A book on the notion of fundamental length, covering issues in the philosophy of math, metaphysics, and the history and the philosophy of modern physics, from classical electrodynamics to current theories of quantum gravity. Published (2014) in Cambridge University Press.

In this paper I present the Discrete Space-Time Thesis, in a way which enables me to defend it against various well-known objections, and which extends to the discrete versions of Special and General Relativity with only minor difficulties. The point of this presentation is not to convince readers that space-time really is discrete but rather to convince them that we do not yet know whether or not it is. Having argued that it is an open question whether (...) or not space-time is discrete, I then turn to some possible empirical evidence, which we do not yet have. This evidence is based on some slight differences between commonly occurring differential equations and their discrete analogs. (shrink)

A general metatheorem is proved which reduces a wide class of statements about continuous time stochastic processes to statements about discrete time processes. We introduce a strong language for stochastic processes, and a concept of forcing for sequences of discrete time processes. The main theorem states that a sentence in the language is true if and only if it is forced. Although the stochastic process case is emphasized in order to motivate the results, they apply to a (...) wider class of random variables. At the end of the paper we illustrate how the theorem can be used with three applications involving submartingales. (shrink)

The ‘continuous’ and the ‘discrete’ in nature and in science live and fight forever. The questionnaires and the Lickert scales are indispensable and widely used tools in social sciences research. Vougiouklis & Vougiouklis bar is a new tool introduced as an alternative to Lickert scales. We believe that such an alternative might offer some solutions to problems that crop up during the fight between continuous and discrete. Nevertheless, the greatest contribution of the V&V bar is that (...) it offers the researchers freedom in all stages of the research procedure using a questionnaire. (shrink)

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 is continuous, 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.

The evolution of a single trapped ion exhibiting intermittent fluorescence and dark periods may be described either as a continuous process, using differential rate equations, or discretely, as a Markov process. The latter models the atom as making instantaneous transitions from one energy eigenstate to another, and is open to the objection that superpositions of energy states will form which are not covered by the Markov process. The superposition objection is replied to, and two new mathematical elements, Markov vectors (...) and Markov matrices, are proposed as additions to quantum theory. The paper concludes by attributing the cause of dark periods in the ion's history to instantaneous transitions in the ion itself, rather than to photon detection or other components of measurement. (shrink)

After reporting in detail Aristotle’s texts and comments on the well-known motion 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 traditional continuous 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 a continuous thing. Classical continuous 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 traditional continuous methods suffice instead of discrete methods, and the handling of discrete versus continuous in physics is briefly reported. (shrink)

Leibovich et al. propose that non-symbolic numerosity abilities develop from the processing of more basic, continuous magnitudes such as size, area, and density. Here I review similar arguments arising in the visual perception field and further propose that the evolvement of discrete representations from continuous stimulus properties may be a fundamental characteristic of cognitive development.

This commentary confronts one of the central tenets advanced in Wilkins & Wakefield's target article: By adopting a very narrow perspective on language, the authors have effectively limited discussion of earlier linguistic capabilities thought to be at least facilitative of, if not prerequisite to language defined as a An alternative conceptualization for describing semiogenesis is offered.

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)

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a (...) class='Hi'>continuous variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order. (shrink)

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. Along the way, I explain why the sense of ‘continuity’ ascribed to smooth experiences is distinct from the sense of ‘continuity’ ascribed to the stream of consciousness. (shrink)

Causal connectionists need to provide an account of causal linkage and of causal direction. This chapter distinguishes between two kinds of causal connection, namely, discrete and continuous. Causal connectionists have a number of options for explaining the linkage between causes and effects in the case of discrete causation. The chapter provides some popular options. If some causation is discrete, and the exercise of causal powers provides a direction to discrete causation, then the causal direction of (...) processes can be derived from the causal direction of discrete interaction. David Hume famously made the claim about the knowledge of causation in his Enquiry Concerning Human Understanding. Hume offers, in defense of his principle, a set of thought‐experiments that seem to show that we can always imagine any event's having a different result from the one we in fact observe. Causal processes like locomotion may provide counterexamples to Hume's Epistemic Principle. (shrink)

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a (...) class='Hi'>discrete and finite set of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism. (shrink)

I review arguments demonstrating how the concept of “particle” numbers arises in the form of equidistant energy eigenvalues of coupled harmonic oscillators representing free fields. Their quantum numbers (numbers of nodes of the wave functions) can be interpreted as occupation numbers for objects with a formal mass (defined by the field equation) and spatial wave number (“momentum”) characterizing classical field modes. A superposition of different oscillator eigenstates, all consisting of n modes having one node, while all others have none, defines (...) a non-degenerate “n-particle wave function”. Other discrete properties and phenomena (such as particle positions and “events”) can be understood by means of the fast but continuous process of decoherence: the irreversible dislocalization of superpositions. Any wave-particle dualism thus becomes obsolete. The observation of individual outcomes of this decoherence process in measurements requires either a subsequent collapse of the wave function or a “branching observer” in accordance with the Schrödinger equation—both possibilities applying clearly after the decoherence process. Any probability interpretation of the wave function in terms of local elements of reality, such as particles or other classical concepts, would open a Pandora’s box of paradoxes, as is illustrated by various misnomers that have become popular in quantum theory. (shrink)

This paper explores the value of skepticism towards the Approximate Number System (ANS). I sketch some of the main arguments levied against ANS-based interpretations of numerical cognition data and argue that there are empirical and conceptual reasons to reject wholesale replacement of the ANS with an Analog Magnitude System (AMS). To simplify the discussion, I focus for the most part on a recent critical review representative of this new wave of revisionist skepticism (Leibovich, T., Katzin, N., Harel, M., & Henik, (...) A., 2017). I start with a brief review of some of the reasons offered to deny that experiments studying our numerical abilities reveal the presence of a system dedicated to representing quantities of discrete objects, before turning briefly to empirical responses to these worries. I then offer a few reflections on why even if the empirical rebuttal were to fail, there are conceptual reasons to doubt that we are only equipped with an AMS. While some of these reasons involve methodological implications of AMS-based theories, other conceptual reasons to doubt AMS skepticism revolve around how ANS-skepticism seems to go against the history of the relation between the continuous and the discrete, and how one cannot be derived from the other. I then end with a potential reply to my worries involving an appeal to the Object-File System (OFS) as a source of discrete content in our numerical abilities and find it wanting. (shrink)

Against Continuity is the first book to demonstrate that the beating heart of Gilles Deleuze's philosophy is a systematic ontology of irreducible, singular entities. This requires a radical break with decades of Deleuzian orthodoxy, according to which Deleuze's metaphysics revolves around the dissolution of discrete entities into a continuous world of flows and events.With reference to all of Deleuze's work, including published and untranslated seminars, as well as the recently published 'Lettres et autres textes', Arjen Kleinherenbrink critically compares (...) Deleuze's ontology to seven related contemporary thinkers: Levi Bryant, Maurizio Ferraris, Markus Gabriel, Manuel DeLanda, Graham Harman, Tristan Garcia and Bruno Latour. These comparisons establish Deleuze as an important precursor to object-oriented speculative realism and open up exciting new avenues of thought for critics and supporters of Deleuze alike. (shrink)

In the context of continuous first-order logic, special attention is often given to theories that are somehow continuous in an ‘essential’ way. A common feature of such theories is that they do not interpret any infinite discrete structures. We investigate a stronger condition that is easier to establish and use it to give an example of a strictly simple continuous theory that does not interpret any infinite discrete structures: the theory of richly branching [Formula: see (...) text]-forests with generic binary predicates. We also give an example of a superstable theory that fails to satisfy this stronger condition but nevertheless does not interpret any infinite discrete structures. (shrink)

Heterocyst spacing in blue -green bacteria is widely assumed to be due to a diffusible inhibitor. The inhibitor, a nitrogen-rich compound, probably glutamine, is produced via the N2-fixing enzymes of the heterocyst and in turn serves to suppress the induction of these enzymes and of the differentiation of vegetative cells to heterocysts. This simple morphogenetic mechanism operating in growing cellular filaments ofAnabaena species is investigated on the basis of a continuous and a discrete cellular model, as well as (...) by cell-by-cell simulation of the inhibitor transport. The resulting distances between heterocysts and kinetics of their production are compared with observations, and the values of physical parameters are estimated from the models. (shrink)

Journal of Mathematical Logic, Ahead of Print. In the context of continuous first-order logic, special attention is often given to theories that are somehow continuous in an ‘essential’ way. A common feature of such theories is that they do not interpret any infinite discrete structures. We investigate a stronger condition that is easier to establish and use it to give an example of a strictly simple continuous theory that does not interpret any infinite discrete structures: (...) the theory of richly branching [math]-forests with generic binary predicates. We also give an example of a superstable theory that fails to satisfy this stronger condition but nevertheless does not interpret any infinite discrete structures. (shrink)

In this article we present symmetric diffusion networks, a family of networks that instantiate the principles of continuous, stochastic, adaptive and interactive propagation of information. Using methods of Markovion diffusion theory, we formalize the activation dynamics of these networks and then show that they can be trained to reproduce entire multivariate probability distributions on their outputs using the contrastive Hebbion learning rule (CHL). We show that CHL performs gradient descent on an error function that captures differences between desired and (...) obtained continuous multivariate probability distributions. This allows the learning algorithm to go beyond expected values of output units and to approximate complete probability distributions on continuous multivariate activation spaces. We argue that learning continuous distributions is an important task underlying a variety of real‐life situations that were beyond the scope of previous connectionist networks. Deterministic networks, like back propagation, cannot learn this task because they are limited to learning average values of independent output units. Previous stochastic connectionist networks could learn probability distributions but they were limited to discrete variables. Simulations show that symmetric diffusion networks can be trained with the CHL rule to approximate discrete and continuous probability distributions of various types. (shrink)

The discrete-time system has more complex and chaotic dynamical behaviors as compared to the continuous-time system. This paper extends a discrete Leslie–Gower predator-prey system with the Allee effect in the predator’s population, whose dynamics are analyzed and explored. We have determined the equilibrium points and studied their local stability properties. We find that the system undergoes flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point by choosing the Allee parameter as a bifurcation parameter. We discuss the (...) stability and direction of both bifurcations with the help of the normal form theory and center manifold theorem. The flip bifurcation and Neimark–Sacker bifurcation are the most common routes to the chaotic orbit in the discrete system. Moreover, we utilize state feedback, pole placement, and hybrid control methods to control the chaos in the system. The work is complete with the numerical simulations to confirm the analytical findings. (shrink)

In this paper, we continue with the study of tense operators on Nelson algebras (Figallo et al. in Studia Logica 109(2):285–312, 2021, Studia Logica 110(1):241–263, 2022). We define the variety of algebras, which we call tense Nelson D-algebras, as a natural extension of tense De Morgan algebras (Figallo and Pelaitay in Logic J IGPL 22(2):255–267, 2014). In particular, we give a discrete duality for these algebras. To do this, we will extend the representation theorems for Nelson algebras given in (...) Sendlewski (Studia Logica 43(3):257–280, 1984) to the tense case. (shrink)

Relations between discrete and continuous complexity models are considered. The present paper is devoted to combine both models. In particular we analyze the 3-Satisfiability problem. The existence of fast decision procedures for this problem over the reals is examined based on certain conditions on the discrete setting. Moreover we study the behaviour of exponential time computations over the reals depending on the real complexity of 3-Satisfiability. This will be done using tools from complexity theory over the integers.

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to (...) argue that quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. (shrink)

Life, whatsoever it is, is a temporal flux. Everything is doomed to change often apparently beyond our awareness. My body appears totally different now, so does my mind. I have gained new attitudes and new ambitions, and a substantial number of old ones have been discarded. But, I am still the same person in an ongoing manner. Besides, recent neuroscientific and psychological evidence has shown that our conscious perception happens as a series of discrete or bounded instants—it emerges in (...) temporally scattered, gappy, and discrete forms. But, if it is so, how does the brain persevere our self-continuity (or continuity of identity) in this gappy setting? How is it possible that despite moment-to-moment changes in my appearance and mind, I am still feeling that I am that person? How can we tackle with this second by second gap and resurrection in our existence which leads to a foundation of wholeness and continuity of ourself? How is continuity of self (collective set of our connected experiences in the vessel of time) that results in a feeling that one’s life has purpose and meaning preserved? To answer these questions, the problem has been comprehended from a philosophical, psychological, and neuroscientific perspective. I realize that first and foremost fact lies in the temporal nature of identity. Having equipped with these thoughts, in this article, it is hypothesized that according to two principles (the principle of reafference or corollary discharge and the principle of a time theory) self-continuity is maintained. It is supposed that there should be a precise temporal integration mechanism in the CNS with the outside world that provides us this smooth, ungappy flow of theSelf. However, we are often taken for granted the importance of self-continuity, but it can be challenged by life transitions such as entering adulthood, retirement, senility, emigration, and societal changes such as immigration, globalization, and in much unfortunate and extreme cases of mental illnesses such as schizophrenia. (shrink)

The linear vector space for the quantum description of a physical system is formulated as the intersection of the domains of Hermiticity of the observables characterizing the system. It is shown that on a continuous interval of its spectrum every Hermitian operator on a Hilbert space of one degree of freedom is a generalized coordinate with a conjugate generalized momentum. Every continuous spectral interval of a Hermitian operator is the limit of a discrete spectrum in the same (...) interval. This result is applied to description of the state of a system by a statistical operator p( $\hat q$ ) following measurement of the probabilities of the eigenvalues of an observable $\hat q$ . Each continuous interval in the spectrum of $\hat q$ is replaced by a discrete spectrum in the same interval with a parameterK; the limitK → ∞ in which the spectrum becomes continuous is not attainable physically. In the linear vector space of a quantum mechanical system every continuous interval in the spectra of the observables is similarly replaced by a discrete spectrum with a finite parameter: it is a space of discrete bases and a set of such parameters. The case of degenerate spectrum is also discussed. (shrink)

We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously (...) variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism. (shrink)

We introduce a smooth mapping of some discrete space-time symmetries into quasi-continuous ones. Such transformations are related with q-deformations of the dilations of the Euclidean space and with the noncommutative space. We work out two examples of Hamiltonian invariance under such symmetries. The Schrödinger equation for a free particle is investigated in such a noncommutative plane and a connection with anyonic statistics is found.

Much of neuroscience is currently dominated by an information processing metaphor which is largely conceptualized in discrete terms. An alternative metaphor conceptualizes information flow as continuous. A qualitative set of hypotheses based on this metaphor, the energy model, is described here. This model considers information transfer in terms of the flow of an abstract variable, energy, between points in a field comprising the extent of the nervous system. Although extremely simple, it generates some intriguing consequences. In particular, it (...) provides a useful way in which to look at consciousness. Traditional problems of consciousness, such as qualia and the unity of consciousness, are briefly addressed, and outlines are sketched of the answers given by the energy model. (shrink)

Much of neuroscience is currently dominated by an information processing metaphor which is largely conceptualized in discrete terms. An alternative metaphor conceptualizes information flow as continuous. A qualitative set of hypotheses based on this metaphor, the energy model, is described here. This model considers information transfer in terms of the flow of an abstract variable, energy, between points in a field comprising the extent of the nervous system. Although extremely simple, it generates some intriguing consequences. In particular, it (...) provides a useful way in which to look at consciousness. Traditional problems of consciousness, such as qualia and the unity of consciousness, are briefly addressed, and outlines are sketched of the answers given by the energy model. (shrink)

Temporal stability and change in neutral contexts can transform continuous experiences into distinct and memorable events. However, less is known about how shifting emotional states influence these memory processes, despite ample evidence that emotion impacts non-temporal aspects of memory. Here, we examined if emotional stimuli influence temporal memory for recent event sequences. Participants encoded lists of neutral images while listening to auditory tones. At regular intervals within each list, participants heard emotional positive, negative, or neutral sounds, which served as (...) “emotional event boundaries” that divided each sequence into discrete events. Temporal order memory was tested for neutral item pairs that either spanned an emotional sound or were encountered within the same auditory event. Encountering a highly arousing event boundary led to faster response times for items encoded within the next event. Critically, we found that highly arousing sounds had different effects on binding ongoing versus ensuing sequential representations in memory. Specifically, highly arousing sounds were significantly more likely to enhance temporal order memory for ensuing information compared to information that spanned those boundaries, especially for boundaries with negative valence. These findings suggest that within aversive emotional contexts, fluctuations in arousal help shape the temporal organisation of events in memory. (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)

Much legal research focuses on understanding how judicial decision-makers exercise their discretion. In this paper we examine the notion of legal or judicial discretion, and weaker and stronger forms of discretion. At all times our goal is to build cognitive models of the exercise of discretion, with a view to building computer software to model and primarily support decision-making. We observe that discretionary decision-making can best be modeled using three independent axes: bounded and unbounded, defined and undefined, and binary and (...) continuous. Examples of legal tasks are given from each of the eight ensuing octants and we conclude by saying what this model shows about current legal trends. We should stress that our taxonomy has been based on our observations of how discretionary legal decisions are made. No claim is made that our model is either complete (providing advice in every domain) or exact, but it does help knowledge engineers construct legal decision support systems in discretionary domains. (shrink)

We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

Biological and epidemiological phenomena are often measured with error or imperfectly captured in data. When the true state of this imperfect measure is a confounder of an outcome exposure relationship of interest, it was previously widely believed that adjustment for the mismeasured observed variables provides a less biased estimate of the true average causal effect than not adjusting. However, this is not always the case and depends on both the nature of the measurement and confounding. We describe two sets of (...) conditions under which adjusting for a non-deferentially mismeasured proxy comes closer to the unidentifiable true average causal effect than the unadjusted or crude estimate. The first set of conditions apply when the exposure is discrete or continuous and the confounder is ordinal, and the expectation of the outcome is monotonic in the confounder for both treatment levels contrasted. The second set of conditions apply when the exposure and the confounder are categorical. In all settings, the mismeasurement must be non-differential, as differential mismeasurement, particularly an unknown pattern, can cause unpredictable results. (shrink)

Corruption cases that result in small state financial losses continue to end up in the Corruption Court without alternative solutions that are faster, simpler and cheaper, even though the Corruption Court is located in the provincial capital and the corruption trial process requires a lot of money. So that it is not commensurate between the costs of law enforcement incurred with the state financial losses incurred due to corruption. The method of this research approach is juridical sociological because the problems (...) studied concern the relationship between juridical factors and sociological factors. The urgency of discontinuing the prosecution of corruption crimes with small state financial losses is that the prosecution of corruption crimes with small state financial losses is inefficient and the Discontinuation of Prosecution of corruption crimes with small state financial losses does not have a legal umbrella This is because there are weaknesses in legal substance, legal structure and legal culture. The reconstruction model for the termination of prosecution of corruption crimes with small state financial losses based on Prosecutorial Discretion by the Public Prosecutor in a Progressive Legal Perspective is the Reconstruction of Legal Substance, namely adding Prosecutorial Discretion by the Public Prosecutor as a manifestation of the Dominus Litis Principle can stop the prosecution of corruption crimes with small state financial losses where this has not been regulated in Article 140 paragraph (2) of the Criminal Procedure Code, Reconstruction of legal structure, namely the delegation of authority from the Attorney General of the Republic of Indonesia to the Head of the High Prosecutor's Office as the controller of the termination of prosecution of corruption crimes with small state financial losses, Reconstruction of legal culture, namely changing the patterns of thought of public prosecutors from positivistic to progressive in exercising their authority. The recommendations in this study encourage the reform of the Criminal Procedure Code (KUHAP) related to the authority to discontinue prosecution by the Public Prosecutor and the need to make a regulation of the Attorney General related to the Discontinuation of Prosecution based on the Discretion of the Public Prosecutor. (shrink)

With the advent of new technology and imaging techniques that measure brain activity and with the development of the computer as a model for human thinking, it is not surprising to find many authors currently addressing issues regarding brain function and themind/body problem. What is perhaps surprising, given the absence of these techniques at the time, is that Merleau-Ponty addresses these same issues with a rigor and insight that equals, and perhaps even exceeds, most current philosophical studies. Merleau-Ponty’s frequently ignored (...) early work, The Structure of Behavior, contains a wealth of analysis still relevant to current biological and neurophysiological studies and to the philosophical consequences frequently drawn from them. Merleau-Ponty critically addresses not only theories that attempt to understand human behavior as the linear calculation of discrete physiological events but also theories that would explain human behavior simply by appealing to abstract conceptual analysis. His theory of emergent materialism focuses on the human body as a concrete organic whole that can be reduced neither to linear physical events nor to abstract conceptual relations. Understanding the human being requires a theory that recognizes the human body as an original whole, that is, that recognizes a body that intimately integrates mind and matter. It is this theory that Merleau-Ponty first articulates in The Structure of Behavior. It is the main themes of this theory that I will attempt to reveal here. (shrink)

The iterated prisoner’s dilemma (IPD) has been widely used in the biological and social sciences to model dyadic cooperation. While most of this work has focused on the discrete prisoner’s dilemma, in which actors choose between cooperation and defection, there has been some analysis of the continuous IPD, in which actors can choose any level of cooperation from zero to one. Here, we analyse a model of the continuous IPD with a limited strategy set, and show that (...) a generous strategy achieves the maximum possible payoff against its own type. While this strategy is stable in a neighborhood of the equilibrium point, the equilibrium point itself is always vulnerable to invasion by uncooperative strategies, and hence subject to eventual destabilization. The presence of noise or errors has no effect on this result. Instead, generosity is favored because of its role in increasing contributions to the most efficient level, rather than in counteracting the corrosiveness of noise. Computer simulation using a single-locus infinite alleles Gaussian mutation model suggest that outcomes ranging from a stable cooperative polymorphism to complete collapse of cooperation are possible depending on the magnitude of the mutational variance. Also, making the cost of helping a convex function of the amount of help provided makes it more difficult for cooperative strategies to invade a non-cooperative equilibrium, and for the cooperative equilibrium to resist destabilization by noncooperative strategies. (shrink)

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)

In this paper I attempt to throw new light on Leibniz's apparently conflicting remarks concerning the continuity of matter. He says that matter is "discrete" yet "actually divided to infinity" and (thus dense), and moreover that it fills (continuous) space. I defend Leibniz from the charge of inconsistency by examining the historical development of his views on continuity in their physical and mathematical context, and also by pointing up the striking similarities of his construal of continuity to the (...) approach taken by 20th century Combinatorial Topology. (shrink)