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Papers in mathematics are generally written as if the major insights suddenly appeared, unbidden, in a notebook on the researcher's desk and then were fleshed out into the final paper.

While this is great for finding out about results, it's terrible for finding out about how they were arrived at.

What I want are papers, books or essays written by researchers about their work on problems, especially if they describe the evolution of their work on a specific problem (Polya's writings on problem-solving are great, but not what I'm interested in). I'd like to know how hundred-page treatises on problems unsolved for decades are born.

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    $\begingroup$ Terence Tao's "What is good mathematics?" isn't a bad place to start (arxiv.org/abs/math.HO/0702396). Actually, I think Terence Tao's and Tim Gowers' blogs, and blogs in general, are a great way to learn about the thought process behind mathematical research. $\endgroup$
    – Qiaochu Yuan
    Commented May 13, 2010 at 18:02
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    $\begingroup$ Kodaira talks about how he and Spences figured out some of their results in "Complex Manifolds and Deformations of Complex Structures". Those comments are intervowen with the text though, so they're not easy to dig out if that's the only thing you're interested in. $\endgroup$
    – Gunnar Þór Magnússon
    Commented May 13, 2010 at 18:44
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    $\begingroup$ Poincare wrote some essays that touch on this topic. Look for his book "Science and Method". $\endgroup$
    – KConrad
    Commented May 13, 2010 at 19:47
  • $\begingroup$ @Qiaochu Yuan Thanks for the great link. :D Why don't you post that link as an answer? $\endgroup$
    – Pratik Deoghare
    Commented May 13, 2010 at 21:21
  • $\begingroup$ Probably "how it was arrived at" is not what most readers want to know, which is the reason it is generally written as it is. $\endgroup$
    – Gerald Edgar
    Commented May 29, 2010 at 12:01

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Not mentioned so far is Bill Thurston's On proof and progress in mathematics (1994). With more than three hundred citations, it surely qualifies as a classic ... it is a permanent left-column link on Terry Tao's weblog, for example.

Thurston's essay is unique, relative to other such essays, in that it describes (in Section 6, "Some Personal Experiences") not one path, but two distinct paths relating to thought processes in mathematical research:

  • a solitary path associated to Thurston's early work on foliations
  • a social path associated to Thurston's later work on the Geometrization Conjecture

Thurston's latter approach is the topic of much research today, under various rubrics that include "social media", "social networks", and "roadmapping".

The foresighted points -- by 17 years -- of Thurston's essay include:

  • social elements of research can be consciously chosen by individuals
  • fundamental mathematics can provide uniquely strong foundations for social enterprises
  • healthy mathematical communities make faster progress, and also, a better environment for nurturing the next generation of young mathematicians.

A recent well-respected essay that amounts to a consensus abstraction of Thurston's ideas is the International Roadmap Committee (IRC) More-than-Moore White Paper. For modern-day systems engineers especially, it is very instructive to read-out the main themes of Thurston's 1994 essay from the IRC's 2010 white paper, and thus to appreciate that Thurston's ideas were far ahead of their time.

In particular, the IRC's five consensus preconditions for successful roadmapping are anticipated with near-perfection by Thurston's essay ... and this is why Thurston's essay no doubt will continue to gather new citations through decades to come.

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  • $\begingroup$ It’s silly to say “anticipated with near-perfection?” I can see thr similar themes, but that phrase falsely suggests that Thurston had a similar list of conditions. $\endgroup$
    – user44143
    Commented Sep 24, 2021 at 9:20
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One of the all time classic talks on the general question of research is "You and your research" by richard hamming pdf at www.cs.virginia.edu/~robins/YouAndYourResearch.pdf and theres other transcripts of this talk everywhere.

GREAT GREAT reading.

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    $\begingroup$ There is also a 41mins long talk by Hamming youtube.com/watch?v=a1zDuOPkMSw if you don't like to read the transcript. $\endgroup$
    – Jernej
    Commented Mar 19, 2013 at 17:52
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"The psychology of invention in the mathematical field" by Jacques Hadamard. A preview of a recent reedition is available on Google Books.

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The introduction to Wiles's famous paper on Fermat's Last Theorem (from the Annals in the mid 1990s) gives an unusually detailed account of the process by which Wiles developed the arguments of the paper.

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"Grothendieck – Serre correspondence" and "Recoltes et semailles", if you are into algebraic geometry and related fields.

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From my comment above:

Terence Tao's What is good mathematics? isn't a bad place to start. Actually, I think Terence Tao's and Tim Gowers' blogs, and blogs in general, are a great way to learn about the thought process behind mathematical research.

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Peter Cameron has a nice blog where he wrote about Doing Research: http://cameroncounts.wordpress.com/2009/11/11/doing-research/.

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I guess you should definitely take a look at

WAERDEN, B. L. van der, "How the proof of Baudet's conjecture was found", Studies in Pure Mathematics (papers presented to R. Rado on the occassion of his 65th birthday, ed. by L. Mirsky, Academic Press), pp. 252-260.

According to N. G. De Bruijn, the note by B. L. van der Waerden

... was partly intendend as an illustration of the author's ideas on the psychology of mathematical invention. The reading of the report is recommended to all those for whom understanding is not just formal verification, but rather a procedure by which intuitive ideas and experiences are linked together to each other in order to build up the final mathematical structure... The reading of van der Waerden's report is also recommended to those who are interested to learn about the discussion (with Artin and Schreier) that preceded van der Waerden's discovery, and to learn about the actual contributions of Artin and Schreier to the solution of the problem.

It is important to add that Baudet's conjecture goes now under the name of van der Waerden's theorem on arithmetic progressions. The proof of this cute result is showcased in A. Y. Khinchin's Three Pearls of Number Theory. Yet, be warned that Khinchin does not present the original approach by van der Waerden, but an attack communicated to him by a M. A. Lukomskaya (cf. my reply to this discussion).

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Paul Seymour gives an account of how they (Chudnovsky, Robertson, Seymour, Thomas) ended up solving the strong perfect graph conjecture here.

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  • $\begingroup$ This paper is really interesting, particularly in the way he describes the sort of "punctuated equilibrium" of his progress. 2.5 years of work divided pretty neatly into three types of research at different times - 1) the futile, which was abandoned with some lessons learned but mostly wasted time; 2) the grueling, the painful building of long and complex proofs that seem needlessly complicated. This type is hard to differentiate from the first until its clear the results will be in hand; 3) the ecstatic, where progress is very concrete, fast and clear. $\endgroup$
    – DoubleJay
    Commented May 15, 2010 at 18:47
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"Théorème vivant" by Cédric Villani

does a great job at explaining the process of research in mathematics to non mathematicians (but it's a great read for mathematicians too), focusing on how he proved his theorem on Landau damping with C. Mouhot.

Right now it's available only in French, but I think an English translation is in the making.

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  • $\begingroup$ As of last month the English version is still to arrive (13/9/2013 blog post by Villani, cedricvillani.org/born-to-be-alive) $\endgroup$
    – David Roberts ♦
    Commented Oct 25, 2013 at 7:27
  • $\begingroup$ Surprising, as the German translation already exists since April under the title "Das lebendige Theorem". $\endgroup$
    – Lennart Meier
    Commented Oct 25, 2013 at 13:19
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David Ruelle's "The mathematician's brain", in particular Chapter 21, The Strategy of Mathematical Invention.

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Manindra Agrawal has talked about the story behind the primality testing algorithm: http://www.cse.iitk.ac.in/~manindra/presentations/GodelTalk.pdf

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Franz Lemmermeyer's paper [1] contains a very interesting account of the truth about how Kummer was led to the invention of his ideal numbers (the popular legend is far from reality). In particular he brings to the fore the key role played by Jacobi sums from Jacobi's lectures on cyclotomy and corrects the following myths:

(1) Kummer’s idea was brilliant and new; there were no traces of it in the number theoretical work of his predecessors: it appeared out of the blue and solved the “problem” of nonunique factorization in a way reminiscent of Alexander the Great’s solution of the Gordian knot.

(2) Kummer’s definition of an ideal prime is difficult to understand and not easy to use in practice.

Also he "tr[ies] to correct the historical picture of the development of Kummer’s ideal numbers by showing that the notion of ideal numbers used by Kummer is perfectly natural, and that it is based to a large degree on ideas put forth by Jacobi in his investigations in cyclotomy. Moreover, a theory of divisibility built on these ideas is hardly more complicated than Dedekind’s approach" and he concludes by "discuss[ing] the relevance of the notion of integral closure for Kummer’s work by looking carefully at the concept of singularity in number theory and algebraic geometry."

[1] Franz Lemmermeyer. Jacobi and Kummer's ideal numbers.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg v. 79, 2, 2009, 165-187.
http://dx.doi.org/10.1007/s12188-009-0020-5

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Here is a really nice collection of essays by mathematicians at the IHES: The unravelers: mathematical snapshots

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I know it is a bit dated, but I am fond of Littlewood's essay "The Mathematician's Art of Work" in his Miscellany.

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Atiyah's Advice to a Young Mathematician is worth a look:

http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf

Michael

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J.H.S. writes: "It is important to add that Baudet's conjecture goes now under the name of van der Waerden's theorem on arithmetic progressions."

In fact, I proved that the conjecture was created independently by Issai Schur and Pierre Joseph Henry Baudet. Therefore, I call this classic result Baudet-Schur-Van der Waerden Theorem.

See details in my "The Mathematical Coloring Book", Springer, 2009:

http://www.springer.com/new+%26+forthcoming+titles+%28default%29/book/978-0-387-74640-1

Alexander Soifer

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van der Waerden's discussion of his (and two colleague's) proof of Baudet's Conjecture describes how the proof was found after a sequence of about half a dozen episodes of trying an approach, seeing that it failed, analysing the failure, and using this analysis to discover a fresh approach that patches the failures in the earlier one. Typical patches are: try a more complex induction rule, generalise the original conjecture, invent and prove an intermediate lemma. If the final proof is presented without the story behind it's invention, it appears to be magic, but actually it's the result of a prolonged trial and error.

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Yves André's interesting talk "La maïeutique mathématique selon Poincaré et selon Grothendieck" can be viewed online.

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A good book in algebraic combinatorics that describes the research process is David M. Bressoud's "Proofs and Confirmations". From the back cover:

This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author tells the story of the search for and discovery of a proof of a formula conjectured in the early 1980s: the number of $n\times n$ alternating sign matrices, objects that generalize permutation matrices. Although it was soon apparent that the conjecture must be true, the proof was elusive....

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"How to solve it" by George Polya addresses problem solving at the school level, but through this book many people got introduced to (some aspects of) doing mathematics. The methodology is quite general.

http://books.google.com/books?id=X3xsgXjTGgoC&dq=inauthor:%22George+Polya%22

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  • $\begingroup$ The question seems to exclude this book specifically. $\endgroup$
    – user5117
    Commented Sep 12, 2012 at 16:05
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    $\begingroup$ I should have made this remark as a comment rather than an answer. Too late... $\endgroup$
    – Margaret Friedland
    Commented Sep 12, 2012 at 18:00
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This less known paper of H. WHITNEY is a joy to read: Letting research come naturally. Just to make you curious, here is the opening of the paper:

The purpose of this paper is to show that creative mathematical work is not just the privilege of a few geniuses; it can be a natural activity of any of us with sufficient desire and freedom.

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  • $\begingroup$ Maybe I'm not searching correctly, but I'm not able to find a copy of the paper, even with access to JSTOR and other resources through my university. Can you comment on where to find it online? $\endgroup$
    – Zev Chonoles
    Commented Oct 26, 2013 at 19:58
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    $\begingroup$ @ZevChonoles Unfortunately, the paper is not available online. I have a scanned copy of the paper and I have just sent it to you. Please let me know if you have received it. Just in case, anybody else needs the paper, gmail me at asghari.amir $\endgroup$
    – Amir Asghari
    Commented Oct 26, 2013 at 20:38
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Berkovich talked about "Non-archimedean analytic geometry: first steps", available at his homepage: here

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Like someone has commented, Poincaré has lectured on this topic. An excerpt is given the "World of Mathematics", which, as you all may know, is a collection of mathematical articles by famous people.

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The comments at the end of Andre Weil's collected papers are quite fascinating in that respect.

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http://www.amazon.com/History-Algebraic-Differential-Topology-1900/dp/0817649069/ref=sr_1_1?ie=UTF8&qid=1347474254&sr=8-1&keywords=history+of+algebraic+and+differential+topology

I am currently reading this book, A history of Algebraic and Differential Topology, by Jean Dieudonne, partially in the same spirit of the question. Very good reference, but more importantly throws very good light on the development of the subject, not just in the mind of one mathematician, but across many of them, and across decades.

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  • $\begingroup$ Dieudonne also wrote history books about other parts of mathematics. I like his history of algebraic geometry. $\endgroup$
    – Lennart Meier
    Commented Oct 25, 2013 at 13:26
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Jean Louis Loday writed about how he discovered, using TeX, a realization of the Associahedron and the primary ideas behind his paper "Realization of the Stasheff polytope" (2004).

The text is in french. He begin the introduction as follow:

"La petite histoire de comment j’ai trouvé l’algorithme simple pour construire l’associaèdre de Stasheff. Ou comment TeX peut influer sur la recherche mathématique."

"The little story of how I found the simplest algorithm to construct the Stasheff's associahedron. Or how TeX can affect mathematic research."

http://www-irma.u-strasbg.fr/~loday/associaedreHistoire.pdf

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The book “Indra's Pearls” by David Mumford, Caroline Series, and David Wright is another good example of a work that explains the thought process behind mathematical research. It covers a broad span of topics, including geometry, number theory, abstract algebra, and computer graphics. Moreover, it shows how these fields are interconnected.

In the authors' own words:

Our dream is that this book will reveal to our readers that mathematics is not alien and remote but just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty - Indra's Pearls p. viii.

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A"Geometric phase memories" by Michael Berry

Michael Berry: The moment of conception of the geometric phase can be pinpointed precisely, but related ideas had been formulated before, in various guises. Not less varied were the ramifications that became clear once the concept was identified formally.

The following book details the milieu of factors that can influence research to a degree I've rarely encountered.

"Masters of Theory: Cambridge and the Rise of Mathematical Physics" by Andrew Warwick addresses aspects of this question (a Google preview) though not written by the actual researchers developing the problems and solutions in mathematical physics. See this review.

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