FRESH BREEZES IN THE PHILOSOPHY OF MATHEMATICS

Since Pythagoras, philosophy of mathematics tried to account for mathe matical existence and the nature of mathematical objects. Numbers, cir cles, n-dimensional manifolds, all are different from everything else we think about. They're neither physical nor mental. Not mental, becau se the Pythagorean theorem or any other well-established mathematical fact is independent of what you or I think. Whether we know it and bel ieve it or don't know it and don't believe it, the Pythagorean theorem is still true. Yet it's not physical either! Plate and Aristotle expl ained that the triangles and circles of the geometer are not physical triangles or circles, but something ''ideal.'' Spiritual, empirical, p sychological, formalist, and logicist explanations have been offered. None give a credible account of what we do when we do mathematics. Pre sently some authors are constructing a humanist answer.