Neo-Fregean Logicism in the Philosophy of Mathematics
2007 (English)Conference paper, (Other academic)
Frege tried to explain our knowledge of the natural numbers by reducing arithmetic to logic. This program, however, could not be carried out. The main reason was that the theory of classes (or extensions) that Frege used in deﬁning the natural numbers turned out to be inconsistent. In addition, the program is in apparent conﬂict with Gödel’s ﬁrst incompleteness theorem according to which every consistent formal system for arithmetic, with sufﬁcient expressive power, contains true arithmetic statements that are not provable in the system. For a long time, therefore, Frege’s philosophy of mathematics came to be regarded as hopelessly passe. Recently, however, the situation has changed, mainly due to the revision of Frege’s program by Crispin Wright and Bob Hale and logical investigations carried out by the late George Boolos, Richard Heck, and others. The aim of the project is to subject the neo-Fregean program (and similar neo-logicist programs) in the philosophy of mathematics to a critical examination. Many of the assumptions and presuppositions of this program can be questioned. What is the epistemic status of the higher-order logic that is assumed? Can all the principles and rules of inference of this logic be justiﬁed on the basis of conceptual connections? Or is it rather,
as critics have claimed, that the Neo-Fregeans have provided substantial mathematical assumptions with an innocent-looking logical disguise? What is the status of Hume’s principle and similar abstraction principles? What about the “bad company objection” against Fregean abstraction principles and the “Caesar objection” against implicit deﬁnitions? The focus will be on questions concerning the limits of logic, the interpretation of
higher order logic, and the status of abstraction principles. The importance of the project lies in the clarity that it can provide concerning philosophically important concepts and problems.
Place, publisher, year, edition, pages
2007. , 16 p.16- p.
Philosophy of Mathematics, logicism, Frege, Crispin Wright, Bob Hale
IdentifiersURN: urn:nbn:se:umu:diva-9133OAI: oai:DiVA.org:umu-9133DiVA: diva2:148804
Föredrag vid SCAS (The Swedish Collegium for Advanced Study) Uppsala, spring semester 2007.