Abstract
The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received adequate justification: the assumption that a proposition invariant under the appropriate group is therefore meaningful, and the assumption that representations should be unique up to a transformation of the appropriate group. A general theory of representational meaningfulness is offered, based on a semantic and syntactic analysis of representational propositions. Two neglected features of representational propositions are formalized and made use of: (a) that such propositions are induced by more general propositions defined for other structures than the one being represented, and (b) that the true purpose of representation is the application of the theory of the representing system to the represented system. On the basis of these developments, justifications are offered for the two problematic assumptions made by the existing theories.
Similar content being viewed by others
References
Adams, E.: 1959, ‘The Foundations of Rigid Body Mechanics and the Derivation of its Laws from those of Particle Mechanics’, in L. Henkin, P. Suppes, and A. Tarski (eds.), The Axiomatic Method, North-Holland, Amsterdam, pp. 250–65.
Borsuk, K. and W. Szmielew: 1960, Foundations of Geometry, North-Holland, Amsterdam.
Carnap, R.: 1966, Philosophical Foundations of Physics, Basic Books, New York. Reprinted as An Introduction to the Philosophy of Science, Basic Books, New York, 1974.
Dorling, J. (a): ‘Special Relativity out of Euclidean Geometry’, to appear in J. Pitt (ed.), HPS IV (Proceedings of IUHPS Blacksburg conference, 1982).
Field, H.: 1980, Science Without Numbers, Princeton University Press, Princeton, NJ.
Hilbert, D.: 1899, Foundations of Geometry, Open Court, La Salle, Illinois, 1971. (Second Open Court edition, translated from the tenth German edition, 1968. First German edition, 1899.)
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.
Iyanaga, S. and Y. Kawada (eds.): 1977, Encyclopedic Dictionary of Mathematics, English translation, MIT Press, Cambridge, Massachusetts, 1977.
Krantz, D., R. Luce, P. Suppes, and A. Tversky: 1971, Foundations of Measurement, vol. 1, Academic Press, New York.
Kunle, H. and K. Fladt: 1974, ‘Erlanger Program and Higher Geometry’, in H. Behnke, F. Bachmann, K. Fladt, and H. Kunle (eds.), Fundamentals of Mathematics, vol. 2: Geometry, MIT Press, Cambridge, Massachusetts, pp. 460–516.
Luce, R. D.: 1978, ‘Dimensionally Invariant Numerical Laws Correspond to Meaningful Qualitative Relations’, Philosophy of Science 45, 1–16.
Luce, R. D.: 1979, ‘Suppes' Contributions to the Theory of Measurement’, in Radu Bogdan (ed.), Patrick Suppes, D. Reidel, Dordrecht, pp. 93–111.
Mundy, B.: 1982, Synthetic Affine Space-Time Geometry, thesis, Department of Philosophy, Stanford University.
Mundy, B.: 1983, ‘Relational Theories of Euclidean Space and Minkowski Space-Time’, Philosophy of Science 50, 205–26.
Mundy, B.: (a), ‘The Physical Content of Minkowski Geometry’, to appear in British Journal for the Philosophy of Science.
Mundy, B.: (b), 1986, ‘Embedding and Uniqueness in Relational Theories of Space’, Synthese 67, 383–390, this issue.
Mundy, B.: (c), ‘Optical Axiomatization of Minkowski Space-Time Geometry’, to appear in Philosophy of Science.
Pfanzagl, J.: 1971, Theory of Measurement, Physica-Verlag, Würzberg-Vienna.
Roberts, F. and C. Franke: 1976, ‘On the Theory of Uniqueness in Measurement’, Journal of Mathematical Psychology 14, 211–18.
Roberts, F.: 1979, Measurement Theory, Encyclopedia of Mathematics and its Applications, vol. 7, Addison Wesley, Reading, Massachusetts.
Robinson, R. E.: 1963, A Set-Theoretical Approach to Empirical Meaningfulness of Measurement Statements, Technical Report No. 55, Institute for Mathematical Studies in the Social Sciences, Stanford University, Stanford, California.
Schutz, J.: 1973, Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time, Lecture Notes in Mathematics, vol. 361, Springer Verlag, Berlin.
Schutz, J.: 1981, ‘An Axiomatic System for Minkowski Space-Time’, Journal of Mathematical Physics 22, 293–302.
Scott, D. and P. Suppes: 1958, ‘Foundational Aspects of Theories of Measurement’, Journal of Symbolic Logic 23, 113–28.
Sneed, J. D.: 1979, The Logical Structure of Mathematical Physics, second edition. D. Reidel, Dordrecht.
Stevens, S. S.: 1946, ‘On the Theory of Scales of Measurement’, Science 103, 677–80.
Suppes, P.: 1951, ‘A Set of Independent Axioms for Extensive Quantities’, Portugalia Mathematica 10, 163–172. Reprinted in P. Suppes, Studies in the Methodology and Foundations of Science, D. Reidel, Dordrecht, 1969, pp. 36–45.
Suppes, P.: 1960, ‘A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences’, Synthese 12, 287–301. Reprinted in P. Suppes, Studies in the Methodology and Foundations of Science, D. Reidel, Dordrecht, 1969, pp. 10–23.
Suppes, P.: 1970, Set-Theoretical Structures in Science, Institute for Mathematical Studies in the Social Sciences, Stanford University, Stanford, California.
Suppes, P.: 1973, ‘Some Open Problems in the Philosophy of Space and Time’, in P. Suppes (ed.), Space, Time, and Geometry, D. Reidel, Dordrecht, pp. 383–401.
Suppes, P. and J. L. Zinnes: 1963, ‘Basic Measurement Theory’, in R. D. Luce, R. R. Bush, and E. Galanter (eds.), Handbook of Mathematical Psychology, vol. 1. Wiley, New York, pp. 1–76.
Tarski, A.: 1954–1955, ‘Contributions to the Theory of Models, I, II, III’, Indagationes Mathematicae 16, 572–88, and 17, 56–64.
Torretti, R.: 1983, Relativity and Geometry, Pergamon Press, Oxford.
Yaglom, I. M.: 1979, A Simple Non-Euclidean Geometry and its Physical Basis, Springer Verlag, New York.
Author information
Authors and Affiliations
Additional information
Material from this paper was presented at a conference on meaningfulness in the theory of measurement held at New York University in December 1984, hosted by J. C. Falmagne. I would like to thank Patrick Suppes for arranging my invitation to this conference, and David Krantz, R. Duncan Luce, and Fred Roberts for helpful comments. I would also like to thank an anonymous referee for extremely detailed and helpful comments and suggestions, the most important of which are acknowledged in footnotes.
Rights and permissions
About this article
Cite this article
Mundy, B. On the general theory of meaningful representation. Synthese 67, 391–437 (1986). https://doi.org/10.1007/BF00485942
Issue Date:
DOI: https://doi.org/10.1007/BF00485942