The long awaited (by me, at least) article by Paolo Mancosu on the Good Company objection has been out in the Review for a few months and I missed it. You probably didn’t, if you care about these things, but anyway:
In a recent article (Mancosu, 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 ‘sizes’ 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. Numerosity assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this paper I generalize some 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 paper is centered around five main parts. The first (§3) 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 (§4) 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 (or so I claim) 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 (§5) connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part (§6) states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section I offer a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, §7 makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.