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)

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)

Infinity is a concept that fascinates everyone from a seven-year-old child to a maths professor. So remarkable and strange is it that contemplating it has driven at least two great mathematicians over the edge into insanity. Where did the concept of infinity come from? Who were the people who originally defined and later refined this paradoxical quantity? Why is infinity, a concept we can never experience or truly grasp, at the heart of science? How can some infinities (...) be bigger than others? An exploration of the most mind-boggling feature of maths and physics, this work examines amazing paradoxes, for example Hilbert's Hotel. This imaginary resort has an infinite number of rooms, which all happen to be occupied. Unfortunately an endless coach turns up carrying an infinite number of new guests. It's not a problem though - it's easy to prove they can all be accommodated. The book also looks at: the people who devised and refined the concept, the many mind-bending paradoxes of infinity, infinity's place at the heart of mathematics and science in processes such as calculus, how dividing by zero brings infinity into view and infinity and cosmos. (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)

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)

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)

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)

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)

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 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)

Reminiscence of the author's education in infinity and how it turned him toward philosophy.

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)

It has long been known that in the context of axiomatic second-order logic (SOL), Hume's Principle (HP) is mutually interpretable with "the universe is Dedekind infinite" (DI). I offer a more fine-grained analysis of the logical strength of HP, measured by deductive implications rather than interpretability. The main result is that HP is not deductively conservative over SOL + DI. That is, SOL + HP proves additional theorems in the language of pure second-order logic that are not provable from SOL (...) + DI alone. Arguably, then, HP is not just a pure axiom of infinity, but rather it carries additional logical content. On the other hand, I show that HP is $\Pi^1_1$ conservative over SOL + DI, and that HP is conservative over SOL + DI + "the universe is well ordered" (WO). Next, I show that SOL + HP does not prove any of the simplest and most natural versions of the axiom of choice, including WO and weaker principles. Lastly, I discuss other axioms of infinity. I show that HP does not prove the Splitting or Pairing principles (axioms of infinity stronger than DI). (shrink)

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)

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)

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)

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)

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.

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)

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)

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)

Explains and gives examples of the mathematical concept of infinity.

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)

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 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)

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.

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)

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)

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.

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)

'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)

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)

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)

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)

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.

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

The article analyses the explication of the infinity in the philosophical courses taught at Kyiv-Mohyla Academy at the 17th and 18th centuries. It examines 12 philosophical courses – since 1645 (the course by Inokentii Gizel) until 1751 (the course by Georgii Konyskyi). It shows how the infinity was defined and in which kinds it was divided in different courses. In general, all the professors, as well as other scholastic philosophers, agree that categorematic infinity exists only in God, (...) but syncategorematic is present in the created world. Regarding the question whether God, being omnipotent, can create a categorematic infinity in the world, the Mogilyans are divided into several camps: (1) Inokentii Gizel, Stefan Yavorskyi, Inokentii Popovskyi, Sylvestr Pinovskyi, Platon Malynovskyi gave a positive answer to the question; (2) Yosyf Volchanskyi, Ilarion Levytskyi, Amvrosii Dubnevych, Sylvestr Kulabka believed that this kind of infinity is in principle impossible, hence God cannot create it; (3) Teofan Prokopovych and Georgii Konyskyi took a sceptical stand and consider that the human mind as such could not solve this problem. The article analyses which arguments were offered by each camp, and gives suggestions what influenced the position of certain Mohylian professors. Most probably, Mohylian philosophers who supported the possibility of actual infinity in the created world were influenced by nominalistically oriented Jesuit philosophers, like Pedro Hurtando de Mendoza and Rodrigo Arriaga. The other Mohylians backed a more traditional idea, supported by Thomas Aquinas, that only one actual infinity can exist and it is God. (shrink)

In their recent work, L. Graham and J.-M. Kantor discuss a remarkable connection between diverging conceptions of the mathematical infinite in Russia and France at the beginning of the twentieth century and the religious convictions of their respective authors. They expand much more on the Russian side of the cultural equation they propose; I do believe, however, that the French (or rather ‘West European’) side is more complex than it seems, and that digging deeper into it is worthwhile. In this (...) paper I shall therefore broaden the path laid out in Graham and Kantor’s work, by connecting two different strands of research concerning the origin of what I loosely call ‘formal’ ideas: firstly, the rich but complex relation between logic and rhetoric throughout European cultural history, and secondly, the impact of religious convictions on the formation of certain mathematical and scientific ideas during Renaissance and Early Modernity, especially but not exclusively in France. (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)

Do actual infinities exist or are they impossible? Does mathematical practice require the existence of actual infinities, or are potential infinities enough? Contrasting points of view are examined in depth, concentrating on Aristotle’s ancient arguments against actual infinities. In the long 19th century, we consider Cantor’s successful rehabilitation of the actual infinite within his set theory, his views on the continuum, Zeno's paradoxes, and the domain principle, criticisms by Frege, and the axiomatisation of set theory by Zermelo, as well as (...) Zermelo’s assertion of the primacy of potential infinity in mathematics. (shrink)

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.

We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of wide sets.

_ Infinity, Causation, and Paradox _, by PrussAlexander. Oxford: Oxford University Press, 2018. Pp. xiii + 207.

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.