The standard theory of computation excludes computations whose completion requires an infinite number of steps. Malament-Hogarth spacetimes admit observers whose pasts contain entire future-directed, timelike half-curves of infinite proper length. We investigate the physical properties of these spacetimes and ask whether they and other spacetimes allow the observer to know the outcome of a computation with infinitely many steps.

We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to (...) the extent of computational limits there. The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second-order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime, which is thus a universal constant of that spacetime. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any spacetime there will be a countable ordinal upper bound, , on the complexity of questions in the Borel hierarchy computable in it. Introduction 1.1 History and preliminaries Hyperarithmetic Computations in MH Spacetimes 2.1 Generalising SADn regions 2.2 The complexity of questions decidable in Kerr spacetimes An Upper Bound on Computational Complexity for Each Spacetime CiteULike Connotea Del.icio.us What's this? (shrink)

This article shows a clear sense in which general relativity allows for a type of ‘machine’ that can bring about a spacetime structure suitable for the implementation of ‘supertasks’. 1Introduction2Preliminaries3Malament–Hogarth Spacetimes4Machines5Malament–Hogarth Machines6Conclusion.

Malament-Hogarth spacetimes are the sort of models within general relativity that seem to allow for the possibility of supertasks. There are various ways in which these spacetimes might be considered physically problematic. Here, we examine these criticisms and investigate the prospect of escaping them.

This paper develops my (BJPS 2009) criticisms of the philosophical significance of a certain sort of infinitary computational process, a hyperloop. I start by considering whether hyperloops suggest that "effectively computable" is vague (in some sense). I then consider and criticise two arguments by Hogarth, who maintains that hyperloops undermine the very idea of effective computability. I conclude that hyperloops, on their own, cannot threaten the notion of an effective procedure.

Positive review of John Earman's *Bangs, Crunches, Shrieks, Whispers*.

In this paper I will argue that Tait’s axiomatic conception of mathematics implies that it is in principle impossible to be justified in believing a mathematical statement without being justified in believing that statement to be provable. I will then show that there are possible courses of experience which would justify acceptance of a mathematical statement without justifying belief that this statement is provable.

A true Turing machine (TM) requires an infinitely long paper tape. Thus a TM can be housed in the infinite world of Newtonian spacetime (the spacetime of common sense), but not necessarily in our world, because our world-at least according to our best spacetime theory, general relativity-may be finite. All the same, one can argue for the "existence" of a TM on the basis that there is no such housing problem in some other relativistic worlds that are similar ("close") to (...) our world. But curiously enough-and this is the main point of this paper-some of these close worlds have a special spacetime structure that allows TMs to perform certain Turing unsolvable tasks. For example, in one kind of spacetime a TM can be used to solve first-order predicate logic and the halting problem. And in a more complicated spacetime, TMs can be used to decide arithmetic. These new computers serve to show that Church's thesis is a thoroughly contingent claim. Moreover, since these new computers share the fundamental properties of a TM in ordinary operation (e.g. intuitive, finitely programmed, limited in computational capability), a computability theory based on these non-Turing computers is no less worthy of investigation than orthodox computability theory. Some ideas about this new mathematical theory are given. (shrink)

In his recent article On Relativity Theory and Openness of the Future (1991), Howard Stein proves not only that one can define an objective becoming relation in Minkowski spacetime, but that there is only one possible definition available if one accepts certain natural assumptions about what it is for becoming to occur and for it to be objective. Stein uses the definition supplied by his proof to refute an argument due to Rietdijk (1966, 1976), Putnam (1967) and Maxwell (1985, 1988) (...) that Minkowski spacetime leaves no room for objective becoming whatsoever. However, Stein's proof does not seem to go far enough. By considering only what events have become from the standpoint of any given event, Stein's uniqueness proof fails from the outset to allow for a more general kind of becoming whereby it is understood to occur from the standpoint of events on the particular worldlines followed by observers. This suggests that there may, after all, be more than one way to define objective becoming in Minkowski spacetime once each observer's worldline is allowed to figure in the definition. This suspicion is further aroused by two recent proposals for objective, worldline-dependent becoming due to Peacock (1992) and Muller (1992) who advocate ways of defining becoming that are not equivalent to the definition Stein's uniqueness proof delivers. Nevertheless, we show that Stein's uniqueness proofcan be extended in a natural way to cover this more general kind of becoming, provided one does not enrich standard Minkowski spacetime by privileging certain sets of worldlines over others in an unwarranted manner. Thus we aim to reinforce Stein's point that standard Minkowski spacetime does make room for objective becoming, but in essentially only one way, despite arguments and proposals to the contrary. (shrink)

This collection of articles on the theme of space and time covers such broad topics as the philosophy of spacetime, spacetime structure, spacetime ontology, the epistemology of geometry, and general relativity.

David Albert claims that classical electromagnetic theory is not time reversal invariant. He acknowledges that all physics books say that it is, but claims they are ``simply wrong" because they rely on an incorrect account of how the time reversal operator acts on magnetic fields. On that account, electric fields are left intact by the operator, but magnetic fields are inverted. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. I argue, to the (...) contrary, that the inversion of magnetic fields makes good sense and is, in fact, forced by elementary geometric considerations. I also suggest a way of thinking about the time reversal invariance of classical electromagnetic theory -- one that makes use of the invariant (four-dimensional) formulation of the theory -- that makes no reference to magnetic fields at all. It is my hope that it will be of interest in its own right, Albert aside. It has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. (The only assumption one needs is temporal orientability.). (shrink)

Presented here is a new result concerning the computational power of so-called SADn computers, a class of Turing-machine-based computers that can perform some non-Turing computable feats by utilising the geometry of a particular kind of general relativistic spacetime. It is shown that SADn can decide n-quantifier arithmetic but not (n+1)-quantifier arithmetic, a result that reveals how neatly the SADn family maps into the Kleene arithmetical hierarchy. Introduction Axiomatising computers The power of SAD computers Remarks regarding the concept of computability.

An interesting difficulty arises if one tries to reconcile Reichenbach's views about "absolute" rotation in general relativity with his commitment to a "causal theory of space-time structure." This difficulty is made precise in the form of a simple theorem about relativistic space-time geometry.

This survey article is divided into two parts. In the first (section 2), I give a brief account of the structure of classical relativity theory. In the second (section 3), I discuss three special topics: (i) the status of the relative simultaneity relation in the context of Minkowski spacetime; (ii) the ``geometrized" version of Newtonian gravitation theory (also known as Newton-Cartan theory); and (iii) the possibility of recovering the global geometric structure of spacetime from its ``causal structure".

Within the framework of general relativity, in some cases at least, it is a delicate and interesting question just what it means to say that an extended body is or is not "rotating". It is so for two reasons. First, one can easily think of different criteria of rotation. Though they agree if the background spacetime structure is sufficiently simple, they do not do so in general. Second, none of the criteria fully answers to our classical intuitions. Each one exhibits (...) some feature or other that violates those intuitions in a significant and interesting way. The principal goal of the paper is to make the second claim precise in the form of a modest no-go theorem. (shrink)

I give two new uniqueness results for the standard relation of simultaneity in the context of metrical time oriented Minkowski spacetime. These results improve on the classic ones due to Malament and Hogarth, for they adopt only minimal uncontroversial assumptions. I conclude addressing whether these results should be taken to definitely refute the general epistemological thesis of conventionalism.

Here we provide a proof that there exist closed timelike curves in Gödel spacetime with total acceleration less than 2π(9 + 6√3)^1/2. This answers a question posed by David Malament.

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin and Malament. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski : a predicate of betwenness and a four (...) place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant, 175–214 1999)—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models \ of our system \ that satisfy second order continuity to the mathematical structure \, called ‘Minkowski spacetime’ in physics textbooks. (shrink)

Here, we show that one may "time travel" in Gödel spacetime with less total acceleration than was previously known. This answers a question posed by Malament.

Malament (Noûs 11:293–300, 1977) proved a certain uniqueness theorem about standard synchrony, also known as Poincaré-Einstein simultaneity, which has generated many commentaries over the years, some of them contradictory. We think that the situation called for some clarification. After reviewing and discussing some of the literature involved, we prove two results which, hopefully, will help clarifying this debate by filling the gap between the uniquess of Malament’s theorem, which allows the observer to use very few tools, and the (...) complete arbitrariness of a time coordinate in full-fledged Relativity theory. In the spirit of Malament’s theorem, and in opposition to most of its commentators, we emphasize explicit definability of simultaneity relations, and give only constructive proofs. We also explore what happens when we reduce to “purely local” data with respect to an observer. (shrink)

Here, we briefly review the notion of observational indistinguishability within the context of classical general relativity. We settle a conjecture given by Malament (1977) concerning the subject and then strengthen the result considerably. The upshot is this: There seems to be a robust sense in which the global structure of every cosmological model is underdetermined.

In a recent paper, Malament (2004) employs a time reversal transformation that differs from the standard one, without explicitly arguing for it. This is a new and important understanding of time reversal that deserves arguing for in its own right. I argue that it improves upon the standard one. Recent discussion has focused on whether velocities should undergo a time reversal operation. I address a prior question: What is the proper notion of time reversal? This is important, for it (...) will affect our conclusion as to whether our best theories are time-reversal symmetric, and hence whether our spacetime is temporally oriented. *Received February 2007; revised March 2008. †To contact the author, please write to: Department of Philosophy, Yale University, P.O. Box 208306, New Haven, CT 06520-8306; e-mail: [email protected]. (shrink)

The claim that distant simultaneity with respect to an inertial observer is conventional arose in the context of a space-and-time rather than a spacetime ontology. Reformulating this problem in terms of a spacetime ontology merely trivializes it. In the context of flat space, flat time, and a linear inertial structure (a purely space-and-time formalism), we prove that the hyperplanes of space for a given inertial observer are determined by a purely spatial criterion that depends for its validity only on the (...) two-way light principle, which is universally regarded as empirically verified. All (empirically determined) “spacetime” entities, such as the conformal structure or light surface equation, are used in a purely mathematical manner that is independent of and hence isneutral with respect to the ontological status that is ascribed to them. In this regard, our criterion is significantly stronger than thespacetime criterion recently advanced by D. Malament, which appeals explicitly to the conformal orthogonality of spacetime vectors and to the invariance of the conformal-orthogonal structure of spacetime under the causal automorphisms of spacetime. Once the hyperplanes of space for a given inertial observer have been determined by our empirical and purely spatial criterion, the following holds: there exists one and only one $\vec \varepsilon $ -synchronization procedure, namely the standard procedure proposed by Einstein, such that the planes of common time are thesame as the nonconventional hyperplanes of space for the inertial observer. It follows that our criterion provides an empirical even if indirect method for determining that the one-way speed of light is the same as the average two-way speed of light. In addition, two inertial observers that are not at rest with respect to each other necessarily havedifferent hyperplanes of space, and consequently their respective spatial views cannot be encompassed in a single three-dimensional space. Hence, our purely spatial criterion provides an empirical motivation for adopting the more comprehensive spacetime ontology. (shrink)

Hilary Putnam argued that the special theory of relativity shows that there can be no temporal becoming. Howard Stein replied by defining a becoming relation in Minkowski spacetime. Clifton and Hogarth extended and sharpened Stein’s results. Game over? To the contrary, it has been argued that the Stein-Clifton-Hogarth theorems actually support Putnam’s contention, in that if an apparently minimal condition is put on the becoming relation, then these theorems entail that the becoming relation must be the universal relation. (...) I recount this dialectic in some detail and then try to define and defend a becoming relation based on a present that does indeed consist of more than one point or event but still satisfies the sort of objectivity requirements that Stein-Clifton-Hogarth require of a becoming relation. This present is not a global hyperplane or surface, however; it is a local structure. I close with some methodological remarks about the relation between the present and the real and about the importance of the specious or psychological present. (shrink)

It has been suggested by Clark Glymour that the spatio-temporal structure of the universe might be underdetermined by all observational data that could ever, theoretically, be gathered. It is possible for two spacetimes to be observationally indistinguishable (OI) yet topologically distinct. David Malament extended the argument for the underdetermination of spacetime structure by showing that under quite general conditions (such as the absence of any closed timelike curves) a spacetime will always have an OI counterpart (at least in (...) weak sense). Because the plight of the cosmologist seemed to be so discouraging in this regard, Malament considered the relationship between global properties and OI spacetimes. This information is helpful to the cosmologist. It allows, in principle, one to reject some spacetime models based on observational evidence. In this paper, I consider the relationship between variants of geodesic incompleteness and different senses (some old and some new) of OI. In light of the findings, it seems that (for the most part) the predicament of the cosmologist is not good. Quite generally, versions of geodesic incompleteness are not conserved even under the strongest formulations of OI. (shrink)

In Philip K. Dick’s Counter-Clock World the direction of time flips in 1986, putting the Earth into what its inhabitants call the ‘Hogarth Phase’. Named after the scientist who predicted that ‘time’s arrow' would change direction, the Hogarth Phase is a period in which entropy decreases instead of increases. During this time the dead call from their graves to be excavated, people clean their lungs by ‘smoking’ stubs that grow into mature cigarettes, coffee separates from cream, and so (...) on. Although such reversals may be familiar from works of fiction, we are utterly unfamiliar with them in experience. The processes of nature behave in a temporally asymmetric manner. Once the cream mixes with the coffee, it stays that way, never to return to its original separated state. Neither cigarettes nor people ever fully reconstitute themselves. This kind of asymmetric behaviour is ubiquitous in thermodynamic, radiative, and quantum mechanical phenomena. But we also find our common-sense impression ofthe world painted with temporally asymmetric concepts. Time feels like it is ‘f|owing’ forward and causation appears to be an asymmetric relation. These phenomena and impressions stand in sharp contrast with the world as perceived through eyes tutored by modem science. Spacetime, rather than.. (shrink)

Book Review for Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, La Salle, IL: Open Court, 2002. Edited by David Malament. This volume includes thirteen original essays by Howard Stein, spanning a range of topics that Stein has written about with characteristic passion and insight. This review focuses on the essays devoted to history and philosophy of physics.

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

Philosophers of science have written at great length about the geometric structure of physical space. But they have devoted their attention primarily to the question of the epistemic status of our attributions of geometric structure. They have debated whether our attributions are a priori truths, empirical discoveries, or, in a special sense, matters of stipulation or convention. lt is the goal of this paper to explore a quite different issue the role played by assumptions of spatial geometry within physical theory, (...) specifically within Newtonian gravitational theory. (shrink)

We propose an "explanation scheme" for why the Gibbs phase average technique in classical equilibrium statistical mechanics works. Our account emphasizes the importance of the Khinchin-Lanford dispersion theorems. We suggest that ergodicity does play a role, but not the one usually assigned to it.

John Norton has recently argued that Newtonian gravitation theory (at least as applied to cosmological contexts where one envisions the possibility of a homogeneous mass distribution throughout all of space) is inconsistent. I am not convinced. Traditional formulations of the theory may seem to break down in cases of the sort Norton considers. But the difficulties they face are only apparent. They are artifacts of the formulations themselves, and disappear if one passes to the so-called "geometrized" formulation of the theory.

It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein's equation (or of the conservation principle that is, itself, a consequence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein's equation (or the conservation (...) principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). (shrink)

Itamar Pitowsky's book, published in the Springer-Verlag Lecture Notes in Physics series, brings together several extremely interesting component investigations concerning the foundations of quantum mechanics. All deal with issues of probability including, in one case, the relation of probability to logic. It is a significant contribution, offering both new, nontrivial mathematical results, and provocative philosophical remarks about their significance.

The paper first tries to explain how the possibility of "time travel" arises in the Godel universe. It then goes on to discuss a technical problem conerning minimal acceleration requirements for time travel. A theorem is stated and a conjecture posed. If the latter is correct, time travel can be ruled out as a practical possibility in the Godel universe.

In my contribution to the Symposium ("On the Vagaries of Determinism and Indeterminism"), I will identify several issues that arise in trying to decide whether Newtonian particle mechanics qualifies as a deterministic theory. I'll also give a mini-tutorial on the geometry and dynamical properties of Norton's dome surface. The goal is to better understand how his example works, and better appreciate just how wonderfully strange it is.

In this book, 13 leading philosophers of science focus on the work of Professor Howard Stein, best known for his study of the intimate connection between ...