Counting

[gfish]

As part of my ongoing book research, I've been reading a lot of mathematical philosophy. Often that ends up being related to logicism, the search for a way to phrase all of math in terms of pure logic. It's not directly relevant to the book, but it's important background so I usually wade through it anyway.

One important detail is the attempt to remove the act of counting from the concept of numbers. It's not enough to say that the set of all sets with three members is itself the number three, as that's circular. The solution is to work with one-to-one relations: for all X there exists a Y in this relation to X, and for all Z in this relation to X, Z=Y. That is, X and Y are in this relation, and they are only in this relation with each other. Russell's example is, if you ignore polygamy and polyandry (yes, he explicitly excludes these, which makes more sense if you've read his autobiography), then you know that the number of wives and the number of husbands are equal, even though you have no way of knowing what the actual count is. Thus you can ask if every member of a set can be matched with a member of a three set with none left over. If so then it is also a three.

That's all well and good, and struck me as quite clever when I first saw it. But I'm increasingly convinced it's faulty. Distinguishing between zero and not zero is still counting. That gets hidden with the use of first order logic, but both the universal quantifier ∀ and the existential quantifier ∃ require counting. To say ∀x f(x) means that you have counted the number of x's for which f(x) is false and found that number to be zero. To say ∃x f(x) is to say that you've counted the number of x's for which f(x) is true and found that number not to be zero.

Of course, logicism is long since dead and doesn't need more nails in its coffin. But as someone who can't help but look for a touch of platonism to explain the unreasonable effectiveness of mathematics, I can't help but wonder if something important was missed here. Like the "between" concept which grounds Hilbert's axiomatization of geometry, could this unavoidable "is/is not" distinction hint at a fundamental feature of the "deep logic" of the universe?

Comments

[juli]

Is the distinction between comparison and counting as accumulation? Like, comparing 0 and 1 is a direct comparison between the two things and does not require any grasp of higher numbers? It seems like sets and numbers encoded in unary form here are equivalent, for instance like Church encoding of numbers in the lambda calculus. Certainly it's computationally equivalent; so the only thing I can think of is that counting requires history, a stack, an accumulator, something; whereas that approach of equivalence allows you to throw out information at each step, the final state is independent of the previous states. This is easy to demonstrate in lambda calculus, but I'm not sure there's a meaningful distinction over counting since, like, the set itself represents all of the information that would be encoded in an accumulator, or on a stack, or whatever. It's the same information in different forms.

(I am wildly out of my depth on these things, but the linkage between numbers in Church encoding and sets is so clear that the faultiness seems evident to me at least in that space. Like, this is no different to a trivial equivalence function which does no math, and a naive one which checks if either is zero, and then decrements each and recurses.)

[gfish]

I think that would be the official response, anyway, if all the relevant people hadn't died before I was born. I guess I'm trying to sort how much counting I think there is in recognizing one (and only one!) of something.

(Honestly I'm completely out of my depth here as well, which I why I ended up cutting the directly relevant chapter out of my book.)

[juli]

Have you spent much time reasoning about how Church encoding of numbers works? I mean, it's just combinatory logic at that point, which feels innately different to counting for me. So it's not so much "is this 1 or 0" as like "T or F?" You can represent higher counting operations with combinatory logic, but not without giving up the concept of numbers: there may be things other than numbers that can be used like numbers, for instance. There may be things that are extensionally-equivalent to numbers (and there may be multiple representations possible for a number, possibly even reflecting the history of all computations done on a number depending on evaluation strategy.)

Do you touch on any combinatory logic type stuff in your book? Is that even in scope?

[gfish]

I haven't read much Church yet, but I recently realized that there are some nice points I could make so it's on the list. Combinatory logic might get a mention, but just as an example of another time that completely abstract math has ended up being surprisingly practical after computers came around.

[randomdreams]

I feel like they're making a pointer to counting and relying on that level of abstraction to claim it's not really counting.