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Neurath Reconsidered pp 241–337Cite as

Neurath and the Legacy of Algebraic Logic

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Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 336))

Abstract

In this paper I introduce a broader context, and sketch an integrated account with the purpose of examining the significance of Neurath’s attention to logic in early works and subsequent positions. The specific attention to algebraic logic is important in integrating his own interest in mathematics and combining, since Leibniz, the ideals of a universal language and of a calculus of reasoning. The interest in universal languages constitutes a much broader, so-called tradition of pasigraphy that extended beyond philosophical projects. I argue (1) that Neurath’s works can be embedded in a richer intellectual landscape that includes developments in logic and their local reception in Vienna, and that his attention to logic developed a sustained symbolic standpoint – with semiotic and typographic expressions –; (2) that specific aspects of the work in algebraic logic became a standard and a resource in subsequent work often thought independent, while its value was steadily challenged by the separate goal of empirical theorizing and practical application in social domains – including in the areas of economics, history and visual communication –; that (3), in particular, the presentation of systems of algebraic logic by Neurath’s sources such as Stanley Jevons and Schröder was not isolated from discussions of political economy; and finally (4) that some of his positions in matters of language, unity and epistemology in the articulation of logical empiricism and its debates are better understood in terms of shared but diversified acquaintance with pasigraphy, formal standards and logical projects.

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Notes

  1. 1.

    I have in mind, for instance, Uebel (1992) and (2007), Cartwright et al. (1996), Nemeth and Stadler (1996), Stadler (2001), Reisch (2005), Vossoughian (2008), Burke et al. (2013) and Sandner (2014).

  2. 2.

    An early valuable discussion by Eckehart Köhler (1991) is otherwise cursory, dismissive and, equally significantly, isolated from his discussion of other work by Neurath.

  3. 3.

    Lewis , expounding on Couturat’s narrative – and his narrow focus on Huntington’s recent formulation –, provided the main source for other authors.

  4. 4.

    Lambert was the most explicit in his analysis; see Lambert (1764) and Lewis (1918, 23).

  5. 5.

    Lewis (1918, 79 n. 121). Lewis cited the title incorrectly as “Some illustrations of the Science of Logic;” the series consists of the essays “The Fixation of Belief” and “How to Make our Ideas Clear,” “The Doctrine of Chances,” “The Probability of Induction,” “The Order of Nature” and “Deduction, Induction and Hypotheses.”

  6. 6.

    I am grateful to Volker Peckhaus for a helpful discussion of the significance of Schröder’s contributions.

  7. 7.

    See Frege (1879), Schröder (1880), van Heijenoort (1967a) and (1967b, 1–2), and Peckhaus (2004, 599).

  8. 8.

    In Principia Mathematica, Whitehead and Russell pointed to both dimensions of Schröder’s contribution: his notation as one, alongside those of Frege and Peano , which they will adopt and modify as convenient, and to his calculus, for instance, his explanation of the proposition that ‘p and not-q imply r’ is equivalent to ‘p implies q or r’ (Whitehead and Russell 1910, vol 1, viii and 123). In the wake of reviews by Venn , Husserl , Peano and Peirce’s students, for instance, Whitehead and Russell’s mention might have drawn an even wider international attention to the work of the isolated teacher at a provincial German technical college.

  9. 9.

    See Couturat (1914), Lewis (1918), Dipert (1991), Grattan-Guinness (2000), Hailperin (2004) and Peckhaus (2004).

  10. 10.

    See Schröder (1877, 1890a, 1890b/1892, 1891, 1892, 1895 and 1898a, b).

  11. 11.

    But see relevant remarks in Grattan-Guinness (2000) and Peckhaus (2004).

  12. 12.

    Yet again, this text was surprisingly appended as a supplement on recent algebra to the Baden-Baden gymnasium program.

  13. 13.

    Peirce (1867, 251) had noted in passing such parallelism between the laws of union, or disjunction, and intersection, or conjunction, and was stressed by his student Christine Ladd (1880).

  14. 14.

    Schröder (1873, 146) noted that the key notion was contained in the statement about the general character of the product of indeterminate quantities, in the Lehrbuch.

  15. 15.

    Peckhaus (2004, 585–588) calls Schröder’s geometry-inspired approach quasi-axiomatics; unlike derived principles, especially inductive ones such as duality, axioms are formal to the extent that they are derived from self-evident intuitions.

  16. 16.

    Schröder presented the proof first at the 1883 meeting of the British Association of the Advancement of Science along with a summary of the laws for algebraic operations he had established in 1877 (see Schröder 1884a and 1884b).

  17. 17.

    See Dipert (1978, 287–318), Grattan-Guinness (2000, 170–72), and Peckhaus (2004, 591–96).

  18. 18.

    By this time, discussions of logic included examples of human relationships and even a calculus such as Alexander Macfarlane’s, extending Leibniz’s combinatorial discussions; see Peckhaus (2004, 596, n.54).

  19. 19.

    See van Heijenoort’s introduction to the translation of the Begriffsschrift in van Heijenoort (1967b, 1 n. b).

  20. 20.

    See Schröder (1890b), which appeared translated in The Open Court in 1892 as “Signs and Symbols,” see Schröder (1892).

  21. 21.

    Here he was following both Lambert and the philologist Max Müller .

  22. 22.

    The essay deserves close comparison with the symbolic, or semiotic, approach in both Mach’s Contributions to the Analysis of Sensations, of 1886, and Peirce’s subsequent writings on semiotics.

  23. 23.

    Also in this respect Schröder’s writings should be compared to Mach’s and his appeal to a principle of economy of thought, rejected by philosophers such as Husserl and Jerusalem as an expression of psychologism, especially in matters of logic, the paradigmatic case of a priori; see Kusch (1995).

  24. 24.

    I am grateful to Margaret Schabas for drawing my attention to the role of logic in connection to mathematics and its application in economics.

  25. 25.

    The project was much in line with the standard set by the optimistic industrial culture of machines and engines prompted by the Industrial Revolution and placed at the heart of the economy. Jevons mentioned attempts by Blaise Pascal , Arthur Colmar , Charles Babbage and Alfred Smee (Jevons 1874, 123–124).

  26. 26.

    See, for instance, Schröder (1890a), Lesson 40, sect 26 and especially Appendix 6.

  27. 27.

    The tension between the roles of mathematics and logic is discussed in Schabas (1990, Ch. 4.), and in Mosselmans (2007, Ch. 4.). The value of Boole’s system, according to Jevons , lied only in an analogy between mathematics and logic, so that logic is the algebra of two terms, 0 and 1 (Jevons 1874, vol. 2, 293).

  28. 28.

    Before his Leibnizian commitment to pasigraphy, Schröder had embraced this symbolic and typographic standpoint in the Lehrbuch der Arithmetik und Algebra (1873).

  29. 29.

    Partly in association with Mill ; see Schröder (1890a, 177).

  30. 30.

    For a general overview see Johnston (1972), Schorske (1979), Janik and Toulmin (1973), Blackmore et al. (2001) and Stadler (2001).

  31. 31.

    For a record of titles for lecture delivered at the Philosophical Society see Reininger (1938) and Blackmore et al. (2001, 283–98).

  32. 32.

    I am indebted to Professor Karl Sigmund for providing the information; see Öffentliche Vorlesungen an der k.k. Universität zu Wien, Archiv der Universität Wien.

  33. 33.

    Migerka was the organizer of a special exhibition on women at work at the Vienna World Fair of 1873 and married Otto Neurath’s future maternal aunt.

  34. 34.

    The emphasis on mathematics is Marie Neurath’s in her note to Neurath’s memories (Neurath and Cohen 1973, 7). In a letter to his son Paul , Neurath placed the emphasis on the natural sciences to tell of his additional interest in the social sciences; see Uebel (2004, 16).

  35. 35.

    See W. Neurath’s autobiographical sketch, in Neurath and Cohen (1973, 2). W. Neurath died in March 1901.

  36. 36.

    I am indebted for this information to Karl Sigmund; see Otto Neurath’s Nationale for winter 1902–3 and summer 1903, Archiv der Universität Wien.

  37. 37.

    Neurath (1903); see especially the translation of the first and last paragraphs in Uebel (2004, 16).

  38. 38.

    See Wilhelm Neurath’s textbook, W. Neurath (1896).

  39. 39.

    Uebel (forthcoming) elaborates on this apparent inconsistency in Neurath’s position and its conflict with the established accounts of logical positivism in Uebel.

  40. 40.

    Petzoldt (1895) and (1900); on its influence on Einstein’s development of the general theory of relativity see Howard (1996).

  41. 41.

    Uebel develops this point in Uebel (2004, 12–15).

  42. 42.

    For the supporting documents, see “Ansuchen um Zulassung zu den Rigorosen”, Rigorosenakt Olga Hahn, Archiv der Universität Wien, Sig. PH RA 3111 and “Beurteilung der Dissertation Olga Hahn (von Adolf Stöhr)”, Rigorosenakt Olga Hahn, Archiv der Universität Wien, Sig. PH RA 3111. I am grateful to Karl Sigmund, Johannes Friedl and Christoph Limbeck for assistance.

  43. 43.

    In his brief and dismissively critical overview, Köhler (1991, 109–112) somewhat misleadingly refers to sets and set-theoretic operations, whereas Schröder’s extensional approach, following Boole , is based on part-whole relations rather than membership.

  44. 44.

    Neurath (1909a/1981, 1–3) and Appendix in this volume, 489–492.

  45. 45.

    Here Köhler (1991, 110) speculates that the absence of the sign motivated Neurath to declare that one can “equate symbolically” ‘ab’ and ‘ba.’

  46. 46.

    Köhler (1991, 111) introduces the symbol ‘&’ for conjunction, which Schröder and Neurath, after him, didn’t use, and he himself has noted the notational role in Neurath’s argument.

  47. 47.

    This typographic standpoint is of course a matter of both types and concrete visible token marks.

  48. 48.

    See Neurath (1910a/1981, 19–21), and this volume 509–511. Köhler (1991, 111) notes, despite the asymmetry in the definitions of ab and ba, that Neurath seems to think that the equality ‘ab = ba’ is the same as ‘ab = a.b’.

  49. 49.

    Neurath (1909a/1981, 3) and this volume, 492.

  50. 50.

    According to Köhler (1991, 111), it undermines the expressive power, namely, for representing the properties of operations. As a criticism, he also points to alternative treatments of the question of equality he considers more enlightening and influential such as Frege’s (1892) “On Sense and Reference.” Later, Carnap (1947), committed to his teacher Frege and to logicism, would offer also an intensional account of asymmetric identities in Meaning and Necessity. A different source is Waissmann’s (1936/1977) Wittgensteinian defense of redundancy in language and the treatment of identity.

  51. 51.

    On the relation between syntacticism and physicalism, see Derek Anderson’s chapter in the present volume.

  52. 52.

    The first volume of the Vorlesungen includes a full section on the topic.

  53. 53.

    Neurath (1909b, 18), and this volume, 506.

  54. 54.

    Neurath (1909b, 18), and this volume, 506.

  55. 55.

    Nevertheless, with Axiom VIx’, below, for instance, Schröder could also define the products asymmetrically in terms of the more fundamental relation of subordination.

  56. 56.

    Neurath (1909b, 17), and this volume, 505. Schröder’s (1909, 30) more general introduction used the example of putatively different null domains.

  57. 57.

    Axiom VIx’ is (x € a)(x € b) € (x € ab) (Schröder 1909, 23).

  58. 58.

    Neurath (1910a/1981, 19–21), and in this volume, 509–11.

  59. 59.

    Neurath (1910a/1981, 19), and this volume, 509.

  60. 60.

    Neurath (1910a/1981, 19), and this volume, 509. See Petzoldt (1895) and Wundt (1894) and (1893/1895).

  61. 61.

    Neurath (1910a/1981, 20), and this volume, 510.

  62. 62.

    Note that as theoretical approach, Neurath had already adopted empirical classifications in his work in the history of economics, namely, with the classification of systems of organization of production and distribution of goods.

  63. 63.

    Neurath (1910a/1981, 21), and this volume, 511.

  64. 64.

    Walter Dubislav (1931) devoted an entire monograph to definitions in 1931, in a book, Die Definition, published in the Erkenntnis series edited by Carnap and Reichenbach , a collaboration between the Berlin and Vienna groups. He distinguished, following Leibniz , the following kinds of definitions: nominal, real, causal and essential (1931, 24). He also listed Neurath’s article in the bibliography, but, as in other authors’ bibliographical mentions, didn’t discuss it (1931, 155).

  65. 65.

    O. Hahn (1909), and this volume, 507–9.

  66. 66.

    O. Hahn (1909, 345), and this volume, 507.

  67. 67.

    See Theorem 31, (a |)| = a, Schröder (1890a, b, 305).

  68. 68.

    O. Hahn (1909, 347), and this volume, 509. However, Schröder’s introduction of the operations in terms of subsumption is in fact based on asymmetric expressions.

  69. 69.

    O. Hahn and Neurath (1909), in this volume, 493–505, and in Haller and Rutte (1981, 5–16).

  70. 70.

    Köhler (1991, 112) objects that Hahn and Neurath fare no better by appealing to a list of theorems too, but that is just as well, since the principle is not an axiom or a theorem, but a metatheoretic statement operating as a proof procedure. For Köhler, again, implicit in their argument is yet another redeeming metatheoretic intuition.

  71. 71.

    Neurath and Schapire-Neurath (1910, vol. 2, p. 168). See also Neurath (1910a/2004, 162).

  72. 72.

    Neurath (1910b/2004, 290 n. 25). He devoted an entire chapter in his textbook, Neurath (1910b, 40–47).

  73. 73.

    In a note Neurath (1910b/2004, 290, n. 27) points out that “the above account required no examination of the motives” and, more radically, that if we ignore moral pleasure or pain, the transition from one distribution of wealth to a more satisfying one represents an improvement since it doesn’t constrain the particular kind of transfer of goods involved and takes into account, or at least makes room for, different possibilities.

  74. 74.

    He introduced the notion of complementary goods as a fundamental concept also in his textbook on economics of the same year (Neurath 1910b, 1).

  75. 75.

    For Schröder (1909, 26), the exhaustive conjunction of two domains of objects (including propositions) may be represented as a + b = 1 (totality, universe), so that negation allows the definition of a complementary (or reciprocal) domain b = not-a.

  76. 76.

    Notice that Schröder’s Vorlesungen bore the subtitle Exakte Logik.

  77. 77.

    On the influence of Neurath and Carnap on Stevens see Hardcastle’s (2003) discussion of Carnap and Stevens at the head of the so-called Harvard Science of Science Discussion Group, in 1940–41.

  78. 78.

    On reasoning with inequalities, see Jevons (1874, vol. 1, 186–188).

  79. 79.

    Neurath’s notation for the indeterminate relation when the inequality sign doesn’t apply was the question mark “?”.

  80. 80.

    See, for instance, Jevons (1874, vol. 1, 122).

  81. 81.

    Note that in the case of symbolic equalities, Neurath also pointed to the typographic artifact of one-dimensionality in our symbolic representations.

  82. 82.

    On the role of analogies and their relation to Mach’s work also on optics, see Stöltzner (1996/1983).

  83. 83.

    See Duhem (1906). Duhem was also concerned with classification as the proper aim of science, without deeper explanations.

  84. 84.

    Neurath’s linguistic treatment of the conceptual blur is the doctrine of Ballungen, see Cat (1995) and Cartwright et al. (1996).

  85. 85.

    On the optical illustration, see Gábor Zemplén’s chapter in the present volume.

  86. 86.

    In the plural scene of nineteenth-century British new empiricism, the view can be traced back especially to Herschel and Mill , also Whewell .

  87. 87.

    He was following Ampère, and indirectly a long literature going back to the seventeenth century. Duhem would adopt a more regulative and metaphysical notion of natural classification (Duhem 1906).

  88. 88.

    This is similar to the structural description Carnap would introduce in his Aufbau later in the same decade.

  89. 89.

    See Burke et al. (2013), also Angélique Groß’s chapter in the present volume.

  90. 90.

    Mosselmans (2007, 50) quotes Neurath merely to identify in Jevons a “logical positivist attitude.”

  91. 91.

    See, for instance, his review of Wundt in Neurath (1910b/2004) and the additional discussion of aggregate pleasure in Neurath (1912/1973).

  92. 92.

    See, among others, Uebel (1992) and (2007) and Cartwright et al. (1996).

  93. 93.

    For an overview of the intellectual and associationist Viennese landscape, see Stadler (2001).

  94. 94.

    Rather than subscribing to overdrawn contrasts between pre-1930, pre-1935 and post-1935 Neurath, drawn for instance by Mormann (1999), and equally between Neurath and Carnap, usefully drawn by Uebel and, especially after 1935, by Mormann, I want to emphasize any differences only in relation to a shared background and interest in symbolic language and logic.

  95. 95.

    For a detailed discussion of Carnap’s involvement in logic in the 1920s, see Reck (2004) and Reck and Awodey (2004).

  96. 96.

    Itelson , Couturat and Lalande made the proposal at the 1904 at the International Congress of Philosophy in Geneva.

  97. 97.

    About the debate between them, see Couturat (1905/1914) and Peckhaus (2004).

  98. 98.

    In all the techniques Carnap was relying, also rhetorically, on modern scientific developments, both empirical, e.g., Gestalt psychology, and formal and foundational, e.g., set theory, axiomatics and logicism.

  99. 99.

    Carus (2007, 103) stresses the Fregean connection.

  100. 100.

    Neurath (1937a/1983, 174) later wrote, in relation to the new discipline of the logic of science, a replacement of philosophy, and linguistic (anti-metaphysical and social) unification, that “progress of the unification of scientific language requires a logical analysis of the sciences” and that “Mach’s far-reaching conclusions did not depend upon new experimental data, but simply upon a rigorous logical analysis of the traditional formulations.”

  101. 101.

    My discussion of this aspect supplements the account in Stadler (2001) and the detailed reconstruction of the manifesto’s writing in Uebel (2008).

  102. 102.

    Carnap had done the same in the Abriss, of the same year.

  103. 103.

    An important question that deserves attention is the distinction between sign and symbol. These authors often lack a systematic distinction, or a consistent statement of their equivalence: they sometimes use the terms interchangeably, or else their use is inconsistent in taking ‘sign’ as the more general term, almost in Peirce’s sense, or else as a concrete token of a symbol, or as an index, in contrast with the more conventional character of a symbol, restricted further by the rules of a specific system of symbols, or language, including mathematics.

  104. 104.

    Carnap would soon add to the list of sources the material mode of expression.

  105. 105.

    Neurath drew attention to the scientific value of general names (variables).

  106. 106.

    Note also that for Peirce (1884), algebraic logic was a contribution to the ‘philosophy of notation.’

  107. 107.

    In the followings sections, the Neurath-Carnap letters are quoted from the Carnap Archive at Pittsburgh (Rudolf Carnap Papers, 1905–1970, ASP.1974.01, Special Collections Department, University of Pittsburgh). All rights reserved.

  108. 108.

    For a discussion of the first stage of the so-called protocol-sentence debate see, Uebel (1992) and (2007).

  109. 109.

    The literature on Carnap’s syntactic turn is vast, but to stay within the context of the Circle, the reader may do well to start with Uebel (1992) and (2007).

  110. 110.

    As he had done in the Aufbau, in the bibliography Carnap listed Gätschenberger’s Symbola (1920).

  111. 111.

    In the tradition of Leibniz , Lambert and turn-of-the-century symbolic logic, Gätschenberger (1920, 126, 133, and 454) declared that sematology contributed to algebraic logic, especially to the calculus of relations sought by Peirce and Schröder, and from that standpoint, it was the mother of mathematics. On Gätschenberger’s relevance to Carnap’s early works, see Haller (1959); on his place in German semiotics, see Eschbach (1987).

  112. 112.

    Neurath to Carnap, 3 November, 1932; RC 029-12-13. In the same letter Neurath also analyzed cases of empirical falsehood as forms of logical contradiction or nonsense.

  113. 113.

    Neurath to Carnap, 5 April 1934; RC 029-10-77.

  114. 114.

    Mormann adopts Hintikka’s terminology, paraphrasing van Heijenort’s focus on logic, language as universal medium and language as calculus.

  115. 115.

    In a number of places, Neurath adopted the characaterization of the encyclopedia as a model, in a Duhemian sense, part of the opposition to the characaterization of unity as a system (Neurath 1936/1983, 145); but he used the term “model” also for a broader category to contrast the encylopedia with the system as models of knowledge (ibid., 156). In relation to Duhem, it must be noted that he adopted the notion of logical unity in the sense of oneness, as a fundamental feature of natural classification.

  116. 116.

    I’m grateful to Adam Tamas Tuboly for drawing my attention to the letter.

  117. 117.

    These are names listed by Neurath on the cover letter from 10 May 1937 with the proposal sent out for circulation. Nachlass K13. The Congress Report included a mention of the committee that listed Alfred Tarski and Olaf Helmer (Erkenntnis 7, 63).

  118. 118.

    See also Adam Tamas Tuboly’s chapter in the present volume.

  119. 119.

    Neurath to Carnap, 25 September, 1943 (RC 102-55-03). See letter 22 in the present volume.

  120. 120.

    Neurath to Carnap, 16 June, 1945 (RC 102-55-11). See letter 31 in the present volume.

  121. 121.

    Neurath to Carnap, 15 January, 1943 (RC 102-55-02). See letter 17 in the present volume.

  122. 122.

    See letter to Carnap, 22 September 1945, unsent (RC 115-07-66). See letter 34 in the present volume.

  123. 123.

    See letter to Carnap, 22 September 1945, unsent (RC 115-07-66). See letter 34 in the present volume.

  124. 124.

    Neurath to Carnap 4 April 1941 (RC 102-55-20), see letter 5 in the present volume.

  125. 125.

    Carnap to Neurath, 5 June 1941 (RC 102-55-19), see letter 6 in the present volume.

  126. 126.

    Neurath to Carnap, 17 July 1942 (RC 102-56-04), see letter 11 in the present volume.

  127. 127.

    Neurath to Carnap, 20 July 1942 (RC 115-07-57). See letter 12 in the present volume.

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    Jordi Cat

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Cat, J. (2019). Neurath and the Legacy of Algebraic Logic. In: Cat, J., Tuboly, A. (eds) Neurath Reconsidered. Boston Studies in the Philosophy and History of Science, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-030-02128-3_11

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