Frege's Theorem

Richard Heck has collected eleven essays on issues surrounding Frege's Theorem in one excellent collection. Even though it is a ‘mere’ collection of papers—which often suggests that the book will involve repetition and no clear red thread—it actually has the feel of a book: Heck offers a detailed introduction in which he surveys the debate and locates each essay within it. Thereby, he provides a kind of narrative to his collection; the addition of numerous cross-references further highlights their interconnectedness. All but two papers have been published before; five contributions now feature a postscript, which in some cases, e.g. ch. 10, could easily be a stand-alone paper. There is some repetition, but it is mainly restricted to the first few pages, ensuring thereby that each chapter can still be treated as a self-contained article. The title of the book ‘Frege's Theorem’ suggests a narrow focus. However, the essays are not merely concerned with the mathematical fact that second-order Peano Arithmetic is derivable from Frege Arithmetic (FA), i.e. Hume's principle (HP) plus full second-order logic—a fact known as Frege's Theorem. Rather, the book covers historical, mathematical, as well as wide-ranging philosophical issues arising from a proper appreciation of Frege's Theorem; and then there are topics that do not quite fit the title: most importantly, the book contains Heck's own abstractionist views developed in the later chapters (chs 7, 9 and 11). His view is motivated and developed on the basis of numerous in-depth discussions of the (Scottish) neo-Fregean programme. In this review, I offer an overview of the collection and some critical comments on selected chapters.

Automated theorem proving · Calculus (dental) · Epistemology · Linguistics · Narrative · Order (exchange) · Peano axioms · Programming language · Classical Philosophy and Thought · Computer Science · Mathematics · Philosophy · Philosophy and Theoretical Science