Quantum sidelights on The Material Theory of Induction

John Norton's The Material Theory of Induction bristles with fresh insights and provocative ideas that provide a much needed stimulus to a stodgy if not moribund field. I use quantum mechanics (QM) as a medium for exploring some of these ideas. First, I note that QM offers more predictability than Newtonian mechanics for the Norton dome and other cases where classical determinism falters. But this ability of QM to partially cure the ills of classical determinism depends on facts about the quantum Hamiltonian operator that vary from case to case, providing an illustration of Norton's theme of the importance of contingent facts for inductive reasoning. Second, I agree with Norton that Bayesianism as developed for classical probability theory does not constitute a universal inference machine, and I use QM to explain the sense in which this is so. But at the same time I defend a brand of quantum Bayesianism as providing an illuminating account of how physicists' reasoning about quantum events. Third, I argue that if the probabilities induced by quantum states are regarded as objective chances then there are strong reasons to think that fair infinite lotteries are impossible in a quantum world.     

Emergent evolutionism,determinism and unpredictability

The fact that there exist in nature thoroughly deterministic systems whose future behavior cannot be predicted, no matter how advanced or fined-tune our cognitive and technical abilities turn out to be, has been well established over the last decades or so, essentially in the light of two different theoretical frameworks, namely chaos theory and (some deterministic interpretation of) quantum mechanics. The prime objective of this paper is to show that there actually exists an alternative strategy to ground the divorce between determinism and predictability, a way that is older than—and conceptually independent from—chaos theory and quantum mechanics, and which has not received much attention in the recent philosophical literature about determinism. This forgotten strategy—embedded in the doctrine called “emergent evolutionism”—is nonetheless far from being a mere historical curiosity that should only draw the attention of philosophers out of their concern for comprehensiveness. It has been indeed recently revived in the works of respected scientists.     

A remark on Fuchs’ Bayesian interpretation of quantum mechanics

Quantum mechanics is a theory whose foundations spark controversy to this day. Although many attempts to explain the underpinnings of the theory have been made, none has been unanimously accepted as satisfactory. Fuchs has recently claimed that the foundational issues can be resolved by interpreting quantum mechanics in the light of quantum information. The view proposed is that quantum mechanics should be interpreted along the lines of the subjective Bayesian approach to probability theory. The quantum state is not the physical state of a microscopic object. It is an epistemic state of an observer; it represents subjective degrees of belief about outcomes of measurements. The interpretation gives an elegant solution to the infamous measurement problem: measurement is nothing but Bayesian belief updating in a analogy to belief updating in a classical setting. In this paper, we analyze an argument that Fuchs gives in support of this latter claim. We suggest that the argument is not convincing since it rests on an ad hoc construction. We close with some remarks on the options left for Fuchs’ quantum Bayesian project.     

Quantum probabilities as degrees of belief

I outline an argument for a subjective Bayesian interpretation of quantum probabilities as degrees of belief distributed subject to consistency constraints on a quantum rather than a classical event space. I show that the projection postulate of quantum mechanics can be understood as a noncommutative generalization of the classical Bayesian rule for updating an initial probability distribution on new information, and I contrast the Bayesian interpretation of quantum probabilities sketched here with an alternative approach defended by Chris Fuchs.     

A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one

Carlton Caves, Fuchs, and Schack (2002) have recently appealed to an argument of mine (Stairs, 1983) to address a problem for their subjective Bayesian account of quantum probability. The difficulty is that on the face of it, quantum mechanical probabilities of one appear to be objective, but in that case, the Born Rule would yield a continuum of probabilities between zero and one. If so, we end up with objective probabilities strictly between zero and one. The authors claim that objective probabilities of one leads to a dilemma: give up locality or fall into contradiction. I argue that this conclusion depends on an overly strong interpretation of objectivism about quantum probabilities.     

Determinism and Chance

It is generally thought that objective chances for particular events different from 1 and 0 and determinism are incompatible. However, there are important scientific theories whose laws are deterministic but which also assign non-trivial probabilities to events. The most important of these is statistical mechanics whose probabilities are essential to the explanations of thermodynamic phenomena. These probabilities are often construed as ‘ignorance’ probabilities representing our lack of knowledge concerning the microstate. I argue that this construal is incompatible with the role of probability in explanation and laws. This is the ‘paradox of deterministic probabilities’. After surveying the usual list of accounts of objective chance and finding them inadequate I argue that an account of chance sketched by David Lewis can be modified to solve the paradox of deterministic probabilities and provide an adequate account of the probabilities in deterministic theories like statistical mechanics.     

Quantum probability and many worlds

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the non-probabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an additional probability rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics.     

Intersubjective corroboration

How are we to understand the use of probability in corroboration functions? Popper says logically, but does not show we could have access to, or even calculate, probability values in a logical sense. This makes the logical interpretation untenable, as Ramsey and van Fraassen have argued.If corroboration functions only make sense when the probabilities employed therein are subjective, however, then what counts as impressive evidence for a theory might be a matter of convention, or even whim. So isn’t so-called ‘corroboration’ just a matter of psychology?In this paper, I argue that we can go some way towards addressing this objection by adopting an intersubjective interpretation, of the form advocated by Gillies, with respect to corroboration. I show why intersubjective probabilities are preferable to subjective ones when it comes to decision making in science: why group decisions are liable to be superior to individual ones, given a number of plausible conditions. I then argue that intersubjective corroboration is preferable to intersubjective confirmation of a Bayesian variety, because there is greater opportunity for principled agreement concerning the factors involved in the former.     

Understanding Deutsch's probability in a deterministic multiverse

Difficulties over probability have often been considered fatal to the Everett interpretation of quantum mechanics. Here I argue that the Everettian can have everything she needs from ‘probability’ without recourse to indeterminism, ignorance, primitive identity over time or subjective uncertainty: all she needs is a particular rationality principle.The decision-theoretic approach recently developed by Deutsch and Wallace claims to provide just such a principle. But, according to Wallace, decision theory is itself applicable only if the correct attitude to a future Everettian measurement outcome is subjective uncertainty. I argue that subjective uncertainty is not available to the Everettian, but I offer an alternative: we can justify the Everettian application of decision theory on the basis that an Everettian should care about all her future branches. The probabilities appearing in the decision-theoretic representation theorem can then be interpreted as the degrees to which the rational agent cares about each future branch. This reinterpretation, however, reduces the intuitive plausibility of one of the Deutsch–Wallace axioms (measurement neutrality).     

States of ignorance and ignorance of states: Examining the Quantum Principal Principle

Earman (2018) has recently argued that the Principal Principle, a principle of rationality connecting objective chance and credence, is a theorem of quantum probability theory. This paper critiques Earman's argument, while also offering a positive proposal for how to understand the status of the Principal Principle in quantum probability theory.     

What could be objective about probabilities?

The basic notion of an objective probability is that of a probability determined by the physical structure of the world. On this understanding, there are subjective credences that do not correspond to objective probabilities, such as credences concerning rival physical theories. The main question for objective probabilities is how they are determined by the physical structure.In this paper, I survey three ways of understanding objective probability: stochastic dynamics, humean chances, and deterministic chances (typicality). The first is the obvious way to understand the probabilities of quantum mechanics via a collapse theory such as GRW, the last is the way to understand the probabilities in the context of a deterministic theory such as Bohmian mechanics. Humean chances provide a more abstract and general account of chances locutions that are independent of dynamical considerations.     

A comparative evaluation of objective and subjective weather forecasts in the united states

This paper is concerned primarily with the evaluation and comparison of objective and subjective weather forecasts. Operational forecasts of three weather elements are considered: (1) probability forecasts of precipitation occurrence, (2) categorical (i.e. non-probabilistic) forecasts of maximum and minimum temperatures and (3) categorical forecasts of cloud amount. The objective forecasts are prepared by numerical-statistical procedures, whereas the subjective forecasts are based on the judgements of individual forecasters. In formulating the latter, the forecasters consult information from a variety of sources, including the objective forecasts themselves. The precipitation probability forecasts are found to be both reliable and skilful, and evaluation of the temperature/cloud amount forecasts reveals that they are quite accurate/skilful. Comparison of the objective and subjective forecasts of precipitation occurrence indicates that the latter are generally more skilful than the former for shorter lead times (e.g. 12–24 hours), whereas the two types of forecasts are of approximately equal skill for longer lead times (e.g. 36–48 hours). Similar results are obtained for the maximum and minimum temperature forecasts. Objective cloud amount forecasts are more skilful than subjective cloud amount forecasts for all lead times. Examination of trends in performance over the last decade reveals that both types of forecasts for all three elements increased in skill (or accuracy) over the period, with improvements in objective forecasts equalling or exceeding improvements in subjective forecasts. The role and impact of the objective forecasts in the subjective weather forecasting process are discussed in some detail. The need to conduct controlled experiments and other studies of this process, with particular reference to the assimilation of information from different sources, is emphasized. Important characteristics of the forecasting system in meteorology are identified, and they are used to describe similarities and differences between weather forecasting and forecasting in other fields. Acquisition of some of these characteristics may be beneficial to other forecasting systems.     

Probability in GRW theory

GRW theory postulates a stochastic mechanism assuring that every so often the wave function of a quantum system is ‘hit’, which leaves it in a localised state. How are we to interpret the probabilities built into this mechanism? GRW theory is a firmly realist proposal and it is therefore clear that these probabilities are objective probabilities (i.e. chances). A discussion of the major theories of chance leads us to the conclusion that GRW probabilities can be understood only as either single case propensities or Humean objective chances. Although single case propensities have some intuitive appeal in the context of GRW theory, on balance it seems that Humean objective chances are preferable on conceptual grounds.     

The Loss of Coherence in Quantum Cosmology

I analyse two different methods for the retrieval of a classical notion of spacetime from the theory of quantum cosmology in terms of the different means they employ to bring about the necessary loss of coherence. One method employs a direct coarse graining of the appropriate phase space, whereas the other method is based on decohering the system by the interaction with an environment. Although these methods are equivalent on a phenomenological level, I argue that conceptually the decoherence approach is superior. The coarse graining approach construes the necessary loss of coherence in epistemic terms, whereas the method based on decohering the system by interaction with an environment provides a dynamical explanation for the emergence of classical notions of spacetime. On the latter account the emergence of classical behaviour is an objective property of the physical system under consideration, in contradistinction with the subjective coarse graining account of the retrieval of a classical spacetime in terms of measurements made by an observer.     

Decision theory and information propagation in quantum physics

In recent papers, Zurek [(2005). Probabilities from entanglement, Born's rule p_k=|ψ_k|² from entanglement. Physical Review A, 71, 052105] has objected to the decision-theoretic approach of Deutsch [(1999) Quantum theory of probability and decisions. Proceedings of the Royal Society of London A, 455, 3129–3137] and Wallace [(2003). Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation. Studies in History and Philosophy of Modern Physics, 34, 415–438] to deriving the Born rule for quantum probabilities on the grounds that it courts circularity. Deutsch and Wallace assume that the many worlds theory is true and that decoherence gives rise to a preferred basis. However, decoherence arguments use the reduced density matrix, which relies upon the partial trace and hence upon the Born rule for its validity. Using the Heisenberg picture and quantum Darwinism—the notion that classical information is quantum information that can proliferate in the environment pioneered in Ollivier et al. [(2004). Objective properties from subjective quantum states: Environment as a witness. Physical Review Letters, 93, 220401 and (2005). Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe. Physical Review A, 72, 042113]—I show that measurement interactions between two systems only create correlations between a specific set of commuting observables of system 1 and a specific set of commuting observables of system 2. This argument picks out a unique basis in which information flows in the correlations between those sets of commuting observables. I then derive the Born rule for both pure and mixed states and answer some other criticisms of the decision theoretic approach to quantum probability.     

The early history of chance in evolution

Work throughout the history and philosophy of biology frequently employs ‘chance’, ‘unpredictability’, ‘probability’, and many similar terms. One common way of understanding how these concepts were introduced in evolution focuses on two central issues: the first use of statistical methods in evolution (Galton), and the first use of the concept of “objective chance” in evolution (Wright). I argue that while this approach has merit, it fails to fully capture interesting philosophical reflections on the role of chance expounded by two of Galton's students, Karl Pearson and W.F.R. Weldon. Considering a question more familiar from contemporary philosophy of biology—the relationship between our statistical theories of evolution and the processes in the world those theories describe—is, I claim, a more fruitful way to approach both these two historical actors and the broader development of chance in evolution.     

On the borderline between Science and Philosophy: A debate on determinism in France around 1880

In the second half of the nineteenth century, a new interest in explosive chemical reactions, sudden release of energy in living beings, physical instabilities, and bifurcations in the solutions of differential equations drew the attention of some scholars. New concepts like triggering actions and guiding principles also emerged. Mathematicians, physicists, physiologists, and philosophers were attracted by this kind of phenomena since they raised a question about the actual existence of a strict determinism in science. In 1878 the mathematical physicist Joseph Boussinesq pointed out a structural analogy among physical instabilities, some essential features of living beings, and singular solutions of differential equations. These developments revived long-lasting philosophical debates on the problematic link between deterministic physical laws and free will. We find in Boussinesq an original and almost isolated attempt to merge mathematical, physical, biological, and philosophical issues into a complex intellectual framework. In the last decades, some philosophers of science rediscovered the connection between physical instabilities and determinism, both in the context of chaos theory, and in the debates on the Norton dome. I put forward a consistent historical reconstruction of the main issues and characters involved.