A single paper everyone should read? - MathOverflow

A single paper everyone should read?

2009-10-23T18:42:05Z

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.

Answer by Dmitri for A single paper everyone should read?

2009-10-23T21:20:17Z

If you are a geometer I would say it is worth to read the paper of Gromov, called "Spaces and Questions", this is not about one single result, but more about a point of view on geometry, which seems very inspiering, at least to me, he is the refference: http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf

Answer by A. Rex for A single paper everyone should read?

2009-10-24T06:17:42Z

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.

Answer by yanzhang for A single paper everyone should read?

2009-10-24T17:16:13Z

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.

-Yan

Answer by subshift for A single paper everyone should read?

2009-10-24T17:56:34Z

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 ;).

Answer by Spinorbundle for A single paper everyone should read?

2009-10-25T14:59:46Z

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

Answer by Spinorbundle for A single paper everyone should read?

2009-10-25T16:52:11Z

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

Answer by Steven Gubkin for A single paper everyone should read?

2009-10-25T17:15:29Z

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.

Answer by Kirill Levin for A single paper everyone should read?

2009-10-25T18:23:52Z

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

Answer by subshift for A single paper everyone should read?

2009-10-25T19:20:12Z

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.

Answer by Andy Putman for A single paper everyone should read?

2009-10-26T14:18:38Z

Cannon's beautiful and accessible paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the original impetuses for geometric group theory. It inspired many people (including me) to become interested in infinite discrete groups. It is available here:

http://www.springerlink.com/content/v6051511338244x2/

Answer by shreevatsa for A single paper everyone should read?

2009-10-26T23:21:49Z

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

Answer by Carter Tazio Schonwald for A single paper everyone should read?

2009-10-27T01:36:57Z

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!

Answer by Jonah Sinick for A single paper everyone should read?

2009-10-27T07:20:32Z

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

available here http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001:1974:20::43&subp=hires

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.]

Answer by Jose Brox for A single paper everyone should read?

2009-11-08T03:22:17Z

Carl's Pomerance "A tale of two sieves", available at

http://www.ams.org/notices/199612/pomerance.pdf

It makes a quick introduction to subexponential factoring algorithms via their development from Fermat's Algorithm and then compares the Quadratic Sieve with Her Majesty the (General) Number Field Sieve, in a thorough, appealing and very understanable manner.

Answer by Theo Johnson-Freyd for A single paper everyone should read?

2009-11-08T04:56:43Z

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.

Answer by Kevin Lin for A single paper everyone should read?

2009-11-08T05:06:50Z

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.

Answer by martin.nikolov for A single paper everyone should read?

2009-11-08T05:16:21Z

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

Answer by Henry Wilton for A single paper everyone should read?

2009-11-08T05:18:45Z

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

Answer by Andrew Niefer for A single paper everyone should read?

2009-11-19T02:10:06Z

"On the Electrodynamics of Moving Bodies", Albert Einstein

Answer by Vladimir Dotsenko for A single paper everyone should read?

2009-11-19T16:18:19Z

Missed Opportunities, Freeman Dyson

Answer by Vincent for A single paper everyone should read?

2009-11-20T14:20:58Z

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

Answer by Gabriel Benamy for A single paper everyone should read?

2009-11-20T19:52:49Z

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.

Answer by Gil Kalai for A single paper everyone should read?

2009-11-20T21:34:22Z

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.)

Answer by Gil Kalai for A single paper everyone should read?

2009-11-21T10:37:12Z

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

Answer by Alasdair McAndrew for A single paper everyone should read?

2009-11-30T13:12:50Z

"On Computable Numbers, with an Application to the Entscheidungsproblem", Alan Turing, 1936. A great mind and a great paper.

Answer by mathphysicist for A single paper everyone should read?

2009-12-30T19:20:08Z

Advice to a Young Mathematician in the Princeton Companion to Mathematics

Answer by mathphysicist for A single paper everyone should read?

2009-12-30T19:23:23Z

Paul Halmos How to Write Mathematics

Answer by kakaz for A single paper everyone should read?

2010-02-15T12:35:58Z

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

Answer by Justin Curry for A single paper everyone should read?

2010-02-15T17:22:25Z

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: http://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.

Answer by fpqc for A single paper everyone should read?

2010-02-15T17:39:07Z

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.

Answer by Darsh Ranjan for A single paper everyone should read?

2010-02-18T17:17:08Z

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.)