We define and study a new notion called k-immunity that lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that θ does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev [7]. We apply the result to show that Φ′ does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new (...) characterization of effectively simple sets and show that simple sets are not btt-cuppable. (shrink)

There is a family of questions in relativized complexity theory--weak analogs of the Friedberg Jump-Inversion Theorem--that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets. i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any (...) set X there is an aw2-generic set G such that $\mathbf{NP}^G \cap co-\mathbf{NP}^G \nsubseteq \mathbf{P}^{G \oplus X}$ . (On the other hand, if G is 1-generic, then $\mathbf{NP}^G \cap co-\mathbf{NP}^G \subseteq \mathbf{P}^{G \oplus \mathrm{SAT}}$ . where SAT is the NP-complete satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets differ radically from those of 1-generic sets: every degree above O' contains an aw2-generic set: no aw2-generic set exists below any incomplete r.e. set; there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0' is the jump of an aw2-generic degree. (shrink)

Higman essentially showed that if A is any language then SUBSEQ(A) is regular, where SUBSEQ(A) is the language of all subsequences of strings in A. Let s1, s2, s3, . . . be the standard lexicographic enumeration of all strings over some finite alphabet. We consider the following inductive inference problem: given A(s1), A(s2), A(s3), . . . . learn, in the limit, a DFA for SUBSEQU). We consider this model of learning and the variants of it that are usually (...) studied in Inductive Inference: anomalies, mind-changes, teams, and combinations thereof. This paper is a significant revision and expansion of an earlier conference version [10]. (shrink)

In this paper, I explore some parallels and dissimilarities between aesthetic appreciation that takes as its focus art objects and that which focuses on natural objects. I cover three areas. The first deals with general approach, whether a paradigm of engagement is more appropriate to environmental aesthetics than one of detachment and disinterest. The second theme is about preservation and whether the appropriate model is static or dynamic. The final theme is about environmental criticism and the application of aesthetic theory (...) to arguments for preservation. (shrink)

ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us (...) to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)

A Brief History of Time: From the Big Bang to Black Holes is a popular-science book on cosmology (the study of the origin and evolution of the universe) by British physicist Stephen Hawking. It was first published in 1988. Hawking wrote the book for readers who have no prior knowledge of the universe and people who are interested in learning.

Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but (...) conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction. (shrink)

_Plain Aesthetics_ is an introduction to philosophical aesthetics and the philosophy of art written for all audiences. While students studying philosophy will find it informative, it is specifically constructed to be accessible to anyone, even those with no background in philosophy. It contains no jargon or technical language, except where such terms are defined at their point of use. Philosophers and theorists are discussed only where appropriate, and their views explained in context. _Plain Aesthetics_ is written as a conversation between (...) the author and the reader, and employs a great many examples of fascinating and influential artworks. Images and other works are presented to the reader both within the text and through an innovative interactive system. This book makes aesthetics accessible to everyone. (shrink)

LEGO and math are both about what one do with the objects. In LEGO, he/she can build sets following the instructions, or alternatively dump a whole bunch of LEGO on the floor and build whatever he/she like. In math, he/she have a similar freedom to create new things, solve problems, and play around. Geometry makes far greater use of pictures and diagrams than tends to be the case for other areas of mathematics. This chapter focuses on diagrammatic proofs as a (...) key case where proofs guide him/her through a series of actions. If one accepts the LEGO account of diagrammatic proofs then he/she has strongly sided with the geometers against Plato. The main principle of departure from Plato is to focus on mathematical activities. The same thought is meant to apply to LEGO: the particular bricks are only important insofar as they facilitate the things we can practically do with them. (shrink)

Adult number representations can belong to either of two types. One is discrete, language-specific, and culturally-derived; the other is analog and language-independent. Quantitative evidence is presented to demonstrate that analog number representations are adult-like in young children. Five- to 7-year-olds accurately estimated rapidly presented groups of 5--11 items. Groups were presented in random order and random arrangements controlling for overall area. Children's data were qualitatively, and to some degree quantitatively, similar to adult data with one exception: the ratio of the (...) standard deviation of estimates to mean estimates decreased with age. (shrink)

What sorts of epistemic virtues are required for effective mathematical practice? Should these be virtues of individual or collective agents? What sorts of corresponding epistemic vices might interfere with mathematical practice? How do these virtues and vices of mathematics relate to the virtue-theoretic terminology used by philosophers? We engage in these foundational questions, and explore how the richness of mathematical practices is enhanced by thinking in terms of virtues and vices, and how the philosophical picture is challenged by the complexity (...) of the case of mathematics. For example, within different social and interpersonal conditions, a trait often classified as a vice might be epistemically productive and vice versa. We illustrate that this occurs in mathematics by discussing Gerovitch’s historical study of the aggressive adversarialism of the Gelfand seminar in post-war Moscow. From this we conclude that virtue epistemologies of mathematics should avoid pre-emptive judgments about the sorts of epistemic character traits that ought to be promoted and criticised. (shrink)

A comprehensive reassessment of the concept of mimesis in the history of ancient Greek aesthetics and philosophy of art, with particular attention to Plato, Aristotle, Hellenistic philosophy, and neoplatonism. There is also a wide-ranging review of arguments pro and contra the idea of artistic mimesis from the Renaissance to modern literar theory. The book challenges standard accounts in numerous respects and builds a new dialectical model with which to make sense of the entire history of mimeticist thinking in aesthetics.

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...) to highlight a potential danger of intellectual enculturation. (shrink)

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on (...) the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour. (shrink)

Aboutness has been studied from any number of angles. Brentano made it the defining feature of the mental. Phenomenologists try to pin down the aboutness-features of particular mental states. Materialists sometimes claim to have grounded aboutness in natural regularities. Attempts have even been made, in library science and information theory, to operationalize the notion. But it has played no real role in philosophical semantics. This is surprising; sentences have aboutness-properties if anything does. Aboutness is the first book to examine through (...) a philosophical lens the role of subject matter in meaning. A long-standing tradition sees meaning as truth-conditions, to be specified by listing the scenarios in which a sentence is true. Nothing is said about the principle of selection--about what in a scenario gets it onto the list. Subject matter is the missing link here. A sentence is true because of how matters stand where its subject matter is concerned. Stephen Yablo maintains that this is not just a feature of subject matter, but its essence. One indicates what a sentence is about by mapping out logical space according to its changing ways of being true or false. The notion of content that results--directed content--is brought to bear on a range of philosophical topics, including ontology, verisimilitude, knowledge, loose talk, assertive content, and philosophical methodology. Written by one of today's leading philosophers, Aboutness represents a major advance in semantics and the philosophy of language. (shrink)

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little (...) attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

Extinction is a concept of rapidly growing importance, with the world currently in the sixth mass extinction event and a biodiversity crisis. However, the concept of extinction has itself received surprisingly little attention from philosophers. I will first argue that in practice there is no single unified concept of extinction, but instead that its usage divides between descriptive, epistemic, and declarative concepts. I will then consider the epistemic challenges that arise in ascertaining whether a species has gone extinct, and how (...) these lead to serious limitations on responsibility and accountability for preventing extinctions. I will propose two conceptual engineering changes to our understanding of extinction. Firstly, to use twin epistemic concepts of extinction, corresponding to high and low epistemic demands. Secondly, to explicitly recognize that in many contexts extinction is a thick concept that contains an evaluative component, which therefore motivates explicit ethical considerations such as the use of precautionary principles. (shrink)

In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis of formality and informality. (...) I use this perspective to respond to the arguments for the inconsistency of mathematics made by Beall and Priest, presenting problems with the assumptions needed concerning formalisation, the unity of informal mathematics and the relation between the formal and informal. (shrink)

In Return to Reason, Stephen Toulmin argues that the potential for reason to improve our lives has been hampered by a serious imbalance in our pursuit of ...

Stephen Hetherington's prominent career within epistemology has been a series of distinctive, bold, varied and provocative arguments and ideas. Bringing together Hetherington's unique body of writing for the first time, this collection features previously published as well as new material that link his approaches to key issues including knowledge, justification, fallibility, scepticism and the Gettier Problem. Advancing our understanding of the systemic nature of Hetherington's thinking, Stephen Hetherington on Epistemology presents his distinctive perspective on some of philosophy's central (...) questions about knowledge - an inviting blend of forensic detail and 'big picture' proposals. (shrink)

Defenders of pragmatic encroachment in epistemology (or what I call practicalism) need to address two main problems. First, the view seems to imply, absurdly, that knowledge can come and go quite easily—in particular, that it might come and go along with our variable practical interests. We can call this the stability problem. Second, there seems to be no fully satisfying way of explaining whose practical interests matter. We can call this the “whose stakes?” problem. I argue that both problems can (...) be addressed in roughly the same terms. More exactly, I argue that by first clarifying the whose stakes? problem an answer to the stability problem naturally falls out. (shrink)

Elizabeth Anscombe famously argued that "modern moral philosophy" centrally involved unsupported notions of obligation and culpability. Modern Moral Philosophy: From Grotius to Kant exhibits, for the first time, resources that modern moral philosophers had to respond to Anscombe's challenge, also enhancing our own philosophical grasp of morality and its foundations.

In this paper I elucidate various ways in which understanding can be seen as an excellence of the mind or intellectual virtue. Along the way, I take up the neglected issue of what it might mean to be an “understanding person”—by which I mean not a person who understands a number of things about the natural world, but a person who steers clear of things like judgmentalism in her evaluation of other people, and thus is better able to take up (...) different perspectives and view them with a sympathetic eye. Being an understanding person in this sense seems to be a character-level virtue that interestingly combines moral and epistemic elements; it also seems to be a virtue particularly needed in our age of deep political division, where it is commonly said that failures of mutual understanding are partly to blame for this problem. (shrink)

Keith Augustine’s critical evaluation of the essay contest sponsored by the Bigelow Institute of Consciousness Studies (BICS) is an interesting but problematic review. It mixes reasonable and detailed criticisms of the contest and many of the winning essays with a disappointing reliance on some of the most trite and superficial criticisms of parapsychological research. Ironically, Augustine criticizes the winning essays for using straw-man arguments and cherry-picked evidence even though many of his own arguments commit these same errors.

Can hybridism about moral claims be made to work? I argue it can if we accept the conventional implicature approach developed in Barker (Analysis 2000). However, this kind of hybrid expressivism is only acceptable if we can make sense of conventional implicature, the kind of meaning carried by operators like ‘even’, ‘but’, etc. Conventional implictures are a form of pragmatic presupposition, which involves an unsaid mode of delivery of content. I argue that we can make sense of conventional implicatures, but (...) doing so requires we embrace a form of pure, non-hybrid expressivism. This is a cognitivist expressivism I have developed elsewhere. We need cognitivist expressivism to make sense of how we evaluate—judge as correct or incorrect—implicature-bearing sentences. Once we embraced the possibility of this pure expressivism, we might as well be pure expressivists about normative discourse too. I show how we can do that. The motivations for a specifically hybrid theory are dialectically undercut. (shrink)

Many experimental philosophers are philosophers by training and professional affiliation, but some best work in experimental philosophy has been done by people who do not have advanced degrees in philosophy and do not teach in philosophy departments. This chapter explains that the experimental philosophy is the empirical investigation of philosophical intuitions, the factors that affect them, and the psychological and neurological mechanisms that underlie them. It explores what are philosophical intuitions, and why do experimental philosophers want to study them using (...) the methods of empirical science. The positive program in experimental philosophy shares the goal of the substantial part of traditional philosophy that is concerned with the analysis of important philosophical concepts. The negative program has implications for philosophical projects whose goal is conceptual analysis. There have been a number of responses to the challenge posed by experimental philosophy's negative program. The chapter also focuses on the expertise defense. (shrink)

Philosophical Perspectives on Art presents a series of essays devoted to two of the most fundamental topics in the philosophy of art: the distinctive character of artworks and what is involved in understanding them as art. In Part I, Stephen Davies considers a wide range of questions about the nature and definition of art. Can art be defined, and if so, which definitions are the most plausible? Do we make and consume art because there are evolutionary advantages to doing (...) so? Has art completed the mission that guided its earlier historical development, and if so, what is to become of it now? Should architecture be classified as an art form? -/- Part II turns to the interpretation and appreciation of art. What is the target and purpose of the critic's interpretation? Is interpretation primarily directed at uncovering artists' intended meanings? Can apparently contradictory interpretations of a given piece both be true? Are interpretative evaluations entailed by descriptions of a work's aesthetic and artistic characteristics? In addition to providing fresh answers to these and other central questions in aesthetics, Davies considers the nature and content of metaphor, and the relation between the expressive qualities of a work of art and the emotions of its creator. (shrink)

In November 2022, OpenAI released ChatGPT, an incredibly sophisticated chatbot. Its capability is astonishing: as well as conversing with human interlocutors, it can answer questions about history, explain almost anything you might think to ask it, and write poetry. This level of achievement has provoked interest in questions about whether a chatbot might have something similar to human intelligence or even consciousness. Given that the function of a chatbot is to process linguistic input and produce linguistic output, we consider the (...) question whether a sophisticated chatbot might have inner speech. That is: Might it talk to itself, internally? We explored this via a conversation with ‘Playground’, a chatbot which is very similar to ChatGPT but more flexible in certain respects. We asked it questions which, plausibly, can only be answered if one first produces some inner speech. Here, we present our findings and discuss their philosophical significance. (shrink)

Why is there anything, rather than nothing at all? This question often serves as a debating tactic used by theists to attack naturalism. Many people apparently regard the question—couched in such stark, general terms—as too profound for natural science to answer. It is unanswerable by science, I argue, not because it’s profound or because science is superficial but because the question, as it stands, is ill-posed and hence has no answer in the first place. In any form in which it (...) is well-posed, it has an answer that naturalism can in principle provide. The question therefore gives the foes of naturalism none of the ammunition that many on both sides of the debate think it does. (shrink)

How is modern psychotherapy impacted when it is approached from the presence and understanding of the unconditioned mind? What happens when therapists are able to function as a sacred mirror for their clients' essential nature, reflecting back not only the contents of awarenessùthoughts, feelings and sensationsùbut awareness itself? Informed by their direct experience as well as by nondual teachings from both eastern and western wisdom traditions, the authors take a fresh look at what psychotherapy can be. These seminal essays will (...) challenge and inspire readers to approach psychotherapy in a new wayùas a potential portal for experiencing their deepest nature as free and joyful beings.Seasoned clinicians, Dan Berkow, Stephan Bodian, Dorothy Hunt, Sheila Krystal, Lynn Marie Lumiere, Richard Miller, John Prendergast, John Welwood, Jennifer Welwood and Bryan Wittine, and innovative western spiritual teachers, Adyashanti and Peter Fenner, explore critical issues at the interface of psychology and spirituality from a nondual perspective. (shrink)

Tracing the history of American sociology since the Civil War, the authors of this important volume explain the field′s diversity, its lack of unifying paradigms, its broad, eclectic research agenda and its general weakness as an institutional force in either academia or the policy arena. They highlight the equivocal and often contradictory missions that sociologists prescribe for themselves and the variable nature of human, financial and intellectual resources available to the profession.

A compelling investigation into one of the most coveted and cherished ideals, "Wisdom" also chronicles the efforts of modern science to penetrate the mysterious nature of this timeless virtue.

Punishment and Reparations are sometimes held to express retaliatory emotions whose object is to strike back against a victimizer. I begin by examining a version of this idea in Mill's writings about natural resentment and the sense of justice in Chapter V of Utilitarianism. Mill's view is that the ?natural? sentiment of resentment or ?vengeance? that is at the heart of the concept of justice is essentially retaliatory, therefore has ?nothing moral in it,? and so must be disciplined or moralized (...) from without by the desire to promote the general welfare. I argue to the contrary that if ?reactive attitudes? like resentment and moral blame are understood as Strawson analyzed them, as essentially ?interpersonal? or ?second personal?, they have a different content and function. They implicitly demand respect in a way that also expresses respect for the victimizer as a member of mutually accountable community of moral equals. Some implications of this idea are discussed for the ?expressive theory of punishment? and ?civil recourse? theories of torts. (shrink)

From 1876, this influential work in the history of ideas focuses on the eighteenth-century deist controversy and its effects.

A controversial life -- School life -- Family life -- Public life/private life -- Love life -- Writing life -- Pirate life -- An armed and neutral life -- A life concluded -- A life continued -- Afterword -- Overviews of the works of Søren Kierkegaard.

"Stretta è la porta e angusta è la via." I fondamentalisti di ogni sorta ispirano la propria vita a questo venerabile motto e quindi devono brandire senza sosta le proprie spade in una continua battaglia mentale contro le opinioni antitetiche degli apostati e dei rivali.