This chapter begins my discussion of probability — generally held to be the most severe difficulty for the Everett interpretation. The chapter argues that since mathematically speaking probability has a natural place in the Everett interpretation, any ‘probability problem’ is philosophical in nature. Accordingly, the chapter's strategy is to review various different approaches to understanding probability — in particular, those based on long-run frequencies, on betting preferences, and on symmetries — and to argue that there are no problems with any of these approaches that are specific to Everettian quantum theory. The chapter concludes by arguing that, if probability is to be understood in terms of symmetry, it actually makes more sense in quantum than in classical physics. The chapter concludes with a sketch proof of the quantum probability rule on the basis of symmetry.
This chapter begins my discussion of probability — generally held to be the most severe difficulty for the Everett interpretation. The chapter argues that since mathematically speaking probability has a natural place in the Everett interpretation, any ‘probability problem’ is philosophical in nature. Accordingly, the chapter's strategy is to review various different approaches to understanding probability — in particular, those based on long-run frequencies, on betting preferences, and on symmetries — and to argue that there are no problems with any of these approaches that are specific to Everettian quantum theory. The chapter concludes by arguing that, if probability is to be understood in terms of symmetry, it actually makes more sense in quantum than in classical physics. The chapter concludes with a sketch proof of the quantum probability rule on the basis of symmetry.
This chapter extends chapter 5's formal proof into a general discussion of how we should assess evidence — and, in particular, how we should test scientific theories — given the possibility that Everettian quantum mechanics is correct. The chapter proves — initially with informal arguments, but in due course with full mathematical rigor — that rational scientists should treat the branch weights assigned by quantum theory to quantum-mechanical branches exactly as if they were probabilities; as such, the chapter argues, probability is actually on firmer conceptual footing in quantum theory than in classical physics. After a brief digression to consider probability in approaches to quantum theory other than the Everett interpretation, the chapter concludes by placing the work in the broader context of recent work on Everett probability.
Abstract Much of the evidence for quantum mechanics is statistical in nature. Relative frequency data summarizing the results of repeated experiments is compared to probabilities calculated from the theory; close agreement between the observed relative frequencies and calculated probabilities is taken as evidence in favour of the theory. The Everett interpretation, if it is to be a candidate for serious consideration, must be capable of doing justice to this sort of reasoning. Since, on the Everett interpretation, all outcomes with non-zero amplitude are actualized on different branches, it is not obvious that sense can be made of ascribing probabilities to outcomes of experiments, and this poses a prima facie problem for statistical inference. It is incumbent on the Everettian either to make sense of ascribing probabilities to outcomes of experiments in the Everett interpretation, or to find a substitute on which the usual statistical analysis of experimental results continues to count as evidence for quantum mechanics, and, since it is the very evidence for quantum mechanics that is at stake, this must be done in a way that does not presuppose the correctness of Everettian quantum mechanics. This requires an account of theory confirmation that applies to branching-universe theories but does not presuppose the correctness of any such theory. This chapter supplies and defends such an account. The account has the consequence that statistical evidence can confirm a branching-universe theory such as Everettian quantum mechanics in the same way in which it can confirm a non-branching probabilistic theory.
After an initial discussion of the general aims of scientific theories, the quantum measurement problem is presented. Non-Everettian solutions to the problem are briefly discussed; the aim of the discussion is to show that these solutions require significant change either to our philosophical approach to science or to quantum theory itself. In this context, the Everett interpretation is introduced and motivated as the result of simply applying our normal approach to scientific theories to the mathematical structure of quantum mechanics and following where it leads. The chapter ends with a introductory discussion of potential problems, and with references to subsequent chapters where they are discussed more fully.
It is often objected that the Everett interpretation of QM cannot make sense of quantum probabilities, in one or both of two ways: either it can't make sense of probability at all, or it can't explain why probability should be governed by the Born rule. David Deutsch has attempted to meet these objections. He argues not only that rational decision under uncertainty makes sense in the Everett interpretation, but also that under reasonable assumptions, the credences of a rational agent in an Everett world should be constrained by the Born rule. David Wallace has developed and defended Deutsch's proposal, and greatly clarified its conceptual basis. In particular, he has stressed its reliance on the distinguishing symmetry of the Everett view, viz., that all possible outcomes of a quantum measurement are treated as equally real. The argument thus tries to make a virtue of what has usually been seen as the main obstacle to making sense of probability in the Everett world. In this note I outline some objections to the Deutsch-Wallace argument, and to related proposals by Hilary Greaves about the epistemology of Everettian QM. (In the latter case, my arguments include an appeal to an Everettian analogue of the Sleeping Beauty problem.) The common thread to these objections is that the symmetry in question remains a very significant obstacle to making sense of probability in the Everett interpretation.
The article discusses approaches to understanding the nature of probability in mathematics and physics with the consequences of these approaches, as well as the difficulties that arise. The ontological status of probability is discussed. It is shown that probability theory, contrary to R. Mises, should be classified as a mathematical science, using the definition formulated by A.N. Kolmogorov. The application of this definition in practice is provided within the framework of the classical approach, provided that the connection with physical reality is defined through the concept of equal opportunity. To do this, the concepts of simple and complex events are introduced, and the method of decomposing complex events into simple ones is explained. The point of view is substantiated, according to which a mathematical probability determined and related to physical reality in this way can be considered an objective probability, that is, existing independently of any tests. Various interpretations of quantum mechanics and approaches to understanding probability are also revealed. It is shown that in physics today the existing approaches to understanding the nature of probability contradict each other. In the Bohm interpretation, probability is an analogue of the classical approach – statistical objective probability, which has a mathematical formalization, and in quantum Bayesianism, probability quantifies the degree of reliability and depends on the subject conducting the experiment. The Copenhagen interpretation of quantum mechanics tries to eliminate the difficulties associated with combining the wave and corpuscular properties of light and electron beams into a single picture by giving the ontological status of probability, which leads to the problem of understanding physical reality. The article argues for the need for a new connection between the mathematical description of probability and physical reality. The empirical (frequency) approach of R. Mises is analyzed, which causes a lot of criticism, as it does not have a clear consistent formalization. The article argues that this approach is inconsistent with quantum mechanics. The probability given by M. Born in the Copenhagen interpretation of quantum mechanics is also considered, which formulates a new definition of probability, which does not exclude the difficulties associated with the physical interpretation of probability. Born’s approach is based on the classical approach and is reduced to a formal axiomatic definition, which assumes the interpretation of probability as a mathematical construct. The connection of probability with physical reality and the consistency of the statistical method of finding probability values remain unresolved problems, which creates a dualism of understanding probability: in mathematics – one thing, in physics – another. If in the mathematical approach, the solution of these problems is carried out through the connection of probability with physical properties, then in quantum mechanics, in the Copenhagen interpretation, problems persist, which is caused by giving the wave function an ontological status.