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.