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 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.

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.

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 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)

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)

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)

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)

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)

Recently, I argued that human subjects in artificial wombs ‘share the same moral status as newborns’ and so, deserve the same treatment and protections as newborns. This thesis rests on two claims: subjects of partial ectogenesis—those that develop in utero for at time before being transferred to AWs—are newborns and subjects of complete ectogenesis—those who develop in AWs entirely—share the same moral status as newborns. In response, Elizabeth Chloe Romanis argued that the subject in an AW is ‘a unique human (...) entity…rather than a fetus or a newborn’. She provides four lines of response to my essay. First, she argues that I have ‘misconstrued’ what birth is. Once we correct that error, it becomes clear that subjects of partial ectogenesis have not been born. Second, she argues that my claims imply that non-implanted embryos ‘would also be “born”’. But that is absurd. Third, she claims I fail to ‘meaningfully respond’ to distinctions she draws between subjects of ectogenesis and neonates. Finally, she criticises my essay for focusing on subjects of AWs rather than focusing on pregnant persons. I respond to each of these charges. In doing so, I reaffirm that some subjects of ectogenesis are newborns and all subjects of ectogenesis—even those that have not been born—share the same moral status as newborns. (shrink)

Our paradigms of aesthetic value condition the philosophical questions we pose and hope to answer about it. Theories of aesthetic value are typically individualistic, in the sense that the paradigms they are designed to capture, and the questions to which they are offered as answers, center the individual’s engagement with aesthetic value. Here I offer some considerations that suggest that such individualism is a mistake and sketch a communitarian way of posing and answering questions about the nature of aesthetic value.

We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

"Ethical thought has no chance of being everything it seems." Bernard Williams offered this cryptic remark in Ethics and the Limits of Philosophy, and in this chapter I argue that understanding it is the key to understanding Williams' skepticism about moral theory and about systematization in ethics. The difficulty for moral philosophy, Williams believed, is that ethics looks one way to embodied, active agents, but looks entirely different when considered from the standpoint of theory. This, in turn, means that following (...) a moral theory is a sure route to practical alienation. I argue that once we see it for what it is, we can see that Williams own skepticism undermines his later attempts to vindicate Truthfulness. Moreover, given the pervasiveness of reflective practical alienation in ordinary life, it leaves us with a difficult question: what would a non-alienated ethical existence even look like for us? (shrink)

This article argues that there can be epistemic dilemmas: situations in which one faces conflicting epistemic requirements with the result that whatever one does, one is doomed to do wrong from the epistemic point of view. Accepting this view, I argue, may enable us to solve several epistemological puzzles.

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)

What is it to know more? By what metric should the quantity of one's knowledge be measured? I start by examining and arguing against a very natural approach to the measure of knowledge, one on which how much is a matter of how many. I then turn to the quasi-spatial notion of counterfactual distance and show how a model that appeals to distance avoids the problems that plague appeals to cardinality. But such a model faces fatal problems of its own. (...) Reflection on what the distance model gets right and where it goes wrong motivates a third approach, which appeals not to cardinality, nor to counterfactual distance, but to similarity. I close the paper by advocating this model and briefly discussing some of its significance for epistemic normativity. In particular, I argue that the 'trivial truths' objection to the view that truth is the goal of inquiry rests on an unstated, but false, assumption about the measure of knowledge, and suggest that a similarity model preserves truth as the aim of belief in an intuitively satisfying way. (shrink)

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)

Epistemologists often appeal to the idea that a normative theory must provide useful, usable, guidance to argue for one normative epistemology over another. I argue that this is a mistake. Guidance considerations have no role to play in theory choice in epistemology. I show how this has implications for debates about the possibility and scope of epistemic dilemmas, the legitimacy of idealisation in Bayesian epistemology, uniqueness versus permissivism, sharp versus mushy credences, and internalism versus externalism.

Subjects of ectogenesis—human beings that are developing in artificial wombs (AWs)—share the same moral status as newborns. To demonstrate this, I defend two claims. First, subjects of partial ectogenesis—those that develop in utero for a time before being transferred to AWs—are newborns (in the full sense of the word). Second, subjects of complete ectogenesis—those who develop in AWs entirely—share the same moral status as newborns. To defend the first claim, I rely on Elizabeth Chloe Romanis’s distinctions between fetuses, newborns and (...) subjects of ectogenesis. For Romanis, the subject of partial ectogenesis ‘is neither a fetus nor a baby’ but is, instead, a ‘new product of human reproduction’. In this essay, I begin by, expanding upon Romanis’s argument that subjects of partial ectogenesis are not fetuses while arguing that those subjects are newborns. Next, I show that the distinction that Romanis draws between subjects of partial ectogenesis and newborns needs to be revised. The former is a kind of the latter. This leads us to an argument that shows why different moral statuses cannot be justifiably assigned to subjects of partial ectogenesis and subjects of complete ectogenesis, respectively. As subjects of partial ectogenesis share the same moral status as newborns, it follows that subjects of complete ectogenesis share the same moral status as newborns as well. I conclude by considering implications that this essay may have for the research and development of AW technology and conceptual links between a subject’s moral status and birth. (shrink)

Are interpersonal affordances a distinct type of affordance, and if so, what is it that differentiates them from other kinds of affordances? In this paper, I show that a hard distinction between interpersonal affordances and other affordances is warranted and ethically important. The enactivist theory of participatory sense-making demonstrates that there is a difference in coupling between agent-environment and agent-agent interactions, and these differences in coupling provide a basis for distinguishing between the perception of environmental and interpersonal affordances. Building further (...) on this foundation for understanding interpersonal affordances, I argue that in line with some enactivist work on social cognition, interpersonal affordances ought to be considered as those that are afforded by agents and are recognized as such. Given this distinction, I also make the point that because our social conventions establish persons as more than mere agents, the direct perception of interpersonal affordances may also involve seeing others as embodied selves. Distinguishing between types of affordances thus also matters ethically: there can be harms done when an agent is not perceived as an agent, and there can be harms done when an agent is not perceived as a self. (shrink)

Non-representational approaches to cognition have struggled to provide accounts of long-term planning that forgo the use of representations. An explanation comes easier for cognitivist accounts, which hold that we concoct and use contentful mental representations as guides to coordinate a series of actions towards an end state. One non-representational approach, ecological-enactivism, has recently seen several proposals that account for “high-level” or “representation-hungry” capacities, including long-term planning and action coordination. In this paper, we demonstrate the explanatory gap in these accounts that (...) stems from avoiding the incorporation of long-term intentions, as they play an important role both in action coordination and perception on the ecological account. Using recent enactive accounts of language, we argue for a non-representational conception of intentions, their formation, and their role in coordinating pre-reflective action. We provide an account for the coordination of our present actions towards a distant goal, a skill we call distal engagement. Rather than positing intentions as an actual cognitive entity in need of explanation, we argue that we take them up in this way as a practice due to linguistically scaffolded attitudes towards language use. (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)

This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at www<dot>beyondspacetime<dot>net.) This chapter investigates the meaning and significance of string theoretic dualities, arguing they reveal a surprising physical indeterminateness to spacetime.

I give an informal presentation of the evolutionary game theoretic approach to the conventions that constitute linguistic meaning. The aim is to give a philosophical interpretation of the project, which accounts for the role of game theoretic mathematics in explaining linguistic phenomena. I articulate the main virtue of this sort of account, which is its psychological economy, and I point to the casual mechanisms that are the ground of the application of evolutionary game theory to linguistic phenomena. Lastly, I consider (...) the objection that the account cannot explain predication, logic, and compositionality. (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)

A theory of aesthetic value should explain what makes aesthetic value good. Current views about what makes aesthetic value good privilege the individual’s encounter with aesthetic value—listening to music, reading a novel, writing a poem, or viewing a painting. What makes aesthetic value good is its benefit to the individual appreciator. But engagement with aesthetic value is often a social, participatory matter: sharing and discussing aesthetic goods, imitating aesthetic agents, dancing, cooking, dining, or making music together. This article argues that (...) we should understand aesthetic value in a way that centers these social forms of aesthetic engagement. To this end, the article argues that there is a social practice of aesthetic valuing, characterized as a participatory practice governed by the value of aesthetic community, which engages us in the social development of our capacities for discretionary valuing and volitional openness. Current theories of aesthetic value have trouble capturing the character of the practice of aesthetic valuing, and this motivates a novel communitarian theory of aesthetic value: aesthetic value is what is worthy of engagement in the social practice of aesthetic valuing. (shrink)

This is an opinionated guide to the literature on epistemic dilemmas. It discusses seven kinds of situations where epistemic dilemmas appear to arise; dilemmic, dilemmish, and non-dilemmic takes on them; and objections to dilemmic views along with dilemmist’s replies to them.

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)

Prior research found correlations between reflection test performance and philosophical tendencies among laypeople. In two large studies (total N = 1299)—one pre-registered—many of these correlations were replicated in a sample that included both laypeople and philosophers. For example, reflection test performance predicted preferring atheism over theism and instrumental harm over harm avoidance on the trolley problem. However, most reflection-philosophy correlations were undetected when controlling for other factors such as numeracy, preferences for open-minded thinking, personality, philosophical training, age, and gender. Nonetheless, (...) some correlations between reflection and philosophical views survived this multivariate analysis and were only partially confounded with either education or self-reported reasoning preferences. Unreflective thinking still predicted believing in God whereas reflective thinking still predicted believing that (a) proper names like ‘Santa’ do not necessarily refer to entities that actually exist and (b) science does reveal the fundamental nature of the world. So some robust relationships between reflection and philosophical tendencies were detected even among philosophers, and yet there was clearly more to the link between reflection and philosophy. To this end, demographic and metaphilosophical hypotheses are considered. (shrink)

Philosophers often characterize discourse in general as aiming at some sort of convergence (in beliefs, plans, dispositions, feelings, etc.), and many views about aesthetic discourse in particular affirm this thought. I argue that a convergence norm does not govern aesthetic discourse. The conversational dynamics of aesthetic discourse suggest that typical aesthetic claims have directive force. I distinguish between dynamic and illocutionary force and develop related theories of each for aesthetic discourse. I argue that the illocutionary force of aesthetic utterances is (...) typically invitational because its dynamic force is influenced by a ‘communal’ norm. I draw on dynamic pragmatics to develop a formal account of this dynamic force that explains why invitation has pride of place in aesthetic conversation. It turns out that the end of aesthetic discourse is not convergence but a distinctive form of community, a kind of harmony of individuality, that is compatible with aesthetic disagreement. If this is right, then convergence theories of aesthetic and normative discourse, and of conversation in general, need to be revised. (shrink)

Epistemologists spend a great deal of time thinking about how we should respond to our evidence. They spend far less time thinking about the ways that evidence can be acquired in the first place. This is an oversight. Some ways of acquiring evidence are better than others. Many normative epistemologies struggle to accommodate this fact. In this article I develop one that can and does. I identify a phenomenon – epistemic feedback loops – in which evidence acquisition has gone awry, (...) with the result that even beliefs based on the evidence are irrational. Examples include evidence acquired under the influence of confirmation bias and evidence acquired under the influence of cognitively penetrated experiences caused by implicit bias. I then develop a theoretical framework which enables us to understand why beliefs that are the outputs of epistemic feedback loops are irrational. Finally, I argue that many popular approaches to epistemic normativity may need to be abandoned on the grounds that they cannot comfortably explain feedback loops. The scope of this last claim is broad: it includes almost all contemporary theories of justified/rational belief and of the epistemology of cognitive penetration. (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.

Cognitive enhancement may be defined as the amplification or extension of core capacities of the mind through improvement or augmentation of internal or external information processing systems. Cognition refers to the processes an organism uses to organize information. These include acquiring information (perception), selecting (attention), representing (understanding) and retaining (memory) information, and using it to guide behavior (reasoning and coordination of motor outputs). Interventions to improve cognitive function may be directed at any of these core faculties.

Daniel Greco (forthcoming) argues that there cannot be epistemic dilemmas. I argue that he is wrong. I then look in detail at a would-be epistemic dilemma and argue that no non-dilemmic approach to it can be made to work. Along the way, there is discussion of octopuses, lobsters, and other ‘inscrutable cognizers’; the relationship between evaluative and prescriptive norms; a failed attempt to steal a Brueghel; epistemic and moral blame and residue; an unbearable guy who thinks he’s God’s gift to (...) women; excuses; stupid games involving hats; radical permissivism; how I’ll never be able to afford to buy a house in Hampstead; and many other exciting topics. (shrink)

Contemporary philosophical attitudes toward beauty are hard to reconcile with its importance in the history of philosophy. Philosophers used to allow it a starring role in their theories of autonomy, morality, or the good life. But today, if beauty is discussed at all, it is often explicitly denied any such importance. This is due, in part, to the thought that beauty is the object of “disinterested pleasure”. In this paper I clarify the notion of disinterest and develop two general strategies (...) for resisting the emphasis on it, in the hopes of getting a clearer view of beauty’s significance. I present and discuss several literary depictions of the encounter with beauty that motivate both strategies. These depictions illustrate the ways in which aesthetic experience can be personally transformative. I argue that they present difficulties for disinterest theories and suggest we abandon the concept of disinterest to focus instead on the special kind of interest beauty fuels. I propose a closer look at the Platonic thought that beauty is the object of love. (shrink)

What is it for a person to have style? Philosophers working in action theory, ethics, and aesthetics are surprisingly quiet on this question. I begin by considering whether theories of artistic style shed any light on it. Many philosophers, artists, and art historians are attracted to some version of the view that artistic style is the expression of personality. I clarify this view and argue that it is implausible for both artistic style and, suitably modified, personal style. In fact, both (...) theories of style crack along the same line, which suggests that they can indeed be mutually illuminating. I articulate and defend a view of personal style according to which, roughly, having style is a matter of expressing one’s ideals. I show how this illuminates the widely neglected value of personal style and propose a new, analogous theory of individual artistic style: artistic style is the expression of the ideals the artist has for her art. (shrink)

This paper details the ways that gender structures our senses of agency on an enactive framework. While it is common to discuss how gender influences higher, narrative levels of cognition, as with the formulation of goals and in considerations about our identities, it is less clear how gender structures our more immediate, embodied processes, such as the minimal sense of agency. While enactivists often acknowledge that gender and other aspects of our socio-cultural situatedness shape our cognitive processes, there is little (...) work on how this shaping takes place. In order to provide such an account, I will first look at the minimal and narrative senses of agency (Gallagher 2012), a distinction that draws from work on minimal and narrative selves (Zahavi 2010). Next I will explain the influence of the narrative sense of agency on the minimal sense of agency through work on intention-formation (Pacherie 2007). After a discussion of the role of gender in the narrative sense of agency, I’ll expand on work by Haslanger (2012) and Young (1980) to offer three ways in which gender influences the minimal sense of agency, showing the effect that gender has on how we perceive our possibilities for interaction in a phenomenologically immediate, pre-reflective manner. (shrink)

What are the sensory individuals of audition? What are the entities our auditory system attributes properties to? We examine various proposals about the nature of the sensory individuals of audition, and show that while each can account for some aspects of auditory perception, each also faces certain difficulties. We then put forward a new conception of sensory individuals according to which auditory sensory individuals are composite individuals. A feature shared by all existing accounts of sounds and sources is that they (...) postulate sensory individuals that are non-composite. They identify the sensory individual of sound hearing or source hearing with one type of entity in the environment, be they sound waves, vibrations, or interactions. We question this assumption and argue that our perceptual systems represent two or more aspects of the environment as a single sensory individual. Finally, we show that taking auditory individuals to be composite sensory individuals allows for an account of audition that is less problematic than its existing alternatives. (shrink)

Analytic moral philosophers have generally failed to engage in any substantial way with the cultural history of morality. This is a shame, because a genealogy of morals can help us accomplish two important tasks. First, a genealogy can form the basis of an epistemological project, one that seeks to establish the epistemic status of our beliefs or values. Second, a genealogy can provide us with functional understanding, since a history of our beliefs, values or institutions can reveal some inherent dynamic (...) or pattern which may be problematically obscured from our view. In this paper, I try to make good on these claims by offering a sketchy genealogy of emancipatory values, or values which call for the liberation of persons from systems of dominance and oppression. The real history of these values, I argue, is both epistemologically vindicatory and functionally enlightening. (shrink)

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)

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)

How should we pursue aesthetic value, or incorporate it into our lives, if we want to? Is there an ideal of aesthetic life? Philosophers have proposed numerous answers to the analogous question in moral philosophy, but the aesthetic question has received relatively little attention. There is, in essence, a single view, which is that one should develop a sensibility that would give one sweeping access to aesthetic value. I challenge this view on two grounds. First, it threatens to undermine our (...) "aesthetic love", or the meaningful attachments we form with aesthetic items, e.g., poems, paintings, songs, or items of design and dress. Second, it fails to accommodate the motivational character of our encounter with beauty, which can diminish our desire to pursue the wider world of aesthetic value. I conclude that whatever the aesthetic ideal is, it must reconcile our desire to broaden our access to aesthetic value with our desire to maintain and cultivate our meaningful aesthetic attachments. I motivate the alternative thought that having style is the aesthetic ideal. (shrink)

The notion of more truth, or of more truth and less falsehood, is central to epistemology. Yet, I argue, we have no idea what this consists in, as the most natural or obvious thing to say—that more truth is a matter of a greater number of truths, and less falsehood is a matter of a lesser number of falsehoods—is ultimately implausible. The issue is important not merely because the notion of more truth and less falsehood is central to epistemology, but (...) because an implicit, false picture of what this consists in underpins and gives shape to much contemporary epistemology. (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)

Philosophy is a reflective activity. So perhaps it is unsurprising that many philosophers have claimed that reflection plays an important role in shaping and even improving our philosophical thinking. This hypothesis seems plausible given that training in philosophy has correlated with better performance on tests of reflection and reflective reasoning has correlated with demonstrably better judgments in a variety of domains. This article reviews the hypothesized roles of reflection in philosophical thinking as well as the empirical evidence for these roles. (...) This reveals that although there are fairly reliable links between reflection and philosophical judgment among both laypeople and philosophers, the role of reflection in philosophical thinking may nonetheless depend in part on other factors, some of which have yet to be determined. So progress in research on reflection in philosophy may require further innovation in experimental methods and psychometric validation of philosophical measures. (shrink)