Abstract
We look at how Anglophone mathematicians have, over the last hundred years or so, presented their activities using metaphors of landscape and journey. We contrast romanticised self-presentations of the isolated genius with ethnographic studies of mathematicians at work, both alone, and in collaboration, looking particularly at on-line collaborations in the “polymath” format. The latter provide more realistic evidence of mathematicians daily practice, consistent with the “growth mindset” notion of mathematical educators, that mathematical abilities are skills to be developed, rather than fixed traits. We place our observations in a broader literature on landscape, social space, craft and wayfaring, which combine in the notion of the production of mathematics as crafting the exploration of an unknown landscape. We indicate how “polymath” has a two-fold educational role, enabling participants to develop their skills, and providing a public demonstration of the craft of mathematics in action.
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References
Barany, M, & Mackenzie, D. (2014). Chalk: Materials and concepts in mathematics research. In Representation in scientific practice revisited (pp. 107–130). MIT Press.
Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Wiley & Sons.
Bourdieu, P. (1984). Distinction: A social critique of the judgement of taste, tr. Richard Nice. Harvard University Press.
Bourdieu, P. (1985). The social space and the genesis of groups. Theory and Society, 14, 723–744.
Bundy, A. (1988). The use of explicit plans to guide inductive proofs. In International Conference on Automated Deduction.
Calegari, F. (2017). galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/.
Crowdmath. (2015). https://artofproblemsolving.com/polymath.
Davis, P. J. & Hersh, R. (1995). The mathematical experience (study ed.). Birkhauser.
De Morgan, A. (1842). The differential and integral calculus (p. vii). Baldwin and Cradock.
du Sautoy, M. (2015). How mathematicians are storytellers and numbers are the characters. The Guardian. www.theguardian.com/books/2015/jan/23/mathematicians-storytellers-numbers-characters-marcus-du-sautoy.
Dweck, C. S. (2006). Mindset: The new psychology of success. New York, NY: Random House Incorporated.
Ehrhardt, C. (2010). A social history of the “Galois Affair” at the Paris academy of sciences. Science in Context, 23(1), 91–119.
Gowers, T. (2000). The two cultures of mathematics. In V.I. Arnold (Ed.), Mathematics: Frontiers and perspectives. AMS. https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf.
Gowers, T. (2009). https://www.gowers.wordpress.com/2009/01/27/ismassively-collaborative-mathematicspossible/. Gowers, T., & Nielsen, M. (2009). Massively collaborative mathematics. Nature, 461, 879–881.
Hardy, G. H. (1929). Mathematical proof. Mind.
Harris, M. (2015). Mathematics without apologies. Princeton.
Heidigger, M. (1962). Being and time, tr J. Macquarrie & E. Robinson. Harper & Row.
Hollings, C., Martin, U., & Rice, A. (2017). The Lovelace–De Morgan mathematical correspondence: A critical re-appraisal. Historia Mathematica, 44(3), 202–231.
Ingold, T. (1993). The temporality of the landscape. World Archaeology, 152–174.
Ingold, T. (2011). Being alive: Essays on movement, knowledge and description. Taylor & Francis.
Jenkins, A. (2007). Space and the ‘March of Mind’: Literature and the physical sciences in Britain 1815–1850. OUP.
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Lane, L. (2017). The bridge between worlds: Relating position and Ddsposition in the mathematical field, Ph.D. thesis, University of Edinburgh
Levi-Strauss, C. (1966). The savage mind. University of Chicago Press.
Mackenzie, D. (2003). An equation and its worlds: Bricolage, exemplars, disunity and performativity in financial economics. Social Studies of Science, 33, 831–868.
Martin, U. (2015). Stumbling around in the dark: Lessons from everyday mathematics. In A. P. Felty & A. Middeldorp (Eds.), Proceedings of CADE-25. Lecture Notes in Artificial Intelligence (Vol. 9195). Springer.
Mason J., & Hanna, G. (2016). Values in caring for proof. In B. Larvor (Ed.), Mathematical cultures. Springer Trends in the History of Science.
Murray-Rust, D., et al. (2015). On wayfaring in social machines. In Proceedings of the 24th International Conference on the World Wide Web (pp. 1143–1148).
Neale, V. (2017). Closing the gap, the quest to understand prime numbers. Oxford University Press.
Polya, G. (1945). How to solve it. Princeton University Press.
Polymath, D. H. J. (2014). The ‘bounded gaps between primes’ polymath project: A retrospective analysis. Newsletter of the European Mathematical Society, 94, 13–23.
Pye, D. (1968). The nature and art of workmanship. Cambridge University Press.
Roberts, S. (2015). John Horton Conway, the world’s most charismatic mathematician. The Guardian. www.theguardian.com/science/2015/jul/23/john-horton-conway-the-most-charismatic-mathematician-in-the-world.
Robinson, A. (2011). Genius, a very short introduction. Oxford University Press.
Ryle, G. (1971). Thinking and self-teaching. Journal of Philosophy of Education, 5, 216–228.
Simon, S. (2009). Newton on the beach: The information order of Principia Mathematica. History of Science, xlvi. Science History Publications Ltd.
Sewell, W. H. (1992). A theory of structure: Duality, agency, and transformation. American Journal of Sociology, 98, 1–29.
Tanswell, F. (2017). Proof, rigour and informality: A virtue account of mathematical knowledge. Ph.D. thesis, University of St Andrews.
Tao, T. (2007). Does one have to be a genius to do maths? terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/.
Tao, T. (2017). Blog comment on “The ABC conjecture has (still) not been proved”. galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4563.
Thomas, R. (2016). Beauty is not all there is to aesthetics in mathematics. Philosophia Mathematica, 25, 116–127.
Turnor, E., Collections for the history of the town and soke of Grantham (London, 1806, Vol. 173, No. 2), where it is claimed this was said by Newton “a little before his death”.
Villani, C. (2015). Birth of a theorem: A mathematical adventure. Farrar, Straus and Giroux.
Wiles, A. (2000). Transcription of interview by PBS. www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html.
Zeitz, P. (2006). The art and craft of problem solving. Wiley.
Acknowledgements
We thank Dave de Roure and Pip Willcocks for helpful discussions, and the referees for their thoughtful comments. Support from the UK Engineering and Physical Sciences Research Council is acknowledged under grants EPSRC EP/K040251/2 (Martin, Lane, Tanswell), EP/J017728/2 (Murray-Rust) and EP/P017320/1 (Pease).
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Lane, L., Martin, U., Murray-Rust, D., Pease, A., Tanswell, F. (2019). Journeys in Mathematical Landscapes: Genius or Craft?. In: Hanna, G., Reid, D., de Villiers, M. (eds) Proof Technology in Mathematics Research and Teaching . Mathematics Education in the Digital Era, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-28483-1_9
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