# A single paper everyone should read? [closed]

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.

• 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
• Andrew Ranicki’s site answers this question well many times over. Search engines accept site:maths.ed.ac.uk/~v1ranick filetype:pdf. – isomorphismes Jan 6 '19 at 3:02

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.

• 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 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. – Steven Gubkin Oct 26 '09 at 22:08

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:

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.)

• Indeed, you get correctly the original intent, though I think some reviews are original enough in their scope and composition to be considered essentially primary sources for some problems. – Ilya Nikokoshev Nov 30 '09 at 15:09

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

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

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.

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.

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.

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.

• As came up in a different question recently, this paper has some missteps. (In fact it has an erratum, albeit published 8 years later.) I agree that it is worth reading, but I would recommend also Smiley's Filters and equivalent nets, American Math. Monthly, 1957. – Pete L. Clark Jul 14 '10 at 20:24

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]"?)

• 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
• Doesn't this "richer" statement follow easily from the law of cosines? – Federico Poloni Jul 6 '11 at 10:23

"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.

http://www.jstor.org/sici?sici=0025-570X%28199112%2964%3A5%3C291%3ARAPIMA%3E2.0.CO%3B2-Z