Affirmative Abstraction

There is a line of argument that takes this:

“There are an odd number of planets” (oddness used for posterity)

to imply this:

“There are numbers”

Fregeans might say that the quantifier represented in English by “there are” is actually about planets, so there is no implication for the existence of numbers. But I think we can convey truths about numbers in different ways that probably commits us to their existence. Say, for example:

“2 is prime”

Seems true enough, and I can follow a rudimentary argument from reference that bullies the constituents of any proposition with truth value into existence. But perhaps this doesn’t settle the question that metaphysicians are interested in, namely, what makes up the world.  Following Jonathan Schaffer, we’ll call views that treat all existence questions as metaphysical questions Quinean or “flat”. I don’t know that there is a good definition or creed that these flat philosophers would universally hold to, but I think it’s safe to say that they are interested in the existential quantifier. This is what does the heavy lifting in your ontology, and if you polish up your quantifier then you will have a good look under the skirt of the universe.

But perhaps you don’t like this view. Something fishy is going on when we move from facts about planets and slide to making numbers a real constituent of the world. So maybe things can exist without being a part of the world. Or, to put it differently, maybe some things can exist without being among the set of fundamentals like particles or propositions or strings or simples or whatever you think is fundamental (really real, not just kind of real).

I’m mostly interested in indispensability and how those kinds of arguments work; what are the Quinean gears pushing our intuitions around, elevating numbers and sets into the Platonic heaven? Following in the off the cuff attitude of this post thus far, I won’t produce any interesting insight into what numbers are. I don’t even think I could provide an historically accurate presentation of Quine’s own position. But it does make a difference if you ditch a flat ontology for a richer hierarchical vision like Schaffer or Heidegger. So maybe if you find arguments like I presented at the top not just wrong but misguided, then you might have a hierarchical view of existence. I’m counting fictionalism as a brand of this view, but that might be inaccurate. Suppose a Platonist asks a fictionalist how mathematical statements can be true without the existence of the referents. I suppose he could answer that he is just piggybacking on the Platonist and qualifying it with “They are true in language P”.  I take it that a hierarchicalist would say that of course numbers exist; they just aren’t constituents of reality. They exist in the linguistic traditions of Platonists. So the above argument:

“2 is a prime number”

Therefore

“2 exists”

Goes through, but it does not enrich our ontology. But back to indispensability. Suppose you think things quantified over in our best scientific theories exist. Well what then? Ditching a flat view of existence, there is still more work to be done. But what would convince these hierarchicalists that numbers are actually constituents of reality? I’m not so sure. Right now I’d have to say that there is no one-size-fits-all method that just ushers in a clean and neat ontology. For numbers, perhaps something like Neil Tennant’s arguments go through, and this would show that numbers are real abstracta existing across all possible worlds. For propositions, there are interesting ersatzean arguments for the necessity of propositions that, if they went through, would probably convince me of their ontological gusto. And so on and so forth.

Anyway, this is probably a topic I’ll be revisiting as I dive headfirst into metametaphysics literature.