Against ontology – part 2

by Neil Rickert

My second example of why I don’t like ontology, is a TEDx talk by Kit Fine (h/t Brian Leiter).  In that talk, Fine discusses what is the fundamental nature of the being of numbers.

It’s a puzzle to me that anyone would suppose that numbers have any fundamental being.  It seems obvious that they do not.

Fine gives three possible versions of the nature of numbers.  The first is due to Frege and Russell, the second to von Neumann, the third to Cantor.  The only one of those that I find useful is von Neumann’s.  But I do not take it as being about the nature of numbers.  Rather, I take it as a useful way to model arithmetic within set theory.  I have always assumed (perhaps wrongly) that was why von Neumann proposed that definition.

Kit Fine seems to think that there are puzzles about numbers and mathematics, that can be resolved by understanding the nature of numbers.  He suggests that there is a puzzle as to why mathematics is so useful in science.  Others apparently also see that as a puzzle.  Fine asks (about numbers):

How can they be so far removed from the familiar world, yet so intimately connected to it?

Presumably, he thinks that understanding the fundamental nature of numbers will answer that question.

Numbers have no fundamental nature.  Perhaps knowing that will help Fine.

The usefulness of numbers and of mathematics is explained by how we use them, not by what they are.  The usefulness of numbers in science is explained by how scientists use them.

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11 Comments to “Against ontology – part 2”

  1. Neil, it may be that you are actually making an ontological claim though when you state that numbers are only a useful way to model arithmetic and nothing more than that.

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    • … when you state that numbers are only a useful way to model arithmetic and nothing more than that.

      But that isn’t what I said. Rather, I said that von Neumann’s characterization of numbers is a useful way …

      However, you are right that criticizing metaphysics is taken to be doing metaphysics. But I don’t see that as a particular concern.

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      • Yeah, I didn’t join your two sentences, sorry about that.

        I’m not really a studied philosopher, but I’m confused a little about your point. Thinking about the kinds of things that exist and whether they exist or not (e.g. numbers) is useless? You are probably saying something different from that so I think it’s best I shut up. 🙂

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    • When I said “Numbers have no fundamental nature,” I guess that was making an ontological claim. My real point is that the fundamental nature of numbers (if there is such a thing) has no relevance to anything important to me.

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      • Yeah, I think I get that. Whether or not numbers exist as an entity somewhere doesn’t really seem to have relevance to anything practical in anyone’s life. It does still seem though that you are saying that it would help people do epistemology better if they realize that numbers don’t have a fundamental nature, so it looks like you even think there may be some importance to at least thinking about the question.

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        • It does still seem though that you are saying that it would help people do epistemology better if they realize that numbers don’t have a fundamental nature, …

          Then I did not explain that very well.

          My actual view is that epistemology would be better if people found a way of doing it that is independent of ontological questions. For example, “knowledge” should be defined in terms of behavior and behavioral abilities, rather than in terms of belief.

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  2. “The usefulness of numbers and of mathematics is explained by how we use them, not by what they are. The usefulness of numbers in science is explained by how scientists use them.”

    Although one must also recognize that in order for humans or scientists to be able to use them in the ways that they do, does suggest that they have certain properties and thus “what they are”, at least in that sense. If we can say what numbers aren’t, then we must at least in theory be able to say what they are.

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