In this reply to Tim Bayne and Yujin Nagasawa, I defend the possibility of a maximal-excellence account of the grounding of the obligation to worship God. I do not offer my own account of the obligation to worship God; rather I argue that the major criticism fails. Thus maximal-excellence can ground an obligation to worship God.

In some games, the products of the probabilities times the payouts result in a series that is conditionally convergent, which means that the sum can vary based on the order in which the products are summed. The purpose of this paper is to address the question: How should such games be valued? We first show that, contrary to widespread belief, summing in the order determined by the mechanism of the game does not lead to the correct value. We then consider (...) other attempts to value such games. We suggest that games with conditionally convergent series are properly valued by summing the products in increasing order of the absolute values of the game’s payouts. (shrink)

In ’The Coherence of Theism’ Richard Swinburne writes that a person cannot be omniscient and perfectly free. In ’The Existence of God’ Swinburne writes that God is a person who is omniscient and perfectly free. There is a straightforward reason why the two passages are not in tension, but recognition of this reason raises a problem for Swinburne’s argument in ’The Existence of God’ (the conclusion of which is that God likely exists). In this paper I present the problem for (...) Swinburne’s argument. I then consider two potential responses and suggest that neither succeeds. (shrink)

In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual infinite. I (...) suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite. (shrink)

Richard Swinburne’s argument in The Existence of God discusses many probabilities, ultimately concluding that God probably exists. Swinburne gives exact values to almost none of these probabilities. I attempted to assign values to the probabilities that met that weak condition that they could be correct. In this paper, I first present a brief outline of Swinburne’s argument in The Existence of God. I then present the problems I encountered in Swinburne’s argument, specifically problems that interfered with my attempt to arrive (...) at values for the probabilities discussed by Swinburne. Finally, I suggest that Swinburne’s argument would be more persuasive if he assigned exact values to his probabilities. (shrink)

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of the non-Cantorian outlook.

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted (...) by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that (...) the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. (shrink)

Swinburne relies on principle P in The Existence of God to argue that God is simple and thus likely to exist. In this paper, I argue that Swinburne does not support P. In particular, his arguments from mathematical simplicity and scientists’ preferences both fail. Given the central role P plays in Swinburne’s overall argument in The Existence of God , I conclude that Swinburne should further support P if his argument that God likely exists is to be persuasive.

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second conception of (...) infinite number is the correct one, and analyze what this means for multiverses. (shrink)

In ‘Infinitism Regained’, Jeanne Peijnenburg argues for a version of infinitism wherein ‘beliefs may be justified by an infinite chain of reasons that can be actually completed’. I argue that Peijnenburg has not successfully argued for this claim, but rather has shown that certain infinite series can be computed.

In this brief reply to Stephen Davis, I argue that Davis’s separationist position, wherein those who remain eternally apart from God do so by choice, is internally contradictory in that it leads to universalism.

The purpose of this paper is to point the way to the money tree. Currently, almost all investment professionals think that outperformance requires an “edge,” that is, the ability to predict the future to some degree. In this paper, I suggest that money can be made in a 0, or even slightly negative, expected value environment by carefully choosing investment/bet sizes. Philosophical considerations are found mainly in Section 4.

In a recent paper, Sylvia Wenmackers and Leon Horsten discuss how the concept of a fair infinite lottery can best be extended to denumerably infinite lotteries. Their paper uses and builds on the alpha-theory of Vieri Benci and Mauro Di Nasso. The purpose of this paper is to demonstrate a potential problem for alpha-theory.

When two omnipotent beings are randomly and sequentially selecting positive integers, the being who selects second is almost certain to select a larger number. I then use the relativity of simultaneity to create a paradox by having omnipotent beings select positive integers in different orders for different observers.

The purpose of this paper is to suggest that we are in the midst of a Cantorian bubble, just as, for example, there was a dot com bubble in the late 1990’s.

In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. I go on to present one reason why this conclusion (that supertasks are (...) impossible) is important, namely that it implies a new verdict on a decision puzzle propounded by Jeffrey Barrett and Frank Arntzenius. (shrink)

In “How an Unsurpassable Being can Create a Surpassable World,” Daniel and Frances Howard-Snyder employ a fascinating thought experiment in anattempt to show that a morally unsurpassable being can create a surpassable world. Imagine that for each positive integer there is a world that a good,omnipotent, omniscient being can create. Jove randomly selects a number and creates the corresponding world; Thor simply creates world 888. The Howard-Snyders argue that it is logically possible that Jove is morally unsurpassable. William Rowe counters (...) that Thor morally surpasses Jove, thus contradictingthe claim that Jove is morally unsurpassable. Does either Jove or Thor morally surpass the other? How do their strategies compare? Could a morally unsurpassable being employ Jove’s strategy? The purpose of this paper is to answer these questions. (shrink)

In this paper, two concepts of completing an infinite number of tasks are considered. After discussing supertasks, equisupertasks are introduced. I suggest that equisupertasks are logically possible.

The purpose of this paper is to dissolve paradoxes of the infinite by correctly identifying the infinite natural numbers.

In ‘The Train Paradox’, I argued that sequential random selections from the natural numbers would grow through time. I used this claim to present a paradox. In response to this proposed paradox, Jon Pérez Laraudogoitia has argued that random selections from the natural numbers do not grow through time. In this paper, I defend and expand on the argument that random selections from the natural numbers grow through time. I also situate this growth of random selections in the context of (...) my overall work on infinite number, which involves two main claims: 1) infinite numbers, properly understood, are the infinite natural numbers in a nonstandard model of the reals, and 2) ω is potentially infinite. (shrink)

In "Religion and Scientific Method," George Schlesinger presented a strikingly original theodicy. In this paper, I explain the strategy underlying Schlesinger's argument. I then present a parallel argument to indicate the weakness of Schlesinger's theodicy. Finally, I show that Schlesinger's theodicy assumes a false principle, and therefore fails.

In this reply to Jesse Steinberg’s ‘God and the possibility of random creation’, I suggest a procedure whereby a being such as God could randomly select a number from an infinite set.

Steven M. Cahn presented a fascinating and recalcitrant puzzle. Imagine that you own stock in a company that has undertaken an immoral course of action. Most people believe that selling the stock is the right thing to do. But if you sell the stock, you thereby abet the buyer in an immoral course of action, namely the buyer’s coming to own tainted stock. Is principled divestiture possible? If so, what are the moral grounds on which principled divestiture stands? These questions (...) have proven difficult. Cahn answered many attempted replies to his puzzle. In this journal, Saul Smilansky also attempted to answer the divestiture puzzle. Smilansky concluded, “Attempting to divest by offering the stock for sale is, however, morally problematic. It is problematic because it involves the bad faith of putting something forward as morally acceptable while claiming it to be morally tainted.” Smilansky continued, “One needs to get rid of the stock and take it off the market at the same time, i.e., make it impossible for others to purchase the stock.” Smilansky thereby implies that one should destroy the stock. However, Cahn had already replied to this move, noting that destroying one’s stock would increase the moral culpability of other owners. (shrink)

Consider a subset, S, of the positive integers. What is the probability of selecting a number in S, assuming that each positive integer has an equal chance of selection? The purpose of this short paper is to provide an answer to this question. I also suggest that the answer allows us to determine the relative sizes of two subsets of the positive integers.

It is commonly believed that when a finite value is received in a game that has an infinite expected value, it is in one’s interest to redo the game. We have argued against this belief, at least in the repeated St Petersburg two-envelope case. We also show a case where repeatedly opting for a higher expected value leads to a worse outcome.

Swinburne relies on principle P in ’The Existence of God’ to argue that God is simple and thus likely to exist. In this paper, I argue that Swinburne does not support P. In particular, his arguments from mathematical simplicity and scientists’ preferences both fail. Given the central role P plays in Swinburne’s overall argument in ’The Existence of God’, I conclude that Swinburne should further support P if his argument that God likely exists is to be persuasive.

In this brief paper, I present a puzzle for consideration.

Recently Timothy Williamson asked ‘How probable is an infinite sequence of heads?’ In this paper, I suggest the probability of an infinite sequence of heads.

Jeremy Gwiazda made two criticisms of my formulation in terms of Bayes’s theorem of my probabilistic argument for the existence of God. The first criticism depends on his assumption that I claim that the intrinsic probabilities of all propositions depend almost entirely on their simplicity; however, my claim is that that holds only insofar as those propositions are explanatory hypotheses. The second criticism depends on a claim that the intrinsic probabilities of exclusive and exhaustive explanatory hypotheses of a (...) phenomenon must sum to 1; however it is only those probabilities plus the intrinsic probability of the non-occurrence of the phenomenon which must sum to 1. (shrink)

In an earlier paper I argued that there are cases in which an infinite probabilistic chain can be completed. According to Jeremy Gwiazda, however, I have merely shown that the chain in question can be computed, not that it can be completed. Gwiazda thereby discriminates between two terms that I used as synonyms. In the present paper I discuss to what extent computability and completability can be meaningfully distinguished.

Jeremy Gwiazda has criticized my claim that God, understood as an omnipotent, omniscient, and perfectly free person is a person ’of the simplest possible kind’ on the grounds that omnipotence, etc., as spelled out by me are omnipotence, etc., of restricted kinds, and so less simple forms of these properties than maximal forms would be. However, the account which I gave of these properties in ’The Christian God’ (although not in ’The Coherence of Theism’) shows that, when they (...) are defined in certain ways, they all follow from one property of ’pure, limitless, intentional power’. I argue here that a person who has these properties so defined is a person ’of the simplest possible kind’. (shrink)

Jeremy Gwiazda has criticized my claim that God, understood as an omnipotent, omniscient, and perfectly free person is a person ‘of the simplest possible kind’ on the grounds that omnipotence etc. as spelled out by me are omnipotence etc. of restricted kinds, and so less simple forms of these properties than maximal forms would be. However the account which I gave of these properties in The Christian God shows that, when they are defined in certain ways, they all (...) follow from one property of ‘pure, limitless, intentional power’. I argue here that a person who has these properties so defined is a person ‘of the simplest possible kind’. (shrink)

Recently, in this journal, Jeremy Gwiazda has offered a critique of our separationist view of hell. His objection relies on two key assumptions, and we show in our reply that both assumptions can be denied.

Bayne and Nagasawa have argued that the properties traditionally attributed to God provide an insufficient grounding for the obligation to worship God. They do so partly because the same properties, when possessed in lesser quantities by human beings, do not give rise to similar obligations. In a recent paper, Jeremy Gwiazda challenges this line of argument. He does so because it neglects the possible existence of a threshold obligation to worship, i.e. an obligation that only kicks in when (...) the value of a parameter has crossed a certain threshold. This article argues that there is a serious flaw in Gwiazda’s proposal. Although thresholds may play an important part in how we think about our obligations, their function is distinct from that envisaged by Gwiazda. To be precise, this article argues that thresholds are only relevant to obligations to the extent that they transform a pre-existing imperfect obligation or act of supererogation into a perfect obligation. Since it is not clear that there is an imperfect obligation to worship any being, and indeed since on a certain conception of moral agency it is highly unlikely that there could be, the search for a rational basis for the obligation to worship must continue. (shrink)

Principles of legislation.--Principles of the civil code.--Principles of the penal code.

Jeremy Sherman distills Terrence Deacon's breakthrough natural science hypothesis for the emergence of agents and agency, selves and aims in an otherwise aimless universe. The theory cuts a new path through the dualistic spirit vs. mechanism debate, unifying the hard and soft sciences and suggesting new solutions to philosophical mysteries.

We characterize a type of functional explanation that addresses why a homologous trait originating deep in the evolutionary history of a group remains widespread and largely unchanged across the group’s lineages. We argue that biologists regularly provide this type of explanation when they attribute conserved functions to phenotypic and genetic traits. The concept of conserved function applies broadly to many biological domains, and we illustrate its importance using examples of molecular sequence alignments at the intersection of evolution and cell biology. (...) We use these examples to show how the study of conserved functions can integrate knowledge of a trait’s causal effects on fitness and its history of natural selection without invoking adaptation. We also show how conserved function provides a novel basis for addressing objections against evolutionary functions raised by Robert Cummins. (shrink)

Although the pedagogy literature indicates significant relationships between cheating intentions and both personal and situational factors, no published research has examined the joint effect of personal moral philosophy and perceived moral intensity components on students’ cheating intentions. Hence, a structural equation model that relates magnitude of consequences, relativism, and idealism to willingness to cheat, is developed and tested. Using data from undergraduate business students, the empirical results provide insight into these relationships and evidence of mediation for magnitude of consequences on (...) idealism and students’ cheating intentions. Implications for educators are offered. (shrink)

In considering the debate about the meaning of ‘disease’, the positions are generally presented as falling into two categories: naturalist, e.g., Boorse, and normativist, e.g., Engelhardt and many others. This division is too coarse, and obscures much of what is going on in this debate. I therefore propose that accounts of the meaning of ‘disease’ be assessed according to Hare’s (1997) taxonomy of evaluative terms. Such an analysis will allow us to better understand both individual positions and their inter-relationships. Most (...) importantly, it will show that it is unlikely that there is a single unique disease-concept at issue. Rather, different authors are, for the most part, considering different concepts. (shrink)

Many philosophers think that free will requires alternative possibilities. Other philosophers deny this. There are plenty of philosophical arguments on both sides of this debate, but here I want to highlight various theological pressures that might push Christians into rejecting the principle of alternative possibilities. In this paper, I explore six cases that might push Christians in that direction: the case of divine foreknowledge, the case of prophecy, the case of the blessed in heaven, the case of Christ's human freedom, (...) the case of Mary's fiat in light of her immaculate conception, and the case of prayers for the past. As I will argue, in each of these cases, given certain other standard theological commitments, it seems that Christians are pushed to admit that the agent in question does indeed act freely but also that he or she did not possess alternative possibilities at the moment of decision. (shrink)

A rigorous, authoritative new anthology which brings together some of the most significant contemporary scholarship on the theory of knowledge Carefully-calibrated and judiciously-curated, this strong and contemporary new anthology builds upon Epistemology: An Anthology, Second Edition (Wiley Blackwell, 2008) by drawing a concise and well-balanced selection of higher-level readings from a large, diverse, and evolving body of research. Includes 17 readings that represent a broad and vital part of contemporary epistemology, including articles by female philosophers and emerging thought leaders Organized (...) into seven thoughtful and distinct sections, including virtue epistemology, practical reasons for belief, and epistemic dysfunctions among others Designed to sit alongside the highly-successful anthology of canonical essays, Epistemology: An Anthology, Second Edition (Wiley Blackwell, 2008) Edited by a distinguished editorial team, including Ernie Sosa, one of the most influential active epistemologists Highlights cutting edge methodologies and contemporary topics for advanced students, instructors, and researchers. (shrink)

"Arguing that debates over legitimacy, political violence, freedom, and justice would benefit greatly from cross-tradition theorizing, this book shows how putting analytic and continental political theory in conversation would help us to overcome these intractable problems"--.

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms (...) from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns. (shrink)

We have two aims. The main one is to expound the idea of renormalization in quantum field theory, with no technical prerequisites. Our motivation is that renormalization is undoubtedly one of the great ideas—and great successes--of twentieth-century physics. Also it has strongly influenced in diverse ways, how physicists conceive of physical theories. So it is of considerable philosophical interest. Second, we will briefly relate renormalization to Ernest Nagel's account of inter-theoretic relations, especially reduction. One theme will be a contrast between (...) two approaches to renormalization. The old approach, which prevailed from ca. 1945 to 1970, treated renormalizability as a necessary condition for being an acceptable quantum field theory. On this approach, it is a piece of great good fortune that high energy physicists can formulate renormalizable quantum field theories that are so empirically successful. But the new approach to renormalization explains why the phenomena we see, at the energies we can access in our particle accelerators, are described by a renormalizable quantum field theory. For whatever non-renormalizable interactions may occur at yet higher energies, they are insignificant at accessible energies. Thus the new approach explains why our best fundamental theories have a feature, viz. renormalizability, which the old approach treated as a selection principle for theories. That is worth saying since philosophers tend to think of scientific explanation as only explaining an individual event, or perhaps a single law, or at most deducing one theory as a special case of another. Here we see a framework in which there is a space of theories. And this framework is powerful enough to deduce that what seemed “manna from heaven” is to be expected: the good fortune is generic. We also maintain that universality, a concept stressed in renormalization theory, is essentially the familiar philosophical idea of multiple realizability; and that it causes no problems for reductions of a Nagelian kind. (shrink)

This essay develops a framework for understanding what I call the ethics of reflexivity, that is, the norms that govern attitudes and actions with respect to one’s own worth. I distinguish five central aspects of the reflexive commitment to living in accordance with one’s personal ideals: the extent to which and manner in which one regards oneself from an evaluative point of view, the extent to which one cares about receiving the respect of others, the degree to which one interprets (...) one’s personal ideals in an individualistic or collective manner, the degree to which one’s commitment to living in accordance with one’s personal ideals is rigid or flexible, and the worthiness of one’s personal ideals. This framework, I argue, illuminates the nature and moral significance of virtuous and vicious forms of the character trait of pride. (shrink)

One of the most important philosophical topics in the early twentieth century and a topic that was seminal in the emergence of analytic philosophy was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of (...) logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant. (shrink)

As our civilization careens toward climate breakdown, ecological destruction, and gaping inequality, people are losing their existential moorings. The dominant worldview of disconnection, which tells us we are split between mind and body, separate from each other, and at odds with the natural world, has been invalidated by modern science. Award-winning author, Jeremy Lent, investigates humanity's age-old questions -- Who am I? Why am I? How should I live? -- from a fresh perspective, weaving together findings from modern systems (...) thinking, evolutionary biology, and cognitive neuroscience with insights from Buddhism, Taoism, and Indigenous wisdom. The result is a breathtaking accomplishment: a rich, coherent worldview based on a deep recognition of connectedness within ourselves, between each other, and with the entire natural world. It offers a compelling foundation for a new philosophical framework that could enable humanity to thrive sustainably on a flourishing Earth. The Web of Meaning is for everyone looking for deep and coherent answers to the crisis of civilization. -- Amazon. (shrink)