Abstraction and Four Kinds of Invariance (Or: What’s So Logical About Counting) †

<span class="paragraphSection">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 fine-grained 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.</span>

from #AlexandrosSfakianakis via Alexandros G.Sfakianakis on Inoreader http://ift.tt/2m1UoFG
via IFTTT