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)

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)

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)

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)

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 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)

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 Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as arguments and (...) dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion. (shrink)

A common description of a mathematical proof is as a logically structured sequence of assertions, beginning from accepted premises and proceeding by standard inference rules to a conclusion. Does this description match the language of proofs as mathematicians write them in their research articles? In this chapter, we use methods from corpus linguistics to look at the prevalence of imperatives and instructions in mathematical preprints from the arXiv repository. We find thirteen verbs that are used most often to form imperatives (...) in proofs, and that these show up significantly more often within proofs than in the surrounding mathematical writing. We also show that there are many more verbs used to form a diverse selection of instructions in proofs. These findings are at odds with the view of proofs as sequences of assertions. Instead, we argue in favour of the recipe model of proofs: that proofs are like recipes, giving instructions for mathematical actions to be carried out. (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...) meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

Mathematicians often conduct aesthetic judgements to evaluate mathematical objects such as equations or proofs. But is there a consensus about which mathematical objects are beautiful? We used a comparative judgement technique to measure aesthetic intuitions among British mathematicians, Chinese mathematicians, and British mathematics undergraduates, with the aim of assessing whether judgements of mathematical beauty are influenced by cultural differences or levels of expertise. We found aesthetic agreement both within and across these demographic groups. We conclude that judgements of mathematical beauty (...) are not strongly influenced by cultural difference, levels of expertise, and types of mathematical objects. Our findings contrast with recent studies that found mathematicians often disagree with each other about mathematical beauty. (shrink)

Records of online collaborative mathematical activity provide us with a novel, rich, searchable, accessible and sizeable source of data for empirical investigations into mathematical practice. In this paper we discuss how the resources of crowdsourced mathematics can be used to help formulate and answer questions about mathematical practice, and what their limitations might be. We describe quantitative approaches to studying crowdsourced mathematics, reviewing work from cognitive history (comparing individual and collaborative proofs); social psychology (on the prospects for a measure of (...) collective intelligence); human–computer interaction (on the factors that led to the success of one such project); network analysis (on the differences between collaborations on open research problems and known-but-hard problems); and argumentation theory (on modelling the argument structures of online collaborations). We also give an overview of qualitative approaches, reviewing work from empirical philosophy (on explanation in crowdsourced mathematics); sociology of scientific knowledge (on conventions and conversations in online mathematics); and ethnography (on contrasting conceptions of collaboration). We suggest how these diverse methods can be applied to crowdsourced mathematics and when each might be appropriate. (shrink)

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)

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)

In this paper, I lay out a case for why those objects of aesthetic attention which are principally characterized as natural objects should be understood not statically, as existing in merely a three-dimensional fixed state, but as dynamic, as existing in a space-time context, complete with change, movement, and flux. After this, I explain why this is important, how the dynamic nature of natural objects raises a concern for aesthetically evaluating natural objects, and how that concern may be addressed.

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.1 (2003) 40-53 [Access article in PDF] Aesthetic Experience and Aesthetic Analysis David E. W. Fenner The "raw data" that aesthetics is meant to explain is the aesthetic experience. People have experiences that they class off from other experiences and label, as a class, the aesthetic ones. Aesthetic experience is basic, and allother things aesthetic — aesthetic properties, aesthetic objects, aesthetic attitudes — (...) are secondary in their importance to aesthetic experiences. 1Considering aesthetic experience as the raw data that philosophical aesthetics seeks to explain is a relatively recent phenomenon. This was certainly not the focus in the seventeenth and eighteenth centuries. Aesthetic judgment was the focus of Shaftesbury, Hutcheson, Hume, and Kant: "How do we make meaningful judgments (hopefully real ones) about the aesthetic quality of (certain) objects and events?" But with George Santayana, John Dewey, and Jerome Stolnitz, the focus changes. Now the interest is in the aesthetic experience: what makes those experiences we label "aesthetic" special? Why do we separate those experiences from others?The movement from the Taste Theories to those focused on aesthetic experience is not a movement that is over and done with — far from it. There is still (and I think there will always be) a tension between these two very basic aspects of philosophical aesthetics. And, although I claim that aesthetic experience is the most basic thing that aesthetics studies, I recognize that this is challengeable and only true from a certain temporal viewpoint.I recently taught the most rewarding undergraduate course in aesthetics. The reason that it was so rewarding was that the students carried the class with deep and insightful discussions, and they were not shy about challenging what was coming out of my mouth. One of the challenges that informed our entire semester focused on the tension between experience and judgment. I lectured comfortably about aesthetic experience as our "raw data," and I lectured equally comfortably about how taking an aesthetic view of an object or event meant focusing primarily, if not exclusively, on [End Page 40] what is available to us through simple sensory acquaintanceship with the object. "Aesthetics," I said, "is about the sensuous aspects of our experiences." And so we could, for instance, take an aesthetic view of a Robert Mapplethorpe photograph which precluded our experiences being mired in the themes that the more famous Mapplethorpe photos take as their content. "Mapplethorpe is a great photographer," I argued, "and one can see this if one is willing to focus strictly on what meets your eye when you look at the picture." In short, I held the position that the aesthetic view is the formal one.Without saying more, what I did was conflate two different things. On the one hand, I argued that aesthetic experience is a natural part of life that aesthetics seeks to explore. On the other, I argued that appreciating something aesthetically was to appreciate its formal qualities, those qualities that one could access simply through looking, hearing, touching, for example. But these really are two different things.Aesthetic experiences, if we are to treat them as "raw data," must be explored without pre-conception, prejudice, or limitation. And, truly enough, the vast majority of aesthetic experiences are not focused exclusively, in terms of their contents, on formal or simple-sensory matters. Aesthetic experiences are, first, experiences. They are complex things, having to do with things as tidy as the formal qualities of the object under consideration and with things as messy as whether one had enough sleep the night before, whether one just had a fight with his roommate, whether one is carrying psychological baggage that is brought to consciousness by this particular aesthetic object. Later in this essay, I want to explore some of this complexity.The other side of what was happening in my class, the formal focus on the sensory as the basis for an aesthetic viewing, is not the substance of "aesthetic experience" per se. It is rather the basis of what we might call "aesthetic analysis." Aesthetic analysis has to do with separating out from our... (shrink)

"In describing Hauerwas' work as Christian ethics, one can allow that phrase its full scope of meaning. It is the work of an ethician who is thoroughly conversant with that branch of philosophy and comes to grips with its major issues. He is also firmly committed to the view that, in modifying the substantive 'ethics' with the adjective 'Christian, ' one is designating a distinct reality. . . . Hauerwas invites us to share an understanding of ethics in general and (...) of Christian ethics in particular that is a great deal subtler and more complicated than most currently popular versions of those subjects. For contemporary Christian ethics to accept his invitation will mean letting itself in for some very rigorous and versatile thinking." --America. (shrink)

This paper identifies and discusses eight perspectives one can take toward preparing to experience a garden, experiencing it, and then reflecting on that experience. The paper does not argue that one is better than another but rather explores the options, encouraging the garden visitor to adopt a plurality as means to enhance their appreciation of gardens.

Jason Stanley's "Knowledge and Practical Interests" is a brilliant book, combining insights about knowledge with a careful examination of how recent views in epistemology fit with the best of recent linguistic semantics. Although I am largely convinced by Stanley's objections to epistemic contextualism, I will try in what follows to formulate a version that might have some prospect of escaping his powerful critique.

Many philosophers believe that there is a fundamental distinction between knowing that something is the case and knowing how to do something. According to Gilbert Ryle, to whom the insight is credited, knowledge-how is an ability, which is in turn a complex of dispositions. Knowledge-that, on the other hand, is not an ability, or anything similar. Rather, knowledge-that is a relation between a thinker and a true proposition.

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role.

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role.

What is film? Why are movies important? Why do we care about them in the way we do? How do we think of the connections between the projected image and what it is actually an image of? Most movie-goers assume that they are entitled to make jugments and come to conclusions about the movies they see--to evaluate how "good" they are, or what they "mean." But what do they base, or what should they base, their judgments on? In this thought-provoking (...) new book, Stanley Cavell, a professor of philosophy and aesthetics at Harvard who has long taken a lively interest in movies, looks closely at these and other questions concerning America's most popular art and the perceptions we make of, by, and through it. His answers are surprising and important--not only to our understanding of film and of ourselves, but to our expectations of life and art. Mr. Cavell's explorations of some of Hollywood's stars, directors, and most famous films--as well as his fresh look at Godard, Bergman, and other great European directors--will be of lasting interest to movie-viewers and intelligent people of all ages. (shrink)

Development and Evolution surveys and illuminates the key themes of rapidly changing fields and areas of controversy that the redefining the theory and philosophy of biology. It continues Stanley Salthe's investigation of evolutionary theory, begun in his influential book Evolving Hierarchical Systems, while negating the implicit philosophical mechanisms of much of that work. Here Salthe attempts to reinitiate a theory of biology from the perspective of development rather than from that of evolution, recognizing the applicability of general systems thinking (...) to biological and social phenomena and pointing towards a non-Darwinian and even a postmodern biology. (shrink)

This book is a major contribution to the study of the philosophy of action, moral philosophy, and political philosophy. Its central idea is a radically unorthodox theory of rational action. Most contemporary Anglo-American philosophers believe that action is motivated by desire. Professor Benn rejects the doctrine and replaces it with a reformulation of Kant's ethical and political theory, in which rational action can be determined simply by principles, regardless of consequences. The book analyzes the way in which value conflicts can (...) be rationally resolved, the objectivity of value, the concept of moral personality, the principles of non-interference and respect of persons, the ideals of autonomy and community and various aspects of individual rights - focusing on the rights to freedom, welfare, and privacy. (shrink)

This handsome new edition of Stanley Cavell's landmark text, first published 20 years ago, provides a new preface that discusses the reception and influence of his work, which occupies a unique niche between philosophy and literary studies.

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)

In an era when much of what passes for debate is merely moral posturing--traditional family values versus the cultural elite, free speech versus censorship--or reflexive name-calling--the terms "liberal" and "politically correct," are used with as much dismissive scorn by the right as "reactionary" and "fascist" are by the left--Stanley Fish would seem an unlikely lightning rod for controversy. A renowned scholar of Milton, head of the English Department of Duke University, Fish has emerged as a brilliantly original critic of (...) the culture at large, praised and pilloried as a vigorous debunker of the pieties of both the left and right. His mission is not to win the cultural wars that preoccupy the nation's attention, but rather to redefine the terms of battle. In There's No Such Thing as Free Speech, Fish takes aim at the ideological gridlock paralyzing academic and political exchange in the nineties. In his witty, accessible dissections of the swirling controversies over multiculturalism, affirmative action, canon revision, hate speech, and legal reform, he neatly eviscerates both the conservatives' claim to possession of timeless, transcendent values, and the intellectual left's icons of equality, tolerance, and non-discrimination. He argues that while conservative ideologues and liberal stalwarts might disagree vehemently on what is essential to a culture, or to a curriculum, both mistakenly believe that what is essential can be identified apart from the accidental circumstances to which the essential is ritually opposed. In the book's first section, which includes the five essays written for Fish's celebrated debates with Dinesh D'Souza, Fish turns his attention to the neoconservative backlash. In his introduction, Fish writes, "Terms that come to us wearing the label 'apolitical'--'common values', 'fairness', 'merit', 'color blind', 'free speech', 'reason'--are in fact the ideologically charged constructions of a decidedly political agenda. I make the point not in order to level an accusation, but to remove the sting of accusation from the world 'politics' and redefine it as a synonym for what everyone inevitably does." Fish maintains that the debate over political correctness is an artificial one, because it is simply not possible for any party or individual to occupy a position above or beyond politics. Regarding the controversy over the revision of the college curriculum, Fish argues that the point is not to try to insist that inclusion of ethnic and gender studies is not a political decision, but "to point out that any alternative curriculum--say a diet of exclusively Western or European texts--would be no less politically invested." In Part Two, Fish follows the implications of his arguments to a surprising rejection of the optimistic claims of the intellectual left that awareness of the historical roots of our beliefs and biases can allow us, as individuals or as a society, to escape or transcend them. Specifically, he turns to the movement for reform of legal studies, and insists that a dream of a legal culture in which no one's values are slighted or declared peripheral can no more be realized than the dream of a concept of fairness that answers to everyone's notions of equality and jsutice, or a yardstick of merit that is true to everyone's notions of worth and substance. Similarly, he argues that attempts to politicize the study of literature are ultimately misguided, because recharacterizations of literary works have absolutely no impact on the mainstream of political life. He concludes his critique of the academy with "The Unbearable Ugliness of Volvos," an extraordinary look at some of the more puzzing, if not out-and-out masochistic, characteristics of a life in academia. Penetrating, fearless, and brilliantly argued, There's No Such Thing as Free Speech captures the essential Fish. It is must reading for anyone who cares about the outcome of America's cultural wars. (shrink)

Explores the meaning of death, how people of different times, regions, and religions have coped with it, and the progress and effects of the war waged against it by researchers, physicians, and surgeons.

In these three lectures, Cavell situates Emerson at an intersection of three crossroads: a place where both philosophy and literature pass; where the two traditions of English and German philosophy shun one another; where the cultures of America and Europe unsettle one another. "Cavell’s ’readings’ of Wittgenstein and Heidegger and Emerson and other thinkers surely deepen our understanding of them, but they do much more: they offer a vision of what life can be and what culture can mean.... These profound (...) lectures are a wonderful place to make [Cavell’s] acquaintance."—Hilary Putnam. (shrink)

These lectures by one of the most influential and original philosophers of the twentieth century constitute a sustained argument for the philosophical basis of romanticism, particularly in its American rendering. Through his examination of such authors as Emerson, Thoreau, Poe, Wordsworth, and Coleridge, Stanley Cavell shows that romanticism and American transcendentalism represent a serious philosophical response to the challenge of skepticism that underlies the writings of Wittgenstein and Austin on ordinary language.

"This is a timely collection of important papers.

Stanley Cavell, one of America's most distinguished philosophers, has written an invaluable companion volume to Walden, a seminal book in our cultural heritage. This expanded edition includes two essays on Emerson.

A philosophical study of popular movies uses the viewer as a point of reference.

Despite increased focus on emotional language, research lacks for the most emotional language: Swearing. We used event-related potentials to investigate whether swear words have content dist...