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.

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)

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)

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

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

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.

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)

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.

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 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 model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets 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 present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...) complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (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)

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)

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)

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)

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.

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)

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.

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)

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)

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)

LandryElaine, * ed. Categories for the Working Philosopher. Oxford University Press, 2017. ISBN 978-0-19-874899-1 ; 978-0-19-106582-8. Pp. xiv + 471.

I defend hedonism about moral value by first presenting an argument for moral skepticism, and then showing that phenomenal introspection gives us a unique way to defeat the skeptical argument and establish pleasure's goodness.

This paper elaborates and defends an expressivist account of the claims of mind-independence embedded in ordinary moral thought. In response to objections from Zangwill and Jenkins it is argued that the expressivist 'internal reading' of such claims is compatible with their conceptual status and that the only 'external reading' available doesn't commit expressivisists to any sort of subjectivism. In the process a 'commitment-theoretic' account of the semantics of conditionals and negations is defended.

Causal theories of action, perception and knowledge are each beset by problems of so-called ‘deviant’ causal chains. For each such theory, counterexamples are formed using odd or co-incidental causal chains to establish that the theory is committed to unpalatable claims about some intentional action, about a case of veridical perception or about the acquisition of genuine knowledge. In this paper I will argue that three well-known examples of a deviant causal chain have something in common: they each violate Yablos proportionality (...) constraint on causation. I will argue that this constraint provides the key to saving causal theories from deviant chains. (shrink)

Black holes are extremely relativistic objects. Physical processes around them occur in a regime where the gravitational field is extremely intense. Under such conditions, our representations of space, time, gravity, and thermodynamics are pushed to their limits. In such a situation philosophical issues naturally arise. In this chapter I review some philosophical questions related to black holes. In particular, the relevance of black holes for the metaphysical dispute between presentists and eternalists, the origin of the second law of thermodynamics and (...) its relation to black holes, the problem of information, black holes and hypercomputing, the nature of determinisim, and the breakdown of predictability in black hole space-times. I maintain that black hole physics can be used to illuminate some important problems in the border between science and philosophy, either epistemology and ontology. (shrink)

I defend ethical hedonism, the view that pleasure is the sole good thing, by arguing that it offers the only answer to an argument for moral skepticism. The skeptical problem arises from widespread fundamental moral disagreement, which entails the presence of enough moral error to undermine the reliability of most processes generating moral belief. We know that pleasure is good through the reliable process of phenomenal introspection, which reveals what our experiences are like. If knowing of pleasure’s goodness through phenomenal (...) introspection is the only reliable way to form our initial moral beliefs, we should accept ethical hedonism. (shrink)

It is well known that the nature of consciousness is elusive, and that attempts to understand it generate problems in metaphysics, philosophy of mind, psychology, and neuroscience. Less appreciated are the important – even if still elusive – connections between consciousness and issues in ethics. In this chapter we consider three such connections. First, we consider the relevance of consciousness for questions surrounding an entity’s moral status. Second, we consider the relevance of consciousness for questions surrounding moral responsibility for action. (...) Third, we consider the relevance of consciousness for the acquisition of moral knowledge. (shrink)

Abstract: This entry elucidates causal and constitutive roles that various forms of imagining play in human action. Imagination influences more kinds of action than just pretend play. I distinguish different senses of the terms “imagining” and “imagination”: imagistic imagining, propositional imagining, and constructive imagining. Each variety of imagining makes its own characteristic contributions to action. Imagistic imagining can structure bodily movement. Propositional imagining interacts with desires to motivate pretend play and mimetic expressive action. And constructive imagination generates representations of possibilities (...) and actions on the basis of which we choose what to do. [Version archived here is a penultimate draft.]. (shrink)

A number of concerns have been raised about the possible future use of pharmaceuticals designed to enhance cognitive, affective, and motivational processes, particularly where the aim is to produce morally better decisions or behavior. In this article, we draw attention to what is arguably a more worrying possibility: that pharmaceuticals currently in widespread therapeutic use are already having unintended effects on these processes, and thus on moral decision making and morally significant behavior. We review current evidence on the moral effects (...) of three widely used drugs or drug types: propranolol, selective serotonin reuptake inhibitors, and drugs that effect oxytocin physiology. This evidence suggests that the alterations to moral decision making and behavior caused by these agents may have important and difficult-to-evaluate consequences, at least at the population level. We argue that the moral effects of these and other widely used pharmaceuticals warrant further empirical research and ethical analysis. (shrink)

Pietro Gori dedicates Nietzsche’s Pragmatism “To the wanderers and Good Europeans,” and Anglophone wanderers into Sarah de Sanctis’s translation will indeed find good European Nietzsche scholarship. The table of contents is a helpful map of the book, with five chapters consisting of twenty-eight sections on a sequence of philosophical and interpretive topics. Perspectival thought, addressed in the subtitle, is the explicit topic of the third chapter. Pragmatism, mentioned in the title, is the explicit topic of the fifth and final chapter. (...) While both topics also are discussed in many other places, the overall focus is on Nietzsche’s views of epistemology and truth.The first chapter discusses Nietzsche’s views of... (shrink)

The relevance of analytic metaphysics has come under criticism: Ladyman & Ross, for instance, have suggested do discontinue the field. French & McKenzie have argued in defense of analytic metaphysics that it develops tools that could turn out to be useful for philosophy of physics. In this article, we show first that this heuristic defense of metaphysics can be extended to the scientific field of applied ontology, which uses constructs from analytic metaphysics. Second, we elaborate on a parallel by French (...) & McKenzie between mathematics and metaphysics to show that the whole field of analytic metaphysics, being useful not only for philosophy but also for science, should continue to exist as a largely autonomous field. (shrink)

Epistemic akrasia can be rational. I consider a lonely pragmatist who believes that her imaginary friend doesn’t exist, and also believes on pragmatic grounds that she should believe in him. She rationally believes that her imaginary friend doesn’t exist, rationally follows various sources of evidence to the view that she should believe in him to end her loneliness, and rationally holds these attitudes simultaneously. Evidentialism suggests that her ambivalent epistemic state is rational, as considerations grounded in the value of truth (...) justify her beliefs. (shrink)

This paper claims that the standard characterization of the motivational role of belief should be supplemented. Beliefs do not only, jointly with desires, cause and rationalize actions that will satisfy the desires, if the beliefs are true; beliefs are also the practical ground of other cognitive attitudes, like imagining, which means beliefs determine whether and when one acts with those other attitudes as the cognitive inputs into choices and practical reasoning. In addition to arguing for this thesis, I take issue (...) with Velleman's argument that belief and imagining cannot be distinguished on the basis of motivational role. (shrink)

This essay presents an ideal for modern Western romantic love.The basic ideas are the following: people want to form a distinctive sort of plural subject with another, what Nozick has called a "We", they want to be loved for properties of certain kinds, and they want this love to establish and sustain a special sort of commitment to them over time.

The rightness and wrongness of actions fits on a continuous scale. This fits the way we evaluate actions chosen among a diverse range of options, even though English speakers don’t use the words “righter” and “wronger”. I outline and defend a version of scalar consequentialism, according to which rightness is a matter of degree, determined by how good the consequences are. Linguistic resources are available to let us truly describe actions simply as right. Some deontological theories face problems in accounting (...) for degrees of rightness, as they don't invoke continuous parameters among the right-making features of action. (shrink)

What is posthumanism and why does it matter? This book offers an introduction to the ways in which humanism's belief in the natural supremacy of the Family of Man has been called into question at different moments and from different theoretical positions. What is the relationship between posthumanism and technology? Can posthumanism have a politics—postcolonial or feminist? Are postmodernism and poststructuralism posthumanist? What happens when critical theory meets Hollywood cinema? What links posthumanism to science fiction. Posthumanism addresses these and other (...) questions in an attempt to come to terms with one of the most pressing issues facing contemporary society. (shrink)

Philosophers from Hart to Lewis, Johnston and Bennett have expressed various degrees of reservation concerning the doctrine of double effect. A common concern is that, with regard to many activities that double effect is traditionally thought to prohibit, what might at first look to be a directly intended bad effect is really, on closer examination, a directly intended neutral effect that is closely connected to a foreseen bad effect. This essay examines the extent to which the commonsense concept of intention (...) supports a reasonably consistent and coherent application of double effect. Two important conclusions are these: (1) a number of traditionally proscribed activities involve a kind of “targeting” of innocents that can be taken to exhibit a direct intention to harm them; (2) a direct intention to harm need not involve a desire to harm in any ordinary sense of the latter expression. (shrink)

This essay defends a strong version of the Humean theory of motivation on which desire is necessary both for motivation and for reasoning that changes our desires. Those who hold that moral judgments are beliefs with intrinsic motivational force need to oppose this view, and many of them have proposed counterexamples to it. Using a novel account of desire, this essay handles the proposed counterexamples in a way that shows the superiority of the Humean theory. The essay addresses the classic (...) objection that the Humean theory cannot explain the feeling of obligation, Stephen Darwall's example of motivationally potent reasoning that is not based on preexisting desires, Thomas Scanlon's criticism that the Humean theory fails to account for the structure and phenomenology of deliberation, and the phenomenon of akrasia as discussed by John Searle. In each case a Humean account explains the data at least as thoroughly as opposing views can, while fitting within a simpler total account of how we deliberate and act. (shrink)

I argue that one intends that ϕ if one has a desire that ϕ and an appropriately related means-end belief. Opponents, including Setiya and Bratman, charge that this view can't explain three things. First, intentional action is accompanied by knowledge of what we are doing. Second, we can choose our reasons for action. Third, forming an intention settles a deliberative question about what to do, disposing us to cease deliberating about it. I show how the desire- belief view can explain (...) why these phenomena occur when they occur, and why they don't when they don't. (shrink)

Causal exclusion arguments are taken to threaten the autonomy of the special sciences, and the causal efficacy of mental properties. A recent line of response to these arguments has appealed to “independently plausible” and “well grounded” theories of causation to rebut key premises. In this paper I consider two papers which proceed in this vein and show that they share a common feature: they both require causes to be proportional to their effects. I argue that this feature is a bug, (...) and one that generalises: any attempt to rescue the autonomy of the special sciences, or the efficacy of the mental, from exclusion worries had better not look to proportionality for help. (shrink)

This paper argues that qualia share their physical correlates' locations. The first premise comes from the theory of relativity: If something shares a time with a physical event in all reference frames, it shares that physical event’s location. The second premise is that qualia share times with their correlates in all reference frames. Having qualia and correlates share locations makes relations between them easier to explain, improving both physicalist and dualist theories.

Propositionalism is the view that the contents of intentional attitudes have a propositional structure. Objectualism opposes propositionalism in allowing the contents of these attitudes to be ordinary objects or properties. Philosophers including Talbot Brewer, Paul Thagard, Michelle Montague, and Alex Grzankowski attack propositionalism about such attitudes as desire, liking, and fearing. This article defends propositionalism, mainly on grounds that it better supports psychological explanations.

In this paper I draw apart two different contexts of Freirean pedagogical practice that I label interactional and institutional. The interactional refers to the immediate learning environment with relation to the interaction between the students and the teacher. In contrast, the institutional refers to how the institutions of education are managed, constructed, and organised and how they relate to the individuals those institutions are composed of. I begin by presenting a brief overview of Freire’s argument in favour of a revolutionary (...) pedagogy. I then highlight how this project is represented at both interactional and institutional levels with a particular focus upon the practice of the revolutionary leaders and their role in the development of the humanisation of the oppressed. In presenting the interactional and institutional elements of Freire’s pedagogy separately the tension between the individual and the state is made plain and the limitations of his politico-educational project are drawn out. I argue that due to an inconsistent application of the relationship between means and ends in Freire’s politico-educational project an incoherence between the interactional and institutional levels of education is highlighted. In responding to this limitation of Freire’s politico-educational project with a thoroughly interconnected relationship of means and ends I show that there remains a necessary conflict between the interests of the individual and the state. As a result of this a door is opened for the development of an argument in favour of the disestablishment of education and state from a Freirean perspective. (shrink)

The ‘doctrine of double effect’ claims that it is in some sense morally less problematic to bring about a negatively evaluated state of affairs as a ‘side effect’ of one’s pursuit of another, morally unobjectionable aim than it is to bring it about in order to achieve that aim. In a first step, this chapter discusses the descriptive difference on which the claim is built. That difference is shown to derive from the attitudinal distinction between intention and ‘acceptance’, a distinction (...) that is in turn claimed to ground in a feature of the decisions that generate the attitudes in question. The resulting analysis is then plugged into two different normative principles that may each be thought to specify the intuitions behind the doctrine of double effect, but which have frequently been conflated. The first concerns the permissibility of bringing about the merely accepted state of affairs, the second its reduced attributability. It is argued that examination of the intuitions behind the two principles supports neither version of the doctrine. Rather, the intuitions are best captured in an attribution principle based on subjective probabilities and a principle of attitude evaluation, neither of which make explicit reference to the attitude of intending. (shrink)