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Extra info for Order and Convexity in Potential Theory, H-Cones
1965], “Frege’s Theory of Number”, in Black : 180–203. Rayo, A. , “Frege’s Unofficial Arithmetic”, Journal of Symbolic Logic 67: 1623–1638, reprinted below as chapter 10. Russell, B. , “Letter to Frege” in van Heijenoort : 124–125. Introduction xxxvii Russell, B. , “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society 4: 29–53. Shapiro, S. , “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis”, Notre Dame Journal of Formal Logic 41: 335–364, reprinted below as chapter 14.
Deriving arithmetic. It might serve as something to be obtained from Basic Law V. It might be used as an example of a beautiful proposition. Etc. etc. But there’s no such thing as the [unique] role of HP. It is certainly true that one of the ways in which HP can be used is to fix the character of a certain concept. Here’s how: lay Hume down. Then the concept the number of . . will have been fixed to be such that numbers belonging to concepts will be the same if and only if the objects falling under one of the concepts are in one–one correspondence with those falling under the other.
Part of the problem with this suggestion is this: in HP, numbers belonging to concepts are themselves among the objects over which the first-order variables on the right-hand side range. Talk of reconceptualizing a state of affairs would be in order only if the objects supposedly introduced by stipulation were new, objects that had not been previously quantified over. Whether old objects can be chosen to be identical or not under the right conditions would not seem to be a matter that it could be up to us to decide.