Number and Existence

I haven't posted much lately, because I've been studying a lot of things unrelated to Scotus and I'm always uncertain how far afield this blog should roam. For the last week, for instance, I've been reading Hegel's Phenomenology of Spirit pretty intensively, along with Kalkavage's commentary, but I don't know that Smithy readers want to read about that.

They may not want to read about this, either, but I've seen a couple of references to Bill Vallicella's post about Inwagen and existence. Brandon Watson has a post on it, for instance, saying that Vallicella went too easy on Inwagen. I agree, and I also think that Watson went to easy on Vallicella, since in my opinion Inwagen's argument is worse than either of them indicate.

Here is the Maverick Philosopher:

Van Inwagen begins by noting that number words such as 'six' or 'forty-three' do not "mean different things when they are used to count objects of different sorts." Surely he is correct: "If you have written thirteen epics and I own thirteen cats, the number of your epics is the number of my cats." So the first premise of the argument is the indisputable:

1. Number-words are univocal in sense: they mean the same regardless of the sorts of object they are used to count.

I am okay with this. But not with this:

"2. "But existence is closely allied to number.". . . Van Inwagen proceeds: "The univocacy [univocity] of number and the the intimate connection between number and existence should convince us that existence is univocal." The conclusion of the argument, then, is:

3. Existence is univocal.

Vallicella is not okay with it either, but to my mind not for the right reason. Vallicella accepts that "existence is closely allied to number", but doesn't give a good reason for thinking so. He says, "to say that unicorns do not exist is equivalent to saying that the number of unicorns is zero, and to say that horses exist is equivalent to saying that the number of horses is one or more." I don't see that this is necessarily true at all. It depends on whether we're already talking about existence or not.

Take the following two statements:

a) The number of cats in the room right now is two.

b) Of the four hobbits that set out for Mount Doom, the number that arrived is two.

Now, the number "two" is univocal in both statements; the number two means the same thing regardless of the sorts of object it is used to count. And the two statements are true: my cats are two and Frodo and Sam are two, and in the same sense. But obviously the two hobbits don't have existence in the way that the cats do: my cats have actual existence and the hobbits don't and never did. Number numbers existing beings, but it can be used equally well to number things without existence. Non-existing things are numbered by the same number as existing things. So whence comes this "existence is closely allied to number"? I would propose that "to say that unicorns do not exist is equivalent to saying that the number of unicorns is zero" is only true when it's already clear that our domain of discourse is the actually existing world, which it often is not.

And what about numbers themselves? Do they exist? Do they exist in just the same sense that cats and dogs do? The number of cats in the room is two; does it make sense to ask what is the number of twos in the room? Numbers can be numbered; the number of primes between 1 and 10 is 4 (2,3,5,7). Do these four primes exist, then? But there are good reasons to claim that there cannot be an actually existing multitude; but the number of numbers is infinite. Do numbers then not exist, or just not all of them? Does a number have to number an existing multitude of things to exist? Call the number of existing particles in the universe (x); do the numbers (x) and (x+1) have the same kind of existence? The number of things that can be numbered by (x) is 1 (the collection of particles in the universe); the number of things that can be numbered by (x+1) is 0. Does this mean that the number (x) exists but that (x+1) doesn't? Do negative numbers etc. have actual existence or are they beings of reason?

Note that I'm not saying these issues can't be resolved, or that (for token Scotus relevance) we don't have a univocal concept of being, but that, while existing things can be counted or more generally quantified, not everything that can be counted or quantified exists. In this sense, the sense that whatever exists can be numbered, though number does not exhaust the being of anything that actually exists, we might say that "existence is closely allied to number." But then whatever exists can be cognized, so we might also say that "existence is closely allied to thought," or any number of similar statements. But we shouldn't infer anything about the nature of existence from this kind of thinking. We might as well argue that color as applied to men and holograms is univocal, since we see color in a hologram of a man and an actually existing man in the same way, so holograms and men have color in the same way, therefore holograms and men have the same kind of being.

Update: On further thought, I think a more useful approach would be to consider the transcendental convertibility between unity and being. Aristotle and his followers all agree that something has being just to the extent that it is one. But this unity is not numerical unity, the unity of counting, because different sorts of beings, e.g. fictional hobbits and real cats, can be counted with the same numbers. Scotus recognizes a less than numerical unity, the unity of universals; there is numerical unity, the unity whereby a thing can be counted as one item; perhaps we should also recognize a more than numerical unity, the unity of a real being, which comes in degrees.