1 Many thanks to John Bell and Michael Hallett for their perceptive comments on an earlier draft — and to Michael Dummett and Anil Gupta for their reading of the penultimate draft and their generous correction of a number of slips. Support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.

2 This perhaps is what Dummett means when he says that had Frege fully developed his ideas, the mathematical difference with Dedekind would have been slender (288).

3 The conjecture is proved in Adeleke, S.A. Dummett, Michael and Neumann, Peter M. ‘On a Question of Frege’s About Right-Ordered Groups,’ Bulletin of the London Mathematical Society 19 (1987) 513-21.CrossRefGoogle Scholar

4 Russell, Bertrand ‘On Some Difficulties in the Theory of Transfinite Numbers and Order Types,’ Proceedings of the London Mathematical Society, Series 2 4 (1906) 29-53Google Scholar. (See §1.) Reprinted in Lackey, D. ed., Bertrand Russell: Essays in Analysis (London: Allen and Unwin 1973)Google Scholar.

5 Quotations from Grundlagen (Die Grundlagen der Arithmetik [Berlin: George Olms Verlagsbuchandlung 1884; 1961 reprint]) are from the translation by Austin, J.L. The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, second revised edition (Evanston, IL: Northwestern University Press 1980)Google Scholar. Quotations from Grundgesetze (Grundgesetze der Arithmetik [Hamburg: George Olms Verlagsbuchandlung 1893; 1903; 1961 reprint]) are from the translation by Furth, Montgomery The Basic Laws of Arithmetic: Exposition of the System (Berkeley: University of California Press 1964)Google Scholar. Citation of both works is either by section or to the page number of the German edition.

6 So called because Frege introduces it with a quote from Hume’s Treatise, I, III, I, para. 5. See George Boolos, ‘The Consistency of Frege’ s Foundations of Arithmetic,’ in Thomson, Judith Jarvis ed., On Being and Saying: Essays for Richard Cartwright (Cambridge: MIT Press 1987)Google Scholar; and ‘The Standard of Equality of Numbers,’ in Boolos, George ed., Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge: Cambridge University Press 1990)Google Scholar.

7 In his Frege’ s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press 1983).

8 §xix, 154-69. In Wright’s book, Hume’s Principle is called N

9 The Philosophical Review 93 (1984) 638-40

10 ‘The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik,’ Journal of Symbolic Logic 58 (1993) 579-60

11 Basic Law V, explicitly formulated in Grundgesetze §20, implies that the extension of the concept F = the extension of the concept G iff everything falling under F falls under G, and conversely, everything falling under G falls under F. In Grundgesetze concepts are treated as a special type of function (after the proposal of Frege’s lecture, ‘Function and Concept’ [1891]) and what we are calling the extension of a concept is there represented by the notion of the value range of the corresponding function. For Frege, concepts satisfy a condition of ‘extensional sameness’: F is the same concept as G if everything falling under F falls under G, and conversely, everything falling under G falls under F. Concepts are not, however, to be identified with the sets or extensions which they determine. (‘Function and Concept,’ Peter Geach, trans., is collected in McGuinness, Brian ed., Gottlob Frege: Collected Papers in Mathematics, Logic, and Philosophy [Oxford: Basil Blackwell 1984]Google Scholar).

Basic Law V is generally regarded as the source of the contradiction exposed by Russell’s paradox. The situation is, however, rather subtle. Thus, Dummett remarks that the theory consisting of Basic Law V and the first-order fragment of Grundgesetze is consistent (219). The model Dummett describes also suffices to show the consistency of the second-order theory consisting of Hume’s principle. (See also Terence Parsons, ‘The Consistency of the First-Order Portion of Frege’s Logical System,’ Notre Dame Journal of Formal Logic 28 [1967]161-8.)

12 For an account of how Frege’s theorem may be incorporated into the conceptual structure of Zermelo Fraenkel set theory, see Demopoulos, William and Bell, John ‘Frege’s Theory of Concepts and Objects and the Interpretation of Second-Order Logic,’ Philosophia Mathematica, Series III 1 (1993) 139-56CrossRefGoogle Scholar. Cf. §1 and the Appendix.

13 Grundlagen §§70-2

14 Grundlagen §§64-7

15 See Wilson, Mark ‘Frege: The Royal Road from Geometry,’ Noûs 26 (1992) 149-80CrossRefGoogle Scholar, for a detailed discussion of this connection.

16 Grundlagen, §64. Although we may perhaps take his remark that while contextual definition ‘seems to be a very odd kind of definition, to which logicians have not paid enough attention, it is not altogether unheard of’ as an allusion to this geometric tradition (§63).

17 Dummett calls §§62-9 ‘the most brilliant and philosophically most fruitful of the book, and the most important for Frege’s philosophy of mathematics, and, indeed, his philosophy generally’ (111).

18 The terminology can be confusing. Thus, in The Principles of Mathematics, Russell attributes to Peano a notion of ‘definition by abstraction’ that corresponds to what we are calling contextual definition (210).

19 Frege’s Conception of Numbers as Objects. Of the secondary literature on or inspired by Frege, Wright’s book receives the most attention from Dummett.

20 See Dummett’s discussion of the translation of Grundlagen §46 (88).

21 In Grundlagen §57 Frege suggests that the second member of this pair of sentences may be ‘converted’ into the first, and that the pair of sentences therefore exemplify what, in §64, is described as different ‘carvings’ of a single ‘content.’ This is a large issue that we cannot go into here. Dummett discusses the matter in chapter 14.

22 What the adjectival strategy requires is the definition of an operator that will map numerical quantifiers onto their ‘successors.’ (Such an operator would be the analogue of a successor operation, defined on numbers.) A complication arises because an operator, in the technical sense of a mapping from functions to functions, will not comply with Frege’s preference for mappings from concepts or functions to objects (rather than to other concepts or functions). Dummett shows how to define the ‘successor’ of a numerical quantifier in such a way that this preference of Frege’ s is met.

23 See Frege’s discussion of Lange, Ludwig in Zeitschrift für Philosophie und Philosophische Kritik 98 (1891) 145-61Google Scholar, translated as ‘On the Law of Inertia,’ Hans Kaal, trans., and collected in McGuinness, Brian ed., Gottlob Frege: Collected Papers in Mathematics, Logic, and Philosophy (Oxford: Basil Blackwell 1984)Google Scholar.

24 It turns out that absolute acceleration (but not absolute space) is required by the derivation. The issue received its definitive clarification by E. Cartan in the 1920s. (For an exposition and discussion see Mark Wilson, ‘There’s a Hole and a Bucket, Dear Leibniz,’ in French, P. et al., eds., Midwest Studies in Philosophy [Minneapolis: University of Minnesota Press, forthcoming]CrossRefGoogle Scholar.) The possibility of developing Newtonian Mechanics with absolute acceleration but without absolute velocity was suggested by F.P. Ramsey in manuscript notes for his posthumously published paper, ‘Theories’ (1929), in Braithwaite, R. B. ed., The Foundations of Mathematics and Other Logical Essays (Paterson: Littlefield Adams 1960)Google Scholar. See Ms. #005-17-01, Hillman Library, University of Pittsburgh.

25 Cf. Dummett, 131ff. and 225f. See Hodes, Harold ‘Logicism and the Ontological Commitments of Arithmetic,’ Journal of Philosophy 81 (1984) 123-49CrossRefGoogle Scholar, for a recent development of the idea of identifying numbers with concepts of higher level.

26 As Dummett points out; cf. 55.

27 The Philosophical Review 74 (1965) 47-73

28 Toward the end of his life, Frege seems to have fallen into some such view. See his posthumously published, ‘Logic in Mathematics’ (1914), in Hermes, H. et al., eds., Gottloli Frege: Posthumous Writings (Chicago: University of Chicago Press 1980)Google Scholar.

29 Regarding the terminology of Grundgesetze, see note 11, above. When quoting from Grundgesetze, I have followed the translation of Montgomery Furth, with the exception that I have replaced’ course of values’ with ‘value range,’ since this is the English expression favored by Dummett.

30 See, for example, the Appendix to the paper of Demopoulos and Bell.

31 As Dummett notes (224).

32 For Dummett’s discussion of Frege’s attempted consistency proof (in §31 of Grundgesetze), see especially the section entitled ‘How the Serpent Entered Eden’ (215-22).

33 Of course Frege held that Euclidian geometry is also true simpliciter. Cf. his correspondence with Hilbert, collected in McGuinness, Brian ed., Gottlob Frege: The Philosophical and Mathematical Correspondence, Kaal, Hans trans. (Chicago: University of Chicago Press 1980)Google Scholar.