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)

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.

Este artigo enfoca a concepção de infinito atual do filósofo judeu medieval Hasdai Crescas, formuladas no livro Or Hashem às três primeiras proposições de Maimônides, tal como são enunciadas na segunda parte do Guia dos Perplexos. As teses de Maimônides têm como objetivo negar a possibilidade do infinito atual como magnitude imaterial ou material, como conjunto infinito de seres finitos e como série infinita de causa e efeito. Após uma breve exposição da trajetória dos conceitos de infinito nas diversas tradições (...) sapienciais judaicas recebidas na Idade Média, indicamos como a argumentação em prol da ideia de infinito atual em Crescas dialoga com elas. Desse diálogo, emerge o conceito de infinito como singularidade atualizada paralelamente no real como vácuo infinito, lugar de coexistência de infinitos universos, e como infinito atual divino como Kavod, Glória, que preenche o universo infinito e como causa imanente da série infinita de causa e efeito que constitui a existência eterna dos seres contingentes. Na crítica à terceira proposição maimonidiana, a causa primeira é descrita como causa ontológica e imanente da série infinita de causas e efeitos. Nessa discussão, Crescas aponta para uma ideia de Deus muito diferente daquela elaborada por Maimônides. Temos aqui o debate judaico medieval entre os defensores da transcendência divina e os defensores da imanência. Esse tema é importante para a compreensão da recepção da obra de Hasdai Crescas por Picco Della Mirandolla, Bruno e Espinosa. (shrink)

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

In “Nonconservation of Energy and loss of Determinism II. Colliding with an Open Set” (2010) Atkinson and Johnson argue in favour of the idea that an actual infinity should be excluded from physics, at least in the sense that physical systems involving an actual infinity of component elements should not be admitted. In this paper I show that the argument Atkinson and Johnson use is erroneous and that an analysis of the situation considered by them is (...) possible without requiring any type of rejection of the idea of infinity. (shrink)

In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveller from reaching his destination. In this paper it is argued, contra Priest and Yablo, that the paradox must be resolved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New Zeno (...) and the paradoxes of truth, and a constructivist interpretation of Kripke’s theory of truth is given. (shrink)

Recent work on Kant’s conception of space has largely put to rest the view that Kant is hostile to actual infinity. Far from limiting our cognition to quantities that are finite or merely potentially infinite, Kant characterizes the ground of all spatial representation as an actually infinite magnitude. I advance this reevaluation a step further by arguing that Kant judges some actual infinities to be greater than others: he claims, for instance, that an infinity of miles (...) is strictly smaller than an infinity of earth-diameters. This inequality follows from Kant’s mereological conception of magnitudes (quanta): the part is (analytically) less than the whole, and an infinity of miles is equal to only a part of an infinity of earth-diameters. This inequality does not, however, imply that Kant’s infinities have transfinite and unequal sizes (quantitates). Because Kant’s conception of size (quantitas) is based on the Eudoxian theory of proportions, infinite magnitudes (quanta) cannot be assigned exact sizes. Infinite magnitudes are immeasurable, but some are greater than others. (shrink)

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would be (...) another two centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

Cosmological arguments attempt to prove the existence of God by appeal to the necessity of a first cause. Schematically, a cosmological argument will thus appear as: (1) All contingent beings have a cause of existence. (2) There can be no infinite causal chains. (3) Therefore, there must be some non-contingent First Cause. Cosmological arguments come in two species, depending on their justification of the second premiss. Non-temporal cosmological arguments, such as those of Aristotle and Aquinas, view causation as requiring explanatory (...) or conceptual priority, and thus insist that there can be no infinite regresses in such priority. Temporal cosmological arguments, also called kalam cosmological arguments due to their historical roots in Islamic kalam philosophers such as Abu Yusuf Ya'qub b. Ishaq al-Kindi and Abu Ali al-Hussain ibn Sina, view causation as requiring temporal priority, and thus insist that there can be no infinite temporal regresses.1 The kalam cosmological argument thus requires some supporting argument showing the incoherence of an infinite temporal regress of causally related events. William Lane Craig, in "The Finitude of the Past and the Existence of God"2, attempts to provide such an argument: (4) An actual infinite cannot exist. (5) An infinite temporal regress of events is an actual infinite. (6) Therefore an infinite temporal regress of events cannot exist. (9) I will not be concerned here with the general status of cosmological arguments, kalam or otherwise, or with contesting Craig's assumption that an infinite past would (unlike an infinite future) constitute a problematic actual infinity. I am rather concerned with Craig's general working principle, embodied in (4) above, that actual infinities are impossible. Craig, of course, is not alone in denying the possibility of the actually infinite. Resistance to such infinities is at least as old as Aristotle (Physics 3.5.204b1 – 206a8), and, as Craig rightly points out, persists through much of modern (i.e., post-scholastic, pre-twentieth-century) philosophy.. (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.

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)

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)

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

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)

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)

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)

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)

This article interprets Józef Maria Hoëné Wronski’s (1776–1853) use of actual infinities in his mathematical work. The interpretation places this usage, which undermined Wronski’s acceptance as a mathematician, in his contemporary mathematical and philosophical context and in the context of his own sociopolitical-philosophical project.

In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of ω? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. Many mathematicians (...) believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity. (shrink)

Hillel Steiner has recently attacked the notion of inalienable rights, basing some of his arguments on the Hohfeldian analysis to show that infinite arrays of legal positions would not be associated with any inalienable rights. This essay addresses the nature of the Hohfeldian infinity: the main argument is that what Steiner claims to be an infinite regress is actually a wholly unproblematic form of infinite recursion. First, the nature of the Hohfeldian recursion is demonstrated. It is shown that infinite (...) recursions of legal positions ensue regardless of whether inalienable rights exist or not. Second, the alleged problems that this might pose for the analysis are discussed. The conclusion is that one should not worry about the recursion as long as one understands correctly the role of the Hohfeldian analysis in normative reasoning. (shrink)

This article was found among the papers left by Prof. Laugwitz (May 5, 1932–April 17, 2000). The following abstract is extracted from a lecture he gave at the Fourth Austrain Symposion on the History of Mathematics (Neuhofen/ybbs, November 10, 1995).About 100 years ago, the Cantor-Veronese controversy found wide interest and lasted for more than 20 years. It is concerned with “actual infinity” in mathematics. Cantor, supported by Peano and others, believed to have shown the non-existence of infinitely small (...) quantities, and therefore he fought against the infinitely large and small numbers in Veronese’s geometry, but also against the non-archimedean systems of Thomae, du Bois-Reymond, and Stolz. As a positive consequence of the controversy the distinction between Cantor’s transfinite arithmetic and the theory of ordered algebraic structures becomes clear. (shrink)

In this note, I consider an argument advanced by William Lane Craig and James D. Sinclair against the possibility of actual infinite collections based onHilbert’s Hotel and alleged problems with inverse operations in transfinite arithmetic. I aim to show that this argument is misguided, since it is based on a mistaken view that the impossibility of defining ℵ0 - ℵ0 entails the impossibility of removing an infinite subcollection from an infinite collection.

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

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

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)

Aristotle was the first not only to distinguish between potential and actual infinity but also to insist that potential infinity alone is enough for mathematics ...

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)

Throughout the long centuries of western metaphysics the problem of the infinite has kept surfacing in different but important ways. It had confronted Greek philosophical speculation from earliest times. It appeared in the definition of the divine attributed to Thales in Diogenes Laertius (I, 36) under the description "that which has neither beginning nor end. " It was presented on the scroll of Anaximander with enough precision to allow doxographers to transmit it in the technical terminology of the unlimited (apeiron) (...) and the indeterminate (aoriston). The respective quanti tative and qualitative implications of these terms could hardly avoid causing trouble. The formation of the words, moreover, was clearly negative or privative in bearing. Yet in the philosophical framework the notion in its earliest use meant something highly positive, signifying fruitful content for the first principle of all the things that have positive status in the universe. These tensions could not help but make themselves felt through the course of later Greek thought. In one extreme the notion of the infinite was refined in a way that left it appropriated to the Aristotelian category of quantity. In Aristotle (Phys. III 6-8) it came to appear as essentially re quiring imperfection and lack. It meant the capacity for never-ending increase. It was always potential, never completely actualized. (shrink)

Salmon was the first to speak explicitly of paradoxes of kinematics. In this short note I introduce a new class of infinity puzzles. Following natural terminology, they should actually be called static paradoxes.

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major (...) motivation for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

It is commonly assumed that Aristotle denies any real existence to infinity. Nothing is actually infinite. If, in order to resolve Zeno’s paradoxes, Aristotle must talk of infinity, it is only in the sense of a potentiality that can never be actualized. Aristotle’s solution has been both praised for its subtlety and blamed for entailing a limitation of mathematic. His understanding of the infinite as simply indefinite (the “bad infinite” that fails to reach its accomplishment), his conception of (...) the cosmos and even its prime mover as finite (in the sense of autarchic/self-contained) have been contrasted with the subsequent claim of God as ens infinitum (understood as a “positive” infinity). The goal of this essay is to reexamine the major texts (notably De caelo) and to demonstrate that (1) Aristotle’s claim according to which there is no actual infinite concerns only substances, not processes. (2) That Aristotle does not deny an actual infinite as such. (3) That when considering time and God (qua eternal) Aristotle acknowledges an actual infinite. (shrink)

The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...) reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspectives on the Actual Infinite”) (1885). (shrink)

In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

What is music? Modern society has come to view music largely as entertainment and commodity. In response, We Are Music: An Existential Journey Toward Infinity provides the reader with a holistic starting point. Music has unlimited potential to transform and enlighten, and is only impeded when bound by materialism, physicalism, and reductionism. We Are Music is an attempt to bring music back to the core of humanity as an agent of positive empowerment, self-actualization, and beyond. Embracing interconnectivity, music is (...) more deeply experienced alongside the arts, science, social sciences, math, philosophy, history, and, above all, spirituality. An endless spiral, self-perception and identity can be vastly expanded, if not questioned and transcended. Music is an infinite field and any attempt to define and describe it is problematically finite and consequently limited. Herein lies the impossibility of the book which both excites and disturbs; the paradox of being and not being. A roadmap for music lovers toward self-realization, We Are Music is for those who desire to delve deeper into the power and illusion of self through music. (shrink)

The Successive Addition Argument (SAA) is one of the arguments proposed by the defenders of the Kalām Cosmological Argument to support the claim that the universe has a beginning. The main premise of SAA states that a collection formed by successive addition cannot be an actual infinite. This premise is challenged by an argument originally proposed by Fred Dretske. According to Dretske’s Argument (DA), the scenario of a counter who starts counting numbers and never stops can provide a counterexample (...) to the main premise of SAA. I argue that neither DA nor its past-oriented counterpart—which discusses the scenario of a counter who has always been counting negative integers from the infinite past—can play a decisive role in our evaluation of the strength of the arguments that are intended to establish the finitude of the past based on the impossibility of an actually infinite number of successive additions. (shrink)

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

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

In the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic, and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen, was not a major motivation for (...) operative logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)