Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only (...) a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

Having been raised as a Catholic and educated in the West, then trained as a monk in India since the 1980s, Canadian author Swami Muktananda of Rishikesh is uniquely positioned to bring the Eastern tradition of Vedanta to Western spiritual seekers. In Awakening to the Infinite, he answers the eternal, fundamental question posed by philosophical seekers, "Who am I?" with straightforward simplicity. Knowing who you are and adopting a spiritual outlook, he counsels, can help solve problems in daily life (...) to do with relationships, work, children, sexuality, illness, and social injustice. In answering the question "Who am I?," Swami Muktananda of Rishikesh draws on the ancient teachings of advaita or nonduality, the truth that there is no division between ourselves and others at a fundamental level. The book is drawn from the many conversations he has throughout the year meeting with individuals from different backgrounds, cultures, and walks of life, as well as public talks and question and answer sessions. Covering subjects ranging from the spiritual path to everyday problems, Muktananda addresses questions with simplicity and compassion, from the standpoint of someone who is anchored in the infinite but fully in touch with the world. His teaching follows the words of Christ and the Old Testament as well as those of the sages of India, who all proclaim a universal truth that transcends all religions: 'You are not the body; you are not the mind; you are the immortal, divine Self. Realize this and be free.'. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less (...) than the ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

Introduction. This paper is a sequel to our paper [3]. In that paper we introduced the notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2]. The results of the paper [3] show relationships between the models of an infinitely long sentence and the models of its finite approximations. In the present paper we shall apply the main result of [3] to prove a number (...) of theorems about ordinary finitely long sentences; these theorems are of a type which one might call “preservation theorems”. The following two known results are typical preservation theorems: A sentence φ is preserved under substructures if and only if φ is equivalent to some universal sentence ; φ is preserved under homomorphic images if and only if φ is equivalent to some positive sentence. An expository account of preservation theorems may be found in Lyndon [9].We shall use freely all of the notation introduced in [3], and shall not attempt to make this paper selfcontained. However, the reader should have no difficulty following this paper after he has read [3]. In § 1 we cover some preliminary topics. In § 2 and § 3 we shall illustrate our method in familiar situations by proving anew the two preservation theorems mentioned above. We shall then obtain four new preservation theorems in §§ 4, 5, and 6, involving respectively direct powers and roots, strong homomorphisms and retracts, and direct factors. The result in § 6 was stated in the abstract [4], while the results in § 5 were stated in the abstract [5]. (shrink)

Le présent article vise à éclairer la nature des activités perceptives et cognitives attribuées au νοῦς d’Anaxagore, en particulier à lever les difficultés liées à l'évaluation de la part des dimensions mécaniques, cognitives et téléologiques dans l'activité du νοῦς cosmique, par une comparaison avec l'usage des verbes γιγνώσκω, νοέω et du substantif νοῦς dans le contexte du champ de bataille homérique. Les rangeurs d'hommes homériques partagent avec le νοῦς d'Anaxagore une description de leurs activités en termes de tri, d'extraction et (...) de mise en ordre, par l'usage commun des verbes διακρίνω, ἀποκρίνω, διακοσμέω. Or l'analyse des opérations perceptives et cognitives de ceux-là révèle un modèle d'action immanente, liant la perception des singularités au sein de touts complexes et mouvants à l'appréhension globale des positions passées, présentes et possibles des troupes en action. Pensé sur ce modèle, le νοῦς d'Anaxagore sera conçu selon un modèle de perception par immersion dont sa nature physique très particulière le rend capable : c'est en faisant intégralement corps avec le mélange matériel que le νοῦς se rend capable d'en faire le tri et d'en anticiper les états possibles. L'illimitation de sa connaissance et de son pouvoir a néanmoins une conséquence qui le différencie du chef homérique : n'avoir aucun choix du fait de la puissance de réaliser toutes les différenciations possibles. (shrink)

Let A and B be subsets of the space 2 N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2 N into 2 N such that A = Φ -1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, $a \subset b$ implies $\Phi (a) \subset \Phi(b)$ . The set A is said to be monotone (...) if a ∈ A and $a \subset b$ imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The ▵ 0 1 sets are all reducible to the (Σ 0 1 but not ▵ 0 1 ) sets, which are in turn all reducible to the strictly ▵ 0 2 sets, which are all in turn reducible to the strictly Σ 0 2 sets. In addition, the nontrivial Σ 0 n sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly Π 0 2 monotone sets which have different monotone degrees. We show that every Σ 0 2 monotone set is actually Σ 0 2 positive. We also consider reducibility for subsets of the space of compact subsets of 2 N . This leads to the result that the finitely iterated Cantor-Bendixson derivative D n is a Borel map of class exactly 2n, which answers a question of Kuratowski. (shrink)

Some environmental ethicists and economists argue that attributing infinite value to the environment is a good way to represent an absolute obligation to protect it. Others argue against modelling the value of the environment in this way: the assignment of infinite value leads to immense technical and philosophical difficulties that undermine the environmentalist project. First, there is a problem of discrimination: saving a large region of habitat is better than saving a small region; yet if both outcomes have (...) infinite value, then decision theory prescribes indifference. Second, there is a problem of swamping probabilities: an act with a small but positive probability of saving an endangered species appears to be on par with an act that has a high probability of achieving this outcome, since both have infinite expected value. Our paper shows that a relative concept of infinite value can be meaningfully defined, and provides a good model for securing the priority of the natural environment while avoiding the failures noted by sceptics about infinite value. Our claim is not that the relative infinity utility model gets every detail correct, but rather that it provides a rigorous philosophical framework for thinking about decisions affecting the environment. (shrink)

In last year’s Review Gregory Brown took issue with Laurence Carlin’s interpretation of Leibniz’s argument as to why there could be no world soul. Carlin’s contention, in Brown’s words, is that Leibniz denies a soul to the world but not to bodies on the grounds that “while both the world and [an] aggregate of limited spatial extent are infinite in multitude, the former, but not the latter, is infinite in respect of magnitude and hence cannot be considered a (...) whole”. Brown casts doubt on this interpretation—or rather, he begins by questioning its adequacy as an interpretation of a central passage, only to concede its essential correctness as an interpretation of Leibniz’s position, and to turn his attack on the latter itself. In this note I shall argue that Brown underestimates the subtlety of Leibniz’s views on the infinite, and that Carlin is basically correct in his suggestion that the infinite magnitude of the world is what precludes it from having even the phenomenal unity that a body does. (shrink)

It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

Positive thinking is one of the most valuable tools that postmodern man possesses for personal development and transforming community life. Positive thinking is based on the harmonious relation of the individual with himself, with the others, and with the surrounding reality. It comes at the end of a process of conscious change which the individual goes through in gradual steps, and becomes nuanced as thinking and personal practices develop. Our starting point is the role of thinking as laid out by (...) Descartes. Descartes’ conception of philosophy as a thought system points to the acquisition of virtue through knowledge of truth, by cultivating measure, and thus highlighting man as a thinking being. The entire effort of achieving knowledge, of acquiring wisdom, has a moral finality. Its fundament is the presence of God as the infinite instance that deems possible the free manifestation of the human person. As a thinking being, the human subject brings himself into existence. He certifies the reality of the world. Everything takes place on the horizon of the infinite self-existence represented by the philosophical idea of God. As a spiritual being, man can use thinking to shape his desires, to master his willpower, but most of all, to integrate himself in the process of conscious personal development. (shrink)

A comment on Paul Schoemaker's target article in Behavioral and Brain Sciences, 14 (1991), p. 205-215, "The Quest for Optimality: A Positive Heuristic of Science?" (https://doi.org/10.1017/S0140525X00066140). This comment argues that the optimizing model of decision leads to an infinite regress, once internal costs of decision (i.e., information and computation costs) are duly taken into account.

In lieu of an abstract, here is a brief excerpt of the content:Bishop Berkeley Exorcises the Infinite: Fuzzy Consequences of Strict Finitism1 David M. Levy Introduction It all began simply enough when Molyneux asked the wonderful question whether a person born blind, now able to see, would recognize by sight what he knew by touch (Davis 1960). After George Berkeley elaborated an answer, that we learn to perceive by heuristics, the foundations ofcontemporarymathematics wereinruin. Contemporary mathematicians waved their hands and (...) changed the subject.2 Berkeley's answer received a much more positive response from economists. Adam Smith,in particular, seizedupon Berkeley's doctrine that welearn to perceive distance tobuild anelaborate system in which one learns to perceive one's self-interest.3 Perhaps because older histories of mathematics are a positive hindrance in helping us understand the importance of Berkeley's argument against infinitesimals,4 its consequences for economics have passed unnoticed. If infinitesimal numbers are ruled out, and we wish to maintain an algebra, then we rule out infinite numbers. This implication of Berkeley's argumenthas a dramatic implication for our understanding ofthe history ofideas about rule-directed action. Let me motivate this claim with a well-known historical problem. The social doctrine set forth by John Locke outside the Two Treatises depends critically upon individual belief about possible states of infinite pain or pleasure. The critical issue for the intolerance of atheism in Locke's system is his claim that without beUefin the infinite bliss of heaven forgone by a criminal, there is no reason to think a calculating individual will avoid crime.5 Locke's discussion is so different from the discussion of religion by David Hume and Smith. Here, questions of the substance of belief—the infinite worth of Heaven—have largely vanished.6 What happened between Locke and Hume-Smith? One answer is Pierre Bayle and Bernard Mandeville. Berkeley's doctrine that only the finite is perceived is, I think, a better answer. Out of sight, out ofmind. In this paper I propose to give a close look at the content and consequences ofBishop Berkeley's once famous Towards a New Theory ofVision. While there are other aspects ofBerkeley's work that attract Volume XVIII Number 2 511 DAVID M. LEVY attention from historians ofeconomics,7 1 shall claim that Berkeley's insight into human perception in the Theory of Vision is central to Adam Smith's attempt to found economic behaviour on the self-awareness of systematic illusion. There is no doubt that Smith finds that much ofhuman behaviour is characterized by illusion.8 The difficulty arises, or so it seems to me, from modern unwilUngness to beheve that illusion can be systematic. There are four pieces to my argument. First, we consider Berkeley's statementofwhat we shall call his doctrine ofstrict finitism in perception.9 Berkeley puts forward two postulates: 1) there exists some minimum perceptible quantity, and 2) distance is not perceived directly but by experience. Second, we define strict finitism and demonstrate that it translates into the claim that perception is fuzzy in a technical sense. If Berkeleian perception is fuzzy, then some modern criticism of Berkeley's strict finitism is based upon a simple misunderstandingofthe propertiesoffuzzyrelations. Thisis quite easy to set right. (Getting right the doctrine of those who implicitly take perception to be fuzzy—Adam Smith is my prime candidate—will be muchharder.)Third, wereconsiderBerkeley's dispute withMandeville on the role ofinfinite gain in choice. Here, we find Mandeville putting forward the Berkeleian position that if infinite amounts mattered, people would not behave as they do. Berkeley's position against Mandeville, however, seems to depend upon the supposition that infinite distance is sensible.10 Earlier, however, the position that Mandeville argued for was defended by Berkeley. Fourth, we ask why all this isn't well known. The debate between Berkeley and Mandeville has been well studied, as has been the link between Berkeley and the Scots. In fact, our work focuses on one aspect of Berkeley's work that led to his challenge of the contemporary foundations of the calculus. If sense is a matter of perception, and infinitesimals cannot be perceived, then they are quite literally nonsense. Armed with this insight, Berkeley refuted contemporary mathematics. Nevertheless, the calculus was too... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Bishop Berkeley Exorcises the Infinite: Fuzzy Consequences of Strict Finitism1 David M. Levy Introduction It all began simply enough when Molyneux asked the wonderful question whether a person born blind, now able to see, would recognize by sight what he knew by touch (Davis 1960). After George Berkeley elaborated an answer, that we learn to perceive by heuristics, the foundations ofcontemporarymathematics wereinruin. Contemporary mathematicians waved their hands and (...) changed the subject.2 Berkeley's answer received a much more positive response from economists. Adam Smith,in particular, seizedupon Berkeley's doctrine that welearn to perceive distance tobuild anelaborate system in which one learns to perceive one's self-interest.3 Perhaps because older histories of mathematics are a positive hindrance in helping us understand the importance of Berkeley's argument against infinitesimals,4 its consequences for economics have passed unnoticed. If infinitesimal numbers are ruled out, and we wish to maintain an algebra, then we rule out infinite numbers. This implication of Berkeley's argumenthas a dramatic implication for our understanding ofthe history ofideas about rule-directed action. Let me motivate this claim with a well-known historical problem. The social doctrine set forth by John Locke outside the Two Treatises depends critically upon individual belief about possible states of infinite pain or pleasure. The critical issue for the intolerance of atheism in Locke's system is his claim that without beUefin the infinite bliss of heaven forgone by a criminal, there is no reason to think a calculating individual will avoid crime.5 Locke's discussion is so different from the discussion of religion by David Hume and Smith. Here, questions of the substance of belief—the infinite worth of Heaven—have largely vanished.6 What happened between Locke and Hume-Smith? One answer is Pierre Bayle and Bernard Mandeville. Berkeley's doctrine that only the finite is perceived is, I think, a better answer. Out of sight, out ofmind. In this paper I propose to give a close look at the content and consequences ofBishop Berkeley's once famous Towards a New Theory ofVision. While there are other aspects ofBerkeley's work that attract Volume XVIII Number 2 511 DAVID M. LEVY attention from historians ofeconomics,7 1 shall claim that Berkeley's insight into human perception in the Theory of Vision is central to Adam Smith's attempt to found economic behaviour on the self-awareness of systematic illusion. There is no doubt that Smith finds that much ofhuman behaviour is characterized by illusion.8 The difficulty arises, or so it seems to me, from modern unwilUngness to beheve that illusion can be systematic. There are four pieces to my argument. First, we consider Berkeley's statementofwhat we shall call his doctrine ofstrict finitism in perception.9 Berkeley puts forward two postulates: 1) there exists some minimum perceptible quantity, and 2) distance is not perceived directly but by experience. Second, we define strict finitism and demonstrate that it translates into the claim that perception is fuzzy in a technical sense. If Berkeleian perception is fuzzy, then some modern criticism of Berkeley's strict finitism is based upon a simple misunderstandingofthe propertiesoffuzzyrelations. Thisis quite easy to set right. (Getting right the doctrine of those who implicitly take perception to be fuzzy—Adam Smith is my prime candidate—will be muchharder.)Third, wereconsiderBerkeley's dispute withMandeville on the role ofinfinite gain in choice. Here, we find Mandeville putting forward the Berkeleian position that if infinite amounts mattered, people would not behave as they do. Berkeley's position against Mandeville, however, seems to depend upon the supposition that infinite distance is sensible.10 Earlier, however, the position that Mandeville argued for was defended by Berkeley. Fourth, we ask why all this isn't well known. The debate between Berkeley and Mandeville has been well studied, as has been the link between Berkeley and the Scots. In fact, our work focuses on one aspect of Berkeley's work that led to his challenge of the contemporary foundations of the calculus. If sense is a matter of perception, and infinitesimals cannot be perceived, then they are quite literally nonsense. Armed with this insight, Berkeley refuted contemporary mathematics. Nevertheless, the calculus was too... (shrink)

In this article we consider some questions raised by F. Benoist, E. Bouscaren, and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mathbb{Z}$, then there are no infinitely $p$-divisible points of order a power of $p$.

The philosophies of Jacques Derrida and Paul Shepard, while rarely encoun- tering the other, nevertheless prove to be surprisingly complementary. Derrida acknowl- edges the impossibility and necessity of the human/animal frontier, thinking the human/ animal relation in a paradigm of seeing and being seen, conceived in particular in the context of a sphere of the intimate. Shepard's not merely biological but ontological interpretation of evolution argues that humans need animals, not only metabolically but for their mental development. From the positive (...) dependence of the human on the animal follows an infinite debt that can never be repaid; but in attempting to do so lies the responsibility and destiny of the human, that most animal animal. (shrink)

This paper explores the place of Hegel in Gadamer's hermeneutics through an analysis of the idea of "infinite dialogue." It is argued that infinite dialogue cannot be understood as a limited Hegelianism, i.e., as the life of spirit in language that does not reach its end. Rather, infinite dialogue can be understood only by taking the Heideggerian idea of radical finitude seriously. Thus, while infinite dialogue has a speculative element, it remains a dialogue conditioned by the (...) occlusion in temporal becoming. This idea is developed further by contrasting Gadamer's position with that of Blanchot, who also stands under the shadow of Hegel. (shrink)

“Wuqiong,” namely the concept of infinite regress has been identified with the hallmark of the Huayan scholasticism, which is dubbed as “chongchong wujin” (repetitive containment ad infinitum). Such an inconceivable perspective is drawn from the Huayan thinkers’ presupposition that a part contains the whole, which is again composed of such parts. But many philosophical traditions, in general, try to avoid the infinite regress as one of the logical fallacies. This paper examines the Buddhist literature that alludes to “wuqiong” (...) as infinite regress, and highlights that the Huayan thinkers also had a critical attitude toward this issue when they discussed the ontological status of the dissociated factors. But they also used the term in a positive sense of inexhaustiveness. This twofold usage of the term seems to show that the Huayan thinkers critically incorporated the elements of previous scholarship even when they created their own unique metaphysical theories. ‘무궁’, 곧 무한소급은 ‘중중무진’으로 표현되는 화엄교학과 동일시되어 왔다. 그것은 부분이 전체를 포함한다는 화엄사상가들의 이론적 전제로부터 각각의 부분들에 전체가 끊임없이 내포된다는 불가사의한 세계관이 도출되기 때문이다. 그러나 불교철학은 물론, 서양철학에서도 무한소급은 논리적 오류로서 회피해야 하는 것으로 간주되어 왔다. 이와 관련하여 본 논문에서는 화엄사상 성립 이전의 몇몇 불교 문헌들에서 특히 유위 4상(相), 득(得)과 같은 불상응행법을 설정할 경우 ‘무궁과’라는 논리적 오류가 발생함을 지적한 사례들을 살펴보았는데, 그러한 비판이 화엄의 문헌들에도 거의 그대로 수용되고 있음을 확인할 수 있다. 그러나 화엄의 문헌에는 ‘무궁’이라는 용어가 ‘무궁진’, ‘무진’, ‘무변’과 동의어로서 긍정적인 맥락에서 사용하고 있는 경우도 자주 발견된다. 이처럼 화엄의 문헌에 이중적인 의미로 사용된 ‘무궁’이라는 개념은 화엄사상가들이 자신들의 독특한 형이상학을 제시하면서도 그들 이전의 여러 교학체계들을 비판적으로 수용했음을 보여주는 증거라고 할 수 있을 것이다. (shrink)

Amid a period of isolation and profound internal conflict, both in his life and in his thought, arises that which, according to Hölderlin, is “the general conflict in the human being”, namely the conflict between the “aspiration to limitation” and “the aspiration to the absolute”. The aim of this article is to analyze and, as much as possible, follow to its fullest extent, this fundamental thought: to see how it molds Hölderlin’s positions on existence and philosophy, how it meets the (...) most poignant philosophical concerns of the time and hence directly influences Hölderlin’s reading of Fichte, and how it leads the young poet to the notion of an infinite progression of philosophy, that is, to an infinite approximation to the ideal of knowledge. (shrink)

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H of a given finite binary string s. In the standard way, H is defined as the length of the (...) shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H is defined as –log2m without reference to the concept of program-size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed.In what follows, we introduce an operator version equation image of H in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (shrink)

ABSTRACT In this paper I will provide a reading of the relation between nature and spirit starting from an analysis of the movement of positing and presupposing, discussed by Hegel in the "Science of Logic" in his discussion of the transition between Being and Essence. 1) I will offer an analysis of the logical context within which this dialectical movement emerges. 2) I will show the role played by this dialectical movement in determining the relation between the spheres of nature (...) and spirit in Hegelʼs "Encyclopedia". 3) I will then claim that a difficulty related to this movement arises in the "Logic", and that this difficulty reverberates in the relation between nature and spirit in Hegel's system. In particular, 4) an ambiguity arising at a logical level generates two different configurations of this movement, depending on whether the latter is meant to describe the relation of 4.1) the finite or of 4.2) the infinite spirit to nature. 5) I will finally focus on some problems deriving from my reconstruction and I will try to suggest - in very general and programmatic terms - in what sense the internal analysis of the relation between nature and spirit in Hegelʼs system can be fruitfully discussed against the dominant framework of the contemporary debate, i.e. naturalism. However, this suggestion does not necessarily imply an endorsement of Hegel's position. RESUMO Neste artigo, proponho-me a fornecer uma leitura da relação entre natureza e espírito a partir de uma análise do movimento de pôr e pressupor descrito por Hegel na "Ciência da Lógica", na passagem do Ser à Essência. 1) Analisarei o contexto lógico dentro do qual surge o movimento de pôr e pressupor. 2) Mostrarei o papel deste movimento dialético na determinação da relação entre as esferas da natureza e do espírito na "Enciclopédia" de Hegel. 3) Destacarei uma dificuldade que emerge na "Lógica" e mostrarei como esta dificuldade tem repercussões sobre a relação entre natureza e espírito no sistema de Hegel. Em particular, 4) uma ambiguidade que emerge no nível lógico gera duas diferentes configurações deste movimento, dependendo de se ele descreve 4.1) a relação do espírito finito ou 4.2) a relação do espírito infinito com a natureza. 5) Na última parte, mostrarei alguns problemas que derivam de minha reconstrução e tentarei sugerir - em termos gerais e programáticos - em que sentido uma análise interna da relação entre natureza e espírito em Hegel pode ser proficuamente discutida com respeito ao paradigma dominante do debate contemporâneo, isto é, o naturalismo. (shrink)

It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain the same (...) agents and some agents are better off in the first world than they are in the second and no agents are worse off than they are in the second, then the first world is better than the second. I show that if we accept four axioms – the Pareto principle, transitivity, an axiom stating that populations of worlds can be permuted, and the claim that if the ‘at least as good as’ relation holds between two worlds then it holds between qualitative duplicates of this world pair – then we must conclude that there is ubiquitous incomparability between infinite worlds. I show that this is true even if the populations of infinite worlds are disjoint or overlapping, and that we cannot use any qualitative properties of world pairs to rank these worlds. Finally, I argue that this incomparability result generates puzzles for both consequentialist and non-consequentialist theories of objective and subjective permissibility. (shrink)

On the rise over the past 20 years has been ‘moderate supernaturalism’, the view that while a meaningful life is possible in a world without God or a soul, a much greater meaning would be possible only in a world with them. William Lane Craig can be read as providing an important argument for a version of this view, according to which only with God and a soul could our lives have an eternal, as opposed to temporally limited, significance, by (...) virtue of our moral choices then making an ultimate difference. I present two major objections to this position. On the one hand, I contend that if God existed and we had souls that lived forever, then, in fact, all our lives would turn out the same. On the other hand, I maintain that, if this objection is wrong, so that our moral choices would indeed make an ultimate difference and thereby confer an eternal significance on our lives (only) in a supernatural realm, then Craig could not capture the view, aptly held by moderate supernaturalists, that a meaningful life is possible in a purely natural world. An eternal significance would be ‘too big’, reducing any meaning possible during an earthly life to nothing by comparison. I conclude that moderate supernaturalists would be wise to avoid appealing to eternal (or infinite) meaning when spelling out the way God and a soul could alone impart a great meaning to our lives; some other notion of greatness should be considered, perhaps one involving a temporary afterlife. (shrink)

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform (...) infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. -/- (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, is unjustified, in general. This means my position is stronger against this objection than the paper suggests, since omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences.). (shrink)

Raphael Lataster and Herman Philipse present an argument which they think decisively demonstrates polytheism over monotheism, if theism is assumed. Far from being decisive, the argument depends on very controversial and likely false assumptions about how to treat infinities in probability. Moreover, these problems are well known. Here, we focus on three objections. First, the authors rely on both countable additivity and the Principle of Indifference, which contradict each other. Second, the authors rely on a particular way of dividing up (...) the possibility space, when there are equally as reasonable alternative divisions which give different answers to the one the authors arrived at. Third, the authors’ argument proves too much, giving us an argument against many positions we should not be able to argue against so easily. (shrink)

En considérant les occurrences orales de différents emplois de la périphrase en aller + infinitif décrits par Bres et Labeau (2012a), cet article illustre la contribution positive des corpus oraux, et particulièrement du nouveau Corpus de français parlé à Bruxelles (CFPB) – au rassemblement d’occurrences authentiques de phénomènes linguistiques peu étudiés, au test de leurs descriptions théoriques, mais aussi à l’enrichissement de celles-ci.

This paper discusses the theoretical relationship between the views of Damascius and those of Pseudo-Dionysius the Areopagite. While Damascius’ De principiis is a bold treatise devoted to investigating the hypermetaphysics of apophatism, it anticipates various theoretical positions put forward by Dionysius the Areopagite. The present paper focuses on the following. First, Damascius is the only ancient philoso­pher who systematically demonstrates the first principle to be infinite. Second, Damascius modifies the concept and in several important passages shows the infinite (...) to be superior and prior to the finite. Third, Damascius’ theory of being is the strongest ancient articulation of the nature of the One which is a clear prefiguration of the negative theology developed by Dionysius the Areopagite. (shrink)

Current axiomatizations for extensive measurement postulate the existence of infinitely small objects. This assumption is neither necessary nor reasonable. This paper develops this theme and presents a more acceptable axiom system. A representation theorem is stated and proved in detail. This work improves some previous results of the author.

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XX, Number 2, November 1994, pp. 219-240 Infinite Divisibility in Hume's First Enquiry DALE JACQUETTE The Limitations of Reason The arguments against infinite divisibility in the notes to Sections 124 and 125 of David Hume's Enquiry Concerning Human Understanding are presented as "sceptical" results about the limitations of reason. The metaphysics of infinite divisibility is introduced merely as a particular, though especially representative (...) problem, among several that Hume discusses. Hume first writes: The chief objection against all abstract reasonings is derived from the ideas of space and time; ideas, which, in common life and to a careless view, are very clear and intelligible, but when they pass through the scrutiny of the profound sciences (and they are the chief object of these sciences) afford principles, which seem full of absurdity and contradiction.1 There follows an intuitive appeal to the apparent incoherence of the consequences of infinite divisibility in the geometry of space and measurement of time. Hume enlists natural belief against the infinite divisibility thesis in the mathematics of extension: But what renders the matter more extraordinary, is, that these seemingly absurd opinions are supported by a chain of reasoning, the Dale Jacquette is at the Department of Philosophy, The Pennsylvania State University, 246 Sparks Building, University Park PA 16802 USA. e-mail: [email protected] 220 Dale Jacquette clearest and most natural; nor is it possible for us to allow the premises without admitting the consequences. Nothing can be more convincing and satisfactory than all the conclusions concerning the properties of circles and triangles; and yet, when these are once received, how can we deny, that the angle of contact between a circle and its tangent is infinitely less than any rectilinear angle, that as you may increase the diameter of the circle in infinitum, this angle of contact becomes still less, even in infinitum, and that the angle of contact between other curves and their tangents may be infinitely less than those between any circle and its tangent, and so on, in infinitum? The demonstration of these principles seems as unexceptionable as that which proves the three angles of a triangle to be equal to two right ones, though the latter opinion be natural and easy, and the former big with contradiction and absurdity. (EHU 156-157) The implied sense of contradiction is then extended by Hume to the problem of infinite divisibility in time. The absurdity of these bold determinations of the abstract sciences seems to become, if possible, still more palpable with regard to time than extension. An infinite number of real parts of time, passing in succession, and exhausted one after another, appears so evident a contradiction, that no man, one should think, whose judgment is not corrupted, instead of being improved, by the sciences, would ever be able to admit of it. (EHU 157) The contradictions involved in traditional concepts of extension and time demarcate the limits of which reason is capable. But Hume's answer is more complex than merely surrendering to skepticism over the prospects of reason achieving understanding in the metaphysics of space and time. He tries to show that what he calls "Pyrrhonian" skepticism is ultimately self-defeating in its application of reason against reason. In place of a negative conclusion, he offers another solution to the paradoxes of infinite divisibility. What is remarkable about Hume's pronouncements against the infinite divisibility of extension and time, that may signal a change in his methodology if not his overall position on divisibility in the Enquiry from that of A Treatise of Human Nature, is his straightforward claim that natural belief rebels against the concept of infinite divisibility as its consequences apply to the metric of space and time.2 As in the Treatise, Hume pits natural disbelief in infinite divisibility against abstract metaphysical reason. But in the Enquiry, he does not invoke reason merely to correct the excesses of natural belief for the skeptical deflation of natural belief, say, in the necessity of causal connection, or the existence of mind-independent continuants. On the contrary, Hume Studies Infinite Divisibility in Hume's First Enquiry 221 Hume here appears instead to invoke natural disbelief in... (shrink)

In his latest book Christopher Bollas uses detailed studies of real clinical practice to illuminate a theory of psychoanalysis which privileges the human impulse to question. From earliest childhood to the end of our lives, we are driven by this impulse in its varying forms, and _The Infinite Question_ illustrates how Freud's free associative method provides both patient and analyst with answers and, in turn, with an ongoing interplay of further questions. At the book's core are transcripts of real (...) analytical sessions, accompanied by parallel commentaries which highlight key aspects of the free associative method in practice. These transcripts are contextualised by further discussion of the cases themselves, as well as a wider theoretical framework which places its emphasis on Freud's theory of the logic of sequence: by learning to listen to this free associative logic, Bollas argues, we can discover a richer and more complex unconscious voice than if we rely solely on Freud's theory of repressed ideas. Bollas demonstrates, in an eloquent and persuasive manner, how the Freudian position of evenly suspended attentiveness enables the analyst's unconscious to catch the drift of the patient's own unconscious. He also shows that to stimulate further questioning is often of more benefit to the analytical process than to jump to an interpretation. Yet whatever fascinating course a session may take, neither the patient nor the analyst can halt the progress of the self-propelling interrogative drive. _The Infinite Question_ will be invaluable to both the new student and the experienced psychoanalyst, read either on its own or as a practice-based extension of the theoretical ideas elaborated in its companion volume, _The Evocative Object World_. (shrink)

Después de haber analizado las razones que indujeron a las antiguas matemáticas griegas y de que Aristóteles sólo admitiera una débil forma de lo infinito, se explora una ampliación de este concepto más allá de sus referencias numéricas y geométricas. El infinito puede expresar la “inagotable” riqueza ontológica de los atributos de las entidades individuales o, en otro sentido, el infinito puede ser entendido como aquello “ilimitado”. En este segundo sentido la “negación” se presenta como una fuerza positiva en la (...) formación del sentido ontológico de lo infinito.After having analyzed the reasons that induced ancient Greek mathematics and Aristotle to admit only a weak form of the infinite a broadening of this notion beyond its numerical and geometrical references is explored. The infinite can express the “inexhaustible” ontological richness of the attributes of individual entities or, in another sense, the infinite can be understood as the “unlimited”. In this second sense the “negation” appears as a positive force in the shaping of the ontological meaning of the infinite. (shrink)

In Levinas’ hands, the problematic of transcendence challenges phenomenological description by positing, as primary, that which is outside intentionality. How, then, to think about this transcendence outside intentionality? This essay explores the possibilities of a description of transcendence through Levinas’ and Marion’s readings of the Cartesian idea of the Infinite. What emerges from these readings of Descartes’ idea of the Infinite is a sense of indication that is fundamentally elliptical, pointing beyond what it can render to presence, but (...) pointing nonetheless. Thinking through this problem of elliptical indication, I argue, is central to generating a phenomenological account of transcendence. (shrink)

This research partially answers the question raised by Goncharov about the size of the class of positive elements of a Roger's semilattice. We introduce a notion of effective infinity of classes of computable enumerations. Then, using finite injury priority method, we prove five theorems which give sufficient conditions to be effectively infinite for classes of all enumerations without repetitions, positive undecidable enumerations, negative undecidable enumerations and all computable enumerations of a family of r.e. sets. These theorems permit to strengthen (...) the results of Pour-El, Pour-El and Howard, Ershov and Khutoretskii about existence of enumerations without repetitions and positive undecidable enumerations. (shrink)

This paper aims to show that a proper understanding of what Leibniz meant by “hypercategorematic infinite” sheds light on some fundamental aspects of his conceptions of God and of the relationship between God and created simple substances or monads. After revisiting Leibniz’s distinction between (i) syncategorematic infinite, (ii) categorematic infinite, and (iii) actual infinite, I examine his claim that the hypercategorematic infinite is “God himself” in conjunction with other key statements about God. I then discuss (...) the issue of whether the hypercategorematic infinite is a “whole”, comparing the four kinds of infinite outlined by Leibniz in 1706 with the three degrees of infinity outlined in 1676. In the last section, I discuss the relationship between the hypercategorematic infinite and created simple substances. I conclude that, for Leibniz, only a being beyond all determinations but eminently embracing all determinations can enjoy the pure positivity of what is truly infinite while constituting the ontological grounding of all things. (shrink)

ABSTRACT This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ and (...) his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

What is wealth? -- Looking at the world from God's perspective -- Changing your attitude to bring success -- Suffering caused by desire -- Accumulating beneficial wealth -- Making progress with love and ideals -- Creating "utopian economics" -- The path to progress -- The definition of progress -- The joy of progress -- The driving force of progress -- Realizing hope -- A life filled with light -- Living with optimism -- Living lightheartedly -- Believing tomorrow will be better (...) -- Expanding the circle of happiness -- Walking with God -- A mind that invites happiness -- The laws of the mind -- A mind that brings others happiness -- In tune with God's vibration -- A humble, accepting mind -- The power of prayer -- The true nature of wealth -- Wealth as a foundation -- Wealth and competition -- Good and evil in relation to wealth -- The right way to use wealth -- Accumulating inner wealth -- The study of human nature leads to success -- Gifts and talents -- The quality and period of effort -- Conditions for receiving inspiration -- I) the basis for inspiration -- II) a loving heart -- III) a positive attitude to life -- The person who is admired -- The goal is to soar -- A steady eye -- Making progress with a loving heart -- Confidence and abilities -- The goal is to soar -- The philosophy of progress -- The pledge to God -- The truth from God's perspective -- I) male and female souls -- II) God's ideal -- III) what is death? -- The true right to know -- I) the truth God created -- II) restoring religion -- Wrongdoing without knowing -- Toward the state of Arhat -- Aim for the world of Bodhisattva -- Postscript. (shrink)

The paper discusses the recent literature on abstraction/idealization in connection with the “paradox of infinite idealization.” We use the case of taking thermodynamics limit in dealing with the phenomena of phase transition and critical phenomena to broach the subject. We then argue that the method of infinite idealization is widely used in the practice of science, and not all uses of the method are the same. We then confront the compatibility problem of infinite idealization with scientific realism. (...) We propose and defend a contextualist position for realism and argue that the cases for infinite idealization appear to be fully compatible with contextual realism. (shrink)

Galileo's Paradox Contradictory or Counter-Intuitive? Imagination v. Conception Infinity as a Positive Idea Do We Experience Anything Infinite? The Sublimity of the Infinite Conclusion Bibliography Notes Appendix: A Crash Course in the Mathematics of Infinite Sets..

In this essay, I provide an overview of the debate on infinite and essential idealizations in physics. I will first present two ostensible examples: phase transitions and the Aharonov– Bohm effect. Then, I will describe the literature on the topic as a debate between two positions: Essentialists claim that idealizations are essential or indispensable for scientific accounts of certain physical phenomena, while dispensabilists maintain that idealizations are dispensable from mature scientific theory. I will also identify some attempts at finding (...) a middle ground between the essentialists and dispensabilists camps. Finally, I will raise questions for future research on essential and infinite idealizations via the notion of exploration. (shrink)

The paper discusses the recent literature on abstraction/idealization in connection with the “paradox of infinite idealization.” We use the case of taking thermodynamics limit in dealing with the phenomena of phase transition and critical phenomena to broach the subject. We then argue that the method of infinite idealization is widely used in the practice of science, and not all uses of the method are the same. We then confront the compatibility problem of infinite idealization with scientific realism. (...) We propose and defend a contextualist position for realism and argue that the cases for infinite idealization appear to be fully compatible with contextual realism. (shrink)

A collection of aphorisms, fragments, and observations on philosophy and pessimism.Composed of aphorisms, fragments, and observations both philosophical and personal, Eugene Thacker's Infinite Resignation traces the contours of pessimism, caught as it is between a philosophical position and a bad attitude. By turns melancholic, misanthropic, and tinged with gallows humor, Thacker's writing tenuously hovers over that point at which the thought of futility becomes the futility of thought.

Given a countable graph, we say a set A of its vertices is universal if it contains every countable graph as an induced subgraph, and A is weakly universal if it contains every finite graph as an induced subgraph. We show that, for almost every graph on, (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on will (...) almost surely contain non‐universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families. (shrink)

In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot (...) escape having bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete. (shrink)