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)

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.

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)

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)

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.

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)

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)

Certainty and Practical Reason is concerned with Kant's practical response to epistemological skepticism and radical doubt. ;It begins from Kant's remark that the concept of freedom is the keystone of the arch of reason, theoretical as well as practical, and sustains reason against skepticism; and from Kant's account of the practical motives of transcendental realism, the source of skepticism, in the First and Second Critiques. The Critiques suggest both that Kant's response to skepticism is practical, and that skepticism is itself (...) both practically motivated and associated with practical error. ;In the first chapter, it is argued that the Transcendental Analytic of the First Critique is not a final refutation of skepticism and radical doubt, and that Kant does not see it as such. The theoretical arguments of the Transcendental Analytic need to be completed with an appeal to practical reason if we are to ground Kant's starting assumption, the fact of experience. ;The second chapter considers epistemological skepticism as involving a practical error answered by the practical cure of the categorical imperative. ;The third chapter begins by considering Cartesian doubt as a practically motivated attempt to pose the problem of practical rationality, and Kant's response to this problem. It then discusses Kant's view that the transcendental realist demand for knowledge of the noumenal, which gives rise to skeptical doubt, is a pursuit of the unconditioned whose hidden meaning is practical; and explains how Kant responds to the doubt by responding to and interpreting transcendental realism as a practical project. ;The fourth chapter discusses the Critique of Judgment as a response to "enthusiasm" or the transcendental realist claim to knowledge of the unconditioned, and thereby to skepticism as well. (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)

The belief that the spirits of the dead can return to haunt the living exists either as a tenet or as a marginal conviction in all civilizations, whether ancient or modern. More often than not, the dead do not return to reunite the living with their loved ones but rather to lead them into some dreadful snare, entrapping them with disastrous consequences. To be sure, all the departed may return, but some are predestined to haunt: the dead who have been (...) shamed during their lifetime or those who took unspeakable secrets to the grave. From the brucolacs, the errant sprits of outcasts in ancient Greece, to the ghost of Hamlet’s vengeful father, and on down to the rapping spirits of modern times, the theme of the dead—who, having suffered repression by their family or society, cannot enjoy, even in death, a state of authenticity—appears to be omnipresent on the fringes of religions and, failing that, in rational systems. It is a fact that the “phantom,” whatever its form, is nothing but an invention of the living. Yes, an invention in the sense that the phantom is meant to objectify, even if under the guise of individual or collective hallucinations, the gap that the concealment of some part of a loved one’s life produced in us. The phantom is, therefore, also a metapsychological fact. Consequently, what haunts are not the dead, but the gaps left within us by the secrets of others.Because the phantom is not related to the loss of a loved one, it cannot be considered the effect of unsuccessful mourning, as is the case of melancholics or of all those who carry a tomb within themselves. It is the children’s or descendants’ lot to objectify these buried tombs through diverse species of ghosts. What comes back to haunt are the tombs of others. The phantoms of folklore merely objectify a metaphor active within the unconscious: the burial of an unspeakable fact within the loved one.Here we are in the midst of clinical psychoanalysis and still shrouded in obscurity, an obscurity, however, that the nocturnal being of phantoms can, paradoxically, be called upon to clarify. The most recently published book of essays by Nicolas Abraham is Rythmes de l’oeuvre, de la traduction et de la psychanalyse . “Notes on the Phantom” is the preliminary statement of his theory of transgenerational haunting. Nicholas Rand, assistant professor of French at the University of Wisconsin—Madison, is the English-language editor of Abraham’s works. (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)

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)

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)

'This is an unusually ambitious book... a considerable achievement. It raises important issues, and affords many valuable insights in the course of its historical reflections.' -Maurice Wiles, Journal of Theological Studies 'Every issue and thinker is expounded clearly and concisely, with attention always drawn to strengths as well as weaknesses. To this non-specialist the argument was always accessible and regularly persuasive.' -The Expository TimesCanon and Criterion in Christian Theology provides an original and important narrative on the significance of canon in (...) the Christian tradition. Abraham shows that the move to treat canon as a criterion of truth has had unsuspecting consequences for the history of theology and philosophy, from the Fathers to modern feminist theology. (shrink)

The massive increase of data generated by heterogeneous sources requires the development of computer tools enabling their semantic interoperability. Applied ontologies aim at fulfilling such needs. We will show in this article the central role that philosophical ontology can play for applied ontology, with a focus on biomedical ontologies; and reciprocally, how applied ontology can enlighten some classical issues in philosophical ontology, by considering the following question: Is disease a natural kind?

In his _Being Known_ Peacocke sets himself the task of answering how we come to know about metaphysical necessities. He proposes a semantic principle-based conception consisting of, first, his Principles of Possibility which pro­vide necessary and sufficient conditions for a new concept 'admissibility', and second, characterizations of possibility and of necessity in terms of that new con­cept. I focus on one structural feature; viz. the recursive application involved in the specification of 'admissibility'. After sketching Peacocke’s proposal, I intro­duce a fictional (...) protagonist, Cautious Peacocke, whose coherence I claim shows that Peacocke s proposal cannot be made good. I conclude with the conjecture that similar failure will attend any such semantic-based attempts to ground the epistemology of metaphysical necessity. -/- Dans son livre intitulé Being Known, C. Peacocke se propose de repondre à la question de savoir comment nous en venons a connaitre des nécessités métaphysiques. Il développe une conception sémantique reposant sur des principes comprenant, d’une part, ses Principes de Possibilité — qui fournissent les conditions nécessaires et suffisantes pour un nouveau concept, « l’admissibilité » — et d’autre part des caractérisations de la possibilité et de la nécessité a l’aide de ce nouveau concept. Je me concentre sur une caractéristique structurelle, a savoir l’application récursive a l’œuvre dans la spécification de «l’admissibilité ». Apres avoir esquisse la proposition de Peacocke, j’introduis un personnage fictif, Peacocke-prudent. Je soutiens que la cohérence de ce personnage montre que la proposition de Peacocke ne peut pas être satisfaite. Je conclus en conjecturant qu’un échec similaire se présentera pour toutes les tentatives de fonder l’épistemologie de la nécessité métaphysique sur de telles bases semantiques. (shrink)

An exploration of the theories, histories, practices, and contradictions of liberalism today. What does it mean to be a liberal in neoliberal times? This collection of short essays attempts to show how liberals and the wider concept of liberalism remain relevant in what many perceive to be a highly illiberal age. Liberalism in the broader sense revolves around tolerance, progress, humanitarianism, objectivity, reason, democracy, and human rights. Liberalism's emphasis on individual rights opened a theoretical pathway to neoliberalism, through private property, (...) a classically minimal liberal state, and the efficiency of “free markets.” In practice, neoliberalism is associated less with the economic deregulation championed by its advocates than the re-regulation of the economy to protect financial capital. Liberalism in Neoliberal Times engages with the theories, histories, practices, and contradictions of liberalism, viewing it in relation to four central areas of public life: human rights, ethnicity and gender, education, and the media. The contributors explore the transformations in as well as the transformative aspects of liberalism and highlight both its liberating and limiting capacities. The book contends that liberalism—in all its forms—continues to underpin specific institutions such as the university, the free press, the courts, and, of course, parliamentary democracy. Liberal ideas are regularly mobilized in areas such as counterterrorism, minority rights, privacy, and the pursuit of knowledge. This book contends that while we may not agree on much, we can certainly agree that an understanding of liberalism and its emancipatory capacity is simply too important to be left to the liberals Contributors Alejandro Abraham-Hamanoiel, Patrick Ainley, Abdullahi An-Na'im, Michael Bailey, Haim Bresheeth, Başak Çalı, David Chandler, William Davies, Costas Douzinas, Natalie Fenton, Des Freedman, Roberto Gargarella, Priyamvada Gopal, Jonathan Hardy, John Holmwood, Ratna Kapur, Gholam Khiabany, Ray Kiely, Monika Krause, Deepa Kumar, Arun Kundnani, Colin Leys, Howard Littler, Kathleen Lynch, Robert W. McChesney, Nivedita Menon, Toby Miller, Kate Nash, Joan Pedro-Carañana, Julian Petley, Anne Phillips, Jonathan Rosenhead, Annabelle Sreberny, John Steel, Michael Wayne, Milly Williamson. (shrink)

I argue that Plato's purpose in the discussion of false belief in the "Theaetetus" is to entertain and then to reject the idea that thinking is a kind of mental grasping. The interpretation allows us to make good sense of Plato's discussion of 'other-judging' (189c-190e), of his remarks about mathematical error (195d-196c), and most importantly, of the initial statement of the puzzle about falsity (188a-c). That puzzle shows that if we insist on conceiving of the relation between thought and its (...) objects on the model of holding or grasping something in our hands, we will be unable to account for the possibility of false identity judgments: For no one who is literally grasping two things in his hands would seriously entertain the idea that one of the things is numerically identical with the other. (shrink)

In this paper, I elaborate a Derridean deconstruction of law through the lens of Lacanian psychoanalysis. Derrida only focuses on jurisprudential law enforcement in his famous ‘Force of Law’ lecture, leaving corporeal law enforcement untouched. In turn, I explore the irresolvable conceptual tensions within corporeal law enforcement from the standpoints of (a) individuals rationalizing their obedience to law enforcement and (b) the legal system rationalizing its circumscription of acceptable law enforcement. To support my analysis, I examine landmark court cases and (...) their deleterious effects on accountability for law enforcers. In particular, I track the progression from the legal standard of ‘shock of conscience’ (i.e., established by Rochin v. California (1952)) to that of ‘objective reasonableness’ (i.e., established by Graham v. Connor (1989)), which reinforces the legal protections for and libidinal features of police brutality. In this sense, I seek to clarify the dimensions of the libidinal economy belonging to policy brutality, namely its components of sadism and superegoic enjoyment. (shrink)

I, an Indian Christian, have been made to feel inferior in Britain on account of my Indian identity and my sex. Something of the past British imperialism still expresses itself in its sexism and racism towards me. This combination I name imperial patriarchy. I find that white feminist theologians often present an analysis that still stems from an imperial/colonial mind-set. I critique Mary Daly’s Gyn/ecology: The Metaethics of Radical Feminism with the help of post-colonial feminists. Daly suffered the same convenient (...) amnesia as the Victorian missionaries who criticized Chinese foot binding while forgetting that a Victorian woman’s torso was bound by a corset or tight lacing. I call upon Western feminists to become aware of the imprint of imperial patriarchy on their own thinking and then to join with their non-Western sisters in combating both sexism and racist imperialism. (shrink)

Everyone agrees that testing hypotheses is important, but Bowers et al. provide scant details about where hypotheses about perception and brain function should come from. We suggest that the answer lies in considering how information about the outside world could be acquired – that is, learned – over the course of evolution and development. Deep neural networks (DNNs) provide one tool to address this question.

Research across a variety of domains has found that people fail to evaluate statistical information in an atheoretical manner. Rather, people tend to evaluate statistical information in light of their pre-existing beliefs and experiences. The locus of these biases continues to be hotly debated. In two experiments we evaluate the degree to which reasoning when relevant beliefs are readily accessible (i.e., when reasoning with Belief-Laden content) versus when relevant beliefs are not available (i.e., when reasoning with Non-Belief-Laden content) differentially demands (...) attentional resources. In Experiment 1 we found that reasoning with scenarios that contained Belief-Laden content required fewer attentional resources than reasoning with scenarios that contained Non-Belief-Laden content, as evidenced by smaller costs on a secondary memory load task for the former than the latter. This trend was reversed in Experiment 2 when participants were instructed to ignore their beliefs when reasoning with Belief-Laden and Non-Belief-Laden scenarios. These findings provide evidence that beliefs automatically influence reasoning, and attempting to ignore them comes with an attentional cost. (shrink)

Evolutionary theory and methods are central to understanding the design of organisms, including their brains. This book does much to demonstrate the value of evolutionary neuroscience. Further work is needed to clarify the ways that neural systems evolved in general (specifically, the interaction between mosaic and coordinated evolution of brain components), and phylogenetic methods should be given a more prominent role in the analysis of comparative data.