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.
CITATION STYLE
Cook, R. T. (2017). Abstraction and four kinds of invariance (Or: What’s so logical about counting). Philosophia Mathematica, 25(1), 3–25. https://doi.org/10.1093/philmat/nkw014
Mendeley helps you to discover research relevant for your work.