Definitions by abstraction: From stable practice to global foundations

Paolo Mancosu

UC Berkeley

One of the most influential programs in contemporary philosophy of mathematics is the neo-logicist program. At the core of neo-logicism are a technical result and a set of philosophical considerations. The technical result is called Frege's theorem: second-order Peano arithmetic can be derived from Hume's principle and second order logic. The cluster of philosophical considerations are aimed at showing that Hume's principle is analytic and that second order logic does not take us out of the realm of analyticity. As is well known, there are abstraction principles that are inconsistent (the notorious basic Law V in Frege's Grundgesetze) and abstraction principles, such as the nuisance principle, that are true on a domain iff Hume's principle is false on that domain. The attempt at differentiating 'good' from 'bad' abstractions has given rise, in the last twenty years, to an extensive literature on abstraction principles. This literature is mathematically sophisticated and philosophically stimulating. It is however surprising that apart from some perfunctory references to Frege's Grundlagen §64, the analytic literature on abstraction principles ignores the extensive discussion on such principles that occupied many mathematicians and philosophers at the end of the nineteenth century and at the beginning of the twentieth century, including Peano, Burali-Forti, Russell, Padoa, and Scholz. The discussion was framed within the context of so-called 'definitions by abstraction" (the term was coined by Peano but the idea occurred earlier). An example of such a definition is given by Frege in §64: for all lines a and b, the direction of line a is equal to the direction of line b if and only if a and b are parallel. But Frege's wording in §64 seems to imply that the practice of introducing definitions by abstraction was still considered quite rare when Frege was writing the Grundlagen. In my presentation I intend to show that the logical discussion of definitions by abstraction is anchored in a stable and widespread mathematical (and physical) practice – including complex analysis, geometry, and vector analysis – and how reflection on the mathematical practice led from local foundational issues to global foundational issues.