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

  • Jordi CatEmail author
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 325)

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.

Keywords

Membership Function Fuzzy Logic Fuzzy Model Classical Logic Scientific Practice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bacciagaluppi, G.: Is logic empirical? In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures. Quantum Logic, pp. 49–78. Elsevier, Amsterdam (2009)CrossRefGoogle Scholar
  2. 2.
    Batterman, R.W.: The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence. Oxford University Press, Oxford (2002)Google Scholar
  3. 3.
    Bellman, R.E., Zadeh, L.A.: Local and fuzzy logics. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic. Reidel, Dordrecht (1977)Google Scholar
  4. 4.
    Bogen, J., Woodward, J.: Saving the phenomena. Philos. Rev. 97(3), 303–352 (1988)CrossRefGoogle Scholar
  5. 5.
    Bowker, G.C., Starr, S.L.: Sorting Things Out: Classification and Its Consequences. MIT Press, Cambridge (1997)Google Scholar
  6. 6.
    Cartwright, N.: Evidence, external validity, and explanatory relevance. In: Morga, G.J. (ed.) Philosophy of Science Matters, pp. 15–28. Oxford University Press, New York (2011)CrossRefGoogle Scholar
  7. 7.
    Cat, J.: Modeling cracks and cracking models: structures, mechanisms, boundary conditions, constraints, inconsistencies and the proper domains of natural laws. Synthese 146, 447–487 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cat, J.: Fuzzy empiricism and fuzzy-set causality: what is all the fuzz about? Philos. Sci. 73(1), 26–41 (2006)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cat, J.: The unity of science. In: Stanford Encyclopedia of Philosophy Online (2013)Google Scholar
  10. 10.
    Cattaneo, G., Dalla Chiara, M.L., Giuntini, R., Paoli, F.: Quantum logic and nonclassical logics. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures. Quantum Logic, pp. 127–226. Elsevier, Amsterdam (2009)CrossRefGoogle Scholar
  11. 11.
    Cooke, R.M.: Experts in Uncertainty: Opinion and Subjective Probability in Science. Oxford University Press, New York (1991)Google Scholar
  12. 12.
    Dalla Chiara, M.L., Giuntini, R.: Quantum logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 6, pp. 129–228. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
  13. 13.
    Daston, L., Galison, P.: Objectivity. Zone Books, Cambridge (2007)Google Scholar
  14. 14.
    Dickson, W.M.: Quantum Chance and Non-Locality. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  15. 15.
    Douglas, H.: Inductive risk and values in science. Philos. Sci. 67, 559–579 (2000)CrossRefGoogle Scholar
  16. 16.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  17. 17.
    Elgin, C.Z.: Between the Absolute and the Arbitrary. Cornell University Press, Ithaca (1997)Google Scholar
  18. 18.
    Galison, P.: The ontology of the enemy: Norbert Wiener and the cybernetic vision. Crit. Inq. 21(1), 228–266 (1994)CrossRefGoogle Scholar
  19. 19.
    Gibbins, P.: Particles and Paradoxes: The Limits of Quantum Logic. Cambridge University Press, Cambridge (1984)Google Scholar
  20. 20.
    Giere, R.: N: Science Without Laws. University of Chicago Press, Chicago (1999)Google Scholar
  21. 21.
    Gigerenzer, G., Todd, P., ABC Research Group: Simple Heuristics that Make Us Smart. Oxford University Press, New York (2000)Google Scholar
  22. 22.
    Gooday, G.J.N.: The Morals of Measurement: Accuracy, Irony, and Trust in Late Victorian Electrical Practice. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  23. 23.
    Haack, S.: Deviant Logic, Fuzzy Logic. University of Chicago Press, Chicago (1996)zbMATHGoogle Scholar
  24. 24.
    Howison, S.: Practical Applied Mathematics: Modeling, Analysis, Approximation. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  25. 25.
    Kadanoff, L.P.: More is the same: phase transitions and mean field theories, J. Stat. Phys. 137, 777–797 (2009). University of Chicago ms: http://pitpas1.phas.ubc.ca/varchive/7pines09/Kadanoff.pdf
  26. 26.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River (1995)zbMATHGoogle Scholar
  27. 27.
    Krieger, M.H.: Constitutions of Matter. Mathematically Modeling the Most Everyday of Physical Phenomena. University of Chicago Press, Chicago (1996)zbMATHGoogle Scholar
  28. 28.
    Laymon, R.: Computer simulations, idealizations and approximations. In: Fine, A., Forbes, M., Wessels, L. (eds.) PSA, vol. 2, pp. 519–534. Philosophy of Science Association, East Lansing (1990)Google Scholar
  29. 29.
    Longino, H.: Science as Social Knowledge. Princeton University Press, Princeton (1998)Google Scholar
  30. 30.
    Longino, H.: What’s so great about an objective concept of evidence? In: Morgan, G.J. (ed.) Philosophy of Science Matters, pp. 124–134. Oxford University Press, New York (2011)CrossRefGoogle Scholar
  31. 31.
    Mayo, D.G.: Error and the Growth of Knowledge. University of Chicago Press, Chicago (1996)CrossRefGoogle Scholar
  32. 32.
    Miller, D.: Out of Error: Further Essays on Critical Rationalism. Ashgate, Aldershot (2006)Google Scholar
  33. 33.
    Nguyen, H.T., Walker, E.A.: A First Cause in Fuzzy Logic, 3rd edn. Chapman & Hall, Boca Raton (2006)Google Scholar
  34. 34.
    Pavičić, M., Megill, N.D.: Is quantum logic logic? In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Logic, pp. 23–48. Elsevier, Amsterdam (2009)Google Scholar
  35. 35.
    Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, vol. 2. Gauthier-Villars, Paris (1893)zbMATHGoogle Scholar
  36. 36.
    Putnam, H.: Is logic empirical? 1968, reprinted as, “The logic of quantum mechanics”. In: Putnam, H. (eds.) Mathematics, Matter and Method: Philosophical Papers, vol 1, pp. 174–197. Cambridge University Press, Cambridge (1979)Google Scholar
  37. 37.
    Putnam, H.: The Collapse of the Fact/Value Dichotomy and Other Essays. Harvard University Press, Cambridge (2002)Google Scholar
  38. 38.
    Putnam, H.: The curious story of quantum logic. In: Putnam, H. (ed.) Philosophy in the Age of Science: Physics, Mathematics, and Skepticism, pp. 162–177. Harvard University Press, Cambridge (2012)Google Scholar
  39. 39.
    Ragin, C.: Fuzzy-Set Social Science. University of Chicago Press, Chicago (2000)Google Scholar
  40. 40.
    Ramsey, J.L.: Towards an expanded epistemology for approximation. In: Hu, D., Forbes, M., Okruhlik, K. (eds.) PSA, vol. 1, pp. 154–164. Philosophy of Science Association, East Lansing (1992)Google Scholar
  41. 41.
    Smith, N.J.J.: Vagueness and Degrees of Truth. Oxford University Press, Oxford (2008)CrossRefGoogle Scholar
  42. 42.
    Scruton, R.: Scientism in the arts and the humanities. New Atlantis 40, 33–47 (2013)Google Scholar
  43. 43.
    Steffens, K.G.: The History of Approximation Theory from Euler to Bernstein. Birkhäuser, Boston (2006)Google Scholar
  44. 44.
    Wise, M.N. (ed.): The Values of Precision. Princeton University Press, Princeton (1995)Google Scholar
  45. 45.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man Cybern. 3, 28–44 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Zadeh, L.A.: A theory of approximate reasoning (AR), 1977, reprinted. In: Hayes, J.E., Michie, D., Mikulich, L.I. (eds.) Machine Intelligence, vol 9, pp. 149–194. Elsevier, New York (1979)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Indiana UniversityBloomingtonUSA

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