Ion Soteropoulos reconciles the contradiction between the finite and infinite and transforms this reconciliation into the founding principle of motion. This book will appeal to readers interested in the logical mechanics of the physical universe, the hidden powers of our finite brain, and the utility of robots in the future.

The importance of infinity for Spinoza's philosophy can hardly be overstated. Understanding Spinoza means understanding (Spinoza's take on) infinity. In this paper, I present a deflationary account of Spinoza's infinity: Infinities across ontological states (modes, attributes, substance) follow the same general trajectory: From an indeterminate essence, infinitely many things follow. And as a consequence, Spinoza's universe is infinite all the way down. Some think that to Spinoza, infinity is indeterminacy (acosmism). Others say that infinity in (...) substance follows from the essence of substance, while infinity in modes does not. I argue that both views are false. Instead, infinity in modes and infinity in substance emerges from the numerical indeterminacy of essences. I show that this view can be traced back to Descartes' famous wax example and that it is manufactured to avoid a particular contradiction. In order to specify my point, I demonstrate that both ‘absolute infinity’ (God) and ‘in suo genere infinity’ (attributes) must be read through the deflationary account of infinity. I also argue that we do not need an additional qualitative indeterminacy and that causation and conception are not always parallel (such as in reductio proofs). (shrink)

This dissertation shows that Proclus provides a consistent reading of Plato's late dialogues, and develops a three level ontology which stands on its own. By augmenting the reserve of Platonist philosophy with Post Platonic developments of Greek mathematics and astronomy and physics, at points where Platonism ceased to provide operating principles, Proclus, reached for formulations which went beyond Plato. His own metaphysics, though sometimes obscured by theurgic allusions, grounds Being in an infinite One. ;One of the problems that Proclus (...) attempts to solve through the posit of three infinities, is that an iterative potential infinity, seen in the interval structure of the physical world, could divide to oblivion if uncolonized by Principle. The 'One Being', which brings Limit and noetic sameness, is itself entrapped in Self Reflexivity. An infinity of a third kind, one beyond all Limit and Unlimit is posed to solve this problem. This is Proclus' NeoPlatonic solution to the fact that Being alone, albeit under the rule of intellect, cannot guarantee itself. ;Platonism assumes a "universe" i.e. One grounding Many, which predetermines the place of all structure. $\rm A\nu\lambda o\gamma\iota\alpha$ is a principle of substitution which governs the visible cosmos and makes it comprehensible by mathematical formulations. Plato's emphasis on the self same as the epitome of Being, with circularity as the highest form for thought, helps Proclus provide a consistent account of: Creation ; Infinity as a Simultaneous Whole ; Soul and Nous, Dialectic is valorized over other forms of thought as most like $$ the eternal truth and a means to assimilation. Procession and reversion forms a circle which demonstrates a metaphysical successor of cosmic revolutions found to exist as the heavenly bodies carry out their noetic patterns. ;Finally Eternity encapsulates Time; but again its infinite circularity is only recouped by a One beyond all Limit which can carry out the Providence of a final cause. (shrink)

John Tomarchio - Aquinas's Concept of Infinity - Journal of the History of Philosophy 40:2 Journal of the History of Philosophy 40.2 163-187 Aquinas's Concept of Infinity John Tomarchio MUCH HAS BEEN WRITTEN of late about Aquinas's concept of divine infinity, but the attention given to his other metaphysical uses of the term 'infinite' has been peripheral -- sometimes to ill effect in the interpretation of his concept of divine infinity. The intent of this article is (...) to offer an explication of Aquinas's analogical concept of infinity. What Aquinas means when he calls beings or their principles infinite is sufficiently complex that it needs to be analyzed in its own right before his arguments for things' actually being infinite can be evaluated. There is no one place in his writings where Aquinas gives a full treatment of the various metaphysical senses of infinity, and so his view has to be reconstructed from the analysis of sundry texts in juxtaposition. As my intent is to reconstruct his overall theory, I will take up the relevant texts in a logical rather than historical order. I will first address the kind of relative infinity Aquinas predicates of certain principles of things, namely, the matter, form, and existence of finite beings, and then the kind of actual infinity he predicates of certain things, namely angelic substances and God. In the course of doing so, I will identify four global distinctions that run through all these.. (shrink)

This paper consists of two parts and has two aims. The first is to elaborate on the historical-philosophical background of Cantor’s notions of infinity in the context of Spinoza’s influence on him. To achieve this aim, in the first part I compare Spinoza’s and Cantor’s conceptions of actual infinity along with their mathematical implications. Explaining the metaphysical, conceptual, and methodological aspects of Cantor’s expansion of the orthodox finitist conception of number of his time, I discuss how he adopts (...) Spinoza’s motifs to overcome the challenges on his way to proposing his transfinite infinites. To achieve the second aim, i.e., to investigate the philosophical foundations of Cantor’s transfinite infinites, I follow a three-step outline analyzing Cantor’s three principles along with their metaphysical and epistemological-linguistic aspects. I analyze the strengths and weak sides of Lakoff and Núñez’s Basic Metaphor of Infinity and Pantsar’s Process → Object Metaphor, then compare their interpretations with mine. I conclude that the metaphors can be improved when their source domains are limited to the target domains they apply to and when supported by a thorough analysis of the historical-philosophical background of their target domains. (shrink)

The aim of this paper is not only to deal with the concept of infinity, but also to develop some considerations about the epistemological status of cosmology. These problems are connected because from an epistemological point of view, cosmology, meant as the study of the universe as a whole, is not merely a physical (or empirical) science. On the contrary it has an unavoidable metaphysical character which can be found in questions like “why is there this universe (or a (...) universe at all)?”. As a consequence, questions concerning the infinity of the universe in space and time can correctly arise only taking into account this metaphysical character of cosmology. Accordingly, in the following paper it will be shown that two different concepts of physical infinity of the universe (the relativistic one and the inflationary one) rely on two different ways of solution of a metaphysical problem. The difference between these concepts cannot be analysed using the classical distinctions between actual/potential infinity or numerable/continuum infinity, but the introduction of a new “modal” distinction will be necessary. Finally, it will be illustrated the role of a philosophical concept of infinity of the universe. (shrink)

As José Benardete observes, "the concept of the infinite is found to impinge on almost the whole schedule of ontological questions" (Infinity, viii). This is especially true for the early moderns, for whom questions like the following were still very much in play: Does the world have a beginning? Are there bounds to the spatial extent of the world? How does an imperfect creation flow from an infinitely perfect creator? How does the infinite divisibility of the continuum relate to (...) the structure of the material world? How are human actions related to the (possibly) infinite chain of causes or influences that shape our behaviors? Far from being an esoteric topic, which is best relegated to the outskirts of metaphysics or mathematics, infinity has tendrils reaching into nearly every philosophical domain. (shrink)

Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of (...) infinity is based on a metaphysical prejudice in favor of numbers as objects — a prejudice that mathematics can get along without. (shrink)

Personal time, as opposed to external time, has a certain role to play in the correct account of death and immortality. But saying exactly what that role is, and what role remains for external time, is not straightforward. I formulate and defend accounts of death and immortality that specify these roles precisely.

This introduction to the roles infinity plays in metaphysics includes discussion of the nature of infinity itself; infinite space and time, both in extent and in divisibility; infinite regresses; and a list of some other topics in metaphysics where infinity plays a significant role.

The two studies comprising this dissertation focus upon the contribution of Dietrich von Freiberg, O.P. to late mediaeval problems concerning identity and change in nature. The first study presents Dietrich's theories of the elements and prime matter, and features an extended historical and philosophical critique of Thomas Aquinas' notion of virtual being. It is with this notion that Aquinas attempts to resolve the apparent tension between Aristotle's theory of the chemical mixture and Aquinas' own doctrine of the unity of substantial (...) form. It is argued that virtual being is not a genuine Aristotelian mode of being and in any case does not resolve this apparent tension. Dietrich's own solution follows that of Averroes, maintaining that the elements in the mixed body are diminished from pure act to some degree of potency. It is argued that Dietrich mis-applies aspects of his theory of substantial generation in arriving at this solution. Such diminishment is possible, argues Dietrich, because of the elements' proximity to prime matter. The doctrine of proper form in its proper matter entails that prime matter must be complex, and those "natures" comprising prime matter be punctually indistinct. Thus are the elements themselves punctually indistinct in the homeomerous mixed body. Dietrich's solution is thus problematic precisely because it requires that the elements be capable of interpenetration. ;The second study evaluates, and attempts to situate in its historical context, Dietrich's theory that numerical identity is not preserved in intensive qualitative change. Rather, each new degree of intensity introduces a numerically distinct quality. This is because qualities lack the interiority of essence that substances possess. A survey of contemporaneous sources shows that while there was much opposition to this position, it was not uncommon. The discussion continues with an analysis of Dietrich's arguments that such change also neither is continuous nor entails an infinity of degrees. Degrees of intensity do not correspond with each "now" in the continuum of time. Rather, there must be an indivisible minimum, and intensive qualitative change must therefore be understood as stepwise and discontinuous. (shrink)

This volume contains essays that examine infinity in early modern philosophy. The essays not only consider the ways that key figures viewed the concept. They also detail how these different beliefs about infinity influenced major philosophical systems throughout the era. These domains include mathematics, metaphysics, epistemology, ethics, science, and theology. Coverage begins with an introduction that outlines the overall importance of infinity to early modern philosophy. It then moves from a general background of infinity up (...) through Kant. Readers will learn about the place of infinity in the writings of key early modern thinkers. The contributors profile the work of Descartes, Spinoza, Leibniz, and Kant. Debates over infinity significantly influenced philosophical discussion regarding the human condition and the extent and limits of human knowledge. Questions about the infinity of space, for instance, helped lead to the introduction of a heliocentric solar system as well as the discovery of calculus. This volume offers readers an insightful look into all this and more. It provides a broad perspective that will help advance the present state of knowledge on this important but often overlooked topic. (shrink)

This article discusses the history of the concepts of potential infinity and actual infinity in the context of Christian theology, mathematical thinking and metaphysical reasoning. It shows that the structure of Ancient Greek rationality could not go beyond the concept of potential infinity, which is highlighted in Aristotle's metaphysics. The limitations of the metaphysical mind of ancient Greece were overcome through Christian theology and its concept of the infinite God, as formulated in Gregory of Nyssa's theology. (...) That is how the concept of actual infinity emerged. However, Gregory of Nyssa's understanding of human rationality went still further. He said that access to the infinity of God was to be found only in an asymptotic ascension, as expressed in apophatic theology, and this view endangered the rationality of theology. Deeply influenced by the apophatic tradition, Nicholas of Cusa avoided this danger by showing that infinity could be accessed by symbolic representation and asymptotic mathematical reasoning. Thus, he made an immense contribution in making infinity rationally accessible. This endeavor was finally realized by Georg Cantor. This rational accessibility revealed discernible structures of infinity, such as the continuum hypothesis, and the cardinal numbers. However, the probable logical inconsistency of Georg Cantor's all-encompassing Absolute Infinity points to an intuitive understanding of infinity that goes beyond its rational structures – as in apophatic theology. (shrink)

‘Substance’ (substantia, zelfstandigheid) is a key term of Spinoza’s philosophy. Like almost all of Spinoza’s philosophical vocabulary, Spinoza did not invent this term, which has a long history that can be traced back at least to Aristotle. Yet, Spinoza radicalized the traditional notion of substance and made a very powerful use of it by demonstrating – or at least attempting to demonstrate -- that there is only one, unique substance -- God (or Nature) -- and that all other things are (...) merely modes or states of God. Some of Spinoza’s readers understood these claims as committing him to the view that only God truly exists, and while this interpretation is not groundless, we will later see that this enticing and bold reading of Spinoza as an ‘acosmist’ comes at the expense of another audacious claim Spinoza advances, i.e., that God/Nature is absolutely and actually infinite. But before we reach this last conclusion, we have a long way to go. So, let me first provide an overview of our plan. In the first section of this paper we will examine Spinoza’s definitions of ‘substance’ and ‘God’ at the opening of his magnum opus, the Ethics. Following a preliminary clarification of these two terms and their relations to the other key terms defined at the beginning of the Ethics, we will briefly address the Aristotelian and Cartesian background of Spinoza’s discussion of substance. In the second section, we will study the properties of the fundamental binary relations pertaining to Spinoza’s substance: inherence, conception, and causation. The third section will be dedicated to a clarification of Spinoza’s claim that God, the unique substance, is absolutely infinite. This essential feature of Spinoza’s substance has been largely neglected in recent Anglo-American scholarship, a neglect which has brought about an unfortunate tendency to domesticate Spinoza’s metaphysics to more contemporary views. The fourth section will study the nature of Spinoza’s monism. It will discuss and criticize the interesting yet controversial views of the eminent Spinoza scholar, Martial Gueroult, about the plurality of substances in the beginning of the Ethics; address Spinoza’s claim in Letter 50 that, strictly speaking, it is improper to describe God as “one”; and, finally, evaluate Spinoza’s kind of monism against the distinction between existence and priority monism recently introduced into the contemporary philosophical literature. The fifth and final section will explain the nature, reality, and manner of existence of modes. We therefore have an ambitious plan; let’s get down to business. (shrink)

The German-Hungarian phenomenologist became one of the few philosophers at the beginning of the 21st century who analyzed the foundations of transcendentalism in terms of the continental tradition. As a philosopher working within the framework of the Cartesian attitude, he posed the same questions as other philosophers after Heidegger - is it possible an alternative to ontotheology, is metaphysics possible after the rejection of metaphysics? But his answer quite accurately reflects both the internal contradictions of phenomenology and the (...) heuristic possibilities of modern non-classical ontology. The purpose of this article is to analyze the idea of L. Tengelyi's metaphysical phenomenology as a methodological project, with an emphasis on the concept of infinity. His work “World and Infinity. Towards the Problem of Phenomenological Metaphysics ”concentrates in itself an analysis of both the prehistory and the prospects of the phenomenological method at the beginning of the 21st century regarding post-metaphysical thinking. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.4 (2000) 539-560 [Access article in PDF] Plenitude, Possibility, and the Limits of Reason: A Medieval Arabic Debate on the Metaphysics of Nature Taneli Kukkonen In a recent article Simo Knuuttila has examined the argumentative patterns of modern cosmology, especially the search in fundamental physics for an "ultimate explanation," a unified "Theory of Everything" that would subsume all more local theories under (...) its aegis. Knuuttila goes on to compare contemporary modes of explanation with previous ways of conceiving of the metaphysics of understanding. As it turns out, on the cosmological borderline where physics shades into metaphysics desires similar at heart have fueled the efforts of thinkers throughout the history of western thought. When what is at stake is the final intelligibility of the world, shared ideals of necessity, transparency, and simplicity come to the fore. More often than not, theological concerns and preconceptions accompany these notions. 1One can react to this finding in one of two ways, either by retorting that the moderns have not gotten very far or by pointing out that earlier discussions on the topic were often quite philosophically sophisticated. In this article I propose to investigate how some of the issues raised by Knuuttila figure in a specific historical context. This is the debate on the pre-eternity of the world initiated by the celebrated Muslim theologian Abû Hâmid al-Ghazâlî (1058-1111) in his polemic arguing for The Incoherence of the Philosophers (Tahâfut al-falâsifa, 1095) and later picked up by the Andalusian philosopher Averroës (1126-98) in [End Page 539] his reply, The Incoherence of the Incoherence (Tahâfut al-tahâfut, 1180). 2 The exceptionally rich cosmological materials could be taken in any number of directions, but I will focus on three main issues: the discussants' construction of the domain of possibilities (i.e. the relation of the possible to the actual), their understanding of the reasonability of creation, and the question of its finitude and infinity. The study is undertaken partly to illustrate the sophistication with which the topics were handled in medieval Islamic philosophy, but in each case I will also draw parallels with certain modern conceptions, and I submit that in each case al-Ghazâlî's and Averroës's debate throws light on the contemporary discussion. In the final section I will develop this contention further by pointing to a connection between finitude and possibilities which I believe has both heuristic and existential value. Plenitude and Possibility Among historians of philosophy it is now acknowledged that in late medieval discussions certain modal theoretical innovations were instrumental in bringing about profound change in the western worldview. A conceptual shift occurred where, roughly speaking, a single intelligible universe was exchanged for several possible worlds. In Knuuttila's estimation this change in worldviews was in some ways more fundamental than anything modern science has produced since. 3 This claim deserves consideration but I would like first to outline what the change was about, for in al-Ghazâlî's and Averroës's debate this shift in modal paradigms is clearly discernible.Representative of the philosophers' point of view, Averroës holds onto the received Arabic Aristotelian interpretation of setting possibilities in a statistical temporal-frequency framework. According to this account, what is truly possible must at some time be realized--this is a technical interpretation of what has after Lovejoy commonly become known as the principle of plenitude. 4 As a [End Page 540] corollary, what always is is necessarily. The cosmological consequences of these assumptions can be outlined as follows. First, lest God's creative possibilities be limited, it must be admitted that the world could have been created however long ago in the past. By extension God must have been able to create the world an infinite time ago. But if it is a genuine possibility that the world is eternally old, then according to the principle of plenitude, this possibility is an actual fact. Once it is... (shrink)

Werkhoven’s ‘A Dispositional Theory of Health’ is an important and original contribution to debates about the disease concept, which persuasively demonstrates that dispositions must play some role in a full account of what it is to be healthy or ill. Unfortunately, as a theory, it cannot as it stands be correct.I first demonstrate what appears to be a significant, and possibly fatal, flaw; the proliferation of dispositions which Werkhoven’s theory requires makes impossible, at least in the absence of significant further (...) metaphysical work, the comparative numerical judgements on which its account of health and illness are based. I then demonstrate two further problems, concerning the exclusion of ‘technological’ dispositions from those under consideration, and a large class of compensatory biological functions which Werkhoven’s theory seems to have overlooked. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:152 BOOK REVIEWS with Weinrih's theory of formalism which Joseph Raz points out in his essay. One of the most serious of these deficiencies in my opinion is the role that is accorded to the judiciary. Weinrih's theory, as Raz shows, requires that when positive law is in conflict with the " form of law," positive law should he disregarded by the courts, and the courts in these cases (...) not only have the legal right to disregard the statutes of law hut even the moral duty to do so. To give such total discretionary power to the courts is certainly to offer an invitation impossible to re· sist, as the recent judicial history of the United States Supreme Court clearly shows, to autocratic and politically motivated judicial activism. The above remarks, inadequate as they are because of the constraints of space, may offer some indication of the rich diversity of natural law theories currently offering themselves for our consideration. Those interested in natural law theory either within the framework of moral theory, or legal theory, or both, will surely find that a careful reading of this volume will amply reward their efforts. St. John's University Jamaica, New York THOMAS A. FAY Metaphysics: An Outline of the History of Being. By MIECZYSLAW ALBERT KRAPIEC, O.P., translated by Theresa Sandok. New York: Peter Lang, 1991. Pp. 539. $69.95 (cloth). This volume is the second in Lang's Catholic Thought from Lublin series edited by Andrew Woznicki. Krapiec's work is a tour de force of Gilsonian Thomism as it founds metaphyscis upon a judgmental grasp of the act of existence in sensible things. The hook is unrelentingly Krapiec. Though there are abundant historical descriptions, there is a minimum of textual citation. Also, the reader can plainly see that Krapiec is intent upon presenting his understanding of metaphysics ad mentem Thomae. Three main parts comprise the hook. Part One is on the object of metaphysics-being as being (as existing). Part One contains the most crucial pages of the hook, viz., pp. 86-100. There Krapiec explains the attainment of this object. This portion is obviously the heart of the hook, and I will treat it in detail later. Part One also includes a lengthy exposition of the transcendentals. Part Two focuses on the structure of being and elaborates the following : act and potency, substance and accidents, matter and form, essence and existence, and the causes of being. There are two noteworthy points here. First, Krapiec's lengthy inclusion (pp. 313-374) of BOOK REVIEWS 153 matter and form within his metaphysics seems to have metaphysics swallowing up natural philosophy. One's misgivings are not assuaged by Krapiec's presentation of the division of the speculative sciences hack in his General Introduction. There the first degree of abstraction is no longer the home of natural philosophy hut the home of the modern humanistic and natural sciences (p. 28). The second degree harbors mathematics, while the third degree is home not only to metaphysics hut "philosophy" (pp. 28-9). The reader is left wondering what has happened to natural philosophy. I see no exigency' for Krapiec's metaphysics to eclipse natural philosophy. Moreover, for some Thomists, the move will create an unfortunate distraction from the good points of Krapiec's metaphysics. Second, by this reader's count, Krapiec presents no less than nine arguments for the real distinction between essence and existence. These arguments include the following: Aquinas's De Ente et Essentia argument that one can know what a phoenix is without knowing that one exists; the necessary character of existence as an essence vs. contingent realities; the monism of existence as an essence vs. the plurality of beings ; the infinity of existence as an essence vs. finite realities; and the reality of the potency/act distinction applied to essence-as·potency and existence-as-act. I cannot engage this involved topic in the short space of a review. I will say, however, that with Joseph Owens, I fail to see why a conceptual distinction between essence and existence is insufficient to deal with the above facts and problems. Underwriting my skepticism is the understanding that meanings can remain... (shrink)

In my review, I survey the main content of L. Tengeli's book World and Infinity, reconstruct its structure and discuss the author’s research results in the field of phenomenology of the world, presented in it. In particular, I note here that Tengelyi conducts a detailed criticism of onto-theology, starting from the problem of the katholo-protological structure of metaphysics in Aristotle. He outlines Husserl’s doctrine of primordial facts and Heidegger’s idea of metontology. In addition, here I give a relatively (...) detailed description of Tengeliy’s understanding of a thing and the world, as well as his interpretation of infinity as a basic characteristic of the world. (shrink)

Kant claimed that it is impossible for us to have a consistent notion of the infinite. I shall concentrate on three versions of the paradoxes of the infinite: Kant’s first antinomy, the paradoxes of Cantorian set theory, and applications of Cantorian arguments to the metaphysics of the world. I shall dare two side-glance looks at Ancient Chinese Philosophy, where analogies to the Western paradoxes can be found. I shall first discuss key passages from the Chinese sophists, and then consider (...) the formulation of the Law of Non-Contradiction in the Moist Canons. I conclude that the paradoxes of the infinite remain a major challenge for reason. (shrink)

This article aims to analyze the views of two pioneering Ḥanafī scholars, Abū Manṣūr al- Māturīdī and Abū Zayd al-Dabūsī, on the meaning of death and life in terms of their general doctrine of religion. In the first part, the general framework of Māturīdī and Dabūsī’s evidentialist conception of religion are drawn. In the second part, Māturīdī's views on the meaning of death and life and are explored. In the third part, the views of Abū Zayd al-Dabūsī on the meaning (...) of death and life, who establishes a striking connection between the divine order and existence, divine prohibition and non-existence, are examined, and a general evaluation is offered in the last section. Al-Māturīdī proposed a very original and systematic example of doing philosophy of religion, which still promising for the modern problems of the field, by opening the phenomenon of religion itself to discussion, and analyzing the nature and elements of religion. Māturīdī aims to place the problems on a universal and theoretical basis and to present a holistic, metaphysical understanding while discussing specific and particular issues in religion. Just as there are main notes in classical Western music that repeat throughout the symphony in baroque style, there are universal principles in al-Māturīdī’s metaphysics that cut all particular debates vertically. The basic principle of the al-Māturīdī’s metaphysics is based on this idea: God is a wise being, he does not do anything wise. Accordingly, it is absurd to create something to destroy; God did not create man to perish. The love of life and the fear of death are the deepest existential problems facing man as a finite being that goes towards death. Religion can be regarded as a powerful alternative to other worldviews to the extent that it provides a reasonable explanation for man’s deep concern about finitude. For al-Māturīdī, neither pessimistic understandings that emphasize only the fear of death nor hedonistic understandings that ignore death offer a reasonable solution. God created death as something that people fear the most and avoid, and life as something they love and desire. The testing of man is about being between the fear of death and the desire to live in the world. According to al-Māturīdī, this dilemma that man is facing also shows him the way out. If people do good and useful deeds because of their love for life, and avoid nonsense and ugly deeds because of their afraid of death, they can make their life eternal with their own efforts and escape from eternal/permanent death. The other Hanafī thinker, a philosopher of law Abū Zayd al-Dabūsī’s views on the meaning of death and life are parallel to Māturīdī's. According to al-Dabūsī, to test man, what God has placed in human nature is not only the soul and reason but also the spirit (al-nafs) and whim (al-hawa). While the spirit and whim invites man to the temporary by ignorance, it calls the soul and reason to the permanent by knowledge. The test for human beings is to choose between the temporary and the eternal life. For al-Dabūsī, man’s saving himself from falling into nothingness and gaining the eternal life depends on his use of reason. In other words, the reason binds human beings to life, and the whim leads them to nothingness. Drawing a parallel between reasoning and existence, ignorance and absence, Dabūsī defines the terms divine command and divine prohibition in a very original way. For him, the divine command is the things that must exist, and the divine prohibition is the things that must not exist. If the man helps exist the things that God wills to exist, he can make his own existence eternal. If the man helps exist the things that God does not will to exist, he puts himself into nothingness. Lastly, both for al-Māturīdī and al-Dabūsī, life is a divine grace bestowed on human being to gain eternal life through the good deeds he does at any moment. In short, reason is necessary in religion because its use is a preference of existence to non-existence. This metaphysical system, which sees goodness as a bridge from this finite world to heaven and eternity and makes a connection between faith and deeds on the ontological and ethical level is the essence of the worldview of the Hanafī-Māturīdī tradition. (shrink)

In the main bulk of this chapter, I offer a Wittgensteinian take on infinity and deduce from this some Wittgensteinian criticisms of Chomsky on ‘creativity’, treating this as one among many examples of how metaphors, following the understanding of Lakoff and Johnson, following Wittgenstein, can delude one into metaphysics. As per my title, ‘metaphysics’ turns out to be, really, nothing other than metaphorics in disguise. Our aim in philosophy, then, is to turn latent metaphors into patent metaphors. (...) When we do this, the charm of metaphysics evaporates. Or again, its charm, if still felt, is properly contextualised: ‘metaphysics’ becomes at best a kind of accidental or un-self-aware poetry, rather than something like a super- or supra-science. (shrink)

This volume contains essays that examine infinity in early modern philosophy. The essays not only consider the ways that key figures viewed the concept. They also detail how these different beliefs about infinity influenced major philosophical systems throughout the era. These domains include mathematics, metaphysics, epistemology, ethics, science, and theology. Coverage begins with an introduction that outlines the overall importance of infinity to early modern philosophy. It then moves from a general background of infinity up (...) through Kant. Readers will learn about the place of infinity in the writings of key early modern thinkers. The contributors profile the work of Descartes, Spinoza, Leibniz, and Kant. Debates over infinity significantly influenced philosophical discussion regarding the human condition and the extent and limits of human knowledge. Questions about the infinity of space, for instance, helped lead to the introduction of a heliocentric solar system as well as the discovery of calculus. This volume offers readers an insightful look into all this and more. It provides a broad perspective that will help advance the present state of knowledge on this important but often overlooked topic. (shrink)

Excerpt from Origin of History as Metaphysic The Muse Clio, carted from Pieria to the museums, can no longer be invoked without a libation to her warders, the numerous scribes, who have been busy since her fall correlating her steps, or her metamorphoses, as some say, for she has proved a difficult subject for classification: She is becoming bigger or better, nay she is growing many; she stations one foot in the beginning, but where is the other? Alas, it is (...) in Asia, and the head is a fake in need of restoration! Thus she has become Clio Tymbuchos in the professional hands of the scribes who find lively occupation exhuming and questioning her relics corked into formaldehyde and coffined into airtight compartments called historical facts. Men have speculated her into a new myth, a sepulchre of forms, and they have named the sepulchre History. But observe her resurrections! She has arisen, a mighty thing-in-herself with her own laws and growths and decays, and disembodied, dismembered, this apparition, this butterfly-concept called Clio, is expanding towards infinity. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

This book must have been a joy "to write": the author relishes playing with variations of Zeno's 'bisection' paradox to vindicate the reality of an Actual Infinite. The Infinite is a "lush" concept and though mathematical rigor forbids it, the world demands it. Benardete traces the development of mathematics through Aristotle, Leibniz, Gauss, Cantor, and Brouwer, and he examines recent developments in hyper-mathematics. Siding with Cantor, he argues that mathematics is no longer a formal discipline. It is teleological and it (...) requires a rich ontology of the Infinite. He criticizes modern philosophy, particularly Kant and the later Wittgenstein, for rejecting the "cosmological horizon," for ruling out the Actual Infinite. Even a non-mathematician will understand his lively discussions of "random numbers," the "long run," nondenumerable and denumerable infinites, and the significance of the Lowenheim-Skolem theorem.—A. B. D. (shrink)

Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large is intimately related to the infinitely small. Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics rest (...) upon some version of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. (shrink)

Three writers who utilize a similar metaphysics to understand the relationship between Ultimate Reality and conventional reality are compared. The metaphysics of what I call an inclusive Infinity is the common thread employed in comparing the thought of Hegel, Hartshorne and Nishida. I contrast the concept of inclusive Infinity with that of radical particularity and argue that people are private centers of conscious awareness who cannot be encompassed within an infinity or totality. Because of the (...) individuality and uniqueness of particulars, there can be no absolutely inclusive Infinite. (shrink)

Aristotle is said to have held that any kind of actual infinity is impossible. I argue that he was a finitist (or "potentialist") about _magnitude_, but not about _plurality_. He did not deny that there are, or can be, infinitely many things in actuality. If this is right, then it has implications for Aristotle's views about the metaphysics of parts and points.

This is a book review of Oppy's "Philosophical Perspectives on Infinity", which is of interest to those in metaphysics, epistemology, philosophy of science, mathematics, and philosophy of religion.

In two rarely discussed passages – from unpublished notes on the Principles of Philosophy and a 1647 letter to Chanut – Descartes argues that the question of the infinite extension of space is importantly different from the infinity of time. In both passages, he is anxious to block the application of his well-known argument for the indefinite extension of space to time, in order to avoid the theologically problematic implication that the world has no beginning. Descartes concedes that we (...) always imagine an earlier time in which God might have created the world if he had wanted, but insists that this imaginary earlier existence of the world is not connected to its actual duration in the way that the indefinite extension of space is connected to the actual extension of the world. This paper considers whether Descartes’s metaphysics can sustain this asymmetrical attitude towards infinite space vs. time. I first consider Descartes’s relation to the ‘imaginary’ space/time tradition that extended from the late scholastics through Gassendi and More. I next examine carefully Descartes’s main argument for the indefinite extension of space and explain why it does not apply to time. Most crucially, since duration is merely conceptually distinct from enduring substance, the end or beginning of the world entails the end or beginning of real time. In contrast, extension does not depend on any enduring substance besides itself. (shrink)

What should philosophical theology look like after the critique of Onto-theology, after Phenomenology, and in the age of Speculative Realism? What does Kabbalah have to say to Philosophy? Since Kant and especially since Husserl, philosophy has only permitted itself to speak about how one relates to God in terms of the intentionality of consciousness and not of how God is in himself. This meant that one could only ever speak to God as an addressed and yearned-for holy Thou, but not (...) to God as infinite creator of all. In this book-length essay, the author argues that reality itself is made up of the Holy Name of God. Drawing upon the set-theoretical ontology of Alain Badiou, the computational theory of Stephen Wolfram, the physics of Frank Tipler, the psychoanalytical theory of Jacques Lacan, and the genius of Georg Cantor, the author works to demonstrate that the universe is a computer processing the divine Name and that all existence is made of information (the bit). As a result of this ontic pan-computationalism, it is shown that the future resurrection of the dead can take place and how it may in fact occur. Along the way, the book also offers compelling critiques of several significant theories of reality, including the phenomenological theologies of Emmanuel Levinas and Jean-Luc Marion, Process Theology, and Object-Oriented Ontology. Reality in the Name of God explores how the concepts of Jewish mysticism can be articulated and deployed as philosophical theses within current metaphysical debates. It provides a new and dynamic Structural Realist ontology of information. Ultimately, the book aims to deal a death blow to the restriction of philosophy and theology in relation to elaborations of a how a believer relates to a God outside the mind and to return thought to a direct encounter with the divine nature of reality itself and its creator"--Website of the publisher, October 19, 2020. (shrink)

Introduction: Wonder and the births of philosophy -- Socrates' small difficulty -- The wound of wonder -- The death and resurrection of Thaumazein -- The Thales dilemma -- Repetition : Martin Heidegger -- Metaphysics small difficulty -- Wonder and the first beginning -- Wonder and the other beginning -- Theaetetus redux : the ghost of the Pseudes Doxa -- Once again to the cave -- Rethinking Thaumazein -- Openness : Emmanuel Levinas -- Passivity and responsibility -- The ethics of (...) the cave -- Infinity and astonishment -- Opening out : from existent to existence -- Closing down : from existence to existent -- Locking up : totality and infinity -- The phantom of the autrement -- Awakening -- Relation : Jean-Luc Nancy -- The problem of Mitsein -- Mitsein as essential inessentiality -- The myth of essentialism -- Unworking -- Interruption -- Repetition -- Decision : Jacques Derrida -- Thaumazein, the irresponsible, and the undecidable -- Hospitality -- Undecidability revisited -- Much madness is divinest sense (or, who comes after the decision?) -- How to avoid the subject (or, that's not my hedgehog!) -- Undecidability, take three : think here of Kierkegaard -- Mysterium tremendum -- Postlude: Possibility. (shrink)

In this paper a simple model in particle dynamics of a well-known supertask is constructed (the supertask was introduced by Max Black some years ago). As a consequence, a new and simple result about creation ex nihilo of particles can be proved compatible with classical dynamics. This result cannot be avoided by imposing boundary conditions at spatial infinity, and therefore is really new in the literature. It follows that there is no reason why even a world of rigid spheres (...) should be eternal, as has been erroneously assumed, especially since the time of Newton. (shrink)

Metaphysics and the Modern World makes the abiding questions of the nature of the self, world, and God available for the modern reader. Donald Phillip Verene presents these questions in both their systematic and historical dimensions, beginning with Aristotle's claim in his Metaphysics that philosophy begins in wonder. The first three chapters concern the origin of metaphysics as the transformation of the conception of reality in ancient Greek mythology, the ontological argument as the basis of Christian (...) class='Hi'>metaphysics, and the Renaissance cosmology of infinite worlds and the coincidence of contraries. The final four chapters present the central issues of the metaphysics of history through the New Science of Vico, the principle of true infinity of Hegel's Logic, the dialectic of spirit and life in Cassirer's Metaphysics of Symbolic Forms, and the conception of actual entities and God in Whitehead's Process and Reality. In these discussions, the reader will find a lively and learned account of a field of philosophy that is often thought difficult to access, but in this work becomes most accessible and a pleasure to read. -- back cover. (shrink)

In this paper I will examine what Blaise Pascal means by “infinite distance”, both in his works on projective geometry and in the apologetics of the Pensées’s. I suggest that there is a difference of meaning in these two uses of “infinite distance”, and that the Pensées’s use of it also bears relations to the mathematical concept of heterogeneity. I also consider the relation between the finite and the infinite and the acceptance of paradoxical relations by Pascal.

Werkhoven’s ‘A Dispositional Theory of Health’ is an important and original contribution to debates about the disease concept, which persuasively demonstrates that dispositions must play some role in a full account of what it is to be healthy or ill. Unfortunately, as a theory, it cannot as it stands be correct.I first demonstrate what appears to be a significant, and possibly fatal, flaw; the proliferation of dispositions which Werkhoven’s theory requires makes impossible, at least in the absence of significant further (...) metaphysical work, the comparative numerical judgements on which its account of health and illness are based. I then demonstrate two further problems, concerning the exclusion of ‘technological’ dispositions from those under consideration, and a large class of compensatory biological functions which Werkhoven’s theory seems to have overlooked. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Flipping the DeckOn Totality and Infinity’s Transcendental/Empirical PuzzleJack Marsh (bio)How does one perceive a transcendental condition?— Martin Kavka... if it is legitimate to hold Levinas to the standards that he himself imposes on certain other philosophers.— Robert BernasconiI do not believe that there is a transparency possible in method. Nor that philosophy might be possible as transparency.— Emmanuel LevinasThe question of the precise methodological status of the face (...) has yet to be satisfactorily resolved. The best evidence for this claim perhaps lies in texts intended to introduce Levinas to a general philosophical audience. If even first-rate philosophers wrestle still with the puzzle Totality and Infinity bequeaths us, we seem to still await a full and precise diagnosis. For example, Michael Morgan holds the “face-to-face and the responsible self — they are one thing characterized from [End Page 79] two perspectives — are” “the transcendental condition for meaningful human life,” yet elsewhere concedes, “it is unavoidable that in some sense or other the face-to-face does occur as an actually lived experience.” If Morgan’s own nuanced treatment of this question still lands him in equivocation, the problem is certainly not Morgan’s own.1 As Atterton notes, Morgan must finally hold that Levinas’s method is a “transcendental enterprise of a certain sort.”2 But of what sort? Indeed, this question has been asked or analyzed in one way or another from the beginning of Totality and Infinity’s English-language reception.3 Of course, Derrida was the first to broach the question of Totality and Infinity’s general intelligibility, alleging that Levinas ultimately succumbs to a form of empiricism. Of course, de Boer was the first to mount a rigorous Cartesian reading of the text as an (apparently) “ethical transcendental philosophy.” Of course, Bernasconi’s response to de Boer, and his creative repetition of Derrida’s critical point, was the first to give the problem a precise formulation.4 Derrida, de Boer, and Bernasconi hand us the now canonical problem of what I will call the text’s “transcendental/empirico-metaphysical puzzle.”5 The issue boils down to Levinas’s dual insistence that the face “cannot... be stated in terms of experience,” yet is nevertheless “reflected within experience” (TI 25, 23, 67, 260). How should we understand Levinas’s references to various “formal structure[s]” — “of language,” “of interpellation,” “of the idea of infinity” — next to talk of “pure” or “concrete moral experience” (195, 79, 64, 53)? Does the Other occur in “pure experience,” or is she “deduced” as a condition of possibility for experience in general? In this essay, I will propose an internal explanation for Totality and Infinity’s longstanding methodological puzzle. By proposing to “explain” this puzzle, I am making at least three distinct claims: to show (1) that the text suffers from genuine incoherence, thus confirming the general criticism of Derrida, Bernasconi, and others, (2) precisely how this incoherence follows from the details of Levinas’s own explicit method, and (3) precisely why, philosophically, he may have constructed and performed his method as he did.6 By proposing to explain the book’s incoherence, I can [End Page 80] also make specific hermeneutic predictions. If Totality and Infinity’s transcendental/empirico-metaphysical puzzle is genuinely aporetic, it follows that no resolution to it is possible on the text’s own terms.7 As such, any attempt to resolve the puzzle must either (1) augment the text’s own descriptions, (2) utilize other philosophical resources to render its claims coherent, or (3) precisely attempt resolution in either a transcendental or empirico-metaphysical direction. As it turns out, extant scholarship reflects just these hermeneutic strategies.Over the years many philosophers have contributed to this longstanding debate, variously arguing for or against the general intelligibility of Levinas’s account of the face. On the one hand, scholarship that questions the text’s intelligibility does so from a variety of methodological perspectives.8 On the other hand, scholarship that mounts a defense of the text seems to proceed in one of three ways: by (1) apparently repeating Totality and Infinity’s methodological problems, (2) anticipating Levinas’s later work, read... (shrink)

Three writers who utilize a similar metaphysics to understand the relationship between Ultimate Reality and conventional reality are compared. The metaphysics of what I call an inclusive Infinity is the common thread employed in comparing the thought of Hegel, Hartshorne and Nishida. I contrast the concept of inclusive Infinity with that of radical particularity and argue that people are private centers of conscious awareness who cannot be encompassed within an infinity or totality. Because of the (...) individuality and uniqueness of particulars, there can be no absolutely inclusive Infinite. (shrink)

This paper explores the metaphysical roots of Cantor’s conception of absolute infinity in order to shed some light on two basic issues that also affect the mathematical theory of sets: the viability of Cantor’s distinction between sets and inconsistent multiplicities, and the intrinsic justification of strong axioms of infinity that are studied in contemporary set theory.

Laszlo Tengelyi’s phenomenological project, as presented in his book “World and infinity. To the problem of the phenomenological metaphysics”, published posthumously, is founded on the idea of the infinity of the world. In rehabilitating the husserlian thesis of the infinity as constitutive feature of the worldly experience, Tengelyi departs from the phenomenology of the finitude, which goes back to Heidegger. The article analyzes the origins of the infinity/finitude dilemma in transcendental phenomenology of the world in (...) Husserl, Heidegger and Tengelyi. It demonstrates that the dilemma is rooted in the ambiguity of the phenomenological concept of transcendence. On the one hand, the transcendence is interpreted as the essential openness of the experience, which renders it infinite. On the other hand, the openness is constituted by horizon as the finitude of the experience. (shrink)

One of the main themes in Spinoza’s Ethics is the issue of human freedom: What does it consist in and how may it be attained? Spinoza’s ethical views crucially depend on his metaphysical theory, and this close connection provides the answer to several central questions concerning Spinoza’s conception of human freedom. Firstly, how can we accommodate human freedom within Spinoza’s necessitarianism—in the context of which Spinoza rejects the notion of a free will? Secondly, how can humans, as merely finite beings, (...) genuinely attain freedom? Can Spinoza defend his claim that we may even attain blessedness? I will argue that these questions are answered by appeal to a twofold in human nature. According to Spinoza, we are finite in infinity. (shrink)

This is a translation of a work, which appeared originally in French in 1923, that exposes in considerable detail the doctrine of Giordano Bruno on various kinds of minima. Bruno is justly famous for his teachings on infinity, but is little known for adumbrating atomic concepts through his finalist approach to the ultimate constituents of matter and mathematical continua. The author proposes to remedy this defect by exposing, in a systematic way, the contents of Bruno’s somewhat confusing Latin poem (...) De triplici minimo et mensura, ad trium speculativarum scientiarum et multarum activarum artium principia, first published in 1591 at Frankfurt. Like others of Bruno’s compositions, this is "a tiresome melange of irregular hexameters," in imitation of Lucretius, interspersed with Latin scholia "occasionally more poetic than the verses themselves". In three chapters the author successively details the various forerunners of Bruno’s doctrine ; makes sense of the three minima ; and offers a critique. (shrink)

While proponents of traditional theism obviously reject the “pantheistic” metaphysic that there is nothingwhich is ultimately distinct from God—i.e., that the Divine Substance exhausts the whole of Reality—it seems tome that the following question is yet properly to be addressed: given that the doctrine of God’s infinity or absolute unlimitedness is no less axiomatic or nonnegotiably foundational to traditional theism than it is to the pantheistic interpretation of reality, how can traditional theists justifiably deny that the Divine Substance exhausts (...) the whole of Reality? What acceptable basis do traditional theists have—given their presupposition that God is absolutely unlimited—for denying the long-standing contention of Spinoza and Spinozists that the Divine Being of canonical Judeo-Christian belief could not ultimately fail to be all-inclusive? (shrink)