In an attempt to learn from COVID-19, this essay features six responses to the question: what did COVID-19 teach us, expose in us, or purge out of us when it comes to spiritual formation in Christ? Each response was written independently of the others by one of the coauthors. Diane J. Chandler focuses in on how COVID-19 exposed grievous inequities for ethnic groups in the American church and broader society. Kelly M. Kapic reminds us of the goodness of human finitude (...) and how COVID restrictions have forced many of us to embrace our limitations. Siang-Yang Tan reflects on eight lessons he has learned during this pandemic year in his role shepherding a local church. James C. Wilhoit calls us to consider the structures that are needed for local church leadership to make wise and godly decisions in times of crisis. Richard Peace draws our attention to what might be learned from the forced monasticism brought about by COVID-19 quarantines. Finally, Ruth Haley Barton pauses to consider the interdependence of human life that has been dramatically illustrated by this pandemic. While these six responses certainly do not exhaust all there is for the church to learn from COVID, we present them in the spirit of “O Lord, teach us what we do not see” and hope they will inspire your own reflections. (shrink)

Summary Popular science journalism flourished in the 1860s in England, with many new journals being projected. The time was ripe, Victorian men of science believed, for an ?organ of science? to provide a means of communication between specialties, and between men of science and the public. New formats were tried as new purposes emerged. Popular science journalism became less recreational and educational. Editorial commentary and reviewing the progress of science became more important. The analysis here emphasizes those aspects of popular (...) science which have been identified by Frank Turner as ?public science? and by Thomas Gieryn as ?boundary-work?. The religious, intellectual, and utilitarian values claimed for science by editors and contributors in their tasks of persuading the public to support science and of distinguishing science from what they often called ?applied science? are discussed. These values are shown to vary among editors and, for the editors examined here, Shirley Hibberd, Henry Slack, James Samuelson, William Crookes, and Henry Lawson, to differ significantly from those of T. H. Huxley, John Tyndall, and Norman Lockyer, on whom much study of the popularization of science in the 1860s has focused. (shrink)

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...) to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions. (shrink)

Let's suppose you think that there are no uncountable sets. Have you adopted a restrictive position? It is certainly tempting to say yes---you've prohibited the existence of certain kinds of large set. This paper argues that this intuition can be challenged. Instead, I argue that there are some considerations based on a formal notion of restrictiveness which suggest that it is restrictive to hold that there are uncountable sets.

Readers of this book are sure to view the ethics of incentives in a new light.

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating an account of numbers as properties of collections, work on the philosophy of absences, and recent (...) work in numerical cognition and ontogenetic studies. (shrink)

In this paper I adduce a new argument in support of the claim that IBE is an autonomous form of inference, based on a familiar, yet surprisingly, under-discussed, problem for Hume’s theory of induction. I then use some insights thereby gleaned to argue for the claim that induction is really IBE, and draw some normative conclusions.

We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...) their usual foundational roles. (shrink)

One of the most lasting contributions of Marr's posthumous book is his articulation of the different “levels of analysis” that are needed to understand vision. Although a variety of work has examined how these different levels are related, there is comparatively little examination of the assumptions on which his proposed levels rest, or the plausibility of the approach Marr articulated given those assumptions. Marr placed particular significance on computational level theory, which specifies the “goal” of a computation, its appropriateness for (...) solving a particular problem, and the logic by which it can be carried out. The structure of computational level theory is inherently teleological: What the brain does is described in terms of its purpose. I argue that computational level theory, and the reverse-engineering approach it inspires, requires understanding the historical trajectory that gave rise to functional capacities that can be meaningfully attributed with some sense of purpose or goal, that is, a reconstruction of the fitness function on which natural selection acted in shaping our visual abilities. I argue that this reconstruction is required to distinguish abilities shaped by natural selection—“natural tasks” —from evolutionary “by-products”, rather than merely demonstrating that computational goals can be embedded in a Bayesian model that renders a particular behavior or process rational. (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)

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power (...) and meta-theoretic properties when comparing first-order and second-order logic. (shrink)

This book considers ethical thinking in ancient Israel in the period from the 8th to the 2nd century BC.

Some have thought that the plausibility of teleosemantics requires that it yield a determinate answer to the question of what the semantic “content” is of the “representation” triggered in the optic nerve of a frog that spots a fly. An outsize literature has resulted in which, unfortunately, a number of serious confusions and omissions that concern the way teleosemantics would have to work have appeared and been passed on uncorrected leaving a distorted and simplistic picture of the teleosemantic position. I (...) will speak to a half dozen of the worst of these misunderstandings and then say just a word about the frog question itself. (shrink)

This article articulates and assesses an imperatival approach to the foundations of mathematics. The core idea for the program is that mathematical domains of interest can fruitfully be viewed as the outputs of construction procedures. We apply this idea to provide a novel formalisation of arithmetic and set theory in terms of such procedures, and discuss the significance of this perspective for the philosophy of mathematics.

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...) reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

This essay explores the contemporary experiences of women who live with pain, given the complex responses they encounter within Western medical systems, including pervasive stigma, bias, clinician disbelief, and poor health outcomes. In response to these realities, as highlighted within recent literature and exemplified in a first-person account provided by the paper’s author, this essay explores the Christian practice of the Stations of the Cross as a faithful response to women living with pain. The Stations provide a distinctive Christian practice (...) that invites women living with pain, as well as their clinicians and loved ones, into faithful care marked by prayer, solidarity, and hospitable listening. Practicing the Stations provides one faithful response that Christian clinicians and those who live with pain might engage in the clinic and beyond. (shrink)

Over one in five adults in the United States and around the world are estimated to live with chronic pain. How are we to attend well to persons living with pain? This is a difficult, pressing question for both healthcare institutions and Christian communities, and it is only made more complex both by the contemporary opioid crisis and by how experiences of pain and addiction are shaped in the American context by race, gender, and class. Attending faithfully to persons in (...) pain demands thoughtful, creative resources on both practical and conceptual levels. In this special issue of Christian Bioethics, eight scholars from different disciplines—Sarah Barton, Farr Curlin, Jaime Konerman-Sease, Brett McCarty, Joel Shuman, Devan Stahl, John Swinton, and Emmy Yang—engage the meaning of attending to persons in pain for Christian bioethics and for faithful Christian practice. (shrink)

I provide an examination and comparison of modal theories for underwriting different non-modal theories of sets. I argue that there is a respect in which the `standard' modal theory for set construction---on which sets are formed via the successive individuation of powersets---raises a significant challenge for some recently proposed `countabilist' modal theories (i.e. ones that imply that every set is countable). I examine how the countabilist can respond to this issue via the use of regularity axioms and raise some questions (...) about this approach. I argue that by comparing them with the `standard' uncountabilist theory, a new approach that brings in arbitrariness rather than the strict controls of forcing is desirable. (shrink)

The feminist sex wars have been characterized by debates between radical feminists and sex radical feminists about women's experiences of empowerment versus oppression in the sex industry. Based on a qualitative study of the experiences of exotic dancers, this article introduces a new theoretical paradigm to the feminist sex wars that values the contributions of both sex radicals and radical feminists. It articulates a twofold temporal dimension of sex workers' experiences: how women's feelings of pleasure and empowerment gradually decline over (...) time and how good and bad experiences rapidly transform into their opposites over the course of a single evening, making the strip bar a volatile and unsettling space for performers. (shrink)

Incentives are typically conceived as a form of trade, and so voluntariness appears to be the only ethical concern. As a consequence, incentives are often considered ethically superior to regulations because they are voluntary rather than coercive. But incentives can also be viewed as one way to get others to do what they otherwise would not; that is, as a form of power. When incentives are viewed in this light, many ethical questions arise in addition to voluntariness: What are the (...) responsibilities of the powerful in using incentives? Can incentives be manipulative or exploitative, even if people are free to refuse them? Like all other forms of power, incentives can be abused. This paper develops criteria for distinguishing their legitimate from their illegitimate uses, viz. legitimacy of purpose, voluntariness, and effect on character. The criteria are then applied to three cases: plea bargaining, recruiting medical research subjects, and motivating children to learn. Thinking of incentives in terms of.. (shrink)

Much of the discussion of set-theoretic independence, and whether or not we could legitimately expand our foundational theory, concerns how we could possibly come to know the truth value of independent sentences. This paper pursues a slightly different tack, examining how we are ignorant of issues surrounding their truth. We argue that a study of how we are ignorant reveals a need for an understanding of set-theoretic explanation and motivates a pluralism concerning the adoption of foundational theory.

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that (...) the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)