Paolo Mancosu.*Abstraction and Infinity

Abstract

Paolo Mancosu’s Abstraction and Infinity is an extended — and, most importantly, historically informed and inflected — investigation into abstraction generally, and abstraction principles in particular. An abstraction principle is a formula of the form: ... The two best known abstraction principles are Basic Law V: ... ... Abstraction principles are central to both Gottlob Frege’s original logicist attempt to reduce all of mathematics to logic (where |${\sf BLV}$| was, for Frege, a logical truth; see [Frege, 1980; 2013]) and more recent neo-logicist attempts to achieve something similar (where other abstraction principles, including |${\sf HP}$|⁠, serve as implicit definitions of mathematical concepts; see [Wright, 1983; Linnebo, 2018]). Of course, Frege’s project collapsed upon the discovery that |${\sf BLV}$| was inconsistent, and neo-logicists continue to struggle to extend the success of |${\sf HP}$| to other mathematical domains. Mancosu’s investigation of abstraction principles does not focus primarily on either the historical project of reconstructing Frege’s logicism in detail or the philosophical project of demonstrating that most or all of contemporary mathematics can be reconstructed in terms of neo-logicistically acceptable abstraction principles. Rather, as the title of the book suggests, Mancosu focuses instead on the connections between mathematical and philosophical work on abstraction and parallel developments in the mathematics and philosophy of the infinite. Nevertheless, the historical narrative that Mancosu constructs, and the philosophical puzzles he raises along the way, have deep implications for both the historical reconstruction of Frege’s original project and more recent attempts to construct Frege-inspired foundations for mathematics.