: In the Cambridge Conferences Lectures of 1898 Peirce defines a continuum as a "collection of so vast a multitude" that its elements "become welded into one another." He links the transinfinity (the "vast multitude") of a continuum to the confusion of its elements by a line of mathematical reasoning closely related to Cantor's Theorem. I trace the mathematical and philosophical roots of this conception of continuity, and examine its unresolved tensions, which arise mainly from difficulties in Peirce's theory of (...) collections. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

In the last decade of his life C.S. Peirce began to formulate a purely geometrical theory of continuity to supersede the collection-theoretic theory he began to elaborate around the middle of the 1890s. I argue that Peirce never succeeded in fully formulating the later theory, and that while that there are powerful motivations to adopt that theory within Peirce’s system, it has little to recommend it from an external perspective.

The Archimedean Axiom is often held to be an intuitively obvious truth about the geometry of physical space. After a general discussion of the varieties of geometrical intuition that have been proposed, I single out one variety which we can plausibly be held to have and then argue that it does not underwrite the intuitive obviousness of the Archimedean Axiom. Generalizing that result, I conclude that the Axiom is not intuitively obvious.

Mathematical naturalism forbids philosophical interventions in mathematical practice. This principle, strictly construed, places severe constraints on legitimate philosophizing about mathematics; it is also arguably incompatible with mathematical realism. One argument for the latter conclusion charges the realist with inability to take a truly naturalistic view of the Gödel Program in set theory. This argument founders on the disagreement among mathematicians about that program's prospects for success. It also turns out that when disagreements run this deep it is counterproductive to take (...) too narrow a view of how philosophers of mathematics may legitimately proceed. (shrink)

Can a scientific naturalist be a mathematical realist? I review some arguments, derived largely from the writings of Penelope Maddy, for a negative answer. The rejoinder from the realist side is that the irrealist cannot explain, as well as the realist can, why a naturalist should grant the mathematician the degree of methodological autonomy that the irrealist's own arguments require. Thus a naturalist, as such, has at least as much reason to embrace mathematical realism as to embrace irrealism.

The philosophy of mathematics plays a vital role in the mature philosophy of Charles S. Peirce. Peirce received rigorous mathematical training from his father and his philosophy carries on in decidedly mathematical and symbolic veins. For Peirce, math was a philosophical tool and many of his most productive ideas rest firmly on the foundation of mathematical principles. This volume collects Peirce’s most important writings on the subject, many appearing in print for the first time. Peirce’s determination to understand matter, the (...) cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic. (shrink)

Reasons are considerations that figure in sound reasoning. This is considered by many philosophers to be little more than a platitude. I argue that it actually has surprising and far-reaching metanormative implications. The view that reasons are linked to sound reasoning seems platitudinous only because we tend to assume that soundness is a normative property, in which case the view merely relates one normative phenomenon (reasons) to another (soundness). I argue that soundness is also a descriptive phenomenon, one we can (...) pick out with purely descriptive terms, and that the connection between normative reasons and sound reasoning therefore provides the basis for a reductive account of reasons. Like all proposed reductions, this one must confront some version of G. E. Moore’s open question argument. I argue that a reductive view rooted in the idea that reasons figure in sound reasoning is well-equipped to meet the open question challenge head on. (shrink)

In Sect. 1 of this chapter, Matthew McGrath examines Sosa's work on the nature of truth. Sosa's chief purpose is to determine what sort of theory of truth is appropriate for truth-centered epistemology -- an epistemology that takes truth to be the goal of inquiry and which explains key epistemic notions in terms of truth. While Sosa refutes arguments from Putnam and Davidson against the correspondence theory, he is hesitant to endorse it because he doubts we have a clear (...) enough grasp of what correspondence amounts to and what the correspondents are. A truth-centered epistemologist, however, is free to work with minimalism about truth and Moorean primitivism. Part of Sosa's case for primitivism, and against minimalism, involves a comparison with Moore's account of goodness. Here McGrath notes an important dissimilarity between the two (i.e., susceptibility to open-question arguments) and suggests that this may be reason to prefer minimalism to primitivism. -/- In Sect. 2, Jeremy Fantl discusses Sosa's work on the role of truth in epistemology. Sosa seems to be motivated by a dilemma facing any account of that role on which true belief is the sole fundamental epistemic value. On the one hand, we want an account of the role of truth in epistemology to explain why we epistemically evaluate beliefs and guide our intellectual lives in the way we do. On the other hand, it should not come out that we have any sort of epistemic obligation to form beliefs about completely boring or trivial matters (e.g., about the first phone number listed on page 356 of the phone book). Sosa's attempt to resolve the dilemma is to, first of all, adopt something like a pluralism about epistemic value and, second of all, move the primary locus of epistemic evaluation from beliefs to faculties. The second part of this chapter investigates the intricacies of these maneuvers. (shrink)

This project is concerned with the attempt to diagnose certain types of deductive inferences as exhibiting failure of transmission of justification. The canonical example of alleged transmission failure is G. E. Moore’s infamous ‘proof’ of the external world, in which Moore reasoned here is a hand, therefore the external world exists. If the transmission failure diagnosis is correct, then this inference is incapable of providing a route to learning of its conclusion on the grounds that it is only (...) if one already has some justification to believe its conclusion that one can acquire justification for its premise. The thesis presents two novel problems for the transmission failure diagnosis. Firstly, this diagnosis is only appealing if it is able to safeguard epistemic closure, but it is unclear whether it is able to do this once we distinguish between propositional and doxastic justification. Secondly, a dilemma is presented for entitlement theory, the traditional framework for understanding how there could be justification for believing the conclusions at issue. The dilemma arises once we consider the question of what the apposite degree of confidence is that is licenced by an entitlement, the upshot of which is that either entitlement theory is unmotivated, or it leads to irrational doxastic attitudes. In light of these problems, an alternative framework for understanding the phenomenon of transmission failure is presented. This framework rejects a common assumption in the literature that we can only accommodate the phenomenon by appealing to internalist notions of justification. The resulting externalist framework is defended against two potentially devastating objections. Finally, implications for how these views regarding the structure of justification bear on issues in the epistemology of testimony are considered, and it is argued that a certain family of views in the literature on testimony are committed to certain problematic views regarding the structure of justification. (shrink)

Assertion is a speech act that stands at the intersection of the philosophy of language and social epistemology. It is a phenomenon that bears on such wide-ranging topics as testimony, truth, meaning, knowledge and trust. It is thus no surprise that analytic philosophers have devoted innumerable pages to assertion, trying to give the norms that govern it, its role in the transmission of knowledge, and most importantly, what assertion is, or how assertion is to be defined. In this thesis I (...) attempt to show that all previous answers to the question “What is assertion?” are flawed. There are four major traditions in the literature: constitutive norm theories of assertion, accounts that treat assertion as the expression of speaker attitudes, accounts that treat assertion as a proposal to add some proposition to the common ground, and accounts that treat assertion as the taking of responsibility for some claim. Each tradition is explored here, the leading theories within the tradition developed, and then placed under scrutiny to demonstrate flaws within the positions surveyed. I follow the work of G.E. Moore and William P. Alston, whilst drawing on the work of Robert Brandom in order to give a new bipartite theory of assertion. I argue that assertion consists in the explicit presentation of a proposition, along with a taking of responsibility for that proposition. Taking Alston's explicit presentation condition and repairing it in order to deal with problems it faces, whilst combining it with Brandom's responsibility condition, provides, I believe, the best account of assertion. (shrink)

Assertion is a speech act that stands at the intersection of the philosophy of language and social epistemology. It is a phenomenon that bears on such wide-ranging topics as testimony, truth, meaning, knowledge and trust. It is thus no surprise that analytic philosophers have devoted innumerable pages to assertion, trying to give the norms that govern it, its role in the transmission of knowledge, and most importantly, what assertion is, or how assertion is to be defined. -/- In this thesis (...) I attempt to show that all previous answers to the question “What is assertion?” are flawed. There are four major traditions in the literature: constitutive norm theories of assertion, accounts that treat assertion as the expression of speaker attitudes, accounts that treat assertion as a proposal to add some proposition to the common ground, and accounts that treat assertion as the taking of responsibility for some claim. Each tradition is explored here, the leading theories within the tradition developed, and then placed under scrutiny to demonstrate flaws within the positions surveyed. -/- I follow the work of G.E. Moore and William P. Alston, whilst drawing on the work of Robert Brandom in order to give a new bipartite theory of assertion. I argue that assertion consists in the explicit presentation of a proposition, along with a taking of responsibility for that proposition. Taking Alston’s explicit presentation condition and repairing it in order to deal with problems it faces, whilst combining it with Brandom’s responsibility condition, provides, I believe, the best account of assertion. (shrink)

G. E. Moore famously argued on the basis of semantic intuitions that moral properties are not reducible to natural properties, and therefore that moral predicates refer to nonnatural properties. This clearly represents a version of “moral realism,” but it is a testament to the strength of naturalist intuitions in contemporary philosophical debate that, insofar as one accepts Moore’s arguments, this is typically seen as a boon for antirealists rather than realists. For many philosophers worry that putative nonnatural properties (...) would be too metaphysically and epistemologically queer to be admissible into our ontology. These philosophers conclude that if moral properties cannot be understood as natural properties, then it is more reasonable to abandon commitment to moral properties than to follow Moore into nonnaturalism. -/- In this dialectical milieu, three positions have become popular. Naturalists argue that Moorean arguments are bunk; moral properties can be understood as natural properties. Constructivists argue that Moore was right that moral properties cannot be understood as objective natural properties—that is, the properties that are logically independent of the attitudes of human agents—however, quasi‐objective moral properties can be constructed out of the attitudes of human agents. And noncognitivists argue that moral predicates are nonreferential, and so the best understanding of moral discourse is one which does not construe it as committed to the existence of moral properties. -/- Russ Shafer‐Landau’s new book is distinctive in that it argues against each of these three popular positions while developing and defending a contemporary nonnaturalist version of moral realism. (shrink)

Sadness is generally seen as a negative emotion, a response to distressing and adverse situations. In an aesthetic context, however, sadness is often associated with some degree of pleasure, as suggested by the ubiquity and popularity, throughout history, of music, plays, films and paintings with a sad content. Here, we focus on the fact that music regarded as sad is often experienced as pleasurable. Compared to other art forms, music has an exceptional ability to evoke a wide-range of feelings and (...) is especially beguiling when it deals with grief and sorrow. Why is it, then, that while human survival depends on preventing painful experiences, mental pain often turns out to be explicitly sought through music? In this article we consider why and how sad music can become pleasurable. We offer a framework to account for how listening to sad music can lead to positive feelings, contending that this effect hinges on correcting an ongoing homeostatic imbalance. Sadness evoked by music is found pleasurable (1) when it is perceived as non-threatening; (2) when it is aesthetically pleasing; and (3) when it produces psychological benefits such as mood regulation, and empathic feelings, caused, for example, by recollection of and reflection on past events. We also review neuroimaging studies related to music and emotion and focus on those that deal with sadness. Further exploration of the neural mechanisms through which stimuli that usually produce sadness can induce a positive affective state could help the development of effective therapies for disorders such as depression, in which the ability to experience pleasure is attenuated. (shrink)

G. E. Moore was a central figure in twentieth-century philosophy. Along with Russell and Wittgenstein, he pioneered analytic philosophy, and his Principia Ethica shaped the contours of twentieth-century ethics. Indeed, until the publication of Rawls's A Theory of Justice, no single book in moral philosophy was to equal Principia's influence. Unfortunately, however, Principia Ethica has so dominated critical discussions of Moore's work that even experts on his moral philosophy have tended to ignore his Ethics, which he published eight (...) years later. But Ethics is Moore's only other book on moral philosophy, and one of only a handful of post-Principia publications dealing with ethics. Its detailed discussions of utilitarianism, free will, and the objectivity of moral judgements find no real counterpart in Principia while its account of right and wrong and of the nature of intrinsic value deepen our understanding of his moral philosophy. The republication of Ethics thus rounds out our understanding of Moore's ethical thought. But the book's value goes beyond its historical or scholarly interest. A short but philosophically rich text, Ethics stands independent of Principia and repays careful study in its own right. By raising a number of fundamental questions in ethics, questions that remain live today, by proffering clear, credible, and often innovative answers to them, and by doing so with a philosophical skill that is still impressive, Moore's short book is a minor classic. Almost a century after its original publication, it still amply rewards those who read it.This new edition of Moore's Ethics includes his essay 'The Nature of Moral Philosophy' as well as editorial notes, an introduction, and a guide to further reading. (shrink)

A personal leadership fable on applying principles of Zen to work and life choices The Shibumi Strategy is a little book about a big breakthrough. It tells the story of a hardworking family man who finds himself in crisis when his company closes. Through his struggle, and guidance from unlikely sources, he learns subtle lessons in the form of "personal zen" principles, coming to understand that it is often the involuntary challenge, the setbacks, that harbor the power to transform. When (...) approached as an opportunity - no easy task-unforeseen trials can sometimes result in an altogether new lease on life. Shows how "personal leadership" can lead to real (and not always easy) breakthroughs Includes key lessons on commitment, preparation, struggle, breakthrough, and transformation Is based on Shibumi, a Japanese word without literal definition the describes the height of personal excellence, elegant performance, and effortless effectiveness For those struggling with personal breakthroughs, The Shubimi Strategy offers a new way to face work and life challenges for balanced solutions. (shrink)

This book presents strategies for using systemic theory and evidence-based practice in schools to support students, the adults in their lives, and their wider communities. Beginning by introducing and explaining the Advocating Students-within-Environments (ASE) theory, each chapter then addresses a specific school-based issue, such as academic achievement, crisis, trauma, and resiliency, from a systemic and environmental lens. Practical and accessible, the chapters are filled with case examples, evidence-base interventions, and helpful tools to show how counselors can incorporate the approach into (...) their practice. Extending beyond school and student problems, this text also explores greater system functioning, such as community outreach and state level intervention, discussing advocacy and political issues. This book is essential for school-based professionals who are looking for new ways to work with students, families, and their communities. It will also be of interest to mental health professionals who work systemically, such as marriage and family therapists and community counselors. (shrink)

To corroborate the music and social bonding hypothesis, we propose that future investigations isolate specific components of social bonding and consider the influence of context. We deconstruct and operationalize social bonding through the lens of social psychology and provide examples of specific measures that can be used to assess how the link between music and sociality varies by context.

G. E. Moore was a central figure in twentieth-century philosophy. Along with Russell and Wittgenstein, he pioneered analytic philosophy, and his Principia Ethica shaped the contours of twentieth-century ethics. Indeed, until the publication of Rawls's A Theory of Justice, no single book in moral philosophy was to equal Principia's influence. Unfortunately, however, Principia Ethica has so dominated critical discussions of Moore's work that even experts on his moral philosophy have tended to ignore his Ethics, which he published eight (...) years later. But Ethics is Moore's only other book on moral philosophy, and one of only a handful of post-Principia publications dealing with ethics. Its detailed discussions of utilitarianism, free will, and the objectivity of moral judgements find no real counterpart in Principia while its account of right and wrong and of the nature of intrinsic value deepen our understanding of his moral philosophy. The republication of Ethics thus rounds out our understanding of Moore's ethical thought. But the book's value goes beyond its historical or scholarly interest. A short but philosophically rich text, Ethics stands independent of Principia and repays careful study in its own right. By raising a number of fundamental questions in ethics, questions that remain live today, by proffering clear, credible, and often innovative answers to them, and by doing so with a philosophical skill that is still impressive, Moore's short book is a minor classic. Almost a century after its original publication, it still amply rewards those who read it.This new edition of Moore's Ethics includes his essay 'The Nature of Moral Philosophy' as well as editorial notes, an introduction, and a guide to further reading. (shrink)