Arguing that the universe is absolutely directioned and that there exist spatial (directional) relations that Leibniz overlooked, H. Scott Hestevold formulates a new relationalist theory of space, exploring its implications for the Special Composition Question, reductivism regarding boundaries and holes, and the nature of spacetime.

In this paper I offer a fresh interpretation of Leibniz’s theory of space, in which I explain the connection of his relational theory to both his mathematical theory of analysis situs and his theory of substance. I argue that the elements of his mature theory are not bare bodies (as on a standard relationalist view) nor bare points (as on an absolutist view), but situations. Regarded as an accident of an individual body, a situation is the complex of its (...) angles and distances to other co-existing bodies, founded in the representation or state of the substance or substances contained in the body. The complex of all such mutually compatible situations of co-existing bodies constitutes an order of situations, or instantaneous space. Because these relations of situation change from one instant to another, space is an accidental whole that is continuously changing and becoming something different, and therefore a phenomenon. As Leibniz explains to Clarke, it can be represented mathematically by supposing some set of existents hypothetically (and counterfactually) to remain in a fixed mutual relation of situation, and gauging all subsequent situations in terms of transformations with respect to this initial set. Space conceived in terms of such allowable transformations is the subject of Analysis Situs. Finally, insofar as space is conceived in abstraction from any bodies that might individuate the situations, it encompasses all possible relations of situation. This abstract space, the order of all possible situations, is an abstract entity, and therefore ideal. (shrink)

The present paper formulates a reconstruction of Leibniz’s theory of space and time from a metaphysical point of view. Leibniz’s theory of space-time is considered a prototype of an “unrealist,” “nonsubstantivalist” and “relationalist” theory of space-time. These attributions to Leibniz’s view are, at face value, well-founded in his corpus.

Cet article est un nouvel examen des objections soulevées par Leibniz dans la controverse avec Clarke contre la théorie absolutiste de l'espace et du temps. Or la plupart de ces objections sont fondées sur le principe de raison suffisante; mais Leibniz utilise aussi le principe de l'identité des indiscernables, qu'il prétend déduire du principe de raison suffisante . Ce qui m'intéresse c'est que Leibniz présente parfois deux versions de la même objection: l'une reposant uniquement sur le principe de raison suffisante, (...) l'autre uniquement sur le principe de l'identité des indiscernables. Se pose alors la question suivante: Leibniz aurait-il pu soulever toutes les objections qu'il voulait contre la théorie absolutiste, en utilisant seulement le principe de l'identité des indiscernables? Il me semble qu'il aurait certainement pu le faire. Et de plus, je crois que si Leibniz avait suivi cette voie, son raisonnement contre la théorie absolutiste aurait été beaucoup plus convaincant. Ainsi donc, je me propose de mener l'attaque de Leibniz contre la théorie absolutiste, en isolant, reformulant, et exposant systématiquement sept arguments dont il se servait ou dont il aurait pu se servir, qui sont tous fondés sur le principe de l'identité des indiscernables. (shrink)

The physicist's conception of space-time underwent two major upheavals thanks to the general theory of relativity and quantum mechanics. Both theories play a fundamental role in describing the same natural world, although at different scales. However, the inconsistency between them emerged clearly as the limitation of twentieth-century physics, so a more complete description of nature must encompass general relativity and quantum mechanics as well. The problem is a theorists' problem par excellence. Experiment provide little guide, and the inconsistency (...) mentioned above is an important problem which clearly illustrates the intermingling of philosophical, mathematical, and physical thought. In fact, in order to unify general relativity with quantum field theory, it seems necessary to invent a new mathematical framework which will generalise Riemannian geometry and therefore our present conception of space and space-time. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new kind of geometry may enter physics, and space-time itself can be reinterpreted as an approximate, derived concept. The main purpose of this article is to show the great significance of space-time geometry in predetermining the laws which are supposed to govern the behaviour of matter, and further to support the thesis that matter itself can be built from geometry, in the sense that particles of matter as well as the other forces of nature emerges in the same way that gravity emerges from geometry. Scientific research is not a process of steady accumulation of absolute truths, which has culminated in present theories, but rather a much more dynamic kind of process in which there are no final theoretical concepts valid in unlimited domains. (David Bohm). (shrink)

It is well known that a central issue in the famous debate between Gottfried Wilhelm Leibniz and Samuel Clarke is the nature of space. They disagreed on the ontological status of space rather than on its geometrical or physical structure. Closely related is the disagreement on the existence of vacuums in nature: while Leibniz denies it, Clarke asserts it. In this paper, I shall focus on Leibniz's position in this debate. In part one, I shall reconstruct the theory (...) of physical space which Leibniz presents in his letters to Clarke. This theory differs from Leibniz's ultimate metaphysics of space, but it is particularly interesting for systematic reasons, and it also gave rise to a lively discussion in modern philosophy of science. In part two, I shall examine whether the existence of vacuums is ruled out by that theory of space, as Leibniz seems to imply in one of his letters. I shall confirm the result of E. J. Khamara that Leibniz's theory of space rules out the existence of a certain kind of vacuum, namely extramundane vacuums, although it does not rule out vacuums within the world. (shrink)

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous (...) lattices and “pointless” topology.

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Hegel’s Philosophy of Nature begins with the concepts of space and time, and all of the concepts and phenomena that follow in the text are spatio-temporal. But the actual content of Hegel’s theory of space and time has remained obscure despite two centuries of interpretation, and obscure for at least two reasons. First, it is unclear how Hegel’s theory relates to those of his immediate predecessors (Newton, Leibniz, Kant) as well as other theories in the history of (...) philosophy. Second, the theoretical function to be played by this theory of space and time has remained unclear. In this paper, we clarify the first issue by interpreting Hegel’s theory as a theory of space-time, and we clarify the second issue by interpreting that theory as securing the theoretical possibility of motion. Of course, it will not be the same theory of space-time as the one that proceeds from modern relativistic physics; but designating Hegel’s theory as a theory of space-time helps to mark it out from its most prominent predecessors and helps to make sense of some of its most distinctive features—particularly Hegel’s insistence that time has three dimensions just as space does. This latter feature—the three-dimensionality of time—serves to distinguish Hegel’s view both from the classical mechanics of Newton and Kant (in which time does not interact with the spatial directions) and from contemporary relativistic physics (in which there is such interaction but time is only a single dimension added to the spatial dimensions). In Hegelian space-time, three-dimensional time is the flip side of three-dimensional space—it is this conception that makes Hegel’s view so distinctive. (shrink)

In a recent paper on Heidegger, Frederick Olafson attacks Hubert Dreyfus for prioritizing our “social” existence (under the notion of das Man) over the individual. In a reply, Taylor Carman, defending Dreyfus, criticizes Olafson for his “subjectivist” notion of Dasein. This paper pursues the implication of this disagreement in the context of Heidegger’s theory of space. Dreyfus’ discussion of Heideggerian spatiality nicely displays the tension between the “public” vs. “individual” domains of being, and consistent with his overall approach, Dreyfus (...) claims that public space should be prioritized. Dreyfus concludes, however, that Heidegger is confused and prioritizes individual space. This paper argues that the categories of “public” and “individual” are inappropriate for analyzing Heidegger’s sense of the shared and personal characters of space. Heidegger’s “indexical” theory of space in fact saves both of these domains without raising the question of priority and without presupposing a “subjectivist” Dasein. On this reading, Olafson’s “individualized” account of presence does not commit him to subjectivism. The confusion Dreyfus attributes to Heidegger is cleared up by an indexical account of spatiality in Heidegger’s text. (shrink)

Jennifer Trusted's new book argues that metaphysical beliefs are essential for scientific inquiry. The theories, presuppositions and beliefs that neither science nor everyday experience can justify are the realm of metaphysics, literally `beyond physics'. These basic beliefs form a framework for our activities and can be discovered in science, common sense and religion. By examining the history of science from the eleventh century to the present, this book shows how religious and mystical beliefs, as well as philosophical speculation have (...) had a considerable role in motivating scientists and inspiring scientific inquiry. _Physics and Metaphysics_ presupposes no technical knowledge of either philosophy or science. It is an ideal introduction to science and the important forces that have shaped its history and ideas. (shrink)

Helmholtz's theory of space had significant impact on Schlick's early ?critical realist? point of view. However, it will be argued in this paper that Schlick's appropriation of Helmholtz's ideas eventually lead to a rather radical transformation of the original Helmholtzian position.

Hume's Treatise arguments concerning space, time, and geometry, especially ones involving his denial of infinite divisibility; have suffered harsh criticism. I show that in the section "Of the ideas of space and time," Hume gives important characterizations of his skeptical approach, in some respects Pyrrhonian, that will be developed in the rest of the Treatise. When that approach is better understood, the force of Hume's arguments can be appreciated, and the influential criticisms of them can be seen to (...) miss the mark. (shrink)

This paper discusses the theory of knowledge based on the idea of dynamical space. The goal of this effort is to comprehend the knowledge that remains beyond the human domain, e.g., of the artificial cognitive systems. This theory occurs in two versions, weak and strong. The weak version is limited to knowledge in which retention and articulation are performed through the discourse. The strong version is general and is not limited in any way. In the weak version, knowledge is (...) represented by the trajectories of discourses in time, in a dynamical space called the discursive space, which has an arbitrary number of dimensions. Given space is used to represent a given part of knowledge. A manifold is introduced to represent knowledge with a wider scope (all knowledge). The strong version is an extrapolation of the weak version to cover all forms of knowledge, not necessarily human or manifesting in language. The use of dynamical space construction allows one to formalize knowledge as such. Such an effort requires us to initially consider knowledge as mainly a social and linguistic phenomenon, which also could be presented as a result of the evolution of the understanding of knowledge that took place in the 20th century. (shrink)

The primary objective of this study is to provide a description of the major ideas about void space within and beyond the world that were formulated between the fourteenth and early eighteenth centuries. The second part of the book - on infinite, extracosmic void space - is of special significance. The significance of Professor Grant's account is twofold: it provides the first comprehensive and detailed description of the scholastic Aristotelian arguments for and against the existence of void (...) class='Hi'>space; and it presents (again for the first time) an analysis of the possible influence of scholastic ideas and arguments on the interpretations of space proposed by the nonscholastic authors who made the Scientific Revolution possible. The concluding chapter of the book is unique in not only describing the conceptualizations of space proposed by the makers of the Scientific Revolution, but in assessing the role of readily available scholastic ideas on the conception of space adopted for the Newtonian world. (shrink)

In the usual description of space and time, particles are represented by continuous world lines. We replace these world lines by discrete rows of points, obtaining a locally finite, partially ordered set. The “distances” between points along these discrete world lines, and also the “distances” between different world lines, are measured not simply as the distances within the space-time manifold in which the partially ordered set happens to be embedded, but rather in terms of the partially ordered set (...) itself. It is found that variations in the density of points of the partially ordered set produce corresponding variations in the scale of “distances” within the set. These variations produce a gravitational effect which is similar to that of the general theory of relativity. (shrink)

Although Descartes’ natural philosophy marked an important advance in the development of modern science, many of his specific concepts of science have been largely discarded, and consequently neglected, since their introduction in the seventeenth century. Many critics over the years, such as Newton (in his early paper De gravitatione), have presented a series of apparently devastating arguments against Descartes' theory of space and motion; a generally negative historical verdict which, moreover, most contemporary scholars accept. Nevertheless, it is also true (...) that most historians and philosophers of science have not sufficiently investigated the viability and effectiveness of the case against Cartesian space and motion. This book explores the success of the arguments against Descartes’ theory of space and motion by determining if it is possible to formulate a Cartesian theory of science that can avoid the problems raised by Newton, as well as by a host of other past and contemporary philosophers either sympathetic or hostile to the overall design of Cartesian science. In particular, if Descartes' theory of motion can be reconciled with his hypotheses on dynamics, then a Cartesian can successfully resolve the difficulties imposed by his critics. This book comprises, therefore, a sustained attempt to construct a consistent “Cartesian” spacetime theory: i.e., a theory of space and time that incorporates Descartes’ various theories of physics and nature without falling into the kinds of problems and contradictions as charged by, for example, Newton. (shrink)

This book, explores the conceptual foundations of Einstein's theory of relativity: the fascinating, yet tangled, web of philosophical, mathematical, and physical ideas that is the source of the theory's enduring philosophical interest. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of (...) the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Newly updated study surveys concept of space from standpoint of historical development. Space in antiquity, Judeo-Christian ideas about space, Newton’s concept of absolute space, space from 18th century to present. Extensive new chapter (6) reviews changes in philosophy of space since publication of second edition (1969). Numerous original quotations and bibliographical references. "...admirably compact and swiftly paced style."—Philosophy of Science. Foreword by Albert Einstein. Bibliography.

In this paper I attempt to trace certain aspects of the evolution of theories of space-time and mechanics as revealed by a brief comparative study of Newtonian theory, Robb's theory, general relativity, and Milne's kinematical relativity. The first object is to emphasise how each theory leaves us in a position in which the succeeding one appears as a perfectly natural next step in the development of ideas. The second object is to show how, in spite of superficial differences (...) in character, the theories all necessarily possess the same general structure constituted by the presence of hypotheses, from which certain general mathematical relations are deduced, which in their turn are used to predict relations between observable quantities. (shrink)

Since antiquity, natural philosophers have struggled to comprehend the nature of three tightly interconnected concepts: space, time, and motion. A proper understanding of motion, in particular, has been seen to be crucial for deciding questions about the natures of space and time, and their interconnections. Since the time of Newton and Leibniz, philosophers’ struggles to comprehend these concepts have often appeared to take the form of a dispute between absolute conceptions of space, time and motion, and relational (...) conceptions. This article guides the reader through some of the history of these philosophical struggles. Rather than taking sides in the (alleged) ongoing debates, or reproducing the standard dialectic recounted in most introductory texts, we have chosen to scrutinize carefully the history of the thinking of the canonical participants in these debates — principally Descartes, Newton, Leibniz, Mach and Einstein. Readers interested in following up either the historical questions or current debates about the natures of space, time and motion will find ample links and references scattered through the discussion and in the Other Internet Resources section below. (shrink)

Regions-based theories of space aim—among others—to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed, one should show that they are points of a geometrical or a topological space constructed in a specific way. This paper intends to show how the development of mathematical tools allows showing that Whitehead’s method of extensive abstraction provides a construction of objects (...) that are fundamental building blocks of specific topological spaces. (shrink)

In considering Kant’s response to the Leibnizian theory of space in the Transcendental Aesthetic, scholars have overwhelmingly emphasized Kant’s response to Leibniz’s relationalism. They have largely missed the metaphysically realistic aspects of Leibniz’s theory with which Kant is primarily concerned. As such, scholars have failed to appreciate the threat Leibniz’s theory poses to Kant’s idealism, a point made publicly as early as 1786 by H. A. Pistorius. I argue that the Aesthetic does indeed contain a compelling argument against the (...) Leibnizian theory of space. This argument crucially depends upon Kant’s claim that space is monistic: that the whole of space is prior to the parts of space and, as such, spatial representations are not decomposable. (shrink)

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed fields (...) is the model-completion of the theory of vector spaces. (shrink)

For much of this century it was widely assumed in philosophical circles that the relational doctrine of space, time, and motion had finally been established beyond the point of reasonable controversy. In large part this was due to a widespread perception that the theory of relativity is itself a relational theory. Indeed, some of Einstein’s own pronouncements foster this impression. For example, in his definitive formulation of general relativity of 1916, he argued the need for a generalization of the (...) principle of special relativity in order to eliminate the “factitious” causal role he saw inertial frames playing in classical mechanics and special relativity. ‘The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion’. Einstein claimed this requirement is fulfilled by satisfaction of the Principle of General Covariance—that the equations expressing the laws of nature remain valid under arbitrary coordinate substitutions. This, he says, ‘takes away from space and time the last remnant of physical objectivity.’. (shrink)

Some Philosophical Prehistory of General Relativity As history, my remarks will form rather a medley. If they can claim any sort of unity (apart from a ...

This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roeper's notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. (...) This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roeper's paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormann's notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained. (shrink)