What Kind Of Thing Is A Number?
JB: Reuben, got an interesting question?
HERSH: What is a number? Like, what is two? Or even three? This is sort of a kindergarten question, and of course a kindergarten kid would answer like this: (raising three fingers). Or two (raising two fingers). That's a good answer and a bad answer. .
It's good enough for most purposes, actually. But if you get way beyond kindergarten, far enough to risk asking really deep questions, it becomes: what kind of a thing is a number?
Now, when you ask "What kind of a thing is a number?" you can think of two basic answers--either it's out there some place, like a rock or a ghost; or it's inside, a thought in somebody's mind. Philosophers have defended one or the other of those two answers. It's really pathetic, because anybody who pays any attention can see right away that they're both completely wrong.
A number isn't a thing out there; there isn't any place that it is, or any thing that it is. Neither is it just a thought, because after all, two and two is four, whether you know it or not.
Then you realize that the question is not so easy, so trivial as it sounds at first. One of the great philosophers of mathematics, Gottlob Frege, made quite an issue of the fact that mathematicians didn't know the meaning of One. What is One? Nobody could answer coherently. Of course Frege answered, but his answer was no better, or even worse, than the previous ones. And so it has continued to this very day, strange and incredible as it is. We know all about so much mathematics, but we don't know what it really is.
Of course when I say, "What is a number?" it applies just as well to a triangle, or a circle, or a differentiable function, or a self-adjoint operator. You know a lot about it, but what is it? What kind of a thing is it? Anyhow, that's my question. A long answer to your short question. JB: And what's the answer to your question?
HERSH: Oh, you want the answer so quick? You have to work for the answer! I'll approach the answer by gradual degrees.
When you say that a mathematical thing, object, entity, is either completely external, independent of human thought or action, or else internal, a thought in your mind--you're not just saying something about numbers, but about existence--that there are only two kinds of existence. Everything is either internal or external. And given that choice, that polarity or dichotomy, numbers don't fit - that's why it's a puzzle. The question is made difficult by a false presupposition, that there are only two kinds of things around.
But if you pretend you're not being philosophical, just being real, and ask what there is around, well for instance there's the traffic ticket you have to pay, there's the news on the TV, there's a wedding you have to go to, there's a bill you have to pay--none of these things are just thoughts in your mind, and none of them is external to human thought or activity. They are a different kind of reality, That's the trouble. This kind of reality has been excluded from metaphysics and ontology, even though it's well-known--the sciences of anthropology and sociology deal with it. But when you become philosophical, somehow this third answer is overlooked or rejected.
Now that I've set it up for you, you know what the answer is. Mathematics is neither physical nor mental, it's social. It's part of culture, it's part of history, it's like law, like religion, like money, like all those very real things which are real only as part of collective human consciousness. Being part of society and culture, it's both internal and external. Internal to society and culture as a whole, external to the individual, who has to learn it from books and in school. That's what math is.
But for some Platonic mathematicians, that proposition is so outrageous that it takes a lot of effort even to begin to consider it.