nt - p h / 04 12 19 4 v 1 2 4 D ec 2 00 4 What is Probability ?

What is probability? Physicists, mathematicians, and philosophers have been engaged with this question since well before the rise of modern physics. But in quantum mechanics, where probabilities are associated only with measurements, the question strikes to the heart of other foundational problems: what distinguishes measurements from other physical processes? Or in more formal terms: when are the unitary dynamical equations suspended in favour of probabilistic ones? This is the problem of measurement in quantum mechanics. The most clearcut solutions to it change the theory: they either add hidden variables (as in the pilot-wave theory), or they give up the unitary formalism altogether (as in state-reduction theories). The two strategies are tied to different conceptions of probability: probability as in classical statistical mechanics (as formulated by Boltzmann, Gibbs and Einstein), and probability as in Brownian motion (with the dynamics given by a stochastic process, as formulated by Einstein and Smulochowski). The former is sometimes called epistemic probability, as classical mechanics is deterministic: probabilities arise as a consequence of incomplete description. It is the latter, stochastic, probability that is usually thought the more fundamental, as it enters directly into the fundamental equations. Indeterminism encouraged a late-comer to the philosophy of probability, Popper’s propensity approach [12], in which probabilities are identified with certain kinds of properties “dispositions” of chance set-ups. But whether thought of in terms of incomplete descriptions, or as propensities, these kinds of probability have puzzled philosophers. By way of contrast, probability as degree of subjective expectation, or subjective weighting, is much clearer: if that is what probability is one can explain why it obeys the rules that it does (I shall come back to this point later). Neither the epistemic probabilities of classical statistical mechanics, nor the objective probabilities of a stochastic dynamics as in Brownian motion, are happily thought of as subjective expectations. They are grounded, surely, in facts about physical states of affairs, independent of persons altogether. It is true that epistemic probability is often thought of in terms of ignorance, or incomplete knowledge, but it is precisely this link with purely subjective considerations that is so puzzling from the point of view of its role in thermodynamics: heat transfers, surely, take place according to thermodynamical laws, independent of whether anyone is looking, and of what anyone knows. The point is all the more evident in the case of stochastic probability. These epistemic and stochastic probabilities of physics are objective rather than subjective quantities. What, then, are the physical facts that make true some probability statements but not others? What are we trying to get right about when we make judgements of objective probabilities? It is not informative to say it is that there are chances that are thus and so; the difficulty, that has long