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.

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)

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)

Many philosophers are aware of the paradoxes of set theory (e.g. Russell's paradox). For many people, these were solved by the iterative conception of set which holds that sets are formed in stages by collecting sets available at previous stages. This Element will examine possibilities for articulating this solution. In particular, the author argues that there are different kinds of iterative conception, and it's open which of them (if any) is the best. Along the way, the author hopes to make (...) some of the underlying mathematical and philosophical ideas behind tricky bits of the philosophy of set theory clear for philosophers more widely and make their relationships to some other questions in philosophy perspicuous. (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.

"In Getting Even, Charles Barton contends that revenge can be a form of justice that is constructive and healing for our society. Our current judiciary system, he explains, denies both victims and the accused an active role in the legal proceedings and resolution of their cases, reducing them to bystanders in what is essentially their own conflict. Barton does not argue for an individual's right to take the law into his own hands, but does show that the courts (...) should recognize the revenge motive as legitimate and rational within the rules of justice."--pub. desc. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor (...) sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

The Continuum Hypothesis featured top of Hilbert's list of 23 problems in 1900. Today, we still consider the question, with various programmes pulling in different directions. This conceptual diversity raises a puzzle: In what sense do we disagree when we talk about it? A standard assumption takes it that the content of our thought about classes and the Continuum Hypothesis has not changed. Assuming a moderate view of how content is determined, I reject this assumption but also argue that whilst (...) the Continuum Hypothesis can have different content for different agents, it can also be determinate for certain programmes. In particular, I suggest that there is a fault line between those who think there are uncountable sets and recent countabilist programmes. (shrink)

John Barton is convinced that in many areas of ethical enquiry the Old Testament has much to teach us, and he argues his case in this new book, which developed out of lectures given both in Canada and in Oxford. During it he looks again at the Ten Commandments; the narrative about David and his children; the ethical issues of ecology, sexual morality and property; human morality as the express command of God; and the motivation for moral conduct. By (...) the end of the book it is clear that although the Old Testament comes to us from a remote, alien and apparently unsophisticated context, the powerful minds who worked on its books still have things to say which remain evocative and suggestive for our own moral enquiries. (shrink)

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)

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)

I investigate whether tobacco health warnings’ interference with autonomy is ethically justifiable in order to deter people from smoking. I dissociate first the informational role and the persuasive role of tobacco health warnings and show that both roles enable typical addicted smokers to better rule themselves, fostering their autonomy. The fact that some messages address people’s non-deliberative faculties is therefore compensated by a larger positive influence on their autonomy. However, misleading messages are not ethically justified and should be avoided. Tobacco (...) health warnings’ effect on autonomy highlights an important difference between libertarian paternalism and classical paternalism. (shrink)

A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, (...) the status of reflection principles as axioms is left open for the Zermelian. (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)

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)

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)

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...) perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures. (shrink)

We review the various factors that limit adaptation by natural selection. Recent discussion of constraints on selection and, conversely, of the factors that enhance “evolvability”, have concentrated on the kinds of variation that can be produced. Here, we emphasise that adaptation depends on how the various evolutionary processes shape variation in populations. We survey the limits that population genetics places on adaptive evolution, and discuss the relationship between disparate literatures. BioEssays 22:1075–1084, 2000. © 2000 John Wiley & Sons, Inc.

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)

Mukti Barton’s 2000 paper, ‘The Skin of Miriam Became as White as Snow: The Bible, Western Feminism and Colour Politics’, reflected on interpretations of the story of Miriam and her sister-in-law Zipporah, that have focused mainly on Miriam becoming white, as indicative of the effect of colour politics on biblical hermeneutics. Updating that article, the focus was on Zipporah and the main text was again Numbers 12. What are the lessons for Feminist Theology?

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)

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

Although widespread throughout the biomedical sciences, the practice of honorary authorship—the listing of authors who fail to merit inclusion as authors by authorship criteria—has received relatively little sustained attention. Is there something wrong with honorary authorship, or is it only a problem when used in conjunction with other unethical authorship practices like ghostwriting? Numerous sets of authorship guidelines discourage the practice, but its ubiquity throughout biomedicine suggests that there is a need to say more about honorary authorship. Despite its general (...) acceptance among many scientists, honorary authorship is unethical. Even if burdensome, responsible researchers are obligated to forgo honorary authorship. (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)

Although popular, I argue that the International Committee of Medical Journal Editors (ICMJE) account of authorship is flawed. It inadvertently allows for practices that it was designed to prevent. In addition, it creates a new category of authorless papers—orphan papers. The original World Association of Medical Editors (WAME) criterion is preferable.

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)

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural' perspective (...) to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

The received view in philosophy of biology is that there is a well-understood, philosophically rigorous account of information—causal information. I argue that this view is mistaken. Causal information is fatally undermined by misinterpretations and conflations between distinct independent accounts of information. As a result, philosophical arguments based on causal information are deeply flawed. I end by briefly considering what a correct application of the relevant accounts of information would look like in the biological context.

Capitalism has been an unprecedented engine of wealth creation for many centuries, leading to sustained productivity gains and long-term growth and lifting an increasing proportion of humanity out of poverty. But its effects, and hence its future, have come increasingly under question: Is capitalism still improving wealth and well-being for the many? Or, is long-term value creation being sacrificed to the pressures of short-termism, with potentially far-reaching consequences for society, the natural environment, prosperity, and global order? Building on a collaboration (...) between the Schulich School of Business and global management consultancy McKinsey & Company, this volume reflects both the urgency of the needed action and the tremendous opportunity to forge consensus and catalyze a lasting movement toward a more responsible, long-term, and sustainable model of capitalism. This unique volume brings together many of the leading proponents for a reformed, re-imagined capitalism from the Âfields of academia, business, and NGOs. Its contributors have been at the forefront of thought and action in regard to the future of capitalism. Both individually and collectively, they provide powerful suggestions of what such a long-term oriented model of capitalism should look like and how it can be achieved. Drawing on their research and professional experience, they write in an accessible style aiming to reach the broad audiences required to turn a re-imagined capitalism into a reality. (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)

In his review of Getting Even: Revenge as a Form of Justice (Open Court: Chicago. 1999). Michael Davis challenges the view put forward in the book that revenge is personal retributive punishment. Davis also claims that “the purpose Barton seeks to achieve under the banner of ‘victims rights’ has no more to do with punishment than with revenge.” In my response, I argue that Davis’s views and conclusions are based partly on a misreading of Getting Even, and partly on (...) mistaken assumptions about the nature of victim rights, justice, punishment, and revenge. (shrink)

The opposition set up between co-ordinated and mosaic brain evolution distracts from the fact that the two go hand-in-hand. Here and elsewhere (Barton & Harvey 2000), I show that the patterns of co- ordinated evolutionary change among brain structures fit a mosaic evolution model. The concept of overarching developmental constraints is unnecessary and is not supported by the data.

The minutiae of F.W.J. Schelling’s Naturphilosophie have been perennially dismissed due to its allegedly infeasible and indefensible assertions about Nature, such as his designation of Nature as “universal organism.” In the realm of post-Kantian German Idealism, such a dismissive attitude toward Schelling’s so-called objective idealism, more often than not, develops itself along the lines of Hegel’s critique of Schelling’s conception of the Absolute. In turn, I aim to accomplish two tasks in the following investigation. First, I intend to clarify Schelling’s (...) characterization of Nature as universal organism through a practical or teleological, instead of a theoretical or metaphysical, approach. Second, I seek to undermine the Hegelian-influenced criticisms of Schelling’s “Absolute” by demonstrating the ways in which the practico-teleological characterization of Schelling’s formulation of Nature extends into his later philosophical works. Through these clarifications, I hope to emphasize the uniqueness and richness of Schelling’s configuration of Nature. (shrink)

The problem of 'no fault' compensation for patients who suffer adverse effects as a result of their participation in clinical trials is discussed in the light of the guidelines issued by the Association of the British Pharmaceutical Industry (ABPI) and our recent experiences in reviewing protocols submitted to the local ethics of surgical research sub-committee. We have found a variety of qualifications being applied by pharmaceutical firms which are not in the spirit of the guidelines, let alone the interests of (...) the patient, and we suggest a means whereby the patients can be assured of fair treatment in the event of 'no fault' injury. (shrink)

Although written consent forms are standard in clinical research, there is little regulatory or empirical guidance regarding how to most effectively review consent forms with potential participants. We developed an algorithm for embedding five questions with corrective feedback while reading consent forms with potential participants, and then applied it in the context of seven clinical research studies. A substantial proportion of participants within each protocol displayed initially inadequate responses to at least one question, but after the protocol elements were explained (...) again, most people demonstrated adequate understanding of them. These data illustrate the need for more attention not just to the content of consent forms, but also to the manner in which the forms are incorporated into the consent process. Embedding explicit inquiries about key protocol elements during consent form review and giving corrective feedback appears to be a simple yet effective way to foster better understanding of disclosed information. (shrink)

Anthony Rudd presents a striking new account of the self as an ethical, evaluative being.