There are many kinds of concepts at our disposal, and one is a concept that refers to an existent. This is why “empty space” is a concept of method, because it does not directly (but parenthetically) refer to an existent.
I think, in a literally sense, concepts don’t exist, but some concepts refer to existents. That is, concept qua concept doesn’t exist, they are mental tools we use to refer to reality.
So, if we have an idea of a physical location in space that does not contain physical entities, our idea does not refer to entities, but for the idea to make any sense, we parenthetically, in the back of our minds, know that only entities are measurable and it is the existents of entities that allow us to discuss “empty space,” for “empty space” is the absence of the entities that we would otherwise have measured.
To answer your last question, location is not the same thing as the concept of location. Concepts can refer to existents, but they are not the existent.
Same goes for a set like the natural numbers–this set qua set doesn’t actually exist, it is a tool that can be used to refer to things that do exist. This doesn’t invalidate the concept, it just defines when and how it can be used interchangeably with actually things that exist.
The first four paragraphs of your post make good sense. And I think it needs to be seen how set theory employs what you call concepts of method and that these concepts of method do not entail the ontological absurdities that detractors claim them to entail.
In the present case, the set of natural numbers is an extremely useful concept of method. But to posit (or conceive, as a concept of method, mind you) the existence of the set of natural number is to posit the existence of an infiinite set. And once you’ve posited the existence of an infinite set, the other (pretty non-controversial) axioms ensure that there are even larger infinite sets. We can’t get around that. It’s a cold stone truth of logic that if we posit just one infinite set, then (with the other non-controversial axioms) we have to accept that there are sets of even larger infinititude than that of the set of natural numbers, even if only as concepts of method. There shoud be no railing against this. There should be no charges of ‘mysticism’ or ‘false ontology’ or any of that.
On the other hand, one is free to reject the axiom of infinity as even a working concept of method. But then it’s not clear how one would derive even the basic mathematics of the real numbers, let alone the basic set theory needed to perform the meta-logic needed for first order logic itself. Also, one may reject classical logic and adopt instead an intuitionist logic or some other system. But these have drawbacks too, and are usually much less intuitive and much more complicated just to get off the ground with the formal semantics than classical logic.
“[…] a set like the natural numbers–this set qua set doesn’t actually exist, it is a tool that can be used to refer to things that do exist.”
-
Aside from the set of natural numbers, which is an infinite set, my question was what do you take to be the ontological status of the numbers themselves. For example, what do you think the number two is? Is it an existing entity or is it a concept of method?
-
If you don’t take the set of natural numbers to be an existing set but only a concept of method, then why take any set to be anything other than a concept in method?