Uninstantiated Universals and ‘Semi-Platonist’ Aristotelianism

Abstract

Aristotelian philosophy of mathematics holds that the objects of mathematics – such properties as symmetry, continuity and order – are realized in the physical world, so that mathematics is a science of aspects of the world, as much as biology is. The principal objection to that thesis is, ‘Some of the objects of mathematics are not realized in the physical world, such as large infinite numbers’. It may be that the world is finite, in which case infinite numbers and very large lengths are not instantiated in the real world. Even more so the higher infinities: ‘set theory is committed to the existence of infinite sets that are so huge that they simply dwarf garden variety infinite sets, like the set of all the natural numbers. There is just no plausible way to interpret this talk of gigantic infinite sets as being about physical objects.’¹ Or as Shapiro writes:

It seems reasonable to insist that there is some limit to the size of the physical universe. If so, then any branch of mathematics that requires an ontology larger than that of the physical universe must leave the realm of physical objects if these branches are not to be doomed to vacuity. Even with arithmetic, it is counterintuitive for an account of mathematics to be held hostage to the size of the physical universe.²