Can the right to private property be claimed as one of the `rights of mankind'? This is the central question of this comprehensive and critical examination of the subject of private property. Jeremy Waldron contrasts two types of arguments about rights: those based on historical entitlement, and those based on the importance of property to freedom. He provides a detailed discussion of the theories of property found in Locke's Second Treatise and Hegel's Philosophy of Right to illustrate this contrast. (...) The book contains original analyses of the concept of ownership, the ideas of rights, and the relation between property and equality. The author's overriding determination throughout is to follow through the arguments and values used to justify private ownership. He finds that the traditional arguments about property yield some surprisingly radical conclusions. (shrink)

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms (...) from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns. (shrink)

"Arguing that debates over legitimacy, political violence, freedom, and justice would benefit greatly from cross-tradition theorizing, this book shows how putting analytic and continental political theory in conversation would help us to overcome these intractable problems"--.

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

"Robert Brandom" is one of the most significant philosophers writing today, yet paradoxically philosophers have found it difficult to get to grips with the details and implications of his work. This book aims to facilitate critical engagement with Brandom's ideas by providing an accessible overview of Brandom's project and the context for an initial assessment. Jeremy Wanderer's examination focuses on Brandom's inferentialist conception of rationality, and the core part of this conception that aims to specify the structure that a (...) set of performances within a social practice must have for the participants to count as sapient beings by virtue of their participation in the practice, and for the performances within the practice to have objective semantic content by virtue of their featuring within the practice. Wanderer's exploration of these two goals forms the structure to the book. It Includes: Part I that provides a structural model of linguistic practice and considers various groups of potential participants in terms of their relationships to this practice; and, Part II that examines the meaning of the performances that are caught up in this gameplaying practice. Brandom's approach to semantics is outlined and the challenge such an approach has in allowing for a representational dimension of language and thought is explored. Wanderer offers readers a valuable framework for understanding the Brandomian system and helps situate Brandom's systematic theorizing within contemporary Anglo-American philosophy. This book will be a sought after aid to reading Brandom for advanced students and philosophers engaging with his challenging body of work. (shrink)

Did Plato intend the laws of the Laws to change? While most scholars agree that there is to be legal change in Magnesia, I contend that this issue has been clouded by confusing three distinct questions: (1) whether there are legal mechanisms for changing the law in Magnesia, (2) what the attitudes of Magnesian citizens towards innovation and legal change are, and (3) whether Plato thinks the law is always the ultimate political authority. Once we separate these issues and look (...) at the relevant texts more closely, we see that the Athenian Visitor is primarily concerned with supplementing the original lawgiver’s laws and that any innovation that occurs should be consistent with this lawcode. This does not settle the question about whether it is always right to act in accordance with the existing laws, though it does provide a valuable framework for understanding a central aspect of Platonic political philosophy. Thus, although Plato’s ideal places knowledge above the law, in practice the laws of Magnesia will be fixed and authoritative for those people living under them. (shrink)

Every branch of mathematics has its subject matter, and one of the distinguishing features of logic is that so many of its fundamental objects of study are rooted in language. The subject deals with terms, expressions, formulas, theorems, and proofs. When we speak about these notions informally, we are talking about things that can be written down and communicated with symbols. One of the goals of mathematical logic is to introduce formal definitions that capture our intuitions about such objects and (...) enable us to reason about them precisely. At the most basic level, syntactic objects can be viewed as strings of symbols. For concreteness, we can identify symbols with particular set-theoretic objects, but for most purposes, it doesn't matter what they are; all that is needed is that they are distinct from one another. (shrink)

I propose and articulate a novel theory of desire, called the Revised Reward Theory. As the name suggests, the theory is based—and expands—on Arpaly and Schroeder’s (2014) Reward Theory of Desire. The initial Reward Theory identifies desires with states of the reward learning system such that for an organism to desire some P is for its reward system to treat P as a reward upon receipt. The Revised Reward Theory identifies desires with a different state of the same system, such (...) that for an organism to desire some P is for its reward system to expect or predict that P will be rewarding (roughly) prior to receipt. The difference amounts to equating desires with what we ultimately find rewarding or satisfying versus those that underlie our motivations to obtain that which we take ourselves to desire. I argue that the structure of the reward system is incompatible with the original Reward Theory but compatible with the Revised Reward Theory. I demonstrate that this difference has important philosophical implications. I focus on moral responsibility and demonstrate Arpaly and Schroeder’s argument, that addiction can mitigate moral responsibility, turns on this precise difference. Arpaly, Schroeder, and I all ascribe to a meta-theory called ‘natural kindism’ which identifies mental kinds with neurocognitive kinds. This discussion, in addition to defending a theory of desire, is intended to act as a proof of concept for natural kindism as offering a powerful framework for relating empirical results to philosophical issues. (shrink)

Most people who defend physician-assisted death (PAD) endorse the Joint View, which holds that two conditions—autonomy and welfare—must be satisfied for PAD to be justified. In this paper, we defend an Autonomy Only view. We argue that the welfare condition is either otiose on the most plausible account of the autonomy condition, or else is implausibly restrictive, particularly once we account for the broad range of reasons patients cite for desiring PAD, such as “tired of life” cases. Moreover, many of (...) the common objections to an Autonomy Only view fail once we understand the extent of the autonomy condition’s requirements—in particular, the importance of one’s values for autonomous choices. If our view is correct, then the scope of permissible PAD is broader than is currently accepted in both the philosophical literature and the law, and therefore poses an important challenge to this widely accepted view on justified PAD. (shrink)

[COMMENTARY on Walter Veit's "A Philosophy for the Science of Animal Consciousness"] The framework Veit introduces for animal consciousness turns on finding and articulating its evolutionary origins. Veit argues that consciousness first evolved as affective experience in the Cambrian period. His argument centers around the plausible need of organisms in the Cambrian for a common currency of subjective valuation. I argue that such an adaptive pressure is unlikely to result in affective experience. I review other processes that instantiate common currencies (...) of subjective valuation: unconscious (non-experiential) affect and the reward learning system and argue that each is likely to have evolved prior to affective experience. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...) standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

An enduring theme of Western philosophy is that we are all one another’s equals. Yet the principle of basic equality is woefully under-explored in modern moral and political philosophy. What does it mean to say we are all one another’s equals? Jeremy Waldron confronts this question fully and unflinchingly in a major new multifaceted account.

Recently, Lisa Feldman Barrett and colleagues have introduced the Theory of Constructed Emotions (TCE), in which emotions are constituted by a process of categorizing the self as being in an emotional state. The view, however, has several counterintuitive implications: for instance, a person can have multiple distinct emotions at once. Further, the TCE concludes that emotions are constitutively social phenomena. In this article, I explicate the TCE*, which, while substantially similar to the TCE, makes several distinct claims aimed at avoiding (...) the counterintuitive implications plaguing the TCE. Further, because of the changes that comprise the TCE*, emotions are not constitutively social phenomena. (shrink)

People tend to perceive ongoing continuous activity as series of discrete events. This partitioning of continuous activity may occur, in part, because events correspond to dynamic patterns that have recurred across different contexts. Recurring patterns may lead to reliable sequential dependencies in observers' experiences, which then can be used to guide perception. The current set of simulations investigated whether this statistical structure within events can be used 1) to develop stable internal representations that facilitate perception and 2) to learn when (...) to update such representations in a self-organizing manner. These simulations demonstrate that experience with recurring patterns enables a system to accurately predict upcoming stimuli within an event, to identify boundaries between such events based on transient increases in prediction error, and to use such boundaries to improve prediction about subsequent activities. (shrink)

“Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).

Can testimony provide reasons to believe some proposition about an artwork’s aesthetic character? Can testimony bring an agent into a position where they can issue an aesthetic judgement about that artwork? What is the epistemic value of aesthetic communication? These questions have received sustained philosophical attention. More fundamental questions about aesthetic communication have meanwhile been neglected. These latter questions concern the nature of aesthetic communication, the criteria that determine when aesthetic communication is successful, and the frequency of communicative success in (...) aesthetic communication. The neglect of these questions is a serious oversight, not least because they bear directly on each of the other questions listed. This paper’s focus is the more fundamental set of questions. I argue for a restricted form of communicative pessimism. Discerning aesthetic communication about an artwork typically fails unless its recipient is both acquainted with that artwork and able to coordinate with the speaker on an aesthetic understanding of it. I arrive at this conclusion by challenging the standard conception of the nature of aesthetic communication that the literature presupposes, as well as an accompanying criterion of communicative success. I introduce an alternative view. In closing I relate my discussion to the former set of questions. (shrink)

Although forgiveness is often taken to bear a close connection to the value of reconciliation, there is a good deal of scepticism about its role in situations where there is no consensus on the moral complexion of the past and no admission of guilt on the part of the perpetrator. This scepticism is typically rooted in the claims that forgiveness without perpetrator acknowledgement aggravates the risk of recidivism; yields a substandard and morally compromised form of political accommodation; and comes across (...) as patronizing and offensive to the recipient, thereby causing further alienation. In this article, my aim is to show, firstly, that none of these arguments is decisive and, secondly, that forgiveness is a suitable object of political concern in the absence of cross-community consensus on the rights and wrongs of a conflict. In this way, I aim to demonstrate that forgiveness deserves to be taken seriously as a means to civic reconciliation in a broader range of situations than many have allowed. (shrink)

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.

Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...) of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical factions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the opposite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. In his textbook on the philosophy of mathematics (Shapiro, 2000), Stewart Shapiro characterizes this attitude as ‘philosophy last, if at all.’. (shrink)

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...) conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. (shrink)

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, (...) we show that, for any nonexpansive linear operator T on a separable Hilbert space and any element f , it is possible to compute a bound on the rate of convergence of Anf from T , f , and the norm f ∗ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2 (X ), then there is a computable bound on the rate of convergence of the sequence Anf. (shrink)

In this paper, I introduce an account of aesthetic understanding. Recent discussions of aesthetic understanding have associated it with aesthetic justification and with understanding why, for example, a given object is aesthetically valuable. I introduce a notion of aesthetic understanding as a form of objectual understanding, which I refer to as ‘appreciative understanding’. Appreciative understanding is related to and partly constituted by an agent’s capacity to comprehend and experience an artwork holistically and to communicate effectively regarding its particular aesthetic character (...) and value. I then argue for the understanding account of aesthetic judgement on which the paradigmatic form of aesthetic judgement is grounded on appreciative understanding. This argument partly consists in demonstrating how the understanding account can explain the autonomy of aesthetic judgement. In closing, I explore the potential of the understanding account to explain the structure of our appreciative practices. That is, I put forward the view that our appreciative practices are structured so as to promote appreciative understanding. (shrink)

is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.

The philosophy of mathematics has long been concerned with deter- mining the means that are appropriate for justifying claims of mathemat- ical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary math- ematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The as- sociated values are often loosely classied as (...) aspects of \mathematical understanding." Meanwhile, in a branch of computer science known as \formal ver- ication," the practice of interactive theorem proving has given rise to software tools and systems designed to support the development of com- plex formal axiomatic proofs. Such eorts require one to develop models of mathematical language and inference that are more robust than the the simple foundational models of the last century. This essay explores some of the insights that emerge from this work, and some of the ways that these insights can inform, and be informed by, philosophical theories of mathematical understanding. (shrink)

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...) of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

Most of us believe that partiality applies in a broad range of relationships. One relationship on which there is much disagreement is co-nationality. Some writers argue that co-national partiality is not justified in certain cases, like killing in war, since killing in defense of co-nationals is intuitively impermissible in other contexts. I argue that this approach overlooks an important structural feature of partiality—namely, that its scope is sometimes restricted. In this essay, I show how some relationships that generate reasons of (...) partiality are restricted in scope—that is, they generate reasons within particular contexts or with respect to particular goods. I then argue that co-national partiality is scope restricted. I then show how this fact helps proponents of co-national partiality overcome the aforementioned objection to its application in cases like war. (shrink)

Abstract below. The published version of this article is available open access at The Monist's website. Part of Plato’s complaint about the cognitive status of art cites the pollution of aesthetic cognition by the affective side of our natures. Schopenhauer, by contrast, takes aesthetic cognition to transcend (some of) the limitations of everyday cognition precisely because in it agents become the “pure, will-less subject of cognition” (WWR I 219). On the orthodox reading of his later philosophy, Nietzsche scorns Plato and (...) Schopenhauer’s association of the value of art with its truthfulness. I challenge the orthodox reading and argue that one strand of art’s value is its penetrative cognitive power. In contrast to Schopenhauer, Nietzsche understands this cognitive power to result from the embracing and exploitation of the affects in the artistic process rather than their extirpation. (shrink)

In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: some (...) deal with the ability of computers to deliver appropriate “evidence” for mathematical assertions, a notion that is explored in Section 4, while others deal with the ability of computers to deliver appropriate mathematical “understanding,” a notion that is considered in Section 5. Final thoughts are provided in Section 6. (shrink)

I argue that addiction is not an appropriate category to support generalizations for the purposes of scientific prediction. That is, addiction is not a natural kind. I discuss the Homeostatic Property Cluster (HPC) theory of kinds, according to which members of a kind share a cluster of properties generated by a common mechanism or set of mechanisms. Leading accounts of addiction in literature fail to offer a mechanism that explains addiction across substances. I discuss popular variants of the disease conception (...) and demonstrate that at least one class of substances that fails to confirm a major prediction of each account. When no mechanism can be found to explain the occurrence of the relevant properties in members of a category, the HPC view suggests that we revise our categories. I discuss options offered by the HPC view, including category revision and category replacement. I then conclude that talk of addiction as a prediction-supporting category should be replaced with categories such as “S-addiction” and “T-addiction,” where S and T are substances or sets of substances of abuse, as these categories are genuine natural kinds. (shrink)

From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.

Since his death in 1986, Michel de Certeau's reputation as a thinker has steadily grown both in France and throughout the English-speaking world. His work is extraordinarily innovative and wide-ranging, cutting across issues in historiography, literary and cultural studies, anthropology, sociology, theology, philosophy and psychoanalysis. This book represents the first full-length study of Certeau's thought. It is organized around the central theme of interpretation and alterity, which Ahearne uses to illuminate Certeau's work as a whole. The author also examines Certeau's (...) theory and practice of historiography; his reflection on the relations between changing historical forms of writing, reading and orality; and his distinction between the "strategic" programmes of the politically powerful and the "tactics" of the relatively powerless. Ahearne places Certeau's work in its general intellectual context, relating it to the views of important contemporary thinkers, such as Pierre Bourdieu and Michel Foucault, and demonstrating the decisive importance to Certeau's thought of the writings of the early modern mystics and travellers. This book constitutes an excellent critical introduction to Certeau's work, while also providing a comprehensive and nuanced reading for those already familiar with his thought. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.

The COVID-19 global pandemic has shone a light on several important ethical questions, ranging from fairness in resource allocation to the ethical justification of government mandates. In addition to these institutional issues, there are also several ethical questions that arise at the interpersonal level. This essay focuses on several of these issues. In particular, I argue that, despite the insistence in public health messaging that avoiding infecting others constitutes ‘saving lives’, virus transmission that results in death constitutes an act of (...) killing. Whether this killing is wrongful depends on several factors. I consider one intuitively plausible view—namely, that in many cases, killing via unintentional transmission is not wrongful, because the parties in question have implicitly waived their rights against this harm, often via reciprocal risk imposition. I argue that this view is mistaken, but that its central insight can be better captured by identifying the appropriate standards of blame that we ought to apply during a pandemic. I conclude by showing how these conclusions can be fruitfully applied to certain institutional questions, such as helping to justify restricting government mandates. (shrink)

We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.

Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as special (...) cases of the cut-elimination theorems for intuitionistic logic. (shrink)

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.

The present edition of 'Of the Limits of the Penal Branch of Jurisprudence' supersedes 'Of Laws in General,' edited by H.L.A. Hart and published by the Athlone Press in 1970, as a volume in The Collected Works of Jeremy Bentham." --P. xi.

At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...) from calculation gave rise to concerns as to whether such methods were meaningful, or appropriate to mathematics. The discovery of paradoxes stemming from overly naive use of set-theoretic language and methods led to even more pressing concerns as to whether the modern methods were even consistent. This led to heated debates in the early twentieth century and what is sometimes called the “crisis of foundations.” In lectures presented in 1922, David Hilbert launched his Beweistheorie, or Proof Theory, which aimed to justify the use of modern methods and settle the problem of foundations once and for all. This, Hilbert argued, could be achieved as follows. (shrink)

We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.

Time Wars is for anyone who has ever wondered why, in a culture so obsessed with efficiency, we seem to have so little time we can call our own. A courageous, thought-provoking challenge to conventional wisdom.

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer\'edi's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength (...) of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the $\omega^{\omega^\omega}$th level. (shrink)

For the first time Entropy has been completely revised and updated to include a new subtitle which reflects the expanded focus on the greenhouse effect--the largest crisis ever to face mankind.

Currently the dominant interpretation of Hobbes in the field of moral and political philosophy is as a social contract theorist: that he legitimates moral rules and sovereign power by arguing that we would agree we are better off obeying a sovereign than living in a state of nature, and that we are best off if that sovereign is an absolute monarch. There are interesting alternatives to this reading of Hobbes—Warrender’s divine-command interpretation and Boonin-Vail’s virtue theory interpretation, to name just two—but (...) it is not my purpose here to debate their relative merits. Rather, I want to comment on one of the main features of the social contract view, namely, the means of maintaining political stability. (shrink)

As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. On the one hand, formal languages fail to account for a number of features of informal mathematical language that are essential to the communicative and inferential goals of the subject. On the other hand, many of these features are independent of the choice of a formal foundation, so grounding their analysis on a (...) particular choice of a formal system introduces unnecessary specificity. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants and their libraries. (shrink)

In this paper I direct attention to one feature of Hacking’s recent work on styles of reasoning and argue that this feature is of far greater philosophical significance than Hacking’s limited discussion of this suggests. The feature in question is his use of ‘legendary beginnings’ in setting out a given style, viz. the method of introducing a style of reasoning by recounting a popular and quasi-mythical narrative that ties the crystallisation of that style to a particular person in a particular (...) place and at a particular time. Whilst Hacking both deploys and discusses this method, his comments suggest that this is primarily a stylistic device employed for reasons of expedience. In contrast, it is argued here that recounting the legendary origins of a style of reasoning affords a distinctive way of vindicating that style, a vindication from within the style itself. (shrink)

Despite a significant amount of literature debating the efficiency of high-risk pools in health insurance, dramatically less has been written about their normative implications. The present article takes the route less traveled by setting aside the question of efficiency to argue that the use of high-risk pools creates some serious normative concerns. The article explores these concerns by dividing them on two fronts. First, as regards the social-recognitional status of those who are forced into the high-risk pool. Second, as regards (...) a general concern of distributive justice, namely fairness in access to resources. The author argues that regardless of the veracity of arguments which laud the efficiency of high-risk pools, their use in health insurance is unjust because of the herein explained implications for social recognition and distributive justice. (shrink)