This chapter contains sections titled: Aristotle on the Infinite (to apeiron): From Cosmological Principle to Mathematical Operation Aristotle on Space: Magnitude (megethos) and Place (topos) Aristotle on Time: The “Number of Motion” and “Ever‐rolling Stream” Bibliography.

The theory of branching space-times is designed as a rigorous framework for modelling indeterminism in a relativistically sound way. In that framework there is room for "funny business", i.e., modal correlations such as occur through quantummechanical entanglement. This paper extends previous work by Belnap on notions of "funny business". We provide two generalized definitions of "funny business". Combinatorial funny business can be characterized as "absence of prima facie consistent scenarios", while explanatory funny business characterizes situations in which no localized (...) explanation of inconsistency can be given. These two definitions of funny business are proved to be equivalent, and we provide an example that shows them to be strictly more general than the previously available definitions of "funny business". (shrink)

What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a "graviton mass term" for gravity, which is _algebraic_ in the potential. Unlike Nordström's "massless" theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning (...) volumes. Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity: matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric in the mass term. What is the 'true' geometry, one might wonder, in line with Poincaré's modal conventionality argument? Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities---indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein's equations along the lines of Einstein's newly re-appreciated "physical strategy" and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity. The Putnam-Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th-21st century space-time philosophy in the light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the 'universal force' influencing the causal structure. Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space-time theory that would have blocked Moritz Schlick's supposed refutation of synthetic _a priori_ knowledge, and how Einstein's false analogy between the Neumann-Seeliger-Einstein modification of Newtonian gravity and the cosmological constant \Lambda generated lasting confusion that obscured massive gravity as a conceptual possibility. (shrink)

Quantum Theory, similar to Relativity Theory, requires a new concept of space-time, imposed by a universal constant. While velocity of lightcnot being infinite calls for a redefinition of space-time on large and cosmological scales, quantization of action in terms of a finite, i.e. non vanishing, universal constanthrequires a redefinition of space-time on very small scales. Most importantly, the classical notion of “time”, as one common continuous time variable and nature evolving continuously (...) “in time”, has to be replaced by an infinite manifold of transition rates for discontinuous quantum transitions. The fundamental laws of quantum physics, commutation relations and quantum equations of motion, resulted from Max Born’s recognition of the basic principle of quantum physics:To each change in nature corresponds an integer number of quanta of action. Action variables may only change by integer values ofh, requiring all other physical quantities to change by discrete steps, “quantum jumps”. The mathematical implementation of this principle led to commutation relations and quantum equations of motion. The notion of “point” in space-time looses its physical significance; quantum uncertainties of time, position, just as any other physical quantity, are necessary consequences of quantization of action. (shrink)

In the correspondence with Clarke, Leibniz proposes to construe physical theory in terms of physical (spatio-temporal) relations between physical objects, thus avoiding incorporation of infinite totalities of abstract entities (such as Newtonian space) in physical ontology. It has generally been felt that this proposal cannot be carried out. I demonstrate an equivalence between formulations postulating space-time as an infinite totality and formulations allowing only possible spatio-temporal relations of physical (point-) objects. The resulting rigorous formulations of (...) physical theory may be seen to follow Leibniz' suggestion quite closely. On the other hand, physical assumptions implicit in the postulation of space-time totalities are made explicit in the reconstruction of the space-time versions from the physical-relation versions. (shrink)

The questions of what represents space-time in GR, the status of gravitational energy, the substantivalist-relationalist issue, and the exceptional status of gravity are interrelated. If space-time has energy-momentum, then space-time is substantival. Two extant ways to avoid the substantivalist conclusion deny that the energy-bearing metric is part of space-time or deny that gravitational energy exists. Feynman linked doubts about gravitational energy to GR-exceptionalism, as do Curiel and Duerr; particle physics egalitarianism encourages realism (...) about gravitational energy. In that spirit, this essay proposes a third possible view about space-time, one involving a particle physics-inspired non-perturbative split that characterizes space-time with a constant background _matrix_, a sort of vacuum value, thus avoiding the inference from gravitational _energy to substantivalism. On this proposal, space-time is, where eta=diag is a spatio-temporally constant numerical signature matrix, a matrix already used in GR with spinors. The gravitational potential, to which any gravitational energy can be ascribed, is g_{\mu\nu}- eta, an _affine_ geometric object with a tensorial Lie derivative and a vanishing covariant derivative. This non-perturbative split permits strong fields, arbitrary coordinates, and arbitrary topology, and hence is pure GR by almost any standard. This razor-thin background, unlike more familiar backgrounds, involves no extra gauge freedom and so lacks their obscurities and carpet lump-moving. After a discussion of Curiel's GR exceptionalist denial of the localizability of gravitational energy and his rejection of energy conservation, the two traditional objections to pseudotensors, coordinate dependence and nonuniqueness, are explored. Both objections are inconclusive and getting weaker. A literal interpretation involving infinitely many energies corresponding by Noether's first theorem to the infinite symmetries of the _action_ largely answers Schroedinger's false-negative coordinate dependence problem. Bauer's false-positive objection has multiple answers. Non-uniqueness might be handled by Nester et al.'s finding physical meaning in multiplicity in relation to boundary conditions, by an optimal candidate, or by Bergmann's identifying the non-uniqueness and coordinate dependence ambiguities as one. (shrink)

Este artículo examina la concepción del espacio infinito y del tiempo en Hasdai Crescas, Gianfrancesco Pico della Mirandola y Giordano Bruno. Si la presencia de Crescas es explícita en el _Examen vanitatis_ (1520) de Pico, su recepción por Bruno, que nunca lo menciona, fue postulada por Harry A. Wolfson en 1929. Más recientemente, David Harari y Mauro Zonta han afirmado el papel intermediario de un autor judío desconocido. Sin embargo, una comparación de la crítica de Aristóteles efectuada por Crescas y (...) Bruno pone de manifiesto que, excepto dos puntos, Bruno pudo alcanzar sus posiciones con independencia de Crescas, a partir de su propia lectura crítica de Aristóteles y de su conocimiento de los desarrollos de la escolástica medieval y del concepto neoplatónico del tiempo. La significativa calificación por Crescas del espacio y del tiempo como atributos de Dios en un sentido puramente metafórico, una cuestión que Pico deja a un lado, deja paso en Bruno a la calificación del espacio infinito, de la duración y también de la materia como atributos reales de Dios, que es a la vez mente e intelecto, espacio y materia, en tanto que unidad indiferente de los opuestos. (shrink)

The Riemannian manifold structure of the classical (i.e., Einsteinian) space-time is derived from the structure of an abstract infinite-dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H of functions of abstract parameters. The space H is associated with the space of states of a macroscopic test-particle in the universe. The spatial localization of state of the particle through its interaction with the environment is associated with the (...) selection of a submanifold M of realization H. The submanifold M is then identified with the classical space (i.e., a space–like hypersurface in space-time). The mathematical formalism is developed which allows recovering of the usual Riemannian geometry on the classical space and, more generally, on space and time from the Hilbert structure on S. The specific functional realizations of S are capable of generating spacetimes of different geometry and topology. Variation of the length-type action functional on S is shown to produce both the equation of geodesics on M for macroscopic particles and the Schrödinger equation for microscopic particles. (shrink)

Not much attention has been paid to Malebranche’s philosophy of time. Scholars who have written on it have typically written about it only in passing, and by and large discuss it only in relation to his philosophy of religion. This is appropriate insofar as Malebranche doesn’t discuss his views of time in isolation from his religious metaphysics. I argue that Malebranche’s conception of how created beings have their properties commits him to saying that God is omnitemporal rather than (...) atemporal. For just as bodies get their spatiality by participating in God’s omnipresence, so all creatures get their temporality by participating in God’s omnitemporality. Moreover, Malebranche is a substantivalist about space and time: infinite space and time are one and the same divinely simple substance, God (partially considered), who contains the world. My exploration of Malebranche’s metaphysics sheds light on his views of eminent containment, participation, and causation. (shrink)

This book provides an account of the unity of Immanuel Kant’s early metaphysics, including the moment he invents transcendental idealism. Matthew Rukgaber argues that a division between “two worlds”—the world of matter, force, and space on the one hand, and the world of metaphysical substances with inner states and principles preserved by God on the other—is what guides Kant’s thought. Until 1770 Kant consistently held a conception of space as a force-based material product of monads that are only (...) virtually present in nature. As Rukgaber explains, transcendental idealism emerges as a constructivist metaphysics, a view in which space and time are real relations outside of the mind, but those relations are metaphysically dependent on the subject. The subject creates the simple “now” and “here,” thus introducing into the intrinsically indeterminate and infinitely divisible continua of nature a metric with transformation rules that make possible all individuation and measurement. (shrink)

Complexified Liénard–Wiechert potentials simplify the mathematics of Kerr–Newman particles. Here we constrain them by fiat to move along Bohmian trajectories to see if anything interesting occurs, as their equations of motion are not known. A covariant theory due to Stueckelberg is used. This paper deviates from the traditional Bohmian interpretation of quantum mechanics since the electromagnetic interactions of Kerr–Newman particles are dictated by general relativity. A Gaussian wave function is used to produce the Bohmian trajectories, which are found to be (...) multi-valued. A generalized analytic continuation is introduced which leads to an infinite number of trajectories. These include the entire set of Bohmian trajectories. This leads to multiple retarded times which come into play in complex space-time. If one weights these trajectories by their natural Bohmian weighting factors, then it is found that the particles do not radiate, that they are extended, and that they can have a finite electrostatic self energy, thus avoiding the usual divergence of the charged point particle. This effort does not in any way criticize or downplay the traditional Bohmian interpretation which does not assume the standard electromagnetic coupling to charged particles, but it suggests that a hybridization of Kerr–Newman particle theory with Bohmian mechanics might lead to interesting new physics, and maybe even the possibility of emergent quantum mechanics. (shrink)

I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of (...) a larger one, but it requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”. (shrink)

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as (...) gravity, electromagnetism, and strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; the Hamilton–Jacobi equation loaded by the quantum potential. This (...) equation in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium ; gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation. (shrink)

This collection focuses on the ontology of space and time. It is centred on the idea that the issues typically encountered in this area must be tackled from a multifarious perspective, paying attention to both a priori and a posteriori considerations. Several experts in this area contribute to this volume: G. Landini discusses how Russell’s conception of time features in his general philosophical perspective;D. Dieks proposes a middle course between substantivalist and relationist accounts of space-time;P. (...) Graziani argues that it is necessary to provide an account of the “synthetic procedures” implicit in the recourse to diagrams in Euclid’s Elements, while E. Mares comes to the conclusion that in Euclid’s Elements we should treat the parallel postulate as empirical and the postulate that space is continuous as a priori. M. Arsenijevi?/M. Adži? present an important formal result concerning two theories of the infinite two-dimensional continua, which sheds new light on the current dispute between gunkologists and pointilists; F. Orilia discusses two problems for presentism, one regarding the duration of the present and the other related to Zeno’s paradoxes. A. Iacona delves deep into logical matters by focusing on the so-called T×W modal frames in order to deal with the deteterminism-indeterminism controversy. D. Mancuso outlines a non-standard temporal model compatible with time travel, andV. Fano/G. Macchia discuss time travels in the light of an important foundational principle of modern cosmology, Weyl’s Principle. (shrink)

. The concepts of space and time are important in physics and geometry, but their definition is not the exclusive prerogative of those sciences. Space and time are important for ordinary human experience, as well as for philosophy and theology. Samuel Clarke, Gottfried Wilhelm Leibniz, Isaac Newton, Immanuel Kant, and Albert Einstein are important figures in shaping our understandings of space, time, and eternity. The author subjects their arguments to critical examination. Space is (...) neither an infinite and empty receptacle nor a system of relations in the mind . Infinite space and time can be interpreted as expressing God's eternity and omnipresence in relating to the creation , but such an interpretation is enhanced by Kant's thinking, to clarify that even though time and space are differentiated in individual events, the whole is at the same time present. Even human experience recognizes this wholeness, and for God eternity is the simultaneous presence and possession of the wholeness. The temporal existence of finite entities is also related to a future participation in God's eternal life. Concepts of contingency are brought into the discussion as well. (shrink)

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)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully (...) shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

Space and time are the most fundamental features of our experience of the world, and yet they are also the most perplexing. Does time really flow, or is that simply an illusion? Did time have a beginning? What does it mean to say that time has a direction? Does space have boundaries, or is it infinite? Is change really possible? Could space and time exist in the absence of any objects or (...) events? What, in the end, are space and time? Do they really exist, or are they simply the constructions of our minds? Robin Le Poidevin provides a clear, witty, and stimulating introduction to these deep questions and many other mind-boggling puzzles and paradoxes. He gives a vivid sense of the difficulties raised by our ordinary ideas about space and time, but he also gives us the basis to think about these problems independently, avoiding large amounts of jargon and technicality. His book is an invitation to think philosophically rather than a sustained argument for particular conclusions, but Le Poidevin does advance and defend a number of controversial views. He argues, for example, that time does not actually flow, that it is possible for space and time to be both finite and yet be without boundaries, and that causation is the key to an understanding of one of the deepest mysteries of time: its direction. Drawing on a variety of vivid examples from science, history, and literature, Travels in Four Dimensions brings to life some of the most profound questions imaginable. (shrink)

This paper analyzes two different proposals, one by Ellis and Brundrit, based on classical relativistic cosmology, the other by Garriga and Vilenkin, based on the DH interpretation of quantum mechanics, both concluding that, in an infinite universe, planets and beings must be repeated an infinite number of times. We point to possible shortcomings in these arguments. We conclude that the idea of an infinite repetition of histories in space cannot be considered strictly speaking a consequence of (...) current physics and cosmology. Such ideas should be seen rather as examples of «ironic science» in the terminology of John Horgan. (shrink)

000000001. Introduction Call a theory of the good—be it moral or prudential—aggregative just in case (1) it recognizes local (or location-relative) goodness, and (2) the goodness of states of affairs is based on some aggregation of local goodness. The locations for local goodness might be points or regions in time, space, or space-time; or they might be people, or states of nature.1 Any method of aggregation is allowed: totaling, averaging, measuring the equality of the distribution, measuring (...) the minimum, etc.. Call a theory of the good finitely additive just in case it is aggregative, and for any finite set of locations it aggregates by adding together the goodness at those locations. Standard versions of total utilitarianism typically invoke finitely additive value theories (with people as locations). A puzzle can arise when finitely additive value theories are applied to cases involving an infinite number of locations (people, times, etc.). Suppose, for example, that temporal locations are the locus of value, and that time is discrete, and has no beginning or end.2 How would a finitely additive theory (e.g., a temporal version of total utilitarianism) judge the following two worlds? Goodness at Locations (e.g. times) w1:..., 2, 2, 2, 2, 2, 2, 2, 2, 2, ..... w2:..., 1, 1, 1, 1, 1, 1, 1, 1, 1, ..... Example 1 At each time w1 contains 2 units of goodness and w2 contains only 1. Intuitively, we claim, if the locations are the same in each world, finitely additive theorists will want to claim that w1 is better than w2. But it's not clear how they could coherently hold this view. For using standard mathematics the sum of each is the same infinity, and so there seems to be no basis for claiming that one is better than the other.3 (Appealing to Cantorian infinities is of no help here, since for any Cantorian infinite N, 2xN=1xN.). (shrink)

This work celebrates the investigative power of phenomenology to explore the phenomenological sense of space and time in conjunction with the phenomenology of intentionality, the invisible, the sacred, and the mystical. It examines the course of life through its ontopoietic genesis, opening the cosmic sphere to logos. The work also explores, on the one hand, the intellectual drive to locate our cosmic position in the universe and, on the other, the pull toward the infinite. It intertwines science (...) and its grounding principles with imagination in order to make sense of the infinite. This book is the second of a two-part work that contains papers presented at the 62nd International Congress of Phenomenology, The Forces of the Cosmos and the Ontopoietic Genesis of Life, held in Paris, France, August 2012. It features the work of scholars in such diverse disciplines as biology, anthropology, pedagogy, and psychology who philosophically investigate the cosmic origins of beingness. Coverage in this second part includes: Communicative Virtues of A-T. Tymieniecka’s Phenomenology of Life, Intentionality of Time and Quantum – Phenomenological Sense of Space, Consciousness of the Cosmos: A Thought Experiment Through Philosophy and Science Fiction, The Cosmos and Bodily Life on Earth Elucidated within the Historicity of Human Existence, Novel as Path - Mamardashvili's Lectures on Proust, and Comments on Max Scheler's Thought and Philosophical Counseling. (shrink)

This study considers Newton's views on space and time with respect to some important ontologies of substance in his period. Specifically, it deals in a philosophico-historical manner with his conception of substance, attribute, existence, to actuality and necessity. I show how Newton links these “features” of things to his conception of God's existence with respect of infinite space and time. Moreover, I argue that his ontology of space and time cannot be understood without (...) fully appreciating how it relates to the nature of Divine existence. In order to establish this, the ontology embodied in Newton's theory of predication is analysed, and shown to be different from the presuppositions of the ontological argument. From the historical point of view Gassendi's influence is stressed, via the mediation of Walter Charleton. Furthermore, Newton's thought on these matters is contrasted with Descartes's and Spinoza's. In point of fact, in his earliest notebook Newton recorded observations on Descartes's version of the ontological argument. Soon, however, he was to oppose the Cartesian conception of the actuality of Divine existence by means of arguments similar to those of Gassendi. Lastly, I suggest that the nature and extent of Henry More's influence on Newton's conception of how God relates to absolute space and time bears further examination. (shrink)

In two rarely discussed passages – from unpublished notes on the Principles of Philosophy and a 1647 letter to Chanut – Descartes argues that the question of the infinite extension of space is importantly different from the infinity of time. In both passages, he is anxious to block the application of his well-known argument for the indefinite extension of space to time, in order to avoid the theologically problematic implication that the world has no beginning. (...) Descartes concedes that we always imagine an earlier time in which God might have created the world if he had wanted, but insists that this imaginary earlier existence of the world is not connected to its actual duration in the way that the indefinite extension of space is connected to the actual extension of the world. This paper considers whether Descartes’s metaphysics can sustain this asymmetrical attitude towards infinite space vs. time. I first consider Descartes’s relation to the ‘imaginary’ space/time tradition that extended from the late scholastics through Gassendi and More. I next examine carefully Descartes’s main argument for the indefinite extension of space and explain why it does not apply to time. Most crucially, since duration is merely conceptually distinct from enduring substance, the end or beginning of the world entails the end or beginning of real time. In contrast, extension does not depend on any enduring substance besides itself. (shrink)

There are few notions in philosophy seen more clearly, and in parallel so laden with confusion, than that of space and time. The subjective nature of analyses is most likely to blame. As it stands, a universal agreement has not yet been reached. My position is simply that the mind, when passive, has no qualms with space and time itself, nor is it concerned with its principles. It is only when our passions are ignited, and our (...) judgment is utilized, i.e. when we begin to actively think about space and time, that the notion becomes confounded. Any further digression into semantics of the issue by trying to give a ‘say all end all’ explanation will always lead to a dead end. Both space and time are ideas of an infinite nature and can never be visited manifestly since the mind itself is undeniably finite. (shrink)

An original and insightful account of nature and our place in it from one of France’s preeminent historians of philosophy. One of France’s preeminent historians of philosophy, Marcel Conche has written and translated more than thirty-five books and is recognized for his groundbreaking and authoritative work in Greek philosophy, as well as on Montaigne. In Philosophizing ad Infinitum, one of his most remarkable and daring books, Conche articulates a unique and powerful understanding of nature, inclusive of humanity, as infinite (...) in time and space—ever self-renewing, eternal, and beyond complete understanding or control. In today’s world the notion of infinity is at the core of the crisis humanity faces understanding nature. For the last two hundred years economies have been running at full speed, fueled by the implicit belief that natural resources are infinite; however, it is clear that they are not and that humanity needs to radically rethink the foundations of environmental and economic systems. Conche seeks to begin this rethinking, illustrating along the way insightful and sometimes unorthodox ideas about Plato, Aristotle, Epicurus, Montaigne, Nietzsche, Bergson, and others. “Philosophizing ad Infinitum: this book paints a perfect picture of the life of Marcel Conche, a natural-born philosopher. His aptitude to question absolutely everything drove him to the pre-Socratics, even at an early age, in his relentless quest for the truth. With Montaigne, the thinker to whom he feels most akin, Marcel Conche stands tall as an exceptional figure in contemporary French philosophy, renewing and enriching a great philosophical tradition.” — Françoise Dastur, University of Nice Sophia Antipolis. (shrink)

Infinity exists as a concept but has no existence in actuality. For infinity to have existence in actuality either time or space have to already be infinite. Unless something is already infinite, the only way to become infinite is by an 'infinity leap' in an infinitely small moment, and this is not possible. Neither does infinitely small have an existence since anything larger than zero is not infinitely small. Therefore infinity has no existence in actuality.

The connection between the infinite nature of Quantum Cosmology and the infinite nature of God is presented here. At the beginning of the creation process, there was a single God/Void that was divided into many Gods/Voids all filled with Dark Energy consisting of photons which were responsible for creating the Multiverses made of matter, antimatter, space, time, charge, and multiple dimensions of space. The one God initially had no material existence which along with the laws (...) of science was a creation of the powers of His infinite mind. Since the process of creation started with the Dark Energy of the Voids, the Conservation of Energy Principle applied to all of them. If the Dark Energy of each Void were to be finite, then its energy would begin to decrease as Multiverses were created, and the only solution for not violating the Conservation of Energy Principle was that the Dark Energy of each of the Voids must be infinite implying that the Voids themselves must be infinite, and with infinite amount of energy be able to create an infinite number of Multiverses. According to Mathematics, all infinite numbers are not equal, some infinities are greater than other infinities. Hence of the many infinite Voids that were initially created, some Voids were greater than others. Each Void has a Supreme Leader that we refer to as God, and therefore there exist many Gods, some Gods more powerful than other Gods. (shrink)

The paper investigates, in the framework of branching space–times, whether an infinite EPR-like correlation which does not involve finite EPR-like correlations is possible.

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any (...) such theory. -/- My goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

For infinite machines that are free from the classical Thomson’s lamp paradox, we show that they are not free from its inverted-in-time version. We provide a program for infinite machines and an infinite mechanism that demonstrate this paradox. While their finite analogs work predictably, the program and the infinite mechanism demonstrate an undefined behavior. As in the case of infinite Davies machines :671–682, 2001), our examples are free from infinite masses, infinite velocities, (...) infinite forces, etc. Only infinite divisibility of space and time is assumed. Thus, the infinite devices considered are possible in a Newtonian Universe and they do not conflict with Newtonian mechanics. Note that the classical Thomson’s lamp paradox leads to infinite velocities which may not be producible in acceptable models of Newtonian mechanics. Finally, it is shown that the “paradox of predictability” is similar to the inverted Thomson’s lamp paradox. (shrink)

How might we extend aggregative moral theories to compare infinite worlds? In particular, how might we extend them to compare worlds with infinite spatial volume, infinite temporal duration, and infinitely many morally valuable phenomena? When doing so, we face various impossibility results from the existing literature. For instance, the view we adopt can endorse the claim that worlds are made better if we increase the value in every region of space and time, or that they (...) are made better if we increase the value obtained by every person. But they cannot endorse both claims, so we must choose. In this paper I show that, if we choose the latter, our view will face serious problems such as generating incomparability in many realistic cases. Opting instead to endorse the first claim, I articulate and defend a spatiotemporal, expansionist view of infinite aggregation. Spatiotemporal views such as this do face some difficulties, but I show that these can be overcome. With modification, they can provide plausible comparisons in cases that we care about most. (shrink)

Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used (...) in modern physics. Of the few who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)

Famously, Kant describes space and time as infinite “given” magnitudes. An influential interpretative tradition reads this as a claim about phenomenological presence to the mind: in claiming that space and time are given, this reading holds, Kant means to claim that we have phenomenological access to space and time in our original intuitions of them. In this paper, I argue that we should instead understand givenness as a metaphysical notion. For Kant, space (...) and time are ‘given’ in virtue of three related facts: (i) they are necessary grounds of the existence of all other spatial and temporal possibilities; (ii) in virtue of being such grounds, they are metaphysically linked to all other represented spatiotemporal things; and (iii) as representations, space and time issue from the nature of our faculty of sensibility rather than from an arbitrary act of choice. Understanding givenness in this non-epistemological way helps us recover what is plausible in both ‘intellectualist’ and ‘anti-intellectualist’ readings of the Transcendental Aesthetic. Anti-intellectualists are correct that space and time are given in mere sensibility, but intellectualists are correct that we depend on the understanding for consciousness of space and time. (shrink)