Abstraction and four kinds of invariance (Or: What's so logical about counting)

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Philosophia Mathematica, ISSN: 1744-6406, Vol: 25, Issue: 1, Page: 3-25

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Cook, Roy T.
Oxford University Press (OUP)
Mathematics; Arts and Humanities
article description
Fine and Antonelli introduce two generalizations of permutation invariance - internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most finegrained abstraction principle that is invariant in each sense. Hume's Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume's Principle, and the comparative lack of success in reconstructing areas of mathematics other than arithmetic based on non-invariant abstraction principles.