Joel David Hamkins: belief

Transcript context

…So ZFC sounds like a super technical thing, but it is the set of axioms that’s the foundation of modern mathematics. Yeah, absolutely. So one should be aware also that there are huge parts of mathematics that pay attention to whether the axiom of choice is being used, and they don’t want to use the axiom of choice, so they work out the consequences that are possible without the axiom of choice or with weakened forms of Zermelo-Fraenkel set theory, and so on. And there’s quite a vibrant amount of work in that area. But going back to the axiom of choice for a bit, it’s maybe interesting to give Russell’s description of how to think about the axiom of choice. So Russell describes this rich person who has an infinite closet. In that closet, he has infinitely many pairs of shoes, and he tells his butler, “Please go and give me one shoe from each pair.” And the butler can do this easily because for any pair of shoes, he can just always pick the left shoe. There’s a way of picking that we can describe. We always take the left one or always take the right one, or take the left one if it’s a red shoe and the right one if it’s a brown shoe, you know. We can invent rules that would result in these kinds of choice functions so we can describe explicit choice functions. For those cases, you don’t need the axiom of choice to know that there’s a choice function. When you can describe a specific way of choosing, then you don’t need to appeal to the axiom to know that there’s a choice function. But the problematic case occurs when you think about the infinite collection of socks that the person has in their closet. And if we assume that socks are indistinguishable within each pair, you know, they match each other, but they’re indiscernible, then the butler wouldn’t have any kind of rule for which sock in each pair to pick. And so it’s not so clear that he has a way of producing one sock from each pair, right? So that’s what’s at stake, is the question of whether you can specify a rule by which the choice function, you know, a rule that it obeys that defines the choice function, or whether there’s sort of this arbitrary choosing aspect to it. That’s when you need the axiom of choice to know that there is such a function. But of course, as a matter of mathematical ontology, we might find attractive the idea that, well, look, I mean, not every way of choosing the socks has to be defined by a rule. Why should everything that exists in mathematical reality follow a rule or a procedure of that sort? If I have the idea that my mathematical ontology is rich with objects, then I think that there are all kinds of functions and ways of choosing. tical reality follow a rule or a procedure of that sort? If I have the idea that my mathematical ontology is rich with objects, then I think that there are all kinds of functions and ways of choosing. Those are all part of the mathematical reality that I want to be talking about, and so I don’t have any problem asserting the axiom of choice. Yes, there is a way of choosing, but I can’t necessarily tell you what it is. But in a mathematical argument, I can assume that I fix the choice function because I know that there is one. So it’s a… The philosophical difference between working when you have the axiom of choice and when you don’t is the question of this constructive nature of the argument. So if you make an argument and you appeal to the axiom of choice, then maybe you’re admitting that the objects that you’re producing in the proof are not going to be constructive. You’re not going to be able to necessarily say specific things about them. But if you’re just claiming to make an existence claim, that’s totally fine. Whereas if you have a constructive attitude about the nature of mathematics, and you think that mathematical claims maybe are only warranted when you can provide an explicit procedure for producing the mathematical objects that you’re dealing with, then you’re probably going to want to deny the axiom of choice and maybe much more.…

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