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Published online by Cambridge University Press: 01 June 1975
Logicism is the doctrine that mathematics is reducible to logic. It is usually presented in two theses: (1) Every mathematical concept is definable in terms of logical concepts. (2) Every mathematical theorem is deducible from logical principles. In this paper, I am not concerned with the truth or falsity of (1) and (2). Rather, I am concerned with the underlying philosophical system. Logicism is connected with the names of Frege, Russell (and Whitehead). But the logicism which is familiar to most philosophers is not the original logicist system of Russell. Instead, we read either Russell's later Introduction to Mathematical Philosophy or articles by Carnap and Hempel. My purpose in this paper is to return to the original logicist system of Russell. This system, at least in essentials, lasts through the publication of the First Edition of Principia Mathematica. I believe that examination of this system will shed light on why certain difficulties arose in later logicism, including the logicist views of the Logical Empiricist movement. Further, the issues are closely connected with general doctrines on the nature of philosophical analysis.
1 R. Carnap, “The Logicist Foundation of Mathematics” and C.G. Hempel “On the Nature of Mathematical Truth”; both reprinted in P. Benacerraf and H. Putnam (editors), Philosophy of Mathematics (Englewood Cliffs, N.J.: Prentice-Hall, 1964).
2 Volume I was published in 1910, Volume II in 1912, Volume III in 1913.
3 The Principles of Mathematics. (London: George Allen and Unwin. 1956; originally published 1903), pp. xviii, 4. Henceforth cited POM. Also see The Philosophy of Bertrand Russell, edited by P.A. Schilpp (LaSalle, 111.: Open Court, 1971), p. 12.
4 In fact, his opposition to Kant was so important to Russell that he later said, “The work that ultimately became my contribution to Principia Mathematicu presented itself to me, at first, as a parenthesis in the refutation of Kant. “The Philosophy of Bertrand Russell, p. 13.
5 Kant, Critique of Pure Reason, A 715 (B743) and following.
6 POM, pp. 157 f.
7 “Principles of Mathematics, draft of 1899–1900”. Book I, Chapter II, p. 9. Manuscript in Bertrand Russell Archives at McMaster University.
8 See especially G.E. Moore, ”Th e Nature of Judgment”, Mind 8 (1899), pp. 176–193. I am assuming that Russell and Moore agreed on major points of metaphysics. Russell frequently mentions his agreement with Moore in POM.
9 POM, p. 112.
10 POM, p. 3.
11 POM, pp. 241 f. The contrast on which I am focussing is that of the philosopher-logician against the practising mathematician. This should not be confused with Russell's distinction between the former philosophical approach to definitions and his new logical approach which he calls ‘mathematical’ as against the old ‘philosophical’ approach. For this latter contrast see POM, p. 111.
12 POM, pp. 449 ff. I am using Moore's word ‘concept’ rather than Russell's word ‘term’ to avoid the linguistic associations of ‘term’.
13 POM, p. xv.
14 “The Nature of Judgment”, p. 178.
15 Principia Mathematica 1, p. 8 (Second Edition pagination).
16 POM, p. 454.
17 “The Philosophical Importance of Mathematical Logic”. The Monist 23 (1913), pp. 485 f.
18 POM, p. 3.
19 POM. pp. 36 ff.
20 “The Philosophical Importance of Mathematical Logic” p. 489.
21 POM, p. 5.
22 POM, p. 373.
23 “Non-Euclidean Geometry”, The Athenaeum No. 4018 (1904) pp. 592 f.
24 POM, p. 430.
25 C.J. Keyser, “The Axiom of Infinity” and Russell, “The Axiom of Infinity”, both in The Hibbert Journal 2, (1903–4). The latter is reprinted in B. Russell, Essays in Analysis, ed. by D. Lackey, New York 1973.
26 “The Nature of Judgment”, p. 180.
27 “[Review of] F.P. Ramsey's The Foundations of Mathematics and otlier Logical Essays”, Mind 40 (1931), p. 477. Also see a second review by Russell in Philosophy 7 (1932), pp. 84–85.
28 An Essay on the Foundations of Geometry. (New York: Dover, 1956; originally published 1897), pp. 57–59.
29 A Critical Exposition of the Philosophy of Leibniz. (London: George Allen and Unwin, 1958; originally published 1900), pp. 16–17, 20. POM, p. 457.
30 The Problems of Philosophy. (London: Oxford University Press, 1959; originally published 1912), p. 84.
In addition to these references to published materials, we know that in correspondence (1904–5), Couturat questioned Russell on the analytic-synthetic distinction. Unfortunately, Russell's replies have not been located, but it is apparent that Russell was not sufficiently interested in the issue to comment in publications. An unpublished paper in the Archives entitled “Necessity and Possibility” from this period may be Russell's earliest expression of his opposition to Kant in terms of disagreement with Kant over the synthetic character of mathematics. Here Russell says that mathematics may be called analytic because it is deducible from the laws of logic, which are, of course, more powerful than in Kant's day.
31 “Introduction to Mathematical Philosophy. (London: George Allen and Unwin, 1960; originally published 1919), pp. 204 f.
32 The Analysis of Matter. (New York: Dover, 1954; originally 1927). pp. 170–172. An Inquiry into Meaning and Truth. (New York: W.W. Norton. 1940), p. 175.
33 POM, p. xii.
