Intuition and Rigor : Some Problems of a ‘Logic Of Discovery’ in Mathematics

In a famous intervention at the Second International Congress of Mathematicians (Paris, 1900), Poincar\ue9 divided research mathematicians, from Euclid onwards, into two groups: those who are rigorous (or \u2018analysts\u2019) and leave nothing to chance, and those who are intuitive (or \u2018geometricians\u2019) and make rapid, but often precarious conquests. While recognizing that \u201cboth analysis and synthesis have their legitimate role\u201d and that consequently mathematical research, in order to make progress, cannot do without either group, he added, however, that \u201cin becoming rigorous, mathematics assumes an artificial character [. . .]; it forgets its historical origins.\u201d2 Regressive results, in research not less than in didactics, could be avoided only by carefully distinguishing between the roles of the two groups: with \u2018analysis\u2019 it is mathematical proof; with \u2018synthesis\u2019, on the other hand, it is mathematical invention