When we teach arithmetic to children, we make use of counterfactuals: "If you had two oranges, and got two more, you would have four. So, two and two makes four." Then, later on, we say "Two plus two is four." There are three steps here. First, the counterfactual. Then, an implicitly universal claim: two and two (always) makes four. Finally, a categorical mathematical claim: two plus two is four. I wonder if it might not be a mistake to focus on the final claim in philosophy of mathematics. Perhaps it is a mere abstraction from the first and second, having no additional content?
3 comments:
FWIW, I think philosophy of mathematics ought to begin from the observation that in ordinary language, the numbers are quantifiers ("two oranges") but in arithmetic-speak, they are singular terms ("two plus two"). This indicates to me that arithmetic is a kind of second-order language for representing patterns in the first-order language. There are probably other ways to go, too, but the linguistic observation is important.
I am inclined to agree.
FWIW I agree with both of you. Neo-logicism was all the rage in Scotland, when I was there recently, and my reaction to it was as in your post.
Hume's principle is analytic insofar as it's conditional - if there were N things and another M things, and those two classes corresponded one-to-one, then N = M - but to use that to build up the numbers from nothing you need to get rid of the conditionality, which seems to me to get rid of the analyticity.
Arithmetic seems to be about the structural possibilities inherent in the concept of an individual. The connection with logic is pretty deep because such a concept is also presupposed by classical logic. Perhaps that's how the neo-logicists get away with playing games with equivocations. If only more philosophers were as perceptive as you two, is what I think.
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