Abstract

Self‐evidently the standard work on the topic its whole title defines, Sir Michael Dummett’s Frege: Philosophy of Mathematics (FPM) is also the most profound and creative discussion in recent decades of the problems confronting the branch of philosophy mentioned after the colon. Chapters 14‐18 and 23‐24 of this book constitute a continuous and challenging diagnosis of these problems.1 They culminate in the proposal that these problems present an impasse that can be escaped only by adopting a constructivist understanding of mathematical generality. Dummett’s case for that conclusion is no less complexly over‐layered than the problems themselves. By contrast my aims in this discussion of his case are limited in various ways, and three of these should be mentioned straightaway. In the first place, I will aim to consider a case that, if sound, would warrant a constructivist understanding of generality in mathematics generally (and so I will not be considering lines of argument specific to set theory, or to those parts of mathematics plausibly dependent on notions intrinsic to set theory). Secondly, I aim to consider a case which, while general in its application within mathematics, is not more general than that (and so would not warrant a broader anti‐realism). Reasons for these first two limitations are discussed in section 1. A third limitation is that I will aim only to understand Dummett’s case, and not to assess it. Perhaps some will think this third limitation calls for explanation or excuse. I think it needs no excuse and that the explanation is obvious. When we are dealing with fundamentally important work by a great philosopher, understanding is often ambition enough. In Michael Dummett’s work, that is what we are dealing with.