Peter Van Inwagen defined his theory of metaphysics by saying that being was either equivalent to or closely tracked what could be numbered. I owe him a debt since I think this idea is perfectly and brilliantly wrong. If we get down to the very bottom of things, numerable being is exactly what metaphysics seems to be about – what it self-evidently seems to be about – but isn’t. The attraction of the idea is both from its simplicity and obviousness and its promise of power. If being is the numerable, then mathematics is metaphysics – what is most intelligible to us is most intelligible in itself, and the ease and power of mathematics becomes the key to being as such. Because of its algebraic manipulation, symbolic logic is seen as taking part in this mathematical power and attracts us with the possibility of resolving all philosophical disputes by Leibnizian calculation. Putting it like this is more triumphalist and universal than an Analytic philosopher would be comfortable with, but it’s true all the same.
I’ve previously argued that this sort of metaphysics is contradictory since it requires that everything that exists (the one of being) is part or source of a larger whole (the one of number), which leaves us having to define existence in opposition to the totality of all things (regardless of whether this whole is finite or not) even while the word “existence” above all is coined in order to mean the totality of all that is since it names what is common to all that is.
Another argument is based not on the division of one but an an account of number itself:
So far as something is potential, it has being derivatively and by another
Things are numbered so far as they are potential.
First, a note on the composition of act and potency:
Aristotle introduces act and potency as intrinsic principles of movable being, and so he considers them parts of the being they constitute. But there is a crucial distinction between act and potency as parts and the integral parts that constitute a whole, like the bricks that make a wall or the organs that make up the nervous system. Act is a principle that constitutes the whole as such and it is only together with potency when the being has constitutive realities (parts) that are distinct from the whole. If such parts exist, they are given being and coordinated by form. Each of the parts is interactive and dependent on the other parts, but form is not dependent on parts – it is a part only in the sense of being a principle of the whole.
But – and I take this as self-evident – everything able to be numbered is a part of some whole*; and to be a part of a whole belongs to a thing so far as it is potential, then numerable beings are only possible to the extent they are potential. This potency divides into three strata:
1.) What is potential both in its being and its activity. Such existence is material, and can be numbered both in its activities and its substance
2.) What is potential in its activity, but not in its being. This is subsistent intellectual activity which uses more than one concept to know all things distinctly. At its summit, this is whatever angel can know all things distinctly either by two concepts. These concepts admit of real enumeration, even in angels, and so are characterized by a discourse and so by a sort of time. This is the sense of Augustine’s observation that God moves all creatures (even angels) in time. It is a reference to the multiplicity of concepts required for all finite intellects. These beings are numbered in their activities but not in their substance.
3.) Neither potential in activity nor in being. At this level, division does not arise from finitude or opposition. Division in God does not result in parts or in individuals that are enumerated. The Father is different from the Son is different from the Holy Spirit, but this division is not into parts but into individuals that are each the entirety of what is in finite beings the class or totality in which they are contained.
Now so far as the Father, Son and Spirit are things known to us or concepts in our mind, then they share in the multiplicity that all such concepts have. In fact, all second intentions (concepts as conceived by us) have this sort of multiplicity. But this is a fact about our understanding and not about things. We might “number” all the angels too, but this is also doing nothing but numbering the second intentions of a human mind. This is the only sense in which divinity might be numbered, whether in substance or activity.
It is this sheer homogeneity of human second intentions that gives us the illusion that being is numerable, and this is at the heart of the error of the “thin theory of being” that Van Inwagen sees as metaphysics. He thinks he is counting beings when in fact he’s only counting thoughts about things in the lowest sort of intellectual being.
*Whether this whole is infinite or finite is of no difference here, only that each of the parts is constitutive of it.