In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howard's majorizability notion; see [T, p. 456].

Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $, into the Esterle algebra of formal power (...) series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on X. (shrink)

Two different kinds of stochastic processes in Hilbert space used to introduce spontaneous localization into the quantum evolution are investigated. In the processes of the first type, finite changes of the wave function take place instantaneously with a given mean frequency. The processes of the second type are continuous. It is shown that under a suitable infinite frequency limit the discontinuous process transforms itself into the continuous one.

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mind-change-bounded learnability is equivalent to finite View the MathML source2-piecewise computability 2 denotes the difference of two View the MathML sourceΠ10 sets), error-bounded learnability is equivalent to (...) class='Hi'>finite View the MathML sourceΔ20-piecewise computability, and learnability is equivalent to countable View the MathML sourceΠ10-piecewise computability . Second, we introduce disjunction-like operations such as the coproduct based on BHK-like interpretations, and then, we see that these operations induce Galois connections between the Medvedev degree structure and associated Medvedev/ Muchnik-like degree structures. Finally, we interpret these results in the context of the Weihrauch degrees and Wadge-like games. (shrink)

Contrasting accounts of physicalism and strong emergentism face two problems. According to the neutrality problem, contrasting supervenience-based formulations of these positions cannot be neutral with respect to certain unrelated metaphysical commitments. According to the collapse problem, emergent properties can be accounted for using an appropriately expansive physical ontology, rendering strong emergentism metaphysically suspect. I argue that both these problems can be solved with a principled distinction between emergent causal laws and physical laws. I propose such a distinction based on a (...) finite discontinuity in the behavior of fundamental physical constituents as a function of complexity. (shrink)

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies (Nash, Proceedings (...) of the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players. (shrink)

The Royal College of Physicians of London published the 2013 national clinical guidelines on prolonged disorders of consciousness in vegetative and minimally conscious states. The guidelines acknowledge the rapidly advancing neuroscientific research and evolving therapeutic modalities in PDOC. However, the guidelines state that end-of-life decisions should be made for patients who do not improve with neurorehabilitation within a finite period, and they recommend withdrawal of clinically assisted nutrition and hydration . This withdrawal is deemed necessary because patients in PDOC (...) can survive for years with continuation of CANH, even when a ceiling on medical care has been imposed, i.e., withholding new treatment such as cardiopulmonary resuscitation for acute life-threatening illness. The end-of-life care pathway is centered on a staged escalation of medications, including sedatives, opioids, barbiturates, and general anesthesia, concurrent with withdrawal of CANH. Agitation and distress may last from several days to weeks because of the slow dying process from starvation and dehydration. The potential problems of this end-of-life care pathway are similar to those of the Liverpool Care Pathway. After an independent review in 2013, the Department of Health discontinued the Liverpool Care pathway in England. The guidelines assert that clinicians, supported by court decisions, have become the final authority in nonconsensual withdrawal of CANH on the basis of “best interests” rationale. We posit that these guidelines lack high-quality evidence supporting: 1) treatment futility of CANH, 2) reliability of distress assessment from starvation and dehydration, 3) efficacy of pharmacologic control of this distress, and 4) proximate causation of death. Finally, we express concerns about the utilitarian-based assessment of what constitutes a person’s best interests. We are disturbed by the level and the role of medical authoritarianism institutionalized by these national guidelines when deciding on the worthiness of life in PDOC. We conclude that these guidelines are not only harmful to patients and families, but they represent the means of nonconsensual euthanasia. The latter would constitute a gross violation of the public’s trust in the integrity of the medical profession. (shrink)

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete (...) functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:119. HUME'S CONDITIONS FOR' CAUSATION: FURTHER TO GRAY AND IMLAY As part of his second proof of the existence of God, Descartes in Meditations III argues a causal premise derived from the nature of time. He argues it follows from the nature of time "that, in order to be conserved in each moment in which it endures, a substance has need of the same power and action as would (...) be necessary to produce and create it anew, supposing it did not yet exist; so that the light of nature shows us clearly that the distinction between creation and conservation is solely a distinction of reason." The nature of time to which he refers allows only a finite divisibility into durationless moments, each of which is entirely independent of every other. In the ultimate analysis, nothing true of one moment is deducible from what is true of any other, even what happens to be eternally true. The doctrine of the creation of the eternal verities is of a piece with this conception of time, and following with it from the conception of the divinity as sheer omnipotence, points to Hume's view that experience is the only guide to the world. It would seem to be Descartes' view, that nothing at a given time can stand in the necessary relation it must have to be a cause of something at another time; or, only something outside time can be a cause, for whose operation things in time can serve as but occasions. To what extent this dialectic in fact led to later occasionalist views is an historical question we need not consider. My only suggestion here concerns Hume's réponse to the occasionalists. With all but psychological necessity eliminated from his analysis of causation, Hume effectively argues the familiar case that occasional causes are the only causes by arguing the finite divisibility of time. In the literature only Laporte has held that for Descartes time is infinitely divisible and in this sense continuous. The separability of the parts of time, he 120. thought, suffice for their independence and contingency. But if time is continuous, its parts are inseparable; as Gueroult correctly observes, by discontinuity is meant con2 tingence, independence and separability. In addition, if a moment had duration, light would not be propagated instant aneously, thus according to Descartes upsetting his entire 3 physics. Gueroult maintains, rather, that Descartes' view of time is determined by his view of motion. Whatever the historical and logical priorities, there clearly is a connection between the above and what Bergson called the cinematographic conception of motion as a succession of static objects. It is a conception that precludes events as such. Motion occurs only in time, never at the moment, and occurs as a result of the re-creation of matter with regularly differing spatial relations. Due to metaphysical consideration expressed in theological terms, especially with reference to the immutability and simplicity of divine ways, this recreation of matter is constrained in certain ways, some of which we pick out as conditions for what was later called occasional causation. Put very simply, only what is constantly conjoined can stand in this relation; and that there should be instances of constant conjunction follows from the metaphysical considerations. With Hume's elimination of these metaphysical considerations, the status of the conditions, which he takes over as conditions for causation simpliciter, becomes problematic. But a recent challenge to Hume's program is still more problematic. The contention is that there is a case in which Hume's conditions necessary fo constant conjunction cannot all be satisfied. Even worse, i is a case in which his analysis ought to be conspicuously successful, viz. collision, which Hume like Descartes con4 ceives in cinematographic terms. The aim of this paper is to show that Hume's analysis stands against the challenge. Consider the following diagram which illustrates the positions (A3, A_...) of two balls (x,y) at successive moments (t_, t.,...)· 121. A3 A2 A1 A B -B1 B2 B3 fc0 T T T ©© © G®©T©T h fc3 fc4 fc5 H Robert Gray's thesis is that in at least one case of causation, collision which results in... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:119. HUME'S CONDITIONS FOR' CAUSATION: FURTHER TO GRAY AND IMLAY As part of his second proof of the existence of God, Descartes in Meditations III argues a causal premise derived from the nature of time. He argues it follows from the nature of time "that, in order to be conserved in each moment in which it endures, a substance has need of the same power and action as would (...) be necessary to produce and create it anew, supposing it did not yet exist; so that the light of nature shows us clearly that the distinction between creation and conservation is solely a distinction of reason." The nature of time to which he refers allows only a finite divisibility into durationless moments, each of which is entirely independent of every other. In the ultimate analysis, nothing true of one moment is deducible from what is true of any other, even what happens to be eternally true. The doctrine of the creation of the eternal verities is of a piece with this conception of time, and following with it from the conception of the divinity as sheer omnipotence, points to Hume's view that experience is the only guide to the world. It would seem to be Descartes' view, that nothing at a given time can stand in the necessary relation it must have to be a cause of something at another time; or, only something outside time can be a cause, for whose operation things in time can serve as but occasions. To what extent this dialectic in fact led to later occasionalist views is an historical question we need not consider. My only suggestion here concerns Hume's réponse to the occasionalists. With all but psychological necessity eliminated from his analysis of causation, Hume effectively argues the familiar case that occasional causes are the only causes by arguing the finite divisibility of time. In the literature only Laporte has held that for Descartes time is infinitely divisible and in this sense continuous. The separability of the parts of time, he 120. thought, suffice for their independence and contingency. But if time is continuous, its parts are inseparable; as Gueroult correctly observes, by discontinuity is meant con2 tingence, independence and separability. In addition, if a moment had duration, light would not be propagated instant aneously, thus according to Descartes upsetting his entire 3 physics. Gueroult maintains, rather, that Descartes' view of time is determined by his view of motion. Whatever the historical and logical priorities, there clearly is a connection between the above and what Bergson called the cinematographic conception of motion as a succession of static objects. It is a conception that precludes events as such. Motion occurs only in time, never at the moment, and occurs as a result of the re-creation of matter with regularly differing spatial relations. Due to metaphysical consideration expressed in theological terms, especially with reference to the immutability and simplicity of divine ways, this recreation of matter is constrained in certain ways, some of which we pick out as conditions for what was later called occasional causation. Put very simply, only what is constantly conjoined can stand in this relation; and that there should be instances of constant conjunction follows from the metaphysical considerations. With Hume's elimination of these metaphysical considerations, the status of the conditions, which he takes over as conditions for causation simpliciter, becomes problematic. But a recent challenge to Hume's program is still more problematic. The contention is that there is a case in which Hume's conditions necessary fo constant conjunction cannot all be satisfied. Even worse, i is a case in which his analysis ought to be conspicuously successful, viz. collision, which Hume like Descartes con4 ceives in cinematographic terms. The aim of this paper is to show that Hume's analysis stands against the challenge. Consider the following diagram which illustrates the positions (A3, A_...) of two balls (x,y) at successive moments (t_, t.,...)· 121. A3 A2 A1 A B -B1 B2 B3 fc0 T T T ©© © G®©T©T h fc3 fc4 fc5 H Robert Gray's thesis is that in at least one case of causation, collision which results in... (shrink)

Quantum Theory, similar to Relativity Theory, requires a new concept of space-time, imposed by a universal constant. While velocity of lightcnot being infinite calls for a redefinition of space-time on large and cosmological scales, quantization of action in terms of a finite, i.e. non vanishing, universal constanthrequires a redefinition of space-time on very small scales. Most importantly, the classical notion of “time”, as one common continuous time variable and nature evolving continuously “in time”, has to be replaced by an (...) infinite manifold of transition rates for discontinuous quantum transitions. The fundamental laws of quantum physics, commutation relations and quantum equations of motion, resulted from Max Born’s recognition of the basic principle of quantum physics:To each change in nature corresponds an integer number of quanta of action. Action variables may only change by integer values ofh, requiring all other physical quantities to change by discrete steps, “quantum jumps”. The mathematical implementation of this principle led to commutation relations and quantum equations of motion. The notion of “point” in space-time looses its physical significance; quantum uncertainties of time, position, just as any other physical quantity, are necessary consequences of quantization of action. (shrink)

The paradox of phase transitions raises the problem of how to reconcile the fact that we see phase transitions happen in concrete, finite systems around us, with the fact that our best theories—i.e. statistical-mechanical theories of phase transitions—tell us that phase transitions occur only in infinite systems. In this paper we aim to clarify to which extent this paradox is relative to the mathematical framework which is used in these theories, i.e. classical mathematics. To this aim, we will explore (...) the philosophical consequences of adopting constructive instead of classical mathematics in a statistical-mechanical theory of phase transitions. It will be shown that constructive mathematics forces certain ‘de-idealizations’ of such theories: talk of actually infinite systems is meaningless, there are no discontinuous functions, and—in a sense which will be clarified—constructive real numbers reflect our imperfect methods of determining the values of physical quantities. As such, so it will be argued, constructive mathematics offers a means to gain insight in the idealizations introduced in classical theories and the philosophical issues surrounding them. (shrink)

The paradox of phase transitions raises the problem of how to reconcile the fact that we see phase transitions happen in concrete, finite systems around us, with the fact that our best theories—i.e. statistical-mechanical theories of phase transitions—tell us that phase transitions occur only in infinite systems. In this paper we aim to clarify to which extent this paradox is relative to the mathematical framework which is used in these theories, i.e. classical mathematics. To this aim, we will explore (...) the philosophical consequences of adopting constructive instead of classical mathematics in a statistical-mechanical theory of phase transitions. It will be shown that constructive mathematics forces certain ‘de-idealizations’ of such theories: talk of actually infinite systems is meaningless, there are no discontinuous functions, and—in a sense which will be clarified—constructive real numbers reflect our imperfect methods of determining the values of physical quantities. As such, so it will be argued, constructive mathematics offers a means to gain insight in the idealizations introduced in classical theories and the philosophical issues surrounding them. (shrink)

A discussion of the problem of contradiction in mechanical motion has been in progress for a long time in the pages of Filosofskie nauki. The attention given that problem is no accident. In our day, problems concerning contradictions involved in rest and motion, the continuous and the discontinuous, the finite and infinite, etc., have moved from the realm of abstract consideration to that of the concrete and current handling of the subject matter of modern physics, mathematics, and other fields. (...) It is natural that the present stage of knowledge of objective reality should occasion a revived interest on the part of logicians and philosophers in these problems, including the question of how objective contradictions may be expressed in scientific theories. (shrink)

The abstract basis of modern computation is the formal description of a finite state machine, the Universal Turing Machine, based on manipulation of integers and logic symbols. In this contribution to the discourse on the computer-brain analogy, we discuss the extent to which analog computing, as performed by the mammalian brain, is like and unlike the digital computing of Universal Turing Machines. We begin with ordinary reality being a permanent dialog between continuous and discontinuous worlds. So it is with (...) computing, which can be analog or digital, and is often mixed. The theory behind computers is essentially digital, but efficient simulations of phenomena can be performed by analog devices; indeed, any physical calculation requires implementation in the physical world and is therefore analog to some extent, despite being based on abstract logic and arithmetic. The mammalian brain, comprised of neuronal networks, functions as an analog device and has given rise to artificial neural networks that are implemented as digital algorithms but function as analog models would. Analog constructs compute with the implementation of a variety of feedback and feedforward loops. In contrast, digital algorithms allow the implementation of recursive processes that enable them to generate unparalleled emergent properties. We briefly illustrate how the cortical organization of neurons can integrate signals and make predictions analogically. While we conclude that brains are not digital computers, we speculate on the recent implementation of human writing in the brain as a possible digital path that slowly evolves the brain into a genuine (slow) Turing machine. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Foucault’s Renaissance Episteme Reassessed: An Aristotelian CounterblastIan MacleanThere seem to me to be two good reasons for looking at Foucault’s Renaissance episteme again, even though specialists of the Renaissance have given it short shrift and Foucault himself does not seem to have set great store by it in his later writings. 1 The first is that in general books on Foucault accounts of it are still given in a (...) neutral or even approving way, as though neither the concept of episteme (which suggests both that mind-sets or paradigms or epistemes are all-constraining) nor the account of the Renaissance episteme in particular (which is given a predominantly Platonic and hermetic character) require modification, if not refutation. 2 I shall suggest that although in some sense we are all prisoners of our mind-sets, there are ways in which we can escape from this prison 3 ; and with respect to the Renaissance in particular, I shall claim that Aristotelianism, not Platonism, is the appropriate context in which to see the dominant universe of discourse of the day and that this philosophy can be shown to possess the resources for escaping its limitations. To do this I shall [End Page 149] give a brief resume of Foucault’s episteme in general and the Renaissance episteme in particular. Then I shall show first, that the Platonist features which he cites as typical are actively refuted at that time and cannot be described as dominant without severe qualification; second, that the versions of Aris-totelianism widely taught in universities and schools give quite different accounts of these features; and third, that this form of thinking can be shown to have the freedom of self-critique in the very areas in which Foucault chooses to see Renaissance philosophy as both constraining and constrained.In 1969 Foucault produced his “theoretical work” L’Archéologie du savoir as the methodological systematization of his earlier historical writings, L’Histoire de la folie à l’âge classique (l961), La Naissance de la clinique (l963), and Les Mots et les choses (1966). In this work man is described as a modern self-invention, fashioned from a particular configuration of knowledge and destined soon to disappear. Foucault’s history of man is a story about the way he constituted himself as a subject and then made his body, his behavior, his reason, and his consciousness the object of his study and a field to be controlled, regulated, trained, and coerced. In this analysis man’s “rationality” becomes not a faculty but an authoritarian straitjacket. What Foucault sets out to investigate is a very fundamental form of ontology but not one associated with profound knowledge; it is a superficial and finite, if largely invisible, field, which is discontinuous, not defined by the agenda of the modern state of any associated discipline, and peopled by “utterances” (énoncés) which are only graspable in their differences and their dispersion. If coagulations of these “utterances” (formations discursives) occur at all, they appear only in the form of consistent relations: the objects to which they “refer” are no more than objectifications whose scientific status cannot be established and whose meaning cannot be recovered by semantic or rhetorical analysis. In calling his previous book Les Mots et les choses, he was being explicitly ironic, for neither the words nor the things of the title have any substance to them: they are not presences but absences. The description made by the historian is remote from his object, indirect, incomplete; it describes a “preconceptual field” in which the “preconcepts” are no more than organizing operators or functions revealing relationships of dependence, processes of validation, contingent theories of truth. The “utterances” which these “preconcepts” inform come together in “discourses” which are neither systems nor mere aggregations of knowledge; archaeological history as practiced by Foucault uncovers not a “mentality” but an “episteme,” which he defines as follows:By episteme we mean... the total set of relations that unite, at a given period, the discursive practices that give rise to epistemological figures, sciences, possibly formalized systems.... The episteme is not a form of knowledge or type of rationality which, crossing the boundaries [End Page 150] of the most... (shrink)

We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown to (...) be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling. (shrink)

In view of the primacy assigned to the 'present' in traditional metaphysics, in terms of the ways in which questions about existence are expressed, the following discussion takes the question of the temporalizing of the present as its theme. This involves unravelling the historical traces of the thought of the present as a finite, closed, objective point of a successive continuum of discrete moments (a real oscillation between the now and the not-now) by returning to the phenomenological sense of (...) the present as the stretching out of an opening – the 'living Present' (lebendige Gegenwart) – which bears its continuity of presence and non-presence within itself (without restriction to linearity). The transition itself suggests something like a quantum-leap and, in another sense, it also extends beyond the bounds of this simile (and the discontinuity that is implied) by evoking the image of a 'twist' or a 'turn.’ In order to grasp the significance of this turn we shall first examine – re turn to – its main obstacle: the concept of time as a linear and corpuscular continuum. The traditional model of time as a succession of 'now-points' (a notion that 1 still infects discourse on temporality) has always undermined our understanding of 'presence' as that which maintains itself (abides) through succession. In effect, presence must be 'maintained' [maintenant] within the 'now.' Yet, if the 'now' is constantly shifting into non-being through its replacement by a new 'now' then presence must be infused with its own negation and a certain discontinuity. How is it possible, then, to speak of the 'persistence' of 'identity' as something unitary (simultaneous with itself) existing through plurality and successive fragmentation into non-being? Furthermore, in reference to motion, what is entailed in the possibility of experiencing the transition of a selfsame (particular) object from one spatial location to another: how is it that the object 'endures' through its spatial and temporal transition? Since antiquity the question of simultaneity has been taken for granted – generally being consigned to mere spatial models.. (shrink)

This study shows firstly that it is necessary to characterize a computer simulation at a finer level than that of formal models: that of symbols and their various modes of reference. This is particularly true for those that integrate models and formalisms of a heterogeneous nature. This study then examines the ontological causes that, consequently, could explain their epistemic success. It is argued that they can be conveniently explained if one adopts a conception of nature that is both discontinuous and (...) finite. The latter thesis supports a naturalistic metaphysical position by invalidating, at its root, the argument made by speculative materialism against a scientific approach to the physical world that proceeds, like simulations, by finite representations and operations. (shrink)

In the chapter of Difference and Repetition entitled ‘Ideas and the synthesis of difference,’ Deleuze mobilizes mathematics to develop a ‘calculus of problems’ that is based on the mathematical philosophy of Albert Lautman. Deleuze explicates this process by referring to the operation of certain conceptual couples in the field of contemporary mathematics: most notably the continuous and the discontinuous, the infinite and the finite, and the global and the local. The two mathematical theories that Deleuze draws upon for this (...) purpose are the differential calculus and the theory of dynamical systems, and Galois’ theory of polynomial equations. For the purposes of this paper I will only treat the first of these, which is based on the idea that the singularities of vector fields determine the local trajectories of solution curves, or their ‘topological behaviour’. These singularities can be described in terms of the given mathematical problematic, that is for example, how to solve two divergent series in the same field, and in terms of the solutions, as the trajectories of the solution curves to the problem. What actually counts as a solution to a problem is determined by the specific characteristics of the problem itself, typically by the singularities of this problem and the way in which they are distributed in a system. Deleuze understands the differential calculus essentially as a ‘calculus of problems’, and the theory of dynamical systems as the qualitative and topological theory of problems, which, when connected together, are determinative of the complex logic of different/ciation. (DR 209). Deleuze develops the concept of a problematic idea from the differential calculus, and following Lautman considers the concept of genesis in mathematics to ‘play the role of model ... with respect to all other domains of incarnation’. While Lautman explicated the philosophical logic of the actualization of ideas within the framework of mathematics, Deleuze (along with Guattari) follows Lautman’s suggestion and explicates the operation of this logic within the framework of a multiplicity of domains, including for example philosophy, science and art in What is Philosophy?, and the variety of domains which characterise the plateaus in A Thousand Plateaus. While for Lautman, a mathematical problem is resolved by the development of a new mathematical theory, for Deleuze, it is the construction of a concept that offers a solution to a philosophical problem; even if this newly constructed concept is characteristic of, or modelled on the new mathematical theory. (shrink)

Philosophy remains ensnared between reifying the isolated individual subject and reducing it to the structuring forces of nature and society. Neither strategy appears suitable to the first-person participant perspective of the lived-experience of being a finite, conditional self within the world. This self is experienced as embodied, social, and other-dependent, and as environmentally and perspectivally “my own” such that it potentially resists, rather than reproducing, structural forces. In this chapter, I reconsider Dilthey’s alternative to the reductive poles of this (...) dialectic. Dilthey is primarily known in subsequent hermeneutics and social theory as a philosopher of reflexive self-awareness (Innewerden), self-reflection (Selbstbesinnung), and structural methodological individualism. In his works, his critics perceive the specter of the idealist subject and romantic individualism. They thereby miss his radicality. The self is an intersection of natural and social processes, as explained in the third-person, impersonal perspective of the natural and structural social sciences. Yet, the self is formed in historically situated autobiographical and reflective individuation, self-awareness, and self-formation. This hermeneutical situation calls for interpretive understanding in everyday self-other relations, life contexts, and forms of inquiry that interpret historical life from out of itself in relation to its own lived participant perspective. Dilthey’s felt and reflexive self-awareness entails, through resistance and relationality, the differentiation of self and others, things, environment, and world; the formation of an autobiographical sense of self through life’s continuities and discontinuities; and the interpretive, social, and material activities through which individuals not only manifest social systems but immanently appropriate, resist, and transform them. (shrink)

Résumé — Le rapport de Simone Weil au marxisme est paradoxal en ce qu’il dévoile une fidélité à Marx par-delà les ruptures et les renonciations à la théorie marxiste. Au premier abord, les liens de Weil au marxisme semblent discontinus puisque après avoir adopté certaines idées révolutionnaires pendant ses premières années d’activité politique, elle critique Marx dès les années 1930, et finit apparemment par l’abandonner dans la dernière partie de sa vie. Ce constat, cependant, loin de révéler la disparition lente (...) et inexorable des concepts de Marx, semble plutôt démontrer leur persistance et leurs métamorphoses dans la philosophie de Weil. Nous voudrions ainsi suggérer que sa critique de la révolution elle-même, ainsi que le " christianisme " développé à la suite de son année d’usine, constituent sa façon singulière d’être marxiste dans le moment même où celui-ci semble avoir perdu pour elle toute utilité.— S. Weil’s relationship to Marxism is paradoxical because it unveils a loyalty to Marx in spite of ruptures and renunciations of the Marxist theory. S. Weil’s ties to Marxism seem discontinuous because after having adopted certain revolutionary ideas during her first years of political activism she criticizes Marx in the 30’s and ends up seemingly abandoning him in the last part of her life. This path, however, far from revealing the slow and inexorable disappearance of Marx’s concepts, rather, demonstrates the persistence and metamorphoses in S. Weil’s philosophy. We suggest then, that a criticism of the revolution followed by a kind of Christianity developed in the wake of a year spent in the factory, constitute S. Weil’s own special manner of being Marxist, even though Marxism seems to have become useless to her. (shrink)

Spinoza distinguishes two orders of knowledge: an order conceived by the intellect, that is, the necessary order of nature and another order conceived by the imagination, that is, the common order of nature in which the contingent and the possible dwell. However, the common order is not only a deprivation of knowledge, but also a reality for the finite mode. Because we cannot exclude the existence of the common order, this paper attempts to understand how we can reconcile it (...) with the necessary order. Spinoza’s critique of imagination is that it introduces fragmentation and discontinuity into nature, since it is a kind of knowledge that does not realize the basis of this continuity: the necessary order. Through an analysis of time as a product of imagination, we will try to show how it introduces the idea of contingency and consequently that of possible. Just as time, the common order can be grasped by reason as an outer frame, it will no longer be an obstacle to the necessary chain of things in the necessary order. It is through the necessary order that we can realize the necessity that underlies the production of time, possible, and contingency as the corruptibility of all particular things. (shrink)

A rDzogs-chen interpretation of the sense of self is presented that is grounded in the disciplined method of shamatha-vipashyana meditation. This model of self/non-self agrees with Strawson's analysis as far as the discontinuity of self, but elaborates the momentary self not as any kind of ‘thing', but as an energy process having both particle-like and field-like aspects. The moment-by-moment appearance of a sense of self is described as arising in stages over a finite duration from a background of (...) non-dual intelligence and energy. There are implications for further scientific research into the structure of self-consciousness as well as for the cultivation of individual wisdom and compassion. (shrink)

Abstract:This paper presents a model of the continuous structure of Cognition based on several theses proposed by Charles S. Peirce in his youth and in his mature period. In this model, cognitions are discontinuous parts on a continuum and a cognitive process becomes “individually-synthetic,” as a hypostatic abstraction from discontinuous transformations of informational fluxes in the continuous course of experience. That is, they are salient regions or neighborhoods on a continuum rather than points, and the relations of succession and precession (...) among them are inferential, fluid, time sensitive, and goal-directed. First, this paper will outline the theses found in the young Peirce’s work, which inspire a conception of continuous Cognition. Two questions will be raised regarding such a conception: (1) at what point does a particular act of cognition conclude? and (2) how should we characterize individual cognitions? To address these questions, the paper will later introduce the concept of continuity that Peirce developed in his mature years. The synthetic character of the continuum leads to the formulation of the concepts of neighborhood and synthetic individuality. These notions support the conception of a continuous model of Cognition in which the relations of succession and precession between individual finite cognitions are explained. The paper ends with a brief reflection regarding the possibility of developing this model of continuous Cognition as a theory of extended cognition. (shrink)

Thermodynamic macro variables, such as the temperature or volume macro variable, can take on a continuum of allowable values, called thermodynamic macro values. Although referring to the same macro phenomena, the macro variables of Boltzmannian Statistical Mechanics (BSM) differ from thermodynamic macro variables in an important respect: within the framework of BSM the evolution of macro values of systems with finite available phase space is invariably modelled as discontinuous, due to the method of partitioning phase space into macro regions (...) with sharp, fixed boundaries. Conceptually, this is at odds with the continuous evolution of macro values as described by thermodynamics, as well as with the continuous evolution of the micro state assumed in BSM. This discrepancy I call the discontinuity problem. I show how it arises from BSM’s framework and demonstrate its consequences, in particular for the foundational project of reducing thermodynamics to BSM: thermodynamic macro values are shown to not supervene on the corresponding BSM macro values. With supervenience being a conditio sine qua non for the kind of reduction envisaged by the foundational project, the latter is in jeopardy. (shrink)

In this article, the idea that the accidents (a’rād) do not have continuity for two consecutive periods of time and that they are recreated in the second moment of their existence is analyzed within the framework of al-Juwayn¬ī's approaches. For this purpose, first of all, the place of the theory of continuous creation in the thought system of mutakallimūn, and then the consequences of al-Juwaynī's acceptance of continuous creation in terms of the effectiveness of divine intervention and the understanding of (...) the universe (al-‘ālam) are touched upon. Continuous re-creation (Tajaddud al-amthal) is an understanding that is considered as the disappearance of accidents moment by moment and their re-creation with similar ones. In this way, mutakallimūn tried to explain the continuity and change in the universe, as well as the constants that do not change in things. This understanding, which is associated with the renewal of accidents at every moment, is handled by al-Juwayn¬ī within the framework of the principles that the accidents are finite and that they cannot subsist by themselves (qāim bi-nafs). Accordingly, the accident cannot carry another accident because it does not occupy space (mātahayyaza); nor does it have continuity because it cannot be self-subsisting (al-qāim) with invariability (baqā) which is an accident like itself. As a matter of fact, if the existence of the accident had continued for the second time, it would not have been possible for its like to exist in the same place. Likewise, if the accident had the characteristic of continuity, its nonexistence would be impossible. Because what is continuous at two consecutive moments would not be able to perish in the third and subsequent moments. However, senses reveal that the accidents have disappeared. Thus, the accident, which has a discontinuous and impermanent structure, becomes the subject of creation at the same time whenever it loses its existence. The understanding of continuous creation, accepted by most of the mutakallimūn, is systematized over accidents and does not include the renewal of substances. It is striking that al-Juwayn¬ī also made a distinction with regard to the renewal of the accidents and the continuity of the substances. However, by associating the persistence of the substance with the continuous creation of accidents, al-Juwayn¬ī explains this issue in a way that does not contradict the absolute power of Allah. According to him, the condition of continuity of the substance depends on the existence of the accident which subsists with it. As long as Allah wishes for the continuity of the existence of the substance, he creates the accident of baqā in it one after the other. If the accident disappears, the substance also loses its existence. Thus, God's intervention by continuous re-creation has a direct effect on the accident; the substance, depending on the accident, which is the condition of its continuity, is in need of God's intervention. Therefore, the universe as a whole is not independent of the Creator even for a moment. This understanding of creation of al-Juwayn¬ī, on the one hand, hinders the idea of necessary causality, on the other hand, it draws a model of God-world (al-‘ālam) relationship that foresees the constant intervention of the Creator, who is the Free Agent (Fā‘il Mukhtār). The idea that everything that exists before and after is dependent on Allah's creation also invalidates the idea that the objects in the world stem from the qualities they have in their own nature. (shrink)

In this article I analyze the themes of temporality and alterity as they were developed over the first three of Descartes’s Meditations. I discuss the temporality of the evil deceiver, as well as the implicit theory of time and time-consciousness in the Second Meditation. I show that this theory of time is purely subjective, continuous, pre-numerical, and independent of local motion and the body, thus making it independent of Aristotle’s theory of time. I then explain God’s continuous creation of time (...) and the discontinuous theory of time Descartes develops in the Third Meditation. Moreover, I show that there is an ontological and temporal priority of the Other over the self, and that temporal self-consciousness is necessarily also consciousness of God, his continuous creation of time, time itself, and other finite substances. (shrink)

The exceptional nature of poetic language – testified by its patent untranslatability – represents a problem for philosophy, and particularly for analytic philosophy, which aims to provide an overarching explanation of ordinary language. It is difficult to explain the peculiar creativity of poetic language, starting from a finite basic vocabulary and a finite set of rules to be used by an interpreter who has finite powers. Poetry seems indeed not to respect the semantic innocence, spreading new meanings (...) not only according to the context, but even to the graphic and phonological form of the words. Mechanisms, such as quotation and conversational implicatures, have been used to respect semantic innocence and to show at the same time that poetic language is not deviant from the ordinary use of language. This article suggests an alternative route, “imagistic”, to account for the specific creativity of poetic language, without violating the principle of semantic innocence and without hypothesizing a discontinuity with ordinary language. Particularly referring to metaphor, it aims to show that poetic language does not convey as much further meanings, implicit in the use of literal terms, but rather images, evoked by and dependent on these terms, which could explain both poetry emergent properties and untranslatability. “Imageability” is not a feature of the “poetic word”, but rather of any word and, particularly, of concrete words. The creativity of poetic language lies in deepening the multimodal nature of concrete language, which is different from the abstract and conceptual nature of philosophical language. (shrink)

It is not unusual in real-life that one has to choose among finitely many alternatives when the merit of each alternative is not perfectly known. Instead of observing the actual utilities of the alternatives at hand, one typically observes more or less precise signals that are positively correlated with these utilities. In addition, the decision-maker may, at some cost or disutility of effort, choose to increase the precision of these signals, for example by way of a careful study or the (...) hiring of expertise. We here develop a model of such decision problems. We begin by showing that a version of the monotone likelihood-ratio property is sufficient, and also essentially necessary, for the optimality of the heuristic decision rule to always choose the alternative with the highest signal. Second, we show that it is not always advantageous to face alternatives with higher utilities, a non-monotonicity result that holds even if the decision-maker optimally chooses the signal precision. We finally establish an operational first-order condition for the optimal precision level in a canonical class of decision-problems, and we show that the optimal precision level may be discontinuous in the precision cost. (shrink)

Fifteen years after the first edition of this comprehensive work, German historicism remains largely and conspicuously in the shadows. The great historico-philological and historico-sociological work produced by, and on the fringes of, this school has given way to specialization. Great polygraphs of the caliber of a Meinecke, a Vossler, a Curtius, a Cassirer, a Croce, or an Auerbach seem to have completely disappeared from the scene. But is the necessity for cultural synthesis that these men stressed any less urgent today (...) than before? One of the major tasks envisioned by thinkers of the early twentieth century was the development of a philosophical anthropology which would consolidate all the preceding work of cultural synthesis. Today this task remains in the far horizon after a series of distinguished but discontinuous achievements. Thus the reappearance of the ambitious work of Professor Rossi is a welcome event. There are now indications that sociology is moving away from the neutral functionalism of the consensus years and is beginning to show some interest in the subconscious and in concrete constellations of values representative of communal forms of consciousness in a situation. Post-industrial society can no longer be examined with the rough tools of micro- or even macro-analysis. Because of their refinements in quantification and their awareness of the most specific features of every particular form of group consciousness, the new middle range methods are akin to the synthetic spirit of the historical school. As middle range research progresses, its results can provide the ontic ground for an ontology of the human condition in its adjustment to the world, and in its communicative-transcendent goals. For this reason, Rossi’s work, though excellent in its expository aspects should be read with care to avoid overemphasizing, as he does, the distance between rationalism and irrationalism that was apparent in the main tendencies of the historical school. Surely it is a worthwhile thing to show the tensions between the relativists such as Simmel and Spengler, the intuitionists such as Windelband and Rickert, and finally the neo-idealists such as Troeltsch and Meinecke. The tension that pulled apart these intellectual currents shows by contrast the great synthetic force operative in the systems of Dilthey and Weber, which provided the dominant inspiration for the whole school. But the fundamental perspective in today’s reconsideration of historicism need not be the tempering of any search for ideals through the insistence on sober methodological limits. A more fruitful emphasis can be placed today on action as a mediator between thought and adjustment, on community as a mediator between society and individuals, and most of all on expression and adventure as mediators between survival and moral absolutes. With his neo-Kantian insistence on the historical method, on the discovery of the historical horizon of the human condition, and on the establishment of the institutional connections within historical periods, Professor Rossi: a) misses the point of the anti-mechanistic efforts of modern philosophy, i.e., the constitution of synthetic a priori judgments in their true home grounds, the field of spontaneous human communication and action; b) restricts the ability on the part of the historical method to penetrate and elucidate the areas of the unconscious and of passive syntheses ; c) both presupposes and trivializes the definition of freedom. Freedom in Rossi’s interpretation of historicism, seems to result from the objective displacement of values, rather than from the dramatic insufficiency of every finite person as it transpires in the highest forms of symbolic communication and interaction.—A. M. (shrink)

Written by a highly respected scholar of Thomas Aquinas's writings, this volume offers a comprehensive presentation of Aquinas's metaphysical thought. It is based on a thorough examination of his texts organized according to the philosophical order as he himself describes it rather than according to the theological order. -/- In the introduction and opening chapter, John F. Wippel examines Aquinas's view on the nature of metaphysics as a philosophical science and the relationship of its subject to divine being. Part One (...) is devoted to his metaphysical analysis of finite being. It considers his views on the problem of the One and the Many in the order of being, and includes his debt to Parmenides in formulating this problem and his application of analogy to finite being. Subsequent chapters are devoted to participation in being, the composition of essence and esse in finite beings, and his appeal to a kind of relative nonbeing in resolving the problem of the One and the Many. Part Two concentrates on Aquinas's views on the essential structure of finite being, and treats substance-accident composition and related issues, including, among others, the relationship between the soul and its powers and unicity of substantial form. It then considers his understanding of matter-form composition of corporeal beings and their individuation. Part Three explores Aquinas's philosophical discussion of divine being, his denial that God's existence is self-evident, and his presentation of arguments for the existence of God, first in earlier writings and then in the "Five Ways" of his Summa theologiae. A separate chapter is devoted to his views on quidditative and analogical knowledge of God. The concluding chapter revisits certain issues concerning finite being under the assumption that God's existence has now been established. -/- John F. Wippel, professor of philosophy at The Catholic University of America, was recently awarded the prestigious Aquinas Medal by the American Catholic Philosophical Association. In addition to numerous articles and papers, Wippel has coauthored or edited several other works, including Metaphysical Themes in Thomas Aquinas and The Metaphysical Thought of Godfrey of Fontaines, both published by CUA Press. (shrink)

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC (...) property is first order. (shrink)

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures (...) with μ (C)=1 and such that L and L ′ define the same queries on C. We carry out a systematic investigation of $\equiv _{\text{a.e.}}$ with respect to the uniform measure and analyze the $\equiv _{\text{a.e.}}$ -equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework. (shrink)

This essay describes some of the basic pragmatic tendencies at work in the world of working and then shows how the finite provinces of meaning of theoretical contemplation and literature act against those pragmatic tendencies. This analysis prepares the way to see how the religious province of meaning in a similar but also distinctive way acts back against these pragmatic tendencies. These three finite provinces of meaning make it possible to see the world from another center of orientation (...) than that of the ego agens and its proclivity to bring transcendences within reach, thereby freeing oneself to a degree from the fundamental anxiety and revealing the uniqueness of oneself and others, who are no longer subjected to working’s pragmatic imperatives. (shrink)

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

his paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way. We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Lowenheim-Skolem theorem. The paper is completely self-contained and includes examples of application to (...) particular many-valued formal systems. (shrink)

There is a long tradition of reading Spinoza as committed, perhaps unwillingly, to the non-reality of finite modes. While acknowledging that Spinoza does seem to rely on the reality of modes in certain places, Michael Della Rocca has called attention to what he labels an “idealist strand.” As a concluding remark in “Steps Toward Eleaticism in Spinoza's Philosophy of Action,” he claims that faced with these two conflicting strands, which are genuinely to be found in the text, it is (...) better to note both rather than artificially imposing a unity on the text. In this paper, I suggest that one need not admit two conflicting strands in Spinoza, on the one hand, nor arbitrarily disregard one strand, on the other. Rather, I argue for a unified view that diagnoses what gives rise to both seemingly conflicting strands. The key to this enterprise is recognizing that finite modes, paradoxically seeming perhaps at first, are not wholly finite. That is, a finite mode is determined by the totality of finite modes—wherein lies its finite aspect, but at the same time, and equally, partially actively determines this totality—wherein lies its infinite aspect. (shrink)

We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized.

A structure has a (finite-string) automatic presentation if the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

Alexander Pruss and I have proposed an analysis of omnipotence which makes no use of the problematic terms 'power' and 'ability'. However, this raises an obvious worry: if our analysis is not related to the notion of power, then how can it count as an analysis of omnipotence, the property of being all-powerful, at all? In this paper, I show how omnipotence can be understood as the possession of infinite power (general, universal, or unlimited power) rather than the possession of (...) particular powers. I then argue that the ordinary notions of particular powers or abilities can be understood as applying vague limitations to the notion of infinite power. The vagueness of these limitations is the source of the well-known difficulties in the analysis of ability. (shrink)

Metaphysical schemata and intellectual traditions -- Laws of nature : the scientific concept -- Natural law : disputed moments of transition -- Natural rights : origins and grounding.

We explore the finite model theory of the characterisation theorems for modal and guarded fragments of first-order logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known tree-like unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraïssé game arguments, these covers allow us to upgrade finite approximations for (...) full bisimulation equivalence towards approximations for elementary equivalence. These techniques are used to prove several ramifications of the van Benthem–Rosen characterisation theorem of basic modal logic for refinements of ordinary bisimulation equivalence, both in the sense of classical and of finite model theory. (shrink)