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What are examples of well received mathematical papers in which the author provides detail on how a surprising solution to a problem has been found.

I am especially looking for papers that also document the dead ends of investigation, i.e. ideas that seemed promising but lead nowhere, and where the motivation and inspiration that lead to the right ideas came from.

By "surprising solution" I mean solutions that feel right at first reading and it isn't clear why they haven't been found earlier.

Manfred Weis
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  • I was given advice to leave all of that sort of thing out of a journal submission. I don't know if this rule is universal though. – Harry Gindi Nov 21 '18 at 07:28
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    @HarryGindi you can make two papers... the second one called My Route to Solving the XYZ Conjecture – Bjørn Kjos-Hanssen Nov 21 '18 at 08:23
  • George Bergman's classic paper "The diamond lemma for ring theory" seems to fit the bill. – Dan Petersen Nov 21 '18 at 08:33
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    Mochizuki has this sort of writing style. His reflections on his research are integrated into his papers and surveys. This is not surprising as the IUT papers are highly self-reflective in their content and nature. But perhaps I'm under-interpreting the question. – Samantha Y Nov 21 '18 at 08:46
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    A long time ago, a referee asked for more information on how we had gone about finding the solution to the problem. We added half a page on that. Then we got a new report saying "the authors carry on like they solved the Riemann Hypothesis". – Brendan McKay Nov 21 '18 at 10:15
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    @BrendanMcKay its too sad that it seems to be imperative in mathematics to keep up the impression mathematicians don't experience emotions when doing research; no wonder that mathematics is judged as dry and the most undesirable thing to do. – Manfred Weis Nov 21 '18 at 10:26
  • Maybe Cartier, P. (1982), "Comment l'hypothèse de Riemann ne fut pas prouvée", Seminar on Number Theory, Paris 1980–81 (Paris, 1980/1981), Progr. Math., 22, Boston, MA: Birkhäuser Boston, pp. 35–48, MR 0693308 would be of interest. – Gerry Myerson Nov 21 '18 at 12:01
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    Another example: van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London. – Andrés E. Caicedo Nov 21 '18 at 14:40
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    I'm surprised JR Stallings' paper "How Not To Prove The Poincaré Conjecture" has not yet been mentioned. – Sam Hopkins Nov 21 '18 at 16:15
  • Although that (famous) paper does seem hard to track down... – Sam Hopkins Nov 21 '18 at 16:18
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    Not a paper, but Villani's "Birth of a Theorem" does that kind of thing in detail. – Piyush Grover Nov 21 '18 at 17:17
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    Dirac in Recollections of an exciting era (1977) recounts how he came up with his matrices while trying to “take the square root of a sum of four squares”. – Francois Ziegler Nov 21 '18 at 18:40
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    I was just writing an answer about Stalling's paper when I saw the comment of @SamHopkins. Anyway, here's a link with publication information. – Lee Mosher Nov 22 '18 at 16:51
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    Enchiridion: Mathematics User's Guides is a blog/journal which invites authors to post "user guides" to a published or soon to be published mathematical research. These user guides are kind of a research report, and albeit not being exactly mathematical papers, they tell a lot about the context and the insights for the papers' theorems. – Tadashi Dec 01 '18 at 18:42

8 Answers8

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Richard P. Stanley's How the Upper Bound Conjecture was proved ends with two morals:

  1. The shortest path may not be the best.
  2. Even if you don’t arrive at your destination, the journey can still be worthwhile.
David Richerby
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Bjørn Kjos-Hanssen
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The paper

Rawnsley, John; Schmid, Wilfried; Wolf, Joseph A., Singular unitary representations and indefinite harmonic theory, J. Funct. Anal. 51, 1-114 (1983). ZBL0511.22005.

contains an unusual “Historical Note” (pp. 102–107). E.g.:

For various reasons one expects to get $\mu_n$ by... That does not work directly because... In 1975, S & W tried... At that point it became clear that an intrinsic higher $L_2$ cohomology theory was needed... In 1977, R & W looked... They did not see how to... This was the point at which S & W had been stopped... During the following academic year B succeeded in... but the method did not extend past... R & W made some progress in... These results were not published formally because... During the summer of 1979, S & W discussed the apparent disparity and clarified... then carried out a computation... then looked at the case... Thus the original S & W problem was settled... At the end of the summer of 1980, S & W saw that... could be simplified... The present version was completed in... There are two important parallel developments which we understood only after... (etc.)

Francois Ziegler
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The prime example is Euler's papers. This style is out of fashion in 20th century. Polya in Mathematics and Plausible reasoning discusses this question at length and even reproduces completely (in English) one of Euler's papers (on partitions).

Of the 20th century examples I can mention

MR1555091 Malmquist, J. Sur les fonctions a un nombre fini de branches définies par les équations différentielles du premier ordre. Acta Math. 36 (1913), no. 1, 297–343.

Martin Sleziak
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Alexandre Eremenko
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The first example that came to mind was

MR0270881 (42 #5764) van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London.

There, van der Waerden describes some of the history as well as his proof of his well-known theorem.

Another example:

MR2245898 (2007j:05091) Seymour, Paul. How the proof of the strong perfect graph conjecture was found. Gaz. Math. No. 109 (2006), 69–83.

From Wilson's review in Mathematical Reviews: "In this interesting and revealing paper, Seymour describes in graphic terms their assaults on the problem, the difficulties they came across, and the means they used to overcome these difficulties."

Andrés E. Caicedo
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Ryan Williams provides a Casual Tour Around a Circuit Complexity Bound (the bound in question being that NEXP lacks nonuniform polysized ACC circuits) which may fit the bill, although I believe that Williams's goal is to give a motivated exposition rather than a 100% historically accurate account of how he came up with his proof.

Timothy Chow
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A nice example is the article "The method of undetermined generalization and specialization, illustrated with Fred Galvin's amazing proof of the Dinitz Conjecture" by Doron Zeilberger in Amer. Math. Monthly 103, no. 3, 233-239, 1996 (see also here for a freely accessible version).

Martin Sleziak
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Abdelmalek Abdesselam
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I enjoy The genesis of the Macdonald polynomial statistics, complete with journal entries, and detailed descriptions of the experimental method.

This paper describes how the researchers came up with a nice formula for the combinatorial (aka modified) Macdonald polynomials.

Per Alexandersson
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Another type of example. Textbooks on non-Euclidean geometry may often begin with a chapter on historical failed attempts to prove the Parallel Postulate.

Gerald Edgar
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