Gordon Belot

Accelerating Expansion explores some of the philosophical implications of modern cosmology, focused on the significance that the discovery of the accelerating expansion of the Universe has for our understanding of time, geometry, and physics. The appearance of the cosmological constant in the equations of general relativity allows one to model universes in which space has an inherent tendency towards expansion. This constant, introduced by Einstein but subsequently abandoned by him, returned to centre stage with the discovery of the accelerating expansion. This pedagogically-oriented essay begins with a study of the most basic and elegant relativistic world that involves a positive cosmological constant, de Sitter spacetime. It then turns to the relatives of de Sitter spacetime that dominate modern relativistic cosmology. Some of the topics considered include: the nature of time and simultaneity in de Sitter worlds; the sense in which de Sitter spacetime is a powerful dynamical attractor; the limited extent to which observation can give us information about the topology of space in a world undergoing accelerated expansion; and cosmologists' favourite sceptical worry about the reliability of evidence and the possibility of knowledge, the problem of Boltzmann brains.

It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one's credence function, when conditionalized on the chance-making facts, should coincide with the objective chance function. It is a shortcoming of a theory of chance if implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped.

Like Lewis, many philosophers hold reductionist accounts of chance (on which claims about chance are to be understood as claims that certain patterns of events are instantiated) and maintain that rationality requires that credence should defer to chance (in the sense that under certain circumstances one's credence in an event must coincide with the chance of that event). It is a shortcoming of an account of chance if it implies that this norm of rationality is unsatisfiable by computable agents. This shortcoming is more common than one might have hoped.

This paper revisits the bearing of underdetermination arguments on scientific realism. First it argues that underdetermination considerations provide good reason to doubt that science is objective in the strong sense that anyone following the its methods will be led closer and closer to the truth about any given question within the purview of those methods, as more relevant data are considered. Then it argues that scientific realism is difficult to maintain in the absence of this sort of objectivity.

This paper is concerned with learners who aim to learn patterns in infinite binary sequences: shown longer and longer initial segments of a binary sequence, they either attempt to predict whether the next bit will be a 0 or will be a 1 or they issue forecast probabilities for these events. Several variants of this problem are considered. In each case, a no-free-lunch result of the following form is established: the problem of learning is a formidably difficult one, in that no matter what method is pursued, failure is incomparably more common that success; and difficult choices must be faced in choosing a method of learning, since no approach dominates all others in its range of success. In the simplest case, the comparison of the set of situations in which a method fails and the set of situations in which it succeeds is a matter of cardinality (countable vs. uncountable); in other cases, it is a topological matter (meagre vs. co-meagre) or a hybrid computational-topological matter (effectively meagre vs. effectively co-meagre).

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. Further, it is often claimed that the Bayesian framework automatically implements Occam's razor—that conditionalizing on data consistent with both a simple theory and a complex theory more or less inevitably favours the simpler theory. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that cannot be captured within the orthodox Bayesian framework and that the automatic razor does not function for such problems.

A vast amount of ink has been spilled in both the physics and the philosophy literature on the measurement problem in quantum mechanics. Important as it is, this problem is but one aspect of the more general issue of how, if at all, classical properties can emerge from the quantum descriptions of physical systems. In this paper we will study another aspect of the more general issue-the emergence of classical chaos-which has been receiving increasing attention from physicists but which has largely been neglected by philosophers of science.

Physicists who work on canonical quantum gravity will sometimes remark that the general covariance of general relativity is responsible for many of the thorniest technical and conceptual problems in their field.1 In particular, it is sometimes alleged that one can trace to this single source a variety of deep puzzles about the nature of time in quantum gravity, deep disagreements surrounding the notion of ‘observable’ in classical and quantum gravity, and deep questions about the nature of the existence of spacetime in general relativity. Philosophers who think about these things are sometimes skeptical about such claims. We have all learned that Kretschmann was quite correct to urge against Einstein that the “General Theory of Relativity” was no such thing, since any theory could be cast in a generally covariant form, and hence the general covariance of general relativity could not have any physical content, let alone bear the kind of weight that Einstein expected it to.2 Friedman’s assessment is widely accepted: “As Kretschmann first pointed out in 1917, the principle of general covariance has no physical content whatever: it specifies no particular physical theory; rather, it merely expresses our commitment to a certain style of formulating physical theories” (1984, p. 44). Such considerations suggest that general covariance, as a technically crucial but physically contentless feature of general relativity, simply cannot be the source of any significant conceptual or physical problems.3 Physicists are, of course, conscious of the weight of Kretschmann’s points against Einstein. Yet they are considerably more ambivalent than their philosophical colleagues. Consider Kuchaˇr’s conclusion at the end of a discussion of this topic.

Stephen Hawking has argued that universes containing evaporating black holes can evolve from pure initial states to mixed final ones. Such evolution is non-unitary and so contravenes fundamental quantum principles on which Hawking's analysis was based. It disables the retrodiction of the universe's initial state from its final one, and portends the time-asymmetry of quantum gravity. Small wonder that Hawking's paradox has met with considerable resistance. Here we use a simple result for C*-algebras to offer an argument for pure-to-mixed state evolution in black hole evaporation, and review responses to the Hawking paradox with respect to how effectively they rebut this argument.

Substantivalists claim that spacetime enjoys an existence analogous to that of material bodies, while relationalists seek to reduce spacetime to sets of possible spatiotemporal relations. The resulting debate has been central to the philosophy of space and time since the Scientific Revolution. Recently, many philosophers of physics have turned away from the debate, claiming that it is no longer of any relevance to physics. At the same time, there has been renewed interest in the debate among physicists working on quantum gravity, who claim that the conceptual problems which they face are intimately related to interpretative questions concerning general relativity . My goal is to show that the physicists are correct--there is a close relationship between the interpretative issues of classical and quantum gravity. ;In the first part of the dissertation I challenge the received view that substantivalism has a commanding advantage over relationalism on grounds internal to GR. I argue that this view is based on a misconception of the relationships between realism and substantivalism, and between empiricism and relationalism. This has led to a narrow conception of relationalism. Once this is relinquished it can be seen that none of the standard arguments in favor of substantivalism are cogent. ;In the second part of the dissertation, I consider the way in which considerations arising out of quantum gravity bear upon the substantival-relational debate. I develop a framework in which to discuss the interpretative problems of gauge theories and place GR in this context. From this perspective, I provide a taxonomy of interpretative options, and show how the hole argument arises naturally as a consequence of gauge freedom. This means that certain substantivalist interpretations of GR render the theory indeterministic. In the final chapter, I argue that, far from being a drawback, this presents an opportunity for substantivalists. Examples from quantum mechanics, quantum field theory, and quantum gravity, are used to demonstrate that the ambiguities inherent in quantization can lead to an interpretative interplay between theories. In the case of quantum gravity, this means that substantivalism and relationalism suggest, and are suggested by, distinct approaches to quantizing GR.

Under so-called primitive ontology approaches, in fully describing the history of a quantum system, one thereby attributes interesting properties to regions of spacetime. Primitive ontology approaches, which include some varieties of Bohmian mechanics and spontaneous collapse theories, are interesting in part because they hold out the hope that it should not be too difficult to make a connection between models of quantum mechanics and descriptions of histories of ordinary macroscopic bodies. But such approaches are dualistic, positing a quantum state as well as ordinary material degrees of freedom. This paper lays out and compares some options that primitive ontologists have for making sense of the quantum state

Intuitively, a classical field theory is background-in- dependent if the structure required to make sense of its equations is itself subject to dynamical evolution, rather than being imposed ab initio. The aim of this paper is to provide an explication of this intuitive notion. Background-independence is not a not formal property of theories: the question whether a theory is background-independent depends upon how the theory is interpreted. Under the approach proposed here, a theory is fully background-independent relative to an interpretation if each physical possibility corresponds to a distinct spacetime geometry; and it falls short of full background-independence to the extent that this condition fails.

This chapter is concerned with the representation of time and change in classical (i.e., non-quantum) physical theories. One of the main goals of the chapter is to attempt to clarify the nature and scope of the so-called problem of time: a knot of technical and interpretative problems that appear to stand in the way of attempts to quantize general relativity, and which have their roots in the general covariance of that theory. The most natural approach to these questions is via a consideration of more clear cases. So much of the chapter is given over to a discussion of the representation of time and change in other, better understood theories, starting with the most straightforward cases and proceeding through a consideration of cases that lead up to the features of general relativity that are responsible for the problem of time.

This paper is concerned with the relation between two notions: that of two solutions or models of a theory being related by a symmetry of the theory and that of solutions or models being physically equivalent. A number of authors have recently discussed this relation, some taking an optimistic view, on which there is a suitable concept of the symmetry of a theory relative to which these two notions coincide, others taking a pessimistic view, on which there is no such concept. The present paper arrives at a cautiously pessimistic conclusion.

ABSTRACTA popular strategy for understanding the probabilities that arise in physics is to interpret them via reductionist accounts of chance—indeed, it is sometimes claimed that such accounts are uniquely well-suited to make sense of the probabilities in classical statistical mechanics. Here it is argued that reductionist accounts of chance carry a steep but unappreciated cost: when applied to physical theories of the relevant type, they inevitably distort the relations of probability that they take as input.