Foundations of Physics

, Volume 30, Issue 10, pp 1569–1585 | Cite as

Invariance, Symmetry and Meaning

  • Patrick Suppes


The role of the concept of invariance in physics and geometry is analyzed, with attention to the closely connected concepts of symmetry and objective meaning. The question of why the fundamental equations of physical theories are not invariant, but only covariant, is examined in some detail. The last part of the paper focuses on the surprising example of entropy as a complete invariant in ergodic theory for any two ergodic processes that are isomorphic in the measure-theoretic sense.


Entropy Ergodic Theory Physical Theory Fundamental Equation Objective Meaning 
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  1. O. Jones, The Grammar of Chinese Ornaments (Gilbert, London, 1867).Google Scholar
  2. A. N. Kolmogorov, “A new metric invariant of transient dynamical systems and automorphisms in Lebesgue space,” Dokl. Akad. Nauk. SSSR 119, 861 (1958); (Russian) MR 21 #2035a.Google Scholar
  3. A. N. Kolmogorov, “Entropy per unit time as a metric invariant of automorphism,” Dokl. Akad. Nauk. SSSR 124, 754 (1959); (Russian) MR 2 #2035b.Google Scholar
  4. A. Lindenbaum and A. Tarski, Über die Beschränktheit der Ausdrucksmittel deductiver Theorien (Ergebnisse eines mathematischen Kolloquium, fascicule 7 (1934/35)), pp. 15-22; translated by J. H. Woodger, in Logic, Semantics, Metamathematics, 2nd edn., J. Corcoran, ed. (Hacket, Indianapolis, 1983), pp. 384-392.Google Scholar
  5. I. Newton, Principia (1686); translated by F. Cajori (University of California Press, Berkeley, 1946).Google Scholar
  6. E. Noether, “Invarianten beliebiger Differentialausdrucke,” Nachrichten von der Gessellschaft der Wissenschaften zu Göttingen (1918), pp. 37–44.Google Scholar
  7. D. S. Ornstein, “Bernoulli shifts with the same entropy are isomorphic,” Adv. Math. 4, 337 (1970).Google Scholar
  8. A. Palladio, The Four Books of Architecture (1570); translated by Isaac Ware (London, 1731).Google Scholar
  9. Y. G. Sinai, “On the notion of entropy of a dynamical system,” Dokl. Akad. Nauk. SSSR 124, 768 (1959).Google Scholar
  10. P. Suppes, “Axioms for relativistic kinematics with or without parity,” in The Axiomatic Method with Special Reference to Geometry and Physics, L. Henkin, P. Suppes, and A. Tarski, eds. (Proceedings of an international symposium held at the University of California, Berkeley, December 16, 1957-January 4, 1958) (North-Holland, Amsterdam, 1959), pp. 291–307.Google Scholar
  11. A. Tarski and S. Givant, A Formalization of Set Theory without Variables (American Mathe-matical Society, Providence, Rhode Island, 1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Patrick Suppes
    • 1
  1. 1.C.S.L.I.Stanford UniversityStanford

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