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Dimensionally Invariant Numerical Laws Correspond to Meaningful Qualitative Relations

Published online by Cambridge University Press:  01 April 2022

R. Duncan Luce
R. Duncan Luce*
Affiliation:
Harvard University
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Abstract

In formal theories of measurement meaningfulness is usually formulated in terms of numerical statements that are invariant under admissible transformations of the numerical representation. This is equivalent to qualitative relations that are invariant under automorphisms of the measurement structure. This concept of meaningfulness, appropriately generalized, is studied in spaces constructed from a number of conjoint and extensive structures some of which are suitably interrelated by distribution laws. Such spaces model the dimensional structures of classical physics. It is shown that this qualitative concept corresponds exactly with the numerical concept of dimensionally invariant laws of physics.

Type
Research Article
Information
Philosophy of Science , Volume 45 , Issue 1 , March 1978 , pp. 1 - 16
Copyright
Copyright © Philosophy of Science Association 1978

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Footnotes

†

A number of people have made helpful comments on an earlier draft of this paper. By far the most substantial contribution is that of William Craig who, on first listening to me discuss the problem in a seminar at Stanford University, strongly suggested that the best way to formulate the idea of meaningfulness is as invariance under automorphisms of the structure (Definition 3). He later gave me detailed criticisms, including pointing out an error, on a written draft. Others to whom I am also indebted are E. W. Adams, M. Bar-Hillel, R. L. Causey, F. Roberts, P. Suppes, and B. Wandell.

This work was supported in part by a National Science Foundation grant to the University of California at Irvine, Louis Narens principal investigator.

References

[1] Adams, E., Fagot, R., & Robinson, R. “A theory of appropriate statistics.” Psychometrika 30 (1965): 99–127.10.1007/BF02289443CrossRefGoogle ScholarPubMed
[2] Birkhoff, G. Hydrodynamics. A study in logic, fact and similitude. Princeton: Princeton University Press, 1960.Google Scholar
[3] Bridgman, P. Dimensional Analysis. New Haven: Yale University Press, 1931.Google Scholar
[4] Causey, R. “Derived measurement, dimensions, and dimensional analysis.” Philosophy of Science 36 (1969): 252–270.10.1086/288255CrossRefGoogle Scholar
[5] Krantz, D., Luce, R., Suppes, P., & Tversky, A. Foundations of Measurement, Vol. I. New York: Academic Press, 1971.Google Scholar
[6] Luce, R. “A”fundamental” axiomatization of multiplicative power relations among three variables.” Philosophy of Science 32 (1965): 301–309.10.1086/288054CrossRefGoogle Scholar
[7] Luce, R. “Similar systems and dimensionally invariant laws.” Philosophy of Science 38 1971: 157–169.10.1086/288351CrossRefGoogle Scholar
[8] McKinsey, J. & Suppes, P. “On the notion of invariance in classical mechanics.” The British Journal for the Philosophy of Science 5 (1955): 290–302.10.1093/bjps/V.20.290CrossRefGoogle Scholar
[9] Narens, L. & Luce, R. “The algebra of measurement.” Journal of Pure and Applied Algebra 8 (1976): 197–233.10.1016/0022-4049(76)90016-5CrossRefGoogle Scholar
[10] Pfanzagl, J. Theory of Measurement. New York: Wiley, 1971 (2nd ed.).10.1007/978-3-662-41488-0CrossRefGoogle Scholar
[11] Robinson, R. A set-theoretical approach to empirical meaningfulness of empirical statements. Institute for Mathematical Studies in the Social Sciences, Stanford University. Technical Report 55 (1963).Google Scholar
[12] Suppes, P. Introduction to logic. Princeton: van Nostrand, 1957.Google Scholar
[13] Suppes, P. “Measurement, empirical meaningfulness, and three-valued logic.” In Measurement: definitions and theories, Churchman, C. & Ratoosh, P. (eds.). New York: Wiley, 1959. pp. 129–143.Google Scholar
[14] Suppes, P. & Zinnes, J. “Basic measurement theory.” In Handbook of Mathematical Psychology, Vol. I, Luce, R., Bush, R., & Galanter, E. (eds.). New York: Wiley, 1963. pp. 1–76.Google Scholar
[15] Whitney, H. “The mathematics of physical quantities.” Part I: “Mathematical models for measurement.” Part II: “Quantity structures and dimensional analysis.” American Mathematical Monthly 75 (1968): 115–138, 227–256.Google Scholar