Philosophy 290-5

Description: Kant claimed that our knowledge of mathematics is synthetic a priori. Frege agreed that our knowledge of arithmetic is a priori but claimed it was analytic rather than synthetic. In addition, Frege was a platonist and held that our knowledge of arithmetic is about a special domain of mind-independent, non-spatiotemporal objects – the natural numbers – and their characteristic properties and relations. Frege thought he could establish the a priori nature of arithmetic and platonism about its subject matter through his logicism, namely the claim that arithmetic is definitionally reducible to logic. This led him to also claim that numbers are logical objects (extensions of concepts).

In the first part of the seminar, we will look at the philosophical underpinnings and technical elaboration of Frege’s logicism. We will see how Frege motivated his logicism against rival empiricist, formalist, and Kantian views of arithmetic, and how he took his logicism to answer the philosophical problems raised by platonism. We will look at some of the details of his technical theory. And we will see how Russell showed Frege’s theory of extensions to be inconsistent, causing Frege to abandon the project.

The second part of the seminar is devoted to an attempt to revive Frege’s project that goes under the name of Neo-logicism or, now more commonly, Neo-Fregeanism or Abstractionism. A mathematical result established by Wright and foreshadowed by Frege, known as Frege’s theorem, shows how little the technical development of Frege’s logicism actually depends on the theory of extensions shown inconsistent by Russell. There has been much debate about the significance of this work. Wright and Hale, who espouse a neologicist position, have argued that it shows that Frege’s logicist program can be salvaged, with slight modifications, into a philosophically satisfying account of arithmetic. Boolos, Heck, Dummett, and others have argued the contrary position. The discussion has been raging since the publication of Wright’s book “Frege’s conception of numbers as objects” (1983) and it is still the focus of great interest in much contemporary philosophy of mathematics. In addition to study some classic sources (including Wright’s 1983 book), we will also discuss several very recent contributions to Neo-logicism.

Prerequisites: The seminar is open to all graduate students in Philosophy and Logic and Methodology of Science. Interested undergraduates, visiting students, etc. should seek the permission of the instructor before enrolling. I will presuppose that everyone has had a good introductory course in first-order logic, at the level of Philosophy 12A. Previous coursework in logical metatheory, philosophy of mathematics, philosophy of language, Kant, and Frege may be helpful but are not required.