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Informal Meditation on Empiricism and Approximation in Fuzzy Logic and Set Theory: Descriptive Normativity, Formal Informality and Objective Subjectivity

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Towards the Future of Fuzzy Logic

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 325))

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Abstract

The paper defends the view that the application and construction of mathematics may prove to be empiricist, subjective, approximative, contextual and normative. These elements are both inseparable and central to the possibility and success of mathematical practice. The cases of fuzzy set theory and fuzzy logic illustrate and support this account. In turn, the framework the account offers also brings out the particular ways in which the noted elements distinctively characterize these related fuzzy projects. Their future will benefit from understanding them more critically and creatively.

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Notes

  1. 1.

    Here I follow [41], esp. Chap. 6.

  2. 2.

    See [16, 26, 45], etc.

  3. 3.

    Synthetic judgments as statements of fact, ampliative and describing and testable only matters of experience (not pure subjective phenomenological states alone); opposed to analytical ones, relations of ideas, explicative and decidable only by rules or conventions (for Frege mathematics was analytic and analyticity was ampliative and grounded on derivation from basic logical principles, not explicative or revealed by self-contradicting negations as on the Kantian view; for logical empiricists mathematics was both analytic and explicative).

  4. 4.

    Elgin makes the case: “factual judgments are not objective unless value judgments are; and value judgments are not relative unless factual judgments are,” in [17, p. 176].

  5. 5.

    Ernst Cassirer, Rudolf Carnap, Moritz Schlick and Hans Reichenbach were early 20th-century champions of accounts of knowledge and objectivity by means of formal conceptual determination by mathematical structures of relations and, especially Cassirer, invariants.

  6. 6.

    [29] An over-idealized alternative is negative talk of neutrality; see [15].

  7. 7.

    See Putnam’s pragmatist position, above.

  8. 8.

    On the limits of modeling and their approximative and inconsistent application see [7].

  9. 9.

    It is, in Roger Scruton’s words, “the organizing principle of first-person awareness”; [42].

  10. 10.

    See for instance [21].

  11. 11.

    See, for instance, [43, p. vi].

  12. 12.

    For a discussions of the episode, see [22].

  13. 13.

    For instance: Local field operator expansions of effective, lower-energy field theory, are considered approximations to real, high-energy exact theories, whether known (top-down approximation) or unknown (bottom-up approximation); then, a higher interpretive autonomy and realism is assigned to bottom-up approximations.

  14. 14.

    For a still adequate discussion of the human case, see also [16, pp. 261–263].

  15. 15.

    As with any data-based testing of formal models, the evidentiary context might also challenge the empirical support. Thus claims to the effect that the empirical linguistic evidence doesn’t support the fuzzy predication of truth by analogy with paradigmatic modified predicates and their opposites, [23, pp. 244–246].

  16. 16.

    Carnap and Quine are standard references drawing conclusions about the challenge to formulate the conditions of factual meaning and justification.

  17. 17.

    I have counted \(15\) references in [16] and \(19\) in [26]. See also [33].

  18. 18.

    See, for instance, [33, pp. 319–320].

  19. 19.

    A related discussion of Norbert Wiener’s early cybernetic model of the enemy airplane pilot as subject can be found in [18].

  20. 20.

    In [16, p. 137].

  21. 21.

    [26, p. 97].

  22. 22.

    See, for instance, [23, 38]. About Putnam, more below.

  23. 23.

    Cited in [16, p. 173].

  24. 24.

    This is Haack’s view in [23].

  25. 25.

    For an imprecise quantum set theory with references to Takeuti’s non-extensive quantum sets and Da Costa’s and Krause’s quasi-sets see, for instance, [12]. Quantum logics will generally have semantic set-theoretic realizations or models that violate also the law of non-contradiction and even of excluded middle.

  26. 26.

    See ibid., Fig. 11.2.

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  1. Indiana University, Bloomington, USA

    Jordi Cat

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  1. Institut für Geschichte der Medizin, Naturwissenschaft und Technik, Friedrich-Schiller-Universität Jena, Berggasse, Jena, Germany

    Rudolf Seising

  2. Edificio de Investigación, 3ª Planta. C Gonzalo Gutiérrez Quirós S/N, European Centre for Soft Computing, Mieres, Asturias, Spain

    Enric Trillas

  3. Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

    Janusz Kacprzyk

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Cat, J. (2015). Informal Meditation on Empiricism and Approximation in Fuzzy Logic and Set Theory: Descriptive Normativity, Formal Informality and Objective Subjectivity. In: Seising, R., Trillas, E., Kacprzyk, J. (eds) Towards the Future of Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-319-18750-1_13

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