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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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    $\begingroup$ 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? $\endgroup$
    – Thierry Zell
    Commented Sep 4, 2010 at 0:23
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    $\begingroup$ Agreed, as Thierry and Tobias say, there are too many recommendations for punditry. $\endgroup$
    – Robin Chapman
    Commented Nov 17, 2010 at 11:48
  • $\begingroup$ Andrew Ranicki’s site answers this question well many times over. Search engines accept site:maths.ed.ac.uk/~v1ranick filetype:pdf. $\endgroup$
    – isomorphismes
    Commented Jan 6, 2019 at 3:02
  • $\begingroup$ By the time a user is finished reading the 43 proposed basic reads, she can come back here for further suggestions; in the meantime, the question can remain closed. However, it is a useful question and I don't see any reason to delete it. Additional attempts to delete it will be resisted. $\endgroup$
    – Mikhail Katz
    Commented Jul 30 at 7:54

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

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    $\begingroup$ It might not be a bad idea to read Scott Aaronson's thoughts on it afterwards, though: scottaaronson.com/blog/?p=410 $\endgroup$
    – Qiaochu Yuan
    Commented Oct 25, 2009 at 16:10
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    $\begingroup$ I wish all math teachers read that link. $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 25, 2009 at 18:40
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    $\begingroup$ I strongly disagree with the "must-read" label. Lockhart's article is mostly composed of half-truths and empty dramatic language. $\endgroup$
    – S. Carnahan ♦
    Commented Nov 7, 2009 at 5:30
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    $\begingroup$ IMHO, one can say "I prefer geometric proofs/arguments to algebraic ones" in shorter space than 25 pages. $\endgroup$
    – Ketil Tveiten
    Commented Nov 17, 2010 at 11:26
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    $\begingroup$ Absolutely, the only way to reintroduce honesty into the prevailing math curriculum is to abolish the requirement that everyone must study math. $\endgroup$
    – Michael Hardy
    Commented Nov 17, 2010 at 13:53
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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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    $\begingroup$ From a post by Tao on G+: "There are extremely few papers that I would tag as "must read" for all research mathematicians, but I would certainly include Thurston's classic article on what progress in mathematical research is, and how this differs from (but is certainly related to) the mere acquisition of proofs of theorems, among this very short list." plus.google.com/114134834346472219368/posts/DPzKcGENyjq $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 31, 2013 at 1:05
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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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    $\begingroup$ 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." $\endgroup$
    – Harrison Brown
    Commented Nov 1, 2009 at 14:44
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    $\begingroup$ 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 $\endgroup$
    – Carter Tazio Schonwald
    Commented Nov 1, 2009 at 14:57
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    $\begingroup$ 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.) $\endgroup$
    – jon
    Commented Nov 19, 2009 at 4:42
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    $\begingroup$ This is my all-time favorite maths paper. $\endgroup$
    – Amritanshu Prasad
    Commented Oct 27, 2010 at 4:24
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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: https://www.jstor.org/stable/37156

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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    $\begingroup$ 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. $\endgroup$
    – Dan Ramras
    Commented Sep 4, 2010 at 4:49
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Missed Opportunities, Freeman Dyson

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    $\begingroup$ Really up to the point. After reading it, I also think any mathematician (or physicist) should do the same. Thanks! $\endgroup$
    – Jose Brox
    Commented Nov 19, 2009 at 19:19
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    $\begingroup$ I wonder if there is a progress since that time in the directions that Dyson mentioned as what should be done. $\endgroup$
    – Sergei Akbarov
    Commented Jul 31, 2013 at 12:17
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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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    $\begingroup$ 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 (ams.org/notices/200906/rtx090600688p.pdf) $\endgroup$
    – Michael Lugo
    Commented Oct 27, 2009 at 15:40
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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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  • $\begingroup$ I have to upvote this one because I am in the acknowledgments! It is a seriously good intro to an often neglected subject area. $\endgroup$
    – Steven Gubkin
    Commented Oct 25, 2009 at 19:33
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    $\begingroup$ 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. $\endgroup$
    – Richard Dore
    Commented Oct 26, 2009 at 20:15
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    $\begingroup$ @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. $\endgroup$
    – Timothy Chow
    Commented May 31, 2010 at 17:20
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    $\begingroup$ what's going on for real I want to interpret that as a pun $\endgroup$
    – David Corwin
    Commented Jul 11, 2012 at 16:47
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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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  • $\begingroup$ Gets my +1 as well. $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 9:44
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    $\begingroup$ Easily one of the best papers about the practice of mathematics I've read. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 25, 2009 at 0:15
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    $\begingroup$ 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... $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 26, 2009 at 23:29
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    $\begingroup$ 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. $\endgroup$
    – Tobias Hartnick
    Commented Nov 17, 2010 at 12:41
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    $\begingroup$ @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. $\endgroup$
    – shreevatsa
    Commented Jan 7, 2012 at 19:22
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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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    $\begingroup$ Or the first phrase: "In this paper we show that it is possible to divide by three." $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 8, 2009 at 13:01
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    $\begingroup$ The last paragraph (and concretely, the last phrase!) is great. I was not aware of Nelson's program, but it has got me gooseflesh, thing that does not happen to me as often as I would like. $\endgroup$
    – Jose Brox
    Commented Nov 8, 2009 at 13:42
  • $\begingroup$ Mm, they're, I guess, partly joking about the collapse of set theory, right? $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 30, 2009 at 15:10
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    $\begingroup$ No joke. Warning Signs of a Possible Collapse of Contemporary Mathematics $\endgroup$
    – Łukasz Lew
    Commented May 3, 2018 at 16:55
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"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 reference: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SpacesandQuestions.pdf

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  • $\begingroup$ It's real cool! Gets my vote. $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 23, 2009 at 21:22
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    $\begingroup$ 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... $\endgroup$
    – Jose Brox
    Commented Nov 19, 2009 at 17:50
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    $\begingroup$ I personally consider the paper to be highly accessible to non-geometers, but that's just me. Anyway, I think there's more fun in the middle of paper even if the beginning bores you! $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 30, 2009 at 15:12
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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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I would have to go with the "Five Worlds" paper by Impagliazzo.1 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.

-Yan

R. Impagliazzo, "A personal view of average-case complexity," Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference, Minneapolis, MN, USA, 1995, pp. 134-147, doi: 10.1109/SCT.1995.514853.

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  • $\begingroup$ Very interesting to read, and funny, e.g. Minicrypt :) $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 17:20
  • $\begingroup$ A good question is where you think we live ;) I prefer to think we are in Cryptomania (as many others), but I think many smart people seriously differ on these (as many smart people also seem to seriously think P = NP, which came as a huge shock to me). A poll at some point may be educational. $\endgroup$
    – Yan X Zhang
    Commented Oct 31, 2009 at 6:38
  • $\begingroup$ dead link :( a google search turned this math.hunter.cuny.edu/olgak/hunter_5worlds_slides.pdf which seems to be the same? $\endgroup$
    – Tshirtman
    Commented Jul 12, 2016 at 20:56
  • $\begingroup$ Link died again. The paper is called A Personal View of Average-case Complexity, I've linked it from the author's website (but it is hosted as a .ps file instead of the more typical .pdf file). $\endgroup$
    – Mark Schultz-Wu
    Commented Dec 18, 2019 at 16:54
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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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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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"On Computable Numbers, with an Application to the Entscheidungsproblem", Alan Turing, 1936. A great mind and a great paper.

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Stallings's How Not To Prove the Poincare Conjecture (cached at Citeseer) is the funniest paper I've ever read.

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  • $\begingroup$ Why is it funny? $\endgroup$
    – lhf
    Commented Nov 21, 2009 at 11:25
  • $\begingroup$ The bit I like is:- "If he patches up all these points, he will have proved the Poincare Conjecture (for we shall show how Theorem 0 for n=2 implies the Poincare Conjecture) incorrectly." $\endgroup$
    – HJRW
    Commented Nov 21, 2009 at 17:47
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Advice to a Young Mathematician in the Princeton Companion to Mathematics

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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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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: https://web.archive.org/web/20100730044517/http://math.berkeley.edu/~strain/118.S10/chernoff.pointwise.convergence.of.fourier.series.pdf

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  • $\begingroup$ +1. Quite an amazing paper! $\endgroup$
    – Ilya Nikokoshev
    Commented Mar 2, 2010 at 10:04
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    $\begingroup$ Pointwise convergence for continuous functions is indeed very difficult and subtle. Chernoff's paper is about the far easier differentiable case. $\endgroup$
    – Richard Borcherds
    Commented Sep 3, 2010 at 16:31
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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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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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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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  • $\begingroup$ just starting reading this. Absolutely fascinating! $\endgroup$
    – Michael Hoffman
    Commented Oct 24, 2009 at 21:30
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"On the Electrodynamics of Moving Bodies", Albert Einstein

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    $\begingroup$ I love Minkowski's rebuttal to that paper. $\endgroup$
    – Ryan Budney
    Commented Nov 19, 2009 at 2:27
  • $\begingroup$ It is such as accessible paper that we've successfully had students in our "physics for people who hate physics" (Conceptual Physics) course read it and understand large portions of it. $\endgroup$
    – Ian Durham
    Commented Feb 15, 2010 at 17:52
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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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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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    $\begingroup$ 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. $\endgroup$
    – The Mathemagician
    Commented Oct 22, 2010 at 19:04
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    $\begingroup$ 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. $\endgroup$
    – Tobias Hartnick
    Commented Nov 17, 2010 at 13:44
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    $\begingroup$ Ah, I can't resist pointing to a personal favorite counter-article to Wigner's: "How effective indeed is present-day mathematics?" It would certainly be in accordance to the OP. I read it back then by chance, out of misdirected curiosity (just because a teacher of mine wrote it), but I think it was what paved the road for me to go on realizing that a kuhnian view on mathematics is not necessarily a preposterous or a blasphemous idea. $\endgroup$
    – Basil
    Commented Jul 2, 2014 at 22:01
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An Elementary Theory of the Category of Sets

http://tac.mta.ca/tac/reprints/articles/11/tr11abs.html

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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    $\begingroup$ 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. $\endgroup$
    – Richard Dore
    Commented Oct 26, 2009 at 19:51
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    $\begingroup$ See my comment on FOM cs.nyu.edu/pipermail/fom/2008-January/012571.html 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. $\endgroup$
    – Steven Gubkin
    Commented Oct 26, 2009 at 22:08
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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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