This paper compares and contrasts the main metaphysical and religious ideas of the eighteenth-century political theorist and chemist Joseph Priestley (1733–1804) with those of his correspondent Ruder Boscovic (1711–1787), astronomer, poet, mathematician, diplomat and Jesuit priest. It points out the theological differences between the two thinkers resulting from the divergent ontological and metaphysical implications of the theory of point atomism that they shared. This theory they placed in a wider context, considering both its limits and its value as a contribution (...) to ongoing speculations concerning the nature of space and time, theological conundrums such as free will and the mind-body problem. Where they differed, however, was that Priestley, a chemist rather than a mathematician, used point atomism mainly to support his campaign to further materialism and discredit Christianity. Boscovic, on the other hand, carefully distinguished the truths of faith and reason and was therefore indignant at Priestley's misuse of his speculations. Boscovic's protest was probably motivated in part by the materialism of atheists such as the Baron d’Holbach (1723–1789). Priestley was no atheist, however. Interestingly, François Arouet de Voltaire (1698–1778), unaware of the works of either Boscovic or Priestley, devised a theistic materialism in his last years that in many respects resembled Priestley's. I begin with brief biographies of the two thinkers, and outline their intermittent relationship. (shrink)

While operators for logical necessity and possibility represent "internal" conditions of propositions (or of their corresponding states of affairs), These conditions will be "formal", As is required by logical atomism, And not "material" in content if from the (pseudo) semantical point of view the modal operators range over "all the possible worlds" of a logical space rather than over arbitrary non-Empty sets of worlds (as is usually done in modal logic). Some of the implications of this requirement are noted (...) and though several variants of realist logical atomism are distinguished and discussed, The theory of logical form developed is nominalist. Many of nominalism's difficulties and inadequacies become transparent in the context of logical atomism and are so noted. (shrink)

Atomism is the programme explaining all changes in terms of invariant units. The development of physics during the 20th century may be treated as a spectacular triumph of atomism. However, paradoxically, changes and conceptual difficulties brought about by quantum mechanics lead to the conclusion that the ontological model provided by classical atomism has become inadequate. Atoms (and elementary particles) are not atomos—indivisible, perfectly solid, unchangeable, ungenerated and indestructible (eternal), and the void is not simply an empty space. According to (...) quantum mechanics and quantum field theory, there is no unchanging substance at all. If we want to understand contemporary notions of matter and develop an ontological model of the world, consistent with contemporary natural sciences, we should probably go beyond the conceptual framework of atomic philosophy. (shrink)

In our recent book (Abraham and Roy 2010) we have repurposed a mathematical model for the quantum vacuum as a model of consciousness. In this model, discrete space and time are derived from a discrete cellular dynamical network. As our model is essentially atomistic, we included in our book a short support chapter on atomism. In this aticle we expand on the few pages of that chapter devoted to the history of atomism, to place the current revival of (...) atomism in a larger context. (shrink)

The late Shlomo Pines (1908-1990) was this century's outstanding historian of Islamic philosophy and science. This volume offers, for the first time in English. Pines' doctoral dissertation on Islamic atomism; the German version appeared in 1936. Pines presents the atomic theories of matter, time and space, as they are found in the literature of kalam, as well their exposition in the writings of Abŭ Bakr al - Râzî; and then investigates in detail possible sources in the Greek, Indian, and (...) other traditions. The present publication incorporates a few revisions which Pines himself had made in a draft translation. A number of the texts, which Pines consulted in the manuscript, have since been published, and some important studies on the kalam have appeared. Nonetheless, it can be stated in confidence that, sixty years after its first publication, Pines' monograph is till today the most significant work on the subject. (shrink)

I argue that Descartes' Second Causal Proof of God in the Third Meditation evidences, and commits him to, the belief that time is "strongly discontinuous" -- that is, that there is actually a gap between each consecutive moment of time. Much of my article attempts to reconcile this interpretation, the "received view," with Descartes' statements about time, space, and matter in his other writings, including his correspondence with various philosophers.

This book provides detailed information about the theories of place and space of the ancient atomists, Plato, Aristotle, Peripatetics, Stoics and others, about ...

My dissertation is concerned with Epicurus' attempt to reconcile libertarianism and atomism. I begin by offering my solution to 'the problem of the swerve,' arguing that Lucretius is claiming that swerves cause volitions 'from the bottom up' and that the attempts of scholars to construct a better position for Epicurus to have held were doomed to fail, since this is the only position open to the libertarian atomist. ;I also examine the swerve's role in cosmogony, arguing that 'the cosmogonic argument' (...) for the swerve is an afterthought, 'the libertarian argument' being Epicurus' chief argument for the swerve. ;I discuss the physics of the swerve as a minimal, oblique deviation of an atom from its course. And I explore its implications: atoms move at a speed greater than one minimal unit of distance per minimal unit of time, and there are no minimal units of space. ;Returning to the question of the swerve's role in action, I argue that a typical action involves a series of volitions each caused by a plurality of swerves by atoms of the mind's 'fourth nature.' And I argue that one's power to move at will is directly sensed, though also known through one's innate $\pi\rho\acute{o}\lambda\eta\psi\iota\varsigma$ of one's freedom. ;In the second half of my dissertation, I turn to the question of responsibility. I examine Epicurus' remarks about 'what is due to us' . I examine Epicurus' argument that Democriteans are self-refuting in espousing determinism and 'eliminative materialism.' I explain that Epicurus rejects only Democritus' ontology, not his epistemology . I examine Epicurus' discussion in On Nature of our 'notion of responsibility' , our responsibility as $\alpha\pi o\gamma\varepsilon\gamma\varepsilon\nu\nu\eta\mu\acute \varepsilon\nu\alpha$ , and our responsibility for our thoughts of the $\tau\acute\varepsilon\lambda o\varsigma$ and 'the criterion' . I explain what Epicurus means by the $\tau\acute\varepsilon\lambda o\varsigma$: pleasure, "katastematic" and "kinetic" . And finally, I argue that Epicurus fails to meet the challenge of 'psychological determinism'. (shrink)

What ultimately exists for Locke is the solid. Reading this ontology in light of the atomist tradition elucidates and relates a number of important issues in the Essay: the analysis of space and related concepts, the distinction between simple and complex ideas, the distinction between primary and secondary qualitie the analysis of power and causation.

What ultimately exists for Locke is the solid. Reading this ontology in light of the atomist tradition elucidates and relates a number of important issues in the Essay: the analysis of space and related concepts, the distinction between simple and complex ideas, the distinction between primary and secondary qualitie the analysis of power and causation.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 27.2 (2003) 304-317 [Access article in PDF] Sovereign Love and Atomism in Racine's Bérénice Ellen Mcclure ALTHOUGH CRITICS HAVE NOTED links between the new science of the seventeenth century and the works of La Fontaine and Molière, 1 a similar influence of Epicureanism or even Cartesianism upon French classical tragedy is harder to trace. No two areas of seventeenth-century cultural life would seem farther apart than (...) the emerging science, with its rejection of Aristotelian categories as inadequate to experienced reality, and classical French tragedy, with its antimaterialism and claims to universal truth often bolstered by references to none other than Aristotle himself. Yet René Pintard reminds us that the 1660s and 1670s—the period when many of the tragedies we regard as "classical" were written—saw a strong renewal of interest in the philosophies articulated in the first half of the century. 2 So widespread was this interest that Pierre Gassendi's friend and follower Bernier was able to publish an edition and translation of the philosopher's works in 1674-1675 which proved to be a resounding success. It is hard to believe that the much-discussed new science somehow missed writers of tragedies, especially given the extensive contacts between Molière, La Fontaine and Racine.In this article, I demonstrate this influence by positing that Racine's play Bérénice, ostensibly a reworking of a well-known love story that Corneille was also producing on the Paris stage at the same time, is in part a reflection on the pressing question of what holds the universe together. Through his exploration of the possibility of satisfactory love between two independent sovereigns, Racine explores issues that the revival of Epicureanism had brought into the philosophical foreground. This reading of the play brings into focus the incredibly [End Page 304] important, yet too-often neglected, 3 role of Antiochus, the character whom Racine invented for his version of the play. I will show that his tragic in-betweenness offers a necessary alternative to Titus' and Bérénice's valorization of self-contained sovereignty. I Before embarking on a close reading of the play itself, however, I would like to offer a brief survey of the debate to which I believe Racine refers. Both Michel Foucault and Timothy Reiss have identified the seventeenth century as a transitional period between the age of correspondences exemplified in Aristotelian philosophy and the age of representation and quantification that characterized eighteenth-century thought. 4 This transition, however, was far from smooth, as philosophers sought to articulate a universe whose separate and radically de-hierarchized elements remained somehow joined together.For most seventeenth-century philosophers, the sacrifice of Aristotelian categories and scholasticism in no way lessened the imperative for a sense of divine order and causation. Gassendi's often self-contradictory efforts to reconcile Epicurus and Christianity provide perhaps the most striking example of this transitional struggle. The atomism posited by Epicurus through Lucretius posed particular difficulties for a Christian worldview. A world governed by the chance interactions of atoms eliminates any sense that things happen for a purpose; God cannot be the benevolent author and overseer of the universe. Gassendi's equal respect for atomism and for Christianity should be read as evidence of the profound struggle between separateness and interconnection (or, as Gassendi would call it, harmony) taking place during this period. 5 For, as Bloch points out, although Gassendi partially reconciled the demands of the two systems by situating atoms in an uncreated, infinite, and contiguous space and time, he continued to hold to the profoundly troubling existence of the void in the universe. Once again, the chief objection to the void implicit in atomism was religious, since if God is everywhere, the void cannot exist, and likewise, God cannot create nothing. The revival of stoicism supplied instead a vision of the universe ultimately comprised and held together by an ethereal pneuma that some neo-Stoics did not hesitate to identify as a sort of divine presence. 6 Descartes himself made heroic attempts, in his Principes... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 27.2 (2003) 304-317 [Access article in PDF] Sovereign Love and Atomism in Racine's Bérénice Ellen Mcclure ALTHOUGH CRITICS HAVE NOTED links between the new science of the seventeenth century and the works of La Fontaine and Molière, 1 a similar influence of Epicureanism or even Cartesianism upon French classical tragedy is harder to trace. No two areas of seventeenth-century cultural life would seem farther apart than (...) the emerging science, with its rejection of Aristotelian categories as inadequate to experienced reality, and classical French tragedy, with its antimaterialism and claims to universal truth often bolstered by references to none other than Aristotle himself. Yet René Pintard reminds us that the 1660s and 1670s—the period when many of the tragedies we regard as "classical" were written—saw a strong renewal of interest in the philosophies articulated in the first half of the century. 2 So widespread was this interest that Pierre Gassendi's friend and follower Bernier was able to publish an edition and translation of the philosopher's works in 1674-1675 which proved to be a resounding success. It is hard to believe that the much-discussed new science somehow missed writers of tragedies, especially given the extensive contacts between Molière, La Fontaine and Racine.In this article, I demonstrate this influence by positing that Racine's play Bérénice, ostensibly a reworking of a well-known love story that Corneille was also producing on the Paris stage at the same time, is in part a reflection on the pressing question of what holds the universe together. Through his exploration of the possibility of satisfactory love between two independent sovereigns, Racine explores issues that the revival of Epicureanism had brought into the philosophical foreground. This reading of the play brings into focus the incredibly [End Page 304] important, yet too-often neglected, 3 role of Antiochus, the character whom Racine invented for his version of the play. I will show that his tragic in-betweenness offers a necessary alternative to Titus' and Bérénice's valorization of self-contained sovereignty. I Before embarking on a close reading of the play itself, however, I would like to offer a brief survey of the debate to which I believe Racine refers. Both Michel Foucault and Timothy Reiss have identified the seventeenth century as a transitional period between the age of correspondences exemplified in Aristotelian philosophy and the age of representation and quantification that characterized eighteenth-century thought. 4 This transition, however, was far from smooth, as philosophers sought to articulate a universe whose separate and radically de-hierarchized elements remained somehow joined together.For most seventeenth-century philosophers, the sacrifice of Aristotelian categories and scholasticism in no way lessened the imperative for a sense of divine order and causation. Gassendi's often self-contradictory efforts to reconcile Epicurus and Christianity provide perhaps the most striking example of this transitional struggle. The atomism posited by Epicurus through Lucretius posed particular difficulties for a Christian worldview. A world governed by the chance interactions of atoms eliminates any sense that things happen for a purpose; God cannot be the benevolent author and overseer of the universe. Gassendi's equal respect for atomism and for Christianity should be read as evidence of the profound struggle between separateness and interconnection (or, as Gassendi would call it, harmony) taking place during this period. 5 For, as Bloch points out, although Gassendi partially reconciled the demands of the two systems by situating atoms in an uncreated, infinite, and contiguous space and time, he continued to hold to the profoundly troubling existence of the void in the universe. Once again, the chief objection to the void implicit in atomism was religious, since if God is everywhere, the void cannot exist, and likewise, God cannot create nothing. The revival of stoicism supplied instead a vision of the universe ultimately comprised and held together by an ethereal pneuma that some neo-Stoics did not hesitate to identify as a sort of divine presence. 6 Descartes himself made heroic attempts, in his Principes... (shrink)

We set out a fundamental ontology of atomism in terms of matter points. While being most parsimonious, this ontology is able to match both classical and quantum mechanics, and it remains a viable option for any future theory of cosmology that goes beyond current quantum physics. The matter points are structurally individuated: all there is to them are the spatial relations in which they stand; neither a commitment to intrinsic properties nor to an absolute space is required. The spatial (...) relations change. All that is needed to capture change is a dynamical structure, namely dynamical relations as expressed in terms of the dynamical parameters of a physical theory. (shrink)

This article documents the general tendency of seventeenth-century natural philosophers, irrespective of whether they were atomists or anti-atomists, to regard space, time and matter as magnitudes having the same internal composition. It examines the way in which authors such as Fromondus, Basson, Sennert, Arriaga, Galileo, Magnen, Descartes, Gassendi, Charleton as well as the young Newton motivated their belief in the isomorphism of space, time and matter, and how this belief reflected on their views concerning the relation between geometry (...) and physics. Special attention is paid to the fact that most of the authors mentioned above regarded rarefaction and condensation, on the one hand, and acceleration and deceleration, on the other hand, as analogous phenomena, which consequently had to be explained in similar terms. (shrink)

The aim of this paper is to show that John Wyclif’s theory of space is at once an interpretation of the Platonic theory of place and a Neopythagorean conception of magnitudes and numbers. The result is an original form of mathematical atomism in which atoms are point-like entities with a particular situation in space. If the core of this view comes from Boethius’ De arithmetica, John Wyclif is also influenced by Robert Grosseteste’s metaphysics, which includes the Boethian number (...) theory within the Christian tale of the creation of the world ex nihilo. John Wyclif, however, adds some novelty to this theory concerning the epistemological status of this hypothetical description of the creation of the world out of atoms. First, according to Wyclif, whereas geometry is concerned with sensible and imaginable beings, arithmetic, which is purely intellectual, has access to the deep mathematical structure of the universe. He then suggests a subordination of geometry under arithmetic, which he considers the most solid basis for his metaphysics. As a result, with the attribution of numbers and units to every level of reality, it becomes possible to reform our natural imagination, so that it can imagine the atomic structure of matter and space. (shrink)

The paper applies the theory presented in A Formal Ontology of Situations (this journal, vol. 41 (1982), no. 4) to obtain a typology of metaphysical systems by interpreting them as different ontologies of situations. Four are treated in some detail: Hume's diachronic atomism, Laplacean determinism, Hume's synchronic atomism, and Wittgenstein's logical atomism. Moreover, the relation of that theory to the situation semantics of Perry and Barwise is discussed.

A quadratic space is a generalization of a Hilbert space. The geometry of certain kinds of subspaces (“closed,” “splitting,” etc.) is approached from the purely lattice theoretic point of view. In particular, theorems of Mackey and Kaplansky are given purely lattice theoretic proofs. Under certain conditions, the lattice of “closed” elements is a quantum proposition system (i.e., a complete orthomodular atomistic lattice with the covering property).

The aim of this paper is to propose a qualitative approach to the theory of conceptual spaces, in contrast to the usual metric framework. This requires qualitative notions of similarity, simple concepts, prototypes and conceptual categorisation. For this purpose, I will introduce three mathematical models for conceptual spaces. The first one is topological and has been proposed by Mormann. The other two are new and are based on atomistic orders and similarity relations. I will discuss how each of them (...) deals with the Design Principles proposed by Douven and Gärdenfors and with further Adequacy Conditions. Despite being apparently different, I will show that these three models are mathematically equivalent. Finally, I will address three objections to the present approach. The first one says that the qualitative notion of a prototype is a bad analogue of the metric one. The second one suggests that, in contrast to the Voronoi construction, the function qualitatively representing the conceptual categorisation process is arbitrary. The last one appeals to Goodman’s companionship and imperfect community problems to show that there is a flaw in defining simple concepts from similarity relations. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea of extension (...) was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea of extension (...) was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

ABSTRACT. Associationist psychologists of the late 19th-century premised their research on a fundamentally Humean picture of the mind. So the very idea of mental science was called into question when T. H. Green, a founder of British idealism, wrote an influential attack on Hume’s Treatise. I first analyze Green’s interpretation and criticism of Hume, situating his reading with respect to more recent Hume scholarship. I focus on Green’s argument that Hume cannot consistently admit real ideas of spatial relations. I then (...) argue that William James’s early work on spatial perception attempted to vindicate the new science of mind by showing how to avoid the problems Green had exposed in Hume’s empiricism. James’s solution involved rejecting a basic Humean assumption—that perceptual experience is fundamentally composed of so-called minima sensibilia, or psychological atoms. The claim that there are no psychological atoms is interesting because James supported it with experimental data rather than (as commentators typically suppose) with introspective description or a priori argument. James claimed to be the real descendant of British empiricism on grounds that his anti-atomistic model of perception fortified what Green had perhaps most wanted to demolish—the prospect of using empirical, scientific methods in the study of mind. (shrink)

On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary (...) relation (‘qualitative probability’) that is tied to _measure_. It expresses that one region is _smaller than or equal in size_ to another. Algebraic models of our theory are _separation_ _σ_-_algebras with qualitative probability_: \((\mathbb {B}, \ll, \preceq )\), where \(\mathbb {B}\) is a Boolean _σ_-algebra, ≪ is a separation relation on \(\mathbb {B}\), and ≼ is a qualitative probability on \(\mathbb {B}\). We show that from algebraic models of this kind we can, in an interesting class of cases, recover a compact Hausdorff topology _X_, together with a countably additive measure _μ_ on a _σ_-field of Borel subsets of that topology, and that \((\mathbb {B}, \ll, \preceq )\) is isomorphic to a ‘standard model’ arising out of the pair (_X_, _μ_). It follows from one of our main results that any closed ball in Euclidean space, \(\mathbb {R}^{n}\), together with Lebesgue measure arises in this way from a separation _σ_-algebra with qualitative probability. (shrink)

Various axiomatic models for unsharp quantum measurements are investigated. These include effect spaces (E-spaces), effect test spaces (E-test spaces), effect algebras, and test groups. It is shown that a test group G is the universal group of an E-test space if and only if G is strongly atomistic. It follows that if G is strongly atomistic, then G is an interpolation group. We then demonstrate that if G is an interpolation group, then G is the universal group (...) of an E-space. Finally, it is shown that an E-space is isomorphic to an E-test space if and only if it is strongly atomistic. (shrink)

Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colour-exclusion problem. In this paper, I shall explain when the atomist picture can be defended in the face of that problem; and, in the light of this, why the atomist picture should be rejected. I outline the atomist picture in Section 1. In Section 2, I present a very simple necessary and sufficient condition for the tenability of the atomist picture. The condition is: (...) logical space is a power of two. In Sections 3 and 4, I outline the colour-exclusion problem, and then show how the cardinality-condition supplies a response to exclusion problems. In Section 5, I explain how this amounts to a distillation of a proposal due to Moss, which goes back to Carruthers. And in Section 6, I show how all this vindicates Wittgenstein’s ultimate rejection of the atomist picture. The brief reason is that we have no guarantee that there are any solutions to a given exclusion problem but, if there are any, then there are far too many. (shrink)

A thorough and well-documented study of the major revolutions in the concept of space from Pythagoras to Einstein. The theories of the ancient atomists, of Plato, and of Aristotle regarding space are analyzed against the background of modern considerations concerning homogeneity, isotropy, continuity, and infinity. The part played by theology in the development of the concept of space, the controversy attending the Newtonian and Leibnizian developments of absolute and relative theories of space, and, finally, the role (...) of geometry as developed by Gauss, Reimann, and Clifford in the physics of Eddington, Einstein, and Weyl are carefully elucidated.--M. F. (shrink)

Weyl's tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called "the mixed account," according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To (...) motivate this account as a genuine solution to Weyl's tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest's (1995) account in response to Weyl's tile argument, which can be considered as a restricted version of the mixed account. (shrink)

This chapter takes a close look at the thought that mereological relations on material objects mirror, and are mirrored by, parallel mereological relations on their exact locations. This hypothesis is made more precise by means of a battery of principles from which more substantive consequences are derived. Mereological harmony turns out to entail, for example, that atomistic space is an inhospitable environment for material gunk or that Whiteheadian space is not a hospitable environment for unextended material atoms.

This paper takes a close look at the thought that mereological relations on material objects mirror, and are mirrored by, parallel mereological relations on their exact locations. This hypothesis is made more precise by means of a battery of principles from which more substantive consequences are derived. Mereological harmony turns out to entail, for example, that atomistic space is an inhospitable environment for material gunk or that Whiteheadian space is not a hospitable environment for unextended material atoms.

Weyl’s tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called the mixed account, according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To (...) motivate this account as a genuine solution to Weyl’s tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest’s (Synthese 103:327–354, 1995) account in response to Weyl’s tile argument, which can be considered as a restricted version of the mixed account. (shrink)

The subject of my dissertation is the structure of continua and, in particular, of physical space and time. Consider the region of space you occupy: is it composed of indivisible parts? Are the indivisible parts, if any, extended? Are there infinitesimal parts? The standard view that space is composed of unextended points faces both \textit{a priori} and empirical difficulties. In my dissertation, I develop and evaluate several novel approaches to these questions based on metaphysical, mathematical and physical (...) considerations. In particular, I develop and evaluate two infinitesimal theories of space based on Robinson's nonstandard analysis. I argue that \textit{Infinitesimal Gunk}, according to which every region is further divisible and some regions have infinitesimal sizes, has distinct advantages over alternative gunky views. I also advance a new account of distance for atomistic space, \textit{the mixed account}, in response to Weyl's tile argument, which is an influential argument against the view that space is composed of indivisible regions. Having these theories in stock, we make progress in discovering the best theory of continua. (shrink)