the 1955 Volume 62 of American Mathematical Monthly. I remember having a friend in the Stanford mathematics honor society who'd published papers by age 20,but had never heard of either nets or filters. I recommended it to him right on the spot.

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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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Toen's course on stacks. I don't know if this counts as a paper, but courses 2,3, and 4 introduce a really interesting approach to geometry using the functor of points approach that I've not seen before.

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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

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

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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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2N Noncollinear Points Determine at Least 2N Directions, by Peter Ungar. This is a beautiful short paper that proves the result in the title.

A general remark: If you have to choose a single paper (or a single paper of a mathematician selected in other answers), I would recommend more strongy to choose original papers of important basic results rather than large survey papers or "meta" paper about mathematics. (This is also closer to the original intention of the question.)

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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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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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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

"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

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

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

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