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Synthese

, Volume 70, Issue 3, pp 373–400 | Cite as

Faithful representation, physical extensive measurement theory and Archimedean axioms

  • Brent Mundy
Article

Abstract

The formal methods of the representational theory of measurement (RTM) are applied to the extensive scales of physical science, with some modifications of interpretation and of formalism. The interpretative modification is in the direction of theoretical realism rather than the narrow empiricism which is characteristic of RTM. The formal issues concern the formal representational conditions which extensive scales should be assumed to satisfy; I argue in the physical case for conditions related to weak rather than strong extensive measurement, in the sense of Holman 1969 and Colonius 1978. The problem of justifying representational conditions is addressed in more detail than is customary in the RTM literature; this continues the study of the foundations of RTM begun in an earlier paper. The most important formal consequence of the present interpretation of physical extensive scales is that the basic existence and uniqueness properties of scales (representation theorem) may be derived without appeal to an Archimedean axiom; this parallels a conclusion drawn by Narens for representations of qualitative probability. It is concluded that there is no physical basis for postulation of an Archimedean axiom.

Keywords

Uniqueness Property Early Paper Formal Method Physical Science Physical Basis 
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.

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References

  1. Adams, E.: 1979, ‘Measurement Theory’, in P. Asquith and H. Kyburg, (eds.), Current Research in Philosophy of Science, Philosophy of Science Association, East Lansing, Michigan, 207–227.Google Scholar
  2. Behrend, F. A.: 1962, ‘A Characterization of Vector Spaces over Fields of Real Numbers’, Mathematische Zeitschrift 78, 298–304.Google Scholar
  3. Bell, J. L. and A. B. Slomson: 1969, Models and Ultraproducts, North-Holland Publ. Co., Amsterdam.Google Scholar
  4. Campbell, N.: 1920, Physics: the Elements, Harvard University Press, Cambridge. (Reprinted by Dover as Foundations of Science, 1957.)Google Scholar
  5. Colonius, H.: 1978, ‘On Weak Extensive Measurement’, Philosophy of Science 45, 303–308.Google Scholar
  6. Field, H.: 1980, Science Without Numbers, Princeton Univ. Press, Princeton.Google Scholar
  7. Fishburn, P. C.: 1985, Interval Orders and Interval Graphs, Wiley, New York.Google Scholar
  8. Fuchs, L.: 1963, Partially Ordered Algebraic Systems, Addison Wesley, Reading, Massachusetts.Google Scholar
  9. Helmholtz, H.: 1887, ‘Numbering and Measuring from an Epistemological Viewpoint’, English translation in Helmholtz, H., Epistemological Writings, Reidel, Dordrecht, 1977.Google Scholar
  10. Hölder, O.: 1901, ‘Die Axiome der Quantität und die Lehre vom Mass’, Berichte über die Verhandlungen der Königliche Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch-Physische Klasse 53, 1–64.Google Scholar
  11. Holman, E. W.: 1969, ‘Strong and Weak Extensive Measurement’, Journal of Mathematical Psychology 6, 286–293.Google Scholar
  12. Krantz, D.: 1967, ‘Extensive Measurement in Semiorders’, Philosophy of Science 34, 348–62.Google Scholar
  13. Krantz, D.: 1973, ‘Fundamental Measurement of Force and Newton's First and Second Laws of Motion’, Philosophy and Science 40, 481–495.Google Scholar
  14. Krantz, D., R. D. Luce, P. Suppes and A. Tversky: 1971, Foundations of Measurement, Vol. 1, Academic Press, New York.Google Scholar
  15. Luce, R. D.: 1956, ‘Semiorders and a Theory of Utility Discrimination’, Econometrica 24, 178–191.Google Scholar
  16. Luce, R. D.: 1971, ‘Similar Systems and Dimensionally Invariant Laws’, Philosophy of Science 38, 157–169.Google Scholar
  17. Luce, R. D.: 1973, ‘Three Axiom Systems for Additive Semiordered Structures’, SIAM Journal of Applied Mathematics 25, 41–53.Google Scholar
  18. Luce, R. D.: 1978, ‘Dimensionally Invariant Numerical Laws Correspond to Meaningful Qualitative Relations’, Philosophy of Science 45, 1–16.Google Scholar
  19. Mundy, B.: (a), 1986, ‘On the General Theory of Meaningful Representation’, Synthese 67, 391–437.Google Scholar
  20. Mundy, B.: (b) ‘Extensive Measurement and Ratio Functions’, forthcoming in Synthese.Google Scholar
  21. Mundy, B.: (c) ‘The Metaphysics of Quantity’, forthcoming in Phil. Studies.Google Scholar
  22. Narens, L.: 1974a, ‘Minimal Conditions for Additive Conjoint Measurement and Qualitative Probability’, Journal of Mathematical Psychology 11, 404–430.Google Scholar
  23. Narens, L.: 1974b, ‘Measurement without Archimedean Axioms’, Philosophy of Science 41, 374–93.Google Scholar
  24. Narens, L.: 1980, ‘On Qualitative Axiomatizations for Probability Theory’, Journal of Philosophical Logic 9, 143–151.Google Scholar
  25. Narens, L.: 1985, Abstract Measurement Theory, MIT Press, Cambridge.Google Scholar
  26. Roberts, F.: 1979, Measurement Theory, Encyclopedia of Mathematics and its Applications, vol. 7, Addison Wesley, Reading, Massachusetts.Google Scholar
  27. Roberts, F. and R. D. Luce: 1968, ‘Axiomatic Thermodynamics and Extensive Measurement’, Synthese 18, 311–326.Google Scholar
  28. Scott, D. and P. Suppes: 1958, ‘Foundational Aspects of Theories of Measurement’, Journal of Symbolic Logic 23, 113–128.Google Scholar
  29. Shoenfield, J. P.: 1967, Mathematical Logic, Addison Wesley, Reading. Mass.Google Scholar
  30. Stevens, S. S.: 1946, ‘On the Theory of Scales of Measurement’, Science 103, 677–680.Google Scholar
  31. Van Fraassen, B.: 1980, The Scientific Image, Oxford.Google Scholar

Copyright information

© D. Reidel Publishing Company 1987

Authors and Affiliations

  • Brent Mundy
    • 1
  1. 1.Department of PhilosophyUniversity of OklahomaNormanUSA

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