Joel David Hamkins: belief

Transcript context

…So is it fair to think of numbers as a kind of pointer to a deep underlying structure? Yeah, I think so, because I guess part of the point of structuralism is that it doesn’t make sense to consider mathematical objects or individuals in isolation. What’s interesting and important about mathematical objects is how they interact with each other and how they behave in a system, and so maybe one wants to think about the structural role that the objects play in a larger system, a larger structure. There’s a famous question that Frege had asked actually when he was looking into the nature of numbers, because in his logicist program, he was trying to reduce all of mathematics to logic. And in that process, he was referring to the Cantor-Hume principle that, whenever two sets are equinumerous, then they have the same number of elements, if and only if. And he founded his theory of number on this principle, but he recognized that there was something that dissatisfied him about that situation, which is that the Cantor-Hume principle does not seem to give you criteria for which things are numbers. It only tells you a kind of identity criteria for when are two numbers equal to each other. Well, two numbers are equal just in case the sets of those sizes are equinumerous, so that’s the criteria for number identity. But it is not a criteria for what is a number. And so this problem has become known as the Julius Caesar problem because Frege said we don’t seem to have any way of telling from the Hume principle whether Julius Caesar is a number or not. So he’s asking about the essence of number and whether… Of course, one has a sense that he picked maybe what he was trying to present as a ridiculous example… …Because maybe you have the idea that, well, obviously Julius Caesar is not a number, and there’s a lot of philosophical writing that seems to take that line also, that obviously the answer is that Julius Caesar is not a number. But the structuralists disagree with that position. The structuralist attitude is, “Look, you give me a number system. If Julius Caesar isn’t a number, then I can just… Let’s take the number 17 out of that system and plug in Julius Caesar for that role, and now I’ve got a new number system, and now Julius Caesar happens to be the number 17.” And that’s totally fine, you know. So the point of structuralism is that the question of whether Julius Caesar is a number or not is irrelevant to mathematics. It is irrelevant because it is not about structure, it’s about this essence of the mathematical objects. So that’s the structuralist criticism of Frege’s point. You’ve kind of made the case that you can say more concrete things about the existence of objects in mathematics than you can in our physical reality, about which to us human brains, things are obvious or not. So what’s more real, the reality we see with our eyes or the reality we can express in mathematical theorems?…

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