Abstract
The key aims of modern scientific work have generally been to find relationships between observed phenomena, construct mathematical formulas that describe these relationships, take measurements of the observables, and define axioms using terms that are as exact as possible. In many circumstances, the exactness of observed phenomena is limited – or can only be measured with less than perfect accuracy. However, another problem arises if the concepts of a scientific theory do not fit with the picture that scientists use to understand their observations and experimental results. How to deal with such situations is a question that has intrigued many scientists and philosophers. In the 20th century two scientific theories appeared that change scientist’s views from classical to non-classical and on what is measurable: quantum mechanics and fuzzy set theory. This paper focuses on these developments.
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References
Zadeh, L.A.: From Circuit Theory to System Theory. Proc. IRE 50(5), 856–865 (1962)
Zadeh, L.A.: Fuzzy Sets and Systems. In: Fox, J. (ed.) System Theory, Microwave Res. Inst. Symp. Series XV, pp. 29–37. Polytechnic Press, Brooklyn (1965)
Born, M.: Zur Quantenmechanik der Stoßvorgänge. Z. F. Physik. 37, 86–867 (1926)
Born, M.: Das Adiabatenprinzip in der Quantenmechanik. Z. F. Physik. 40, 167–191 (1926)
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press, Princeton (1955)
Birkhoff, G., von Neumann, J.: The Logic of Quantum Mechanics. Annals of Mathematics, Series 2 37, 823 (1936)
Mackey, G.W.: Mathematical Foundations of Quantum Mechanics. W. A. Benjamin, New York (1963)
Kolmogorov, A.: Foundations of the Theory of Probability, 2nd edn. Chelsea, New York (1956); Grundbegriffe der Wahrscheinlichkeitsrechnung. Julius Springer, Berlin (1933)
Suppes, P.: Probability Concepts in Quantum Mechanics. Phil. of Sci. 28, 378–389 (1961)
Suppes, P.: The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanics. Philosophy of Science 33, 14–21 (1966)
Gudder, S.P.: Quantum Probability. Academic Press, San Diego (1988)
Pitowski, I.: Quantum Probability - Quantum Logic. LNP, p. 321. Springer, Berlin (1989)
Seising, R. (guest ed.): Special Issue: Fuzzy and Quantum Systems. Int. J. of General Systems 40(1), 1–9 (2011)
Menger, K.: Statistical Metrics. Proc. Natl. Acad. Sci., U.S.A. 28, 535–537 (1942)
Menger, K.: Probabilistic geometry. Proc. Natl. Acad. Sci. 37, 226–229 (1951)
Menger, K.: Probabilistic Theories of Relations. Proc. Natl. Acad. Sci. 37, 178–180 (1951)
Menger, K.: Ensembles flous et fonctions aléatoires. Comptes Rendus Académie des Sciences 37, 226–229 (1951)
Menger, K.: Geometry and Positivism. A Probabilistic Microgeometry. In: Menger, K. (ed.) Selected Papers in Logic and Foundations, Didactics, Economics, pp. 225–234. D. Reidel Publ. Comp, Dordrecht (1979)
Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik. 43, 172–198 (1927)
Seising, R.: The Fuzzification of Systems. In: The Genesis of Fuzzy Set Theory and Its Initial Applications - Its Development to the 1970s. Springer, Berlin (2007)
Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)
Watanabe, S.: Modified concepts of logic, probability, and information based on generalized continuous characteristic function. Information and Control 15, 1–21 (1969)
Watanabe, S.: Knowing and guessing. John Wiley & Sons, New York (1969)
Termini, S.: Interview with R. Seising. Newsletter Philosophy and Soft Computing 2(2), 7–14 (2010), http://www.eusflat.org/research_wg_phil/Newsletter_Phil_3.pdf
de Luca, A., Termini, S.: A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory. Information and Control 20(4), 301–312 (1972)
Trillas, E.: Interview with R. Seising. Newsletter Philosophy and Soft Computing 2(1), 7–11 (2009), http://www.eusflat.org/research_wg_phil/Newsletter_Phil_1.pdf
Sugeno, M.: Interview with R. Seising. Newsletter Philosophy and Soft Computing 6(1) (to appear, 2012)
Sugeno, M.: Fuzzy Measure and Fuzzy Integral. Transactions of the Society of Instrument and Control Engineers 8(2), 218–226 (1972) (Japanese)
Sugeno, M.: Theory of Fuzzy Integrals and its applications. Ph. D. Diss., Tokyo (1974)
Klir, G., Wang, Z.: Fuzzy measure Theory. Plenum Press, New York (1992)
Choquet, G.: Theory of capacities. Annales de l’institut Fourier, tome 5, 131–295 (1954)
Garmendia, L.: The Evolution of the Concept of Fuzzy Measure. Studies in Computational Intelligence (SCI) 5, 185–200 (2005)
Zi-Xiao, W.: Fuzzy measures and Measures of Fuzziness. J. Math. Analysis and Appl. 104(2), 589–601 (1984)
Mesiar, R.: Fuzzy Measures and Integration. Fuzzy Sets and Systems 156(3), 365–370 (2005)
Dombi, J., Porkoláb, L.: Measures of Fuzziness. Ann. Univ. Sci. Budapest, Sect. Comp. 12, 69–78 (1991)
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Seising, R. (2012). Measures of Observables and Measures of Fuzziness. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31715-6_7
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DOI: https://doi.org/10.1007/978-3-642-31715-6_7
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