Notes Introduction 1 . Not that I am the first to discover this. The early ‘realist’ or ‘post-idealist’ Russell has been the subject of much first-rate scholarship in recent years: see, e.g., Hylton (1990a), Landini (1998), Makin (2000), Stevens (2005) and such collections of papers as those of Irvine and Wedeking (1993), Monk and Palmer (1996), Griffin (2003) and Griffin and Jacquette (2009). These works are examples of the scholarly interest that has arisen in ‘early analytic philosophy’. See Floyd (2009) for an overview of this development. 2 . I borrow the term ‘analytical modernism’ from Skorupski (1993). 3 . I owe this characterization of Russell’s and McTaggart’s relationship to Nicholas Griffin. 4 . This story is told in detail in Griffin (1991). 5 . Bradley (1911, 74). 1 Russell’s Early Logicism: What Was It About? 1 . See Frege (1880/1; 1882). 2 . Cf. Tiles (1980, 158). 3 . The question became more acute with the discovery of paradoxes and the complications it necessitated in the logic underlying the reductionist ambi- tions. Indeed, it seems that the reason why Frege gave up on logicism was that he found himself unable to regard the ensuing complications as a matter of logic . Russell was in this respect much more flexible, and at least in the case of the theory of types, he occasionally pressed the point that the theory was really ‘plain common sense’ (1924, 334). However, the question has been there from the beginning of modern logic. 4 . See Russell (1944, 13; 1959, 74–5). 5 . It is good to remember that the term ‘logicism’ – or its German equivalent, ‘Logizismus’ – was introduced in something like its modern sense only in the 1920s, when Abraham Fraenkel and Carnap used it, independently of one another, to denote a certain position in the philosophy of mathematics (this information derives from Grattan-Guinness 2000, 479, 501). This use became commonplace when Rudolf Carnap (1931) drew the threefold distinction involving logicism, intuitionism and formalism, applying the first term to Frege and Principia Mathematica . The German term ‘Logizismus’ occurs also in Theodor Ziehen (1920), but there it is used to refer to the view that there is a world of abstract objects; logicists in this sense would include such figures as Bernard Bolzano, Franz Brentano, Alexius Meinong and Edmund Husserl as well as Frege. 6 . To speak of the logical empiricists’ views on mathematics as one logicism is to commit oneself to an oversimplification that is itself yet another instance 238 Notes 239 of the one that is discussed in the text. Ayer, for instance, was interested not so much in logicism and its prospects as in maintaining the analyticity (and hence apriority) of mathematics. Indeed, the truth or otherwise of logicism was of little importance to him, as he claimed that on his conception of analy- ticity – the criterion for the analyticity of a proposition being that ‘its validity should follow simply from the definitions of the terms contained in it’ (Ayer 1936, 109) – the propositions of pure mathematics will be analytic whether or not logicism in the sense of Whitehead and Russell turns out to be correct. 7 . Kant’s Critique of Pure Reason (KrV ) will be referred to in the standard manner, by citing the relevant page or pages of the appropriate edition, A (1781) or B (1787). 8 . To make this explanation complete and convincing, one would have to explain the nature of this ‘cannot fail to recognize’. A familiar suggestion is to argue that the condition for possessing a certain concept is the accept- ance as true of a range of propositions involving the concept. If viable, this formulation would explain in a non–ad hoc manner why there can be no gap between possessing a concept and recognizing a certain truth. As we shall see in Chapter 3 , Kant applied to synthetic a priori propositions an explanation that bears some similarity to this view. 9 . The explanation was partial, because it had to be complemented by an account of the nature of language and our knowledge of it that can be deemed acceptable by the logical empiricists’ standards. But here they could refer to conventionalism and behaviourism. 10 . It is significant, though, that Frege uses the concept of analyticity only in his Foundations of Arithmetic (1884). 11 . This lack of interest has been noted by a number of authors, including Taylor (1981), Coffa (1982) and Hylton (1990a, 197; 1990b, 204). 12 . This point is made very clearly by Goldfarb (1982, 693). 13 . This development is explored in some detail in Detlefsen (1996). 14 . The term ‘epistemic logicism’ is borrowed from Irvine (1989). 15 . See Wood (1959). 16 . For example, according to Kant, geometry is a synthetic science, but there are a few ‘fundamental propositions’, such as the proposition that ‘the whole is greater than its part’, which are ‘identical propositions’ and serve therefore only as ‘links in the chain of method’ (KrV , B16–7). 2 Kant and Russell on the Mathematical Method 1 . The term ‘semantic’ must be construed here in a broad sense (cf. Coffa 1991). In this sense, anyone with an interest in ‘representation’ or ‘content’ is engaged in semantics. Semantics, as here understood, is not necessarily tied to any language, even less to any formal language; such associations emerged only in the 1920s and 1930s, when certain philosophers and logi- cians effected the so-called linguistic turn. Instead of ‘semantic’, I will occa- sionally use the more neutral term ‘representation-theoretic’, introduced by Richardson (1998) for a similar purpose. 2 . The ‘external’ reading can be found, for example, in Hintikka (1967; 1969) or Brittan (1978). 240 Notes 3 . Admittedly, there is in Kant a different use of ‘content’ that relates to objec- tive validity; concepts have contents only insofar as they can be given objects in experience, a claim that applies to mathematics, too ( KrV , A239/B299). Such usage by Kant, however, does not justify the omission of the semantic or representation-theoretic sense of ‘content’. In this sense the function of ‘content’ is comparable, for example, to Frege’s notion of ‘judgeable content’ (Frege 1879) or the notion of ‘proposition’ which Russell introduced in PoM 4 . For similar passages, see KrV (B154–5) and Kant (1783, §12). 5 . A detailed treatment of this idea is found in Friedman (1992a). It is quite essential to Kant’s conception of geometrical proof that it is based on a general rule rather than an observation of a single figure; that it operates with a schema , a universal procedure for generating a figure, rather than the figure, which is nothing but an image of a geometrical concept and, as such, an empirical object (KrV , A140–1/B180). One can refute Kant by ignoring the role of rules, which leads to the allegation that he confused geometrical concepts with subjective mental images (cf. Couturat 1904). On the other hand, having observed the role of rules, a critic may press the objection that this makes geometrical reasoning analytic rather than synthetic. When we prove a geometrical theorem, we do not observe the diagram and draw conclusions from this; we look at the rule to see what must hold for all objects falling under the generated concept. Thus the rule may look more conceptual than intuitive, at least in the sense that it is a general representa- tion; for this kind of objection, see Bolzano (1810, Appendix, §7); Couturat (ibid., 348–52); Kitcher (1976, 124). This, though, overlooks the fact that the only way to ‘look at’ a geometrical rule is to draw a figure and perform a series of further constructions on it. 6 . See Kant’s notes on Kästner in Allison (1973, Appendix B, 175–6). 7 . J. G. Schulze, Prüfung der Kantischen Critik der reinen Vernunft (‘Examination of Kant’s Critique of Pure Reason ’); quoted in Allison (1983, 94). 8 . As Friedman points out, Kant’s notion of pure intuition, understood in kinematic terms, ‘contains the seeds of its own destruction’ (2000, 202). The kinematic conception of pure intuition amounts to the view that the formal structure of spatial intuition (possible spatial motions) is expressed by the conditions of free mobility and that these conditions are uniquely captured by Euclidean geometry, which is why it is a priori. After the discovery of non-Euclidean geometries, however, it became clear that these conditions do not yield the specifically Euclidean space but the three classical cases of space of constant curvature. It follows that, insofar as ‘spatial intuition’ or ‘the a priori element in geometry’ is tied up with free mobility, Euclidean geometry can no longer be regarded as a priori. This is one of the conclusions that Russell argued for in An Essay on the Foundations of Geometry , although his reasons were quite different. 9 . Kant’s constructive semantics not only covers geometry but extends to arith- metic, analysis and algebra as well (although it is often not very clear what is going on in these extensions). The case of arithmetic will be mentioned below. For a general survey of Kant’s theory of mathematics, see Shabel (2003). She argues convincingly that Kant’s theory is best understood when it is considered against the background of what may be called ‘early modern mathematics’. Notes 241 10 . ‘Platonic atomism’ is the term used by Hylton (1990a). 11 . Some of the issues that are relevant here will be explored in Chapter 4 . 12 . Russell’s letter to Moore, 16 August 1900, quoted in Russell (1992b, 202). 13 . In the Paris Congress Russell attended a session in which Peano and his colleague Mario Pieri gave talks about definitions in mathematics.