> OK, but, like, which one? There are lots of sets that this describes.
Any, and all. One can do the thinking without specifying what. I believe this is the notion of a 'universal property' in category theory.
> If 0 and 3 aren't sets then what are they?
I mean, if you want to give me a construct I would use, I'd say the following
0 = \s -> \z -> z
3 = \s -> \z -> s s s z
If you say that these are akin to church numerals, that is correct if you look at it as a lambda calc term. However I'm saying something much stronger than just a lambda calculus term. That's just my notation.
I'm saying that zero is the concept of taking two things (one an initial thing and the other a method to take one thing and get the next thing) and then returning the initial thing. While three is the concept of taking the same two things and then doing the method three times on the first thing. I used lambda notation because that is an easy way to convey the idea, not because that is what three is.
> Then what is it? If you say it's a set with those properties, then it should be a set and we can ask set questions about it. If you say it's somehow all of the sets with those properties, or some other concept that somehow encompasses them, then you still haven't explained what it actually is.
It's a set in the colloquial sense, not in the ZFC sense.
There's an idea of a thing, and then the definition of the thing. Both are valid in my mind.
For example, heat is the movement of atoms. It's also the energy in a thing. But it's also the feeling of hot that I cannot describe. It's also an idea in its own right. Things can be many things. There's an infinite hierarchy of description that one may apply to concepts, and, in my philosophy, at some level the description becomes unable to be expressed.
> One can do the thinking without specifying what.
After one has figured out a rigorous approach, one can generally ignore it in day-to-day work. But you need to have that foundation there or you can go horribly wrong. (Analogy: we don't generally think about limits when doing day-to-day calculus. But when people tried to do calculus pre-Weierstrass, unless they were geniuses they "proved" all sorts of nonsense and got crazy results).
> I believe this is the notion of a 'universal property' in category theory.
That's one of the good answers to this problem. But you still need the constructions (if only to prove that an object with the required properties exists at all) and you need a lot of the kind of work the article is talking about, where you establish that these different constructions are equivalent and you can lift all your theorems along those equivalences.
> I'm saying that zero is the concept of taking two things (one an initial thing and the other a method to take one thing and get the next thing) and then returning the initial thing. While three is the concept of taking the same two things and then doing the method three times on the first thing. I used lambda notation because that is an easy way to convey the idea, not because that is what three is.
OK but then what kind of things are these "concepts"? What kind of questions can and can't we ask about them? If you care about "what the natural numbers are" then these kind of questions are worth asking.
Any, and all. One can do the thinking without specifying what. I believe this is the notion of a 'universal property' in category theory.
> If 0 and 3 aren't sets then what are they?
I mean, if you want to give me a construct I would use, I'd say the following
0 = \s -> \z -> z 3 = \s -> \z -> s s s z
If you say that these are akin to church numerals, that is correct if you look at it as a lambda calc term. However I'm saying something much stronger than just a lambda calculus term. That's just my notation.
I'm saying that zero is the concept of taking two things (one an initial thing and the other a method to take one thing and get the next thing) and then returning the initial thing. While three is the concept of taking the same two things and then doing the method three times on the first thing. I used lambda notation because that is an easy way to convey the idea, not because that is what three is.
> Then what is it? If you say it's a set with those properties, then it should be a set and we can ask set questions about it. If you say it's somehow all of the sets with those properties, or some other concept that somehow encompasses them, then you still haven't explained what it actually is.
It's a set in the colloquial sense, not in the ZFC sense.
There's an idea of a thing, and then the definition of the thing. Both are valid in my mind.
For example, heat is the movement of atoms. It's also the energy in a thing. But it's also the feeling of hot that I cannot describe. It's also an idea in its own right. Things can be many things. There's an infinite hierarchy of description that one may apply to concepts, and, in my philosophy, at some level the description becomes unable to be expressed.