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)

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

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)

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)

According to Johansson (2009: 22) an infinite regress is vicious just in case “what comes first [in the regress-order] is for its definition dependent on what comes afterwards.” Given a few qualifications (to be spelled out below (section 3)), I agree. Again according to Johansson (ibid.), one of the consequences of accepting this way of distinguishing vicious from benign regresses is that the so-called Russellian Resemblance Regress (RRR), if generated in a one-category trope-theoretical framework, is vicious and that, therefore, (...) the existence of tropes only makes sense if trope-theory is understood (minimally) as a two-category theory which accepts, besides the existence of tropes, also the existence of at least one universal: resemblance.1 I disagree. But how can that be? How can Johansson and I agree about what distinguishes a vicious from a benign regress, yet disagree about which regresses are vicious and which are benign? In this paper I attempt to answer that question by first setting out and defending the sense of viciousness which both Johansson and I accept, only to then argue that to be able to determine if a particular regress is vicious in this sense, more than features intrinsic to the regress itself must be taken into account. This is why, although the RRR as originally set out by Russell is vicious, the seemingly identical resemblance regress which ensues in a one-category (standard) trope-theoretical context is not (provided, that is, that we accept certain views about how the nature of tropes relates to the resemblance between tropes, and given that we set our theory in a truthmaker theoretical framework – all of which are standard assumptions for proponents of (the standard-version of) the trope-theory).2. (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)

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)

We pose and resolve several vexing decision theoretic puzzles. Some are variants of existing puzzles, such as 'Trumped' (Arntzenius and McCarthy 1997), 'Rouble trouble' (Arntzenius and Barrett 1999), 'The airtight Dutch book' (McGee 1999), and 'The two envelopes puzzle' (Broome 1995). Others are new. A unified resolution of the puzzles shows that Dutch book arguments have no force in infinite cases. It thereby provides evidence that reasonable utility functions may be unbounded and that reasonable credence functions need not be (...) countably additive. The resolution also shows that when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin. Finally, the resolution reveals a new way in which the ability to make binding commitments can save perfectly rational agents from sure losses. (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.

This paper uses a schema for infinite regress arguments to provide a solution to the problem of the infinite regress of justification. The solution turns on the falsity of two claims: that a belief is justified only if some belief is a reason for it, and that the reason relation is transitive.

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)

We tell a story where an agent who chooses in such a way as to make the greatest possible profit on each of an infinite series of transactions ends up worse off than an agent who chooses in such a way as to make the least possible profit on each transaction. That is, contrary to what one might suppose, it is not necessarily rational always to choose the option that yields the greatest possible profit on each transaction.

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)

We describe in some detail how to build an infinite computing machine within a continuous Newtonian universe. The relevance of our construction to the Church-Turing thesis and the Platonist-Intuitionist debate about the nature of mathematics is also discussed.

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)

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.

Suppose we wish to decide which of a pair of actions has better consequences in a case in which both actions result in infinite utility. Peter Vallentyne and others have proposed that one action has better consequences than a second if there is a time after which the cumulative utility of the first action always outstrips the cumulative utility of the second. I argue against this principle, in particular I show how cases may arise in which up to any (...) point of time action a1 produces more utility than action a2, yet for each individual involved a2 produces more utility. (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!

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.

Wolff’s relation to Leibniz and Kant’s relation to both are notoriously vexed questions. First, this paper argues that Wolff’s most serious departure from Leibniz consists in his (so far overlooked) rejection of the latter’s infinitism. Second, it contends that the controversies that surrounded Wolff’s early acceptance of infinite causal regress and prompted his conversion to finitism played a prominent role in shaping the theses of Kant’s Antinomies. Whereas Leibniz and the early Wolff considered infinite regress to provide support (...) for the contingency of the world and the existence of God, Wolff’s enemies condemned it as Spinozistic. After Wolff, the claim that an infinite chain of causes is impossible became the standard view among both Wolffian and anti-Wolffian metaphysicians. (shrink)

It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain the same (...) agents and some agents are better off in the first world than they are in the second and no agents are worse off than they are in the second, then the first world is better than the second. I show that if we accept four axioms – the Pareto principle, transitivity, an axiom stating that populations of worlds can be permuted, and the claim that if the ‘at least as good as’ relation holds between two worlds then it holds between qualitative duplicates of this world pair – then we must conclude that there is ubiquitous incomparability between infinite worlds. I show that this is true even if the populations of infinite worlds are disjoint or overlapping, and that we cannot use any qualitative properties of world pairs to rank these worlds. Finally, I argue that this incomparability result generates puzzles for both consequentialist and non-consequentialist theories of objective and subjective permissibility. (shrink)

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

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.

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.

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)

'Chains of Being' argues that there can be infinite chains of dependence or grounding. Cameron also defends the view that there can be circular relations of ontological dependence or grounding, and uses these claims to explore issues in logic and ontology.

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)

One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on (...) the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (shrink)

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.

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)

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)

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.

Drawing upon the practice of caregiving and the insights of feminist care ethics, I offer a phenomenology of caregiving through the work of Eva Feder Kittay and Emmanuel Lévinas. I argue that caregiving is a material dialectic of embodied response involving moments of leveling, attention, and interruption. In this light, the Levinasian opposition between responding to another's singularity and leveling it via parity-based principles is belied in the experience of care. Contra much of response ethics’ and care ethics’ respective literatures, (...) this dialectic suggests that they are complementary in ways that productively illuminate themes of each. I conclude by suggesting that when response and care ethics are thought together through the experience of caregiving, such labors produce finite responsibility with infinite hope. (shrink)

We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time (...) generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely $\Delta^1_2$ reductions, and hence to the infinite time computable reductions. (shrink)

In this dissertation we discuss infinite regress arguments from both a historical and a logical perspective. Throughout we deal with arguments drawn from various areas of philosophy. ;We first consider the regress generating portion of the argument. We find two main ways in which infinite regresses can be developed. The first generates a regress by defining a relation that holds between objects of some kind. An example of such a regress is the causal regress used in some versions (...) of the Cosmological Argument. The second sort of regress is generated by defining a certain procedure that can be repeated indefinitely many times. This will often be a procedure of analysis or of explanation. ;We then argue that for purposes of discussing regressive viciousness it is helpful to divide regresses into two main types. One type raises primarily epistemological issues. In this class we find the regress of justification often used in foundational arguments in epistemology. We also discuss certain arguments that treat theories of predication as attempts to explain predication. Our other class contains those regresses that raise primarily metaphysical problems. This class includes the causal regress as well as those arguments concerned with predication as an ontological problem. ;We agree with some philosophers that one problem that arises in the case of vicious regression is circularity. We show how this problem can be generalized to include other apparently non-circular regresses. The other main problem in epistemological regresses arises as a result of certain conditions that are placed on the relation involved. We look in detail at the epistemological justification regress to illustrate this last point. ;With respect to metaphysical regresses we again find several types of viciousness arguments. One problem is whether the infinite collection required for the regress is even possible. Another problem is whether an infinite number of events can occur in a finite time. These we deal with only briefly. The main difficulty with these regresses, considered as regresses and not just as infinite sets, again turns out to be certain conditions built into the relation involved. We focus on some arguments about causal regression to illustrate this point. ;Finally, we find no absolutely general conditions for regressive viciousness. There is, however, a certain limited range of ways in which a regress may be vicious. We make these ways explicit. (shrink)

Infinite idealizations appear in our best scientific explanations of phase transitions. This is thought by some to be paradoxical. In this paper I connect the existing literature on the phase transition paradox to work on the concept of indispensability, which arises in discussions of realism and anti-realism within the philosophy of science and the philosophy of mathematics. I formulate a version of the phase transition paradox based on the idea that infinite idealizations are explanatorily indispensable to our best (...) science, and so ought to attract a realist attitude. I go on to offer a solution to the paradox by drawing a distinction between two types of indispensability: constructive and substantive indispensability. I argue that infinite idealizations are constructively indispensable to explanations of phase transitions, but not substantively indispensable. This helps to resolve the paradox, I maintain, since realist commitment tracks substantive, and not constructive, indispensability. (shrink)