A research mathematician's work consists largely of solving problems that arise within mathematics or beyond it, whether in other scientific disciplines or in everyday life. General problem-solving methods are regularly discussed in our pages (see the "Mathematics and psychology" feature published in Tangente 159, 2014, or the feature on "Different ways of solving problems" published in Tangente 183, 2018). Broadly speaking, these heuristics are based, explicitly or implicitly, on René Descartes's (1596–1650) famous Discourse on the Method: he believed that "geometers [that is, mathematicians] are accustomed to using [only four precepts] to arrive at their most difficult proofs" (see box); in particular, he recommended "dividing each of the difficulties […] into as many parts […] as might be required to solve them better".
Breaking with Descartes?
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In his research, the British mathematician Sir William Timothy Gowers (born in 1963, Fields Medal 1998) departs from Cartesian precepts because he often encounters mathematical problems that, at first sight, cannot be broken down into several easy-to-solve subproblems. At the very beginning of 2009, he therefore wondered whether it might be more effective to change the traditional way of working in mathematics and tackle difficult questions in collaboration with a larger number of colleagues. On his blog (***), he reasoned as follows: "Luck is needed to have the idea that solves a problem. [So], if many people think about it, there is a greater chance that one of them will get lucky." He thus posed the question online of whether "massively collaborative mathematics is possible". He returned to this issue in an article written with Michael Aaron Nielsen, a specialist in "open science" (or open science), and published in the journal Nature*.
