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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?
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?