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)

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)

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)

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.

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)

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)

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)

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.

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)

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)

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)

Working within the tradition of continental philosophy, this article argues in favour of a phenomenological understanding of language as a crucial component of bioethical inquiry. The authors challenge the ‘commonsense’ view of language, in which thinking appears as prior to speaking, and speech the straightforward vehicle of pre-existing thoughts. Drawing on Maurice Merleau-Ponty's (1908–1961) phenomenology of language, the authors claim that thinking takes place in and through the spoken word, in and through embodied language. This view resituates bioethics as a (...) matter of bodies that speak. It also refigures the meaning of ethical self-reflexion, and in so doing offers an alternative view on reflexivity and critique. Referring to the Kantian critical tradition and its reception by Hannah Arendt and Michel Foucault, we advance a position we call ‘critical ethical reflexivity’. We contend that Merleau-Ponty's phenomenology of language offers valuable insight into ethical reflexivity and subject formation. Moreover, his understanding of language may foster new qualitative empirical research in bioethics, lead to more nuanced methods for interpreting personal narratives, and promote critical self-reflexion as necessary for bioethical inquiry. (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)

We argue that the physics of complex materials and self-organizing processes should be made central to the biology of form. Rather than being encoded in genes, form emerges when cells and certain of their molecules mobilize physical forces, effects, and processes in a multicellular context. What is inherited from one generation to the next are not genetic programs for constructing organisms, but generative mechanisms of morphogenesis and pattern formation and the initial and boundary conditions for reproducing the specific traits of (...) a taxon. There is no inherent antagonism between this “physicalist” perspective and genetics, since physics acts on matter, and gene products are essential material components of living systems whose variability affects the systems’ parameters. We make this notion concrete by summarizing the concept of “dynamical patterning modules” (DPMs; Newman and Bhat, Phys Biol 5:1–14, 2008; Int J Dev Biol 53:693–705, 2009), an explicit physico-genetic framework for the origin and evolution of multicellular form in animals, as well as (when differences in interaction toolkit genes and applicable physical processes are taken into account) in multicellular plants (Hernández-Hernández et al., Int J Dev Biol 56:661–674, 2012). DPMs provide the missing link between development and evolution by revealing how genes acting in concert with physics can generate and transform morphology in a heritable fashion. (shrink)

In 2008, researchers reported that Timothy Brown , a man with HIV infection and leukemia, received a stem-cell transplant that removed HIV from his body as far as can be detected. In 2013, an infant born with HIV infection received anti-retroviral treatment shortly after birth, but was then lost to the health care system for the next six months. When tested for HIV upon return, the child had no detectable viral load despite cessation of treatment. These remarkable clinical developments have (...) helped reinvigorate the field of ‘HIV cure’ research. (shrink)

In his article ‘The case against ethics review in the social sciences’, Schrag asserts that the social sciences should not be subject to ethical review. He recounts a number of examples where ethical review has seemingly failed. He further suggests some alternative models for dealing with ethical review in the social sciences. Finally, he concludes, and we concur, that there is a lack of empirical evidence as to the benefit of research ethics review.

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)

A long tradition in theology holds that the divine is in some sense incomprehensible, ineffable, or indescribable. This is mirrored in the set-theoretic literature by those who hold that the universe of sets is incomprehensible, ineffable, or indescribable. In this latter field, set theorists often study reflection principles; axioms that posit indescribability properties of the universe. This paper seeks to examine a theological reflection principle, which can be used to deliver a very rich ontology. I argue that in analogy with (...) set-theoretic reflection principles, we should understand theological reflection via schematic commitment. (shrink)

How does our language relate to reality? This is a question that is especially pertinent in set theory, where we seem to talk of large infinite entities. Based on an analogy with the use of models in the natural sciences, we argue for a threefold correspondence between our language, models, and reality. We argue that so conceived, the existence of models can be underwritten by a weak notion of existence, where weak existence is to be understood as existing in virtue (...) of language. (shrink)

With a mass digitisation programme underway and the addition of non-print legal deposit and web archive collections, the National Library of Scotland is now both producing and collecting data at an unprecedented rate, with over 5PB of storage in the Library’s data centres. As well as the opportunities to support large scale analysis of the collections, this also presents new challenges around data management, storage, rights, formats, skills and access. Furthermore, by assuming the role of both creators and collectors, libraries (...) face broader questions about the concepts of ‘collections' and ‘heritage', and the ethical implications of collecting practices. While the ‘collections as data’ movement has encouraged cultural heritage organisations to present collections in machine-readable formats, new services, processes and tools also need to be established to enable these emerging forms of research, and new modes of working need to be established to take into account an increasing need for transparency around the creation and presentation of digital collections. This commentary explores the National Library of Scotland's new digital scholarship service, the implications of this new activity and the obstacles that libraries encounter when navigating a world of Big Data. (shrink)

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

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)

Newborn screening is the programme through which newborn babies are screened for a variety of conditions shortly after birth. Programmes such as this are individually oriented but resemble traditional public health programmes because they are targeted at large groups of the population and they are offered as preventive interventions to a population considered healthy. As such, an ethical tension exists between the goals of promoting the high uptake of supposedly ‘effective’ population-oriented programmes and the goal of promoting genuinely informed decision-making. (...) There is, however, a lack of understanding with regard to how parents experience the tension between promoting uptake and facilitating informed choice. This paper addresses this issue, and data are presented to show how aspects of the timing, presentation of information and procedural routinisation of newborn screening serves to impact on the decisions made by parents. (shrink)

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)

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)

During vertebrate limb development, various genes of the Hox family, the products of which influence skeletal element identity, are expressed in specific spatiotemporal patterns in the limb bud mesenchyme. At the same time, the cells also exhibit ‘self‐organizing’ behavior – interacting with each other via extracellular matrix and cell‐cell adhesive molecules to form the arrays of mesenchymal condensations that lead to the cartilaginous skeletal primordia. A recent study by Yokouchi et al.(1) establishes a connection between these phenomena. They misexpressed the (...) product of the Hoxa‐13 gene in chick limb buds and demonstrated both skeletal pattern perturbations and changes in cell‐cell adhesivity in mesenchyme aberrantly expressing this protein. (shrink)

Let mathematical justification be the kind of justification obtained when a mathematician provides a proof of a theorem. Are Gettier cases possible for this kind of justification? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true mathematical proposition. In this paper, I argue that Gettier cases are possible (and indeed actual) in mathematical (...) reasoning. By analysing these cases, I suggest that the Gettier phenomenon indicates some upshots for actual mathematical practice. (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)

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views.

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 argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. (...) I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles. (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)

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)

Proposed changes to the Common Rule are proffered to save almost 7,000 reviews annually and consequently vast amounts of investigator and IRB-member time. However, the proposed changes have been subject to criticism. While some have lauded the changes as being imperfect, but nevertheless as improvements, others have contended that ‘neither the scientific community nor the public can be confident that improved practices will emerge from the regulatory changes mandated by the NPRM.’ In the present article, I discuss an important aspect (...) that has been overlooked: the question of whether benefits will emerge is demonstrably empirical, yet data upon which to draw conclusions are conspicuous by their absence. This is thrown into sharp relief when we consider the current environment in which health research is increasingly focused on providing evidence of need or benefit, where there is greater emphasis on evidence-based practice, and when we have the nascent field of implementation science. (shrink)

Initiated almost 200 years ago, the accusation that G.W.F. Hegel’s philosophy qualifies as Gnostic has stood the test of time. Beginning with Ferdinand Christian Baur’s 1835 Die christliche Gnosis, thinkers have attempted to inextricably bind Hegel’s philosophical endeavors to the ancient form(s) of religious knowledge production known as ‘Gnosticism’. Two additional figures have surfaced more recently who also champion the Gnostic accusation, namely Eric Voegelin and James Lindsay. Voegelin’s 1968 Science, Politics, and Gnosticism as well as his 1972 ‘On Hegel: (...) A Study in Sorcery’ seek to justify and legitimize the Gnostic accusation in unequivocal terms. Lindsay’s 2022 best-seller, Race Marxism, further perpetuates the charge of Gnosticism against Hegel. I aim to deal with each of these figures and their respective accusations in reverse chronological order, demonstrating how each of them must artificially construct Hegel as a Gnostic in order to successfully defend their accusations. (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.

The UK has a long established programme of newborn bloodspot screening. This operates under a model of informed choice. Understanding is central to the `informed’ element of an informed choice yet it is rarely assessed. To date most research within the context of newborn bloodspot screening has focussed on parental recall of information. In this paper I argue that simplistic assessments of knowledge through recall fail to reflect more complex notions of understanding. In support of this contention I draw on (...) qualitative interviews with parents of children who have undergone newborn bloodspot screening. (shrink)