I argue that the best spacetime setting for Newtonian gravitation (NG) is the curved spacetime setting associated with geometrized Newtonian gravitation (GNG). Appreciation of the ‘Newtonian equivalence principle’ leads us to conclude that the gravitational field in NG itself is a gauge quantity, and that the freely falling frames are naturally identified with inertial frames. In this context, the spacetime structure of NG is represented not by the flat neo-Newtonian connection usually made explicit in formulations, but (...) by the sum of the flat connection and the gravitational field. 1 Introduction2 Newtonian Gravity: The Orthodox Approach3 Newtonian Gravity: Additional Symmetries4 Cosmological Considerations5 A Newtonian Equivalence Principle: Inertial Frames in Newtonian Gravitation6 Theory Equivalence?7 Conclusion. (shrink)

This exploration of the global structure of spacetime within the context of general relativity examines the causal and singular structures of spacetime, revealing some of the curious possibilities that are compatible with the theory, such as `time travel' and `holes' of various types. Investigations into the epistemic and modal structures of spacetime highlight the difficulties in ruling out such possibilities, unlikely as they may seem at first. The upshot seems to be that what counts as a (...) `physically reasonable' spacetime structure in modern physics is far from clear. (shrink)

I discuss several issues related to "classical" spacetime structure. I review Galilean, Newtonian, and Leibnizian spacetimes, and briefly describe more recent developments. The target audience is undergraduates and early graduate students in philosophy; the presentation avoids mathematical formalism as much as possible.

This paper makes an observation about the ``amount of structure'' that different classical and relativistic spacetimes posit. The observation substantiates a suggestion made by Earman and yields a cautionary remark concerning the scope and applicability of structural parsimony principles.

In this essay, I consider the ontological status of spacetime from the points of view of the standard tensor formalism and three alternatives: twistor theory, Einstein algebras, and geometric algebra. I briefly review how classical field theories can be formulated in each of these formalisms, and indicate how this suggests a structural realist interpretation of spacetime.

Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions. The structure of the space of SUSY generators is such that the distinction between internal and spacetime symmetries is blurred. As a result, there are two viable candidates for the correct spacetime setting for a flat supersymmetric field theory---Minkowski spacetime and superspace. an extension of four- dimensional Minkowski spacetime to include (at least) four new dimensions, coordinatised by mathematical objects known as supernumbers. These objects are, (...) in one significant way, quite different from real or complex number---some of them have the property that their order of multiplication makes a difference; in mathematical terms, they are said to have nontrivial commutation properties. This paper argues for two theses: first, that one standard set of arguments, related to universality of symmetry behaviour, that motivate a particular choice of spacetime structure in familiar spacetime theories motivates the choice of superspace as the appropriate spacetime for SUSY field theories. And second, that the metaphysical utility of the concept of spacetime requires more than just the satisfaction of this universality condition; in supersymmetric theories, the spacetime concept is not as useful as in special relativity. (shrink)

I examine the debate between substantivalists and relationalists about the ontological character of spacetime and conclude it is not well posed. I argue that the hole argument does not bear on the debate, because it provides no clear criterion to distinguish the positions. I propose two such precise criteria and construct separate arguments based on each to yield contrary conclusions, one supportive of something like relationalism and the other of something like substantivalism. The lesson is that one must fix (...) an investigative context in order to make such criteria precise, but different investigative contexts yield inconsistent results. I examine questions of existence about spacetime structures other than the spacetime manifold itself to argue that it is more fruitful to focus on pragmatic issues of physicality, a notion that lends itself to several different explications, all of philosophical interest, none privileged a priori over any of the others. I conclude by suggesting an extension of the lessons of my arguments to the broader debate between realists and instrumentalists. 1 Introduction2 The Hole Argument3 Limits of Spacetimes4 Pointless Constructions5 The Debate between Substantivalists and Relationalists6 Existence and Physicality: An Embarassment of Spacetime Structures7 Valedictory Remarks on Realism and Instrumentalism, and the Structure of Our Knowledge of Physics. (shrink)

This paper presents a novel argument against one theoretically attractive form of panpsychism. I argue that “idealist panpsychism” is false since it cannot account for spacetime’s structure. Idealist panpsychists posit that fundamental reality is purely experiential. Moreover, they posit that the consciousness at the fundamental level metaphysically grounds and explains both the facts of physics and the facts of human consciousness. I argue that if idealist panpsychism is true, human consciousness and the consciousness at the fundamental level will (...) have the same metrical structure. However, as I demonstrate, human consciousness does not exhibit the same metrical structure as spacetime. Consequently, the idealist panpsychist faces an explanatory gap between the fundamental consciousness she posits and spacetime. Idealist panpsychism is incompatible with the existence of such an explanatory gap. Thus, idealist panpsychists must close this explanatory gap (which I argue they lack the resources to do), or idealist panpsychism is false. (shrink)

The debate between substantivalists and relationists about spacetime was given a new lease of life approximately twenty years ago, when John Earman and John Norton published an argument for the conclusion that, in the light of general relativity, substantivalism is untenable. Responses to Earman and Norton’s argument generated a proliferation of ‘substantivalisms’, and a debate between them that was, to the ears of at least some, distinctively metaphysical in character.

A categorical ontology of space and time is presented for emergent biosystems, super-complex dynamics, evolution and human consciousness. Relational structures of organisms and the human mind are naturally represented in non-abelian categories and higher dimensional algebra. The ascent of man and other organisms through adaptation, evolution and social co-evolution is viewed in categorical terms as variable biogroupoid representations of evolving species. The unifying theme of local-to-global approaches to organismic development, evolution and human consciousness leads to novel patterns of relations that (...) emerge in super- and ultra- complex systems in terms of colimits of biogroupoids, and more generally, as compositions of local procedures to be defined in terms of locally Lie groupoids. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ may be found with the help of generalized van Kampen theorems in algebraic topology such as the Higher Homotopy van Kampen theorem (HHvKT). Primordial organism structures are predicted from the simplest metabolic-repair systems extended to self-replication through autocatalytic reactions. The intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution is investigated in terms of categories of many-valued, Łukasiewicz–Moisil logic algebras and then compared with those obtained for (non-commutative) quantum logics. The claim is defended in this essay that human consciousness is unique and should be viewed as an ultra-complex, global process of processes. The emergence of consciousness and its existence seem dependent upon an extremely complex structural and functional unit with an asymmetric network topology and connectivities—the human brain—that developed through societal co-evolution, elaborate language/symbolic communication and ‘virtual’, higher dimensional, non-commutative processes involving separate space and time perceptions. Philosophical theories of the mind are approached from the theory of levels and ultra-complexity viewpoints which throw new light on previous representational hypotheses and proposed semantic models in cognitive science. Anticipatory systems and complex causality at the top levels of reality are also discussed in the context of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. The presence of strange attractors in modern society dynamics gives rise to very serious concerns for the future of mankind and the continued persistence of a multi-stable biosphere. A paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics is suggested to unfold now in physics, mathematics, life and cognitive sciences, thus leading to the realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

We give an analysis over a variation of causal sets where the light cone of an event is represented by finitely branching trees with respect to any given arbitrary dynamics. We argue through basic topological properties of Cantor space that under certain assumptions about the universe, spacetime structure and causation, given any event x, the number of all possible future worldlines of x within the many-worlds interpretation is uncountable. However, if all worldlines extending the event x are ‘eventually (...) deterministic’, then the cardinality of the set of future worldlines with respect to x is exactly ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}, i.e., countably infinite. We also observe that if there are countably many future worldlines with respect to x, then at least one of them must be necessarily ‘decidable’ in the sense that there is an algorithm which determines whether or not any given event belongs to that worldline. We then show that if there are only finitely many worldlines in the future of an event x, then they are all decidable. We finally point out the fact that there can be only countably many terminating worldlines. (shrink)

Coordinate-based approaches to physical theories remain standard in mainstream physics but are largely eschewed in foundational discussion in favour of coordinate-free differential-geometric approaches. I defend the conceptual and mathematical legitimacy of the coordinate-based approach for foundational work. In doing so, I provide an account of the Kleinian conception of geometry as a theory of invariance under symmetry groups; I argue that this conception continues to play a very substantial role in contemporary mathematical physics and indeed that supposedly ``coordinate-free'' differential geometry (...) relies centrally on this conception of geometry. I discuss some foundational and pedagogical advantages of the coordinate-based formulation and briefly connect it to some remarks of Norton on the historical development of geometry in physics during the establishment of the general theory of relativity. (shrink)

Many philosophers of physics take the failure of the laws of physics to be invariant under the time reversal transformation to give us good reason to think that spacetime is temporally anisotropic, yet the details of this inference are rarely made explicit. I discuss two reasonable ways of filling in the details of this inference, the first of which utilizes a symmetry principle proposed by John Earman and the second of which utilizes Harvey Brown’s account of spacetime. I (...) contend that neither of the resulting arguments is sound. (shrink)

This essay explores the possibility of constructing a structural realist interpretation of spacetime theories that can resolve the ontological debate between substantivalists and relationists. Drawing on various structuralist approaches in the philosophy of mathematics, as well as on the theoretical complexities of general relativity, our investigation will reveal that a structuralist approach can be beneficial to the spacetime theorist as a means of deflating some of the ontological disputes regarding similarly structured spacetimes.

In this paper we argue that structural explanations are an effective way of explaining well known relativistic phenomena like length contraction and time dilation, and then try to understand how this can be possible by looking at the literature on scientific models. In particular, we ask whether and how a model like that provided by Minkowski spacetime can be said to represent the physical world, in such a way that it can successfully explain physical phenomena structurally. We conclude by (...) claiming that a partial isomorphic approach to scientific representation can supply an answer only if supplemented by a robust injection of pragmatic factors. (shrink)

Debates about the ontological implications of the general theory of relativity have long oscillated between spacetime substantivalism and relationism. I evaluate such debates by claiming that we need a third option, which I refer to as “structural spacetime realism.” Such a tertium quid sides with the relationists in defending the relational nature of the spacetime structure, but joins the substantivalists in arguing that spacetime exists, at least in part, independently of particular physical objects and events, (...) the degree of “independence” being given by the extent to which geometrical laws exist “over and above” physical events exemplifying them. By showing that structural spacetime realism is the natural outcome of a semantic, model-theoretic approach to the nature of scientific theories, I conclude by arguing that the notion of partial isomorphic representation is the most plausible candidate to connect spacetime models with reality. (shrink)

Spacetime substantivalists insist that spatiotemporal points are fundamental entities and thus are ontologically independent from the physical objects occupying these points.This chapter argues that the best metaphysical option for the substantivalist is to adopt a radical version of structural essentialism, according to which participation in a certain relational structure is both a necessary and sufficient condition of identity for the elements characterized by this structure.It comparesthe author’sproposed variant of essentialism with some alternative conceptions, such as Tim Maudlin’s (...) metric essentialism and David Glick’s Minimal Structural Essentialism. It discusses how the author’sRadical Structural Essentialism deals with the challenge of the hole argument as well as some less well-known challenges, such as the problem of contingent counterfactuals regarding the material contents of spacetime, and the ‘mole’ argument. Later the chapter explains why the structure consisting of the metric tensor is a genuinely relational structure, and itdefend the claim that in order to preserve the identity of its elements the actual structure may be merely embeddable in (and not necessarily isomorphic with) an alternative structure. (shrink)

The paper defends two claims; Viewed from the perspective of the substantivalism/relationism debate, structural spacetime realism is a form of relationism; However, if we managed to reinforce Rynasiewicz’s point that the general theory of relativity makes the substantivalism/relationism dispute “outdated”, the re-elaboration of Stein ’s 1967 version of structural spacetime realism to be proposed here proves to be a good, antimetaphysical solution to the problem of the ontological status of spacetime.

Motivated by examples from general relativity and Newtonian gravitation, this essay attempts to distinguish between the dynamical structure associated with a theory in physics, and its kinematical structure. This enables a distinction to be made between a structural realist interpretation of a theory based on its dynamical structure, and a structural realist interpretation of spacetime, as described by a theory, based on its kinematical structure. I offer category-theoretic formulations of dynamical and kinematical structure and (...) indicate the extent to which such formulations deflect recent criticism of the radical ontic structural realist's conception of structure as "relations devoid of relata". (shrink)

The position of moderate ontic structural realism, presenting a version of spacetime structuralism, runs into several problems. One of them is how to discern spacetime points. This was demonstrated in an argument presented by Christian Wüthrich. In defending their position, proponents of moderate ontic structural realism utilize a sort of monistic interpretation of their structuralism. I argue that this interpretation generates further complications. I make a case for a dualistic version of moderate ontic structural realism with respect to (...) the nature of spacetime. I show how this version is less problematic than the other forms of the position discussed. (shrink)

This essay develops a theoretical model of consciousness grounded in the interaction between real and imaginary light cones. Drawing on insights from relativity, quantum mechanics, and phenomenology, the model proposes that consciousness emerges from iterative transitions between actuality and potentiality. Imaginary light cones represent the space of possibilities, while real light cones embody manifest events. Iterative operator dynamics stabilize this exchange, in a manner reminiscent of the Quantum Zeno Effect and neurophysiological gamma oscillations. Philosophically, the model resonates with Aristotle’s actus (...) and potentia, Bergson’s durée, Husserl’s temporal phenomenology, and Jung’s collective unconscious. By linking these traditions with modern physics—particularly Penrose’s twistor theory—it suggests that consciousness is not merely an epiphenomenon of neural complexity but a structural feature of spacetime itself: a continuous mediation between what is and what could be. (shrink)

This paper proposes a functional framework for defining causal order independently of spacetime geometry. By grounding causality in directed variation across a differentiable non-metric structure, we show how causal relations can emerge from asymmetries in a functional field, without reference to temporal or spatial metrics. This model supports a relational ontology and aligns with background-independent approaches in quantum gravity, structural realism, and metaphysics of time.

This paper develops a non-metric model of causal structure based on the directed variation of a scalar field F(x)F. Departing from traditional frameworks that tie causality to spacetime metrics, we propose that causal precedence arises from internal asymmetries in F, independently of any geometric or temporal background. We formalize this idea via acyclic graphs, gradient structures, and threshold actualization, and show how regions of local stabilization in F give rise to emergent metric properties. The model supports a new (...) type of modal logic based on functional flow, with possibility and necessity defined relative to the direction of variation. We discuss the philosophical implications for structural realism, the direction of time, and the ontology of causation, and suggest that this framework offers a viable foundation for causal reasoning in background-independent physical theories such as quantum gravity. (shrink)

This paper offers an alternative view of spacetime different from both substantivalism and relationism. Using basic ideas underlying the fiber bundle formulation of field theories, it illustrates the function of spacetime in individuating local fields. As the system of numerical identities for entities that we can individually refer to, spacetime is an intrinsic, indispensable, and inalienable structure of the physical world with distinct entities.

I argue that the singular structure of spacetime is best conceptualized as an extrinsic, non-local, but localizable, property of spacetime regions (of any size), and that there is no unique, best definition of a singular region of spacetime. There is, however, a schema that any adequate definition instantiates. Moreover, the question of what kind of incomplete curves witness singular structure is intimately connected to the question of what kind of curves represent possible trajectories of observers (...) in "physically reasonable" motion. (shrink)

We propose to model spacetime by a differential space rather than by a differential manifold. A differential space is the pair (M, C), where M is any set, and C a family of real functions on M, satisfying certain axioms; C is called a differential structure of a corresponding differential space. This concept suitably generalizes the manifold concept. We show that C can be chosen in such a way that it contains all information about the causal structure (...) of spacetime. This information can be read out of C with the help of only one postulate, namely that physical signals travel along piecewise smooth curves in (M, C). We effectively construct the Minkowski spacetime, with its cone structure, in this way. Some comments are made. (shrink)

Could spacetime be derived rather than fundamental? The question is pressing because attempts to quantize gravity have led to theories in which (arguably) there are either no, or only extremely thin, spacetime structures. Moreover, recent proposals for the interpretation of quantum mechanics have suggested that 3-dimensional space may be an ‘appearance’ derived from the 3N-dimensional space in which an N-particle wavefunction lives (cross- reference). In fact, I will largely assume a positive answer, and investigate how it could be; (...) in particular, I want to explicate the role of philosophy in producing a satisfactory explanation of spacetime, providing a roadmap for philosophical engagement with quantum gravity. (shrink)

We give a precise and modern mathematical characterization of the Newtonian spacetime structure (ℕ). Our formulation clarifies the concepts of absolute space, Newton's relative spaces, and absolute time. The concept of reference frames (which are “timelike” vector fields on ℕ) plays a fundamental role in our approach, and the classification of all possible reference frames on ℕ is investigated in detail. We succeed in identifying a Lorentzian structure on ℕ and we study the classical electrodynamics of Maxwell (...) and Lorentz relative to this structure, obtaining the important result that there exists only one intrinsic generalization of the Lorentz force law which is compatible with Maxwell equations. This is at variance with other proposed intrinsic generalizations of the Lorentz force law appearing in the literature. We present also a formulation of Newtonian gravitational theory as a curve spacetime theory and discuss its meaning. (shrink)

Spacetime functionalism is the view that spacetime is a functional structure implemented by a more fundamental ontology. Lam and Wüthrich have recently argued that spacetime functionalism helps to solve the epistemological problem of empirical coherence in quantum gravity and suggested that it also (dis)solves the hard problem of spacetime, namely the problem of offering a picture consistent with the emergence of spacetime from a non-spatio-temporal structure. First, I will deny that spacetime functionalism (...) solves the hard problem by showing that it comes in various species, each entailing a different attitude towards, or answer to, the hard problem. Second, I will argue that the existence of an explanatory gap, which grounds the hard problem, has not been correctly taken into account in the literature. (shrink)

I argue that the need to understand spacetime structure as emergent in quantum gravity is less radical and surprising it might appear. A clear understanding of the link between general relativity's geometrical structures and empirical geometry reveals that this empirical geometry is exactly the kind of thing that could be an effective and emergent matter. Furthermore, any theory with torsion will involve an effective geometry, even though these theories look, at first glance, like theories with straightforward spacetime (...) geometry. As it's highly likely that there will be a role for torsion in quantum gravity, it's also highly likely that any theory of quantum gravity will require us to get to grips with emergent spacetime structure. (shrink)

This article proposes a reformulation of the ontological status of the spacetime metric within a stratified framework of physical structure. Rather than treating gμν as a fundamental entity, we argue that it should be interpreted as a projection Πgeo(A) conditionally defined from a more primitive dynamical field A(x) over a differentiable manifold (M, C∞). We examine several physical formalisms, Palatini gravity, causal set theory, and topological field theories, that support this non-metric starting point, and show that general relativity (...) can be viewed as a regime of stabi- lized projection coherence. In this framework, the Einstein field equations become a compatibility condition between co-emergent projections of geometry and matter. We conclude by exploring im- plications for background independence, structural realism, and quantum gravity, where geometry is no longer a foundational backdrop but the codified result of dynamic field stabilization. (shrink)

Several different quantum gravity research programmes suggest, for various reasons, that spacetime is not part of the fundamental ontology of physics. This gives rise to the problem of empirical coherence: if fundamental physical entities do not occupy spacetime or instantiate spatiotemporal properties, how can fundamental theories concerning those entities be justified by observation of spatiotemporally located things like meters, pointers and dials? I frame the problem of empirical coherence in terms of entailment: how could a non-spatiotemporal fundamental theory (...) entail spatiotemporal evidence propositions? Solutions to this puzzle can be classified as realist or antirealist, depending on whether or not they posit a non-fundamental spacetime structure grounded in or caused by the fundamental structure. These approaches place different constraints on our everyday concepts of space and time. Applying lessons from the philosophy of mind, I argue that only realism is both conceptually plausible and suitable for addressing the problem at hand. I suggest a role functionalist version of realism, which is consistent with both grounding and causation, and according to which our everyday concepts reveal something of the true nature of emergent spacetime. (shrink)

According to a number of approaches in theoretical physics, spacetime does not exist fundamentally. Rather, spacetime exists by depending on another, more fundamental, non-spatiotemporal structure. A prevalent opinion in the literature is that this dependence should not be analyzed in terms of composition. We should not say, that is, that spacetime depends on an ontology of non-spatiotemporal entities in virtue of having them as parts. But is that really right? On the contrary, we argue that a (...) mereological approach to dependent spacetime is not only viable, but promises to enhance our understanding of the physical situation. (shrink)

Michel Janssen and Harvey Brown have driven a prominent recent debate concerning the direction of an alleged arrow of explanation between Minkowski spacetime and Lorentz invariance of dynamical laws in special relativity. In this article, I critically assess this controversy with the aim of clarifying the explanatory foundations of the theory. First, I show that two assumptions shared by the parties—that the dispute is independent of issues concerning spacetime ontology, and that there is an urgent need for a (...) constructive interpretation of special relativity—are problematic and negatively affect the debate. Second, I argue that the whole discussion relies on a misleading conception of the link between Minkowski spacetime structure and Lorentz invari-ance, a misconception that in turn sheds more shadows than light on our understand-ing of the explanatory nature and power of Einstein’s theory. I state that the arrow connecting Lorentz invariance and Minkowski spacetime is not explanatory and uni-directional, but analytic and bidirectional, and that this analytic arrow grounds the chronogeometric explanations of physical phenomena that special relativity offers. (shrink)

This book discusses the notion that quantum gravity may represent the "breakdown" of spacetime at extremely high energy scales. If spacetime does not exist at the fundamental level, then it has to be considered "emergent", in other words an effective structure, valid at low energy scales. The author develops a conception of emergence appropriate to effective theories in physics, and shows how it applies (or could apply) in various approaches to quantum gravity, including condensed matter approaches, discrete (...) approaches, and loop quantum gravity. (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.

In prior work, we have argued that spacetime functionalism provides tools for clarifying the conceptual difficulties specifically linked to the emergence of spacetime in certain approaches to quantum gravity. We argue in this article that spacetime functionalism in quantum gravity is radically different from other functionalist approaches that have been suggested in quantum mechanics and general relativity: in contrast to these latter cases, it does not compete with purely interpretative alternatives, but is rather intertwined with the physical (...) theorizing itself at the level of quantum gravity. Spacetime functionalism allows one to articulate a coherent realist perspective in the context of quantum gravity, and to relate it to a straightforward realist understanding of general relativity. (shrink)

This essay will investigate the spatial ontology debate with regards to those quantum gravity theories that posit non-spatiotemporal elements from which spacetime emerges. Whereas substantivalism and relationism both fail to capture the ontology of these non-spatiotemporal theories, such as causal set theory or loop quantum gravity, the largely neglected property theory of space (spacetime) stands out as the best ontological classification, and it also accords with the standard ontological division into substances, properties, and relations. In addition, the property (...) theory will be shown to offer advantages over other alternative approaches, such as structural realism or spacetime functionalism. (shrink)

In this paper we analyse scalar-tensor theories—specific instances of which include mainstream inflation and dark energy models—in light of the spacetime-matter dichotomy. We argue that it is difficult to categorise the scalar fields as either a pure aspect of the spacetime structure or a pure form of matter, by focusing on the Jordan vs Einstein frames of these theories. We present and evaluate various interpretational options available, concluding that the spacetime—matter dichotomy becomes untenable in this context. (...) At the same time, the ontological and conceptual category of spacetime can be decoupled from that of gravity, with the latter remaining viable in the context of scalar-tensor theories. (shrink)

Eleanor Knox has argued that our concept of spacetime applies to whichever structure plays a certain functional role in the laws (the role of determining local inertial structure). I raise two complications for this approach. First, our spacetime concept seems to have the structure of a cluster concept, which means that Knox's inertial criteria for spacetime cannot succeed with complete generality. Second, the notion of metaphysical fundamentality may feature in the spacetime concept, in (...) which case spacetime functionalism may be uninformative in the absence of answers to fundamental metaphysical questions like the substantivalist/relationist debate. (shrink)