Set theory, on the other hand, wasn't invented till the 1870's, not put into its current form till fifty years later in the 19020's, and today goes far far beyond anything that refers to the physical world.
Now of course set theory in its naive form does tell us about how collections behave. Unions, intersections, one-to-one correspondence, and the like, are useful ideas in the real world. But set theory goes very far beyond that to infinitary reasoning that has no reference to the physical world. So no, to answer your question, I do not believe modern set theory tells us much about tallying. The naive set theory of finite sets perhaps does, but modern theory begins with infinite sets and does not refer to the real world. As far as we know, of course. All human knowledge is historically contingent.
Again, finite combinatorics. Not even taught in set theory class. Set theory begins with infinite sets and goes far beyond, to study axiomatic systems whose very meaning even in the world of abstractions is far from clear, let alone the real world.
Finite combinatorics again. I'm perfectly willing to agree that finite combinatorics refers to the real world.
Set theory? Not a bit of it.
ps -- I might have been more clear originally if I'd carefully distinguished modern set theory from basic finite combinatorics and tallying. We'd then be in basic agreement that finite combinatorics tells us something real about the world; whereas with modern set theory, the question is at the very least highly debatable.
pps -- On the other hand I raised my doubts using the examples of the empty sets and singletons, which are part of basic, elementary set theory. So perhaps instead of invoking higher set theory I should stick with the basics.
Do you think the empty set refers to something in the physical world? It's a good question, there's no right answer. Similarly, if we both agree that there's an apple sitting on teacher's desk, is the set containing the apple sitting on the desk as well? I say no. Only in abstract set theory does such a thing exist. Not in the physical world.
Mathematics is abstract. It's not even meant to refer to anything concrete in the physical world, such as a particular set of apples or a particular set of cardinals. In that, maths works very much like a language does.
Words, most of them at least, are not thought of a referring to any concrete thing. The word "
apple" doesn't refer to any concrete thing, such as a particular apple. However, speakers can obviously use words to refer to concrete objects. Like in the utterance, "
Give me this apple" or, "
Give me those five apples". Yet, language is getting more like mathematics if we say instead, "
Give me one apple" or "
Give me five apples". These are very concrete utterances that people make everyday around the world, yet, in these two last examples, "
one apple" and "
five apples" don't refer to anything concrete like a particular apple or a particular collection of five apples. Instead, they specify what is expected in abstract terms so that the speaker leaves it to the discretion of the hearer to decide which particular apples he will pick to make up one apple or five apples. In the end, the speaker will end up with one apple or five apples, as expected, and yet without ever specifying which particular apples to wanted. This way of doing things obviously works very well. It's very concrete. It's an ordinary, everyday situation not involving highfalutin mathematicians and yet it involves the use of very abstract language.
This demonstrate I think that abstraction is at the heart of our relation with the physical world and that it's no going to go away.
So, no, to answer your question, the
empty set doesn't refer to anything concrete. Yet, we can use it to communicate just as we routinely use the expression "
five apples", which also doesn't refer to anything concrete, to communicate about the physical world, and specify what we want. And it works.
As I see it, it's just a category error to think of the empty set as meant to refer to something. Words don't refer. Not by themselves. We can use them to refer to something but then we'll use a complete description of what we mean. If I say, "
I want those five apples", it specifies clearly, at least in context, what I want. So, it's me in this context who is referring to five particular apples.
The notion of empty set can be used to specify for example the result of a process as having produced nothing. So, it works much like the word "nothing". And like nothing, there's just one. It's a mistake to talk of "
empty sets" in the plural. There's just one, even if in practice we may use the plural. Same thing for "two" or "one thousand". There's just one number 2 and there's just one number 1000. This is because these notions are indeed all abstract and so an empty set of apples is the same thing as an empty set of dinosaurs, just as the number of apples in a pair is the same as the number of dinosaurs in a pair.
So, if the empty set isn't the kind of thing that could refer, do we have empty sets in the physical world? Well, in the same sense as we have sets of apples and sets of dinosaurs. A set of apple is nothing beyond the apples themselves considered collectively. Since we certainly consider that each apple exists as a concrete, physical object, we're left we only the option of considering that the set of apples doesn't exist as a set. All we have is the set of apples if by that we mean a particular collection of apples.
I think this shows we can't separate our consideration of abstractions, such as sets and numbers, from how we use them in practice, which is to communicate about the physical world. Yet, the job of mathematicians is precisely to forget about that and consider mathematical abstractions independently of how we might get to use them. It's their job and it seems they are good at it.
I think the problem only arrise when we try to think of mathematical concepts, such as sets and numbers, but that's true of all mathematical concepts, as possibly referring to something concrete. The number 2 just doesn't refer. We can use it to refer concretely to two particular apples or more abstractedly to any two apples. But 2 doesn't refer. There's no concrete thing that's a 2 in the physical world. Yet, we all understand what people mean when they talk of there being two apples in one basket. Similarly, we can use the empty set to talk about there being no apples in this one basket. We all understand, I hope.
So, obviously, if we understand and are able to act on what we understand, this suggests that there's something real in the physical world that somehow corresponds to the abstract notions we use to talk about it. The very abstract notion of apple can be used to refer to a very concrete object, there on the table in front of me, and we will all understand what it is. Yet, the notion of apple is abstract. The notion is no particular apple. It's not concrete. It's nothing physical. It's not physical yet we can't stop using it. We effectively need abstract notions to talk about the physical world.
EB
