Frege and Other Philosophers, and: Frege: Philosophy of Mathematics (review)

Abstract

BOOK REVIEWS 303 Michael Dummett. Frege and OtherPhilosophers. New York: Oxford University Press, x991. Pp. xii + 33o- Cloth, $69.oo. Michael Dummett. Frege:PhilosophyofMathematics. Cambridge, MA: Harvard University Press, 199 I. Pp. xiii + 33 I. Cloth, $34.95. Dummett's writings on Frege have already set a standard of accomplishment to which all philosophical interpretation of the history of philosophy aspires. The works under review do not depart from that standard. This is especially true of Frege:Philosophyof Mathematics (hereafter FPM), the long-awaited completion of a project begun with Frege:PhilosophyofLanguage (hereafter FPL). Unlike FPL, considered by many as one of the most original works in the philosophy of language in the last twenty years, FPM is primarily exegetical. Hence, in FPM the central outlines of Dummett's reading of Frege are much more accessible; one hopes that the sorts of difficulties of interpretation that beset FPL, and that are partially addressed in Frege and OtherPhilosophers (hereafter FOP), can be avoided. FPM is an interpretation of Frege's two main works in the philosophy of mathematics : Grundlagen der Arithmetik (hereafter GO and Grundgesetzeder Arithmetik (hereafter Gg)? Gl announces the aim of Frege's philosophy of mathematics: to show that arithmetical truths are analytic and a priori. Dummett explains Frege's use of these traditional philosophical categories in terms of the justification of propositions: a proposition is analytic a priori if it is justified from general laws that are logical, the mark of which is universal applicability. Thus Frege's project requires a system of definitions of primitive arithmetical vocabulary from which the basic laws of arithmetic are proved. Dummett criticizes Frege for not giving a clear account of the conditions in which a definition is correct. In Gl the criterion seems to be faithfulness to the existing senses of the terms, rather than merely capturing their references, and Dummett takes this to be the criterion that Frege should accept. But, as Dummett notes, in later texts Frege writes of "constructive" definitions that fix the senses of terms with which no clear preexisting senses are associated. So, given Frege's view of the confusion surrounding the concept of number, why could he not have taken his project as providing constructive definitions of primitive arithmetical expressions? I Dummett claims that, apart from giving a justification of the basic laws of arithmetic , Frege has two other arguments for the analyticity of arithmetic: the universal applicability of arithmetical truths and the incoherenceof their denial. The interest of the former is that, according to Dummett, it shows that arithmetic is alreadylogic; the execution of Frege's logicist program merely displays this fact explicitly. Arithmetical expressions occur in two forms: as adjectives, in what Dummett and Frege call "ascriptions of number," and as nouns, in most pure number theoretic propositions. Frege prefaces his actual definitions by arguing that the latter is the basic form of numerical expressions, and hence that which must be defined; the former is explained in terms of the latter. In Chapter 9 Dummett argues that Frege had no good 1Respectively, (Oxford: Blackwell, 1953), and (Hildesheim: Olm, 1962). ' See, e.g., Joan Weiner, FregeinPerspective(Ithaca: Cornell University Press, 199o). 304 JOURNAL OF THE HISTORY OF PHILOSOPHY 31:9 APRIL 1993 reason for adopting this "substantival" strategy of definition in favor of taking the adjectival t'orms as basic, and sketches, using higher-order quantification, how the 9"adjectival" strategy could be carried out.~ However, as he notes in Chapter I l, if the adjectival strategy is adopted, the infinity of the number series cannot be proved without an axiom of infinity. In light of Frege's argument that arithmetic is already logic, this seems to provide a strong reason for the substantival strategy: few would take an axiom of infinity to be a logical law. Frege's project is not merely to show ~at arithmetic is analytic or a part of logic, but also to show that numbers are objects, that arithmetic is about objects. This, of course, has been taken to be the main thesis of Frege's platonism. But, on Dummett's reading, Frege takes semantic categories to be prior to ontological ones; hence his platonism is not the fashionable...