All physical theories, from classical Newtonian mechanics to relativistic quantum field theory, entail propositions concerning the geometric structure of spacetime. To give an example, the general theory of relativity entails that spacetime is curved, smooth, and four-dimensional. In this dissertation, I take the structural commitments of our theories seriously and ask: how is such structure instantiated in the physical world? Mathematically, a property like 'being curved' is perfectly well-defined insofar as we know what it means for a mathematical space to (...) be curved. But what could it mean to say that the physical world is curved? Call this the problem of physical geometry. The problem of physical geometry is a plea for foundations--a request for fundamental truth conditions for physical-geometric propositions. My chief claim is that only a substantival theory of spacetime--a theory according to which spacetime is an entity in its own right, existing over and above the material content of the world--can supply the necessary truth conditions. (shrink)

This essay explores the use of platonist and nominalist concepts, derived from the philosophy of mathematics and metaphysics, as a means of elucidating the debate on spacetime ontology and the spatial structures endorsed by scientific realists. Although the disputes associated with platonism and nominalism often mirror the complexities involved with substantivalism and relationism, it will be argued that a more refined three-part distinction among platonist/nominalist categories can nonetheless provide unique insights into the core assumptions that underlie spatial ontologies, but it (...) also assists in critiquing alternative uses of nominalism, platonism, and both ontic and epistemic structural realism. (shrink)

Mundy (1983) presented the formal apparatus of certain relationist theories of space and space-time taking quantitative relations as primitive. The present paper discusses the philosophical and physical interpretation of such theories, and replies to some objections to such theories and to relationism in general raised in Field (1985). Under an appropriate second-order naturalistic Platonist interpretation of the formalism, quantitative relationist theories are seen to be entirely comparable to spatialist ones in respect of the issues raised by Field. Moreover, it appears (...) that even if accepted as sound, Field's general line of criticism would not diminish the significance of relationism for philosophy of science, since this derives primarily from its connection to physical rather than to mathematical or philosophical ontology. (shrink)

The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received adequate justification: the (...) assumption that a proposition invariant under the appropriate group is therefore meaningful, and the assumption that representations should be unique up to a transformation of the appropriate group. A general theory of representational meaningfulness is offered, based on a semantic and syntactic analysis of representational propositions. Two neglected features of representational propositions are formalized and made use of: (a) that such propositions are induced by more general propositions defined for other structures than the one being represented, and (b) that the true purpose of representation is the application of the theory of the representing system to the represented system. On the basis of these developments, justifications are offered for the two problematic assumptions made by the existing theories. (shrink)

Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as involving (...) arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement. (shrink)

Relationist theories of space or space-time based on embedding of a physical relational system A into a corresponding geometrical system B raise problems associated with the degree of uniqueness of the embedding. Such uniqueness problems are familiar in the representational theory of measurement (RTM), and are dealt with by imposing a condition of uniqueness of embeddings up to composition with an "admissible transformation" of the space B. Friedman (1983) presents an alternative treatment of the uniqueness problem for embedding relationist theories, (...) developed independently of RTM. Friedman's approach differs from that of RTM in securing uniqueness by adding new primitives to the physical system A in contrast to the RTM approach which adds new axioms. Friedman's proposal has recently been developed and defended by Catton and Solomon (1988). This method of solving the uniqueness problem is here argued to be substantially inferior to the RTM method, both in practice and in principle. In practice we find that in none of the concrete examples offered to illustrate the method is the uniqueness problem actually solved in general. Moreover we find that in the most interesting case (addition to the system A of a finite number of relations of finite degree) the method is in principle incapable of success for mathematical reasons. In addition to these technical difficulties there are compelling methodological reasons for preferring the RTM method to the method of adding primitives. (shrink)

Relative motion of particles is examined in the context of relational space-time. It is shown that de Broglie waves may be derived as a representation of the coordinate maps between the rest-frames of these particles. Energy and momentum are not absolute characteristics of these particles, they are understood as parameters of the coordinate maps between their rest-frames. It is also demonstrated the position of a particle is not an absolute, it is contingent on the frame of reference used to observe (...) the particle. (shrink)

Think of “locations” very abstractly, as positions in a space, any space. Temporal locations are positions in time; spatial locations are positions in (physical) space; particulars are locations in quality space. Should we reify locations? Are locations entities? Spatiotemporal relation- alists say there are no such things as spatiotemporal locations; the fundamental spatial and temporal facts involve no locations as objects, only the instantiation of spatial and temporal relations. The denial of locations in quality space is the bundle theory, according (...) to which particulars do not exist; facts apparently about particulars really concern relations between universals. A “space”, in our abstract sense, consists of a set of objects, together with properties and relations defined on those objects. The objects are the locations of the space, and the distribution of the properties and relations over the locations defines the space’s structure. All spaces are thus quality spaces; when the relations are thought of as spatiotemporal then the space is also a spatiotemporal space. By not reifying locations one denies that these abstract spaces isomorphically represent the real world. The real world does in some sense have a structure that can be non-isomorphically represented by a space (or, more likely, a class of spaces), but the locations in those spaces do not correspond to anything real. We will examine modal considerations on reifying locations. Denying the existence of spatiotemporal locations excludes certain possibilities for spatiotemporal reality. Denying the existence of qualitative locations excludes certain possibilities for qualitative space. In each case the excluded possibilities are pre-analytically possible. Some of the possibilities can be reinstated by modifying the locationless theories, but at the cost of an unattractive holism. Do these modal considerations mandate postulating locations? That depends on whether modal intuition can teach us about the actual world.. (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)

In what sense does empty space feature in visual experience? In the first part of this essay I sketch a view advanced by Soteriou and Richardson on which one's visual awareness of empty space is explained by appeal to ‘structural’ features of the phenomenology of visual experience, in particular the phenomenology of experiencing one's visual field as bounded. I suggest that although this ‘structuralist’ view is silent on whether empty space has a phenomenal appearance, the very appeal to structural features (...) seems a natural foil to some such thought. In the second part, I outline a view on which it can be granted that empty space does, after all, have a phenomenal appearance and, so, is not best cast as a species of absence perception as the structuralist avows, at least on a certain construal of ‘absence perception’. After O'Shaughnessy , I contend that although some privations have a phenomenal appearance, ‘absence perception’ should single out the putative perception of phenomena that lack phenomenal reality. I finally consider how the descriptive phenomenology that the structuralist points to, and which I urge should be embraced, can nonetheless be reconciled with the view I spell out. (shrink)

I will discuss only one of the several entwined strands of the philosophy of space and time, the question of the relation between the nature of motion and the geometrical structure of the world.1 This topic has many of the virtues of the best philosophy of science. It is of long-standing philosophical interest and has a rich history of connections to problems of physics. It has loomed large in discussions of space and time among contemporary philosophers of science. Furthermore, there (...) is, I think, widespread agreement that recent insights here have lead to a genuine deepening of our understanding. (shrink)

Substantivalists believe that spacetime and its parts are fundamental constituents of reality. Relationalists deny this, claiming that spacetime enjoys only a derivative existence. I begin by describing how the Galilean symmetries of Newtonian physics tell against both Newton's brand of substantivalism and the most obvious relationalist alternative. I then review the obvious substantivalist response to the problem, which is to ditch substantival space for substantival spacetime. The resulting position has many affinities with what are arguably the most natural interpretations of (...) special and general relativity. I move on to consider and reject two recent antisubstantivalist lines of thought. The interim conclusion is that the best argument for relationalism is an appeal to Ockham's razor. However, for this to be successful there must be genuine relationalist theories that share the theoretical virtues of their substantivalist rivals but without the additional ontological commitment. The bulk of the paper is therefore an investigation of various concrete relationalist proposals. I distinguish three options for the relationalist in the face of the success of Galilean invariant physics and trace how these generalise to relativistic physics. One of the options is particularly promising but, since its basic objects end up being spacetime points, this does not help the prospects of relationalism as traditionally conceived. I end with some reflections on the fate of substantivalism in the aftermath of the Hole Argument, concluding that we have as yet to be given good reasons to abandon the natural, substantivalist interpretation of current physics. (shrink)

We should see the debate over the existence of spacetime as a debate about the fundamentality of spatiotemporal structure to the physical world. This is a non-traditional conception of the debate, which captures the spirit of the traditional one. At the same time, it clarifies the point of contention between opposing views and offsets worries that the dispute is stagnant or non-substantive. It also unearths a novel argument for substantivalism, given current physics. Even so, that conclusion can be overridden by (...) future physics. I conclude that this debate is a substantive one, which the substantivalist is currently winning. (shrink)

When distinguishing absolute, true, and mathematical time from relative, apparent, and common time, Newton wrote: “absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly” [Newton 2004b: 64]. Newton thought that the temporal metric is intrinsic. Many philosophers have argued—for empiricist reasons or otherwise—that Newton was wrong about the nature of time. They think that the flow of time does involve “reference to something external.” They think that the temporal (...) metric is extrinsic. Among others, Mach, Poincaré, and Grünbaum seem to accept this view.1 And these are not the only two views available. Perhaps both Newton and his opponents are wrong and there is no temporal metric at all. (shrink)