The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the (...) last hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics. (shrink)

Let a smooth experience be an experience with perfectly gradual changes in phenomenal character. Consider, as examples, your visual experience of a blue sky or your auditory experience of a rising pitch. Do the phenomenal characters of smooth experiences have continuous or discrete structures? If we appeal merely to introspection, then it may seem that we should think that smooth experiences are continuous. This paper (1) uses formal tools to clarify what it means to say that an (...) experience is continuous or discrete, and (2) develops a discrete model of the phenomenal characters of smooth experiences. As a result, I'll argue that introspection leaves open whether smooth experiences are continuous or discrete. Yet I’ll also argue—perhaps surprisingly—that the discrete theory may better fit our introspective evidence. (shrink)

This paper presents a theoretical study of the binary oppositions underlying the mechanisms of natural computation understood as dynamical processes on natural information morphologies. Of special interest are the oppositions of discrete vs. continuous, structure vs. process, and differentiation vs. integration. The framework used is that of computing nature, where all natural processes at different levels of organisation are computations over informational structures. The interactions at different levels of granularity/organisation in nature, and the character of the phenomena that (...) unfold through those interactions, are modeled from the perspective of an observing agent. This brings us to the movement from binary oppositions to dynamic networks built upon mutually related binary oppositions, where each node has several properties. (shrink)

Advocates of dynamic systems have suggested that higher mental processes are based on continuous representations. In order to evaluate this claim, we first define the concept of representation, and rigorously distinguish between discrete representations and continuous representations. We also explore two important bases of representational content. Then, we present seven arguments that discrete representations are necessary for any system that must discriminate between two or more states. It follows that higher mental processes require discrete representations. (...) We also argue that discrete representations are more influenced by conceptual role than continuous representations. We end by arguing that the presence of discrete representations in cognitive systems entails that computationalism (i.e., the view that the mind is a computational device) is true, and that cognitive science should embrace representational pluralism. (shrink)

The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced (...) by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps. (shrink)

This paper presents a new modified quantum mechanics, Critical Complexity Quantum Mechanics, which includes a new account of wavefunction collapse. This modified quantum mechanics is shown to arise naturally from a fully discrete physics, where all physical quantities are discrete rather than continuous. I compare this theory with the spontaneous collapse theories of Ghirardi, Rimini, Weber and Pearle and discuss some implications of these theories and CCQM for a realist view of the quantum realm.

One of the oldest questions in science and philosophy is whether reality is fundamentally discrete, or fundamentally continuous. In this short essay I show that this question should first be classified into two main categories. 1) Intrinsic existence, 2) Behavioral existence. I show that the first category should be continuous by definition, while the second category could probably be either continuous or discrete.

Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and explain their relation to (...) reasoning fail, so that ignorance of their nature is profound. (shrink)

The ancient Greek philosophers – like Parmenides – reasoned that observable reality cannot exist by itself. It has to be a creation of an underlying reality. An all-in­clusive existence that has a structure because observable reality shows structure at every scale size. Although observable reality is involved in a continuous transformation too. If our concept about the relation between phenomenological reality and the creating underlying reality is correct, the unification of the properties of phenomenological reality is part of an (...) enveloping mathematical model. DOI: 10.5281/zenodo.5457368. (shrink)

A recently proposed model of sensory processing suggests that perceptual experience is updated in discrete steps. We show that the data advanced to support discrete perception are in fact compatible with a continuous account of perception. Physiological and psychophysical constraints, moreover, as well as our awake-primate imaging data, imply that human neuronal networks cannot support discrete updates of perceptual content at the maximal update rates consistent with phenomenology. A more comprehensive approach to understanding the physiology of (...) perception (and experience at large) is therefore called for, and we briefly outline our take on the problem. (shrink)

The original conception of atomism suggests “atoms”, which cannot be divided more into composing parts. However, the name “atom” in physics is reserved for entities, which can be divided into electrons, protons, neutrons and other “elementary particles”, some of which are in turn compounded by other, “more elementary” ones. Instead of this, quantum mechanics is grounded on the actually indivisible quanta of action limited by the fundamental Planck constant. It resolves the problem of how both discrete and continuous (...) (even smooth) to be described uniformly and invariantly in thus. Quantum mechanics can be interpreted in terms of quantum information. Qubit is the indivisible unit (“atom”) of quantum information. The imagery of atomism in modern physics moves from atoms of matter (or energy) via “atoms” (quanta) of action to “atoms” (qubits) of quantum information. This is a conceptual shift in the cognition of reality to terms of information, choice, and time. (shrink)

Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.

Throughout these last few decades, phenomenology and modern physics have slowly started to approach each other in order to bridge the gap between subjective and objective. In this thesis I aim to show an approach done with the help of Karen Barad's agential realism; a quantum interpretation enabling us to better understand and analyse our complex world, as well as our perception of it. Taking inspiration from new materialism, phenomenology, and physics, I see a need to properly leave discrete (...) dualism behind, in order to be able to describe cultural and material structures as continuous manifolds. Instead of endlessly searching for their binary and changeless parts. (shrink)

Niels Bohr gave us the model of the atom as having a central nucleus around which electrons were circulating in stable orbits. He also gave us the complementarity principle that states that the mutually exclusive wave and corpuscular nature of light were not merely contradictory but complementary descriptions. Field theory considers light as a continuous wave phenomenon with a wavelength and frequency, while quantum theory considers its corpuscular nature as a discrete packet of energy called a photon. Thus (...) we actually have an opposition of a continuous-discontinuous description concerning the fundamental nature of light. In line with what we have been discussing about the nature of reality as having an intrinsic polar nature, we have yet another confirmation at even the atomic level of investigation. This harks back to Kant’s “Critique of Pure Reason” where in his second antinomy he tried to show that the continuous and discontinuous descriptions of the cosmos were both possible although they were mutually exclusive of each other. Kant tried to demonstrate that this was a limitation of the way we think about the world, i.e. a defect of reason since the world is obviously content to go on as a single reality regardless of how we understand it. But we are showing here that our understanding of the world does not have to be antagonistic to it if we understand it properly. For Kant, opposites merely exclude one another. We are claiming that not only do they exclude each other but also they depend upon each other for either to exist at all. (shrink)

In this paper, I want to look at Deleuze’s philosophical heritage in two different senses. In the first part of the paper, I explore his relationship to perhaps the most influential philosopher of the twentieth century, Martin Heidegger. Heidegger plays a central role in Deleuze’s early philosophy, and even when in his later collaborations with Guattari their explicit references to Heidegger are dismissive, Heidegger’s influence can clearly be detected, particularly in their critiques of other philosophers. In the second part of (...) the paper, I look at Deleuze’s own contribution to philosophy, and to see how this contribution has been assessed by one of the most influential contemporary French philosophers, Alain Badiou. For Heidegger, Deleuze, and Badiou, perhaps the central problem for philosophy emerges from thinking about totality. For all three, the traditional metaphysical view of totality, derived from Aristotle’s concept of paronymy, occludes rather than solves the problem of how we characterise our most general concepts. As we shall see, Heidegger’s diagnosis of metaphysics, as constituted by what he calls onto-theology, is shared by all three philosophers, while their responses to this diagnosis differ. Deleuze and Badiou both reject Heidegger’s poetics of being in favour of the language of mathematics, but the question I want to explore in the final part of the paper is, which mathematics? The mathematics of the continuous, or the mathematics of the discrete? (shrink)

A generalized and unifying viewpoint to both general relativity and quantum mechanics and information is investigated. It may be described as a generaliztion of the concept of reference frame from mechanics to thermodynamics, or from a reference frame linked to an element of a system, and thus, within it, to another reference frame linked to the whole of the system or to any of other similar systems, and thus, out of it. Furthermore, the former is the viewpoint of general relativity, (...) the latter is that of quantum mechanics and information. Ciclicity in the manner of Nicolas Cusanus (Nicolas of Cusa) is complemented as a fundamental and definitive property of any totality, e.g. physically, that of the universe. It has to contain its externality within it somehow being namely the totality. This implies a seemingly paradoxical (in fact, only to common sense rather logically and mathematically) viewpoint for the universe to be repesented within it as each one quant of action according to the fundamental Planck constant. That approach implies the unification of gravity and entanglement correspondiing to the former or latter class of reference frames. An invariance, more general than Einstein's general covariance is to be involved as to both classes of reference frames unifying them. Its essence is the unification of the discrete and cotnitinuous (smooth). That idea underlies implicitly quantum mechanics for Bohr's principle that it study the system of quantum microscopic entities and the macroscopic apparatus desribed uniformly by the smmoth equations of classical physics. (shrink)

In this work we consider the problem of the approximate hedging of a contingent claim in the minimum mean square deviation criterion. A theorem on martingale representation in case of discrete time and an application of the result for semi-continuous market model are also given.

The photon is typically regarded as a unitary object that is both particle-discrete and wave-continuous. This is a paradoxical position and we live with it by making dualism a fundamental feature of radiation. It is argued here that the photon is not unitary; rather it has two identities, one supporting discrete behavior and the other supporting continuous (wave) behavior. There is photon kinetic energy that is always discrete/localized on arrival; it never splits (on half-silvered mirrors) (...) or diffracts (in pinholes or slits). Then there is the photon’s probability wavefront that is continuous and diffractable. Acknowledging that the photon has two identities explains the photon’s dual nature. And wave-particle duality is central to quantum mechanics. Understanding it leads to new insights into the photon’s constant velocity and its entanglement with another photon. (shrink)

Like elsewhere in Nigeria and Africa, the imposition of colonial rule on Batombuland and the incursion of western ideas produced profound socio-cultural, economic and political changes in the Batombu society. However, unlike several Nigerian and African peoples whose histories have received extensive scholarly attention, the history of the Batombu has attracted very little scholarly attention. Thus virtually neglected, the Batombu occupies a mere footnote position in the extant historiography of Nigeria. This is the gap this article seeks to fill. It (...) examines the impact of colonialism and western civilisation on Batombu’s political, social, economic and cultural institutions and concludes that as profound and far-reaching as these changes were some important aspects of the indigenous institutions and traditional practices of the people survived. (shrink)

Craig (1981) presents and defends several different kalam cosmological arguments. The core of each of these arguments is the following ur argument.

Fifty years of effort in artificial intelligence (AI) and the formalization of legal reasoning have produced both successes and failures. Considerable success in organizing and displaying evidence and its interrelationships has been accompanied by failure to achieve the original ambition of AI as applied to law: fully automated legal decision-making. The obstacles to formalizing legal reasoning have proved to be the same ones that make the formalization of commonsense reasoning so difficult, and are most evident where legal reasoning has to (...) meld with the vast web of ordinary human knowledge of the world. Underlying many of the problems is the mismatch between the discreteness of symbol manipulation and the continuous nature of imprecise natural language, of degrees of similarity and analogy, and of probabilities. (shrink)

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...) of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given. (shrink)

In the philosophy of the analytical tradition, set theory and formal logic are familiar formal tools. I think there is no deep reason why the philosopher’s tool kit should be restricted to just these theories. It might well be the case—to generalize a dictum of Suppes concerning philosophy of science—that the appropriate formal device for doing philosophy is mathematics in general; it may be set theory, algebra, topology, or any other realm of mathematics. In this paper I want to employ (...) elementary topological considerations to shed new light on the intricate problem of the relation of qualities and similarity. Thereby I want to make plausible the general thesis that topology might be a useful device for matters epistemological. (shrink)

We propose that all actual causes are simultaneous with their direct effects, as illustrated by both everyday examples and the laws of physics. We contrast this view with the sequential conception of causation, according to which causes must occur prior to their effects. The key difference between the two views of causation lies in differing assumptions about the mathematical structure of time.

Physics has been slowly and reluctantly beginning to address the role and fundamental basis of the ‘observer’ which has until now also been considered metaphysical and beyond the mandate empirical rigor. It is suggested that the fundamental premise of the currently dominant view of ‘Cognitive Theory’ - “Mind Equals Brain” is erroneous; and the associated belief that the ‘Planck scale, ‘the so-called basement level of reality’, as an appropriate arena from which to model psycho-physical bridging is also in error. In (...) this paper we delineate a simple, inexpensive experimental design to ‘crack the so-called cosmic egg’ thereby opening the door to largescale extra dimensions (LSXD) tantamount to the regime of the unified field and thus awareness. The methodology surmounts the quantum uncertainty principle in a manner violating Quantum Electrodynamics, (QED), a cornerstone of modern theoretical physics, by spectrographic analysis of newly theorized Tight-Bound State (TBS) Bohr orbits in ‘continuous-state’ transition frequencies of atomic hydrogen. If one wonders why QED violation in the spectra of atomic hydrogen relates to solving the mind-body (observer) problem; consider this a 1st wrench in a forthcoming tool box of Unified Field Mechanics, UF that will soon enough in retrospect cause the current tools of Classical and Quantum Mechanics to appear as stone axes. Max Planck is credited as the founder of quantum mechanics with his 1900 quantum hypothesis that energy is radiated and absorbed discretely by the formulation, E = hv. Empirically implementing this next paradigm shift utilizing parameters of the long sought associated ‘new physics’ of the 3rd regime (classical-quantum- unified) allows access to LSXD of space; thus pragmatically opening the domain of mental action for the 1st time in history. This rendering constitutes a massive paradigm shift to Unified Field Theory creating a challenge for both the writer and the reader! (shrink)

Norbert Wiener’s idea of “cybernetics” is linked to temporality as in a physical as in a philosophical sense. “Time orders” can be the slogan of that natural cybernetics of time: time orders by itself in its “screen” in virtue of being a well-ordering valid until the present moment and dividing any totality into two parts: the well-ordered of the past and the yet unordered of the future therefore sharing the common boundary of the present between them when the ordering is (...) taking place by choices. Thus, the quantity of information defined by units of choices, whether bits or qubits, describes that process of ordering happening in the present moment. The totality (which can be considered also as a particular or “regional” totality) turns out to be divided into two parts: the internality of the past and the externality of the future by the course of time, but identifiable to each other in virtue of scientific transcendentalism (e.g. mathematical, physical, and historical transcendentalism). A properly mathematical approach to the “totality and time” is introduced by the abstract concept of “evolutionary tree” (i.e. regardless of the specific nature of that to which refers: such as biological evolution, Feynman trajectories, social and historical development, etc.), Then, the other half of the future can be represented as a deformed mirror image of the evolutionary tree taken place already in the past: therefore the past and future part are seen to be unifiable as a mirrorly doubled evolutionary tree and thus representable as generalized Feynman trajectories. The formalism of the separable complex Hilbert space (respectively, the qubit Hilbert space) applied and further elaborated in quantum mechanics in order to uniform temporal and reversible, discrete and continuous processes is relevant. Then, the past and future parts of evolutionary tree would constitute a wave function (or even only a single qubit once the concept of actual infinity be involved to real processes). Each of both parts of it, i.e. either the future evolutionary tree or its deformed mirror image, would represented a “half of the whole”. The two halves can be considered as the two disjunctive states of any bit as two fundamentally inseparable (in virtue of quantum correlation) “halves” of any qubit. A few important corollaries exemplify that natural cybernetics of time. (shrink)

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows: 1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist. 2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and (...) especially that of conservation of energy: This is the conservation of action or information. 3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete. 4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole. 5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality. 6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice. The list is not intended to be exhausted or disjunctive, but only to give an introductory idea. (shrink)

In 1922, Thoralf Skolem introduced the term of «relativity» as to infinity от set theory. Не demonstrated Ьу Zermelo 's axiomatics of set theory (incl. the axiom of choice) that there exists unintended interpretations of anу infinite set. Тhus, the notion of set was also «relative». We сan apply his argurnentation to Gödel's incompleteness theorems (1931) as well as to his completeness theorem (1930). Then, both the incompleteness of Реапо arithmetic and the completeness of first-order logic tum out to bе (...) also «relative» in Skolem 's sense. Skolem 's «relativity» argumentation of that kind сan bе applied to а vету wide range of problems and one сan spoke of the relativity of discreteness and continuity or, of finiiteness and infinity, or, of Cantor 's kinds of infinities, etc. The relativity of Skolemian type helps us for generaIizing Einstein 's principle of relativity from the invariance of the physical laws toward diffeomorphisms to their invariance toward anу morphisms (including and especiaIly the discrete ones). Such а kind of generalization from diffeomorphisms (then, the notion of velocity always makes sense) to anу kind of morphism (when 'velocity' mау оr maу not make sense) is an extension of the general Skolemian type оГ relativity between discreteness and continuity от between finiteness and infinity. Particularly, the Lorentz invariance is not valid in general because the notion of velocity is limited to diffeomorphisms. [п the case of entanglement, the physical interaction is discrete0. 'Velocity" and consequently, the 'Lorentz invariance'"do not make sense. Тhat is the simplest explanation ofthe argurnent EPR, which tums into а paradox оnJу if the universal validity of 'velocity' and 'Lогелtz invariance' is implicitly accepted. (shrink)

The book is devoted to the contemporary stage of quantum mechanics – quantum information, and especially to its philosophical interpretation and comprehension: the first one of a series monographs about the philosophy of quantum information. The second will consider Be l l ’ s inequalities, their modified variants and similar to them relations. The beginning of quantum information was in the thirties of the last century. Its speed development has started over the last two decades. The main phenomenon is entanglement. (...) The subareas are quantum computer, quantum communication (and teleportation), and quantum cryptography. The book offers the following main conceptions, theses and hypotheses: – dualistic Phythagoreanism as a new kind among the interpretations of quantum mechanics and information: arithmetical, logical, and metamathematical one; – Gödel ’ s first incompleteness theorem is an undecidable proposition, and consequently the second one,too. – a partial rehabilitation of Hilbert ’ s program for the self-foundation of mathematics; – the dual-foundation of mathematics; – Skolemian relativity between: Cantor ’s kinds of infinity, finiteness and infinity, discreteness and continuity, completeness and incompleteness, etc.; – information is a physical quantity representing the non-reducibility of a system to its parts, particularly nonaddtivity; – there exist pure relations «by itself», which cannot be reduced to predications; – energy conservation can and should be generalized; – Einstein’ s «general covariance» or «principle of relativity» can and should be generalized to cover discrete morphisms where the notion of velocity does not make sense. (shrink)

Despite its central role in his important theories of self and external world, Hume’s account of numerical identity has been neglected or misunderstood. The account is designed as a response to a difficulty concerning identity apparently original with Hume. I argue that the problem is real, crucial, and remains unresolved today. Hume’s response to the difficulty enlists his idiosyncratic, empiricist views on time: time consists of discrete, partless moments, some of which coexist with successions of others. Time is more (...) like a wall of variously sized bricks than like a continuous line. Hume’s arguments that time (and space) are not infinitely divisible have met with literal contempt. I show that his unusual views are motivated and consistent. The topic of identity leads naturally to Hume’s account of personal identity and his later retraction--one of the most widely discussed topics in Hume scholarship. I give a new, straightforward explanation of the retraction, by arguing that Hume’s views on consciousness preclude his prior account of the self as a fiction. I then suggest that Hume’s fundamental problem for personal identity is his general difficulty concerning identity. Discussing Hume’s metaphysics raises perhaps the most central and difficult topic in Hume scholarship--how to reconcile the constructive, theoretical Hume with the skeptical Hume. The prevailing view for the last century has been that Hume’s skepticism is limited, leaving room for his theorizing. I argue, rather, that Hume is an unlimited Pyrrhonian skeptic in relevant respects, and that this interpretation of him best reconciles his two sides. (shrink)

While the substantiation of “best interests” has received much attention, the question of how “the child” is conceptualised to ensure any action taken or decision made is in the particular child’s best interests has been largely neglected. In this paper, I argue that the lack of robust understanding of who “the child” is means that we continue to make many generalisations and category-based assumptions in determining the child’s best interests. In addressing the challenge of doing right by the individual child, (...) I propose a three-step approach based on a theoretical model of the child that avoids presumptions about child-typical needs and insists on an assessment of the child’s individual characteristics, needs, qualities, and circumstances, making it the only conceptualisation fully meeting the child centrism criterion required by children’s rights as determined by the United Nations Convention on the Rights of the Child. (shrink)

This is a metaphysical and conceptual analysis of the concepts ‘change’ and ‘continuity’. The Buddhists are in agreement with Heraclitus that all are flowing and nothing remains. However, the Buddhists have a much more elaborate theory about change and continuity, and this theory is a key element in the entire Buddhist system of related doctrines, viz., that of karma and rebirth, the possibility of Liberation and others. Simply put, the Buddhist emphasizes that change is there in every aspect of reality. (...) According to a later developed form of the Buddhist teaching, change is absolutely pervasive, and even these particles are subject to change as they are nothing more than putative objects which are conceptualized to be such and such, and without the conceptualization they are ‘nothing’ at all.. Hence it seems that continuity is not possible. But in fact according to the later theory, change is not only possible, it is accepted as part and parcel of everyday life. The fact that nothing at all remains the same does not imply that continuity is not possible, since continuity does not always have to be that of an inherently existing object. A changing object can be continued also, in roughly the same sense as we say that an event, like a drama, continues even though everything in it is changing. The thread that ties the disparate elements of the event together in this case lies within our own conceptual imputation. This does not imply that everything is subjective, since the distinction between subject and object presupposes the idea of an absolutely existing individual self, which all Buddhist schools rejects. So in this later theory, absolute change is not possible because there is, ultimately speaking nothing to change, and when there is no change there is no continuation either. This is not to deny the empirical fact of changes and continuities that are present to us; things are there and they are indeed changing. What is being denied here is the belief that that there are essences to things which endure through all the changes. Since things are empty they can change, and continuity is only possible, not because there is something that exists and endures, but because there is change. A drama that does not move cannot be continued. The foregoing discussions of the Buddhist theories have many implications for the dialog between science and religion. One point is that science still seems to subscribe to the object/subject distinction. But if change and continuity are not real in the ultimate sense, then perhaps the distinction should be reconsidered. Another point concerns how to find continuity amidst all the change. But perhaps in some important sense continuity depends on us. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by explaining in (...) simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

The early twentieth century witnessed the emergence of "objective relativism," a distinctly American school of metaphysical realism inspired by the works of John Dewey and A.N. Whitehead. Largely forgotten, objective relativism provided a metaphysical framework, based upon an ontology of events and relations rather than substances and discrete properties, that has continued relevance for contemporary metaphysical discussions. In this thesis, I attempt to chart the boundaries and pathways of this ontology, outlining what Dewey calls the "ground-map of the province (...) of criticism." In particular, the ground-map of objective relativism is invoked to situate and analyze the model of psycho-physical emergence outlined in Dewey's Experience and Nature. Because it is a relational ontology, objective relativism avoids problems with emergence common to substantival models. Additional analyses of its ontological premises, both in Dewey's writings and elsewhere, demonstrate how compelling accounts of causation, consciousness, and meaning may be formulated within this model. (shrink)

Grief is, and has always been, technologically supported. From memorials and shrines to photos and saved voicemail messages, we engage with the dead through the technologies available to us. As our technologies evolve, so does how we grieve. In this paper, we consider the role chatbots might play in our grieving practices. Influenced by recent phenomenological work, we begin by thinking about the character of grief. Next, we consider work on developing “continuing bonds” with the dead. We argue that for (...) some, chatbots may play an important role in establishing these continuing bonds by helping us develop what we term “habits of intimacy”. We then turn to the “ick factor” some may feel about this prospect, focusing especially on ethical concerns raised by Patrick Stokes and Adam Buben about the risk of replacing our dead with chatbots. We argue that replacement worries are not as pressing as Stokes and Buben suggest. We resist these replacement worries by appealing to the “thin reciprocity”, as we refer to it, that such bots offer, as well as the fictionalist stance that we think users of the bots adopt when engaging with them. We conclude by briefly raising some additional concerns and highlighting future research questions. (shrink)

What is the state of play for science advice to the government and Parliament? After almost ten years with a prime minister’s chief science advisor, are there lessons to be learnt? How can we continue to ensure that science advice is effective, balanced, transparent and rigorous, while at the same time balancing the need for discretion and confidentiality? In this article, we suggest that the hallmarks of good science – transparency and peer review – can be balanced against the need (...) to provide confidential advice in an Aotearoa New Zealand context. To complement the advice to the prime minister, an expanded role for the Royal Society Te Apärangi would support public and parliamentary understanding of science and science issues relevant to policy. (shrink)

We present in this paper the neutrosophic randomized variables, which are a generalization of the classical random variables obtained from the application of the neutrosophic logic (a new nonclassical logic which was founded by the American philosopher and mathematical Florentin Smarandache, which he introduced as a generalization of fuzzy logic especially the intuitionistic fuzzy logic ) on classical random variables. The neutrosophic random variable is changed because of the randomization, the indeterminacy and the values it takes, which represent the possible (...) results and the possible indeterminacy. Then we classify the neutrosophic randomized variables into two types of discrete and continuous neutrosophic random variables and we define the expected value and variance of the neutrosophic random variable then offer some illustrative examples. -/- . (shrink)

Radical and autopoietic enactivists disagree concerning how to understand the concept of sense-making in enactivist discourse and the extent of its distribution within the organic domain. I situate this debate within a broader conflict of commitments to naturalism on the part of radical enactivists, and to phenomenology on the part of autopoietic enactivists. I argue that autopoietic enactivists are in part responsible for the obscurity of the notion of sense-making by attributing it univocally to sentient and non-sentient beings and following (...) Hans Jonas in maintaining a phenomenological dimension to life-mind continuity among all living beings, sentient or non-sentient. I propose following Merleau-Ponty instead, who offers a properly phenomenological notion of sense-making for which sentience is a necessary condition. Against radicalist efforts to replace sense-making with a deflationary, naturalist conception of intentionality, I discuss the role of the phenomenological notion of sense-making for understanding animal behavior and experience. (shrink)

Among the many problems facing u.s. immigration law is a crisis of discretion and judicial deference. Through two recently passed laws, the United States Congress and the President have seriously limited judicial review o f discretionary immigration decisions o f the Board ofImmigration Appeals. This Article focuses on this preclusion of judicial review of discretionary agency decisions. The Article begins with an examination ofdiscretion from a theoretical perspective, and goes on to examine areas o f immigration law in which discretion (...) was traditionally most imponant. That examination is followed by a discussion of immigration-law scholarship and critique. The Article concludes that what is needed in U.S. immigration law is greater restraint, and calls on the Judiciary to provide more oversight regarding discretionary agency decisions. (shrink)

This Essay analyzes an essay by H. L. A. Hart about discretion that has never before been published, and has often been considered lost. Hart, one of the most significant legal philosophers of the twentieth century, wrote the essay at Harvard Law School in November 1956, shortly after he arrived as a visiting professor. In the essay, Hart argued that discretion is a special mode of reasoned, constrained decisionmaking that occupies a middle ground between arbitrary choice and determinate rule application. (...) Hart believed that discretion, soundly exercised, provides a principled way of coping with legal indeterminacy that is fully consistent with the rule of law. This Essay situates Hart’s paper – Discretion – in historical and intellectual context, interprets its main arguments, and assesses its significance in jurisprudential history. In the context of Hart’s work, Discretion is notable because it sketches a theory of legal reasoning in depth, with vivid examples. In the context of jurisprudential history, Discretion is significant because it sheds new light on long-overlooked historical and theoretical connections between Hart’s work and the Legal Process School, the American jurisprudential movement dominant at Harvard during Hart’s year as a visiting professor. Hart’s Discretion is part of our jurisprudential heritage, advancing our understanding of legal philosophy and its history. (shrink)

Aristotle argues that time depends on soul to count it, but adds that motion, which makes time what it is, may be independent of soul. The claim that time depends on soul or mind implies that there is at least one measurable property of natural beings that exists because of the mind’s activity. This paper argues that for Aristotle time depends partly on soul, but more importantly on motion, which defines a continuum. This argument offers a robust metaphysics of time. (...) In contrast to modern philosophy of physics, for Aristotle the continuum of motion is prior in being to time, while time is a hybrid of the real continuum of motion and the activity of mind. (shrink)

Leibniz argues against Descartes’s conception of material substance based on considerations of unity. I examine a key premise of Leibniz’s argument, what I call the Plurality Thesis—the claim that matter (i.e. extension alone) is a plurality of parts. More specifically, I engage an objection to the Plurality Thesis stemming from what I call Material Monism—the claim that the physical world is a single material substance. I argue that Leibniz can productively engage this objection based on his view that matter is (...) discrete. The discreteness of matter provides two aspects of support for the Plurality Thesis. First, it indicates that the parts of matter do not share boundaries and are, therefore, independent in an important sense. Second, it indicates that the parts of matter are determinate and are, therefore, ontologically prior to the wholes they compose. (shrink)

A number of cases involving self-locating beliefs have been discussed in the Bayesian literature. I suggest that many of these cases, such as the sleeping beauty case, are entangled with issues that are independent of self-locating beliefs per se. In light of this, I propose a division of labor: we should address each of these issues separately before we try to provide a comprehensive account of belief updating. By way of example, I sketch some ways of extending Bayesianism in order (...) to accommodate these issues. Then, putting these other issues aside, I sketch some ways of extending Bayesianism in order to accommodate self-locating beliefs. Finally, I propose a constraint on updating rules, the "Learning Principle", which rules out certain kinds of troubling belief changes, and I use this principle to assess some of the available options. (shrink)

The widely accepted “continuing bonds” model of grief tells us that rather than bereavement necessitating the cessation of one’s relationship with the deceased, very often the relationship continues instead in an adapted form. However, this framework appears to conflict with philosophical approaches that treat reciprocity or mutuality of some form as central to loving relationships. Seemingly the dead cannot be active participants, rendering it puzzling how we should understand claims about continued relationships with them. In this article, we resolve this (...) tension by highlighting two fundamental aspects of paradigmatic loving relationships that can, and often do, continue in an adapted form following bereavement: love and mutual shaping of interests, choices, and self-concepts. Attention to these continuing features of relationships helps to capture and clarify the phenomenological and behavioral features of continuing bonds. However, love and mutual shaping must also change in important ways following bereavement. Love becomes unreciprocated, and although the dead continue to shape our interests, choices, and self-concepts, we predominantly shape their legacies and memories in return. These changes place important constraints upon the nature of our interpersonal connections with the dead. (shrink)

Street-level bureaucrats are a fundamental part of the implementation process of any policy. This study provides an examination of the factors that shape the behavior of street-level bureaucrats at the frontlines of policy implementation. This study investigates how rebelliousness generates an impact on the discretion of street-level bureaucrats and to what extent client meaningfulness plays a moderating factor. It utilizes a survey questionnaire distributed among inspectors of the Department of Labor in the Ministry of Human Resources of Malaysia (n=241). The (...) result of this study demonstrates that rebelliousness has a negative relationship with discretion, and client meaningfulness has a direct positive correlation with discretion. However, the relationship between rebelliousness and discretion is stronger with a high level of client meaningfulness. The objective of this study is to examine street-level bureaucrat behavior through the lens of Lipsky's theory which will provide an answer to the broad question of the factors that contribute to the existence of an imperfect implementation process. This study sheds light on the importance of client meaningfulness in moderating the behavior of street-level bureaucrats while interacting with inspectees. The novelty of this study is by highlighting two main constructs that are likely contributing to the implementation process by directly and indirectly impacting bureaucrats’ discretion: rebelliousness, and client meaningfulness. (shrink)

Poincaré tries to tackle the question "Can science sway us into materialism?". In other words, he analyzes if science - in particular Physics - and its theories are dependent on a materialistic ontological view of the world. His final answer appears to be "no". However, in order to reach this conclusion he resorts to a brief history of Physics, providing an insightful account on the evolution of the concept of atom from Democritus to the, then, recent discoveries on the composition (...) of the atom. In his opinion, science swung and will keep swinging between two opposing approaches over the constitution of nature: the discrete and the continuous approach. (shrink)

A strongly independent preorder on a possibly in finite dimensional convex set that satisfi es two of the following conditions must satisfy the third: (i) the Archimedean continuity condition; (ii) mixture continuity; and (iii) comparability under the preorder is an equivalence relation. In addition, if the preorder is nontrivial (has nonempty asymmetric part) and satisfi es two of the following conditions, it must satisfy the third: (i') a modest strengthening of the Archimedean condition; (ii') mixture continuity; and (iii') completeness. Applications (...) to decision making under conditions of risk and uncertainty are provided. (shrink)

Whilst in recent years sports studies have addressed the calls ‘to bring the body back in’ to theorisations of sport and physical activity, the ‘promise of phenomenology’ remains largely under-realised with regard to sporting embodiment. Relatively few accounts are grounded in the ‘flesh’ of the lived sporting body, and phenomenology offers a powerful framework for such analysis. A wide-ranging, multi-stranded, and interpretatively contested perspective, phenomenology in general has been taken up and utilised in very different ways within different disciplinary fields. (...) The purpose of this article is to consider some selected phenomenological threads, key qualities of the phenomenological method, and the potential for existentialist phenomenology in particular to contribute fresh perspectives to the sociological study of embodiment in sport and exercise. It offers one way to convey the ‘essences’, corporeal immediacy and textured sensuosity of the lived sporting body. The use of Interpretative Phenomenological Analysis (IPA) is also critically addressed. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – (...) Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)