Paolo Mancosu, Willis S. and Marion Slusser Professor of Philosophy
"In good company? On Hume’s principle and the assignment of numbers to infinite concepts"
Abstract: In a recent article (RSL 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this talk I generalize some specific worries, raised by Richard Heck, emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s principle (HP). The talk has four main parts. The first takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second-order arithmetic. All the principles I present agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection as a generalization of Heck's objection to the analyticity of HP based on the theory of numerosities. I then give a taxonomy of possible neo-logicist responses to the ‘good company’ objection.