Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only (...) a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

For over two thousand years thought about the infinite was dominated by Aristotelian hostility to the idea that the infinite could be a legitimate object of mathematical study. Then Cantor's work late in the nineteenth century seemed to overturn this orthodoxy. However, by highlighting ways in which infinitude still could not be brought under the control of mathematicians, Cantor's work may in fact have reinforced the orthodoxy.

This chapter contains sections titled: The Saying and the Said Subjectivity as Sensibility Ethics of Difference or an Ethics of Truths? Reading Levinas with Badiou: Impossibly Demanding? An Impossibly Demanding Education Notes.

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...

ABSTRACT This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ and (...) his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.

Después de haber analizado las razones que indujeron a las antiguas matemáticas griegas y de que Aristóteles sólo admitiera una débil forma de lo infinito, se explora una ampliación de este concepto más allá de sus referencias numéricas y geométricas. El infinito puede expresar la “inagotable” riqueza ontológica de los atributos de las entidades individuales o, en otro sentido, el infinito puede ser entendido como aquello “ilimitado”. En este segundo sentido la “negación” se presenta como una fuerza positiva en la (...) formación del sentido ontológico de lo infinito.After having analyzed the reasons that induced ancient Greek mathematics and Aristotle to admit only a weak form of the infinite a broadening of this notion beyond its numerical and geometrical references is explored. The infinite can express the “inexhaustible” ontological richness of the attributes of individual entities or, in another sense, the infinite can be understood as the “unlimited”. In this second sense the “negation” appears as a positive force in the shaping of the ontological meaning of the infinite. (shrink)

How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.

This third edition of The Infinite includes a new part 'Infinity Superseded' which contains two new chapters refining Moore's ideas through a re-examination of the ideas of Spinoza, Hegel, and Nietzsche. Much of this is heavily influenced by the work of Deleuze. There is also a new technical appendix on still unresolved issues about different infinite sizes.

I thank the editors for inviting me to contribute to this issue on critical views of logic. Kant invented the critical philosophy. He fashioned its doctrines (Understanding versus Reason, synthetic...

Erasmus and Verhoef suggest that a promising response to the infinite God objection to the Kalām cosmological argument include showing that abstract objects do not exist; actually infinite knowledge is impossible; and redefining omniscience as : for any proposition p, if God consciously thinks about p, God will either accept p as true if and only if p is true, or accept p as false if and only if p is false. I argue that there is insufficient motivation (...) for showing and and that is problematic as a definition of omniscience. (shrink)

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully (...) adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

Some philosophers follow mctaggart in holding that there is a vicious infinite regress of tensed predications. Other philosophers claim there is no regress. The author argues that there is a regress, But it is benign.

An infinite future thus threatens to paralyze utilitarianism. Utilitarians need principled ways to determine which possible infinite futures are better or worse. In this article, I discuss a recent suggestion of Peter Vallentyne and Shelly Kagan. I conclude that the best way forward for utilitarians is, in fact, to by-pass the infinite utility debate altogether. (edited).

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of (...) the non-Cantorian outlook. (shrink)

In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural (...) quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, ifMPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects. (shrink)

Mathematics professor from Brown University uses colorful illustrations and cartoons to display the concepts of infinity and large numbers.

I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of a larger one, but (...) it requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”. (shrink)

Erasmus and Verhoef suggest that a promising response to the infinite God objection to the Kalām cosmological argument include showing that abstract objects do not exist; actually infinite knowledge is impossible; and redefining omniscience as : for any proposition p, if God consciously thinks about p, God will either accept p as true if and only if p is true, or accept p as false if and only if p is false. I argue that there is insufficient motivation (...) for showing and and that is problematic as a definition of omniscience. (shrink)

ABSTRACT: I argue that when Spinoza describes substance and its attributes as he means that they are utterly indeterminate. That is, his conception of infinitude is not a mathematical one. For Spinoza, anything truly infinite eludes counting s conception is closer to a grammatical one. I conclude by considering a number of arguments against this account of the Spinozan infinite as indeterminate.

In this article we analyze the claim that a probabilistic interpretation of the infinite epistemic regress problem leads to a unique solution, the so called “completion” of the regress. This claim is implicitly based on the assumption that the standard Kolmogorov axioms of probability theory are suitable for describing epistemic probability. This assumption, however, has been challenged in the literature, by various authors. One of the alternatives that have been suggested to replace the Kolmogorov axioms in case of an (...) epistemic interpretation of probability, are belief functions, introduced by Shafer in 1976. We show that when one uses belief functions to describe the infinite epistemic regress problem, it is no longer the case that the solution is unique. We also argue that this complies with common sense. (shrink)

The infinite-valued semantics was introduced in Rondogiannis and Wadge (2005) as a purely logical way for capturing the meaning of well-founded negation in logic programming. The purpose of this paper is threefold: first, to give a non-technical introduction to the infinite-valued semantics; second, to discuss the applicability of the infinite-valued approach to syntactically richer extensions of logic programming; and third, to present the main open problems whose resolution would further enhance the applicability of the technique.

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point but “in” the point. We are interested in our research in the diagrams (...) which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role, and an unveiling role. In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

In _Althusser, The Infinite Farewell_—originally published in Spanish and appearing here in English for the first time—Emilio de Ípola contends that Althusser’s oeuvre is divided between two fundamentally different and at times contradictory projects. The first is the familiar Althusser, that of _For Marx_ and _Reading Capital_. Symptomatically reading these canonical texts alongside Althusser’s lesser-known writings, de Ípola reveals a second, subterranean current of thought that flows throughout Althusser’s classic formulations and which only gains explicit expression in his later (...) works. This subterranean current leads Althusser to move toward an aleatory materialism, or a materialism of the encounter. By explicating this key aspect of Althusser’s theoretical practice, de Ípola revitalizes classic debates concerning major theoretico-political topics, including the relationship between Marxism, structuralism, and psychoanalysis; the difference between ideology, philosophy, and science; and the role of contingency and subjectivity in political encounters and social transformation. In so doing, he underscores Althusser’s continuing importance to political theory and Marxist and post-Marxist thought. (shrink)

The aesthetic techniques used in The Expanse are indicative of the infinite space that is an essential and ever‐present character in the show. The cinematography and set design of The Expanse make extensive use of chiaroscuro—a famous artistic technique in the history of painting. For some reason, the infinity of The Expanse attracts us. The look and design of the show indulges us in an experience of the sublime. The dynamically sublime is an experience of infinite power, but (...) not where our survival is threatened by it. The experience of the sublime is one proof of freedom. The absence of the ability to know the significance of what one does in the context of the infinite does not diminish the significance of what they do. The characters of The Expanse know this. They act because they know they must, and they have faith that what they do matters. (shrink)

This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed ‘niceness’ by the mathematician Michiel Hazewinkel, a concept we compare to the ‘robustness’ described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but (...) rather whether the choice of our concepts is forced upon us. (shrink)

In this article, I consider the importance of philosophy in the dialogue between religious believers and non-believers. I begin by arguing that a new epistemology of epistemic peer disagreement is required if the dialogue is to progress. Rather than viewing the differences between the positions as due to a deficit of understanding, I argue that differences result from the existential anchoring of such enquiries in life projects and the under-determination of interpretations by experience. I then explore a central issue which (...) is often implicit in these dialogues, namely the ontological status of God-world relations. Drawing on the reflections of Hegel on the infinite and the finite, I argue that his version of panentheism provides an insightful way to conceptualise God-world relations that avoids both dualistic and monistic approaches and helps to explicate a holistic ontology of transcendence from within the world of experience. (shrink)

By “Brentanian inner consciousness” I mean the conception of inner consciousness developed by Franz Brentano. The aim of this paper is threefold: first, to present Brentano’s account of inner consciousness; second, to discuss this account in light of the mereology outlined by Brentano himself; and third, to decide whether this account incurs an infinite regress. In this regard, I distinguish two kinds of infinite regress: external infinite regress and internal infinite regress. I contend that the most (...) plausible reading of Brentano’s account is the so-called fusion thesis, and I argue that internal infinite regress turns out to be inherent to Brentanian inner consciousness. (shrink)

The connection between the infinite nature of Quantum Cosmology and the infinite nature of God is presented here. At the beginning of the creation process, there was a single God/Void that was divided into many Gods/Voids all filled with Dark Energy consisting of photons which were responsible for creating the Multiverses made of matter, antimatter, space, time, charge, and multiple dimensions of space. The one God initially had no material existence which along with the laws of science was (...) a creation of the powers of His infinite mind. Since the process of creation started with the Dark Energy of the Voids, the Conservation of Energy Principle applied to all of them. If the Dark Energy of each Void were to be finite, then its energy would begin to decrease as Multiverses were created, and the only solution for not violating the Conservation of Energy Principle was that the Dark Energy of each of the Voids must be infinite implying that the Voids themselves must be infinite, and with infinite amount of energy be able to create an infinite number of Multiverses. According to Mathematics, all infinite numbers are not equal, some infinities are greater than other infinities. Hence of the many infinite Voids that were initially created, some Voids were greater than others. Each Void has a Supreme Leader that we refer to as God, and therefore there exist many Gods, some Gods more powerful than other Gods. (shrink)

Levinas approaches the Infinite as beyond all possible ideas and totalities.

Attempts a faithful account of Bruno's thought as expressed in his writings and to give an analysis of his thought as he developed it in regard to the issue of the infinitive.

Attempts a faithful account of Bruno's thought as expressed in his writings and to give an analysis of his thought as he developed it in regard to the issue of the infinitive.

I show the inadequacy of whitrow's recent argument ("british journal for the philosophy of science", Volume 29, Pages 39-45) against the possibility of an infinite past. I argue that it is impossible to prove "a priori" the non-Existence of an infinite past or future.

The philosophies of Jacques Derrida and Paul Shepard, while rarely encoun- tering the other, nevertheless prove to be surprisingly complementary. Derrida acknowl- edges the impossibility and necessity of the human/animal frontier, thinking the human/ animal relation in a paradigm of seeing and being seen, conceived in particular in the context of a sphere of the intimate. Shepard's not merely biological but ontological interpretation of evolution argues that humans need animals, not only metabolically but for their mental development. From the positive (...) dependence of the human on the animal follows an infinite debt that can never be repaid; but in attempting to do so lies the responsibility and destiny of the human, that most animal animal. (shrink)

Paradoxes of the Infinite presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19 th century: Dr Bernard Bolzano’s Paradoxien . This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

I show that the recursive structure of Leibniz's Law requires agents to perform infinitely many operations to psychologically identify the referents of phenomenal and physical concepts, even though the referents of ordinary concepts (e.g. Hesperus and Phosphorus) can be identified in a finite number of steps. The resulting problem resembles the hard problem of consciousness in the fact that it appears (and indeed is) unsolvable by anyone for whom it arises, and in the fact that it invites dualist and eliminativist (...) responses. Moreover, if this is the hard problem then we can predict that regardless of the strength of the argument for physicalism, and regardless of physicalism's truth, an ineliminable dissatisfaction is bound to accompany any physicalist theory of consciousness. Accordingly, I suggest that this is the hard problem of consciousness, and therefore that the hard problem arises from a recursively degenerate application of Leibniz's Law. (shrink)

The paper concerns the algebraic structure of the set of cumulative distribution functions as well as the relationship between the resulting algebra and the infinite-valued Łukasiewicz algebra. The paper also discusses interrelations holding between the logical systems determined by the above algebras.

For thousands of years voyagers of inner space - spiritual seekers, shamans and mystics - have returned from their inner travels reporting another level of reality that is more real than the one we inhabit in 'waking life'. Others have claimed that under the influence of mysterious substances, known as entheogens, the everyday human mind can be given glimpses of this multidimensional realm of existence that is usually hidden from us by our five basic senses. Using information from the leading (...) edges of modern science, this book presents a startling new hypothesis that these 'inner worlds' are as real, or possibly even more real, than the 'reality' we experience in waking life. The Infinite Mindfield uses as its starting point the widespread historical belief that the pineal gland - the 'third eye' - is a profoundly important organ. It links this to the various myths, originating in ancient Sumer, that 'dragons' or 'serpents' have guided humanity and presents evidence that these beings are symbolic of DNA. It is now known that DNA gives off a form of light known as bioluminescence. This information-rich 'inner light' needs an organ of sight to process it - that organ is the pineal gland. It is through this small organ that we "perceive" the inner worlds of lucid dreaming, out-of-body experiences, hypnagogic imagery, near-death experiences, astral travel and the kundalini experience. The book ends with the mind-blowing proposition that all living beings are one unitary consciousness experiencing itself subjectively. This book is genuinely revolutionary and will cause both interest and controversy in equal measure. Its science is watertight as is its logic. Be ready to be stunned - you will never have read anything like this before. (shrink)

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and computer science as an illustration of the way mathematical frameworks may be transformed. Originating in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent entities of interest. In particular, many important infinitely large entities can be characterised in terms of such (...) axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework, despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry in the relationship between algebra and coalgebra. (shrink)