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)

In this paper, I present a discrete solution for the paradox of Achilles and the tortoise. I argue that Achilles overtakes the tortoise after a finite number of steps of Zeno’s argument if time is represented as discrete. I then answer two objections that could be made against this solution. First, I argue that the discrete solution is not an ad hoc solution. It is embedded in a discrete formulation of classical mechanics. Second, I show that the discrete solution cannot (...) be falsified experimentally. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability (...) of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)

The present study provides evidence that valence focus and arousal focus are important processes in determining whether a dimensional or a discrete emotion model best captures how people label their affective states. Individuals high in valence focus and low in arousal focus fit a dimensional model better in that they reported more co-occurrences among like-valenced affective states, whereas those lower in valence focus and higher in arousal focus fit a discrete model better in that they reported fewer co-occurrences between like-valenced (...) affective states. Taken together, these findings suggest that one static, nomothetic theory may not accurately describe the subjective affective experience of all individuals. (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)

We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.

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.

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 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)

This paper argues that liberal tenats that justify intervention to promote the welfare of an incompetent do not suffice as a basis for analyzing parent-child relationships, and that this inadequacy is the basis for many of the problems that arise when thinking about the state's role in resolving family conflicts, particularly when monitoring parental discretion in medical decision-making on behalf of a child. The state may be limited by the best interest criterion when dealing with children, but parents are not. (...) The state's relation with the child is formal while the parental relation is intimate, having its own goals and purposes. While the liberal canons insist on the incompetent one's best interest, parents are permitted to compromise the child's interests for ends related to these familial goals and purposes. Parents decisions should be supervened, in general, only if it can be shown that no responsible mode of thinking warrants such treatment of a child. Keywords: proxy medical consent, children's rights, state's protection of children, parental authority and the state's intervention, paternalism, liberalism, parental values CiteULike Connotea Del.icio.us What's this? (shrink)

In this field questions arise which are certainly difficult; but as I listened last time to members of the group, I felt that the main difficulty perhaps lay in determining precisely what questions we are trying to answer. I have the conviction that if we could only say clearly what the questions are, the answers to them might not appear so elusive. So I have begun with a simple list of questions about discretion which in one form or another were, (...) as it seemed to me, expressed by the group last time. I may indeed have omitted something and inserted something useless: if so, no doubt I shall be informed of this later. The central questions then seem to me to be the following: 1. What is discretion, or what is the exercise of discretion? 2. Under what conditions and why do we in fact accept or tolerate discretion in a legal system? 3. Must we accept discretion or tolerate discretion, and if so, why? 4. What values does the use of discretion menace, and what values does it maintain or promote? 5. What can be done to maximize the beneficial operation of the use of discretion and to minimize any harm that it does? (shrink)

Recent discussions in the philosophy of science have devoted considerable attention to the analysis of conceptual issues relating to the methodology of explanation and prediction in the sciences. Part of this literature has been devoted to clarifying the very ideas of explanation and prediction. But the discussion has also ranged over various related topics, including the status of laws to be used for explanatory and predictive purposes, the logical interrelationships between explanatory and predictive reasonings, the differences in the strategy of (...) explanatory argumentation in different branches of science, the nature and possibility of teleological explanation, etc. The aim of the present article is to examine the issues involved in such questions from the specialized perspective afforded by one particular kind of physical systems--namely, systems, here to be characterized as discrete state systems, whose behavior has been studied extensively in the scientific literature under the general heading of Markov chains. These systems have been chosen as our focus because their behavior over time can be analyzed at once with great ease and with extraordinary precision. (shrink)

Prosecutors in the US legal system have great power to interfere at their discretion in the lives of citizens, and face relatively few checks on the exercise of this discretion. The vast scope of the criminal law provides a pretext for prosecuting nearly anyone. Meanwhile, other features of the legal system, such as the way plea bargains are structured and the doctrine of prosecutorial immunity, further increase prosecutorial power. And existing institutional restraints on prosecutorial abuses, such as democratic accountability, the (...) grand jury system, and the possibility of a selective prosecution defense, are mostly ineffectual. I draw on republican political theory, including insights from Philip Pettit and Elizabeth Anderson, to argue that this state of affairs gives prosecutors dominating, and therefore unjust, power over vast swathes of the public. I then survey some potential institutional changes which might help ameliorate the problem. (shrink)

The discretionary powers of welfare state professionals are in tension with the requirements of the democratic Rechtsstaat. Extensive use of discretion can threaten the principles of the rule of law and relinquish democratic control over the implementation of laws and policies. These two tensions are in principle ineradicable. But does this also mean that they are impossible to come to grips with? Are there measures that may ease these tensions? We introduce an understanding of discretion that adds an epistemic dimension (...) (discretion as a mode of reasoning) to the common structural understanding of discretion (an area of judgment and decision). Accordingly we distinguish between structural and epistemic measures of accountability. The aim of the former is to constrain discretionary spaces or the behaviour within them while the aim of the latter is to improve the quality of discretionary reasoning. The focus in this article is on epistemic measures that are internally related to the main characteristic of accountability, namely justified use of discretionary power. (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)

Rousseau's understanding of legislation as the expression of the general will implies a constitutional principle of legislative supremacy. In turn, this should translate to a narrow, mechanical account of adjudication, lest creative judicial interpretation subvert the primacy of legislative power. Yet in his constitutional writings, Rousseau recommends open-textured and vague legislative codes, which he openly admits will require judicial development. Thus he apparently trusts a great deal in judicial discretion. Ostensibly, then, he overlooks the problem of how legislative indeterminacy—and correspondingly, (...) judicial discretion—may undermine the authority of the general will. However, I argue that Rousseau aims to check judicial subversion of legislative supremacy simply by extending his broader social politics—and specifically, his peculiar concept of republican virtue—to the domain of law. His main concern is that the law should not develop as a mystifying expert practice; therefore, he necessarily rejects any understanding of judicial virtue as lying in principled discourse. Instead, he envisages that judicial power will be checked by a more generic sense of republican virtue. In turn this echoes his apprehension of social differentiation and social complexity as sources of domination and hierarchy. (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)

The present study provides evidence that valence focus and arousal focus are important processes in determining whether a dimensional or a discrete emotion model best captures how people label their affective states. Individuals high in valence focus and low in arousal focus fit a dimensional model better in that they reported more co-occurrences among like-valenced affective states, whereas those lower in valence focus and higher in arousal focus fit a discrete model better in that they reported fewer co-occurrences between like-valenced (...) affective states. Taken together, these findings suggest that one static, nomothetic theory may not accurately describe the subjective affective experience of all individuals. (shrink)

Examining the role of dispositions (potentials and propensities) in both physics and psychology reveals that they are commonly derivative dispositions, so called because they derive from other dispositions. Furthermore, when they act, they produce further propensities. Together, therefore, they appear to form discrete degrees within a structure of multiple generative levels. It is then constructively hypothesized that minds and physical nature are themselves discrete degrees within some more universal structure. This gives rise to an effective dualism of mind and nature, (...) but one according to which they are still constantly related by causal connections. I suggest a few of the unified principles of operation of this more complicated but universal structure. (shrink)

H.L.A. Hart’s lost and found essay ‘Discretion’ has provided new insight into the issue of how legal systems can cope with indeterminacy in the law. The so-called ‘open texture’ of law calls for the exercise of judicial discretion, which, I argue, renders judges susceptible to the problem of dirty hands. To show this, I frame the problem as being open to an array of appropriate emotional responses, namely, various senses of guilt. With these responses in mind, I revise an example (...) from Michael Walzer’s original analysis in a way that highlights purely personal sacrifices in solutions to dirty hands situations. I then turn to an account of moral emotions in legal decision-making and show how judges—in failing to advance all interests—might be left with a unique sense of guilt. With an application of this account to Hart’s legal positivism, it can be seen that a judge’s hands are often dirtied in resolving borderline cases. If discretion leaves judges in situations where they must do wrong in order to do right, Hart’s endorsement of a closure view of wrongdoing will lead to difficulties in how he can explain the presence of moral remainders in jurisprudence. (shrink)

In various guises (usually referred to as the “basic emotion” or “discrete emotion” approach), scientists and philosophers have long argued that certain categories of emotion are natural kinds. In a recent paper, Colombetti (2009) proposed yet another natural kind account, and in so doing, characterized and critiqued psychological constructionist approaches to emotion, including our own Conceptual Act Model. In this commentary, we briefly address three topics raised by Columbetti. First, we correct several common misperceptions about the discrete emotion approach to (...) emotion. Second, we discuss misconceptions of our Conceptual Act Model. Finally, we briefly comment on Columbetti's Dynamical Discrete Emotion model. (shrink)

Automated decision-making takes up an increasingly significant place in the administrative state. This article presents a conception of discretion that is helpful for evaluating the proper place of algorithms in public decision-making. I argue that the algorithm itself is not a site of discretion. The threat is that automated decision-making alters the relationships between traditional actors in a way that can cut down discretion and human commitment. Algorithmic decision-makers can serve to fetter the discretion that the legislature and the populace (...) expect to be exercised. We must strive to maintain discretion, moral agency, deliberative ideals, and human commitment through the system that surrounds the use of an algorithm and to develop a new expertise that can retain and exercise the expected discretion. Backing this argument are traditional legal constraints, public expectations, and administrative law principles, tied together through the organizing principle of discretion. (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.

Democratic institutions and practice depend on trust, in two ways. Citizens must trust each other to abide by shared rules and norms that together govern a political community; it is a feature of democratic states that they direct their resources not to enforcement of rule abidingness, but rather towards providing collective and public goods. Instead, states rely on the semi-voluntary compliance of citizens with these shared norms and laws. Citizens must also trust their political representatives, who via their election are (...) granted discretion to make decisions that will have significant impact on the lives of those they represent. Yet, according to many scholars across many disciplines, this trust is at risk of being undermined, if it hasn’t yet been so. The possible reasons for weakened and broken trust are many, ranging from escalating wealth inequalities, to increasing cultural and ethnic diversity, to political incompetence of both voters and political representatives, to corruption among political representatives. In this article, I examine the trust that underpins the discretion accorded to elected representatives, to identify when the improper use of this discretion may reasonably be thought to be a source of distrust. In particular, I argue that where discretion is used arbitrarily, it may threaten to undermine the trust relations between citizens and their representatives, and therefore the efficient and effective running of democratic states. (shrink)

Bekenstein proposed that the spectrum of horizon area of quantized black holes must be discrete and uniformly spaced. We examine this proposal in the context of spherically symmetric charged black holes in a general class of gravity theories. By imposing suitable boundary conditions on the reduced phase space of the theory to incorporate the thermodynamic properties of these black holes and then performing a simplifying canonical transformation, we are able to quantize the system exactly. The resulting spectra of horizon area, (...) as well as that of charge are indeed discrete. Within this quantization scheme, near-extremal black holes (of any mass) turn out to be highly quantum objects, whereas extremal black holes do not appear in the spectrum, a result that is consistent with the postulated third law of black hole thermodynamics. (shrink)

In this essay, I argue against a way of approaching the issue of Judicial discretion that finds its clearest exposition and highest development in recent works by Ronald Dworkin. This approach is too narrow. It ignores a kind of Judicial discretion whose existence has been maintained by jurists with discretionist sympathies, and which is philosophically significant. The kind of discretion it ignores raises the issue of the justification of adjudication as clearly as does the kind of discretion that it recognizes. (...) Moreover, discussion of the kind of discretion ignored is in some respects the natural starting place for a discussion of judicial discretion in general. (shrink)

Managers often have discretion in interpreting their ethical requirements, and they should seek democratic guidance in doing so. The undemocratic nature of managerial ethical discretion is shown to be a recurring problem in business ethics. Joseph Heath’s market failures approach (MFA) is introduced as a theory better positioned to deal with this problem than other views. However, due to epistemic uncertainty and conceptual indeterminacy, the MFA is shown to allow a much wider range of managerial discretion than initially appears. The (...) paper explores how this range can be narrowed down with democratic input, comparing models based on formal state institutions and on the informal public sphere. A case study from the pharmaceutical industry illustrates the merits of the informal public sphere approach. (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)

Elected prosecutors have pledged not to enforce abortion laws, in response to state-level abortion bans. For their pledges to be meaningful, prosecutors must exercise their discretion in cases of individuals who face legal risk, including people who help others self-manage their abortions. With a harm-reduction approach to improving abortion access, prosecutors should aim to reduce abortion helpers’ involvement with the criminal justice system.

Carroll Izard completed his dissertation in 1952, beginning a career spanning more than six decades that coincided with clinical psychology maturing as a profession, and the birth of clinical science and cognitive neuroscience. Izard’s focus on discrete emotions as evolved systems that organize information, prepare responses, and shape the development of personality and relationships persisted through his career, despite “emotions” often being overshadowed by psychodynamic, behavioral, or cognitive perspectives. His theoretical work anticipated and now integrates contemporary neuroscience and relational perspectives. (...) Exploration of discrete emotions has kept lines of inquiry open that enrich our understanding of psychopathology. Izard also embraced clinical science, combining basic research with effectiveness studies addressing the unmet need for mental health services via emotion-focused interventions. (shrink)

Camras and Shutter review evidence suggesting that infants’ facial expressions do not represent discrete emotions and cannot easily be matched to the facial expressions of adults. This raises the important question of whether infants have a notion about the meanings of discrete emotions at all. The authors do not discuss whether infants are sensitive to discrete emotional expressions when perceiving others. In our commentary we discuss evidence for the perception of discrete emotional facial expressions in infancy.

As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from (...) each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system.In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationship between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system. (shrink)

Philip Pettit’s conception of freedom as nondomination is modally robust in that it requires not simply reducing the probability of uncontrolled interference by others but entirely eliminating that possibility. In this article, I consider whether freedom as nondomination provides an attractive analysis of official discretion, particularly in the context of the criminal law, an area of recurring interest for Pettit. I argue that not only does the modally robust character of freedom as nondomination have some rather unattractive implications in the (...) criminal law, but that it sits poorly with Pettit’s more general ambitions to provide a consequentialist framework for the evaluation of social institutions. Drawing on recent work by Niko Kolodny, I develop an ‘anti-deference’ interpretation of nondomination and contrast it to Pettit’s modally robust conception. (shrink)

Attempts to create a coherent scientific picture of the world as a whole on the basis of quantum physics has sped up at the turn of the millennium. There particularly seem to be expectations that the development of a new kind of quantum mechanics could make it possible to describe both matter and consciousness in one frame of reference (“dual aspect approach”). These ideas are often results of brilliant intuitive visions but as yet not able to produce testable hypotheses. Maybe (...) “wave mechanics” is not very suitable in the study of consciousness from the quantum mechanical point of view. The aim of this article is to show how both the matter and the mind systems can be described with one coherent mathematical model if we assume both space and time to be discrete. (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)

We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑ i a i · x i ≥ b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP- inequalities, we prove a monotone interpolation theorem (...) obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Theorem 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantifier elimination for such theories. We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP). (shrink)

Energy nonconservation is a serious problem of dynamical collapse theories. In this paper, we propose a discrete model of energy-conserved wavefunction collapse. It is shown that the model is consistent with existing experiments and our macroscopic experience.

: How might we locate originality as emerging from within the "discrete" work of commentary? Because many women have engaged with philosophy in forms (including commentary) that preclude their work from being seen as properly "original," this question is a feminist issue. Via the work of selected contemporary French women philosophers, the author shows how commentary can reconfigure the philosophical tradition in innovative ways, as well as in ways that change what counts as philosophical innovation.

Three classes of decidable discrete linear orders with varying degrees of effectiveness are investigated. We consider how a classical order type may lie in relation to these three classes, and we characterize by their order types elements of these classes that have effective nontrivial self-embeddings.