This book is a volume in the Penn Press Anniversary Collection. To mark its 125th anniversary in 2015, the University of Pennsylvania Press rereleased more than 1,100 titles from Penn Press's distinguished backlist from 1899-1999 that had fallen out of print. Spanning an entire century, the Anniversary Collection offers peer-reviewed scholarship in a wide range of subject areas.

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 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)

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 paper, I consider the problem of omission for the counterfactual comparative account of harm. A given event harms a person, on this account, when it makes her worse off than she would have been if it had not occurred. The problem arises because cases in which one person merely fails to benefit another intuitively seem harmless. The account, however, seems to imply that when one person fails to benefit another, the first thereby harms the second, since the second (...) person would have been better off if the first had benefited her. I argue that the cases of failing to benefit at issue are in fact cases of harming. They are cases of preventive harm. I also argue that we can explain away the intuition that no harm occurs in these cases, and that the relevant implication of the counterfactual comparative account is consistent with a variety of plausible views about the moral significance of harm. (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.

The Continuum Hypothesis featured top of Hilbert's list of 23 problems in 1900. Today, we still consider the question, with various programmes pulling in different directions. This conceptual diversity raises a puzzle: In what sense do we disagree when we talk about it? A standard assumption takes it that the content of our thought about classes and the Continuum Hypothesis has not changed. Assuming a moderate view of how content is determined, I reject this assumption but also argue that whilst (...) the Continuum Hypothesis can have different content for different agents, it can also be determinate for certain programmes. In particular, I suggest that there is a fault line between those who think there are uncountable sets and recent countabilist programmes. (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)

This book offers a defense of the Property Theory of Content, according to which properties rather than propositions are the contents of our beliefs, desires, and other cognitive attitudes. New arguments for the theory are offered, objections are answered, and applications to problems in the philosophy of mind are discussed.

This article gives an explicit presentation of Newtonian gravitation on the backdrop of Maxwell space-time, giving a sense in which acceleration is relative in gravitational theory. However, caution is needed: assessing whether this is a robust or interesting sense of the relativity of acceleration depends on some subtle technical issues and on substantive philosophical questions over how to identify the space-time structure of a theory.

In recent years there has been an outpouring of work at the intersection of social movement studies and organizational theory. While we are generally in sympathy with this work, we think it implies a far more radical rethinking of structure and agency in modern society than has been realized to date. In this article, we offer a brief sketch of a general theory of strategic action fields (SAFs). We begin with a discussion of the main elements of the theory, describe (...) the broader environment in which any SAF is embedded, consider the dynamics of stability and change in SAFs, and end with a respectful critique of other contemporary perspectives on social structure and agency. (shrink)

There are well-known problems associated with the idea of gravitational energy in general relativity. We offer a new perspective on those problems by comparison with Newtonian gravitation, and particularly geometrized Newtonian gravitation. We show that there is a natural candidate for the energy density of a Newtonian gravitational field. But we observe that this quantity is gauge dependent, and that it cannot be defined in the geometrized theory without introducing further structure. We then address a potential response by showing that (...) there is an analogue to the Weyl tensor in geometrized Newtonian gravitation. (shrink)

In this paper, I consider the role of exact symmetries in theories of physics, working throughout with the example of gravitation set in Newtonian spacetime. First, I spend some time setting up a means of thinking about symmetries in this context; second, I consider arguments from the seeming undetectability of absolute velocities to an anti-realism about velocities; and finally, I claim that the structure of the theory licences us to interpret models which differ only with regards to the absolute velocities (...) of objects as depicting the same physical state of affairs. In defending this last claim, I consider how ideas and resources from the philosophy of language may usefully be brought to bear on this topic. (shrink)

I respond here to an argument given recently in this journal by Erik Carlson, Jens Johansson, and Olle Risberg. The authors object to the counterfactual comparative account of harm. They argue that, on this account, an action that would harm the agent might leave her better off than would some alternative action that would benefit her, and they object to this implication. By appealing to group or plural harm, I argue that their objection fails.

This paper critically assesses whether quantum entanglement can be made compatible with Humean supervenience. After reviewing the prima facie tension between entanglement and Humeanism, I outline a recently-proposed Humean response, and argue that it is subject to two problems: one concerning the determinacy of quantities, and one concerning its relationship to scientific practice.

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)

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)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...) their usual foundational roles. (shrink)

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)

This paper assessed whether belief in conspiracy theories was associated with a particularly cognitive style (worldview). The sample comprised 223 volunteers recruited via convenience sampling and included undergraduates, postgraduates, university employees and alumni. Respondents completed measures assessing a range of cognitive-perceptual factors (schizotypy, delusional ideation and hallucination proneness) and conspiratorial beliefs (general attitudes towards conspiracist thinking and endorsement of individual conspiracies). Positive symptoms of schizotypy, particularly the cognitive-perceptual factor, correlated positively with conspiracist beliefs. The best predictor of belief in conspiracies (...) was delusional ideation. Consistent with the notion of a coherent conspiratorial mindset, scores across conspiracy measures correlated strongly. Whilst findings supported the view that belief in conspiracies, within the sub-clinical population, was associated with a delusional thinking style, cognitive-perceptual factors in combination accounted for only 32% of the variance. (shrink)

Jerome Wakefield’s harmful dysfunction analysis of medical disorder is an influential hybrid of naturalist and normative theories. In order to conclude that a condition is a disorder, according to the HDA, one must determine both that it results from a failure of a physical or psychological mechanism to perform its natural function and that it is harmful. In a recent issue of this journal, I argued that the HDA entails implausible judgments about which disorders there are and how they are (...) individuated. The same arguments apply to other views that incorporate a harm criterion. More recently, David G. Limbaugh has modified the HDA by providing a novel account of the way in which a disorder must be harmful. Here, I briefly review the relevant issues and then critically assess Limbaugh’s account. I argue in the end that Limbaugh’s revisions do not succeed in making accounts like the HDA more attractive. (shrink)

Troubadour of truth, by R. E. Brennan.--Reflections on necessity and contingency, by Jacques Maritain.--Intellectual cognition, by Rudolf Allers.--The problem of truth, J. K. Ryan.--The ontolgical roots of Thomism, by Hilary Carpeuter.--The role of habitus in the Thomistic metaphysics of potency and act, by V. J. Bourke.--The nature of the angels, by J. O. Riedl.--The dilemma of being and unity, by A. C. Pegis.--Prudence, the incommunicable wisdom, by C. J. O'Neil.--A question about law, by M. J. Adler.--The economic philosophy of (...) Aquinas, by J. A. Ryan.--Beyond the crisis of liberalism, by Y. R. Simon.--The fate of representative government, by Walter Farrell.--The Thomistic concept of education, by R. J. Slavin.--The perennial theme of beauty, by Immanuel Chapman.--Epilogue, by H. T. Schwartz.--Bibliography (p.[363]-419). (shrink)

This book focuses on the nature and importance of harm by providing a sustained defense of the counterfactual comparative account, in particular by extending the account to allow for a certain kind of plural or collective harm. According to the counterfactual comparative account, an event harms a person provided that she would have been better off had it not occurred. On the account defended in this book, there are cases in which some events harm a given individual even though none (...) of them by itself harms the individual. This account aims to solve the most serious problem facing the counterfactual comparative account, a problem involving cases of preemption. In such cases, an intuitively harmful event does not make the victim worse off, since if it had not occurred, another event would have had a similar adverse influence on the victim’s well-being. The book also contains discussion of various competing accounts of harm, and it replies to other challenges to the counterfactual comparative account. It argues, moreover, that an action that would harm a person does not thereby have a strong moral reason against it; if there is a strong moral reason against a harmful act, the reason is due primarily to some of the act’s other features. (shrink)

If categorical equivalence is a good criterion of theoretical equivalence, then it would seem that if some class of mathematical structures is represented as a category, then any other class of structures categorically equivalent to it will have the same representational capacities. Hudetz (2019a) has presented an apparent counterexample to this claim; in this note, I argue that the counterexample fails.

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)

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.

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)

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)

How may we characterize the intrinsic structure of physical quantities such as mass, length, or electric charge? This article shows that group-theoretic methods—specifically, the notion of a free and transitive group action—provide an elegant way of characterizing the structure of scalar quantities, and uses this to give an intrinsic treatment of vector quantities. It also gives a general account of how different scalar or vector quantities may be algebraically combined with one another. Finally, it uses this apparatus to give a (...) simple intrinsic treatment of Newtonian gravitation. (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)

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.

Affective forecasting often drives decision-making. Although affective forecasting research has often focused on identifying sources of error at the event level, the present investigation draws upon the “realistic paradigm” in seeking to identify factors that similarly influence predicted and actual emotions, explaining their concordance across individuals. We hypothesised that the personality traits neuroticism and extraversion would account for variation in both predicted and actual emotional reactions to a wide array of stimuli and events (football games, an election, Valentine's Day, birthdays, (...) happy/sad film clips, and an intrusive interview). As hypothesised, individuals who were more introverted and neurotic anticipated, correctly, that they would experience relatively more unpleasant emotional reactions, and those who were more extraverted and less neurotic anticipated, correctly, that they would experience relatively more pleasant emotional reactions. Personality explained 30% of the concordance between predicted and actual emotional reactions. Findings suggest three purported personality processes implicated in affective forecasting, highlight the importance of individual-differences research in this domain, and call for more research on realistic affective forecasts. (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.

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 prevailing view of Advaita Vedānta as world negating and disengaged with worldly activity provides little space for an ethic of environmental care, or a psychology for eco-resilience beyond passive indifference. However, many sources for environmentalism within the Advaita Vedānta tradition and its canon of texts remain untapped. In this paper, I explore the ritual ecology found in chapter three of the Bhagavadgītā as the ground to construct an Advaitin ecotheology and ecopsychology. This all-encompassing ritual ecology, described as a sacrificial (...) wheel, provides meaning to the world and views individuals as essential causative participants within it. Individuals have the religious and moral responsibility to perpetuate the wheel through a relationship of mutual reciprocity with the gods based on an economy of food. I extend this ritual ecology to include moral responsibility to the environment and explain how it provides a religious psychology and practice for handling our anxiety of imminent ecological catastrophe. (shrink)

Should we be Roschians about the concept of disease, rather than taking a classical approach? A classical concept of disease defines disease in terms of necessary and sufficient conditions; any things and only things which meet this definition are members of the class. In Roschian concepts of disease, it is supposed that degree of similarity to a prototype determines membership in the class of diseases. In this paper, the two approaches are pitched against one another in a series of tests (...) which appear on first sight to favor Roschian accounts.These tests are 1) the capacity to accommodate the variety of the class of disease, 2) the capacity to explain controversies about disease attribution, and 3) the capacity to... (shrink)

Abstract:In this paper, I apply John Cage’s wide musical embrace of sound to the field of music education and explore its curricular and practical implications. In particular, I ask music teachers to consider themselves teachers of sound, or “sound teachers.” I argue that privileging sound as our chief concern leads us to reconsider the ways we speak about music, the offerings we include in our music curricula, and the ways we teach (about) sound. In particular, I suggest that application of (...) Cage’s ideas compels us to permit still more genres and styles in the classroom and curriculum, to emphasize activities that allow students to manipulate a diverse palette of sound types (for example, electronic composition), and to teach in ways that expand and diversify rather than narrow and limit students’ relationships with sound. I close by considering how Cage’s ideas of “purposeless play” and “purposeful purposelessness” orient our goals toward making students more attentive to and invested in the world of sound and the sound of the world. (shrink)

The opening of the fourth session of The Neutral — the course given by Roland Barthes at the Collège de France in 1977–8 — is marked by a dramatic spillage of ink. Rather than take this as an incidental, trivial moment, I read it as one of the many ‘ink blots’ that colour the work of Barthes. Tracing his ‘almost obsessive relation to writing instruments’ and the material act of inscription, this essay relates the ‘ink blots’ to the development of (...) a ‘non-arrogant’, non-expressive, non-idealistic theory of language in The Neutral and other texts by Barthes. (shrink)

A central issue in the philosophical debates over general relativity concerns the status of the metric field: should it be regarded as part of the background arena in which physical fields evolve, or as a physical field itself? In this paper, we approach this debate through its relationship to the so-called "Problem of Space": the problem of determining which abstract, mathematical geometries are candidate descriptions of physical space. In particular, we explore the way that Hermann Weyl tackled the Problem of (...) Space in the wake of general relativity, and argue that Weyl’s proposed solution reveals a “middle way” between bare-manifold and manifold-plus-metric accounts of spacetime. (shrink)

Discourse, in general, and gist reasoning specifically, are valuable tools to explore cognitive brain health across the life span. Gist reasoning is a higher-order cognitive function that entails a constructive/integrative process in which explicit content of the stimuli is combined with personal knowledge to generate meaning that is transformed and personally salient. In this article, we discuss gist reasoning ability as a marker of cognitive brain health and its potential in differentiating normal cognitive brain health from brain diseases. We highlight (...) the benefits of gist reasoning training as an approach to strengthening cognitive health across the life span. (shrink)