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)

In a recent article, I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided (...) by Cantor’s cardinality assignments. In this talk, I generalize some specific worries emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The talk has four main parts. The first takes a historical look at 19th-century attributions of equality of numbers in terms of one-one correlations and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions. (shrink)

This book examines the philosophy of the nineteenth-century Indian mystic Sri Ramakrishna and brings him into dialogue with Western philosophers of religion, primarily in the recent analytic tradition. Sri Ramakrishna’s expansive conception of God as the impersonal-personal Infinite Reality, Maharaj argues, opens up an entirely new paradigm for addressing central topics in the philosophy of religion, including divine infinitude, religious diversity, the nature and epistemology of mystical experience, and the problem of evil.

The attempt to formulate a viable empiricist and non-foundationalist epistemology of science faces four problems here confronted. The first is an apparent loss of objectivity in science, in the conditions of use of models in applied science. The second derives from the theory-infection of scientific language, with an apparent loss of objective conditions of truth and reference. The third, often cited as objection to The Scientific Image, is the apparent theory-dependence of the distinction between what is and is not observable. (...) The fourth and last is the loss of the possibility of objective evaluation of rationality in scientific methodology. It is argued that each of these problems is illusory. (shrink)

Infinite Paths to Infinite Reality, by Ayon Maharaj, is a book whose significance is highly likely to reverberate throughout the fields of both the philosophy of religion and the study of Indian philosophy for years to come. It will certainly revolutionize, or at the very least raise important questions for, any future studies of the teachings of the nineteenth century Bengali sage and mystic, Sri Ramakrishna Paramahamsa, whose religious thought is this book's primary focus.Maharaj--who, on February 26, 2020, became Swami (...) Medhananda --has taken a highly original approach, in this book, to reconstructing the philosophy of Sri... (shrink)

This article focuses on how early modern astronomers and historians conceptualised the course of the history of astronomy. The aim is to describe the transition from the idea of closed historical cycles to the theory of infinite progress in astronomy. The cyclical Renaissance concept of the history of astronomy is addressed, highlighting in particular the emphasis placed by Protestant astronomers on the reliance of the history of astronomy on God. This is followed by a discussion of the theory of the (...) origin of astronomy as proposed by 17th century astronomers; and then a clarification of the unusual combination of the cyclical and linear approaches, which emerged in the historiography of astronomy in the 17th century. The article concludes with a brief outline of the transition to the concept of infinite progress that occurred at the end of the 18th century. (shrink)

This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort to make sense of Cantor’s troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday (...) mathematical practice. (shrink)

The blame for the semantic and set-theoretic paradoxes is often placed on self-reference and circularity. Some years ago, Yablo [1985; 1993] challenged this diagnosis, by producing a paradox that's liar-like but does not seem to involve circularity. But is Yablo's paradox really non-circular? In a recent paper, Beall [2001] has suggested that there are no means available to refer to Yablo's paradox without invoking descriptions, and since Priest [1997] has shown that any such description is circular, Beall concludes that Yablo's (...) paradox itself is circular. In this paper, we argue that Beall's conclusion is unwarranted, given that (1) descriptions are not the only way to refer to Yablo's paradox, and (ii) we have no reason to believe that because the description involves self-reference, the denotation of the description is also circular. As a result, for all that's been said so far, we have no reason to believe that Yablo's paradox is circular. (shrink)

Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi'}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$ . We also prove that there exists A such that, for each n (...) , the real $\Omega^A_M$ is $\textrm{high}_n$ for some universal Turing machine M by using the extended van Lambalgen's Theorem. (shrink)

Infinite Thought brings together a representative selection of the range of Alain Badiou's work, illustrating the power and diversity of his thought.

An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough.

Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes (...) the mathematician's fascination with infinity--a fascination mingled with puzzlement. "Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates."--Los Angeles Times "[Eli Maor's] enthusiasm for the topic carries the reader through a rich panorama."--Choice "Fascinating and enjoyable.... places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics."--Science. (shrink)

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.

Having been raised as a Catholic and educated in the West, then trained as a monk in India since the 1980s, Canadian author Swami Muktananda of Rishikesh is uniquely positioned to bring the Eastern tradition of Vedanta to Western spiritual seekers. In Awakening to the Infinite, he answers the eternal, fundamental question posed by philosophical seekers, "Who am I?" with straightforward simplicity. Knowing who you are and adopting a spiritual outlook, he counsels, can help solve problems in daily life to (...) do with relationships, work, children, sexuality, illness, and social injustice. In answering the question "Who am I?," Swami Muktananda of Rishikesh draws on the ancient teachings of advaita or nonduality, the truth that there is no division between ourselves and others at a fundamental level. The book is drawn from the many conversations he has throughout the year meeting with individuals from different backgrounds, cultures, and walks of life, as well as public talks and question and answer sessions. Covering subjects ranging from the spiritual path to everyday problems, Muktananda addresses questions with simplicity and compassion, from the standpoint of someone who is anchored in the infinite but fully in touch with the world. His teaching follows the words of Christ and the Old Testament as well as those of the sages of India, who all proclaim a universal truth that transcends all religions: 'You are not the body; you are not the mind; you are the immortal, divine Self. Realize this and be free.'. (shrink)

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n (...) + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language‐specific counting routines (e.g., the rules in English that represent base‐10 structure). We tested 4‐ and 5‐year‐old children’s knowledge of counting with three tasks, which we then related to (a) children’s belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end. (shrink)

In this paper the notion of unifier is extended to the infinite set case. The proof of existence of the most general unifier of any infinite, unifiable set of types (terms) is presented. Learning procedure, based on infinite set unification, is described.

My essay focuses on the relation between consciousness and reality. I argue, following Georg Cantor, that consciousness is a potential infinite driven by care to invest itself in the world historically. As human beings develop an authentic sense of self they are driven by good faith to postulate an Absolute infinite, which, following Spinoza, we may call God. Thus God, as the Absolute infinite, constitutes our reality, which I characterize as implicate information explicating itself through time. To live authentically is (...) to recognize the nature of God and commit to Him through faith. Faith means to understand one’s self as a potentially infinite consciousness attempting to authentically participate in, and understand, the Absolute of God. (shrink)

This paper is devoted to a consequence relation combining the negation of Classical Logic ) and a paraconsistent negation based on Graham Priest’s Logic of Paradox ). We give a number of natural desiderata for a logic \ that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. (...) We devise the logic \ by means of an axiomatization and three equivalent semantical characterizations. By showing that this logic is maximally paraconsistent, we prove that \ is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations. (shrink)

In this paper is presented and discussed the notion of true finite by opposition to the notion of theoretical finite. Examples from mathematics and physics are given. Fermat’s infinite descent principle is challenged.

It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

"One of Michael Blay's many fine achievements in Reasoning with the Infinite is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion. ...

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)

We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition result (...) involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”. (shrink)

I will explore the paradoxical nature of epistemic access. By critiquing the traditional conception of mental states that are labelled as ’knowledge’, I demonstrate the susceptibility of these states to an infinite regress, thus, challenging their existence and validity. I scrutinise the assumption that an epistemic agent can have complete epistemic access to all facts about a given object while simultaneously being ignorant of certain truths that impact the very knowledge claims about the object. I further analyse the implications of (...) this paradox when attempting to fine-grain the parameters of epistemic access, which happens to reveal the vacuity of such definitions and the inevitability of the regress. (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)

Scientific methods may be viewed as procedures for converging to the true answer to a given empirical question. Typically, such methods converge to the truth only if certain empirical presuppositions are satisfied, which raises the question whether the presuppositions are satisfied. Another scientific method can be applied to this empirical question, and so forth, occasioning an empirical regress. So there is an obvious question about the point of such a regress. This paper explains how to assess the methodological worth of (...) a methodological regress by solving for the strongest sense of single-method performance that can be achieved given that such a regress exists. Several types of regresses are. (shrink)

Alexandre Koyré. of the fixed stars is infinite commit a contradiction in adjecto. In truth, an infinite body cannot be comprehended by thought. For the concepts of the mind concerning the infinite are either about the meaning oftheterm "infinite,"  ...

In this paper I argue -- pace J. L. Schellenberg -- that it remains the case for Kierkegaard that infinite striving, properly understood, is essential to the relationship with God, who remains the Infinite Subject, one necessarily hidden for defensible logical, ontological, and existential reasons. Thus Kierkegaard’s arguments for the hiddenness of God as a logically required ingredient in the relationship that human beings are called to undertake with God can withstand Schellenberg’s criticisms.

In lieu of an abstract, here is a brief excerpt of the content:Transitions to a Modern Cosmology: Meister Eckhart and Nicholas of Cusa on the Intensive InfiniteElizabeth BrientThe Epochal Transition from the late medieval to the early modern world has long been thought in terms of the gradual “infinitization” of the cosmos. Traditionally this process has been studied by focusing on the pre-history and the aftermath of the Copernican revolution, that is, by describing the transition from the finite, hierarchically ordered (...) medieval cosmos to the infinite and homogeneous universe of the new astronomy. I would like to situate this paper within the context of a more general reading of the process of the infinitization of the medieval cosmos. That is, I want to argue that this process of “infinitization” must be understood intensively as well as extensively. Nature, in the modern age, is thought of not only as infinitely extended in space, but also as exhibiting an infinite richness in all of its parts. Each particular, each individual being is grasped as utterly unique, as infinitely rich and consequently as conceptually inexponable. Thus the infinitization of the real leads to an infinitization of the knowable—the radical shift in ontology grounds a corresponding shift in epistemology, so that the progress of human knowledge is understood as an unending project infinitely extended over time. In order to underscore both aspects of the infinitization of the medieval cosmos (intensive as well as extensive) I will focus on the way in which a single metaphor (that of the infinite sphere) is unfolded historically in the thought of Meister Eckhart and Nicholas of Cusa.1 This paper may thus be viewed as a case study of [End Page 575] the way in which the modern notion of “intensive infinity” may be traced back to late medieval speculation on the nature of God’s omnipresence in the world.1. EARLY SOURCES OF THE INFINITE SPHERE METAPHORThe metaphor of the infinite sphere makes its first appearance in the pseudo-Hermetic Liber XXIV philosophorum, a small book in the literary form of an “opus propositionum” which, in twenty-four propositions, describes the ineffable essence of God in the mystical language of Neoplatonic theology. It is in the second proposition of this book that we find for the first time the formulation, “God is an infinite sphere, whose center is everywhere, whose circumference is nowhere.”2 This formulation represents a significant development in the history of the Neoplatonic conception of intensive unity, and in the Christian understanding of divine infinity. The Neoplatonic One is here identified with God as infinite being. As a result the relationship between the One and the many, between God and the world, comes to be viewed in a new light, through the lens of “infinite unity.” The metaphor of the infinite sphere is a paradoxical formulation which pictures the coincidence of divine immanence with divine transcendence. That is, it expresses a double infinity: both intensive and extensive. God’s center is located at every point, as it were, indicating the plenitude and infinite richness of God’s being, while there is no limit, no boundary to the extent of God’s all-encompassing reach. The infinite sphere figures the divine nature as an intensive plenitude without end or bounds. The third and next proposition in this same work goes on to strengthen this sense of divine intensity, by asserting that God in his entirety is present in every part (Deus est totus in quolibet sui). The divine nature is not spread out compositionally, as it were, but is wholly present throughout. Proposition 18 describes God as a sphere which has just as many circumferences as points (Deus est sphaera cuius tot sunt circumferentiae, quot sunt puncta), once again, an exploding image which can only make sense in the case of an infinite sphere. These three propositions taken together provide us with a springboard for considering the nature of infinite unity. The exploding paradox of the infinite sphere [End Page 576] provides a rubric for thinking the coincidence of transcendence and immanence, a way of conceptualizing infinite unity, which cannot be “grasped” with finite concepts or through discursive reasoning. The metaphor of the infinite sphere finds its meaning precisely in the way... (shrink)

Amid a period of isolation and profound internal conflict, both in his life and in his thought, arises that which, according to Hölderlin, is “the general conflict in the human being”, namely the conflict between the “aspiration to limitation” and “the aspiration to the absolute”. The aim of this article is to analyze and, as much as possible, follow to its fullest extent, this fundamental thought: to see how it molds Hölderlin’s positions on existence and philosophy, how it meets the (...) most poignant philosophical concerns of the time and hence directly influences Hölderlin’s reading of Fichte, and how it leads the young poet to the notion of an infinite progression of philosophy, that is, to an infinite approximation to the ideal of knowledge. (shrink)

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...) in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II. (shrink)

Porpora offers an a priori argument for the conclusion that there are infinitely many thoughts that it is physically possible for us to think. That there should be such an a priori argument is astonishing enough. That the argument should be simple enough to teach to a first-year undergraduate class in about 20 min, as Porpora’s is, is more astonishing still. Porpora’s main target is Max Tegmark’s recent argument for the claim that if current physics is right, then there are (...) mental duplicates of us in far flung regions of the Universe. His argument is directed against Tegmark’s assumption that mental facts supervene upon physical facts. So, if Porpora’s argument is sound then not only is Tegmark’s argument unsound, but physicalism is also false. So, Porpora’s argument is powerful indeed. Who would have thought that a simple a priori argument, together with the physical facts, could solve the issue of whether physicalism is true?. Not I. In this paper I take a closer look at Porpora’s argument and show that it is fallacious. I also consider the other reasons Porpora gives for thinking there are infinitely many thinkable thoughts and find them similarly lacking. (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)

The Philosophy of the Infinite, With Special Reference to the Theories of Sir William Hamilton and M. Cousin by Henry Calderwood. This book is a reproduction of the original book published in 1854 and may have some imperfections such as marks or hand-written notes.

The long-periodic/infinite discrete Gabor transform is more effective than the periodic/finite one in many applications. In this paper, a fast and effective approach is presented to efficiently compute the Gabor analysis window for arbitrary given synthesis window in DGT of long-periodic/infinite sequences, in which the new orthogonality constraint between analysis window and synthesis window in DGT for long-periodic/infinite sequences is derived and proved to be equivalent to the completeness condition of the long-periodic/infinite DGT. By using the property of delta function, (...) the original orthogonality can be expressed as a certain number of linear equation sets in both the critical sampling case and the oversampling case, which can be fast and efficiently calculated by fast discrete Fourier transform. The computational complexity of the proposed approach is analyzed and compared with that of the existing canonical algorithms. The numerical results indicate that the proposed approach is efficient and fast for computing Gabor analysis window in both the critical sampling case and the oversampling case in comparison to existing algorithms. (shrink)

Until the Scientific Revolution, the nature and motions of heavenly objects were mysterious and unpredictable. The Scientific Revolution was revolutionary in part because it saw the advent of many mathematical tools—chief among them the calculus—that natural philosophers could use to explain and predict these cosmic motions. Michel Blay traces the origins of this mathematization of the world, from Galileo to Newton and Laplace, and considers the profound philosophical consequences of submitting the infinite to rational analysis. "One of Michael Blay's many (...) fine achievements in _Reasoning with the Infinite_ is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion."—Margaret Wertheim, _New Scientist_. (shrink)