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I examine the reasons Aristotle presents in Physics VIII 8 for denying a crucial assumption of Zeno’s dichotomy paradox: that everymotion is composed of sub-motions. Aristotle claims that a unifiedmotion is divisible into motions only in potentiality (δυνάμει). If it were actually divided at some point, the mobile would need to have arrived at and then have departed from this point, and that would require some interval of rest. Commentators have generally found Aristotle’s reasoning unconvincing. Against (...) David Bostock and Richard Sorabji, inter alia, I argue that Aristotle offers a plausible and internally consistent response to Zeno. I defend Aristotle’s reasoning by using his discussion of what to say about the mobile at boundary instants, transitions between change and rest. There Aristotle articulates what I call the Changes are Open, Rests are Closed Rule: what is true of something at a boundary instant is what is true of it over the time of its rest. By contrast, predications true of something over its period of change are not true of the thing at either of the boundary instants of that change. I argue that this rule issues from Aristotle’s general understanding of change, as laid out in Phys. III. It also fits well with Phys. VI, where Aristotle maintains that there is a first boundary instant included in the time of rest, but not a “first in which the mobile began to change.” I then show how this rule underlies Aristotle’s argument that acontinuousmotion cannot be composed of actual sub-motions. Aristotle distinguishes potential middles, points passed through en route to a terminus, from actual middles. The Changes are Open, Rests are Closed Rule only applies to actual middles, because only they are boundaries of change that the mobile must arrive at and then depart from. On my reading, Aristotle argues that the instant of arrival, the first instant at which the mobile has come to be at the actual middle, cannot belong to the time of the subsequentmotion. If it did, the mobile would already be moving towards the next terminus and thus, per Phys. VI 6, would have already left. But it cannot have moved away from the midpoint at the very same moment it has arrived there. This means that the instant of arrival must be separated from the time of departure by an interval of rest. I show how Aristotle’s reasoning applies generally to rule out anycontinuous reflexivemotion orcontinuous complex rectilinearmotion. On my interpretation, however, the argument does not apply to every change of direction. When, as in the case of projectilemotion, multiple movers and their relative powers explain why the mobile changes directions, distinct sub-motions are not involved. Aristotle holds that such motions cannot becontinuous, not because they involve intervals of rest, but because they involve multiple causes ofmotion. My interpretation of the Changes are Open, Rests are Closed Rule allows us to make better sense of Aristotle’s argument than any previous interpretation. (shrink)

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It is widely accepted that continuity is the most essential characteristic ofmotion; themotion of macroscopic objects is apparentlycontinuous, and classical mechanics, which describes suchmotion, is also based on the assumption ofcontinuousmotion. But ismotion reallycontinuous in reality? In this paper, I will try to answer this question through a new analysis of the cause ofmotion. It has been argued that the standard velocity in (...) classical mechanics cannot fulfill the causal role required for explainingcontinuousmotion in a deterministic way. However, there is a hot debate over the solution to this causal explanation problem. The existing solutions, i.e. the "at-at" theory, the impetus view and the "no instants" view, have serious drawbacks. After reviewing their drawbacks and presenting my own objections, I propose a new solution to this causal explanation problem. It is argued that the relativity ofmotion or the equivalence betweenmotion and rest demands thatmotion has no deterministic cause, and thus no causal explanation is needed formotion. Based on this result, I further argue that the uncausedmotion is not deterministic andcontinuous but essentially random and discontinuous. Moreover, objects have a potential that determines the probabilities of their future positions. This suggests a possible connection with quantum mechanics. Lastly, I give a brief explanation for the apparent inconsistency between the suggested potential theory ofmotion and the appearance ofcontinuousmotion in the macroscopic world. (shrink)

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The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...) which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. (shrink)

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After reporting in detail Aristotle’s texts and comments on the well-knownmotion paradoxes Arrow, Dichotomy, Achilles and Stadium, tracking back to the 5th century BCE and credited by Aristotle to Zeno of Elea, we next explain and dis-cuss traditionalcontinuous solutions of the paradoxes, based on Cauchy’s limit concept. Afterward, the heated philosophical debate on supertasks and infinity machines is reported before the paradoxes are examined within the context of modern quantum theory. Already in 1905, Einstein concluded that (...) matter could not be acontinuous thing. Classicalcontinuous spacetime is replaced by ‘granular’ spacetime, and the concept of distance in granular spacetime is discussed. This is followed by a detailed presentation of modern discrete solutions in granular spacetime, several of which are published here for the first time. Finally, a procedure is presented to determine when traditionalcontinuous methods suffice instead of discrete methods, and the handling of discrete versuscontinuous in physics is briefly reported. (shrink)

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Margaret Cavendish is widely known as a materialist. However, since Cavendishian matter is always inmotion, “matter” and “motion” are equally important foundational concepts for her natural philosophy. In Philosophical Letters (1664), she takes to task her materialist rival Thomas Hobbes by assaulting his account of accidents in general and his concept of “rest” in particular. In this article, I argue that Cavendish defends hercontinuous-motion view in two ways: first, she claims that her account avoids (...) seeing accidents as capable of generation and annihilation, which she argues is inconceivable; and second, she contends that according to Hobbes’s own view “rest” is an absurd conception since it cannot be drawn from experience. Beyond its function as a defense, I claim that Cavendish’s focused criticism of “rest” shows that she is a perceptive reader of Hobbes’s natural philosophy, insofar as her criticisms undercut the two a priori principles of Hobbesian physics. Finally, I show how her views developed in more detail in Philosophical and Physical Opinions (1663) and Observations upon Experimental Philosophy ([1666] 2001) avoid the worries she raises for Hobbesian materialism. (shrink)

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Wholes have parts, and wholes are prior to parts according to Aristotle. Aristotle’s accounts of continuity, in _Phys_. V 3 (plus sections in Metaph. Δ 6 and Ι 1) on the one hand and in _Phys_. VI on the other, are specified in terms of ways in which wholes are related to parts. The synthesis account in Phys. V 3 etc. applies primarily to bodies (in, e.g., anatomy). It indicates a variety of ways in which parts of a body are (...) kept together by a common boundary and are thereby combined into a mostly inhomogeneous, functional whole. Only the analysis account in _Phys_. VI applies primarily to linear continua such as movements, paths of movements, and time. The structure it indicates is only superficially described as indefinite divisibility: what matters is the transfer of potential divisions from path to movement and time (and conversely) which, surprisingly, requires an equivalent to Dedekind’s continuity principle to be tacitly presupposed. – In the present paper, my agenda will focus on the exposition of the relevant theories offered by Aristotle in _Phys_. V 3 and _Phys_. VI 1-2, respectively, with a view to the applications envisaged by Aristotle and to the mathematics involved. (shrink)

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This article denies that Hegel upheld the objective truth of any contradictory statements. Yet he did admit objective contradictions in the sense of intersubjectively held contradictory beliefs at the basis of some institutions, most famously lordship and bondage. He also shared the belief of Zeno, the inventor of dialectic, thatcontinuousmotion is self-contradictory but is an objective contradiction more widely shared by all institutions presupposing continuants (people and ordinary things).

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Apparentmotion is an illusion in which two sequentially presented and spatially separated stimuli give rise to the experience of one moving stimulus. This phenomenon has been deployed in various philosophical arguments for and against various theories of consciousness, time consciousness and the ontology of time. Nevertheless, philosophers have continued working within a framework that does not reflect the current understanding of apparentmotion. The main objectives of this paper are to expose the shortcomings of the explanations provided (...) for apparentmotion and to offer an alternative explanation for the phenomenon. (shrink)

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This paper enquires into a paradigmatic change concerning the concept ofmotion: from a phenomenological conception ofmotion understood as acontinuous and finite process of translation, to a physical conception ofmotion as rectilinear, uniform andcontinuous, that is, as an inertial state that if unhindered, can extend infinitely – the former held by Aristotle, the latter by Koyré, a shift that is evidenced by their contrasting treatment of Zeno’s paradoxes. I argue that both (...) ontologies ofmotion can be understood in phenomenological terms:Motion is accomplished by the real itself. However, while Aristotle’scontinuousmotion as actualization of the form of a given body involves its formal determination and constitution of sense, Koyré’s conceives ofmotion and rest as kinds of being andmotion itself as exemplifcation of the mathematization of nature that Husserl described in the Crisis. Taken together, these changes presuppose a transformation of metaphysical and epistemic frameworks. (shrink)

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Human infants begin very early in life to take advantage of multisensory information by extracting the invariant amodal information that is conveyed redundantly by multiple senses. Here we addressed the question as to whether infants can bind multisensory moving stimuli, and whether this occurs even if themotion produced by the stimuli is only illusory. Three- to four-month-old infants were presented with two bimodal pairings: visuo-tactile and audio-visual. Visuo-tactile pairings consisted of apparently vertically moving bars (the Barber Pole illusion) (...) moving in either the same or opposite direction with a concurrent tactile stimulus consisting of strokes given on the infant’s back. Audio-visual pairings consisted of the Barber Pole illusion in its visual and auditory version, the latter giving the impression of acontinuous rising or ascending pitch. We found that infants were able to discriminate congruently (same direction) vs. incongruently moving (opposite direction) pairs irrespective of modality (Experiment 1). Importantly, we also found that congruently moving visuo-tactile and audio-visual stimuli were preferred over incongruently moving bimodal stimuli (Experiment 2). Our findings suggest that very young infants are able to extractmotion as amodal component and use it to match stimuli that only apparently move in the same direction. (shrink)

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As Aristotle classically defined it, continuity is the property of being infinitely divisible into ever-divisible parts. How has this conception been affected by the process of mathematization ofmotion during the 14th century? This paper focuses on Nicole Oresme, who extensively commented on Aristotle’s Physics, but also made decisive contributions to the mathematics ofmotion. Oresme’s attitude about continuity seems ambivalent: on the one hand, he never really departs from Aristotle’s conception, but on the other hand, he uses (...) it in a completely new way in his mathematics, particularly in his Questions on Euclidean geometry, a tantamount way to an atomization ofmotion. If the fluxus theory of naturalmotion involves that continuity is an essential property of realmotion, defined as a res successiva, the ontological and mathematical structure of this continuity implies that continuum is in some way “composed” of an infinite number of indivisibles. In fact, Oresme’s analysis opened the path to a completely new kind of mathematical continuity. (shrink)

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Self-motion through space generates a visual pattern called optic flow. It can be used to determine one’s direction of self-motion (heading). Previous studies have already shown thatthis perceptual ability, which is of critical importance during everyday life, changes with age. In most of these studies subjects were asked to judge whether they appeared to be heading to the left or right of a target. Thresholds were found to increase continuously with age. In our current study, we were interested (...) in absolute rather than relative heading judgements and in the question about a potential neural correlate of an age-related deterioration of heading perception. Two groups, older test subjects and younger controls, were shown optic flow stimuli in a virtual-reality setup. Visual stimuli simulated self-motion through a 3-D cloud of dots and subjects had to indicate their perceived heading direction after each trial. In different subsets of experiments we varied individuallyrelevant stimulus parameters: presentation time, number of dots in the display, stereoscopic vs. non-stereoscopic stimulation, andmotion coherence. We found decrements in heading performance with agefor each stimulus parameter. In a final step we aimed to determine a putative neural basis of this behavioural decline. To this end we modified a neural network model which previously has proven to be capable of reproduce and predict certain aspects of heading perception. We show that the observed data can be modeled by implementing an age related neuronal cell loss in this neural network. We conclude that acontinuous decline of certain aspects ofmotion perception, among them heading, might directly be based on anage-related progressive loss of groups of neurons being activated by visualmotion. (shrink)

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Following Aristotle, Averroes rejects atomism and the infinite division of geometric lines. Thus, his arguments dealt with the continuity and contiguity of the non-atomic parts ofmotion. He vindicates the perceptual aspect of physical movement that shows itself like in-progress-path between two edge points A and B, in which there are middle parts where qualitative, local, or quantitative changes occur. Ramon Llull took the lines’ geometrical points as “motion parts.” Points are intermediate divisions that represent physical phenomena by (...) the continuity of geometrical lines, surfaces, and figures. Also, he appeals to relational logic to spot the middle parts between A and B into the in-progress-path ofmotion. Those middle parts are signified by a dynamic vocabulary, called: correlative language. This contribution focuses on the conceptual environment of Llull’s assumptions, in which Averroes’ Latin readers explored the geometry and the vocabulary ofmotion intermediate parts. (shrink)

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In recent years, with the rapid development of computer storage capabilities and network transmission capabilities, users can easily share their own video and image information on social networking sites, and the amount of multimedia data on the network is rapidly increasing. With thecontinuous increase of the amount of data in the network, the establishment of effective automated data management methods and search methods has become an increasingly urgent need. This paper proposes a retrieval method of humanmotion (...) data based onmotion capture in index space. By extracting key frames from the originalmotion to perform horizontal dimensionality reduction and defining features based on Labanmotion analysis, themotion segment is subjected to vertical feature dimensionality reduction. After extracting features from the inputmotion segment,motion matching is performed on the index space. This paper designs the optimization method of the phased dynamic time deformation algorithm in time efficiency and analyzes the optimization method of the phased dynamic time deformation algorithm in time complexity. Considering the time efficiency redundancy, this paper optimizes the time complexity of the phased dynamic time deformation method. This improves the time efficiency of the staged dynamic time warping algorithm, making it suitable for larger-scale humanmotion data problems. Experiments show that the method in this paper has the advantage of speed, is more in line with the semantics of humanmotion, and can meet the retrieval requirements of humanmotion databases. (shrink)

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Book 1 of Aristotle’s De Anima extensively discusses two characteristics of the soul: the soul as the source ofmotion of the living being, and the soul as the seat of perception and cognition. The following conclusions are drawn on the nature and function of the soul. The soul is not a magnitude and not material; it is a substance and not an attribute; it is a unity, and the principle of unity is not material continuity. The soul is (...) the origin of perception andmotion, and of psychological processes such as emotions and desires. An adequate account of how the soul causes perception,motion, and the like must not attributemotion to the soul. (shrink)

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A diagram that places two planets in orbit around the sun was inserted into the textual space of a Timaeus manuscript of the late 11th century as well as three more in the 12th century. The diagram derives from a Carolingian tradition of study of Martianus Capella’s astronomy and shows his continued authority into the twelfth century. By way of Capella and through similarly-inspired commentaries on Macrobius’ Commentary on the Dream of Scipio, the idea of circumsolarmotion for Mercury (...) and Venus became a well-known and authoritative theme for William of Conches and other scholars in the eleventh and twelfth centuries wo interpreted Plato in terms of current scientific knowledge. (shrink)

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We solve the problem of formulating Brownianmotion in a relativistically covariant framework in 3+1 dimensions. We obtain covariant Fokker–Planck equations with (for the isotropic case) a differential operator of invariant d’Alembert form. Treating the spacelike and timelike fluctuations separately in order to maintain the covariance property, we show that it is essential to take into account the analytic continuation of “unphysical” fluctuations.

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We present a theory of discontinuousmotion of particles incontinuous space-time. We show that the simplest nonrelativistic evolution equation of suchmotion is just the Schroedinger equation in quantum mechanics. This strongly implies what quantum mechanics describes is discontinuousmotion of particles. Considering the fact that space-time may be essentially discrete when considering gravity, we further present a theory of discontinuousmotion of particles in discrete space-time. We show that its evolution will naturally result (...) in the dynamical collapse process of the wave function, and this collapse will bring about the appearance ofcontinuousmotion of objects in the macroscopic world. (shrink)

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White presents an analysis of three ancient conceptions of spatial magnitude, time, andmotion, namely, Aristotle's, the Stoics', and the quantum views. The greater part of the book deals with Aristotle, according to whom one cannot get magnitude from points. The alleged mistakes in his theory ofmotion melt away if one agrees with his ontology. In the second chapter White discusses Aristotle's conception of time and "a time". Despite the lack of adequate mathematical tools Aristotle had an (...) amazing intuitive grasp of the concept ofcontinuousmotion. White sees a contrast between Aristotle and modern mathematics insofar as in the latter the concept of infinity is extended and magnitudes are sets of points. This difference of view has metaphysical roots. (shrink)

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The text is concerned with a class of two-sided stochastic processes of the form. Here is a two-sided Brownianmotion with random initial data at time zero and is a function of. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when is a jump process. Absolute continuity of under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure,, and (...) on with we verify i.e. where the product is taken over all coordinates. Here is the divergence of with respect to the initial position. Crucial for this is the temporal homogeneity of in the sense that,, where is the trajectory taking the constant value. By means of such a density, partial integration relative to a generator type operator of the process is established. Relative compactness of sequences of such processes is established. (shrink)

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This paper defends Plotinus’ reading ofSophist248e-249d as an expression of the togetherness or unity-in-duality of intellect and intelligible being. Throughout the dialogues Plato consistently presents knowledge as a togetherness of knower and known, expressing this through the myth of recollection and through metaphors of grasping, eating, and sexual union. He indicates that an intelligible paradigm is in the thought that apprehends it, and regularly regards the forms not as extrinsic “objects” but as the contents of living intelligence. A meticulous reading (...) ofSophist248e-249d shows that the “motion” attributed to intelligible being is not temporal change but the activity of intellectual apprehension. Aristotle’s doctrines of knowledge as identity of intellect and the intelligible, and of divine intellect as thinking itself, are therefore in continuity with Plato, and Plotinus’ doctrine of intellect and being iscontinuous with both Plato and Aristotle. (shrink)

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The purpose of the present study was two-fold. First we examined whether visiblemotion appearance was altered by the spatial interaction between invisible and visiblemotion. We addressed this issue by means of simultaneousmotion contrast, in which a horizontal test grating with a counterphase luminance modulation was seen to have the oppositemotion direction to a peripheral inducer grating with unidirectional upward or downwardmotion. Using a mirror stereoscope, observers viewed the inducer and test (...) gratings with one eye, andcontinuous flashes of colorful squares forming an annulus shape with the other eye. Thecontinuous flashes rendered the inducer subjectively invisible. The observers’ task was to report whether the test grating moved upward or downward. Consequently, simultaneousmotion contrast was observed even when the inducer was invisible . Second, we examined whether the observers could correctly respond to the direction of invisiblemotion: It was impossible. (shrink)

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Mr William Charlton ) quotes Russell onMotion. It has been pointed out to me that the book referred to in line 5 as Principles of Philosophy should in all probability be The Principles of Mathematics , and on looking into the matter I find that this is indeed the case. On p. 473 of the latter book I read: 447. It is to be observed that … we must entirely reject the notion of a state ofmotion. (...)Motion consists merely in the occupation of different places at different times, subject to continuity as explained in Part V. (shrink)

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The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation is obtained.

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There is general agreement in mathematics about what continuity is. In this paper we examine how well the mathematical definition lines up with common sense notions. We use a recent paper by Hud Hudson as a point of departure. Hudson argues that two objects moving continuously can coincide for all but the last moment of their histories and yet be separated in space at the end of this last moment. It turns out that Hudson’s construction does not deliver mathematically (...) class='Hi'>continuousmotion, but the natural question then is whether there is any merit in the alternative definition of continuity that he implicitly invokes. (shrink)

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The explanation of the phenomenon of Brownianmotion, given by Einstein in 1905 and based on the kinetic–molecular conception of matter, is considered one of the fundamental pillars supporting atomism in its victorious struggle against phenomenological physics in the early years of this century. Despite the importance of the subject, there exists no specific study on it of sufficient depth. Generally speaking, most histories of physics repeat the following scheme: the discovery made by Robert Brown in 1827 , of (...) thecontinuous movement of small particles suspended in a fluid did not arouse interest for a long time. Finally, at the close of the century, Gouy's research brought it to the attention of the physicists. Gouy was convinced that Brownianmotion constituted a clear demonstration of the existence of molecules incontinuous movement. Nevertheless, he did not work out any mathematized theory that could be subjected to quantitative confirmation. All nineteenth-century research remained at the qualitative level and yet it was able to clarify some general characteristics of the phenomenon: the completely irregular, unceasing,motion of the particles is not produced by external causes. It does not depend on the nature of the particles but only on their size. The first significant measurements, carried out by Felix Exner in 1900, appeared to deny the possibility of reconciling the kinetic theory with Brownianmotion. The discovery of the ultra-microscope then allowed Zsigmondy to perceive the presence of movements, which were completely analogous to Brown's, in the particles of the colloids; these movements were rather smaller in size than those invesigated up to then. Thus Zsigmondy aroused interest in the phenomenon. Finally, in 1905, Einstein succeeded in stating the mathematical laws governing the movements of particles on the basis of the principles of the kinetic–molecular theory. The following year Smoluchowski arrived at conclusions which corresponded to Einstein's. These laws received a first, rough confirmation in the years immediately following by the work of The Svedberg, Seddig and, for some historians, Henri. Then in 1908 Jean Perrin gave it a definitive confirmation. (shrink)

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The thermodynamics of averagedmotion treats the asymptotic spatiotemporal evolution of nonlinear irreversible processes. Dissipative and conservative actions are associated with short and long spatiotemporal scales, respectively. Themotion of asymptotically stable systems is slow, monotonic, andcontinuous, so that the microscopic state variable description of rapidmotion can be supplanted by an analysis of the macroscopic variable equations ofmotion of amplitude and phase. Rapidmotion is associated with instability, and the direction of (...) systemmotion is determined by thermodynamic criteria, in place of an analysis of the microscopic equations ofmotion. The characteristic structural configurations, deduced from the extremum principles of partial differential equations, are compared with the thermodynamic criteria. As a result of the nature of asymptoticmotion, variational principles exist which characterize the asymptotic states of the system. (shrink)

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Thomas Lynch, called "a cross between Garrison Keillor and William Butler Yeats" (New York Times), reminds us not only of how we die but also of how we live. "The facts of life and death remain the same. We live and die, we love and grieve, we breed and disappear. And between these existential gravities, we search for meaning, save our memories, leave a record for those who will remember us." So writes Thomas Lynch, poet and funeral director, and author (...) of the highly praised The Undertaking, winner of an American Book Award and finalist for the National Book Award, as he continues to examine the relations between the "literary and mortuary arts." The essays assembled here explore the human condition at the intersection of millennia, beleaguered by choices and changes, encumbered by merger and acquisition, numbed by maths and technologies, in search of the meaning of life and time, our lives and times. Lynch tenders life and times--sextons, muckrakers, clergy, caskets, condoms, loved poems, a hated cat, the mall, Main Street. In an age that seeks to define human experience in retail, high-tech, or pop-psyche terms, these essays speak to the existentials: between being human and ceasing to be, between birth and death, we are bodies inmotion and at rest. (shrink)

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This book presents a detailed analysis of three ancient models of spatial magnitude, time, and localmotion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in (...) its discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments. (shrink)

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It is hard for most people to appreciate how little they know of themselves in regards to basic aspects of living, or how these simple and apparently uninteresting aspects of ourselves can have an influence on the higher aspects of human life and culture. As Alain Berthoz, in his groundbreaking book, The Brain's Sense of Movement, points out, 'Plato forgot the body.' It is a huge omission that continues into today and affects thinking in all our attempts to understand such (...) aspects of ourselves as perception, cognition, emotion, and that major topic of this journal, consciousness. (shrink)

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Saccadic eye movements create (at least) two related but distinct problems for the visual system: they cause rapid imagemotion and a displacement of the retinal image. Although it is often assumed that themotion is too fast to be resolved, this is certainly not the case for low-spatial-frequency images. Recent experiments have suggested that the reason we are unaware of themotion during saccades is becausemotion channels are selectively suppressed, possibly by suppression of the (...) magno-cellular (but not the parvocellular) pathway. This suppression may explain why there is no sensation ofmotion during saccades, but it leaves open the problem of perceiving a stable world despite continual image displacements. (shrink)

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In his early discussion of the structure of matter andmotion, Leibniz quite explicitly appeals to Aristotle's characterization of continuity, and seems to adopt something like it as his own. Commentators usually assume that Leibniz continues to understand the notion of continuity in this way for the rest of his life. This paper argues that although he does continue to use something like the Aristotelian conception well into the mature period of his thought, he articulates a second sense of (...) continuity in his later writings that proves to be a greater importance to the exposition of his mature metaphysics. (shrink)

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Van Bendegem has recently offered an argument to the effect that, if time is discrete, then there should exist a correspondence between the motions of massive bodies and a discrete geometry. On this basis, he concludes that, even if space iscontinuous, it should nonetheless appear discrete. This paper examines the two possible ways of making sense of that correspondence, and shows that in neither case van Bendegem’s conclusion logically follows.

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Proper interpretation of the problem of contradiction inmotion has acquired pressing importance in recent years in conjunction with the development of dialectical materialist concepts ofmotion, space and time, continuity and discontinuity, infinity, and the like, and also in relations to problems of dialectical logic. In a number of writings , we have attempted to demonstrate that internal contradiction is inherent inmotion that inseparably connected with it are such contradictory factors as variability and stability, movement (...) and rest, with variability playing the leading role. Moreover, generally speaking, the concept of preservation, stability or rest means the preservation or maintenance of the very state ofmotion. (shrink)

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The optimal investment problem is studied for acontinuous time incomplete market model. It is assumed that the risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they are independent from the driving Brownianmotion, and they are currently observable. It is shown that some weakened version of Mutual Fund Theorem holds for this market for general class of utilities. It is shown that the supremum of expected utilities can be achieved on a (...) sequence of strategies with a certain distribution of risky assets that does not depend on risk preferences described by different utilities. (shrink)

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I argue in this paper that Descartes's goal with his doctrine of thecontinuous recreation of the world is to offer a unified and ultimate causal explanation for the possibility ofmotion and duration in the world, the permanence of created things, and the continuation of theirmotion and duration. This unified explanation seems to be the only one which, according to Descartes, satisfies the two basic requirements any ultímate cause should meet: the cause must be active (...) and not being inmotion itself. God's recreations of the world is Descartes's solution to this search. I also show in this article, on the one hand, that this doctrine successfully overcomes, in particular, the four major conflicts which threaten its consistency, and, on the other, the new meaning which the laws of nature acquire under the doctrine of thecontinuous recreation of the world. /// En este artículo defiendo que la pretensión principal de Descartes al proponer la doctrina de la recreación continua del mundo es encontrar una explicación unificada y última de la posibilidad de movimiento y duración en el mundo, de la permanencia de las cosas creadas y de la continuación del movimiento. Esta explicación unificada es la única que, según Descartes, satisface los dos requisitos que parece buscar en la explicación causal última del movimiento: la causa debe ser activa y no debe estar ella misma en movimiento. Las recreaciones del mundo constituyen esa causa. Por una parte, también muestro cómo esta solución supera, en concreto, los cuatro conflictos mayores que amenazan su consistencia, y, por otra, el nuevo significado que las leyes de la naturaleza adquieren en consonancia con esta doctrina de la recreación continua del mundo. (shrink)

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This study explores the relationship between language and thought in similarity judgments by testing how monolingual children who speak languages with partial typological differences inmotion description respond to visualmotion event stimuli. Participants were either Chinese- or English-speaking, 3-year-olds, 8-year-olds and adults who judged the similarity between causedmotion scenes in a match-to-sample task. The results suggest, first of all, that the two younger groups of 3-year-olds are predominantly path-oriented, irrespective of language, as evidenced by their (...) significantly longer fixation on path-match videos rather than manner-match videos in a preferential looking scheme. Using categorical measurement of overt choices, older children and adults also showed a shared tendency of being more path-oriented. However, the analysis usingcontinuous measurement of reaction time revealed significant variations in spatial cognition that can be related to linguistic differences: English speakers tended to be more manner-oriented while Chinese speakers were equally manner- and path-oriented. On the whole, our findings indicate a likelihood that children’s non-linguistic thought is similar prior to internalising the lexicalisation pattern ofmotion events in their native languages, but shows divergences after such habitual use, thus suggesting that the pattern of non-linguistic thought may be linked, among other things, to linguistic structure. (shrink)

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Human articulatedmotion can be readily recognized robustly even from impoverished so-called point-light displays. Such sequence information is processed by separate visual processing channels recruiting different stages at low and intermediate levels of the cortical visual processing hierarchy. The different contributions thatmotion and form information make to form articulated, or biological,motion perception are still under investigation. Here we investigate experimentally whether and how specific spatio-temporal features, such as extrema in themotion energy or maximum (...) limb expansion, indicated by the lateral and longitudinal extension, constrain the formation of the representations of articulated bodymotion. In order to isolate the relevant stimulus properties we suggest a novel masking technique, which allows to selectively impair the ankle information of the body configuration while keeping themotion of the point-light locations intact. Our results provide evidence that maxima in feature channel representations, e.g., the lateral or longitudinal extension, define elemental features to specify key poses of biologicalmotion patterns. These findings provide support for models which aim at automatically building visual representations for the cortical processing of articulatedmotion by identifying temporally localized events in acontinuous input stream. (shrink)

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The debate about cinematicmotion revolves around the question of whether the movement of cinematic images is real. That the movement we perceive in film should be construed as the movement of images is taken for granted. But this is a mistake. There is no reason to suppose that cinematic images of moving objects are themselves perceived to be moving. All that is necessary is to perceive these images as continuously changing images of one and the same object.

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The path-integral approach to quantum theory ofcontinuous measurements has been developed in preceding works of the author. According to this approach the measurement amplitude determining probabilities of different outputs of the measurement can be evaluated in the form of a restricted path integral (a path integral “in finite limits”). With the help of the measurement amplitude, maximum deviation of measurement outputs from the classical one can be easily determined. The aim of the present paper is to express this (...) variance in a simpler and transparent form of a specific uncertainty principle (called the action uncertainty principle, AUP). The most simple (but weak) form of AUP is δS≳ℏ, whereS is the action functional. It can be applied for simple derivation of the Bohr-Rosenfeld inequality for measurability of gravitational field. A stronger (and having wider application) form of AUP (for ideal measurements performed in the quantum regime) is |∫ ′ t″ (δS[q]/δq(t))Δq(t)dt|≃ℏ, where the paths [q] and [Δq] stand correspondingly for the measurement output and for the measurement error. It can also be presented in symbolic form as Δ(Equation) Δ(Path) ≃ ℏ. This means that deviation of the observed (measured)motion from that obeying the classical equation ofmotion is reciprocally proportional to the uncertainty in a path (the latter uncertainty resulting from the measurement error). The consequence of AUP is that improving the measurement precision beyond the threshold of the quantum regime leads to decreasing information resulting from the measurement. (shrink)

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In Book 1 of the Principia, Newton presented two different descriptions of orbitalmotion under the action of a central force. In Prop. 1, he described thismotion as a limit of the action of a sequence of periodic force impulses, while in Prop. 6, he described it by the deviation from inertialmotion due to acontinuous force. From the start, however, the equivalence of these two descriptions has been the subject of controversies. Perhaps the (...) earliest one was the famous discussion from December 1704 to 1706 between Leibniz and the French mathematician Pierre Varignon. But confusion about this subject has remained up to the present time. Recently, Pourciau has rekindled these controversies in an article in this journal, by arguing that “Newton never tested the validity of the equivalency of his two descriptions because he does not see that his assumption could be questioned. And yet the validity of this unseen and untested equivalence assumption is crucial to Newton’s most basic conclusions concerning one-bodymotion” (Pourciau in Arch Hist Exact Sci 58:283–321, 2004, 295). But several revisions of Props. 1 and 6 that Newton made after the publication in 1687 of the first edition of the Principia reveal that he did become concerned to provide mathematical proof for the equivalence of his seemingly different descriptions of orbitalmotion in these two propositions. In this article, we present the evidence that in the second and third edition of the Principia, Newton gave valid demonstrations of this equivalence that are encapsulated in a novel diagram discussed in Sect. 4. (shrink)

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The relationship between the continuity equation and the HamiltonianH of a quantum system is investigated from a nonstandard point of view. In contrast to the usual approaches, the expression of the current densityJ ψ is givenab initio by means of a transport-velocity operatorV T, whose existence follows from a “weak” formulation of the correspondence principle. Once given a Hilbert-space metricM, it is shown that the equation ofmotion and the continuity equation actually represent a system in theunknown operatorsH andV (...) T, due to the arbitrariness on the initial condition of the quantum state. The general solution is given in some cases of special interest and a straightforward application to relativistic quantum mechanics is performed. (shrink)

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Leibniz viewed the principle of continuity, the principle that all natural changes are produced by degrees, as a useful heuristic for evaluating the truth of a theory. Since the Cartesian laws ofmotion entailed discontinuities in the natural order, Leibniz could safely reject it as a false theory. The principle of continuity has similar implications for analyses of Leibniz's theory of consciousness. I briefly survey the three main interpretations of Leibniz's theory of consciousness and argue that the standard account (...) entails a discontinuity that Leibniz could not allow. I argue that the principle of continuity and the textual data favor an interpretation according to which a conscious mental state just is a perception that is distinct to a sufficient degree. (shrink)

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It is shown that the following three common understandings of Newton’s laws ofmotion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisiblecontinuous body (...) with finite mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible. (shrink)

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: Descartes provides an original and puzzling argument for the traditional theological doctrine that the world is continuously created by God. His key premise is that the parts of the duration of anything are "completely independent" of one another. I argue that Descartes derives this temporal independence thesis simply from the principle that causes are necessarily simultaneous with their effects. I argue further that it follows from Descartes's version of thecontinuous creation doctrine that God is the instantaneous and (...) total cause of everything that happens, and that this is just what his physics demands. But although God is the total cause of everything, he is not the only cause, since Descartes considers it obvious that finite minds have the power to move bodies. In the face of this apparent paradox, several recent commentators have suggested that Descartes accepted the late scholastic view that God and finite causes somehow collaborate or concur in the production of numerically identical effects. But close examination reveals that the case for Cartesian concurrentism is weak. Fortunately, there is a simpler and more fruitful solution to the problem of reconciling divine and human action. I argue that that in Descartes's world certain motions, such as voluntary movements of our bodies, are causally overdetermined This account allows Descartes to avoid occasionalism without embracing an elaborate metaphysics of secondary causality. Finally, I argue that the overdeterminist interpretation sheds new light on two longstanding problems of Cartesian metaphysics. First, it resolves a serious difficulty with a standard reading of Descartes's conception of human freedom. Second, it offers a novel approach to the old problem of reconciling genuine human action with the principle of the conservation of total quantity ofmotion. (shrink)

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Ben Woodard's book, exemplarily erudite and so not for the faint of heart, guides the reader through all periods of F.W.J. Schelling's thought—a feat important in its own right given the burgeoning Schelling renaissance in the English-speaking philosophical universe. Woodard follows this trajectory not in order to periodize this giant of German philosophy but rather in order to demonstrate the continuity of Schelling's thought through the leitmotiv of naturalism. Schelling's naturalism, however, is not of the sort to which we have (...) become accustomed. As Woodard rightly insists, for Schelling nature is "not some local part of the universe" but instead names the universal processes productive of... (shrink)

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