Limitations of Panmathematism

Abstract

The confident belief in the unrestricted applicability of logico-mathematical formalism to experience has its historical roots in metaphysical panmathematism, according to which the world itself consists of mathematical entities and their relations. This was the belief of the Pythagoreans when they claimed that things are made of numbers; also of Plato when, in the Timaeus, he constructed matter from elementary plane triangles. In such an extreme form panmathematism, or rather pangeometrism, hardly occurred in the modern era, even though Descartes in his ambitious attempt at reducing matter to geometrical space came very near to it. But once the distinction between homogeneous causally inert space and its material content had been made, pangeometrism in its radical form was all but impossible. Even the homogeneous Cartesian matter was not, as Leibniz pointed out, completely reducible to space since its impenetrability, not to speak of its dynamical manifestations, was not a geometrical property. Similarly, the atoms of Democritus, Gassendi and Dalton are not equivalent to the places they occupy; the former are indivisible and moveable, the latter are mathematically continuous (infinitely divisible) and motionless, being the eternally unmoveable portions of static Euclidian space.