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)

This article presents important properties of standard discrete distributions and its conjugate densities. The Bernoulli and Poisson processes are described as generators of such discrete models. A characterization of distributions by mixtures is also introduced. This article adopts a novel singular notation and representation. Singular representations are unusual in statistical texts. Nevertheless, the singular notation makes it simpler to extend and generalize theoretical results and greatly facilitates numerical and computational implementation.

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.

Mental representations remain the central posits of psychology after many decades of scrutiny. However, there is no consensus about the representational format(s) of biological cognition. This paper provides a survey of evidence from computational cognitive psychology, perceptual psychology, developmental psychology, comparative psychology, and social psychology, and concludes that one type of format that routinely crops up is the language-of-thought (LoT). We outline six core properties of LoTs: (i) discrete constituents; (ii) role-filler independence; (iii) predicate–argument structure; (iv) logical operators; (v) (...) inferential promiscuity; and (vi) abstract content. These properties cluster together throughout cognitive science. Bayesian computational modeling, compositional features of object perception, complex infant and animal reasoning, and automatic, intuitive cognition in adults all implicate LoT-like structures. Instead of regarding LoT as a relic of the previous century, researchers in cognitive science and philosophy-of-mind must take seriously the explanatory breadth of LoT-based architectures. We grant that the mind may harbor many formats and architectures, including iconic and associative structures as well as deep-neural-network-like architectures. However, as computational/representational approaches to the mind continue to advance, classical compositional symbolic structures – that is, LoTs – only prove more flexible and well-supported over time. (shrink)

In this paper I outline a comprehensive theory of legal interpretation based on an assumption that legal text, understood as the aggregate of texts of all legal acts in force at a particular time and place, describes one rational and coherent possible world. The picture of this possible world is decoded from the text by interpreters and serves as a holistic model to which the real world is adjusted when the law is applied. From the above premise I will limit (...) myself to drawing two conclusions for how legal interpretation should be carried out. First, I argue that the possible world described by the legal text has to be ‘accessible’ from the real world, i.e. it has to be feasible to transform the actual world into the described one. Were it otherwise, the possible world could not serve as a model for adjustment. The accessibility requirement imposes obligations on the interpreters to secure the rationality of the possible world decoded from the text, amongst other to secure that the description of this world is not contradictory and – as a consequence – the law of excluded middle is obeyed in the possible world described by the legal text. Secondly, I argue for the inevitability of interpretative discretion arising from the requirement to decode a sufficiently ‘saturated’ picture of the possible world., i.e. possessing enough properties to resemble the actual world. As texts have a limited number of sentences and worlds have an unlimited number of properties, interpreters have to supplement the picture of a possible world to achieve its coherence. This involves the inclusion of some additional, non-predetermined features that integrate with the properties of the world predefined by the legal text. This process of saturation consists of filling in so-called ‘places of indeterminacy’ (Roman Ingarden) with content implicated by other features of the possible world. I also argue that the discretion resulting from the necessity of filling in the places of indeterminacy is justified by the requirement of fulfilling the intention of the lawmaker to make the possible world described by the legal text real. The theory presented here is based on contemporary theories of discourse representation and so-called ‘text-world theory’ by J. Gavins. Phenomenalism and causal (historical) theories of reference provide its philosophical background. (shrink)

Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type of Alexandroff spaces was (...) used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology. (shrink)

The ability to perceive the shapes of things as enduring through changes in how they stimulate our sense organs is vital to our sense of stability in the world. But what sort of capacity is shape constancy, and how is it reflected in perceptual experience? This paper defends a pluralist account of shape constancy: There are multiple kinds of shape constancy centered on geometrical properties at various levels of abstraction, and properties at these various levels feature in the content of (...) perceptual experience, governing patterns of apparent shape similarity. I propose that the varieties of shape constancy are subserved by the syntactic complexity of perceptual shape representations. By assigning discrete constituents to various abstract shape parameters, these representations attune us to the preservation of certain abstract shape properties through changes in more determinate shape properties. Finally, I draw broader lessons concerning the nature and function of perceptual constancy. (shrink)

Quilty-Dunn et al.'s wide-ranging defense of the Language of Thought Hypothesis (LoTH) argues that vision traffics in abstract, structured representational formats. We agree: Vision, like language, is compositional – just as words compose into phrases, many visual representations contain discrete constituents that combine in systematic ways. Here, we amass evidence extending this proposal, and explore its implications for how vision interfaces with the rest of the mind.

The Morris water maze has been put forward in the philosophy of neuroscience as an example of an experimental arrangement that may be used to delineate the cognitive faculty of spatial memory (e.g., Craver and Darden, Theory and method in the neurosciences, University of Pittsburgh Press, Pittsburgh, 2001; Craver, Explaining the brain: Mechanisms and the mosaic unity of neuroscience, Oxford University Press, Oxford, 2007). However, in the experimental and review literature on the water maze throughout the history of its use, (...) we encounter numerous responses to the question of “what” phenomenon it circumscribes ranging from cognitive functions (e.g., “spatial learning”, “spatial navigation”), to representational changes (e.g., “cognitive map formation”) to terms that appear to refer exclusively to observable changes in behavior (e.g., “water maze performance”). To date philosophical analyses of the water maze have not been directed at sorting out what phenomenon the device delineates nor the sources of the different answers to the question of what. I undertake both of these tasks in this paper. I begin with an analysis of Morris’s first published research study using the water maze and demonstrate that he emerged from it with an experimental learning paradigm that at best circumscribed a discrete set of observable changes in behavior. However, it delineated neither a discrete set of representational changes nor a discrete cognitive function. I cite this in combination with a reductionist-oriented research agenda in cellular and molecular neurobiology dating back to the 1980s as two sources of the lack of consistency across the history of the experimental and review literature as to what is under study in the water maze. (shrink)

I give a defense of the Massive Modularity hypothesis: the view that the mind is composed of discrete, encapsulated, informationally isolated computational structures dedicated to particular problem domains. This view contrasts with Psychological Rationalism: the view that mental structures take the form of unencapsulated representational items, all available as inputs to one domain-general computational processor. I argue that although Psychological Rationalism is in principle able to overcome the `intractability objection', the view must borrow many features of a massively modular (...) architecture in order to do so, that although it can, in principle, overcome the `optimality objection', the way it does so does not correlate with the way we think, and that although it can, in principle, respond to the `argument from biology', it cannot do so without advancing an unrealistic and unsupported account of cognitive evolution. (shrink)

The text is a continuation of the article of the same name published in the previous issue of Philosophical Alternatives. The philosophical interpretations of the Kochen- Specker theorem (1967) are considered. Einstein's principle regarding the,consubstantiality of inertia and gravity" (1918) allows of a parallel between descriptions of a physical micro-entity in relation to the macro-apparatus on the one hand, and of physical macro-entities in relation to the astronomical mega-entities on the other. The Bohmian interpretation ( 1952) of quantum mechanics proposes (...) that all quantum systems be interpreted as dissipative ones and that the theorem be thus derstood. The conclusion is that the continual representation, by force or (gravitational) field between parts interacting by means of it, of a system is equivalent to their mutual entanglement if representation is discrete. Gravity (force field) and entanglement are two different, correspondingly continual and discrete, images of a single common essence. General relativity can be interpreted as a superluminal generalization of special relativity. The postulate exists of an alleged obligatory difference between a model and reality in science and philosophy. It can also be deduced by interpreting a corollary of the heorem. On the other hand, quantum mechanics, on the basis of this theorem and of V on Neumann's (1932), introduces the option that a model be entirely identified as the modeled reality and, therefore, that absolutely reality be recognized: this is a non-standard hypothesis in the epistemology of science. Thus, the true reality begins to be understood mathematically, i.e. in a Pythagorean manner, for its identification with its mathematical model. A few linked problems are highlighted: the role of the axiom of choice forcorrectly interpreting the theorem; whether the theorem can be considered an axiom; whether the theorem can be considered equivalent to the negation of the axiom. (shrink)

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive (...) formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

Peter Fitzpatrick’s writings prove once and for all that it is possible for a law professor to write in beautiful English. His work also proves once and for all that the dominating tradition of Anglo-American legal philosophy and of law teaching has been barking up the wrong tree: namely, that the philosopher and professional law teachers can understand justice as nested in empty forms, better known as rules, doctrines, principles, policies, and other standards. The more rigorous our analysis or decomposition (...) of the forms, we have believed, the more closely do we access the identity of laws. Justice has been assumed to be a matter of intellectually accessing such analysed forms. Fitzpatrick’s articles and books embody an implicit critique of the analytic view of law and of justice. My entry point into this critique is his preoccupation with Jacques Derrida’s theory of laws as universals and with Derrida’s theory of justice as an inaccessible immediacy or presence in context-specific or concrete experienced events. Each event is experienced in an official’s decision. Such a decision represents what Derrida, Fitzpatrick, and Hegel call ‘individuality’. -/- Derrida’s theory of law presents a conundrum. Derrida misses the possibility that law may exist by virtue of its content rather than its form. Derrida misses this possibility because, heavily influenced by Kant (in Derrida’s theory of law), Derrida associates law with universals. This is so because Kant (and Derrida) are preoccupied with the identity of what counts as a law (lois) rather than with a law’s legitimacy. A universal cannot exist unless it is legitimate, and it is legitimate, I claim, by virtue of its content. In his association of law with universals, Derrida presupposes that legal knowledge exists with reference to a territorial-like boundary. The forms are represented or signified by signs (signifiers) within a boundary of the ultimate form (the state, the nation, or humanity). This ultimate form as a universal, like the discrete rules or forms, lacks socially contingent content. A boundary separates knowable universals from the unknowable world on the exteriority of the boundary. The unknowable world is constituted by concrete events experienced in context-specific circumstances. In his legal theory Derrida hones in upon the decision as the experienced event. In a decision, one is present or immediate with the event. Derrida considers such immediacy as justice. The immediacy, however, can only be represented as a sign (sometimes called a signifier). The sign, in turn, represents an empty signified or form, according to Derrida. Because the immediacy remains a representation rather than a presentation of the experienced event, laws as universals cannot be just. The rupture between the inaccessible immediacy of a decision on the one hand and the represented empty forms on the other is critical to Derrida’s theory of law. -/- I claim that this rupture permeates Derrida’s writings about law because Derrida possesses a territorial-like sense of legal knowledge. I shall argue to this effect as follows. In the first section I shall explain the importance of Fitzpatrick’s exposure of the vacuity of the foundation of the system or structure of universals. In the second section I shall flesh out two elements of Derrida’s legal theory: law as form and the ipseity or concrete event that the form excludes from law. This takes me to the third section, where I shall elaborate how Derrida’s legal theory presupposes knowledge as territorial. I shall argue in the final section that this very sense of territorial knowledge prevents justice from accessing law and law from accessing justice. I conclude with the hint of a very different sense of law, one that draws from experiential knowledge in contradistinction to territorial knowledge. (shrink)

Examining the historical development of scientific concepts is important for understanding the structured routines within which these concepts are currently used as goal-directed tools in experiments. To illustrate this claim, I will outline how the concepts of mental imagery and hallucinations each draw on an older interdependent set of associations that, although nominally-discarded, continues to structure their current independent uses for pursuing discrete experimental goals. In doing so, I will highlight how three strands of literature offer mutually instructive insights (...) for understanding how the uses of our current scientific concepts contribute to experimental practices. The first strand of literature includes recent scholarship examining how the uses of scientific concepts can enable scientific practices (e.g., Boon 2012; Brigandt 2012; Feest 2010; Steinle 2012). In this context, scientific concepts have been described as tools that function in ways that contribute to empirical knowledge; contributions that extend beyond their traditionally recognized roles in mental and linguistic representation. The second strand of literature can be glimpsed within the technoscientific studies focus on non-human agency; a focus that supports an account of the disciplined routines within which concepts are used as acting in analogous ways to the routinized participation of machines in experiments (Pickering 1995). The third strand of the literature draws attention to how the functions of concepts are grounded by the set of historically-contingent experimental practices (e.g., Canguilhem 2008; Tiles 1984). Although maintaining valuable differences, the focus of these three strands of literature are mutually instructive. Together, they offer a view of the uses of concepts as taken-for-granted tools that function within networks of subterranean associations; associations that are grounded by historically contingent experimental practices that structure how they act (in concert with human and material participants) within the temporally dynamic processes of scientific practice. Through examples drawn from a comparison of the concepts of mental imagery and hallucinations, I aim to demonstrate the value that the converging insights of these three strands of literature offer for studying the role of scientific concepts in experimental practices. (shrink)

In the monograph Philosophy and Psychedelics: Frameworks for Exceptional Experience, Hauskeller raises the important subject of individualization and alienation in psychedelic psychotherapy. Under the prevailing conditions of neoliberalism, Hauskeller contends that psychedelic-assisted psychotherapy appropriates Indigenous knowledges in an oppressive fashion, may be instrumentalised to the ends of productivity gain and symptom suppression, and may be utilised to mask societal systems of alienation. Whilst offering a valuable socio-political critique of psychedelics' clinical uptake, we suggest that Hauskeller's view does not adequately acknowledge (...) the ways in which psychedelics offer a challenge to the Western reductive bio-medical understanding of healing and wellbeing. It is contended herein that Indigenous knowledges, in alliance with a range of emerging sciences, offer both an engagement with ethnomedicines in a less harmfully appropriative fashion, and a renewed understanding of the means by which psychedelics achieve therapeutic change. With this understanding, what becomes apparent are the potential ways in which psychedelic medical usage may produce positive feedback upon the oppressive systems in which we are embedded. That is, transpersonal experience through encounters with the ineffable may offer a revisioning of Western psychology and cognitive science. Indeed, if psychedelics are approached with an understanding of the actual means by which they produce therapeutic outcomes—changing mental representations of the self, or self-insight derived through non-ordinary states of consciousness—then psychedelic psychotherapy offers a reimagining of psychiatric nosology, challenging conventional understandings of both pathology and wellbeing through an overturning of specified and discrete deficit models of psychopathology. This may provide both a critique of the prevailing categories used to describe madness and an expansion of our understanding of the mind-body relation, as well as an increased recognition of positive psychology grounded in cross-cultural contemplative traditions. This provides an implicit challenge to the pharmaceutical industrial-complex and its profit motives; and the corresponding neoliberalist, globalising tendencies which Hauskeller seeks to address. -/- . (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)

Background Men with antisocial personality disorder show lifelong abnormalities in adaptive decision making guided by the weighing up of reward and punishment information. Among men with antisocial personality disorder, modifi cation of the behaviour of those with additional diagnoses of psychopathy seems particularly resistant to punishment. Methods We did a case-control functional MRI (fMRI) study in 50 men, of whom 12 were violent off enders with antisocial personality disorder and psychopathy, 20 were violent off enders with antisocial personality disorder but (...) not psychopathy, and 18 were healthy non-off enders. We used fMRI to measure brain activation associated with the representation of punishment or reward information during an event-related probabilistic response-reversal task, assessed with standard general linear-model-based analysis. Findings Offenders with antisocial personality disorder and psychopathy displayed discrete regions of increased activation in the posterior cingulate cortex and anterior insula in response to punished errors during the task reversal phase, and decreased activation to all correct rewarded responses in the superior temporal cortex. This finding was in contrast to results for off enders without psychopathy and healthy non-off enders. Interpretation Punishment prediction error signalling in off enders with antisocial personality disorder and psychopathy was highly atypical. This finding challenges the widely held view that such men are simply characterised by diminished neural sensitivity to punishment. Instead, this fi nding indicates altered organisation of the information processing system responsible for reinforcement learning and appropriate decision making. This difference between violent offenders with antisocial personality disorder with and without psychopathy has implications for the causes of these disorders and for treatment approaches. (shrink)

Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: algorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their cyclic and divergent properties.

The electronic and muonic hydrogen energy levels are calculated very accurately [1] in Quantum Electrodynamics (QED) by coupling the Dirac Equation four vector (c ,mc2) current covariantly with the external electromagnetic (EM) field four vector in QED’s Interactive Representation (IR). The c -Non Exclusion Principle(c -NEP) states that, if one accepts c as the electron/muon velocity operator because of the very accurate hydrogen energy levels calculated, the one must also accept the resulting electron/muon internal spatial and time coordinate operators (...) (ISaTCO) derived directly from c without any assumptions. This paper does not change any of the accurate QED calculations of hydrogen’s energy levels, given the simplistic model of the proton used in these calculations [1]. The Proton Radius Puzzle [2, 3] may indicate that new physics is necessary beyond the Standard Model (SM), and this paper describes the bizarre, and very different, situation when the electron and muon are located “inside” the spatially extended proton with their Centers of Charge (CoCs) orbiting the proton at the speed of light in S energy states. The electron/muon center of charge (CoC) is a structureless point that vibrates rapidly in its inseparable, non-random vacuum whose geometry and time structure are defined by the electron/muon ISaTCO discrete geometry. The electron/muon self mass becomes finite in a natural way due to the ISaTCOs cutting off high virtual photon energies in the photon propagator. The Dirac-Maxwell-Wilson (DMW) Equations are derived from the ISaTCO for the EM fields of an electron/muon, and the electron/muon “look” like point particles in far field scattering experiments in the same way the electric field from a sphere with evenly distributed charge “e” “looks” like a point with the same charge in the far field (Gauss Law). The electron’s/muon’s three fluctuating CoC internal spatial coordinate operators have eight possible eigenvalues [4, 5, 6] that fall on a spherical shell centered on the electron’s CoM with radius in the rest frame. The electron/muon internal time operator is discrete, describes the rapid virtual electron/positron pair production and annihilation. The ISaTCO together create a current that produce spin and magnetic moment operators, and the electron and muon no longer have “intrinsic” properties since the ISaTCO kinematics define spin and magnetic moment properties. (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)

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)

This book is devoted to the presentation of the new quantum mechanical formalism based on the probability representation of quantum states. In the 20s and 30s it became evident that some properties in quantum mechanics can be assigned only to the quantum mechanical system, but not necessarily to its constituents. This led Einstein, Podolsky and Rosen (EPR) to their remarkable 1935 paper where they concluded that quantum mechanics is not a complete theory of nature (EPR paradox). In order to (...) avoid the contradiction which arises from instantaneous action at a distance mentioned above we introduce an extension of the canonical relativity by using measure algebra of physical events in Minkowski space-time. The canonical QM formalism is extended by additional new postulate of EPRB nonlocality for continuous and discrete observables, chapter I. The postulate of EPRB nonlocality is supported by new quantum mechanical formalism based on the probability representation of quantum states. Chapter II is devoted to the new quantum mechanical formalism based on the probability representation of quantum states. Chapter III is devoted to the Einstein's 1927 gedanken experiment resolution. Chapter IV is devoted to the EPR paradox resolution. Chapter V is devoted to the EPR-B paradox resolution. Chapter VI is devoted to the Schrödinger's cat (measured spin) paradox resolution. Chapter VII is devoted to the Bell inequalities revisited. Remind that the canonical arguments which were presented by many authors, namely, that violations of Bell type inequalities signal us that the classical Kolmogorovian model of probability is inapplicable to quantum phenomena. We claimed that the canonical assumption, under which Bell type inequalities were derived, is not supported by real physical nature of the EPRB experiments. The fundamental physical nature violations of the canonical Bell type inequalities explained by Postulate of EPR-Nonlocality and Heisenberg noise-disturbance uncertainty relations. (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)

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As (...) a result, we should reject these claims, and lay the foundations of decision theory on firmer ground. (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)

How can we think about things in the outside world? There is still no widely accepted theory of how mental representations get their meaning. In light of pioneering research, Nicholas Shea develops a naturalistic account of the nature of mental representation with a firm focus on the subpersonal representations that pervade the cognitive sciences.

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)

In this article, network science is discussed from a methodological perspective, and two central theses are defended. The first is that network science exploits the very properties that make a system complex. Rather than using idealization techniques to strip those properties away, as is standard practice in other areas of science, network science brings them to the fore, and uses them to furnish new forms of explanation. The second thesis is that network representations are particularly helpful in explaining the properties (...) of non-decomposable systems. Where part-whole decomposition is not possible, network science provides a much-needed alternative method of compressing information about the behavior of complex systems, and does so without succumbing to problems associated with combinatorial explosion. The article concludes with a comparison between the uses of network representation analyzed in the main discussion, and an entirely distinct use of network representation that has recently been discussed in connection with mechanistic modeling. (shrink)

Whenever debt is itself money, this money becomes a self-inflating debt principal by already being its own interest. Hence modern inflation, deflation, and eventual monetary crises. Yet why does money become debt? The concept of representational monetary identity answers to precisely this question.

This chapter discusses recent attempts to clarify the notion of practical representation and its theoretical fruitfulness. The ultimate goal is not just to show that intellectualists are on good grounds when they appeal to practical representation in their theories of know-how. Rather, it is to argue that ​ any plausible theory of skill and know-how has to appeal to the notion of practical representation developed here. §1 explains the notion of a mode of presentation and introduces practical (...) modes of presentation. §2 illustrates practical representation by discussing models of motor control in current theories of sensori-motor psychology; §3 puts forward an argument for positing practical representation. §4 goes from practical non-conceptual representations to practical conceptual representations — to practical concepts. §5 concludes. (shrink)

According to a standard representationalist view cognitive capacities depend on internal content-carrying states. Recent alternatives to this view have been met with the reaction that they have, at best, limited scope, because a large range of cognitive phenomena—those involving absent and abstract features—require representational explanations. Here we challenge the idea that the consideration of cognition regarding the absent and the abstract can move the debate about representationalism along. Whether or not cognition involving the absent and the abstract requires the positing (...) of representations depends upon whether more basic forms of cognition require the positing of representations. (shrink)

Much recent cognitive neuroscientific work on body knowledge is representationalist: “body schema” and “body images”, for example, are cerebral representations of the body (de Vignemont 2009). A framework assumption is that representation of the body plays an important role in cognition. The question is whether this representationalist assumption is compatible with the variety of broadly situated or embodied approaches recently popular in the cognitive neurosciences: approaches in which cognition is taken to have a ‘direct’ relation to the body and (...) to the environment. A “direct” relation is one where the boundaries between the body and the head, or between the environment and the animal are not theoretically important in the understanding of cognition. These boundaries do not play a theoretically privileged role in cognitive explanations of behavior. But representationalism appears to put a representational veil between the locus of cognition and that which is represented, making cognitive relations to the body and to the environment be indirect, with a high associated computational load. For this reason, direct approaches have tried to minimize the use of internal representations (Suchman 1987; Barwise 1987; Agre and Chapman 1987; Brooks 1992; Thelen and Smith 1994; van Gelder 1995; Port and van Gelder 1995; Clark 1997, 1999; Rupert 2009, p. 180). Does a cognitive neuroscience committed to direct relations rule out a representationalist approach to body knowledge? Or is direct representationalism possible? (shrink)

The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal (...) representations’, which are representations that capture all that is relevant to explain a physical phenomenon at a specified level of description. In the end, because the role of mathematics in MEPPs is ultimately representational, they cannot be used to support mathematical Platonism. I finish the paper by addressing the claim, advanced by many EIA defendants, that quantification over mathematical objects results in explanations that have more theoretical virtues, especially that they are more general and modally stronger than alternative explanations. I will show that the EIA cannot be successfully defended by appealing to these notions. (shrink)

Many philosophers and psychologists have attempted to elucidate the nature of mental representation by appealing to notions like isomorphism or abstract structural resemblance. The ‘structural representations’ that these theorists champion are said to count as representations by virtue of functioning as internal models of distal systems. In his 2007 book, Representation Reconsidered, William Ramsey endorses the structural conception of mental representation, but uses it to develop a novel argument against representationalism, the widespread view that cognition essentially involves (...) the manipulation of mental representations. Ramsey argues that although theories within the ‘classical’ tradition of cognitive science once posited structural representations, these theories are being superseded by newer theories, within the tradition of connectionism and cognitive neuroscience, which rarely if ever appeal to structural representations. Instead, these theories seem to be explaining cognition by invoking so-called ‘receptor representations’, which, Ramsey claims, aren’t genuine representations at all—despite being called representations, these mechanisms function more as triggers or causal relays than as genuine stand-ins for distal systems. I argue that when the notions of structural and receptor representation are properly explicated, there turns out to be no distinction between them. There only appears to be a distinction between receptor and structural representations because the latter are tacitly conflated with the ‘mental models’ ostensibly involved in offline cognitive processes such as episodic memory and mental imagery. While structural representations might count as genuine representations, they aren’t distinctively mental representations, for they can be found in all sorts of non-intentional systems such as plants. Thus to explain the kinds of offline cognitive capacities that have motivated talk of mental models, we must develop richer conceptions of mental representation than those provided by the notions of structural and receptor representation. (shrink)

Despite its centrality in the philosophy of cognitive science, there has been little prior philosophical work engaging with the notion of representation in contemporary NLP practice. This paper attempts to fill that lacuna: drawing on ideas from cognitive science, I introduce a framework for evaluating the representational claims made about components of neural NLP models, proposing three criteria with which to evaluate whether a component of a model represents a property and operationalising these criteria using probing classifiers, a popular (...) analysis technique in NLP (and deep learning more broadly). The project of operationalising a philosophically-informed notion of representation should be of interest to both philosophers of science and NLP practitioners. It affords philosophers a novel testing-ground for claims about the nature of representation, and helps NLPers organise the large literature on probing experiments, suggesting novel avenues for empirical research. (shrink)

Structural representations are increasingly popular in philosophy of cognitive science. A key virtue they seemingly boast is that of meeting Ramsey's job description challenge. For this reason, structural representations appear tailored to play a clear representational role within cognitive architectures. Here, however, I claim that structural representations do not meet the job description challenge. This is because even our most demanding account of their functional profile is satisfied by at least some receptors, which paradigmatically fail the job description challenge. Hence, (...) the functional profile typically associated with structural representations does not identify representational posits. After a brief introduction, I present, in the second section of the paper, the job description challenge. I clarify why receptors fail to meet it and highlight why, as a result, they should not be considered representations. In the third section I introduce what I take to be the most demanding account of structural representations at our disposal, namely Gładziejewski's account. Provided the necessary background, I turn from exposition to criticism. In the first half of the fourth section, I equate the functional profile of structural representations and receptors. To do so, I show that some receptors boast, as a matter of fact, all the functional features associated with structural representations. Since receptors function merely as causal mediators, I conclude structural representations are mere causal mediators too. In the second half of the fourth section I make this conclusion intuitive with a toy example. I then conclude the paper, anticipating some objections my argument invites. (shrink)

This dissertation argues that mental representation is identical to phenomenal consciousness, and everything else that appears to be both mental and a matter of representation is not genuine mental representation, but either in some way derived from mental representation, or a case of non-mental representation.

This paper engages critically with anti-representationalist arguments pressed by prominent enactivists and their allies. The arguments in question are meant to show that the “as-such” and “job-description” problems constitute insurmountable challenges to causal-informational theories of mental content. In response to these challenges, a positive account of what makes a physical or computational structure a mental representation is proposed; the positive account is inspired partly by Dretske’s views about content and partly by the role of mental representations in contemporary cognitive (...) scientific modeling. (shrink)

Recently there is some new interest in understanding the physical reality behind the formalism of quantum mechanics. This paper relates the known “quantum mysteries” of QM with the properties of the underlying structure of discrete space. DOI: 10.5281/zenodo.5236617.

If emotion is not an arbitrary compilation of fixed types of (descriptive, conceptual, conative, prescriptive) content, nor a state that can be reduced to other types of pre-existing (perceptual, cognitive, behavioral) states, then what sort of thing is it really? Tom Cochrane has proposed that emotions are valent representations of situated concerns. Valent representation is a type of mental content whose function is to detect the presence or absence of certain conditions; what makes that type of content valent is (...) that without needing the mediation of any other state, a response is triggered that is disposed to increase or decrease the presence of the condition represented. I raise doubts about the plausibility of that account in light of recent work in philosophy of biology and philosophy of mind. (shrink)

The main task of this paper is to understand if and how static images like photographs can represent and/or depict temporal extension (duration). In order to do this, a detour will be necessary to understand some features of the nature of photographic representation and depiction in general. This important detour will enable us to see that photographs (can) have a narrative content, and that the skilled photographer can 'tell a story' in a very clear sense, as well as control (...) and guide the attention of the spectator of the photograph. The understanding and defence of this claim is a secondary aim of this paper, and it will then allow us to provide a good treatment of the particular case of photographic representation and depiction of temporal extension. (shrink)

This is a chapter from my introductory book *Perception* covering the representational view of experience. I use the Ramsey-Lewis method to define the theoretical term "experiential representation". I clarify and discuss various questions for representationalists, for instance, "how rich is the content of experience?" and "is the content of visual experience singular or general?" Finally, I address some objections to representationalism - in particular, that it cannot explain perceptual presence (John Campbell), and that it cannot explain the "laws of (...) appearance" (constraints on how things can appear). (shrink)

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a (...) remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced. (shrink)

A satisfactory solution to the problem of consciousness would take the form of a simple yet fully general model that specifies the precise conditions under which any given state of consciousness occurs. Science has uncovered numerous correlations between consciousness and neural activity, but it has not yet come anywhere close to this. We are still looking for the Newtonian laws of consciousness. -/- One of the main difficulties with consciousness is that we lack a language in which to formulate illuminating (...) generalizations about it. Philosophers and scientists talk about "what it’s like", sensations, feelings, and perceptual states such as seeing and hearing. This language does not allow a precise articulation of the internal structures of conscious states and their inter-relations. It is inadequate to capture relations of the kind we are looking for between conscious states and physical states. -/- In this thesis I refine and defend a theory of consciousness which promises to solve this regimentation problem: the representational theory of consciousness. I argue that the representational theory can solve the regimentation problem and smooth out other important obstacles to a fruitful study of consciousness. I also make a case for the theory independently of its payoffs, and I discuss the leading opposing theories at some length. (shrink)

The idea that pictorial art can have cognitive value, that it can enhance our understanding of the world and of our own selves, has had many advocates in art theory and philosophical aesthetics alike. It has also been argued, however, that the power of pictorial representation to convey or enhance knowledge, in particular knowledge with moral content, is not generalized across the medium.

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power (...) and meta-theoretic properties when comparing first-order and second-order logic. (shrink)

The standard representation theorem for expected utility theory tells us that if a subject’s preferences conform to certain axioms, then she can be represented as maximising her expected utility given a particular set of credences and utilities—and, moreover, that having those credences and utilities is the only way that she could be maximising her expected utility. However, the kinds of agents these theorems seem apt to tell us anything about are highly idealised, being always probabilistically coherent with infinitely precise (...) degrees of belief and full knowledge of all a priori truths. Ordinary subjects do not look very rational when compared to the kinds of agents usually talked about in decision theory. In this paper, I will develop an expected utility representation theorem aimed at the representation of those who are neither probabilistically coherent, logically omniscient, nor expected utility maximisers across the board—that is, agents who are frequently irrational. The agents in question may be deductively fallible, have incoherent credences, limited representational capacities, and fail to maximise expected utility for all but a limited class of gambles. (shrink)

An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be (...) found already in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

This paper has a two-fold aim. First, it reinforces a version of the "syntactic argument" given in Aizawa (1994). This argument shows that connectionist networks do not provide a means of implementing representations without rules. Horgan and Tlenson have responded to the syntactic argument in their book and in another paper (Horgan & Tlenson, 1993), but their responses do not meet the challenge posed by my formulation of the syntactic argument. My second aim is to describe a kind of cognitive (...) architecture. This architecture might be called a computational architecture, but it is not a rules and representations architecture nor the representations without rules architecture that Horgan and Tlenson wish to endorse. (shrink)

Arguably the most foundational principle in perception research is that our experience of the world goes beyond the retinal image; we perceive the distal environment itself, not the proximal stimulation it causes. Shape may be the paradigm case of such “unconscious inference”: When a coin is rotated in depth, we infer the circular object it truly is, discarding the perspectival ellipse projected on our eyes. But is this really the fate of such perspectival shapes? Or does a tilted coin retain (...) an elliptical appearance even when we know it’s circular? This question has generated heated debate from Locke and Hume to the present; but whereas extant arguments rely primarily on introspection, this problem is also open to empirical test. If tilted coins bear a representational similarity to elliptical objects, then a circular coin should, when rotated, impair search for a distal ellipse. Here, nine experiments demonstrate that this is so, suggesting that perspectival shapes persist in the mind far longer than traditionally assumed. Subjects saw search arrays of three-dimensional “coins,” and simply had to locate a distally elliptical coin. Surprisingly, rotated circular coins slowed search for elliptical targets, even when subjects clearly knew the rotated coins were circular. This pattern arose with static and dynamic cues, couldn’t be explained by strategic responding or unfamiliarity, generalized across shape classes, and occurred even with sustained viewing. Finally, these effects extended beyond artificial displays to real-world objects viewed in naturalistic, full-cue conditions. We conclude that objects have a remarkably persistent dual character: their objective shape “out there,” and their perspectival shape “from here.”. (shrink)