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)

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)

infinite, and offer several arguments in sup port of this thesis. I believe their arguments are unsuccessful and aim to refute six of them in the six sections of the paper. One of my main criticisms concerns their supposition that an infinite series of past events must contain some events separated from the present event by an infinite number of intermediate events, and consequently that from one of these infinitely distant past events the present could never have been reached. I (...) introduce.. (shrink)

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)

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)

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)

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.

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

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)

I am a theoretical physicist, and, following Aristotles' injunction (Aristotle, Physics III, 202b 34), I do consider it my responsibility to discuss the problem of the notion of infinity in the world--in particular, to "inquire whether there is such a thing or not." I will do so here by illustrating some aspects of the notion of infinity in the natural sciences.

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)

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)

The ontological and epistemological priority of the infinite has been extensively dealt with in Spinoza scholarship. However, Spinoza’s widely debated understanding of the infinite has not figured to the same extent in accounts of his therapy of the passions, the topic which this essay sets out to explore. My reasoning consists of six steps. First, I introduce Spinoza’s cognitive therapy, which claims that we can be healed from our passions by acquiring adequate ideas of them; second, I show that Spinoza’s (...) cognitive therapy is subject to the Affective Force Condition, i.e., the claim that only a stronger affect can defeat another affect ; third, I suggest that the one way an affect can be reliably weakened or strengthened is by us considering its intentional object as necessary or free, respectively; fourth, I argue that we can only adequately ascribe freedom to things if we regard them as absolutely infinite, and necessity to finite things if we can have an adequate idea of them as inscribed in an infinite causal nexus. This presupposes having an adequate idea of the infinite, and so fifth, I ask by which capacity, or from which perspective, we can acquire such an idea, on Spinoza’s view. Sixth, I finally ask whether the infinity of the causal history of a finite thing qualifies as the kind of infinity which we can adequately conceive, and give a tentative answer to this question by situating this infinity in Spinoza’s taxonomy of infinities. (shrink)

Towards Infinity is Ram Chandra’s seminal work on the chakras of the human system and the soul’s journey back to the Source. Its implications are far-reaching – for the first time in thousands of years he sheds new light on human spiritual anatomy by going beyond the seven traditional chakras. The author does not discuss the lower chakras, instead, he starts with the heart and its qualities of love, compassion, courage and empathy as the centre of our humanity. He guides (...) us through the experience of thirteen chakras, from the heart up to the final chakra at the back of the head. He describes the level of consciousness at each chakra and reveals the way to expand our consciousness to embrace each new stage along the way. Towards Infinity is a map of the journey to the ultimate destination, which sincere seekers of spirituality can then experience through their own practice with the support of Yogic Transmission. (shrink)

Infinity and the Brain proposes a logical and scientific way to resolve the paradox of mind and matter -- by explaining how the perception of a finite image is dependent upon the contrasting infinitude of God. The theory holds that awareness is equal to a tension between existence and nonexistence, such that the self is illuminated to itself (becomes conscious) to the exact measure that it anticipates the infinitude of its own nonexistence. This "anticipation" is actually a "tendency toward" a (...) loss of structure in accordance with the second law of thermodynamics, a process that is restrained by effort and the consequent wholesness of an image. An understanding of how effort is linked to an image and how both are dependent upon neural systems inseparable from "infinity anticipation" sets the stage for a new worldview -- in which mind is seen as dependent upon the immanence of an infinite God. And because "mind" can be defined by the presence of an image which intrinsically restrains disorder, we see by extrapolation that the entire universe must be fundamentally personal, not objective as currently viewed by modern science. Book jacket. (shrink)

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed.

The article discusses why the notion of infinity is indispensable in contemporary phenomenology. I will heavily rely on some of L(szl) Tengelyi’s ideas, who in his turn uses Marc Richir’s early works on Cantor and Kant. Although – similarly to Richir and Tengelyi – I also examine some of Cantor’s ideas, I concentrate on a different and more fundamental part of his theory than Tengelyi and Richir.

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 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.

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)

In lieu of an abstract, here is a brief excerpt of the content:Approaching Infinity:Dignity in Arthur Koestler's Darkness at NoonRoger BerkowitzIn his allegorical novel Darkness at Noon, Arthur Koestler tells of Rubashov, a founding father of an unnamed Party in an unnamed state.1 Jailed by the current Party leader, "Number One," and pressed to recant his deviationist views, Rubashov resists. At first, he resolves to go to his death to preserve his integrity. Later, Rubashov recognizes that to hold to his (...) own truth when it endangers the goals of political reform is politically irresponsible. He decides to recant.The aristocratic soldier in the neighboring cell is appalled by Rubashov's self-betrayal. "Honour," the aristocrat insists, "is to live and die for one's belief." Rubashov disagrees with the aristocratic idea of honor and responds that honour is "to be useful without vanity," which provokes his neighbor to erupt: "honour is decency—not usefulness." Rubashov answers: "We have replaced decency by reason" (Koestler, pp. 177–78).Today, we rationalist readers of Darkness at Noon are left with the question of the competing demands of usefulness and decency, of reason and dignity. Reason makes its arguments in familiar terms: to prevent a genocide, we will bomb the aggressors; in the name of rationalized health care, some will not get the care they need; for reasons of state, some will be tortured. In politics, as Max Weber teaches, rational ends demand a slow, powerful boring through of hard boards (Weber, p. 93).2 Thus reason—and Koestler unfolds reason as calculating rationality in its most Machiavellian garb—requires the willingness to employ sober and rational means to achieve desired ends. In common parlance, the end justifies the means. Against the nostalgia for a politics of decency [End Page 296] and dignity, reason counsels that to make an omelet, you need to break a few eggs.But what lies behind the opposing claim to decency? What is it that pulls us to resist the claim of reason when reason counsels war, torture, or even bureaucratic neutrality? What, to paraphrase Hannah Arendt, is to be said for the eggs? (Arendt, pp. 270–84).3We may invoke conscience, duty, and decency. We might invoke civil and human rights. We condemn realpolitik and affirm a Judeo-Christian world order. In short, we invoke the dignity of man and affirm one's humanity as an inalienable core that cannot be breached. The limitations of such invocations are by now clear. At a time of recurrent genocides, democratically sanctioned torture, and suicide bombings of civilians, the stirring claims of the dignity of mankind ring hollow. Yet the collapse of the Western tradition of dignity does not necessitate the loss of dignity itself. The failure to secure dignity on religious or rationalist grounds need not send us fleeing, our eyes covered in horror and our heads between our legs. What is needed is a confrontation with our situation, one that seeks to comprehend both the impulse to affirm an inviolable core to humanity and our failure to do so.The ambition of Darkness at Noon lies in its bald effort to affirm the truth of human decency even as it recognizes that man's humanity is a bald-faced fiction. When Rubashov teeters on the precipice between reason and decency, his "eye" tooth aches. He rubs his eyeglasses. He steps only on whole squares in the floor, avoiding the cracks. Tooth, glass, and wholeness refer to the problem of the singular "I," an entity so mysterious that Rubashov follows Nietzsche in naming it the "grammatical fiction." What is the "I," the conscience, the self? How is it that the fictional "I" can and must have such an essential role in modern politics? Where, if anywhere, lies the relevance of the "I" for politics?Darkness at Noon is acclaimed as one of the most important books of the twentieth century—it is number eight on the Modern Library's list of the Hundred Best Novels of the twentieth century. Yet, given its enormous impact, its philosophical ruminations, and its political theme, the novel has garnered surprisingly little academic consideration. It is worth asking, therefore, why Darkness at Noon has not been taken seriously by the academic... (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.

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)

Explains and gives examples of the mathematical concept of infinity.

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)

Abraham Robinson is well-known as the inventor of nonstandard analysis, which uses nonstandard models to give the notions of infinitesimal and infinitely large magnitudes a precise interpretation. Less discussed, although subtle and original–if ultimately flawed–is Robinson's work in the philosophy of mathematics. The foundational position he inherited from David Hilbert undermines not only the use of nonstandard analysis, but also Robinson's considerable corpus of pre-logic contributions to the field in such diverse areas as differential equations and aeronautics. This tension emerges (...) from Robinson's disbelief in the existence of infinite totalities (any mention of them is ‘literally meaningless’) and the fact that much of his work involves them. I argue that he treats infinitary avenues of mathematics as useful tools to avoid this difficulty, but that this is not successful to the extent that these tools must be justified by a conservative extension from finitary mathematics. While Robinson provides a compelling and unorthodox pragmatic justification for the role of formal systems in mathematical practice despite their apparent infinitary presuppositions, he deflates mainstream mathematics to a collection of games that occasionally produces meaningful results. This amounts to giving up on a commitment to reconciling his finitism with his mathematical practice. (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)

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)

A bold and all-embracing exploration of the nature and progress of knowledge from one of today's great thinkers. Throughout history, mankind has struggled to understand life's mysteries, from the mundane to the seemingly miraculous. In this important new book, David Deutsch, an award-winning pioneer in the field of quantum computation, argues that explanations have a fundamental place in the universe. They have unlimited scope and power to cause change, and the quest to improve them is the basic regulating principle not (...) only of science but of all successful human endeavor. This stream of ever improving explanations has infinite reach, according to Deutsch: we are subject only to the laws of physics, and they impose no upper boundary to what we can eventually understand, control, and achieve. In his previous book, The Fabric of Reality, Deutsch describe the four deepest strands of existing knowledge-the theories of evolution, quantum physics, knowledge, and computation-arguing jointly they reveal a unified fabric of reality. In this new book, he applies that worldview to a wide range of issues and unsolved problems, from creativity and free will to the origin and future of the human species. Filled with startling new conclusions about human choice, optimism, scientific explanation, and the evolution of culture, The Beginning of Infinity is a groundbreaking book that will become a classic of its kind. (shrink)

In this article I concentrate on three issues. First, Graham Oppy’s treatment of the relationship between the concept of infinity and Zeno’s paradoxes lay bare several porblems that must be dealt with if the concept of infinity is to do any intellectual work in philosophy of religion. Here I will expand on some insightful remarks by Oppy in an effort ot adequately respond to these problems. Second, I will do the same regarding Oppy’s treatment of Kant’s first antinomy in the (...) first critique, which deals in part with the question of whether the world had a beginning in time or if time extends infinitely into the past. And third, my examination of these two issues will inform what I have to say regarding a key topic in philosophy of religion: the question regarding the proper relationship between the infinite and the finite in the concept of God. (shrink)

This book focuses on the three main categories of Badiou's philosophy-being, truth, and subject- which are elaborated according to three encounters: structure, real, and mathematical infinity. It articulates an underlying theory, "discipline," constituted based on these encounters, which reveals the inner logic of Badiou's method.

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)

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 seeks to recover a neglected chapter in the historical and theoretical background of social theory in general and critical theory in particular with a view to refining the understanding of the presuppositions of a cognitively enhanced critical social science appropriate to our troubled times. For this purpose, it offers a brief reconstruction of the mathematical-philosophical tradition from ancient to modern times by extrapolating that part of it that is marked by the ideas of infinity, infinite processes and limit (...) concepts. It gives suggestive indications not only of how these ideas relate to extant social-theoretical thought, but especially also of the two related cognitive directions – the weak-naturalistic and the socio-cultural – in which such thought is currently challenged to go. (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)

Es werden zwei Bedeutungen von „Infinitesimal“ unterschieden und zwei Thesen verteidigt: (1) Leibniz glaubte, das Infinitesimale in einer der beiden Bedeutungen sei nicht nur eine nützliche Erdichtung, sondern es sei sogar notwendig fur die Differentialrechnung; (2) die moderne Nichtstand-Analysis rechtfertigt weder Leibniz's Griinde fur die Einführung des Infinitesimalen noch seinen Gebrauch desselben.

It is shown that the following three common understandings of Newton’s laws of motion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisible continuous body with finite (...) mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible. (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)

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)

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)

I conceive of Spinoza’s substance monism as a response to Aristotle’s prohibition against actual infinity for one key reason: nature, being all things, is necessarily infi nite. Spinoza encapsulates his substance monism with the phrase, “Deus sive Natura,” implying that there is only one infinite substance, which also possesses an infi nity of attributes, of which we are but modes. These logical delineations of substance never actually break up God’s reality. Aristotle’s well-known argument against the reality of an actual infinity (...) in his Physics prohibits the existence of an actually infinite bodily substance because it would necessarily “destroy” (Physics 204b26-27) all other elements or bodies. On Aristotle’s view, there is a fundamental and concrete distinction between things: each substance is primarily a this (Categories 3b10). I maintain that Spinoza’s rationalism and radicalization of the principle of sufficient reason lends him greater explanatory potential than Aristotle to justify the (non) existence of actual infinity. (shrink)

Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.

This article explores the consecutive modifications that phenomenology underwent in the works of Heidegger and Levinas. In particular, it discusses their importance for contemporary attempts to expand — and transcend — phenomenology in philosophy and the social sciences. Heidegger and Levinas responded to the problem of subjectivity — and intersubjectivity — in diametrically opposed ways and consequently the exposition of their thoughts involves focusing on conceptual dichotomies like finitude and infinity, time and eternity. Ultimately, it is argued that the very (...) conceptions of the human being that the two thinkers furnish under the guise of Dasein and Autrui reveal the impasses associated with western metaphysics and pave the path for the development of more adequate theorizations of subjectivity, radical otherness and communal being. (shrink)