Abstract

For centuries, the highest level of mathematics has been seen as an isolated creative activity, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers, using programs designed to verify hardware, check proofs that are just too long and complicated for any human to comprehend.

Mathematical practice is am emerging interdisciplinary field which draws on philosophy, social science and ethnography, and the input of mathematicians themselves, to understand how mathematics is produced. Online mathematical activity provides a rich source of data for empirical investigation of mathematical practice – for example the community question answering system mathoverflow contains around 40,000 mathematical conversations, and polymath collaborations provide transcripts of the process of discovering proofs. Such investigations show the importance of “soft” aspects such as analogy and creativity, alongside formal deduction, in the production of mathematics, and give us new ways to think about the possible complementary roles of people and machines in creating new mathematical knowledge

Social machines are new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.