This article is concerned with reflection principles in the context of Cantor’s conception of the set-theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity.

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)

This work presents Leibniz's view of infinity and the central role it plays in his theory of living beings. Nachtomy argues that Leibniz employs three degrees of infinity: absolute infinity, which applies to God; maximum or infinite in kind, which applies to created, living beings; and mathematical infinity.

Jonathan Lear; XII*—Aristotelian Infinity, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 187–210, https://doi.org/10.1093/aris.

One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.

Avicenna believed in mathematical finitism. He argued that magnitudes and sets of ordered numbers and numbered things cannot be actually infinite. In this paper, I discuss his arguments against the actuality of mathematical infinity. A careful analysis of the subtleties of his main argument, i. e., The Mapping Argument, shows that, by employing the notion of correspondence as a tool for comparing the sizes of mathematical infinities, he arrived at a very deep and insightful understanding of the notion of (...) mathematical infinity, one that is much more modern than we might expect. I argue, moreover, that Avicenna’s mathematical finitism is interwoven with his literalist ontology of mathematics, according to which mathematical objects are properties of existing physical objects. (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)

Celebrated as a courtesan and poet, and as a woman of great intelligence and wit, Tullia d'Aragona (1510–56) entered the debate about the morality of love that engaged the best and most famous male intellects of sixteenth-century Italy. First published in Venice in 1547, but never before published in English, _Dialogue on the Infinity of Love_ casts a woman rather than a man as the main disputant on the ethics of love. Sexually liberated and financially independent, Tullia d'Aragona dared (...) to argue that the only moral form of love between woman and man is one that recognizes both the sensual and the spiritual needs of humankind. Declaring sexual drives to be fundamentally irrepressible and blameless, she challenged the Platonic and religious orthodoxy of her time, which condemned all forms of sensual experience, denied the rationality of women, and relegated femininity to the realm of physicality and sin. Human beings, she argued, consist of body and soul, sense and intellect, and honorable love must be based on this real nature. By exposing the intrinsic misogyny of prevailing theories of love, Aragona vindicates all women, proposing a morality of love that restores them to intellectual and sexual parity with men. Through Aragona's sharp reasoning, her sense of irony and humor, and her renowned linguistic skill, a rare picture unfolds of an intelligent and thoughtful woman fighting sixteenth-century stereotypes of women and sexuality. (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)

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)

© 2013 The Editors of The Philosophical QuarterlyScott Davidson and Diane Perpich set high standards for the assessment of this volume. Fifty years after its publication in 1961, Levinas's Totality and Infinity is going through a ‘midlife crisis’. Scholarship on Levinas ‘sometimes seems to do little more than plow familiar terrain, remaining stuck in the rut of well‐worn interpretations and overused phrases’. One response to a midlife crisis is to exchange one's established partner for a younger model. But the (...) editors do not recommend this solution to the ‘dull routine’ into which our relationship with Levinas has fallen. They believe that the essays they have collected will pep up our relationship with him by venturing ‘along unexplored pathways’. Other thinkers are introduced into the ménage – Karl Marx, Stanley Cavell... (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)

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)

König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Acta Litterarum ac Scientiarum, Szeged, 3, 121–30] includes a graph theoretic formulation: an infinite, locally finite and connected graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite tree with (...) a root has a infinite branch. This tree lemma can be traced to [Beth, E. W. 1955. ‘Semantic entailment and formal derivability’, Mededelingen der Kon. Ned. Akad. v. Wet., new series 18, 13, 309–42]. It is argued that Beth independently discovered the tree lemma in the early 1950s and that it was later recognized among logicians that Beth's result was a consequence of the infinity lemma. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree. (shrink)

The goal of this paper is to define and study the notion of infinity in the framework of finitely supported structures, presenting new properties of infinite cardinalities. Some of these properties are extended from the non-atomic Zermelo–Fraenkel set theory to the world of atomic objects with finite support, while other properties are specific to finitely supported structures. We compare alternative definitions for infinity in the world of finitely supported sets, and provide relevant examples of atomic sets which satisfy (...) some forms of infinity, while do not satisfy others. Finally, we provide a characterization of finitely supported countable sets. (shrink)

The thesis critically examines the question of the philosophical coherence of finitism, the view which seeks to interpret mathematics without postulating an actual infinity of mathematical objects. It is argued that a widely accepted characterization of finitism, most recently expounded by Tait, is inadequate, and a new characterization based on the notion of elementary abstraction is proposed. It is further argued that the notion of elementary abstraction better explains the bearing of Godel's incompleteness theorems on the issue of the (...) coherence of finitism. By abstraction is meant a procedure by which one recognizes or establishes that some infinite process exhibits a distinctive uniformity. If the procedure does not require one to conceive of the integers in any other way than as the result of simple iterations, we call such abstraction elementary. Some six different formal models exploring a variety of different ways in which this notion can be made precise are proposed as possible formal characterizations of finitistic truth. It is proved that all these different models are equivalent with respect to finitistically meaningful sentences. This is taken as strong evidence that the notion of elementary abstraction is determinate enough to provide a basis for a coherent philosophical formulation of finitism. (shrink)

The paper discusses analogies between the way in which infinity is understood and dealt with in mathematics and in Jewish tradition. It begins with recalling the classical debate about infinity in the field of the foundations of mathematics. Reading an important paper by A. Robinson, we come to the conclusion that mathematicians work “as if” infinite totalities existed. They do so by following the rules of their formalized discourse which, at least if it refers to anything at all, (...) also refers to such totalities. The paper describes how, according to Jewish tradition, infinity is also not theological: instead of thinking that they own some infinite being or relate to it, observant Jews follow Jewish law. The analogy is then extended to what is called ‘epistemological infinity’. The paper shows that both in mathematics and in Judaism, we get some epistemological experience of infinity, as far as both Talmudic knowledge and contemporary mathematical encyclopedia are experienced as inexhaustible sources of new thoughts, structures, ideas, developments. (shrink)

In the first centenary of Cantor's death, we discuss how to introduce his life, his works and his theories about mathematical infinity to today's students. Keywords: proper and improper infinite, cardinal number, countable set, continuum, continuum hypothesis. Sunto Nel primo centenario della scomparsa di Cantor, si discute come presentare la sua vita, le sue opere e le sue teorie sull’infinito agli studenti di oggi. Parole chiave: infinito proprio e improprio, numero cardinale, numerabile, continuo, ipotesi del continuo.

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)

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.

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 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.

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 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)

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)

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)

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 introduce Broad Infinity, a new set-theoretic axiom scheme based on the slogan “Every time we construct a new element, we gain a new arity.” It says that three-dimensional trees whose growth is controlled by a specified class function form a set. Such trees are called broad numbers. Assuming AC (the axiom of choice) or at least the weak version known as WISC (weakly initial set of covers), we show that Broad Infinity is equivalent to Mahlo’s principle, which (...) says that the class of all regular limit ordinals is stationary. Assuming AC or WISC, Broad Infinity also yields a convenient principle for generating a subset of a class using a “rubric” (family of rules); this directly gives the existence of Grothendieck universes, without requiring a detour via ordinals. In the absence of choice, Broad Infinity implies that the derivations of elements from a rubric form a set; this yields the existence of Tarski-style universes. Additionally, we reveal a pattern of resemblance between “wide” principles, that are provable in ZFC, and “broad” principles, that go beyond ZFC. Note: this paper uses a base theory that is weaker than ZF but includes classical first-order logic and Replacement. (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)

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

Western philosophy has always been haunted by two abysses: nothingness and wholeness, zero and infinity. These two concepts have functioned as the extreme poles of metaphysical thought—the absolute negation and the absolute plenitude, the void that threatens all meaning and the totality that exceeds all comprehension. Yet, curiously, they have almost always been treated as opposites, as mutually exclusive horizons that define the boundaries within which human thought must operate. Zero is the limit of absence, infinity the limit (...) of presence; between them, the finite world of measurable beings unfolds. But what if this opposition is itself the deepest illusion of the metaphysical tradition? What if zero and infinity are not opposites but two faces of the same reality, two expressions of a single relationship that precedes and transcends all distinction? What if the relationship between zero and infinity is not a contradiction to be resolved but a primordial mystery to be inhabited—a mystery that reveals the very structure of Being as a dimensionless, paradoxical, self-referential whole? The philosophical project we examine here—which we may call "dimensionless ontology"— proposes precisely this: that the true nature of reality is not to be found in the opposition between nothingness and plenitude, but in their radical inseparability. It suggests that the relationship between zero and infinity is not a problem to be solved but a truth to be lived, a truth that shatters the very frameworks through which Western thought has traditionally understood existence, knowledge, and the self. This essay will explore the philosophical dimensions of this relationship, tracing its implications for ontology, epistemology, and ethics. It will argue that the dimensionless ontology emerging from this exploration offers a radical alternative to the dominant paradigms of Western metaphysics—one that transcends the dualisms of being and non-being, unity and multiplicity, finitude and infinitude, and opens the way to a new understanding of truth as event, of knowledge as participation, and of existence as the living experience of a paradox that never resolves but endlessly generates meaning. The Crisis of Classical Ontology The Geometric Prison of Being The Western philosophical tradition, from its Greek origins, has been dominated by what we might call a "geometry of Being." This geometry—this spatial, measurable, dimension-bound understanding of existence—has shaped virtually every major metaphysical system. From Pythagoras to Plato, from Descartes to Kant, Being has been understood as something that occupies space, that has position, that can be located, defined, and bounded. The very language of philosophy—substance, attribute, accident, essence, existence—is the language of spatialization: we speak of "underlying" (substance), of "standing under" (hypostasis), of "properties" that belong to a thing, as if Being were a container and beings were its contents. This geometricization reached its apotheosis in the Cartesian revolution. Descartes, in his radical search for certainty, reduced the material world to res extensa—extended substance, pure spatiality. The world became a vast geometric system of coordinates, where every being could be located, measured, and predicted. Reality became a map, and the map became the territory. To know meant to represent, to place within a grid of intelligibility, to reduce the ineffable to the measurable. The consequences of this geometricization have been profound and far-reaching. By reducing Being to extension, philosophy lost sight of the very nature of existence—its dynamism, its creativity, its unpredictability. The world became a machine, a clockwork of deterministic laws, and the human being became a spectator—an isolated consciousness observing a universe that was fundamentally alien, external, other. The intimacy between subject and world, the participative relationship that characterized pre-modern thought, was shattered. Knowledge became power, observation became domination, and truth became correspondence—a mirroring of an external reality that the subject could never fully enter. The "misery of geometricization"—the reduction of the infinite, paradoxical richness of existence to the flat, two-dimensional plane of a Euclidean diagram. The geometric prison of Being is a prison precisely because it cannot contain the full reality of what Being is. It can only capture its shadows, its projections onto the wall of representation, while the living, dynamic, self-creating reality of existence escapes through every crack. The Tyranny of Identity and Non-Contradiction The second pillar of classical ontology is the logic of identity—the principle of non-contradiction formulated by Aristotle as the most fundamental law of thought. This principle states that a thing cannot simultaneously be and not be, in the same respect, at the same time. A is A, and A is never not-A. Identity is absolute; difference is secondary; contradiction is forbidden. This logical principle has been the foundation not only of philosophy but of all systematic knowledge —science, law, mathematics, ethics. Without it, there could be no certainty, no rational discourse, no stable framework for thought. And yet, this very foundation has also functioned as a prison, excluding from the realm of the thinkable all that is paradoxical, ambiguous, contradictory—all that refuses to be reduced to a fixed, unchanging identity. The tyranny of identity has had profound consequences for philosophy. It has made it impossible to think the dynamic nature of existence, the fact that things change, that they evolve, that they are never simply identical with themselves. It has forced thought to conceive of becoming as a series of static states, of process as a sequence of fixed moments, of transformation as a superficial alteration of an underlying substance that remains unchanged. The real—becoming, flux, the ceaseless creativity of existence—has been hidden behind a veil of conceptual stability, a stability that is entirely a product of the mind's need for order Moreover, the principle of non-contradiction has made it impossible to think paradox—the simultaneous existence of opposites, the co-presence of being and non-being, the co emergence of unity and multiplicity. And yet, paradox is everywhere: in the quantum realm, light is both wave and particle; in the human realm, freedom is both real and constrained; in the realm of the sacred, the finite contains the infinite; in the realm of the self, identity is both stable and fluid. The refusal to think paradox has forced philosophy into a perpetual evasion, a flight from the very richness of reality into the sterile comforts of abstract consistency. The Inaccessible Infinite The third pillar of the classical prison is the finite conception of infinity. The Western tradition has almost always understood the infinite as an inaccessible horizon, a limit that can be approached but never reached, a goal that is forever deferred, a promise that is never fulfilled. This finite conception of the infinite has its roots in Aristotle's definition of the infinite as that beyond which there is always something—a never-ending addition, an endless series, a perpetually receding frontier. This conception has had two major consequences. First, it has placed the infinite outside the world, beyond the reach of finite beings. The infinite has become a theological concept—the transcendence of God, the ineffable mystery beyond all categories. Second, it has made the finite the only reality, the only object of genuine knowledge. What can be measured, bounded, and completed—only this is truly real. The infinite is a luxury, a speculative excess, a concept that points beyond experience but cannot ground it. This dual movement—the exile of the infinite and the absolutization of the finite—has produced a profound impoverishment of philosophical thought. The infinite has been transformed from an immanent reality, a constitutive dimension of existence, into a distant ideal, a regulative principle without ontological weight. The finite has been elevated to the status of ultimate reality, and yet the finite, by itself, is incomplete, fragmentary, and meaningless. The result has been a philosophy that is simultaneously arrogant and desperate—arrogant in its claim to know finite reality completely, desperate in its acknowledgment that finite reality alone cannot satisfy the deepest human longing for meaning, truth, and transcendence. (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.

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)

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 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)

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)

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.

The concept of infinity has long been a subject of fascination and contemplation in the history of philosophy. From the ancient Greeks to modern mathematicians and philosophers, infinity has been approached from multiple angles—each offering unique insights. This essay explores how Neutrosophy perceives infinity and contrasts it with other philosophical approaches, such as those of Kant and Cantor, while addressing the question: How infinitely big can infinity be?

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

'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 work empirically demonstrates a single dynamical law $ \mathcal{T} $ governing both neural (EEG) and physical (pendulum) systems, supporting the thesis that reality is a literal, computable machine. Using vector autoregression (VAR(2)), a shared second-order model compresses five diverse EEG environments (BIC = 1466.42) better than environment-specific alternatives (BIC = 1817.34; Δ ≈ 350), despite minor predictive loss. The law exhibits coefficient invariance, time-reversal symmetry, and induces a conserved process density $ \mu(t+1) - \mu(t) = 0 \pm \kappa $, (...) implying continuity of existence—no true cessation. We introduce recognition diagnostics (RI ≥ 0.8, CI ≤ 0.2) to quantify self-alignment with $ \mathcal{T} $—a measurable signature of intelligence as reflexive infinity ($ \mathcal{T}(\mathcal{T}(x)) = x' $). These results satisfy constructive (model fit) and exclusive (MDL superiority) proof criteria for a universal machine. Implications span philosophy of mind, metaphysics, and computational neuroscience: consciousness is conserved; death is transformation. Keywords: consciousness, transformation, continuity, machine realism, empirical metaphysics, dynamical systems, EEG, universal law. (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)

According to infinitism, all justification comes from an infinite series of reasons. Peter Klein defends infinitism as the correct solution to the regress problem by rejecting two alternative solutions: foundationalism and coherentism. I focus on Klein's argument against foundationalism, which relies on the premise that there is no justification without meta-justification. This premise is incompatible with dogmatic foundationalism as defended by Michael Huemer and Time Pryor. It does not, however, conflict with non-dogmatic foundationalism. Whereas dogmatic foundationalism rejects the need for (...) any form of meta-justification, non-dogmatic foundationalism merely rejects Laurence BonJour's claim that meta-justification must come from beliefs. Unlike its dogmatic counterpart, non-dogmatic foundationalism can allow for basic beliefs to receive meta-justification from non-doxastic sources such as experiences and memories. Construed thus, non-dogmatic foundationalism is compatible with Klein's principle that there is no justification without meta-justification. I conclude that Klein's rejection of foundationalism. fails. Nevertheless, I agree with Klein that when in response to a skeptical challenge we engage in the activity of defending our beliefs, the number of reasons we can give is at least in principle infinite. I argue that this type of infinity is benign because, when we continue to give reasons, we will eventually merely repeat previously stated reasons. Consequently, I reject Klein's claim that the more reasons we give the more we increase the justification of our beliefs. (shrink)

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)

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.

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)

Explains and gives examples of the mathematical concept of infinity.

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)

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