Functions and Generality of Logic ― Reflections on Dedekind's and Frege's Logicisms

This book examines three connected aspects of Frege’s logicism: the differences between Dedekind’s and Frege’s interpretation of the term ‘logic’ and related terms and reflects on Frege’s notion of function, comparing its understanding and the role it played in Frege’s and Lagrange’s foundational programs. It concludes with an examination of the notion of arbitrary function, taking into account Frege’s, Ramsey’s and Russell’s view on the subject. Composed of three chapters, this book sheds light on important aspects of Dedekind’s and Frege’s logicisms. The first chapter explains how, although he shares Frege’s aim at substituting logical standards of rigor to intuitive imports from spatio-temporal experience into the deductive presentation of arithmetic, Dedekind had a different goal and used or invented different tools. The chapter highlights basic dissimilarities between Dedekind’s and Frege’s actual ways of doing and thinking. The second chapter reflects on Frege’s notion of a function, in comparison with the notions endorsed by Lagrange and the followers of the program of arithmetization of analysis. It remarks that the foundational programs pursued by Lagrange and Frege are crucially different and based on a different idea of what the foundations of mathematics should be like. However, despite this contrast, the notion of function plays similar roles in the two programs, and this chapter emphasizes the similarities. The third chapter traces the development of thinking about Frege’s program in the foundations of mathematics, and includes comparisons of Frege’s, Russell’s and Ramsey’s views. The chapter discusses earlier papers written by Hintikka, Sandu, Demopoulos and Trueman. Although the chapter’s main focus is on the notion of arbitrary correlation, it starts out by discussing some aspects of the connection between this notion and Dedekind Theorem.

Hourya Benis Sinaceur is Research Director at the CNRS. Her publications include Corps et Modeles, Paris, Vrin, 1991, second ed. 1999; Le labyrinthe du continu (co-ed. with avec J.-M. Salanskis), Springer-Verlag France,1992, Cavailles. Philosophie mathematique, Paris, PUF, 1994; “Tarski's Address at the Princeton University Bicentennial Conference on problems of Mathematics (December 17-19, 1946)”, typescript ed. with additional material and an Introduction, The Bulletin of Symbolic Logic, vol. 6, n° 1 (Mars 2000), p. 1-44; “Alfred Tarski : Semantic shift, heuristic shift in Metamathematics”, Synthese 126, pp. 49-65, 2001. She is member of the Comite National d'Histoire et Philosophie des Sciences de l'Academie des Sciences de Paris and Membre correspondant de l’Academie Internationale d’Histoire des Sciences. Marco Panza is Research Director at the CNRS. He is the author of several book and paper (published in several idioms) concerning history and philosophy of mathematics. The former include: Newton et les orgines de l’Analyse: 1664-1666, Blanchard, Paris, 2005; Plato’s Problem. Introduction to Mathematical Platonism, Palgrave MacMillan, Bsingstoke (UK), 2013 (co-authored with Andrea Sereni). He is member of the Steering Committee of the Association for the Philosophy of Mathematical Practice (APMP), which he contributed to found. Gabriel Sandu is professor of theoretical philosophy whose main contributions are in logic, game-theoretical semantics, IF logic, and truth theories. He also published on Frege and Ransey. His publications includes: Independence-friendli Logic: A game-theoretic approach, CUP, Cambridge, 2011 (Coauthored with A. Mann and M. Sevenster); On the methodology of lingustics: A case study, Blackwell, Oxford, 1991 (Coauthored with J. Hintikka).