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Arnošt Kolman’s Critique of Mathematical Fetishism

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The Vienna Circle in Czechoslovakia

Part of the book series: Vienna Circle Institute Yearbook ((VCIY,volume 23))

Abstract

Arnošt Kolman (1892–1979) was a Czech mathematician, philosopher and Communist official. In this paper, we would like to look at Kolman’s arguments against logical positivism which revolve around the notion of the fetishization of mathematics. Kolman derives his notion of fetishism from Marx’s conception of commodity fetishism. Kolman is aiming to show the fact that an entity (system, structure, logical construction) acquires besides its real existence another formal existence. Fetishism means the fantastic detachment of the physical characteristics of real things or phenomena from these things. We identify Kolman’s two main arguments against logical positivism. In the first argument, Kolman applied Lenin’s arguments against Mach’s empiricism-criticism onto Russell’s neutral monism, i.e. mathematical fetishism is internally related to political conservativism. Kolman’s second main argument is that logical and mathematical fetishes are epistemologically deprived of any historical and dynamic dimension. In the final parts of our paper we place Kolman’s thinking into the context of his time, and furthermore we identify some tenets of mathematical fetishism appearing in Alain Badiou’s mathematical ontology today.

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Notes

  1. 1.

    Arnošt Kolman, “Úkoly soudobé filosofie”, in: Tvorba, Vol. 17, No. 33, 1948, p. 647. English translation: Arnošt Kolman, “Tasks of Contemporary Philosophy”, in: Russell : the Journal of Bertrand Russell Studies, Vol. 36, No. 2, 2016. The congress is covered in Russell’s Collected Papers, Volume 11: Last Philosophical Testament 1943–68, ed. by John G. Slater, London and New York: Routledge, 1997, pp. 115–116. All other translations from Czech and Russian are ours.

  2. 2.

    For details of this exchange see Jakub Mácha, “Arnošt Kolman and Bertrand Russell at the 1948 International Congress of Philosophy”, in: Russell: the Journal of Bertrand Russell Studies, Vol. 36, No.2, 2017, pp. 128–138.

  3. 3.

    Karl Marx , Capital. A Critique of Political Economy. Moscow: Progress Publishers, 1977. The passage on commodity fetishism is Section 4 “The fetishism of commodities and the secret of thereof”.

  4. 4.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky [The Critical Exposition of the Symbolic Method of Modern Logic]. Praha: Orbis, 1948, p. 280.

  5. 5.

    Kolman had already written in 1931 that the roots of mathematical fetishism go back to Hegel . Kolman appreciated that “by coming out against the fetishisation of quantity, which after all is only a reflection of the abstract money-trading relations of the bourgeois order, Hegel in this case actually burst apart the framework of bourgeois philosophy.” (Ernst Kolman, Sofya Yanovskaya, “Гегель и математика” [“Hegel and Mathematics”], Под знаменем марксизма [Under the Banner of Marxism], No. 11–12, 1931, pp. 107–120. English translation in: Mathematical Manuscripts of Karl Marx. London: New Park Publications, 1983, pp. 235–255, p. 242.)

  6. 6.

    Ibid., p. 251.

  7. 7.

    Ibid., p. 19.

  8. 8.

    Aristotle , Metaphysics. Trans. A. Armstrong. London: Heinemann 1933, I.5, 985b–986a.

  9. 9.

    Arnošt Kolman, “O podstatě a původu pythagoreismu” [On the Nature and Origin of Pythagoreanism], in: Česká mysl, No. 40, 1947, p. 148.

  10. 10.

    Cf. Aristotle, Metaphysics, XIII.9, 1085a–b.

  11. 11.

    Arnošt Kolman, “O podstatě a původu pythagoreismu”, op. cit., p. 149.

  12. 12.

    Max Weber also thought that Pythagoreanism had a socio-political origin. But for him, this movement does not express or mirror the ideology of a ruling class, but rather “emerge[s] when the ruling strata, noble or middle class, have lost their political power to a bureaucratic-militaristic unitary state.” (Max Weber, Economy and Society: An Outline of Interpretive Sociology. Trans. and ed. G. Roth, C. Wittich et al. Berkeley : University of California Press 1978, Vol. 1, p. 503.)

  13. 13.

    Arnošt Kolman, Logika [Logic]. Praha: Svoboda 1947, p. 161.

  14. 14.

    Ernst Kolman, Sofya Yanovskaya, “Гегель и математика” [“Hegel and Mathematics”], Под знаменем марксизма [Under the Banner of Marxism], No. 11–12, 1931, pp. 107–120.

  15. 15.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky, op. cit., p. 7.

  16. 16.

    Ibid., p. 276.

  17. 17.

    Ibid., p. 277.

  18. 18.

    Ludwig Wittgenstein , Tractatus Logico-Philosophicus. Trans. by D. Pears and B. McGuinness. London: Routledge 1961, §1.1.

  19. 19.

    This argument can be applied to any a priori law or structure. See in this context the chapter “The Criticism of Kantianism from the Left and From the Right” from Lenin’s Materialism and Empirio-criticism. (Vladimir Ilyich Lenin, Materialism and Empirio-criticism: Critical Comments on a Reactionary Philosophy. Lenin Collected Works. Vol.14. Trans. A. Fineberg. Moscow: Progress Publishers 1972.)

  20. 20.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky, op. cit., pp. 212 & 219.

  21. 21.

    Ibid., p. 18.

  22. 22.

    Ibid., p. 220.

  23. 23.

    Ibid., pp. 199 & 203.

  24. 24.

    Ibid., p. 205.

  25. 25.

    Ibid., pp. 253–4.

  26. 26.

    Ibid., p. 230.

  27. 27.

    Bertrand Russell , The Analysis of Mind. London: George Allen & Unwin 1921, p. 25.

  28. 28.

    Bertrand Russell, The Philosophy of Logical Atomism. London: Routledge 2010, p. 147.

  29. 29.

    Ibid.

  30. 30.

    Ibid., p. 124.

  31. 31.

    Ludwig Wittgenstein , Tractatus Logico-Philosophicus, op. cit., §6.53.

  32. 32.

    Ibid., Preface.

  33. 33.

    Merrill Hintikka and Jaakko Hintikka, Investigating Wittgenstein. Oxford: Basil Blackwell 1986, pp. 51ff. maintain that the primary language of the Tractatus is phenomenological. The objects of the Tractatus are very close to Russell’s objects of acquaintance, i.e. to sense-data. See also Andreas Blank, “Wittgenstein’s Tractatus and the Problem of a Phenomenological Language”, in: Philosophia, Vol. 29, No. 1–4, 2002, p. 327.

  34. 34.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky, op. cit., p. 204.

  35. 35.

    Ludwig Wittgenstein , Tractatus Logico-Philosophicus, op. cit., §6.124.

  36. 36.

    Rudolf Carnap , The Logical Structure of the World. Trans. R. George. Chicago: Open Court Classics 2003.

  37. 37.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky, op. cit., p. 254.

  38. 38.

    Rudolf Carnap, The Logical Structure of the World, op. cit., p. 178; Carnap’s italics.

  39. 39.

    Ludwig Wittgenstein , Tractatus Logico-Philosophicus, op. cit., §6.124.

  40. 40.

    Vladimir Ilyich Lenin , Materialism and Empirio-criticism: Critical Comments on a Reactionary Philosophy, op. cit.

  41. 41.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky, op. cit., p. 277.

  42. 42.

    See Shaun Gallagher, Hege l, History, and Interpretation. SUNY Press: Albany, New York 1997, p. 9.

  43. 43.

    Curiously enough, there has been developed quite opposed arguments claiming that there is a substantial connection between logical positivism and Marxism or communism. Heidegger is the most striking case. He wrote in a reply to Carnap’s paper “Überwinding der Metaphysik durch logische Analyse der Sprache”: “It is also no accident that this kind of ‘philosophy’ stands in both internal and external connection with Russian communism.” His argument, for the internal relatedness, is that in the mathematical philosophy, truth is diverted into a certainty, which leads to the profaning [Entgötterung] of the world. If we get over Heidegger’s too quickly equating Russian communism and Marxism in general, we see he is not the only one holding this view. The goal of Neurath’s scientific philosophy was social and political progress in broadly Marxist perspective. Also Carnap , at least partly, admitted this internal connection when he reported his “Marxist views on how metaphysics will be overcome through reformation [Umgestaltung] of the substructure.” (The quotations are taken from Michael Friedman, A Parting of the Ways. Carnap, Cassirer, and Heidegger. Chicago and La Salle: Open Court, 2000, pp. 22, 21. See for more discussion therein.) It seems, thus, that logical positivism and Marxism are allies rather than enemies. All these facts pose serious problems for Kolman’s view.

  44. 44.

    Arnošt Kolman, Kritický výklad symbolické metody moderní logiky, op. cit., p. 211.

  45. 45.

    Arnošt Kolman, Logika, op. cit., p. 167.

  46. 46.

    Vladimir Ilyich Lenin , Materialism and Empirio-criticism, op. cit., p. 142.

  47. 47.

    David Guest , “The Machian Tendency in Modern British Philosophy”, in: Carmel Haden Guest (Ed.), David Guest: A Scientist Fights for Freedom (1911–1938). A Memoir. London: Lawrence & Wishart 1939, pp. 219–249. First published in Russian in: Under the Banner of Marxism, No. 5, 1934, pp. 31–45.

  48. 48.

    See the discussion of Kolman’s paper “Hegel and Mathematics” above.

  49. 49.

    Bertrand Russell , “The Relation of Sense-data to Physics”, in: Mysticism and Logic. London: George Allen & Unwin 1917, p. 115.

  50. 50.

    Carnap took this maxim as the epigraph of The logical Structure of the World (1928).

  51. 51.

    David Guest , “The Machian Tendency in Modern British Philosophy”, op. cit., p. 224.

  52. 52.

    Ibid., p. 226.

  53. 53.

    Vladimir Ilyich Lenin , Materialism and Empirio-criticism, op. cit., p. 49.

  54. 54.

    Bertrand Russell, “The Relation of Sense-data to Physics”, op. cit., p. 123.

  55. 55.

    David Guest, “The Machian Tendency in Modern British Philosophy”, op. cit., p. 227.

  56. 56.

    Ibid., p. 232.

  57. 57.

    Ibid., p. 238.

  58. 58.

    It is a historical irony that some contemporary interpretations take the Tractatus to be dialectical, historical and anti-metaphysical. See, e.g., Mathew Ostrow, Wittgenstein’s Tractatus: A Dialectical Interpretation. Cambridge: Cambridge University Press 2002 and Ben Ware, Dialectic of the Ladder. Wittgenstein, the ‘Tractatus’ and Modernism. London: Bloomsbury 2015.

  59. 59.

    Our claim is that there is a significant substantial affinity between Kolman’s notion of mathematical fetishism and Wittgenstein’s notion of sublimation of logic. Moreover, there might have been a two-way influence. There are some hints at least. Wittgenstein had visited Russia in 1935. His main contact and host there was Sofya Yanovskaya, Kolman’s long-time collaborator, who coauthored Kolman’s early paper ‘Hegel and Mathematics’ (see footnote 5). Wittgenstein was familiar with the main tenets of dialectical materialism and had acquaintance with parts of Marx’s Capital. (See John Moran, “Wittgenstein and Russia”, in: New Left Review, No. 73, 1972, pp. 85–96.)

  60. 60.

    Ludwig Wittgenstein , Philosophical Investigations. G. E. M. Anscombe and R. Rhees (eds), trans. G. E. M. Anscombe, P. M. S. Hacker and J. Schulte, revised by P. M. S. Hacker and Joachim Schulte, revised fourth edition. London: Blackwell 2009, §38.

  61. 61.

    Ibid., §89.

  62. 62.

    David Andrews , “Commodity Fetishism as a Form of Life”, in: Gavin Kitching and Nigel Pleasants (eds.), Marx and Wittgenstein . Knowledge, Morality and Politics. London: Routledge 2002, p. 85.

  63. 63.

    Ibid., p. 88.

  64. 64.

    Ibid., p. 89.

  65. 65.

    This remark does not mean that mathematical fetishism is to be found only in Pythagoreanism and logical positivism. One could also mention Galileo at least.

  66. 66.

    Alain Badiou , Being and Event. Trans. by Oliver Feltham. London: Continuum 2005, p.xiii.

  67. 67.

    Ibid., p. 14. In Logic of Worlds: Being and Event II. London: Continuum, 2009, Badiou provides a theory of appearance (which he calls phenomenology) based on category theory. For the sake of simplicity, we restrict the present discussion to Badiou’s treatment of set theory.

  68. 68.

    Alain Badiou, Being and Event, op. cit., p. 25.

  69. 69.

    Ricardo L. Nirenberg and David Nirenberg , “Badiou’s Number: A Critique of Mathematics as Ontology”, in: Critical Inquiry, Vol. 37, No. 4 (Summer 2011), pp. 606–7. The authors also mention Kolman and his role in the Luzin affair.

  70. 70.

    Alain Badiou, “To Preface the Response to the ‘Criticisms’ of Ricardo Nirenberg and David Nirenberg”, in: Critical Inquiry, Vol. 38, No. 2, 2012, p. 363.

  71. 71.

    Alain Badiou , Being and Event, op. cit., p. 17.

  72. 72.

    Ibid., p. 340.

  73. 73.

    On the other hand, Badiou thinks, in agreement with Kolman, that analytic philosophy is inherently non-dialectical and it is internally related to conservativism. (Cf. Alain Badiou, “What is Philosophy?”. In: European Graduate School Video Lectures, YouTube, 2 May 2011. Retrieved from https://www.youtube.com/watch?v=Z6FQkTajudY, 27 July 2015.)

  74. 74.

    The fact that for Badiou (political) economy is not the generic procedure (or is not among the generic procedures) has been criticized by various commentators like Žižek or Livingston. See Paul M. Livingston, The Politics of Logic. Badiou, Wittgenstein , and the Consequences of Formalism. New York and London: Routledge, 2012, p. 300.

  75. 75.

    The fact that there are exactly four generic procedures (love, art, etc.) is only an example of how Badiou applies mathematical formalism onto our praxis. All his “textual mediations” in Being and Event involve a certain kind of application of his mathematical and ontological discourse.

  76. 76.

    This is the main thesis of Paul Livingston’s critique of Badiou’s project: “Badiou’s application of mathematical formalisms to the diverse questions of social and political life repeatedly involves fundamental gestures of projection, whereby formal and mathematical structures bear the weight of the theorization of such diverse political and ontological concepts” (Paul M. Livingston, The Politics of Logic. Badiou, Wittgenstein, and the Consequences of Formalism, op. cit., p. 10).

Acknowledgement

We would like to thank Ken Blackwell, Bernard Linsky, Jim Klagge, Michael Hauser, Davide Spagnoli, Saman Pushpakumara and the participants of the conferences The Vienna Circle in Czechoslovakia (Pilsen, February 2015) and SSAHP 2015 (Dublin, June 2015) for their valuable suggestions.

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  1. Department of Philosophy, Masaryk University, Brno, Czech Republic

    Jakub Mácha & Jan Zouhar

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Correspondence to Jakub Mácha .

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  1. Department of Philosophy, University of West Bohemia, Pilsen, Czech Republic

    Radek Schuster

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Mácha, J., Zouhar, J. (2020). Arnošt Kolman’s Critique of Mathematical Fetishism. In: Schuster, R. (eds) The Vienna Circle in Czechoslovakia. Vienna Circle Institute Yearbook, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-030-36383-3_7

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