What are numbers?

Prof. Derden gave a talk on math philosophy at the HSU math colloquium. He was addressing the issue of what numbers really are and his talk was a response to a commonly held belief (in philosophy) that numbers can’t be sets. Besides being sets, he wants to argue that numbers are a particular type of sets called Russell sets. Below is an excerpt of an email I sent the professor. I’ve yet to receive a reply:

As I saw it, your talk consisted of one of two things. There was a strong argument in favor of using Russell sets as a model for the natural numbers OR there was a relatively weak claim that sets, in general, can’t be in the set of “things numbers aren’t.”

I’m particularly fond of the stronger argument. That there would be a correspondence between all sets of pairs and the things we count as two seems to jive with my intuition of what numbers are. When I say “two”, I just mean all those things that I can pair together. Two is, in a sense, a pronoun for all pairs of things (or a synonym maybe?). And when I say “two plus two equals four”, I’m saying something like “two, for example, sheep, plus two, for example, rattle snakes, equals four things.”

Given the title of the talk was “What Natural Numbers Must Be,” its clear you intended to make the stronger claim. In any case, you at least argued the weaker claim. That there is a plausible argument at all for Russell sets as a model for numbers, means that sets COULD be such a model.

The word ‘model’ is tripping me up. Isn’t a model just an approximation of the real thing? Models are built to mimic the behavior of what is being modeled. As long as the model behaves as the thing being modeled (at least in the area of concern), it is said to be a good model. In this sense, the flight of fixed wing airplanes is a good model for the flight of birds. Of course, physics has shown that they are far from the same thing. In the same sense, the natural numbers can be modeled by sets, but that doesn’t mean that they are sets.

This is the same point Benacerraf makes in his essay. I couldn’t find the Benacerraf’s essay online but I found this article discussing the article. From that secondary article, “Benacerraf explains that to characterise the numbers is only to describe the structure, without any identification of the individual elements, and that this is why numbers are not objects at all.” So the model of the numbers helps us describe the structure of the numbers, but it shouldn’t be confused for the real deal.

Also, I can think of one more problem with trying to understand what the numbers really are. In your talk, you mentioned that the Platonists ‘push’ the location of ideas into the mind of god. Oh yeah, where’s he?

Similarly, if we ever find out what the number really are, won’t we just be pushing the problem? For example, let’s say, at your talk, we decided that the numbers were Russell sets. But then an intrepid undergrad would ask, in his required email follow-up, “Thanks Prof. Derden for letting us

know what numbers are. Ok, so what are Russell sets?” To which you

spend another colloquium talk discussing and then narrowing the possibilities. Suppose, at the end of that talk we all decided that Russell Sets are really something called blarbs. Then we’d have to explain what the blarbs are, then what those things are, and so on into an infinite regress. Where would this all end?

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