Abstract
We examine the notions of negative, infinite and hotter than infinite temperatures and show how these unusual concepts gain legitimacy in quantum statistical mechanics. We ask if the existence of an infinite temperature implies the existence of an actual infinity and argue that it does not. Since one can sensibly talk about hotter than infinite temperatures, we ask if one could legitimately speak of other physical quantities, such as length and duration, in analogous terms. That is, could there be longer than infinite lengths or temporal durations? We argue that the answer is surprisingly yes, and we outline the properties of a number system that could be employed to characterize such magnitudes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aristotle, Physics, in Richard McKeon (ed.): 1941, The Basic Works of Aristotle, Random House, New York.
Abragam, A., and W. G. Proctor: 1958, ‘Spin temperature’, The Physical Review 109, 1441–1458.
Bazarov, Ivan: 1964, Thermodynamics, The Macmillan Company, New York, Ch. 10.
Benardette, J.: 1964, Infinity: An Essay in Metaphysics, Clarendon Press, Oxford.
Bergmann, Gustav, and Kenneth W. Spence: 1944, ‘Logic of psycho-physical measurement’, Psychological Review 51, 1–24.
Bloembergen, N.: 1973, ‘The concept of temperature in magnetism’, American Journal of Physics 41, 325–31.
Campbell, Norman: 1957, Foundations of Science, Dover Publications, Inc., New York.
Cantor, G.: 1976, ‘Foundations of the theory of manifolds’, translated by U. Parpart, The Campaigner 9 (January and February), 69.
Casimir, H. B. G., and F. K. DuPre: 1938, ‘Note on the thermodynamic interpretation of paramagnetic relaxation phenomena’, Physica 5, 507–11.
Cohen, Morris, and Ernest Nagel: 1934, An Introduction to Logic and Scientific Method, Harcourt, Brace and World, Inc., New York.
Conway, J. H.: 1976, On Numbers and Games, Academic Press, London.
Dauben, J.: 1979, Georg Cantor, His Mathematics and Philosophy of the Infinite, Harvard University Press, Cambridge, Massachusetts.
DeBroglie, Louis: 1955, Physics and Microphysics, Harper and Brothers, New York.
Desloge, E. A., and W. A. Barker: 1957, ‘Chemical potentials, spin temperatures and the Overhauser effect’, The Physical Review 108, 924.
Duhem, Pierre: 1962, The Aim and Structure of Physical Theory, Atheneum, New York.
Dunning-Davies, J.: 1976, ‘Negative absolute temperatures and Carnot cycles’, Journal of Physics A, Math, Gen. 9, 605.
Earman, J.: 1977, ‘Till the end of time’, in J. Earman, C. Glymour, and J. Stachel (eds.), Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science, Vol. 8, University of Minnesota Press, Minneapolis.
Eddington, Arthur: 1946, Fundamental Theory, Cambridge University Press, London.
Edwards, S. F., and J. B. Taylor: 1974, ‘Negative temperature states of two-dimensional plasmas and vortex fluids’, Proceedings of the Royal Society A336, 257.
Ehrlich, P.: 1981, ‘The concept of temperature and its dependence on the laws of thermodynamics’, American Journal of Physics 49, 622–32.
Ellis, Brian: 1966, Basic Concepts of Measurement, Cambridge University Press, London.
Fraenkel, A. A.: 1976, Abstract Set Theory, Fourth Edition, North-Holland Pub. Co., New York and Amsterdam.
Giauque, W. F.: 1954, ‘Review of Magnetic Cooling by C. C. B. Garret’, Journal of the American Chemical Society 76, 5577.
Gorter, C.: 1947, Paramagnetic Relaxation, Elsevier, Amsterdam.
Helmholtz, Herman von: 1887, Zählen und Messen erkenntnistheoretisch betrachtet. English translation by C. L. Bryan: 1930, Counting and Measuring, Van Nostrand, New Jersey.
Hempel, Carl: 1952, Fundamentals of Concept Formation in Empirical Science, The University of Chicago Press, Chicago.
Holder, O.: 1901, ‘Die Axiome der Quantität und die Lehre vom Mass’, Berichte Über die Verhandlugen der Königlich Sächischen Gesellschaft der Wissenschaften zu Leipzig (Math.-Phys. Klasse) 53, 1–64.
Huntington, E.: 1955, The Continuum and Other Types of Serial Order, Dover Publications, Inc., New York.
Joyce, G., and D. Montgomery: 1973, ‘Negative temperature states for the two-dimensional guiding-center plasma’, Journal of Plasma Physics 10, 107.
Klein, M.: 1956, ‘Negative absolute temperatures’, The Physical Review 104, 589.
Knuth, D. E.: 1974, Surreal Numbers, Addison-Wesley Pub. Co., Reading, Massachusetts.
Krantz, David, R. D. Luce, Patrick Suppes, and Amos Tversky: 1971, Foundations of Measurement, Academic Press, New York.
Landsberg, Peter T.: 1961, Thermodynamics, Interscience Pub., New York.
Landsberg, Peter T.: 1977, ‘Heat engines and heat pumps at positive and negative absolute temperatures’, Journal of Physics 10, 1773.
Landau, Lev, and Evgenii Lifshitz: 1979, Statistical Physics, Pergamon, London.
Machlup, Stefan: 1975, ‘Negative temperatures and negative dissipation’, American Journal of Physics 43, 991–94.
Marvin, Milan: 1966, Negative Absolute Temperatures, Iliffe Books Ltd., London.
Middleton, W. E. K.: 1966, A History of the Thermometer, Johns Hopkins Press, Baltimore.
Nakagomi, T.: 1980, ‘Mathematical formulation of the heat engine theory of thermodynamics including negative absolute temperatures’, Journal of Physics 13, 291.
Onsager, L.: 1949, ‘Statistical hydrodynamics’, Nuovo Cimento Supplemento Series 9 6, 279.
Poincaré, H.: 1952, Science and Hypothesis, Dover Publications, Inc., New York.
Pound, R. V., and E. M. Purcell: 1951, ‘A nuclear spin system at negative temperature’, The Physical Review 81, 279.
Powles, J. G.: 1963, ‘Negative absolute temperatures and rotating temperatures’, Contemporary Physics 4, 338–55.
Proctor, W. G.: 1978, ‘Negative absolute temperatures’, Scientific American (August), 90–99.
Ramsey, Norman: 1956, ‘Thermodynamics and statistical mechanics at negative absolute temperatures’, The Physical Review 103, 20–28.
Ramsey, Norman: 1970, ‘Spin temperatures and negative absolute temperatures’, in Benjamin Gal-Or (ed.), A Critical Review of Thermodynamics, Mono Book Corp., Baltimore, pp. 217–29.
Roberts, F. S. and R. D. Luce: 1968, ‘Axiomatic thermodynamics and extensive measurement’, Synthese 18, 311–26.
Robinson, A.: 1974, Non-Standard Analysis, North-Holland Pub. Co., Amsterdam.
Royden, H. L.: 1968, Real Analysis, Macmillan Pub. Co., New York.
Salmon, W.: 1970, Zeno's Paradoxes, The Bobbs-Merrill Co., New York.
Simon, F. E.: 1955, ‘The concept of temperature near absolute zero’, in Temperature, Its Measurement and Control in Science and Industry, Volume. Compiled under the auspices of the American Institute of Physics. Reinhold Pub. Co., New York, pp. 9–17.
Stuart, F. B., A. J. Brainard, and B. Gal-Or (eds.): 1970, A Critical Review of Thermodynamics, Mono Book Corp., Baltimore.
Suppes, P.: 1951, ‘A set of independent axioms for extensive quantities’, Portugaliae Mathematica 10, 163–72.
Thomson, William: 1882, ‘On an absolute thermometric scale founded on Carnot's theory of the motive power of heat, and calculated from Regnault's observations’, in Sir William Thomson (ed.), Mathematical and Physical Papers, Volume 1, Cambridge University Press, London.
Thomson, William, and P. J. Joule: 1882, ‘On the thermal effects of fluids on motion’, in Sir William Thomson (ed.), Mathematical and Physical Papers, Volume 1, Cambridge University Press, London, pp. 333–455.
Tremblay, A.: 1976, Comment on ‘Negative Kelvin temperatures: some anomalies and a speculation’. Americal Journal of Physics 44, 994–95.
Weinreich, G.: 1968, Fundamental Thermodynamics, Addison-Wesley Pub. Co., Reading, Massachusetts.
Author information
Authors and Affiliations
Additional information
This paper grew out of a talk I gave at Northwestern University during the summer of 1977. Thanks are owed to Arthur Fine, Neal Grossman, David Malament, Brian Skyrms and Paul Teller, each of whom attended the talk and offered helpful comments as well as support. Of them I owe a special thank you to Arthur Fine with whom I have discussed many of the ideas in this paper. Finally, I am greatly indebted to Howard Stein, who brought J. H. Conway's Surreal Numbers to my attention and who suggested their relevance to my work.
Rights and permissions
About this article
Cite this article
Ehrlich, P. Negative, infinite, and hotter than infinite temperatures. Synthese 50, 233–277 (1982). https://doi.org/10.1007/BF00416903
Issue Date:
DOI: https://doi.org/10.1007/BF00416903