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), Poincaré divided research mathematicians, from Euclid onwards, into two groups: those who are rigorous (or ‘analysts’) and leave nothing to chance, and those who are intuitive (or ‘geometricians’) and make rapid, but often precarious conquests. While recognizing that “both analysis and synthesis have their legitimate role” and that consequently mathematical research, in order to make progress, cannot do without either group, he added, however, that “in becoming rigorous, mathematics assumes an artificial character [. . .]; it forgets its historical origins.”2 Regressive results, in research not less than in didactics, could be avoided only by carefully distinguishing between the roles of the two groups: with ‘analysis’ it is mathematical proof; with ‘synthesis’, on the other hand, it is mathematical invention.