Bad company tamed

The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts there are. One of the most serious problems facing the neo-Fregean project of basing mathematics on abstraction principles is the so-called “the bad company problem.” The problem is that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. A classic example of such a “bad companion” is Frege’s inconsistent Basic Law V, which is logically very similar to the neo-Fregeans’ favorite abstraction principle, namely the consistent and philosophically attractive Hume’s Principle. The bad company problem shows that a deeper understanding is needed of the conditions under which abstraction is permissible. The aim of this paper is to explore a new attempt to provide such an understanding, based on the idea that individuation must take the form of a well-founded process. Since an abstraction principle is naturally regarded as a device for individuating one class of entities in terms of another, this idea can be used to motivate restrictions on our theory of abstraction. In the first and less technical half of the paper (Sections 1 6) I develop the idea of individuation as a well-founded process in a modal framework. A surprising consequence of this development is a maximally permissive line on abstraction, according to which any form of abstraction on any concept is permitted and where paradox is instead avoided by restricting what concepts there are. In the second and more technical half (Sections 7 9) I articulate a natural and