Hostname: page-component-5c6d5d7d68-ckgrl Total loading time: 0 Render date: 2024-08-16T12:56:06.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Randomness and determinism in the interplay between the continuum and the discrete†

Published online by Cambridge University Press:  01 April 2007

FRANCIS BAILLY  and
GIUSEPPE LONGO
FRANCIS BAILLY
Affiliation:
Physique, CNRS, Meudon Email: bailly@cnrs-bellevue.fr
GIUSEPPE LONGO
Affiliation:
LIENS, CNRS ENS and CREA, Paris Web site: http://www.di.ens.fr/users/longo
Get access Rights & Permissions [Opens in a new window]

Extract

This paper provides a conceptual analysis of the role of the mathematical continuum versus the discrete in the understanding of randomness as a notion with a physical meaning or origin. The presentation is ‘informal’ as we will not write formulas; however, we will refer to non-obvious technical results from various scientific domains, and we will also propose a conceptual framework for understanding randomness (and predictability), which we believe is, essentially, original. As a matter of fact, unpredictability and randomness may be conveniently identified in various physico-mathematical contexts. This will allow us to explore these concepts in continuous versus discrete frameworks, with particular emphasis on the relationships and differences between classical approaches and quantum theories in Physics.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 2 , April 2007 , pp. 289 - 305
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, R. L. (1979) Topological entropy and equivalence of dynamical systems, American Mathematical Society.CrossRefGoogle Scholar
Alligood, K., Sauer, T. and Yorke, J. (2000) Chaos: an introduction to Dynamical Systems, Springer.Google Scholar
Anandan, J. (2002) Causality, Symmetries and Quantum Mechanics. Foundations of Physics Letters 15 (5)415–438.CrossRefGoogle Scholar
Aspect, A., Grangier, P. and Roger, G. (1982) Experimental Realization of the Einstein–Podolsky–Rosen–Bohm Gedankenexperiment: A New Violation of Bell's Inequalities. Phys. Rev. Let. 49 91.CrossRefGoogle Scholar
Bailly, F. and Longo, G. (2006) Mathématiques et Sciences de la Nature. La singularité physique du vivant, Hermann, Paris. (English introduction available at http://www.di.ens.fr/users/longo.)Google Scholar
Bell, J. S. (1964) On the Einstein–Podolsky–Rosen Paradox. Physics 1 195.CrossRefGoogle Scholar
Bitbol, M. (2000a) L'aveuglante proximité du réel, Flammarion.Google Scholar
Bitbol, M. (2000b) Physique et philosophie de l'esprit, Flammarion.CrossRefGoogle Scholar
Bohm, D. (1951) The Paradox of Einstein, Rosen and Podolsky. Quantum Th. 611–623.Google Scholar
Bohm, D. (1987) La plénitude de l'univers, Le Rocher, Paris.Google Scholar
de Broglie, L. (1961) Introduction à la nouvelle théorie des particules de M. Jean-Pierre Vigier et de ses collaborateurs, Gauthier-Villars.Google Scholar
Devaney, R. L. (1989) An Introduction to chaotic dynamical systems, Addison-Wesley.Google Scholar
Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality be Considered complete? Phys. Rev. 41 777.CrossRefGoogle Scholar
van Fraassen, B. (1994) Lois et symétrie, Vrin.Google Scholar
Laskar, J. (1990) The chaotic behaviour of the solar system. Icarus 88 266–291.CrossRefGoogle Scholar
Laskar, J. (1994) Large scale chaos in the solar system. Astron. Astrophys. 287 L9L12.Google Scholar
Lebowitz, J. (1999) Microscopic Origins of Irreversible Macroscopic Behavior. Physica A CCLXIII 516.CrossRefGoogle Scholar
Lighthill, J. (1986) The recent recognized failure of predictability in Newtonian dynamics. Proc. R. Soc. Lond. A 407 35–50.Google Scholar
Longo, G. (2007) Laplace, Turing and the ‘imitation game’ impossible geometry: randomness, determinism and programs in Turing's test. In: Epstein, R., Roberts, G. and Beber, G. (eds.) The Turing Test Sourcebook, Kluwer.Google Scholar
Mugur-Schachter, M. (2006) Sur le tissage des connaissances, Hermès-Lavoisier.Google Scholar
Pilyugin, S. Yu. (1999) Shadowing in dynamical systems, Springer.Google Scholar
Poincaré, H. (1902) La Science et l'Hypothèse, Flammarion, Paris.Google Scholar
Sauer, T. (2003) Shadowing breakdown and large errors in dynamical simulations of physical systems. Preprint, George Mason University.CrossRefGoogle Scholar
Weyl, H. (1918a) Das Kontinuum, 1918a.CrossRefGoogle Scholar
Weyl, H. (1918b) Raum–Zeit–Materie, Springer.CrossRefGoogle Scholar
Zurek, W. H. (1991) Decoherence and the Transition from the Quantum to the Classical. Physics Today 44 36–44.CrossRefGoogle Scholar