We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if κ≤λ, then sup ‖M‖=λ|Sκ|=|)κ. We show that this holds for any abstract elementary class with λ-amalgamation. No such calculation is possible for nonalgebraic types. However, we introduce a subclass of nonalgebraic types for which the same upper bound holds.

Given a countable graph, we say a set A of its vertices is universal if it contains every countable graph as an induced subgraph, and A is weakly universal if it contains every finite graph as an induced subgraph. We show that, for almost every graph on, (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on (...) class='Hi'>will almost surely contain non‐universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families. (shrink)

For a fixed natural number \, the Hart–Shelah example is an abstract elementary class with amalgamation that is categorical exactly in the infinite cardinals less than or equal to \. We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good \-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the (...) way, we develop new tools to build and analyze good frames. (shrink)

When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and ${\mathcal {L}}_{\omega _1, \omega }$ -elementary? We find that these are exactly the classes (...) that can be defined by an infinitary formula that has no infinitary disjunctions. (shrink)

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...) potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

The interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects. Chapter 1. Is a logicist bound to the claim (...) that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined. Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.Abstract prepared by Will Stafford extracted partially from the dissertation. (shrink)

Proof-theoretic validity has proven a useful tool for proof-theoretic semantics, because it explains the harmony found in the introduction and elimination rules for the intuitionistic calculus. However, the demonstration that a rule of proof is proof-theoretically valid requires checking an infinite number of cases, which raises the question of whether proof-theoretic validity is decidable. It is proven here that it is for the most prominent formulations in the literature for propositional logic.

Every set can been thought of as a directed graph whose edge relation is ∈. We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ1 that are distinct (...) up to siblinghood. (shrink)

The two essays included here are parts of a longer study of temporality, and the genesis of the “religious.” The first part, “Multiple Nows,” depicts a universe in which a present to past relation is establishable from any and every point in consciousness. The resulting perspective differs from that offered by the linear timeline of chronological history. Remembering where I put my glasses is an historicizing act, as fully as is remembering when the Battle of Zama was fought or who (...) won there. On this alternate view of temporality the genesis of the historical perspective is the historicizing subject. The second essay, “The History of a House,” places the observer before an historical structure, then asks where the historicity in the structure is. We discover that the historicity is put there by the observer/subject. This discovery resembles our earlier discovery that historicity is generated by an infinite sequence of nows. The two essays converge on a description of historical cognition as subject-generated. (shrink)

The two essays included here are parts of a longer study of temporality, and the genesis of the “religious.” The first part, “Multiple Nows,” depicts a universe in which a present to past relation is establishable from any and every point in consciousness. The resulting perspective differs from that offered by the linear timeline of chronological history. Remembering where I put my glasses is an historicizing act, as fully as is remembering when the Battle of Zama was fought or who (...) won there. On this alternate view of temporality the genesis of the historical perspective is the historicizing subject. The second essay, “The History of a House,” places the observer before an historical structure, then asks where the historicity in the structure is. We discover that the historicity is put there by the observer/subject. This discovery resembles our earlier discovery that historicity is generated by an infinite sequence of nows. The two essays converge on a description of historical cognition as subject-generated. (shrink)

This paper investigates Descartes’s understanding of the imago Dei, that it is above all in virtue of the will that we bear the image and likeness of God. I challenge the key assumption of arguments that hold that Descartes’s comparison between the human will and the divine will is problematic—that in his conception of the imago Dei Descartes is alluding to Scholastic conceptions of analogy available to him at the time, which would place particular constraints on the (...) legitimacy of the comparison. I argue instead that Descartes is evoking a different tradition regarding the nature of image and imitation, stemming from Augustine and Aquinas, and thus, that those constraints do not apply. I then argue that Descartes thinks the likeness between the human will and God’s will is that both are infinite in “extent.” This means that human will can “extend itself” not only to things that can be the object of some other will, but to things that can be the object of God’s will. This is notable because Descartes famously thinks that absolutely anything can be the object of God’s will. I explain why this interpretation is not implausible, contrary to first appearances. (shrink)

We can have credences in an infinite number of propositions—that is, our opinion set can be infinite. Accuracy-first epistemologists have devoted themselves to evaluating credal states with the help of the concept of ‘accuracy’. Unfortunately, under several innocuous assumptions, infinite opinion sets yield several undesirable results, some of which are even fatal, to accuracy-first epistemology. Moreover, accuracy-first epistemologists cannot circumvent these difficulties in any standard way. In this regard, we will suggest a non-standard approach, called a relativistic approach, to (...) accuracy-first epistemology and show that such an approach can successfully circumvent undesirable results while having some advantages over the standard approach. (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)

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.

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)

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)

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!

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)

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)

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)

Once one accepts that certain things metaphysically depend upon, or are metaphysically explained by, other things, it is natural to begin to wonder whether these chains of dependence or explanation must come to an end. This essay surveys the work that has been done on this issue—the issue of grounding and infinite descent. I frame the discussion around two questions: (1) What is infinite descent of ground? and (2) Is infinite descent of ground possible? In addressing the second question, I (...) will consider a number of arguments that have been made for and against the possibility of infinite descent of ground. When relevant, I connect the discussion to two important views about the way reality can be structured by grounding: metaphysical foundationalism and metaphysical infinitism. (shrink)

In this essay, I provide an overview of the debate on infinite and essential idealizations in physics. I will first present two ostensible examples: phase transitions and the Aharonov– Bohm effect. Then, I will describe the literature on the topic as a debate between two positions: Essentialists claim that idealizations are essential or indispensable for scientific accounts of certain physical phenomena, while dispensabilists maintain that idealizations are dispensable from mature scientific theory. I will also identify some attempts (...) at finding a middle ground between the essentialists and dispensabilists camps. Finally, I will raise questions for future research on essential and infinite idealizations via the notion of exploration. (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.

Using infinite time Turing machines we define two successive extensions of Kleene’s ${\mathcal{O}}$ and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call ${\mathcal{O}^{+}}$ —has height equal to the supremum of the writable ordinals, and that the other extension—which we will call ${\mathcal{O}}^{++}$ —has height equal to the supremum of the eventually writable ordinals. Next we prove that ${\mathcal{O}^+}$ is Turing computably isomorphic to the halting problem of infinite time (...) Turing computability, and that ${\mathcal{O}^{++}}$ is Turing computably isomorphic to the halting problem of eventual computability. (shrink)

In his latest book Christopher Bollas uses detailed studies of real clinical practice to illuminate a theory of psychoanalysis which privileges the human impulse to question. From earliest childhood to the end of our lives, we are driven by this impulse in its varying forms, and _The Infinite Question_ illustrates how Freud's free associative method provides both patient and analyst with answers and, in turn, with an ongoing interplay of further questions. At the book's core are transcripts of real analytical (...) sessions, accompanied by parallel commentaries which highlight key aspects of the free associative method in practice. These transcripts are contextualised by further discussion of the cases themselves, as well as a wider theoretical framework which places its emphasis on Freud's theory of the logic of sequence: by learning to listen to this free associative logic, Bollas argues, we can discover a richer and more complex unconscious voice than if we rely solely on Freud's theory of repressed ideas. Bollas demonstrates, in an eloquent and persuasive manner, how the Freudian position of evenly suspended attentiveness enables the analyst's unconscious to catch the drift of the patient's own unconscious. He also shows that to stimulate further questioning is often of more benefit to the analytical process than to jump to an interpretation. Yet whatever fascinating course a session may take, neither the patient nor the analyst can halt the progress of the self-propelling interrogative drive. _The Infinite Question_ will be invaluable to both the new student and the experienced psychoanalyst, read either on its own or as a practice-based extension of the theoretical ideas elaborated in its companion volume, _The Evocative Object World_. (shrink)

According to Leibniz’s infinite-analysis account of contingency, any derivative truth is contingent if and only if it does not admit of a finite proof. Following a tradition that goes back at least as far as Bertrand Russell, several interpreters have been tempted to explain this biconditional in terms of two other principles: first, that a derivative truth is contingent if and only if it contains infinitely complex concepts and, second, that a derivative truth contains infinitely complex concepts if and only (...) if it does not admit of a finite proof. A consequence of this interpretation is that Leibniz’s infinite-analysis account of contingency falls prey to Robert Adams’s Problem of Lucky Proof. I will argue that this interpretation is mistaken and that, once it is properly understood how the idea of an infinite proof fits into Leibniz’s circle of modal notions, the problem of lucky proof simply disappears. (shrink)

In 1991 Mark Nelson argued that if time is infinitely long towards the future, then under certain easily met conditions traditional utilitarianism is unable to discriminate among actions. For under these conditions, each action produces the same infinite amount of utility, and thus it seems that utilitarianism must judge all actions permissible, judge all actions impermissible, or remain completely silent. In response to this criticism of utilitarianism, I argued that utilitarianism had the resources for dealing with at least some cases (...) of infinite utility. More specifically, I defended the following principle as being part of the "spirit" of traditional (total) utilitarianism: -/- PMU*: An action a1 has better consequences than an action a2 if and only if there is a time t such that for any later time t' the cumulative amount of utility produced by a1 up to t' is greater than that produced by action a2 up to t'. -/- For the usual finite cases, this principle agrees exactly with the standard approach of comparing totals. For infinite cases, however, it can distinguish among two actions each of which produces an infinite amount of utility. It says, for example, that an action that produces 2 units of utility at each time has better consequences than an action that produces 1 unit of utility at each time -- even though they both produce the same infinite amount of utility. Of course, cases where one action produces more utility than a second at each time are going to be rather rare. PMU* is not limited to such cases, however; it also has bite in cases such as the following: -/- Time a1: 1, 1, 1, 1, 2, 2, 2, 2, 2, ... a2: 3, 3, 3, 3, 1, 1, 1, 1, 1, ... -/- For although the cumulative utility of a1 up to and including the fourth time is only 4 utiles and a2's is 12, a1 starts catching up after that, ties a2 in the 12th time, and stays ahead after that. Thus, PMU* judges a1 as having better consequences than a2. I claimed that PMU* better captured the "spirit" of traditional utilitarianism than the usual sum total view -- on the grounds that almost anyone inclined to defend traditional utilitarianism would, upon reflection, want to hold, for example, that 2 utiles at every time is indeed better than 1 utile at each time. Two responses to my view have since been given in this journal. Luc Van Liederkerke has argued that PMU* violates an anonymity (neutrality, impartiality) condition that is central to utilitarianism. And James Cain has argued that PMU* is not person-centered, and that being so is central to utilitarianism. Here I will reply to the criticisms. Basically, I will acknowledge that each is right about an important point, but deny that this establishes that the core idea underlying PMU* is less plausible than the standard sum-ranking versions of utilitarianism. (shrink)

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

I will argue for two main points. First, the regress imbedded in infinitism need not be subject to the Structural Objection; and second, the Structural Objection does not pose a real problem for any regress. I will not be arguing for the correctness of my proposal directly. That is, as will become apparent soon, my proposal rests on two principles of reasoning which together entail infinitism and I will not present my arguments for those principles here. (...) The purpose of this paper is to show that the objections raised by Gillett do not pose a grave problem for infinitism. (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)

I will argue for two main points. First, the regress imbedded in infinitism need not be subject to the Structural Objection; and second, the Structural Objection does not pose a real problem for any regress. I will not be arguing for the correctness of my proposal directly. That is, as will become apparent soon, my proposal rests on two principles of reasoning which together entail infinitism and I will not present my arguments for those principles here. (...) The purpose of this paper is to show that the objections raised by Gillett do not pose a grave problem for infinitism. (shrink)

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)

In his article “Directed Duties and Inalienable Rights,” Hillel Steiner advances an argument to show that there cannot be inalienable rights. This “impossibility theorem,” as well as providing an interesting result by itself, could break the theoretical deadlock in the debate between proponents of interest theory, on the one hand, and proponents of will theory, on the other. In this article, I comment on Steiner’s argument, and I try to show why it does not work. I then expound a (...) paradoxical outcome of the Hohfeldian analytical framework and examine how it might be amended to avoid this outcome. (shrink)

Aggregative moral theories face a series of devastating problems when we apply them in a physically realistic setting. According to current physics, our universe is likely _infinitely large_, and will contain infinitely many morally valuable events. But standard aggregative theories are ill-equipped to compare outcomes containing infinite total value so, applied in a realistic setting, they cannot compare any outcomes a real-world agent must ever choose between. This problem has been discussed extensively, and non-standard aggregative theories proposed to overcome (...) it. This paper addresses a further problem of similar severity. Physics tells us that, in our universe, how remotely in time an event occurs is _relative_. But our most promising aggregative theories, designed to compare outcomes containing infinitely many valuable events, are sensitive to how remote in time those events are. As I show, the evaluations of those theories are then relative too. But this is absurd; evaluations of outcomes must be absolute. So we must reject such theories. Is this objection fatal for all aggregative theories, at least in a relativistic universe like ours? I demonstrate here that, by further modifying these theories to fit with the physics, we can overcome it. (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)

To reach the conclusion that the universe is infinite, physicists (a) make some observations; (b) fit those observations to some mathematical model; (c) find that the neatest model that accommodates the data extrapolates to an infinite universe; (d) conclude that the universe is infinite. In my presentation I will examine the logic by which physicists reach this conclusion. Specifically, I will show that there is no way to empirically justify the move from (b) to (c). An infinite universe (...) should therefore properly be viewed as a metaphysical hypothesis consistent with certain physical theories but hardly mandated by them. By contrast, I will argue that the hypothesis of intelligent design—that a designing intelligence has left clear marks of intelligence in the biophysical universe—is not a metaphysical hypothesis at all but a fully scientific one. In particular, I will argue that whereas an infinite universe does not (and indeed cannot) admit empirical evidence, intelligent design can. Finally, I will indicate why an infinite universe, though sometimes introduced to get around the problem of design, in fact cannot get around it. (shrink)

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

The single greatest challenge facing managers in the developed countries of the world is to raise the productivity of knowledge and service workers. This challenge, which will dominate the management agenda for the next several decades, will ultimately determine the competitive performance of companies. Even more important, it will determine the very fabric of society and the quality of life of every industrialized nation. … Unless this challenge is met, the developed world will face increasing social (...) tensions, increasing polarization, increasing radicalization, possibly even class war. (shrink)

Basarab Nicolescu’s book What is Reality? is an attempt to enlighten the contradictory logic promoted by Stéphane Lupasco. However, it also proves to be the solid ground necessary for the axiomatic crystallization of the transgressive-integratory view of the world, a vision based on transdisciplinary thought. For this reason our text will focus not only on the connection between the Included Third and the Hidden Third but also on the “fusion between horizons” in an Infinite Third, one which belongs to (...) Meaning. (shrink)

It is sometimes argued that contemporary media technologies push individuals into collective action on an industrial scale, without them necessarily being aware of it. Yet what if the problem is not that we are all synchronized to the same affective networks and moments, but rather dispersed into countless different networks and moments? What if the effect of so-called social media is to calibrate the interactive spectacle so that we never fully feel the same way as other potential allies at the (...) same moment? While one person is fuming about economic injustice or climate change denial, another is giggling at a cute cat video, and, two hours late, vice versa. The nebulous indignation which constitutes the very fuel of true social change can be redirected safely around the network, avoiding any dangerous surges of radical activity In this short and provocative book Dominic Pettman examines the deliberate deployment of what he calls ‘asynchronous hypermodulation’, as a key strategy encoded into the contemporary media environment. By focusing on ‘we, the mediated’, this radical intervention challenges the narratives that portray the new media environment as a monolithic media apparatus, rendering us all into the same libidinal grey glue, subsequently channelled at will through the pleasure palace of the internet. The unprecedented power of the Internet requires us to rethink the potential for infinite distractions that accompany it. (shrink)

The case for longtermism depends on the vast potential scale of the future. But that same vastness also threatens to undermine the case for longtermism: If the universe as a whole, or the future in particular, contain infinite quantities of value and/or disvalue, then many of the theories of value that support longtermism (e.g., risk-neutral total utilitarianism) seem to imply that none of our available options are better than any other. If so, then even apparently vast effects on the far (...) future cannot in fact make the world morally better. On top of this, some strategies for avoiding this problem of “infinitarian paralysis” (e.g., exponential pure time discounting) yield views that are much less supportive of longtermism. In this chapter, we explore how the potential infinitude of the future affects the case for longtermism. We argue that (i) there are reasonable prospects for extending risk-neutral totalism and similar views to infinite contexts and (ii) many such extension strategies will still support the case for longtermism, since they imply that when we can only effect (or only predictably affect) a finite, bounded part of an infinite universe, we can ignore the unaffectable rest of the universe and reason as if the finite, affectable part were all there is. (shrink)

It has been suggested that the measuring apparatus used to measure quantum systems ought to be idealized as consisting of an infinite number of quantum systems. Let us call this the infinity assumption. The suggestion that we ought to make the infinity assumption has been made in connection with two closely related but distinct problems. One is the problem of determining the importance of the limitations on measurement incorporated into the Wigner-Araki-Yanase quantum theory of measurement. The other is the measurement (...) problem. These problems are not internal to the quantum mechanical formalism. They arise when we obtain particular formal results in applying the quantum mechanical formalism to measurement interactions. The formal results are problematic because they resist explanation. Various authors have claimed that by making the infinity assumption we can neglect the limitations on measurement. Bub (1988) and (1989) claims it allows us to solve the measurement problem. I will argue that both claims are unjustified. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

The best accuracy arguments for probabilism apply only to credence functions with finite domains, that is, credence functions that assign credence to at most finitely many propositions. This is a significant limitation. It reveals that the support for the accuracy-first program in epistemology is a lot weaker than it seems at first glance, and it means that accuracy arguments cannot yet accomplish everything that their competitors, the pragmatic (Dutch book) arguments, can. In this paper, I investigate the extent to which (...) this limitation can be overcome. Building on the best arguments in finite domains, I present two accuracy arguments for probabilism that are perfectly general—they apply to credence functions with arbitrary domains. I then discuss how the arguments’ premises can be challenged. We will see that it is particularly difficult to characterize admissible accuracy measures in infinite domains. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

You would be hard-pressed to find someone who categorically opposes protecting the environment, yet most people would agree that the environmentalist movement has been ineffectual and even misguided. Some argue that its agenda is misplaced, oppressive, and misanthropic—a precursor to intrusive government, regulatory bungles, and economic stagnation. Others point out that its alarmist rhetoric and preservationist solutions are outdated and insufficient to the task of galvanizing support for true reform. In this impassioned and judicious work, R. Bruce Hull argues that (...) environmentalism will never achieve its goals unless it sheds its fundamentalist logic. The movement is too bound up in polarizing ideologies that pit humans against nature, conservation against development, and government regulation against economic growth. Only when we acknowledge the infinite perspectives on how people should relate to nature will we forge solutions that are respectful to both humanity and the environment. Infinite Nature explores some of these myriad perspectives, from the scientific understandings proffered by anthropology, evolution, and ecology, to the promise of environmental responsibility offered by technology and economics, to the designs of nature envisioned in philosophy, law, and religion. Along the way, Hull maintains that the idea of nature is social: in order to reach the common ground where sustainable and thriving communities are possible, we must accept that many natures can and do exist. Incisive, heartfelt, and brimming with practical solutions, Infinite Nature brings a much-needed and refreshing voice to the table of environmental reform. (shrink)

Recently, several authors have explored the connections between NP-complete problems for finite objects and the complexity of their analogs for infinite objects. In this paper, we will categorize infinite versions of several problems arising from finite complexity theory in terms of their recursion theoretic complexity and proof theoretic strength. These infinite analogs can behave in a variety of unexpected ways.

Pulier (2000, Theory and Decision 49: 291) and Machina (2000, Theory and Decision 49: 293) seek to dissolve the Barrett–Arntzenius infinite decision puzzle (1999, Theory and Decision 46: 101). The proposed dissolutions, however, are based on misunderstandings concerning how the puzzle works and the nature of supertasks more generally. We will describe the puzzle in a simplified form, address the recent misunderstandings, and describe possible morals for decision theory.

Did God create life? Or did life arise via naturalistic processes, along the lines of random mutation and natural selection as suggested by Darwin? Intelligent design proponents attempt to use William Dembski’s design inference to argue that the existence of intelligent life is due to design. I will argue that the design inference is flawed, because it does not take into account the fact that the universe is spatially infinite.

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