Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories (...) of infinity. It discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, (...) Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

Many historical and philosophical studies treat infinity as an exclusively quantitative notion, whose proper domain of application is mathematics and physics. The main aim of this paper is to disentangle, by critically examining, three notions of infinity in the early modern period, and to argue that one—but only one—of them is quantitative. One of these non-quantitative notions concerns being or reality, while the other concerns a particular iterative property of an aggregate. These three notions will emerge through examination (...) of three central figures in the period: Locke, Descartes, and Leibniz. (shrink)

A philosophical exploration of the origin and limits of the modern world.

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of (...) infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. (shrink)

Alexander R. Pruss examines a large family of paradoxes to do with infinity - ranging from deterministic supertasks to infinite lotteries and decision theory. Having identified their common structure, Pruss considers at length how these paradoxes can be resolved by embracing causal finitism.

This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature (...) of part/whole relations, mathematical theories of the infinite, and infinite regression and principles of sufficient reason. (shrink)

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all collisions. (...) I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

When Derrida translated and commented on Husserl’s manuscript The Origin of Geometry in 1962, he gave a central place to what Husserl called the Idea “in the Kantian sense”. This article reflects on the use and function of this Idea in Derrida’s reading of Husserl. It critically interrogates the relationship between the Idea in the Kantian sense and mathematical ideality, as well as the use of this Idea in the interpretation of the Thing (Ding) and the stream of experience (Erlebnisstrom). (...) In the centre of the discussion stands what Derrida calls “pure thinking”, on which he tries to ground the Idea in the Kantian sense. (shrink)

What is infinity? It's reading the last line of a book and imagining the rest. No, wait, it's the instruction manual for the machine that operates the sun and the stars. In unexpected observations, captivating images, and even some equations, celebrated Argentinian author-illustrator Pablo Bernasconi, finalist for the 2018 Hans Christian Andersen Award, offers up verses about what infinity could mean to all of us. Winner of the Grand Prize from the Asociación de Literatura Infantil y Juvenil de (...) la Argentina (ALIJA) in 2018, Infinity is a book children and adults alike will find endlessly fascinating. (shrink)

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds (...) to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”. (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)

The concept of linguistic infinity has had a central role to play in foundational debates within theoretical linguistics since its more formal inception in the mid-twentieth century. The conceptualist tradition, marshalled in by Chomsky and others, holds that infinity is a core explanandum and a link to the formal sciences. Realism/Platonism takes this further to argue that linguistics is in fact a formal science with an abstract ontology. In this paper, I argue that a central misconstrual of formal (...) apparatus of recursive operations such as the set-theoretic operation merge has led to a mathematisation of the object of inquiry, producing a strong analogy with discrete mathematics and especially arithmetic. The main product of this error has been the assumption that natural, like some formal, languages are discretely infinite. I will offer an alternative means of capturing the insights and observations related to this posit in terms of scientific modelling. My chief aim will be to draw from the larger philosophy of science literature in order to offer a position of grammars as models compatible with various foundational interpretations of linguistics while being informed by contemporary ideas on scientific modelling for the natural and social sciences. (shrink)

In this paper two paradoxes of infinity are considered through the lens of counterfactual logic, drawing heavily on a result of Fine (2012b). I will argue that a satisfactory resolution of these paradoxes will have wide ranging implications for the logic of counterfactuals. I then situate these puzzles in the context of the wider role of counterfactuals, connecting them to indicative conditionals, probabilities, rationality and the direction of causation, and compare my own resolution of the paradoxes to alternatives inspired (...) by the theories of Lewis and Fine. Here is a quick overview of the paper. Sections 17.1 and 17.2 introduce two paradoxes of infinity that rest on certain principals concerning the logic of counterfactuals. Section 17.3 considers three possible ways to weaken the counterfactual logic that would resolve these paradoxes, and examines three concrete theories of conditionals which make those weakenings — Lewis (1973), Fine (2012a) and Bacon (2015). With the first two theories found wanting, Sect. 17.4 develops my preferred solution in (Bacon, 2015) a little further, and provides some simple models to show that the paradoxes can indeed be resolved in that framework. (shrink)

A mathematician and scientist in residence at the School of the Art Institute of Chicago helps readers explore the concept of infinity through unique concepts including chessboards, a chicken-sandwich sandwich and the creation of infinite cookies from an infinite dough ball.

Critically debates the distinction of different types of boredom and its impact on Williams’s argument, as well as the question of why personal identity should be threatened by eternally having new ground projects.

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (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)

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)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical (...) matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

'The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.' David Hilbert (1862-1943). This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than (...) a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: Can infinity be found in the real universe? Finally, the book returns to questions of philosophical and theological aspects of infinity"--Provided by publisher. (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)

Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to (...) avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism. (shrink)

In this article we investigate similarities between the role that ineffability of Absolute Infinity plays in class theory and in theology.

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.

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the (...) core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally (...) unsolvable within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies. The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics. (shrink)

It amazes children, as they try to count themselves out of numbers, only to discover one day that the hundreds, thousands, and zillions go on forever—to something like infinity. And anyone who has advanced beyond the bounds of basic mathematics has soon marveled at that drunken number eight lying on its side in the pages of their work. Infinity fascinates; it takes the mind beyond its everyday concerns—indeed, beyond everything—to something always more. Infinity makes even the infinite (...) universe seem small; yet it can also be infinitesimal. Infinity thrives on paradox, and it turns the simplest arithmetic on its head, with 1 seeming feasibly to equal 0, after all. Infinity defies common sense. The contemplation of it has relieved at least two great mathematicians of their sanity. Thoroughly readable and entirely accessible, science writer Brian Clegg's lively history explores infinity in its many intriguing facets, from its ancient origins to its place today at the heart of mathematics and science. He examines infinity's paradoxes and profiles the people who first grappled with and then defined and refined them, offering information, mystery, and poetry to conceive the inconceivable and define the indefinable. (shrink)

Bartha (2012) conjectures that, if we meet all of the other objections to Pascal’s wager, then the many-Gods objection is already met. Moreover, he shows that, if all other objections to Pascal’s wager are already met, then, in a choice between a Jealous God, an Indifferent God, a Very Nice God, a Very Perverse God, the full range of Nice Gods, the full range of Perverse Gods, and no God, you should wager on the Jealous God. I argue that his (...) requirement of [strongly] stable equilibrium is not well-motivated. There are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, which are intuitively no worse wagers than the Jealous God. In particular, I have suggested that one does no worse to wager on a jealous cartel than one does to wager on a Jealous God. Moreover, I argue that there are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, that make trouble for Pascal’s wager, but not because one would do better to wager on them rather than on a Jealous God. Finally, I argue that, if we suppose that infinitesimal credences are in no worse standing the infinite utilities, then we cannot accept the assumption—built into the relative utilities framework—that there cannot be infinitesimal credences. (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)

Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed in (...) terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many. (shrink)

Puzzles can arise in value theory and deontic (permissibility) theory when infinity is involved. These puzzles can arise for ethics, for prudence, or for any normative perspective. For the sake of simplicity, we focus on the ethical versions of these problems. We start by addressing problems that can arise in determining what is permissible, either in a given choice situation when there are an infinite number of options or in infinite sequence of choice situations, each with only finitely many (...) options. There is a common theme here: standard and plausible decision rules, such as the dominance principle or maximising principles, run into conflict with other plausible principles. We then address problems that can arise in determining whether one option is more valuable than another, when the options have infinite, or undefined, value. This includes cases where a single bearer of value has infinite or undefined value, as well as cases where each bearer of value is finite but the total for the collection is infinite or undefined totals. Our focus is on introducing the puzzles, rather than canvassing potential solutions. (shrink)

A reassessment of Aquinas’s doctrine of divine infinity, particularly in the light of the previous history of the concept within Western philosophy and theology. From the critical perspective provided by this history the central place which has been claimed for it in Aquinas’s thinking is questioned, as are also its originality and coherence. The notion that the doctrine of divine infinity was introduced to Western thought by Judaeo‐Christianity is rejected; from Anaximander onwards it had been a central concept (...) in Greek philosophy. Aristotle however had rejected it so effectively that, for several centuries afterwards, it seems to have led an ‘underground’ existence until it finally surfaces again in ‘Gnostic’ and related currents of thought. It is from these circles that it finally, and after some resistance, patchily entered orthodox Christian thinking. Likewise Plotinus was not the source of the doctrine, as some have claimed; historical precedence must be given to the Gnostics. Neoplatonism never, despite the prestige of Plotinus, fully accepted the doctrine; in Proclus it is a subordinate emanation from the One, which is beyond Infinity and Infinitude. Nor, because of the influence of Aristotle, does Islamic Neoplatonized Aristotelianism endorse the doctrine. Aquinas’s commitment to it seems to stem from the impact of John of Damascus’s stress upon the doctrine, the translation of whose work greatly influenced Western theology from the late twelfth century onward, together with his need to distance himself from the restrictions of the divine power and freedom found in the Arabic Aristotelian philosophers, whom, in general, he regarded as philosophical authorities. But his position on the doctrine is flawed by equivocation and self‐contradiction, flowing from his attempt to reconcile Christian personalist theism with Neoplatonized Aristotleain necessitarian monism. (shrink)

ABSTRACT According to an influential line of argument, the past must be finite because no infinite series can be formed by successive addition. The present paper pinpoints the non sequitur at the heart of this argument, disentangles the ambiguities that disguise it, and dismantles the misleading picture of ‘traversing the infinite’ that gives the argument so much of its allure. Finally, the paper critically explores the related argument that a beginningless series of past events is impossible because there could be (...) no explanation of its having been ‘completed’ at one time rather than another. (shrink)

The role of "infinite" (opposed to "indefinite") in Descartes philosophy. The character of being infinite is reserved for God alone, while extension and mathematics are strictly indefinitely large. The paper presents possible reasons behind this distinction.

I advance a novel interpretation of Kant's argument that our original representation of space must be intuitive, according to which the intuitive status of spatial representation is secured by its infinitary structure. I defend a conception of intuitive representation as what must be given to the mind in order to be thought at all. Discursive representation, as modelled on the specific division of a highest genus into species, cannot account for infinite complexity. Because we represent space as infinitely complex, the (...) spatial manifold cannot be generated discursively and must therefore be given to the mind, i.e. represented in intuition. (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.

Many philosophic arguments concerned with infinite series depend on the mutual inconsistency of statements of the following five forms: (1) something exists which has R to something; (2) R is asymmetric; (3) R is transitive; (4) for any x which has R to something, there is something which has R to x; (5) only finitely many things are related by R. Such arguments are suspect if the two-place relation R in question involves any conceptual vagueness or inexactness. Traditional sorites arguments (...) show that a statement of form (4) can fail to be true even though it has no clear counter-example. Conceptual vagueness allows a finite series not to have any definite first member. I consider the speculative possibilities that there have been only finitely many non-overlapping hours although there has been no first hour and that space and time are only finitely divisible even though there are no smallest spatial or temporal intervals. (shrink)

In his Pensées, Blaise Pascal gives vivid voice to both the wonder and anxiety that many early modern thinkers felt towards infinity. Contemplating our place between the infinite expanse of space and the infinite divisibility of matter, Pascal writes.

Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book (...) describes the mathematician's fascination with infinity--a fascination mingled with puzzlement. "Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates."--Los Angeles Times "[Eli Maor's] enthusiasm for the topic carries the reader through a rich panorama."--Choice "Fascinating and enjoyable.... places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics."--Science. (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.

This article examines mixed ℋ -infinity and passivity synchronization of Markovian jumping neutral-type complex dynamical network models with randomly occurring coupling delays and actuator faults. The randomly occurring coupling delays are considered to design the complex dynamical networks in practice. These delays complied with certain Bernoulli distributed white noise sequences. The relevant data including limits of actuator faults, bounds of the nonlinear terms, and external disturbances are available for designing the controller structure. Novel Lyapunov–Krasovskii functional is constructed to verify (...) the stability of the error model and performance level. Jensen’s inequality and a new integral inequality are applied to derive the outcomes. Sufficient conditions for the synchronization error system are given in terms of linear matrix inequalities, which can be analyzed easily by utilizing general numerical programming. Numerical illustrations are given to exhibit the usefulness of the obtained results. (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)

Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor’s set theory. Vopěnka criticised Cantor’s approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets (...) are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopěnka’s theory reverses the process: he models the finite in the infinite. (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)

It is shown that a notion of natural place is possible within modern physics. For Aristotle, the elements—the primary components of the world—follow to their natural places in the absence of forces. On the other hand, in general relativity, the so-called Carter–Penrose diagrams offer a notion of end for objects along the geodesics. Then, the notion of natural place in Aristotelian physics has an analog in the notion of conformal infinities in general relativity.