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WILLIAM DEMOPOULOS Logicism and Its Philosophical Legacy

This volume collects together eleven essays, most previously published (including two in this journal) but some new. Some are directly concerned with logicism as normally understood—the thesis, very roughly, that mathematics is logic—but most are concerned more with what can reasonably regarded as parts of its philosophical legacy, with a good deal of focus on matters which are likely to be of interest to readers of this journal, including Russell’s structuralism, related work of Ramsey and Carnap, and more besides. It will be convenient to discuss the essays in two groups. Three chapters (Chapters 1, 8, 9) focus on Frege’s logicism, and a fourth (Chapter 11) on his understanding of logic. What was Frege trying to show in Grundlagen? The obvious, and indisputably correct, answer is: that arithmetic is analytic. Demopoulos emphasizes two other aims: showing that arithmetic knowledge doesn’t depend on intuition or knowledge of spatial and temporal notions; and showing that it is a priori. In Chapter 1 (Frege’s analysis of arithmetical knowledge), the second point is developed into an extended response to a criticism of Tyler Burge ([2005]) who argues that, without a viable theory of extensions of concepts, Frege has no effective account of the apriority of arithmetic. In this chapter, Demopoulos also argues—plausibly, and with reference to much supporting text—that Frege’s primary interest in rigorous, gap-free proofs was primarily driven not by a desire to put arithmetic on a secure foundation, but by his desire to show that arithmetical truths can be derived from general logical laws together with suitable definitions of their primitive terms, and thus they do not depend upon Kantian intuition and are independent of knowledge of spatial and temporal notions. This theme is taken up again in Chapter 8 (Frege and Brit. J. Phil. Sci. 66 (2015), 459–463