Stumbling Around in the Dark: Lessons from Everyday Mathematics

Abstract

The growing use of the internet for collaboration, and of numeric and symbolic software to perform calculations it is impossible to do by hand, not only augment the capabilities of mathematicians, but also afford new ways of observing what they do. In this essay we look at four case studies to see what we can learn about the everyday practice of mathematics: the polymath experiments for the collaborative production of mathematics, which tell us about mathematicians attitudes to working together in public; the minipolymath experiments in the same vein, from which we can examine in finer grained detail the kinds of activities that go on in developing a proof; the mathematical questions and answers in math overflow, which tell us about mathematical-research-in-the-small; and finally the role of computer algebra, in particular the GAP system, in the production of mathematics. We conclude with perspectives on the role of computational logic.

Further Reading

Barany, M., Mackenzie, D.: Chalk: Materials and Concepts in Mathematics Research. Representation in Scientific Practice Revisited. MIT Press (2014)

Frenkel, E.: Love and Math: The Heart of Hidden Reality. Basic Books (2014)

Gorenstein, D.: Finite Simple Groups: An Introduction to their Classification. Plenum Press, New York (1982)

Hadamard, J.: The Psychology of Invention in the Mathematical Field. Princeton (1954)

Mackenzie, D.: Mechanizing Proof: Computing, Risk, and Trust. MIT Press (2001)

Steingart, A.: A group theory of group theory: collaborative mathematics and the ‘uninvention’ of a 1000-page proof. Soc. Stud. Sci. 42, 185–213 (2014)

Villani, C.: Birth of a Theorem. Random House (2015)

Wiles, A.: Transcript of interview on PBS. www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html

The Power of Collaboration: polymath

Gowers, T., Nielsen, M.: Massively collaborative mathematics. Nature 461, 879–881 (2009)

The polymath Blog. polymathprojects.org

The polymath wiki. michaelnielsen.org/polymath1

“Is massively collaborative mathematics possible?”, Gowers’s Weblog, gowers. wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible

Nielsen, M.: Reinventing Discovery. Princeton (2012)

D H J Polymath: The ‘bounded gaps between primes’ polymath project: a retrospective analysis’. Newslett. Eur. Math. Soc. 94, 13–23

Examples, Conjectures, Concepts and Proofs: minipolymath

Mini-polymath3 discussion. terrytao.wordpress.com/2011/07/19/mini-polymath 3-discussion-thread/

Minipolymath3 project. polymathprojects.org/2011/07/19/minipolymath3- project-2011-imo

Martin, U., Pease, A.: Seventy four minutes of mathematics: an analysis of the third Mini-Polymath project. In: Proceedings of AISB/IACAP 2012, Symposium on Mathematical Practice and Cognition II (2012)

Questions and Answers: mathoverflow

Martin, U., Pease, A.: What does mathoverflow tell us about the production of mathematics? arxiv.org/abs/1305.0904

Tausczik, Y.R., Pennebaker, J.W.: Participation in an online mathematics community: differentiating motivations to add. In: Proceedings CSCW 2012, pp. 207–216. ACM (2012)

Mendes Rodrigues, E., Milic-Frayling, N.: Socializing or knowledge sharing?: Characterizing social intent in community question answering. In: Proceedings CIKM 2009, pp. 1127–1136. ACM (2009)

Everyday Calculation: GAP

Vaughan-Lee, M.: Groups of order $p^8$ and exponent p. Int. J. Group Theor. Available Online from 28 June 2014

Bailey, D.H., Borwein, J.M.: Exploratory experimentation and computation. Not. Am. Math. Soc. 58, 1410–1419 (2011)

GAP – Groups, Algorithms, and Programming, Version 4.7.7. The GAP Group (2015). www.gap-system.org

Harvey, J.A., Rayhaun, B.C.: Traces of Singular Moduli and Moonshine for the Thompson Group. arxiv.org/abs/1504.08179

Conclusions

Donaldson, S., Kontsevich, M., Lurie, J., Tao, T., Taylor, R.: Panel discussion at the 2014 award of the Breakthrough Prize. experimentalmath.info/ blog/2014/11/breakthrough-prize-recipients-give-math-seminar-talks/

du Sautoy, M.: How mathematicians are storytellers and numbers are the characters. www.theguardian.com/books/2015/jan/23/mathematicians-storytellers-numbers-characters-marcus-du-sautoy

Lakatos, I.: Proofs and Refutations. CUP, Cambridge (1976)

Obua, S., Fleuriot, J., Scott, P., Aspinall, D.: ProofPeer: Collaborative Theorem Proving. arxiv.org/abs/1404.6186