Many assume that we can be responsible only what is voluntary. This leads to puzzlement about our responsibility for our beliefs, since beliefs seem not to be voluntary. I argue against the initial assumption, presenting an account of responsibility and of voluntariness according to which, not only is voluntariness not required for responsibility, but the feature which renders an attitude a fundamental object of responsibility (that the attitude embodies one’s take on the world and one’s place in it) also guarantees (...) that it could not be voluntary. It turns out, then, that, for failing to be voluntary, beliefs are a central example of the sort of thing for which we are most fundamentally responsible. (shrink)

I hope to show that, although belief is subject to two quite robust forms of agency, "believing at will" is impossible; one cannot believe in the way one ordinarily acts. Further, the same is true of intention: although intention is subject to two quite robust forms of agency, the features of belief that render believing less than voluntary are present for intention, as well. It turns out, perhaps surprisingly, that you can no more intend at will than believe at will.

We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

A good number of people currently thinking and writing about reasons identify a reason as a consideration that counts in favor of an action or attitude.1 I will argue that using this as our fundamental account of what a reason is generates a fairly deep and recalcitrant ambiguity; this account fails to distinguish between two quite different sets of considerations that count in favor of certain attitudes, only one of which are the “proper” or “appropriate” kind of reason for them. (...) This ambiguity has been the topic of recent discussion, under the head “the wrong kind of reasons problem.”2 I will suggest that confusion about “the wrong kind of reason” will be dispelled by changing our account of what a reason is. While I agree both that reasons are considerations and that certain.. (shrink)

In this paper I consider fairness of blaming a wrongdoer. In particular, I consider the claim that blaming a wrongdoer can be unfair because blame has a certain characteristic force, a force which is not fairly imposed upon the wrongdoer unless certain conditions are met--unless, e.g., the wrongdoer could have done otherwise, or unless she is someone capable of having done right, or unless she is able to control her behavior by the light of moral reasons. While agreeing that blame (...) has a characteristic force, I am skeptical of this charge of unfairness. My skepticism concerns itself less with the particular conditions of fairness proposed than with the idea that blame can be rendered unfair by its characteristic force. If to blame a person were simply to perform certain intentional actions, then, as we will see, blame could be rendered unfair by its force. But to blame a person is not just to act in certain ways. It is, at least in large part, to make certain judgments or adopt certain attitudes. However, it is unclear how these attitudes or judgments carry "force"? and also unclear whether they can be rendered unfair by their force. Examining these issues, I will suggest that much of the force of blame is found in a set of judgments--most centrally, the judgment that one person failed to show proper regard for others. But, I will argue, once it is granted that such judgments are true, their characteristic force cannot render them unfair. (shrink)

I argue to a conclusion I find at once surprising and intuitive: although many considerations show trust useful, valuable, important, or required, these are not the reasons for which one trusts a particular person to do a particular thing. The reasons for which one trusts a particular person on a particular occasion concern, not the value, importance, or necessity of trust itself, but rather the trustworthiness of the person in question in the matter at hand. In fact, I will suggest (...) that the degree to which you trust a particular person to do a particular thing will vary inversely with the degree to which you must rely, for the motivation or justification of your trusting response, on reasons that concern the importance, or value, or necessity of having such a response. (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.

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)

Donald Davidson opens ‘Actions, Reasons, and Causes’ by asking, ‘What is the relation between a reason and an action when the reason explains the action by giving the agent's reason for doing what he did?’ His answer has generated some confusion about reasons for action and made for some difficulty in understanding the place for the agent's own reasons for acting, in the explanation of an action. I offer here a different account of the explanation of action, one that, though (...) minimal and formal, preserves the proper role for the agent's own reasons for acting. (shrink)

Here I defend my solution to the wrong-kind-of-reason problem against Mark Schroeder’s criticisms. In doing so, I highlight an important difference between other accounts of reasons and my own. While others understand reasons as considerations that count in favor of attitudes, I understand reasons as considerations that bear (or are taken to bear) on questions. Thus, to relate reasons to attitudes, on my account, we must consider the relation between attitudes and questions. By considering that relation, we not only solve (...) the wrong-kind-of-reason problem, but we also bring into view rational agency—the use of reasons in thought. (shrink)

I here defend an account of the will as practical reason —or, using Kant's phrase, as " reason in its practical employment"—as against a view of the will as a capacity for choice, in addition to reason, by which we execute practical judgments in action. Certain commonplaces show distance between judgment and action and thus seem to reveal the need for a capacity, in addition to reason, by which we execute judgment in action. However, another ordinary fact pushes in the (...) other direction: the activities of the will are activities for which the person is answerable in a very particular sort of way. This answerability is most easily understood if willing involves settling a question. Settling a question seems to be a capacity of reason. Thus it can seem that activities of will are activities of our capacity for reasoning. I will suggest that we can accommodate the commonplaces while still understanding the will as reason in its practical employment, by abandoning the assumption that practical reasoning concludes in a judgment. Rather, reasoning which concludes in a judgment—reasoning directed at the question of whether p—is theoretical reasoning. In its practical employment, reason is directed at the question of whether to x; it concludes, not in a judgment about x-ing, but rather in an intention to x. (shrink)

According to ‘Excluders’, descriptive uncertainty – but not normative uncertainty – matters to what we ought to do. Recently, several authors have argued that those wishing to treat normative uncertainty differently from descriptive uncertainty face a dependence problem because one's descriptive uncertainty can depend on one's normative uncertainty. The aim of this paper is to determine whether the phenomenon of dependence poses a decisive problem for Excluders. I argue that existing arguments fail to show this, and that, while stronger ones (...) can be found, Excluders can escape them. (shrink)

A common line of thought claims that we are responsible for ourselves and our actions, while less sophisticated creatures are not, because we are, and they are not, self-aware. Our self-awareness is thought to provide us with a kind of control over ourselves that they lack: we can reflect upon ourselves, upon our thoughts and actions, and so ensure that they are as we would have them to be. Thus, our capacity for reflection provides us with the control over ourselves (...) that grounds our responsibility. I argue that this thought is subtly, but badly, confused. It uses, as its model for the control that grounds our responsibility, the kind of control we exercise over ordinary objects and over our own voluntary actions: we represent to ourselves what to do or how to change things, and then we bring about that which we represent. But we cannot use this model to explain our responsibility for ourselves and our actions: if there is a question about why or how we are responsible for ourselves and our actions, it cannot be answered by appeal to a sophisticated, self-directed action. There must be some more fundamental account of how or why we are responsible. I replace the usual account with a novel but natural view: responsible mental activity can be modeled, not as an ordinary action, but as the settling of a question. This shift will require abandoning the tempting but troublesome thought that responsible activity involves discretion and awareness—which, I argue, we must abandon in any case. (shrink)

There seems to be widespread agreement that to be responsible for something is to be deserving of certain consequences on account of that thing. Call this the “merited-consequences” conception of responsibility. I think there is something off, or askew, in this conception, though I find it hard to articulate just what it is. The phenomena the merited-consequences conception is trying to capture could be better captured, I think, by noting the characteristic way in which certain minds can rightly matter to (...) other such minds—the way in which certain minds can carry a certain kind of importance, made manifest in certain sorts of responses. Mattering, not meriting, seems to me central. However, since I cannot yet better articulate an alternative, I continue in the merit-consequences framework. I focus on a particular class of consequences: those that are non-voluntary, in a sense explained. The non-voluntariness of these reactions has two important upshots. First, questions about their justification will be complex. Second, they are not well thought of as consequences voluntarily imposed upon the wrongdoer by the responder. By focusing on merited consequences and overlooking non-voluntariness, we risk misunderstanding the significance of moral criticism and of certain reactions to moral failure. (shrink)

In the classical Turing test, participants are challenged to tell whether they are interacting with another human being or with a machine. The way the interaction takes place is not direct, but a distant conversation through computer screen messages. Basic forms of interaction are face-to-face and embodied, context-dependent and based on the detection of reciprocal sensorimotor contingencies. Our idea is that interaction detection requires the integration of proprioceptive and interoceptive patterns with sensorimotor patterns, within quite short time lapses, so that (...) they appear as mutually contingent, as reciprocal. In other words, the experience of interaction takes place when sensorimotor patterns are contingent upon one’s own movements, and vice versa. I react to your movement, you react to mine. When I notice both components, I come to experience an interaction. Therefore, we designed a “minimal” Turing test to investigate how much information is required to detect these reciprocal sensorimotor contingencies. Using a new version of the perceptual crossing paradigm, we tested whether participants resorted to interaction detection to tell apart human from machine agents in repeated encounters with these agents. In two studies, we presented participants with movements of a human agent, either online or offline, and movements of a computerized oscillatory agent in three different blocks. In each block, either auditory or audiovisual feedback was provided along each trial. Analysis of participants’ explicit responses and of the implicit information subsumed in the dynamics of their series will reveal evidence that participants use the reciprocal sensorimotor contingencies within short time windows. For a machine to pass this minimal Turing test, it should be able to generate this sort of reciprocal contingencies. (shrink)

In "Rational Capacities" Michael Smith outlines the sense of capacity he believes to be required before blame is appropriate. I question whether this sense of capacity is required. In so doing, I consider different ways in which blame might be conditioned.

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 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)

In 1982, when T. M. Scanlon published “Contractualism and Utilitarianism,” he noted that, despite the widespread attention to Rawls’ A Theory of Justice, the appeal of contractualism as a moral theory had been under appreciated. In particular, the appeal of contractualism’s account of what he then called “moral motivation” had been under appreciated.1 It seems to me that, in the intervening quarter century, despite the widespread discussion of Scanlon’s work, the appeal of contractualism, in precisely this regard, has still been (...) under appreciated—even though Scanlon makes what he once called “moral motivation” central, throughout his work. My first aim, then, is to do my best to draw out and make vivid this appeal. I will do this by first considering the two questions that Scanlon thinks must be addressed by any moral theory, what he once called “the question of subject matter” and “the question of motivation.” I will spend some time first locating and explicating the second question, of motivation, and then displaying Scanlon’s answer to it—it is this answer which provides contractualism with its under-appreciated appeal. I will then return to the question of subject matter—which will, by that point, have been revealed as not wholly distinct from the question of motivation, as Scanlon understands it. But it is as an answer to this question that Scanlon’s theory is most often.. (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)

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)

Many think of reasons as facts, propositions, or considerations that stand in some relation (or relations) to attitudes, actions, states of affairs. The relation may be an explanatory one or a “normative” one—though some are uncomfortable with irreducibly “normative” relations. I will suggest that we should, instead, see reasons as items in pieces of reasoning. They relate, in the first instance, not to psychological states or events or states of affairs, but to questions. That relation is neither explanatory nor “normative.” (...) If we must give it a label, we could call it “rational”—but that will mean, I think, only that the consideration bears on the question. By thus putting reasoning first, we not only avoid a handful of difficulties that have plagued thinking about reasons, but we also bring back to center-stage the importance of rational agency. (shrink)

Wright's _Truth and Objectivity_ seeks to systematise a variety of anti-realist positions. I argue that many objections to the system are avoided by transposing its talk of truth into talk of warrant. However, a problem remains about debates involving 'direction-of-fit'. -/- Dummett introduced 'anti-realism' as a philosophical view informed by mathematical intuitionism. Subsequently, the term has been associated with many debates, ancient and modern. _Truth and Objectivity_ proposes that truth admits of different characteristics; these various debates then concern which characteristics (...) truth has, in a given area. This pluralism of truth is at odds with deflationism. I find fault with Wright's argument against deflationism. However, transmission of warrant across the Disquotational Schema suffices to ground Wright's proposal, which survives as a pluralism of classes of warrant. -/- The two main debates concern whether truths are always knowable (Epistemic Constraint) and whether disagreements in an area must be down to some fault of one of those involved (Cognitive Command). I introduce Assertoric Constraint, relating to Epistemic Constraint, where truths cannot outstrip the availability of warrant for their assertion. I solve a structural problem by a comparison with a constitutive analysis of Moore's Paradox. The relativism of blameless disagreement is problematic. Wright's response invokes a sort of ignorance which he calls `Quandary'. I criticise this before proposing an alternative. -/- I agree with Wright that Dummett's original anti-realism does not belong among the positions which Wright seeks to systematise. However, two candidates show that the proposal suffers a weakness. Wright thinks Expressivism misguided, and implicitly rules out his earlier non-cognitivism about necessity. I argue that Expressivism has promise, and I endorse Wright's Cautious Man argument for non-cognitivism about necessity; both involve play with `direction-of-fit'. I conclude that this sort of anti-realist debate needs to be accommodated by the proposal. (shrink)

I here press an often overlooked question: Why does the fairness of a sanction require an adequate opportunity to avoid it? By pressing this question, I believe I have come to better understand something that has long puzzled me, namely, what philosophers (and others) might have in mind when they talk about “true moral responsibility,” or the “condemnatory force” of moral blame, or perhaps even “basic desert.” In presenting this understanding of “condemnation” or of “basic desert,” I am presenting an (...) idea that I do not, myself, endorse—one to which I am, myself, opposed. I present and defend it, hoping that, if what I say accurately captures what people have in mind, it will also show how condemnation or desert, in this sense, can be left behind. (shrink)

I first present what Peter Strawson calls his “Social Naturalism,” as applied to ethics. I then briefly present the way in which his Naturalism allows Strawson to resist skepticism about moral responsibility and free will, as argued in “Freedom and Resentment.” His way of resisting this kind of skepticism opens his Naturalism to another challenge: it can seem objectionably relativistic. I have provided a response to this challenge, on Strawson’s behalf, in the final chapter of my _Freedom, Resentment, and the (...) Metaphysics of Morals_ (2020). In this paper I expand upon that response—in particular, on the way in which the introduction of admittedly contested ideals might provide sufficient critical purchase to avoid objectionable relativism while remaining broadly naturalistic. (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 consider Paulina Sliwa’s fruitful account of “taking responsibility” as “owning the normative footprint” of a wrong. Unlike most, Sliwa approaches the topic without concern for what I call “responsible agency.” I raise the possibility that this is virtue. I then question whether the “footprint” is simply given with the wrong or whether it must instead be made determinate through subsequent interaction, perhaps through conversation. I next distinguish two different kinds of conversation: a cooperative negotiation and a low-level power struggle. (...) The later is often part of what I call “the blame game.” I suggest that the “blame” of the blame game is the blame of ordinary life, to be distinguished from the technical term in the current philosophical literature, and I give an account of it. I note that one may sometimes opt out of the blame game simply by owning a generous interpretation of the footprint, something Sliwa discusses. I note that doing so is sometimes, but not always, virtuous. (shrink)

This chapter presents two contrasting pictures of the will. On the first, “the will” is a psychological structure or module within a person that originates spontaneous or endogenous activity, independently of external influence. On the second, “the will” is that collection of ordinary states of mind (cares, concerns, beliefs, desires, commitments, fears, etc.) that generates intentional, or voluntary, or responsible activity—it is the functioning together of those aspects of mind that account for human activity. A challenge is posed for each. (...) The challenge for the first is to say how the outputs of this module are the activity of the person. The challenge for the second is to say how psychological activity that might be entirely explained by a person’s history and environment can nonetheless be free and responsible activity. The replies take up an important question about the role of consciousness, in each picture. (shrink)

This chapter presents four different senses of “voluntary” that might be in play. First, voluntary1 movement contrasts with bodily movement not guided by the person—such as blinking or digesting, which are involuntary1. Second, you might move voluntarily1, and yet make a mistake—you might send an email to the wrong person—you then act involuntarily2. In contrast, voluntary2 action is successful. Third, you might purposely and even successfully do something you didn’t want to do—through the cargo overboard during the storm. In such (...) cases, you act involuntarily3. In contrast, you might act happily, as a volunteer, so to speak. You then act voluntarily3. Finally, sometimes you can be manipulated or deceived into acting as a volunteer—in which case you lack an important form of freedom. We might say that you act involuntarily4. I suggest that the first two senses might be fruitfully studied by neuroscience, while the final sense marks a distinction draw by ethical reflection. (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)

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)

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)

I suggest that Fischer concedes too much to the consequence argument when he grants that we may not make a difference. I provide a broad sketch of (my take on) the dispute between compatibilists and incompatibilists, while suggesting that some of the discussion may have confused the freedom required for moral responsibility with a very different notion of autonomy. I introduce that less usual notion of autonomy and suggest that those who are autonomous, in this sense, do make a difference.

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 document I survey my work to date (i.e., to September 2010) and connect it to the larger themes that have been animating it.

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)

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.

This paper proposes an intermediate account of personhood, based on the capacity to participate in intersubjective interactions. We articulate our proposal as a reply to liberal and restrictive accounts, taking Mark Rowlands’ and Stephen Darwall’s proposals as contemporary representatives of each view, respectively. We argue that both accounts fall short of dealing with borderline cases and defend our intermediate view: The criteria of personhood based on the second-person perspective of mental state attribution. According to it, a person should be able (...) to participate in intersubjective interactions. We apply our proposal to the borderline cases of non-human primates and human infants. (shrink)

I first sketch the different things we might have in mind, when thinking about responsibility. I then relate each of those to possible investigations of human agency. The most interesting such relation, in my opinion, is that between agency and what I call “responsibility as mattering.” I offer some hypotheses about that relation.

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)

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

The book intends to interpret how different religions articulate their territories and manage the relationship with other religions, understanding systems and multiple everyday spaces, in a dynamic that is not only a component of contemporary reality, but is central to living it. The underlying thesis is that religion is the great geopolitical issue of our time, but an interpretation is only possible in terms of religious plurality and how ideas, symbolism, subjectivities and practices are incorporated in the daily life of (...) those who share the religious times and spaces, producing territorial identities and imaginary concrete cartographies as world projects. The volume has articles that focus on the religious space being gestated, expanded, contracted, celebrated or represented as a foundation for common life, as well as inter-religious conflicts arising in a world of diverse and differential coexistence. It is not a debate about classical geopolitics, but how geopolitical relations cross cultural-religious experience. (shrink)

Recently, there has been an insurgence of corporations that bioprospect in Third World countries (going into these areas in hopes of utilizing traditional knowledge about local natural resources so as to eventually develop a synthetic alternative that they can then market). Although this type of bioprospecting does not encounter the problem of depleting environmental resources, other problems arise. Two primary problems are: (1) determining who has legal ownership of these resources, and (2) who should share in the profits that were (...) derived, in part, from these resources. Despite the attention that these problems have received, there has been little attention focused on the issue that I believe underlies these problems -- the differing views of nature between corporations (who consider nature to be an individually owned resource) and Third World communities (many of whom consider nature to be a communally owned resource). In this paper, I present what I believe is an additional useful tool for policy makers in resolving the current problems regarding bioprospecting in Third World countries. The tool I have in mind is a set of method components that can be used as guidelines as well as a test for inclusiveness. To arrive at these components, I will look to James D. Proctor's methodological proposal for resolving conflicts between those with differing views of nature. Because he presents this methodology in a limited context (as a means of resolving the Ancient Forest debate), I will clarify and expand his methodology so as to arrive at a set of methods that I believe can be helpful in arriving at a fair global policy regarding bioprospecting in Third World countries. (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.

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)

La pertinence de la métaphysique analytique a fait l'objet de critiques : Ladyman et Ross, par exemple, ont suggéré d'abandonner ce domaine. French et McKenzie ont défendu la métaphysique analytique en affirmant qu'elle développe des outils qui pourraient s'avérer utiles pour la philosophie de la physique. Dans cet article, nous montrons dans un premier temps que cette défense heuristique de la métaphysique peut être étendue au domaine scientifique de l'ontologie appliquée, qui utilise des théories et outils issus de la métaphysique (...) analytique. Dans un deuxième temps, nous développons le parallèle que font French et McKenzie entre les mathématiques et la métaphysique pour montrer que l'ensemble du domaine de la métaphysique analytique, étant donné son utilité non seulement pour la philosophie mais également pour la science, devrait continuer à exister en tant que domaine largement autonome. (shrink)