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)

How could physicalism be true of a world in which there are no fundamental physical phenomena? A familiar answer, due to Barbara Gail Montero and others, is that physicalism could be true of such a world if that world does not contain an infinite descent of mentality. Christopher Devlin Brown has produced a counterexample to that solution. We show how to modify the solution to accommodate Brown’s example: physicalism could be true of a world without fundamental physical phenomena if (...) that world does not contain an infinite descent of mentally constituted mentality. This solution is independently plausible and is available to physicalists of virtually all significant varieties. (shrink)

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.

Alexander Pruss and I have proposed an analysis of omnipotence which makes no use of the problematic terms 'power' and 'ability'. However, this raises an obvious worry: if our analysis is not related to the notion of power, then how can it count as an analysis of omnipotence, the property of being all-powerful, at all? In this paper, I show how omnipotence can be understood as the possession of infinite power (general, universal, or unlimited power) rather than the possession (...) of particular powers. I then argue that the ordinary notions of particular powers or abilities can be understood as applying vague limitations to the notion of infinite power. The vagueness of these limitations is the source of the well-known difficulties in the analysis of ability. (shrink)

A philosophical memoir about becoming a father in an increasingly terrible world – can I hope the child growing in my partner's womb will have a good-enough life? For Kant, philosophy boiled down to three key questions: “What can I know?”, “What ought I do?”, and “What can I hope for?” In philosophy departments, that third question has largely been neglected at the expense of the first two – even though it is crucial for understanding why anyone might ask them (...) in the first place. In Infinitely Full of Hope, as he prepares to become a father for the first time, the philosopher Tom Whyman attempts to answer Kant’s third question, trying to make sense of it in the context of a world that increasingly seems like it is on the verge of collapse. Part memoir, part theory, and part reflection on fatherhood, Infinitely Full of Hope asks how we can cling to hope in a world marked by crisis and disaster. (shrink)

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.

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call (...) *Countable Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

In this paper I argue that the infinite regress of resemblance is vicious in the guise it is given by Russell but that it is virtuous if generated in a (contemporary) trope theoretical framework. To explain why this is so I investigate the infinite regress argument. I find that there is but one interesting and substantial way in which the distinction between vicious and virtuous regresses can be understood: The Dependence Understanding. I argue, furthermore, that to be able (...) to decide whether an infinite regress exhibits a dependence pattern of a vicious or a virtuous kind, facts about the theoretical context in which it is generated become essential. It is precisely because of differences in context that he Russellian resemblance regress is vicious whereas its trope theoretical counterpart is not. (shrink)

BioShock Infinite is a piece of fiction that lets one peer into a world where this linearity seems overridden by a multiverse where all the possibilities exist. Stories are important for video games. Its story is one of the reasons BioShock Infinite resonates with audiences all around the world. The field of philosophy that deals with art is called aesthetics. If one think that it's even worth asking the question of whether BioShock Infinite is art or not, (...) then one might want to consider this: What is it we're asking about in this case? When we ask whether Infinite is art, we're asking about the experience of playing it. BioShock Infinite is still one of the greatest video game stories and worlds of all time. In this, it represents a big step towards artful video game design. BioShock succeeds in showing that the possibilities for artful storytelling are infinite. (shrink)

X and Y are desperate to bake infinite pie! With the help of quirky and uber-smart Aunt Z, X and Y will use math concepts to bake their way to success!

In the massive plot twist at the end of BioShock Infinite, the writers beautifully put forth a hypothesis that individuals might exist in more than one possible world. In philosophy, the idea that an individual can exist in more than one world is called transworld identity. An important rival to transworld identity theory is counterpart theory, the idea that individuals cannot exist in more than one possible world and are therefore “world bound.” Modal realism is the thesis according to (...) which there is an infinite number of possible worlds that really exist in the most robust sense, relative to themselves. The world of Columbia is one possible world. The world of Rapture is another possible world. Both the actualist and the possibilist agree on these claims. Most important to the themes present in BioShock, transworld individuality has profound implications for differing notions of freedom. (shrink)

For aggregative theories of moral value, it is a challenge to rank worlds that each contain infinitely many valuable events. And, although there are several existing proposals for doing so, few provide a cardinal measure of each world's value. This raises the even greater challenge of ranking lotteries over such worlds—without a cardinal value for each world, we cannot apply expected value theory. How then can we compare such lotteries? To date, we have just one method for doing so (proposed (...) separately by Arntzenius, Bostrom, and Meacham), which is to compare the prospects for value at each individual location, and to then represent and compare lotteries by their expected values at each of those locations. But, as I show here, this approach violates several key principles of decision theory and generates some implausible verdicts. I propose an alternative—one which delivers plausible rankings of lotteries, which is implied by a plausible collection of axioms, and which can be applied alongside almost any ranking of infinite worlds. (shrink)

Infinite regress arguments play an important role in many distinct philosophical debates. Yet, exactly how they are to be used to demonstrate anything is a matter of serious controversy. In this paper I take up this metaphilosophical debate, and demonstrate how infinite regress arguments can be used for two different purposes: either they can refute a universally quantified proposition (as the Paradox Theory says), or they can demonstrate that a solution never solves a given problem (as the Failure (...) Theory says). In the meantime, I show that Black’s view on infinite regress arguments (1996, this journal) is incomplete, and how his criticism of Passmore can be countered. (shrink)

In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room (...) for a type of fundamentalism combined with infinite descent as long as this descent is “boring,” that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics. (shrink)

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any such theory. (...) -/- My goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)

This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The (...) chapter challenges this, arguing that we have no reason to think there are infinite sets of different sizes. It begins with an initial argument against Cantor’s claim that there are infinite sets of different sizes and then proceeds, by way of an analogy between Cantor’s mathematical result and Russell’s paradox, to a more direct argument. (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.

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 book on infinite regress arguments provides (i) an up-to-date overview of the literature on the topic, (ii) ready-to-use insights for all domains of philosophy, and (iii) two case studies to illustrate these insights in some detail. Infinite regress arguments play an important role in all domains of philosophy. There are infinite regresses of reasons, obligations, rules, and disputes, and all are supposed to have their own moral. Yet most of them are involved in controversy. Hence the (...) question is: what exactly is an infinite regress argument, and when is such an argument a good one? (shrink)

Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian (...) “infinite book copies” objection, and that this provides an interesting way to defuse some cosmological arguments for the existence of God and to give a non-theistic but complete explanation of the Universe. In the course of my argumentation, I also show that circular explanations can be “self-explanatory” as well: explaining two items by each other can explain the couple of items tout court. (shrink)

How might we extend aggregative moral theories to compare infinite worlds? In particular, how might we extend them to compare worlds with infinite spatial volume, infinite temporal duration, and infinitely many morally valuable phenomena? When doing so, we face various impossibility results from the existing literature. For instance, the view we adopt can endorse the claim that worlds are made better if we increase the value in every region of space and time, or that they are made (...) better if we increase the value obtained by every person. But they cannot endorse both claims, so we must choose. In this paper I show that, if we choose the latter, our view will face serious problems such as generating incomparability in many realistic cases. Opting instead to endorse the first claim, I articulate and defend a spatiotemporal, expansionist view of infinite aggregation. Spatiotemporal views such as this do face some difficulties, but I show that these can be overcome. With modification, they can provide plausible comparisons in cases that we care about most. (shrink)

A collection of aphorisms, fragments, and observations on philosophy and pessimism.Composed of aphorisms, fragments, and observations both philosophical and personal, Eugene Thacker's Infinite Resignation traces the contours of pessimism, caught as it is between a philosophical position and a bad attitude. By turns melancholic, misanthropic, and tinged with gallows humor, Thacker's writing tenuously hovers over that point at which the thought of futility becomes the futility of thought.

We analyze recent criticisms of the use of hyperreal probabilities as expressed by Pruss, Easwaran, Parker, and Williamson. We show that the alleged arbitrariness of hyperreal fields can be avoided by working in the Kanovei–Shelah model or in saturated models. We argue that some of the objections to hyperreal probabilities arise from hidden biases that favor Archimedean models. We discuss the advantage of the hyperreals over transferless fields with infinitesimals. In Paper II we analyze two underdetermination theorems by Pruss and (...) show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined. (shrink)

Infinite Hope, rich with inspirational true stories and breathtaking artwork, will bless you with the kind of hope that never fades and always brightens the darkest paths of life. This is no ordinary hope, but instead a life-transforming hope. It is a hope that will fill you with confidence and inspire you to find peace with yourself and your circumstances. Stories and insights about suffering and the goodness of God, along with illustrations from Joni Eareckson Tada and Jill DeHaan, (...) will give you a richer, deeper love for Christ, the Blessed Hope."--Provided by publisher. (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.

Infinite Phenomenology builds on John Russon’s earlier book, Reading Hegel’s Phenomenology, to offer a second reading of Hegel’s Phenomenology of Spirit. Here again, Russon writes in a lucid, engaging style and, through careful attention to the text and a subtle attunement to the existential questions that haunt human life, he demonstrates how powerfully Hegel’s philosophy can speak to the basic questions of philosophy. In addition to original studies of all the major sections of the Phenomenology, Russon discusses complementary texts (...) by Hegel, namely, the Philosophy of Spirit, the Philosophy of Right, and the Science of Logic. He concludes with an appendix that discusses the reception and appropriation of Hegel’s Phenomenology in twentieth-century French philosophy. As with Russon’s earlier work, Infinite Phenomenology will remain essential reading for those looking to engage Hegel’s essential, yet difficult, text. (shrink)

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon and Rabin and Rabern.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

This paper discusses an infinite regress that looms behind a certain kind of historical explanation. The movement of one barbarian group is often explained by the movement of others, but those movements in turn call for an explanation. While their explanation can again be the movement of yet another group of barbarians, if this sort of explanation does not stop somewhere we are left with an infinite regress of barbarians. While that regress would be vicious, it cannot be (...) accommodated by several general views about what viciousness in infinite regresses amounts to. This example is additional evidence that we should prefer a pluralist approach to infinite regresses. (shrink)

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

From Geoffrey A. Moore, author of Crossing the Chasm, which has sold more than 1 million copies, The Infinite Staircase is a bold new book that combines science and philosophy to answer two fundamental questions for humanity: the metaphysical "where do I fit in the grand scheme of things?" and the ethical "how should I behave?".

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)

Modern East Asian philosophy faced a difficulty in endowing objective knowledge with its adequate location in the traditional Eastern view of mind. This led some philosophers to reconsider intellectual intuition and the relevant question of things themselves from an Eastern perspective, and among them most notably are Nishida Kitarō and Mou Zongsan. Although these philosophers have recently been comparatively studied, their core concepts such as ＂absolute nothing＂ and ＂infinite mind＂ have not been sufficiently discussed from the perspective of objective (...) knowledge. This paper examines the question of grasping things themselves or facts themselves by reconsidering two philosophers' middle works on the theory of knowledge. The result is the synthetic view of their seemingly divergent theories, based on the pursuit of self-awareness and the principle of bidirectional transcendence; according to this principle, the outward transcendence is correlated to the inward or immanent transcendence, so that the knowledge of facts themselves are epistemologically correlated to absolute nothing or infinite mind, which gives foundation to objective knowledge. This view not only explains the meaning of the ＂knowledge＂ of facts themselves, but also strongly implies the epistemic value of the two philosophers' seemingly esoteric concepts of mind and nothingness. (shrink)

This paper responds to commentaries by Kaiserman and Magidor, and Hawthorne. The case of the infinite clowns can teach us several things.

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

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

000000001. Introduction Call a theory of the good—be it moral or prudential—aggregative just in case (1) it recognizes local (or location-relative) goodness, and (2) the goodness of states of affairs is based on some aggregation of local goodness. The locations for local goodness might be points or regions in time, space, or space-time; or they might be people, or states of nature.1 Any method of aggregation is allowed: totaling, averaging, measuring the equality of the distribution, measuring the minimum, etc.. Call (...) a theory of the good finitely additive just in case it is aggregative, and for any finite set of locations it aggregates by adding together the goodness at those locations. Standard versions of total utilitarianism typically invoke finitely additive value theories (with people as locations). A puzzle can arise when finitely additive value theories are applied to cases involving an infinite number of locations (people, times, etc.). Suppose, for example, that temporal locations are the locus of value, and that time is discrete, and has no beginning or end.2 How would a finitely additive theory (e.g., a temporal version of total utilitarianism) judge the following two worlds? Goodness at Locations (e.g. times) w1:..., 2, 2, 2, 2, 2, 2, 2, 2, 2, ..... w2:..., 1, 1, 1, 1, 1, 1, 1, 1, 1, ..... Example 1 At each time w1 contains 2 units of goodness and w2 contains only 1. Intuitively, we claim, if the locations are the same in each world, finitely additive theorists will want to claim that w1 is better than w2. But it's not clear how they could coherently hold this view. For using standard mathematics the sum of each is the same infinity, and so there seems to be no basis for claiming that one is better than the other.3 (Appealing to Cantorian infinities is of no help here, since for any Cantorian infinite N, 2xN=1xN.). (shrink)

I have argued for a picture of decision theory centred on the principle of Rationally Negligible Probabilities. Isaacs argues against this picture on the grounds that it has an untenable implication. I first examine whether my view really has this implication; this involves a discussion of the legitimacy or otherwise of infinite decisions. I then examine whether the implication is really undesirable and conclude that it is not.

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform (...) infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. -/- (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, is unjustified, in general. This means my position is stronger against this objection than the paper suggests, since omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences.). (shrink)

A common argument for atheism runs as follows: God would not create a world worse than other worlds he could have created instead. However, if God exists, he could have created a better world than this one. Therefore, God does not exist. In this paper I challenge the second premise of this argument. I argue that if God exists, our world will continue without end, with God continuing to create value-bearers, and sustaining and perfecting the value-bearers he has already created. (...) Given this, if God exists, our world—considered on the whole—is infinitely valuable. I further contend that this theistic picture makes our world's value unsurpassable. In support of this contention, I consider proposals for how infinitely valuable worlds might be improved upon, focusing on two main ways—adding value-bearers and increasing the value in present value-bearers. I argue that neither of these can improve our world. Depending on how each method is understood, either it would not improve our world, or our world is unsurpassable with respect to it. I conclude by considering the implications of my argument for the problem of evil more generally conceived. (shrink)