The fact numbers ain’t for counting themselves implies that the ‘existence’ of numbers presupposes the prior existence of some other more elementary class of things. Numbers-in-themselves are absurd. What do they number? That implies there exists no interpretation of first order logic in which the numbers can be taken as the values of the elementary variables. That implies the falsehood of Platonism, since Platonism is nothing other than the taking of the numbers as the values of (a subset of) the elementary variables. This also implies type theory must be true, since some sets are more elementary than others, the notion of whose elementary existence is really nonsensical. The absurdity of the Von Neumann and Zermelo constructions probably consists at least partly in the fact that they treat the numbers as counting themselves. The Frege-Russell definition is preferable for precisely this sort of reason.

I’m leaning increasingly to the view that intuitionism is true of mathematical reasoning. I think intuitionism is consistent with a bifurcation of logic : there exists a qualitative logic for reasoning about phenomenal states of affairs, and a quantitative (intuitionist) logic for reasoning in mathematics. This way we can accept that a proposition like the continuum hypothesis is meaningless; even uninteresting propositions like Goldbach’s conjecture lack a truth value, until such time as a proof is constructed. I might even concede to the intuitionists that their mathematical intuitions are pre-logical, such that intuitionist logic is developed secondarily as the logic of the immediacy of those intuitions. Classical logic will then remain as the synchronic structure of any possible quality-language. I remain uneasy about this view because it fails to do justice to the stunning fact of the almost complete reducibility of mathematics to classical logic, and also because the immediacy of our supposed mathematical intuitions seems deniable : it has long been my view that an a-arithmetic linguistic community is possible, and it seems highly likely that such communities have existed. This fact constitutes a strong challenge to intuitionism. Still, the implausibility of notions such as ‘there are cardinalities higher than aleph-null’, and ‘the continuum hypothesis has a truth value’, does speak in favor of intuitionism. I think I still somewhat favor reductionist, intensionalist logicism. Classical logic is transcendental and is in essence the intensional logic of phenomenal reasoning. Classes are reducible to intensional objects, and numbers reducible to classes. All talk about even finite numbers is talk about logical constructions, carrying no existential commitment. Surely it’s preferable to retain the univocity of logic.