Question

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.

Answer 1

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.

Answer 2

"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.

Answer 3

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.

Answer 4

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

Answer 5

"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

Answer 6

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

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

Michael

Answer 7

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

Answer 8

Berkovich talked about "Non-archimedean analytic geometry: first steps", available at his homepage: here

Answer 9

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.

Answer 10

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

Answer 11

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....

Answer 12

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

Answer 13

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.

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

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.

Answer 19

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.

Answer 20

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).

Answer 21

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

Answer 22

David Ruelle's "The mathematician's brain", in particular Chapter 21, The Strategy of Mathematical Invention.

Answer 23

"The psychology of invention in the mathematical field" by Jacques Hadamard. A preview of a recent reedition is available on Google Books.

Answer 24

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.

Answer 25

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