What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue that (...) the explanatory criteria offered by earlier accounts can all be thought of as features that make it more likely that a mathematical proof will generate understanding. On the functional account, features such as characterising properties, unification, and salience correlate with explanatoriness, but they do not define explanatoriness. (shrink)

Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both research-active mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we (...) suggest that empirical studies can make a useful contribution to our understanding of mathematical practice. (shrink)

Through a selection of primary sources, this article demonstrates the political and legal languages which articulated monarchist ideas in interwar Yugoslavia. Variations on the theme emerged in different periods. First, the national and so democratic character of the monarch and monarchy was a prevalent image at the end of the First World War and in the first decade of the Yugoslav state’s existence. During the domestic political crises in the second half of the 1920s, the language of monarchism shifted toward (...) discourses of stability and public order. After the declaration of the royal dictatorship in January 1929, the language of monarchism became fully invested in expressing the monarch’s absolute political authority, legally inviolable character, and the resulting ‘unity of state and nation’. For the political Right, the king embodied the spirit of integral Yugoslavism. While the language of monarchism could serve disparate political ideologies – as in the liberal monarchist emigration after spring 1941 – it was rather primarily linked to the political visions of the Right in the final decade of interwar Yugoslavia. (shrink)

Nonsymbolic comparison tasks are commonly used to index the acuity of an individual's Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an adaptation of the (...) standard model, and suggest that this finding has significant methodological implications for numerical cognition research. (shrink)

Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate (...) that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians’ aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed. (shrink)

This article calls for an end to the use of the term ‘invasive species’, both in the scientific and public discourse on wildlife conservation. There are two broad reasons for this: the first problem with the invasive species narrative is that this demonisation of ‘invasives’ is morally wrong, particularly because it usually results in the unjust killing of the animals in question. Following on from this, the second problem is that the narrative is also incoherent, both from scientific and philosophical (...) perspectives. At the heart of both of these issues is the problem that the invasive species narrative oversimplifies what are in fact very complex biological processes. As a result, the policies carried out with the stated aim of ‘controlling’ these animals are often unethical. In light of the current global species decline, this article asserts that the way we think and talk about these animals should be changed and the term ‘invasive species’ should be discontinued, in the hope that this leads to changes in practice. (shrink)

Aquinas is often presented as following Aristotle's "Nicomachean Ethics" when treating moral virtue. Less often do philosophers consider that Aquinas's conception of the highest good and its relation to the functional character of human activity led him to break with Aristotle by replicating each of the acquired moral virtues on an infused level. The author suggests that we can discern reasons for this move by examining Aquinas's commentary on the "Sententiae" of Peter the Lombard and the "Summa theologiae" within their (...) historical context. The author's thesis is that Dominican pastoral and intellectual concerns led Aquinas to argue that moral virtue must necessarily be ordered toward the highest good. Understanding this purpose helps to explain his presentation of moral virtue and its implications for standard philosophical interpretations of his work. (shrink)

v. 1. Classical contributions to cosmopolitanism -- v. 2. Key contemporary analyses of cosmopolitanism -- v. 3. Cosmopolitans and cosmopolitanisms -- v. 4. Contested cosmopolitanisms.

The ancient Greek historian and political scientist Polybius is not as well known in the present day as figures such as Herodotus, Thucydides, Plato and Aristotle. This is in part due to his having lived in the Hellenistic period, an epoch often thought to be characteristic of Greek cultural and political decline, rather than in the earlier ‘golden age’ of Greek intellectual life in the 5th and 4th centuries BCE. Yet Polybius’s ideas have been of profound importance in modern western (...) thought, in areas as diverse as historiography, philosophy of history and the theorization of political institutions. This article illustrates the main contours of how subsequent thinkers have received and made use of Polybian ideas and themes, and argues for regarding Polybius as an important precursor of contemporary social scientific analyses of ‘globalization’. The article first excavates and identifies some of the main forms of appropriation of Polybius’s ideas that have occurred in the West over the last 500 years. Secondly, it delineates the most important appropriation of Polybius in the human sciences that has been effected in recent times, namely the use made of Polybian themes by Michael Hardt and Antonio Negri in Empire, their influential diagnosis of the contemporary state of world affairs. Thirdly, the article proffers a critique of these authors’ use of Polybius, and in its stead offers an alternative mode of appropriation of his work for the purposes of analytically reorienting the human sciences in light of present-day concerns with globalization and conditions of globality. It is argued that, far from being a figure of only antiquarian interest, Polybius continues to be of much relevance for human scientists today, for he may be seen as a foundational figure in efforts to think about the ‘global’ level in human affairs. (shrink)

Of all sociology's `strange others', cultural studies is perhaps the least unfamiliar to many sociologists. Yet cultural studies exists in one of the most ambiguous relationships with sociology of any academic discipline. In this article, it is argued that the complicated nature of the relationship is compelled by the very closeness of the two participants in it. What often seems to be an ongoing state of ritualized antagonism between them flows not from their ostensible differences but in fact from their (...) striking underlying similarities. Their symbiotic bond both compels, and is hidden by, the rhetorical displays of selfhood in which they both often engage. The article reviews and assesses this state of affairs, and argues that the similarities between the two disciplines are actually based on a number of shared epistemological assumptions, a priori ways of thinking that in fact are very much open to question. A critique of these assumptions is outlined, and it is suggested that by recognizing their shared nature, both sociology and cultural studies, and the relationship between them, would enter into a greater state of intellectual maturity. (shrink)

The article discusses the philosophy of sportive existence, as put forward by Spanish philosopher José Ortega y Gasset. Ortega is widely recognized as the major figure in Hispanic philosophy in the 20th century. Sports are an integral aspect of Ortega's philosophical output, both as aids toward understanding more general issues in ontology and philosophical anthropology and as explicit topics for reflection and analysis in and of themselves. Issues to do with sports and the sportive aspects of life were central to (...) Ortega's reflections on the nature of the human condition. Ortega argued that in many ways, the figure of the sportsperson represented more than any other symbol the human being's quintessence. (shrink)

Many authors have commented on the relative frequency of the present tense—and the relative infrequency of the past tense—in mathematical writing. However, none (to our knowledge) have provided an estimate for the size of this effect or explored how universal it is. In this short note we report an analysis of corpora of mathematical and day-to-day English. We conclude that the present-to-past ratio of tenses is at least 3:1 in mathematical English, compared to approximately 5:7 in day-to-day English. Further, we (...) show that this tendency to favour the present tense is almost universally present in written mathematics. (shrink)

As the millennium approaches, the United States is on the verge of major health care reform. While swallowing scarce national resources, our health care system produces unenviable results and major inconsistencies. In 1992, $838.5 billion were spent on health care, biting more than 14 percent out of our gross national product. From 35 to 37 million Americans, or approximately 14 percent of the populationn, are uninsured. Our health care system is inherently inconsistent: We have the highest birthweight-specific survival rate of (...) any country in the world, yet we rank 19th worldwide in infant mortality rate, i.e., state-of-the-art medical technology allows us to save a 500-gram infant, yet the mother of that infant may not have had access to basic, minimal prenatal care. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:“First the Bow is Bent in Study …” Dominican Education before 1350 by M. Michèle MulchaheyJohn InglisM. Michèle Mulchahey. “First the Bow is Bent in Study …” Dominican Education before 1350. Studies and Texts, vol. 132. Toronto: Pontifical Institute of Medieval Studies, 1998. Pp. xxi + 618. Cloth, $110.00.In his The Setting of the Summa theologiae of Saint Thomas, Leonard Boyle represents one of the more interesting directions (...) in the recent historiography of Latin medieval philosophy (Toronto: Pontifical Institute of Mediaeval Studies, 1982). His claim is that if we are to understand Aquinas, we need to pay closer attention to the Dominican intellectual context. With a focus on the moral component of Aquinas’s Summa theologiae, Boyle establishes that Aquinas did not converse alone with Aristotle and Averroes in some philosophical ivory tower as historians often imply, but that he wrote and disputed in light of Dominican goals. This point helps to clarify the context of Aquinas’s texts—a type of historical context not found in the standard philosophical accounts. In the book under review, the first detailed history of Dominican medieval education, M. Michèle Mulchahey supplies a more extended treatment of this intellectual context.Commonly, intellectuals in the high middle ages are presented as spending their lives at universities studying philosophy, conducting disputations, and lecturing on the Sentences of Peter the Lombard. Anthony Kenny and Jan Pinborg present such a picture in their helpful contribution to The Cambridge History of Later Medieval Philosophy ([Cambridge: Cambridge University Press, 1982], 11–42). Mulchahey argues that this sort of picture is rather simplistic when applied to the Dominican intellectual life. From the early years of the order, there was to be a “doctor” in each and every local Dominican house (132). One of Mulchahey’s more interesting conclusions is that the schools associated with these local houses were more advanced than has often been thought. For example, each house was to include lectures on the Sentences of Peter the Lombard and on Scripture, that is, on the very texts which historians commonly associate only with the universities (134–136). Furthermore, disputations were conducted frequently at local houses (134, 167) and by the middle of the thirteenth century, the works of Aristotle were often taught (220). Her point is that if we are to understand the intellectual world in which a Dominican like Aquinas spent his life, then we need to be clear about this context and not just about that of the universities.Even at the university, Dominicans studied and lectured at Dominican houses and not at institutions of the secular clergy. For instance, even when Aquinas was a Bachelor of the Sentences and a Master of the Sacred Page at the University of Paris, he lived and taught at the Dominican convent according to the distinct traditions and practices [End Page 361] of the order. Furthermore, select Dominican schools of higher learning, including those associated with universities, produced teachers for local Dominican communities. Mulchahey argues convincingly that the structures of Dominican education came to exist from the bottom up within a Dominican context and not as an extension of the statutes of the universities (351–378).This Dominican context helps to provide a response to an objection that John Jenkins raises to Boyle’s view of the audience of Aquinas’s Summa theologiae—an issue that has important implications for how to read this work. Boyle argued that the Summa arose out of the pastoral needs of ordinary Dominicans and reflects this institutional purpose. Jenkins “rejects” this claim, arguing that Aquinas’s text is so dense and conceptually difficult that it could only have been written for those who were prepared to study the Sentences of Peter the Lombard at the University of Paris. (See John I. Jenkins, Knowledge and faith in Thomas Aquinas [Cambridge: Cambridge University Press, 1997], 81, 89–90.) Mulchahey provides important support for Boyle’s view by establishing that it was the policy of the order to prepare ordinary Dominicans to study the Sentences of Peter the Lombard at the local houses. At a time when there were few universities, the Dominicans established hundreds of sites of higher education at... (shrink)

The first volume of the Mediaeval Commentaries on the Sentences of Peter Lombard (=MCS1) edited by G. R. Evans in 2002 provided the first comprehensive study of those works that house much Latin medieval philosophy from the middle of the twelfth century to Martin Luther in the sixteenth century. Philipp Rosemann rounded out this project in 2007 with The Story of a Great Medieval Book: Peter Lombard's Sentences (Peterborough, ON: Broadview), which serves as an introduction to the second volume he (...) has now edited of the MCS (=MCS2). These volumes provide much of the context for Latin philosophical work in the later Middle Ages, which is arguably the most understudied period of Western thought. Steven Livesey gives .. (shrink)

The Islamic philosophical tradition was the privileged site for the study and continuation of the Classical philosophical tradition in the Middle Ages. An initial chapter on the history of Islamic philosophy sets the stage for sixteen articles on issues across the Islamic, Jewish and Christian traditions. The goal is to see the Islamic tradition in its own richness and complexity as the context of much Jewish intellectual work. Taken together, these two traditions provide the wider context to which Latin Christian (...) intellectuals would turn. The articles are grouped under six topics relevant both to the period and to current philosophical interest: the Islamic philosophical context, the nature of philosophy in the Middle Ages, Neoplatonism and the activity of the soul, creation, virtue, and the Latin reception. Since the nineteenth century Islamic and Jewish philosophy have been neglected in the standard histories of medieval philosophy. The time is right to begin to write a more balanced history of medieval philosophy. In order to begin to write this history, this book focuses on the Islamic, Jewish, and Christian use of - and reaction to - Classical philosophy during the Middle Ages. (shrink)

ON MEDIEVAL PHILOSOPHY presents a concise overview of the key elements of medieval philosophy, this practical and affordable philosophy text will help you understand and identify key ideas so that you can easily succeed in this course. With coverage of Christian, Islamic, and Jewish traditions, this volume aims to draw attention to the implications of medieval philosophy for the present age.

(1997). Philosophical autonomy and the historiography of medieval philosophy. British Journal for the History of Philosophy: Vol. 5, No. 1, pp. 21-53.

This volume contains the seminal articles that define the influence of Aquinas within legal philosophical thought. A comprehensive reference for those new to the field, it covers such topics as the relation of virtue to law, the common good, natural law, natural rights and property rights; together with social and political issues like abortion, feminism, homosexuality, environment, civil disobedience and just war. Attention is devoted to the new natural law theory and its limitations, as well as the place of historical (...) context in the recovery of social thought. (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)

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative (...) relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd's hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. (shrink)

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such (...) as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book. (shrink)

In recent times developments in the natural sciences and in the sphere of environmental politics have compelled social scientists, and also some natural scientists, to rethink the relations that hitherto have been held, in Western thought generally and within particular disciplines, to characterize ‘nature’ on the one side and ‘culture’ on the other. This article considers the history of this conceptual boundary and looks at new conceptualizations of nature/culture, stimulated by developments both in biotechnology and in the ongoing controversies about (...) environmental degradation. It argues that while some of the contributions to reconfiguring, or abolishing, the nature/ culture division have been productive and stimulating of new ways of conceiving the world, there has nonetheless been an unfortunate tendency for social scientists to bring to bear inherited analytic dispositions on biotechnological and environmental matters. Instead of using these issues as means of challenging social scientific disciplinary dogmas and of engaging in constructive rapprochement with natural scientists, social scientists have in their analyses of such matters often merely asserted the hegemony of ‘culture’ over ‘nature’, and thus in effect the superiority of social scientific over natural scientific conceptualizations of the world. Far from overcoming the nature/culture boundary, social scientists have too often merely asserted the primacy of the sorts of subject matters and analytic techniques they feel comfortable with, rather than subjecting their practices to fully reflexive self-scrutiny. (shrink)

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge (...) that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 439-440 [Access article in PDF] John F. Wippel. The Metaphysical Thought of Thomas Aquinas: From Finite Being to Uncreated Being. Monographs of the Society for Medieval and Renaissance Philosophy, No. 1. Washington, D.C.: Catholic University Press, 2000. Pp. xxvii + 630. Cloth, $59.95. Paper, $39.95. In this weighty volume, John Wippel brings together much of the important research that he has (...) carried out on the metaphysics of Aquinas over the last three decades in order to produce as he calls it, a "Summa metaphysicae" based on the original texts of Aquinas (xxvii). Throughout much of the twentieth century metaphysics has been thought to be Aquinas's greatest philosophical achievement. The historical research of Etienne Gilson and Joseph Owens together with the philosophical reflections of Jacques Maritain did much to place this topic at the heart of Aquinas scholarship. Therefore, this volume both represents Wippel's central research interest and brings to a head an entire school of Aquinas studies of the twentieth century. It is the clearest and most detailed account of the subject in the English language, bearing the hallmarks of Wippel's scholarship: clarity in reference to the arguments and attention to textual chronology.Metaphysics here does not mean so much a focus on Aquinas's treatment of Aristotle's four causes or his act and potency as much as a clear presentation of Aquinas's reflections on existence (esse) and the implications that this has for his thought. After an introductory discussion of the nature and subject of metaphysics for Aquinas, Wippel devotes the first part of the book to Aquinas's view of existence in order to unpack what it means to talk about individual things as beings, being in general, and the catagories of Aristotle (73-74). Masterfully, he explains Aquinas's analogical approach to being and guides the reader through the interpretive thicket of Cornelio Fabro, G. P. Klubertanz, and Ralph McInerny. An interesting parallel is presented between Parmenides' problem of the one and the many and Aquinas's philosophically interesting view of the composite manner in which concrete individuals exist (161-170). Since no individual creature is the author of its own existence, but does in fact exist, each finite being must receive existence from a single being that exists necessarily. Wippel highlights this and other approaches that Aquinas takes to clarify relations between the many individual beings and the single divine being that lies beyond the study of metaphysics.Part two is devoted to many of the ways in which Aquinas talks about concrete individuals. Here Wippel explains Aquinas's distinctions between substance and accident as well as between matter and form and how they relate to existence. Part three concludes the work with a treatment of what Aristotle and Aquinas call divine science. This consists of more than a hundred pages on Aquinas's famous arguments for the existence of God (379-500) and a discussion of what human beings can understand about God (501-575). Not only does Aquinas conclude that God exists, but he indicates the limited manner in which human beings can understand the divine. The problem is that if we acquire language in relation to concrete individuals, how can we use words to discuss a being who is not bound by space or time? The solution is to speak of words like "good" and "wise" in an analogical sense. Words like these when applied to creatures are used in a sense that is both similar to but different from the way they are used to [End Page 439] speak of the divine. Reiterating the theme of the one and the many, the name "good" has a single and most excellent meaning when applied to God and a limited sense when applied to creatures.This is an excellent and accessible book for the educated reader and specialist alike which can be supplemented by works such as Wayne Hankey's God in Himself: Aquinas' Doctrine of God as Expounded in theSumma Theologiae (Oxford: Oxford... (shrink)

‘Globalization’ has become in recent years one of the central themes of social scientific debates. Social theories of globalization may be regarded as specific academic and analytic manifestations of wider forms of ‘global consciousness’ to be found in the social world today. These are ways of thinking and perceiving which emphasize that the whole world should be seen as ‘one place’, its various geographically disparate parts all being interconnected in various complex ways. In this article we set out how both (...) a general ‘ecumenical sensibility’ involving such ways of thinking, and its specifically academic variant, an ‘ecumenical analytic’, are simultaneously responses to ‘globalizing conditions’ and also products of the latter. We demonstrate how social theories of globalization, locatable in the overall set we call the ‘ecumenical analytic’, are reflexive thought-products, in that they both seek to investigate, and are made possible by, ‘globalizing conditions’. Rejecting the view that both an ecumenical sensibility and an ecumenical analytic are solely products of modernity, we show how such modes of consciousness and analysis were present in the period of Greek history between the death of Alexander the Great and the rise of the Roman Empire. We show how at that period a socio-political situation akin in certain ways to modern ‘globalizing conditions’ generated both an ecumenical sensibility and an ecumenical analytic, the latter most forcefully represented by a revolutionary new genre in historiography called Universal History. By examining the ideas of its most famous practitioner, the historian Polybius, we demonstrate that Universal History both provided a framework for understanding what we today would call ‘globalization’, and exhibited a remarkable degree of reflexive awareness about its own conditions of possibility. We seek to show that an attentiveness to ‘global’ processes and a reflexive understanding of what makes that form of thinking possible in the first place, are not solely confined to modernity but were identifiable features of intellectual production in the Hellenistic period of ancient Greece. In this way we argue that scholars today should not imagine that the contemporary ‘global turn’ in social thought is either unique or wholly historically unprecedented. (shrink)

People can be disgusted by the concrete and by the abstract -- by an object they find physically repellent or by an ideology or value system they find morally abhorrent. Different things will disgust different people, depending on individual sensibilities or cultural backgrounds. In _Yuck!_, Daniel Kelly investigates the character and evolution of disgust, with an emphasis on understanding the role this emotion has come to play in our social and moral lives. Disgust has recently been riding a swell (...) of scholarly attention, especially from those in the cognitive sciences and those in the humanities in the midst of the "affective turn." Kelly proposes a cognitive model that can accommodate what we now know about disgust. He offers a new account of the evolution of disgust that builds on the model and argues that expressions of disgust are part of a sophisticated but largely automatic signaling system that humans use to transmit information about what to avoid in the local environment. He shows that many of the puzzling features of moral repugnance tinged with disgust are by-products of the imperfect fit between a cognitive system that evolved to protect against poisons and parasites and the social and moral issues on which it has been brought to bear. Kelly's account of this emotion provides a powerful argument against invoking disgust in the service of moral justification. (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)

General discussion of grounding, including its formal features, relations to other notions, and applications.

In the mathematics education research literature, there is a growing body of scholarship on how mathematicians practice their craft. The purpose of this chapter is to survey some of this literature and explain how it can contribute to the philosophy of mathematical practice. We first describe how mathematics educators use empirical methodologies to investigate the behaviors of mathematicians and argue that findings from these studies can inform the philosophy of mathematical practice. We then illustrate this by summarizing research on mathematicians’ (...) understandings and mathematicians’ proof reading. (shrink)