Neo-Fregean Logicism and the A Priori Nature of Arithmetic
2007 (English)In: Volume of Abstracts of the 13th International Congress of Logic, Methodology, and Philosophy of Science (LMPS Beijing), 2007Conference 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 extensions (or classes) that Frege used in defining the natural numbers turned out to be inconsistent. In addition, the program is in conflict with Godel´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 passé. 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. It was pointed out that Frege in Grundgesetze makes only one essential use of his inconsistent axiom for classes, Basic Law V, namely to establish Hume’s principle, i.e., the statement that the cardinal numbers of the concepts F and G are equal if and only if the F’s and the G’s are equinumerous (can be put in a one-to-one correspondence with each other). Using this principle and appropriate definitions he then proved the fundamental axioms of arithmetic. In other words, Frege had established Frege’s theorem, i.e. the axioms of arithmetic due to Dedekind and Peano are provable from Hume’s Principle in second-order logic. Wright and Hale argue that Hume’s principle is analytically true of the concept of cardinal number and that it is knowable a priori. Thus, by using this principle instead of Frege’s inconsistent Basic Law V one can achieve a substantial part of Frege’s epistemological goals. Wright’s and Hale’s general program is to develop other branches of mathematics, like real and complex analysis and set theory, on the basis abstraction principles similar to Hume’s principle.
In this paper I subject the neo-Fregean program of Hale and Wright to a critical examination.
In particular, I concentrate on two questions:
(i) Hale and Wright describe Hume’s principle as a stipulation about the meaning of “cardinal number”. A subject who is not in possession of the notion of a cardinal number can come to understand that concept by means of Hume’s principle. At the same time it is a very strong principle that implies the existence of infinitely many objects (cardinal numbers). How can one argue that such a strong principle is at the same time a stipulation and a priori true? In what sense can one introduce new entities by stipulation? The vague idea is that the left hand side (implying the existence of numbers) is just a “reconceptualization” of the right hand side (speaking of equinumerosity). I will analyse this idea that statements about numbers express nothing but facts about equinumerosity.
(ii) If the derivation of Dedekind-Peano’s axioms from Hume’s principle preserves the property of being a priori, then we can also gain a priori knowledge of the fundamental laws of arithmetic. But how can we justify the claim that second-order logical consequence preserves apriority? This in turn leads to the question about the interpretation of second-order logic and its “logicality”.
Place, publisher, year, edition, pages
IdentifiersURN: urn:nbn:se:umu:diva-9259OAI: oai:DiVA.org:umu-9259DiVA: diva2:148930
13th International Congress of Logic, Methodology and Philosophy of Science, 9-15.8, 2007, Beijing, China.
Föredrag vid 13th International Congress of Logic, Methodology and Philosophy of Science, 9-15.8, 2007, Beijing, China.2008-03-162008-03-162013-08-22Bibliographically approved