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The aim of Russell’s early logicism: a reinterpretation

Abstract

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from Gödel’s incompleteness theorem and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable.

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Notes

  1. 1.

    Russell ([1903] 1937, p. xv). The logicist project had dawned on Russell in late 1900 or early 1901, and was publicized for the first time in Russell’s “On Recent Work on the Principles of Mathematics” from 1901 (reprinted as “Mathematics and the Metaphysicians” in Russell 1917), before receiving its first full statement in Principles. The term “logicism” (from the German Logizismus) as a name for Russell’s project was coined by Rudolf Carnap in 1929. For more on this, see Grattan-Guinness (1990, pp. 304–328, 501–503).

  2. 2.

    Russell ([1903] 1937, p. 3) (my emphasis).

  3. 3.

    Russell ([1903] 1937, p. 4) (my emphasis).

  4. 4.

    Russell ([1903] 1937, p. 5) (my emphasis).

  5. 5.

    See e.g. Hempel (1945, p. 552); Quine (1994, p. 15); and Pollock (citePo70, pp. 388–389).

  6. 6.

    Carnap ([1931] 1964, p. 31).

  7. 7.

    Carnap ([1931] 1964, pp. 31, 34).

  8. 8.

    For “expressibility” see e.g. Jager (1972, pp. 193, 207), and Sainsbury (1979, p. 272); for “definability” see e.g. Grayling (1996, p. 28), and Carey and Ongley (2009, p. 133); for “constructability” see e.g. Hempel (1945, p. 556n8).

  9. 9.

    For more on this, see e.g. Hylton (1990, pp. 171–179), and Landini (1998, p. 31).

  10. 10.

    Russell ([1903] 1937, p. xviii). Russell says that propositions will be “assumed” to be “independen[t] of any knowing mind,” and that this assumption is “indispensable to any even tolerably satisfactory philosophy of mathematics.”

  11. 11.

    Putnam ([1967] 1975, p. 20).

  12. 12.

    Putnam ([1967] 1975, pp. 13, 20).

  13. 13.

    Putnam ([1967] 1975, p. 13).

  14. 14.

    Putnam ([1967] 1975, p. 20).

  15. 15.

    Putnam ([1967] 1975, p. 20).

  16. 16.

    Musgrave (1977, p. 101). For Russell’s paper, see Russell (1917).

  17. 17.

    Musgrave (1977, p. 109).

  18. 18.

    Musgrave (1977, p. 112).

  19. 19.

    Musgrave (1977, p. 113).

  20. 20.

    Irvine (1989, pp. 311–315). See also Irvine and Godwyn (2003).

  21. 21.

    Irvine (1989, pp. 311–313).

  22. 22.

    Irvine (1989, p. 313).

  23. 23.

    Hylton (1990, p. 322).

  24. 24.

    This objection would not hold if Irvine’s interpretation of the aim of Russell’s logicism were delimited to Russell’s post-1907 writings as opposed to the pre-1907 writings. But Irvine applies the interpretation also to Russell’s pre-1907 writings; see e.g. Irvine (1989, pp. 306–307).

  25. 25.

    Hochberg (1970, pp. 396–397).

  26. 26.

    Klement (2012–2013, pp. 145–146).

  27. 27.

    For an account of Russell’s conception of propositions in Principles, see e.g. Hylton (1990, pp. 171–178).

  28. 28.

    Russell ([1903] 1937, p. 42).

  29. 29.

    Russell ([1903] 1937, p. 47).

  30. 30.

    For another example of this non-technical use of the term “proposition” in Principles, selected more or less at random, see e.g. Russell ([1903] 1937, p. 13): “An expression such as ‘\(x\) is a man’ is therefore not a proposition, for it is neither true nor false. If we give to \(x\) any constant value whatever, the expression becomes a proposition.” The claim that the relevant expression “becomes a proposition” means that the relevant expression “becomes a sentence which expresses a proposition.”

  31. 31.

    Russell ([1903] 1937, p. 5) (my emphasis).

  32. 32.

    Cf. Russell’s claim in ([1901] 1917, p. 76): “formal logic [\(\ldots \)] has thus at last shown itself to be identical with mathematics”.

  33. 33.

    Russell ([1903] 1937, p. 9).

  34. 34.

    Russell ([1903] 1937, p. 11). The view that Russell’s conception of logic involves as a central idea the claim that logical propositions have complete generality or are universally valid is of course not original with me. For elaborations of this idea, see e.g. Pelham (1993).

  35. 35.

    Russell ([1903] 1937, pp. 5–6, 11). Cf. Russell ([1901] 1917, p. 75).

  36. 36.

    Russell ([1903] 1937, pp. 5–6, 11). Cf. Russell ([1901] 1917, p. 75). An illustration of how Russell sought to show that mathematical truths are implications is provided by the famous *54.43 of Russell and Whitehead’s Principia Mathematica (originally intended as part II of Principles), where “1\(+\)1\(=\)2” is (in effect) regarded as definable in terms of the implication “\((\alpha ,\beta \in 1) \rightarrow ((\alpha \cap \beta = \emptyset ) \leftrightarrow (\alpha \cup \beta \, \in \, 2))\)”; see Russell and Whitehead (1910, p. 379) (I have modernized Russell and Whitehead’s notation).

  37. 37.

    Russell ([1903] 1937, p. 4) (my emphasis). Cf. also Russell ([1901] 1917, p. 96) (“The proof that all pure mathematics, including Geometry, is nothing but formal logic, is a fatal blow to the Kantian philosophy. [\(\ldots \)] Kant’s theory [\(\ldots \)] has to be abandoned. The whole doctrine of a priori intuitions, by which Kant explained the possibility of pure mathematics, is wholly inapplicable to mathematics in its present form”); and Russell (1944, p. 13) (“[Logicism was] a parenthesis in the refutation of Kant”). For Kant’s view, see e.g. Kant, Critique of Pure Reason, pp. 286–289 (A162/B202-A166/B207).

  38. 38.

    See e.g. Pollock (1970, pp. 388–395); Grattan-Guinness (1990, p. 565); and Hintikka (1996, p. 183). Cf. also Quine ([1969] 1994, p. 16).

  39. 39.

    A similar response to the relevant Gödelian objection is found in Landini (1998, p. 16).

  40. 40.

    See e.g. Kilmister (1996, pp. 279–280).

  41. 41.

    See e.g. Quine ([1969] 1994, pp. 15–16).

  42. 42.

    Russell ([1912] 1992, p. 54).

  43. 43.

    Russell (1919, pp. 204–205).

  44. 44.

    Russell (1937, p. xii).

  45. 45.

    Russell (1937, p. v).

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Acknowledgments

Many thanks to Hilary Putnam, Kaj B. Hansen, Alan Musgrave, Andrew D. Irvine, and to two anonymous reviewers, for helpful feedback on earlier versions of this paper.

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Correspondence to Anders Kraal.

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Kraal, A. The aim of Russell’s early logicism: a reinterpretation. Synthese 191, 1493–1510 (2014). https://doi.org/10.1007/s11229-013-0342-9

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Keywords

  • Bertrand Russell
  • Logicism
  • Philosophy of mathematics