Perhaps nowhere better than, "On the Names of God," can readers discern Laclau's appreciation of theology, specifically, negative theology, and the radical potencies of political theology. // It is Laclau's close attention to Eckhart and Dionysius in this essay that reveals a core theological strategy to be learned by populist reasons or social logics and applied in politics or democracies to come. // This mode of algorithmically informed negative political theology is not mathematically inert. It aspires to relate a fraction (...) or ratio to a series ... It strains to reduce the decided determinateness of such seriality ever condemned to the naive metaphysics of bad infinity. // It is worth considering that it is the specific 'number' of Dionysius in differential identification with an ineffable god (and, as such, a singular becoming between theology and numbers) that is floating in at least two dimensions [of signification] (be it political Demand on the horizontal dimension or theological Desire on [a] floating dimension) that cannot but *perform the link that relinks* names of god with any political life, populist reason, social justice, or radical democracy straining toward peace. (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)

Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary (...) meta-ontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

This paper develops transfinite extensions of transitivity and acyclicity in the context of population ethics. They are used to argue that it is better to add good lives, worse to add bad lives, and equally good to add neutral lives, where a life's value is understood as personal value. These conclusions rule out a number of theories of population ethics, feed into an argument for the repugnant conclusion, and allow us to reduce different-number comparisons to same-number ones. Challenges to these (...) arguments are addressed, including the issue of comparing existence and non-existence in terms of personal value, the possibility of minimal quanta of time and life, and the meaningfulness of measuring closeness between outcomes with different population sizes. An asymmetry is uncovered between transfinite cycles of worseness and betterness, supporting a version of the weak procreative asymmetry. Transfinite transitivity principles are also favourably compared to the better-known principles of continuity. (shrink)

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all collisions. (...) I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

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)

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (shrink)

This introduction to the roles infinity plays in metaphysics includes discussion of the nature of infinity itself; infinite space and time, both in extent and in divisibility; infinite regresses; and a list of some other topics in metaphysics where infinity plays a significant role.

Levy (2021 Levy, N. (2021). Bad beliefs: Why they happen to good people. Oxford University Press.[Crossref], [Google Scholar]) argues that bad beliefs predominately stem from automatic (albeit rational) updating in response to testimonial evidence. To counteract such beliefs, then, we should focus on ridding our epistemic environments of misleading testimony. This paper responds as follows. First, I argue that the suite of automatic processes related to bad beliefs extends well beyond the deference-based processes that Levy identifies. Second, I push back (...) against Levy’s claim that bad beliefs stem from wholly rational processes, suggesting that, in many cases, such processes are better characterized as arational. Finally, I note that Levy is too quick to dismiss the role that individuals can play in cleaning up their own epistemic environments, and I suggest one route through which this is possible. (shrink)

It is widely accepted that consent is a normative power. For instance, consent can make an impermissible act permissible. In the words of Heidi Hurd, it “turns a trespass into a dinner party... an invasion of privacy into an intimate moment.” In this chapter, I argue against the assumption that consent has such robust powers for moral transformation. In particular, I argue that there is a wide range of sex that harms or wrongs victims despite being consensual. Moreover, these cases (...) are not limited to those where con- sent is vitiated by background conditions. I start by calling this category of consensual sex Bad Sex. I then distinguish subspecies of this category, including psychological pressure, social coercion, and epistemically unsafe sex. I end by responding to an objection on which we should treat at least some subspecies of Bad Sex as rape. Though this alternative proposal is often motivated by ameliorative and strategic considerations, I argue that such considerations actually count against collapsing the categories of Bad Sex and rape. (shrink)

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...) part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems. (shrink)

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds (...) to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”. (shrink)

This chapter defends an axiological theory of pain according to which pains are bodily episodes that are bad in some way. Section 1 introduces two standard assumptions about pain that the axiological theory constitutively rejects: (i) that pains are essentially tied to consciousness and (ii) that pains are not essentially tied to badness. Section 2 presents the axiological theory by contrast to these and provides a preliminary defense of it. Section 3 introduces the paradox of pain and argues that since (...) the axiological theory takes the location of pain at face value, it needs to grapple with the privacy, self-intimacy and incorrigibility of pain. Sections 4, 5 and 6 explain how the axiological theory may deal with each of these. (shrink)

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional (...) methodologies studying infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid’s Common Notion no. 5 ‘The whole is greater than the part’ and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor’s and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann’s result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used. (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.

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

I tried to describe Infinity as a major natural conundrum known to man. The booklet also contains answers to some eternal questions, such as the meaning of life, faith, etc. I am especially proud of my Morality section.

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

I advance a novel interpretation of Kant's argument that our original representation of space must be intuitive, according to which the intuitive status of spatial representation is secured by its infinitary structure. I defend a conception of intuitive representation as what must be given to the mind in order to be thought at all. Discursive representation, as modelled on the specific division of a highest genus into species, cannot account for infinite complexity. Because we represent space as infinitely complex, the (...) spatial manifold cannot be generated discursively and must therefore be given to the mind, i.e. represented in intuition. (shrink)

The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face (...) of plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink)

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the (...) recent debate on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

(1st draft for review) -/- WARNING: -/- Reading this book comes with certain dangers to be mindful of; Please consider the following suggestions to avoid them: -/- 1: Do not try to perceive infinity; Any kind of success here leads to psychosis. 2: Do not try to resolve the paradoxes; To understand the greater truth of this book, paradoxes must be accepted as true. 3: Do not read this book if your faith is unstable and having it challenged could (...) potentially put you into existential crisis. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical (...) matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. Moreover, Cantor envisioned his transfinite set theory (Mengenlehre) as providing the analytical methods and techniques necessary for a comprehensive, organic, non-reductive description of nature, a Naturphilosophie. Thus Cantor’s novel mathematics is presented as part of a long tradition, to which Cusanus, Bruno, Spinoza, Leibniz and others belong, in which the infinity and infinite character of organic life forms is appreciated, and is in some sense a mirror or symbol of the divine. The doctrine of symbolism and their different approach to the laws of logic present in both the work of Cusanus and Cantor enables these thinkers to articulate a transcendental apophatic approach to divinity. (shrink)

Politics abounds with bad language: lying and bullshitting, grandstanding and virtue signaling, code words and dogwhistles, and more. But why is there so much bad language in politics? And what, if anything, can we do about it? In this paper I show how these two questions are connected. Politics is full of bad language because existing social and political institutions are structured in such a way that the production of bad language becomes rational. In principle, by modifying these institutions we (...) can reduce the prevalence of bad language. However, as I show, such practical efforts are fraught with difficulties. After first outlining an account of bad language (Section 1), I examine the rationality of three different types of bad language: inaccurate language, insincere language, and unclear language (Section 2). Next, I discuss the possibility of implementing institutional reforms to improve the quality of political discourse (Section 3). However, I then outline and discuss two serious complications for institutional reforms—namely, they create risk of abuse, and they could preclude instances of seemingly bad language that, in fact, are socially beneficial (Section 4). I conclude with some thoughts about how to pursue institutional reform in an appropriately circumspect manner (Section 5). (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)

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)

The going-on problem (GOP) is the central concern of Wittgenstein's later philosophy. It informs not only his epistemology and philosophy of mind, but also his views on mathematics, universals, and religion. In section I, I frame this issue as a matter of accounting for intentionality. Here I follow Saul Kripke's lead. My departure therefrom follows: first, a criticism of Wittgenstein's “straight” conventionalism and, secondly, a defense of a solution Kripke rejects. I proceed under the assumption, borne out in the end, (...) that statements of rule-following have truth-conditions and are not, as Kripke seems willing to concede, merely "assertible" in circumstances of a specified sort. Ultimately, my goal is to demonstrate that intending can be understood in terms of an individual's dispositions rather than those of the community to which she belongs. (shrink)

infinite, and offer several arguments in sup port of this thesis. I believe their arguments are unsuccessful and aim to refute six of them in the six sections of the paper. One of my main criticisms concerns their supposition that an infinite series of past events must contain some events separated from the present event by an infinite number of intermediate events, and consequently that from one of these infinitely distant past events the present could never have been reached. I (...) introduce.. (shrink)

Must the best friends necessarily be good people? On the one hand, as Aristotle puts it, ‘people think that the same people are good and also friends’. But on the other hand, friendship sometimes seems to require that one behave badly. For example, a normally honest person might lie to corroborate a friend's story. What I will call closeness, which I take to include sensitivity to friends' subjective values and concerns as well as an inclination to take their subjective interests (...) as reasons for action, is characteristic of friendship. But this seems to require that good friends should be morally flexible, more so than is compatible with a virtuous character. This would imply tension between ideals of friendship and ideals of character. But there is an important connection between virtue and friendship which arises precisely from friends' closeness, when concern for wellbeing, another important feature of friendship, is also taken into account. This helps mitigate the tension and shows how friendship and virtue are interconnected. The connection in turn provides friendship-based reason to think the best friends must be good people, even though concerns of friendship may occasionally clash with other moral concerns. (shrink)

Despite Diotima’s irresistible virtues and attractiveness across the millennia, she spells trouble for philosophy. It is not her fault that she has been misunderstood, nor is it Plato’s. Rather, I suspect, each era has made of Diotima what it desired her to be. Her malleability is related to the assumption that Plato invented her, that she is a mere literary fiction, licensing the imagination to do what it will. In the first part of my paper, I argue against three contemporary (...) ‘majority views’ about Diotima that I regard as false. The first is that we can be certain she is fictional;1 a second is that Diotima is our best evidence for Plato’s feminism; the third, that she is Plato’s mouthpiece for the higher mysteries in the Symposium. After I have set aside what I regard as false, I proceed in the second half of my paper to develop Diotima’s positive contribution to philosophical psychology, her naturalistic account of the psyche as mortal, unified, and developmental. Whether the view Plato assigns to her is one that he held I do not pretend to know; but it is a powerful, defensible, and coherent view that inspired positive aspects of Freudian psychology in the twentieth century. Freud’s insight, in fact, makes clear how erōs can be developed in relation to the bad as well as the good. (shrink)

It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, at least (...) as articulated here, would not have satisfied Cantor himself. (shrink)

This chapter contains sections titled: * Brief History * How We Talk * Science and Infinity * Religion and Infinity * Concluding Remarks * Notes * References * Further Reading.

I argue that death can be (and sometimes is) bad for cattle because it destroys relationships that are valuable for cattle for their own sake. The argument relies on an analogy between valuable human relationships and relationships cattle form with conspecifics. I suggest that the reasons we have for thinking that certain rich and meaningful human relationships are valuable for their own sake should also lead us to think that certain cattle relationships are valuable for their own sake. And just (...) as death is bad for us when it destroys our valuable relationships, so death is bad for cattle when it destroys their valuable relationships. This argument is important because it pinpoints something that is bad about death for cattle that is overlooked by popular accounts of the badness of death for non-human animals that focus exclusively on the impact of death on lifetime well-being levels. Thus, the argument reveals an overlooked moral cost of some of our farming practices. (shrink)

This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness (...) and something having no end. -/- My concern is that many of the philosophers arguing for or against the ontology (or possibility) of an actual infinite set are unaware or unfamiliar with the mathematical literature attempting to clearly and rigorously define these terms. I believe it is a mistake to leave mathematicians out of this conversation, as analysts in particular have defined (and used) infinity in a way that is relevant to the ongoing debate between philosophers. -/- For this paper, I have selected examples from Principles of Analysis by Walter Rudin, which has become a standard text for Real Analysis classes taken by pure mathematicians and engineers. I will also restrict the scope of examples to Euclidean spaces for simplicity. Finally, I will be using addition, + and multiplication * in their usual way. (shrink)

Bartha (2012) conjectures that, if we meet all of the other objections to Pascal’s wager, then the many-Gods objection is already met. Moreover, he shows that, if all other objections to Pascal’s wager are already met, then, in a choice between a Jealous God, an Indifferent God, a Very Nice God, a Very Perverse God, the full range of Nice Gods, the full range of Perverse Gods, and no God, you should wager on the Jealous God. I argue that his (...) requirement of [strongly] stable equilibrium is not well-motivated. There are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, which are intuitively no worse wagers than the Jealous God. In particular, I have suggested that one does no worse to wager on a jealous cartel than one does to wager on a Jealous God. Moreover, I argue that there are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, that make trouble for Pascal’s wager, but not because one would do better to wager on them rather than on a Jealous God. Finally, I argue that, if we suppose that infinitesimal credences are in no worse standing the infinite utilities, then we cannot accept the assumption—built into the relative utilities framework—that there cannot be infinitesimal credences. (shrink)

Philosophers have recently come to focus on explaining the phenomenon of ​bad beliefs,​ beliefs that are apparently true and well-evidenced but nevertheless objectionable. Despite this recent focus, a consensus is already forming around a particular explanation of these beliefs’ badness called ​moral encroachment​, according to which, roughly, the moral stakes engendered by bad beliefs make them particularly difficult to justify. This paper advances an alternative account not just of bad beliefs but of bad attitudes more generally according to which bad (...) beliefs’ badness originates not in a failure of sufficient evidence but in a failure to respond adequately to reasons. I motivate this alternative account through an analogy to recent discussions of moral worth centered on the well-known grocer case from Kant’s ​Groundwork​, and by showing that this analogy permits the proposed account to generalize to bad attitudes beyond belief. The paper concludes by contrasting the implications of moral encroachment and of the proposed account for bad attitudes’ blameworthiness. (shrink)

This paper claims that the argument that Aristotle seems to ascribe to Melissus in Physics III.6 about infinity is different from Melissus’ original argument. On scrutiny, it turns out that the Aristotelian version of the argument takes Melissus to suppose that being is unlimited because it is not in contact with anything else. I claim that this is not Melissus’ notion of unlimitedness for being, and that the Aristotelian version hinges on a reversal of Melissus’ own reasoning.

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)

I have recently published a book to which I gave the title: Is Bad Philosophy Responsible for the Climate Crisis? But this proved to be too inflammatory for Palgrave Macmillan, and they changed it to the anodyne The Philosophy of Inquiry and Global Problems: The Intellectual Revolution Needed to Create a Better World. In the book I argue that academic philosophy has a certain responsibility for the failure of humanity to put a stop to the climate and nature crises, in (...) failing even to notice that a bad philosophy of inquiry dominates academic inquiry – one that prevents universities from engaging actively with the public domain to ensure action is taken to prevent the climate and nature crises from developing, or intensifying. The Philosophy of Inquiry and Global Problems spells out what has gone wrong that has reduced academic philosophy to its present pitiful state, and what needs to be done to restore philosophy so that it becomes again a discipline with profoundly fruitful intellectual and humanitarian implications. This Opinion Piece is the Preface to the published book. (shrink)

This paper draws on Claudia Card’s discussions of moral luck to consider the complicated moral life of people—described as pessimists—who accept the heavy knowledge of the predictability of the bad moral luck of oppression. The potential threat to ethics posed by this knowledge can be overcome by the pessimist whose resistance to oppression, even in the absence of hope, expresses a sense of still having a ‘‘claim’’ on flourishing despite its unattainability under oppression.

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.

As it is standardly conceived, Inference to the Best Explanation (IBE) is a form of ampliative inference in which one infers a hypothesis because it provides a better potential explanation of one’s evidence than any other available, competing explanatory hypothesis. Bas van Fraassen famously objected to IBE thus formulated that we may have no reason to think that any of the available, competing explanatory hypotheses are true. While revisionary responses to the Bad Lot Objection concede that IBE needs to be (...) reformulated in light of this problem, reactionary responses argue that the Bad Lot Objection is fallacious, incoherent, or misguided. This paper shows that the most influential reactionary responses to the Bad Lot Objection do nothing to undermine the original objection. This strongly suggests that proponents of IBE should focus their efforts on revisionary responses, i.e. on finding a more sophisticated characterization of IBE for which the Bad Lot Objection loses its bite. (shrink)

One of the common claims of the eternalists is that the "actual" infinite is possible and the universe is eternal. They are trying to refute the Kalam argument. What I wanted to show in this paper is that the "actual" infinite is impossible for logical reasons, and I have shown further that infinity has an effect and application over time, and that there is no way to deny the beginning of the universe for existence. The paper points out the (...) problems of infinity and points to the beginning of the universe. (shrink)

Abstract -/- The concept of infinity is of ancient origins and has puzzled deep thinkers ever since up to the present day. Infinity remains somewhat of a mystery in a physical world in which our comprehension is largely framed around the concept of boundaries. This is partly because we live in a physical world that is governed by certain dimensions or limits – width, breadth, depth, mass, space, age and time. To our ordinary understanding, it is a seemingly (...) finite world under those dimensions and we may find it difficult to comprehend something that by definition can have no beginning and no end, no limit or boundary. The article argues that this concept can have a meaning different from that normally envisaged in science, philosophy or mathematics, a meaning that transcends all boundaries and which proceeds from a non-material or metaphysical perspective. It examines the features and implications of that concept. -/- . (shrink)

This study examined correlations between moral value judgments on a 17-item Moral Intuition Survey (MIS), and participant scores on the Short-D3 “Dark Triad” Personality Inventory—a measure of three related “dark and socially destructive” personality traits: Machiavellianism, Narcissism, and Psychopathy. Five hundred sixty-seven participants (302 male, 257 female, 2 transgendered; median age 28) were recruited online through Amazon Mechanical Turk and Yale Experiment Month web advertisements. Different responses to MIS items were initially hypothesized to be “conservative” or “liberal” in line with (...) traditional public divides. Our demographic data confirmed all of these hypothesized categorizations. We then tested two broad, exploratory hypotheses: (H1) the hypothesis that there would be “many” significant correlations between conservative MIS judgments and the Dark Triad, and (H2) the hypothesis that there would be no significant correlations between liberal MIS judgments and Machiavellianism or Psychopathy, but “some” significant correlations between liberal MIS judgments and Narcissism. Because our hypotheses were exploratory and we ran a large number of statistical tests (62 total), we utilized a Bonferroni Correction to set a very high threshold for significance (p = .0008). Our results broadly supported our two hypotheses. We found eleven significant correlations between conservative MIS judgments and the Dark Triad—all at significance level of p < .00001—but no significant correlations between the Dark Triad and liberal MIS judgments. We believe that these results raise provocative moral questions about the personality bases of moral judgments. In particular, we propose that because the Short-D3 measures three “dark and antisocial” personality traits, our results raise some prima facie worries about the moral justification of some conservative moral judgments. (shrink)

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our (...) perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)

We investigate the idea of sleep as the protostate, and posit the idea of dark consciousness where dark is a 2-fold hybrid. We model dark consciousness as a 2-topos in p-adic time, and outline perfectoid and diamond-like versions. We then introduce and illustrate implications of Dark Imaginarium, which is a higher order Curiosity Artificial Intelligence, an Infinity-Curiosity Type, that thinks in infinity categories.

The Rachels–Temkin spectrum arguments against the transitivity of better than involve good or bad experiences, lives, or outcomes that vary along multiple dimensions—e.g., duration and intensity of pleasure or pain. This paper presents variations on these arguments involving combinations of good and bad experiences, which have even more radical implications than the violation of transitivity. These variations force opponents of transitivity to conclude that something good is worse than something that isn’t good, on pain of rejecting the good altogether. That (...) is impossible, so we must reject the spectrum arguments. (shrink)