How random is a coin toss?
DOI: 10.1063/1.2915570
Probabilistic and deterministic Descriptions of macroscopic phenomena have coexisted for centuries. During the period 1650–1750, for example, Newton developed his calculus of determinism for dynamics while the Bernoullis simultaneously constructed their calculus of probability for games of chance and various other many‐body problems. In retrospect, it would appear strange indeed that no major confrontation ever arose between these seemingly contradictory world views were it not for the remarkable success of Laplace in elevating Newtonian determinism to the level of dogma in the scientific faith. Thereafter, probabilitistic descriptions of classical systems were regarded as no more than useful conveniences to be invoked when, for one reason or another, the deterministic equations of motion were difficult or impossible to solve exactly. Moreover, these probabilistic descriptions were presumed derivable from the underlying determinism, although no one ever indicated exactly how this feat was to be accomplished.
References
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2. A. S. Wightman has also lamented this seeming conspiracy of silence in Perspectives in Statistical Physics, H. J. Raveche, ed., North‐Holland, Amsterdam (1981).
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V. M. Alekseev, M. V. Yakobson, Phys. Repts. 75, 287 (1981).https://doi.org/PRPLCM9. It perhaps should be mentioned here that entries in a cell‐number sequence need not be statistically independent despite any impression to the contrary created by the text itself. See Martin‐ Lof, reference 8.
10. Our definition of incalculable number is not the same as the computer theorist’s definition of uncomputable number, although the terms are related.
11. Semi‐private barroom conversations held at various conference watering holes around the world.
12. Readers with a historical bent may have already recognized many parallels of fact or spirit between the present paper and numerous earlier articles dating back at least to Maxwell. But perhaps the most complete and striking parallel of all is to the paper “Is Classical Mechanics In Fact Deterministic?” by Max Born, Physics In My Generation, Springer‐Verlag, New York (1969), page 78.
More about the Authors
Joseph Ford. Georgia Institute of Technology.
