Jeffrey Barrett

There is a long tradition of trying to find a satisfactory interpretation of Everett's relative-state formulation of quantum mechanics. Albert and Loewer recently described two new ways of reading Everett: one we will call the single-mind theory and the other the many-minds theory. I will briefly describe these theories and present some of their merits and problems. Since both are no-collapse theories, a significant merit is that they can take advantage of certain properties of the linear dynamics, which Everett apparently considered to be important, to constrain their statistical laws.

We are concerned here with explaining how successful rule-following behavior might evolve and how an old evolved rule might come to be successfully used in a new context. Such rule-following behavior is illustrated in the transitive judgments of pinyon and scrub-jays (Bond et al., Anim Behav 65:479–487, 2003). We begin by considering how successful transitive rule-following behavior might evolve in the context of Skyrms–Lewis sender–receiver games (Lewis, Convention. Harvard University Press, Cambridge, 1969; Skyrms, Philos Sci 75:489–500, 2006). We then consider two ways that an agent might come to use an old evolved rule in a new context. The first involves the agent evolving successful dispositions for one concrete type of experience, then associating a new type of experience with the old evolved dispositions. The second involves the agent evolving dispositions that represent a general inferential schema, then composing these dispositions with others in a way that allows the agent to make inferences concerning a new concrete type of experience.

We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this situation leaves one no choice but to restrict the use of a certain class of implicational rules including modus ponens.

This article concerns how rule-following behavior might evolve, how an old evolved rule might come to be appropriated to a new context, and how simple concepts might coevolve with rule-following behavior. In particular, we consider how the transitive inferential rule-following behavior exhibited by pinyon and scrub jays might evolve in the context of a variety of the Skyrms-Lewis signaling game, then how such a rule might come to be appropriated to carry out inferences regarding stimuli different from those involved in the original evolution of the rule, and how this appropriation involves a step toward the evolution of basic ordinal concepts.

α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [10]. Idealized computational models for α-recursion analogous to Turing machine models for classical recursion have been proposed and studied [4] and [5] and are applicable in computational approaches to the foundations of logic and mathematics [8]. They also provide a natural setting for modeling extensions of the algorithmic logic described in [1] and [2]. On such models, an α-Turing machine can complete a θ-step computation for any ordinal θ < α. Here we consider constraints on the physical realization of α-Turing machines that arise from the structure of physical spacetime. In particular, we show that an α-Turing machine is realizable in a spacetime constructed from R only if α is countable. While there are spacetimes where uncountable computations are possible and while they may even be empirically distinguishable from a standard spacetime, there is good reason to suppose that such nonstandard spacetimes are nonphysical. We conclude with a suggestion for a revision of Church’s thesis appropriate as an upper bound for physical computation.

There is good reason to suppose that our best physical theories are false: In addition to its own internal problems, the standard formulation of quantum mechanics is logically incompatible with special relativity. There is also good reason to suppose that we have no concrete idea concerning what it might mean to claim that these theories are approximately or vaguely true. I will argue that providing a concrete understanding the approximate or vague truth of our current physical theories is not a task for traditional epistemology; rather, this is only possible in the context of ongoing empirical inquiry. [Note #1].

In this paper I describe the history of the surreal trajectories problem and argue that in fact it is not a problem for Bohm's theory. More specifically, I argue that one can take the particle trajectories predicted by Bohm's theory to be the actual trajectories that particles follow and that there is no reason to suppose that good particle detectors are somehow fooled in the context of the surreal trajectories experiments. Rather than showing that Bohm's theory predicts the wrong particle trajectories or that it somehow prevents one from making reliable measurements, such experiments ultimately reveal the special role played by position and the fundamental incompatibility between Bohm's theory and relativity. They also provide a striking example of the theory-ladenness of observation.

The bare theory is the standard von Neumann·Dirac formulation of quantum mechanics without the collapse postulate but with the eigenvalueeigenstate link. Albert (1992, 1i6-125) presented the bare theory as one way of understanding EverettRi7;s relative-state interpretation. At first glance, it looks as if the bare theory cannot possibly account for our experience. After all, at the end of a measurement an observer will typically be in a superposition of having recorded mutually incompatible results, which on the standard interpretation of states (the interpretation provided by the eigenvalue-eigenstate..

Hugh Everett III proposed his relative-state formulation of pure wave mechanics as a solution to the quantum measurement problem. He sought to address the theory’s determinate record and probability problems by showing that, while counterintuitive, pure wave mechanics was nevertheless empirically faithful and hence empirical acceptable. We will consider what Everett meant by empirical faithfulness. The suggestion will be that empirical faithfulness is well understood as a weak variety of empirical adequacy. The thought is that the very idea of empirical adequacy might be renegotiated in the context of a new physical theory given the theory’s other virtues. Everett’s argument for pure wave mechanics provides a concrete example of such a renegotiation.

Given Hugh Everett III's understanding of the proper cognitive status of physical theories, his relative-state formulation of pure wave mechanics arguably qualifies as an empirically acceptable physical theory. The argument turns on the precise nature of the relationship that Everett requires between the empirical substructure of an empirically faithful physical theory and experience. On this view, Everett provides a weak resolution to both the determinate record and the probability problems encountered by pure wave mechanics, and does so in a way that avoids unnecessary metaphysical complications. Taking Everett's goal to be showing the empirical faithfulness of the relative-state formulation agrees well with his characterization of his project as one of seeking a model for observation in the correlation structure described by pure wave mechanics and seeking a measure of typicality over this empirical substructure that covaries with our empirically warranted expectations. 1 Pure Wave Mechanics and Relative States2 Everett and Frank3 Everett on the Nature of Physical Theories4 Conditions for Empirical Faithfulness5 The Empirical Faithfulness of Pure Wave Mechanics6 Conclusion.

On Bohm's formulation of quantum mechanics particles always have determinate positions and follow continuous trajectories. Bohm's theory, however, requires a postulate that says that particles are initially distributed in a special way: particles are randomly distributed so that the probability of their positions being represented by a point in any regionR in configuration space is equal to the square of the wave-function integrated overR. If the distribution postulate were false, then the theory would generally fail to make the right statistical predictions. Further, if it were false, then there would at least in principle be situations where a particle would approach an eigenstate of having one position but in fact always be somewhere very different. Indeed, we will see how this might happen even if the distribution postulate were true. This will help to show how loose the connection is between the wave-function and the positions of particles in Bohm's theory and what the precise role of the distribution postulate is. Finally, we will briefly consider two attempts to formulate a version of Bohm's theory without the distribution postulate.

Everett proposed resolving the quantum measurement problem by dropping the nonlinear collapse dynamics from quantum mechanics and taking what is left as a complete physical theory. If one takes such a proposal seriously, then the question becomes how much of the predictive and explanatory power of the standard theory can one recover without the collapse postulate and without adding anything else. Quantum mechanics without the collapse postulate has several suggestive properties, which we will consider in some detail. While these properties are not enough to make it acceptable given the usual standards for a satisfactory physical theory, one might want to exploit these properties to cook up a satisfactory no-collapse formulation of quantum mechanics. In considering how this might work, we will see why any no-collapse theory must generally fail to satisfy at least one of two plausible-sounding conditions.

Belief-revision models of knowledge describe how to update one’s degrees of belief associated with hypotheses as one considers new evidence, but they typically do not say how probabilities become associated with meaningful hypotheses in the first place. Here we consider a variety of Skyrms–Lewis signaling game (Lewis in Convention. Harvard University Press, Cambridge, 1969; Skyrms in Signals evolution, learning, & information. Oxford University Press, New York, 2010) where simple descriptive language and predictive practice and associated basic expectations coevolve. Rather than assigning prior probabilities to hypotheses in a fixed language then conditioning on new evidence, the agents begin with no meaningful language or expectations then evolve to have expectations conditional on their descriptions as they evolve to have meaningful descriptions for the purpose of successful prediction. The model, then, provides a simple but concrete example of how the process of evolving a descriptive language suitable for inquiry might also provide agents with conditional expectations that reflect the type and degree of predictive success in fact afforded by their evolved predictive practice. This illustrates one way in which the traditional problem of priors may simply fail to apply to one’s model of evolving inquiry.

Signaling games with reinforcement learning have been used to model the evolution of term languages (Lewis 1969, Convention. Cambridge, MA: Harvard University Press; Skyrms 2006, “Signals” Presidential Address. Philosophy of Science Association for PSA). In this article, syntactic games, extensions of David Lewis’s original sender–receiver game, are used to illustrate how a language that exploits available syntactic structure might evolve to code for states of the world. The evolution of a language occurs in the context of available vocabulary and syntax—the role played by each component is compared in the context of simple reinforcement learning.

Skyrms-Lewis sender-receiver games with invention allow one to model how a simple mathematical language might be invented and become meaningful as its use coevolves with the basic arithmetic competence of primitive mathematical inquirers. Such models provide sufficient conditions for the invention and evolution of a very basic sort of arithmetic language and practice, and, in doing so, provide insight into the nature of a correspondingly basic sort of mathematical knowledge in an evolutionary context. Given traditional philosophical reflections concerning the nature and preconditions of mathematical knowledge, these conditions are strikingly modest.

argued that there are two options for what he called a realistic solution to the quantum measurement problem: (1) select a preferred set of observables for which definite values are assumed to exist, or (2) attempt to assign definite values to all observables simultaneously (1810–1). While conventional wisdom has it that the second option is ruled out by the Kochen-Specker theorem, Vink nevertheless advocated it. Making every physical quantity determinate in quantum mechanics carries with it significant conceptual costs, but it also provides a way of addressing the preferred basis problem that arises if one chooses to pursue the first option. The potential costs and benefits of a formulation of quantum mechanics where every physical quantity is determinate are herein examined. The preferred-basis problem How to solve the preferred-basis problem Relativistic constraints Conclusion.

I argue that a strong mind–body dualism is required of any formulation of quantum mechanics that satisfies a relatively weak set of explanatory constraints. Dropping one or more of these constraints may allow one to avoid the commitment to a mind–body dualism but may also require a commitment to a physical–physical dualism that is at least as objectionable. Ultimately, it is the preferred basis problem that pushes both collapse and no-collapse theories in the direction of a strong dualism in resolving the quantum measurement problem. Addressing this problem illustrates how the construction and evaluation of explanatorily rich physical theories are inextricably tied to the evaluation of traditional philosophical issues.

On Bohm''s formulation of quantum mechanics particles always have determinate positions and follow continuous trajectories. Bohm''s theory, however, requires a postulate that says that particles are initially distributed in a special way: particles are randomly distributed so that the probability of their positions being represented by a point in any regionR in configuration space is equal to the square of the wave-function integrated overR. If the distribution postulate were false, then the theory would generally fail to make the right statistical predictions. Further, if it were false, then there would at least in principle be situations where a particle would approach an eigenstate of having one position but in fact always be somewhere very different. Indeed, we will see how this might happen even if the distribution postulate were true. This will help to show how loose the connection is between the wave-function and the positions of particles in Bohm''s theory and what the precise role of the distribution postulate is. Finally, we will briefly consider two attempts to formulate a version of Bohm''s theory without the distribution postulate.

Quantum mechanics without the collapse postulate, the bare theory, was proposed by Albert (1992) as a way of understanding Everett's relative-state formulation of quantum mechanics. The basic idea is to try to account for an observer's beliefs by appealing to a type of illusion predicted by the bare theory. This paper responds to some recent objections to the bare theory by providing a more detailed description of the sense in which it can and the sense in which it cannot account for our experience.