FRESH BREEZES IN THE PHILOSOPHY OF MATHEMATICS

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