Abstract
Questions concerning the meaning of probability and its applications in physics are notoriously subtle. In the philosophy of the exact sciences, the conceptual analysis of the foundations of a theory often lags behind the discovery of the mathematical results that form its basis. The theory of probability is no exception. Although Kolmogorov’s axiomatization of the theory [1] is generally considered definitive, the meaning of the notion of probability remains a matter of controversy.1 Questions pertain both to gaps between the formalism and the intuitive notions of probability and to the inter-relationships between the intuitive notions. Further, although each of the interpretations of the notion of probability is usually intended to be adequate throughout, independently of context, the various applications of the theory of probability pull in different interpretative directions: some applications, say in decision theory, are amenable to a subjective interpretation of probability as representing an agent’s degree of belief, while others, say in genetics, call upon an objective notion of probability that characterizes certain biological phenomena. In this volume we focus on the role of probability in physics. We have the dual goal and challenge of bringing the analysis of the notion of probability to bear on the meaning of the physical theories that employ it, and of using the prism of physics to study the notion of probability.
We thank Orly Shenker for comments on an earlier draft.
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Notes
- 1.
As far as interpretation is concerned, the drawback of the axiom system is that, in formalizing measure theory in general, it captures more than the intuitive notion of probability, including such notions as length and volume.
- 2.
This does not apply to Bohmian quantum mechanics, which is deterministic. See also Earman [2] for an analysis of determinism and for an unorthodox view about both classical and quantum mechanics regarding their accordance with determinism.
- 3.
- 4.
It has been shown that the subjective probabilities of an agent who updates her beliefs in accordance with Bayes’ theorem, converge on the observed relative frequencies no matter what her prior subjective probabilities are. But this is different from converging on the chances or the relative frequencies in the infinite limit. Similar considerations apply also to the so-called ‘logical’ approach to probability on which probabilities are quantitative expressions of the degree of support of a statement conditional on the evidence.
- 5.
See van Fraassen [7], p. 83. Note again that the condition of Dutch-book coherence is of no help here since the objective probabilities in the situation we consider are unknown.
- 6.
- 7.
In fact, there are other arguments, which show that Boltzmann’s deterministic approach in deriving the H-theorem could not be consistent with the classical dynamics, e.g. the historically famous objection by Zermelo based on the Poincaré recurrence theorem. It was later discovered that one of the premises in Boltzmann’s proof was indeed time-asymmetric.
- 8.
- 9.
See Maudlin [15] for a more detailed discussion of this problem.
- 10.
- 11.
The situation, however, is somewhat disappointing, since in quantum mechanics the question of how to account for the past is notoriously hard due to the problematic nature of retrodiction in standard quantum mechanics.
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Ben-Menahem, Y., Hemmo, M. (2012). Introduction. In: Ben-Menahem, Y., Hemmo, M. (eds) Probability in Physics. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21329-8_1
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