Spatial relations are central to geometrical thinking. With respect to the classical elementary geometry of Euclid’s Elements, a distinction between co-exact, or qualitative, and exact, or metric, spatial relations has recently been advanced as fundamental. We tested the universality of intuitions of these relations in a group of Senegalese and Dutch participants. Participants performed an odd-one-out task with stimuli that in all but one case display a particular spatial relation between geometric objects. As the exact/co-exact distinction is closely related to (...) Kosslyn’s categorical/coordinate distinction, a set of stimuli for testing all four types was used. Results suggest that intuitions of all spatial relations tested are universal. Yet, culture has an important effect on performance: Dutch participants outperformed Senegalese participants and stimulus layouts affect the categorical and coordinate processing in different ways for the two groups. Differences in level of education within the Senegalese participants did not affect performance. (shrink)

The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, we (...) tested whether the processing of metric vs topological relations yielded the same hemispheric specialization as the processing of coordinate vs categorical relations. In the second part, we investigated the specific performance patterns for the processing of five pairs of exact/co-exact relations, where stimuli for the co-exact relations were divided into three categories depending on their distance from the exact case. Regarding the processing of metric vs topological relations, hemispheric differences were found for only a few of the stimuli used, which may indicate that other processing mechanisms might be at play. Regarding the processing of exact vs co-exact relations, results show that the level of agreement among participants in judging co-exact relations decreases with the distance from the exact case, and this for the five pairs of exact/co-exact relations tested. The philosophical implications of these empirical findings for the epistemological analysis of Euclid's diagram-based geometric practice are spelled out and discussed. (shrink)

Information can be public among a group. Whether or not information is public matters, for example, for accounts of interdependent rational choice, of communication, and of joint intention. A standard analysis of public information identifies it with (some variant of) common belief. The latter notion is stipulatively defined as an infinite conjunction: for p to be commonly believed is for it to believed by all members of a group, for all members to believe that all members believe it, and so (...) forth. This analysis is often presupposed without much argument in philosophy. Theoretical entrenchment or intuitions about cases might give some traction on the question, but give little insight about why the identification holds, if it does. The strategy of this paper is to characterize a practical-normative role for information being public, and show that the only things that play that role are (variants of) common belief as stipulatively characterized. In more detail: a functional role for “taking a proposition for granted” in non-isolated decision making is characterized. I then present some minimal conditions under which such an attitude is correctly held. The key assumption links this attitude to beliefs about what is public. From minimal a priori principles, we can argue that a proposition being public among a group entails common commitment to believe among that group. Later sections explore partial converses to this result, the factivity of publicity and publicity from the perspective of outsiders to the group, and objections to the aprioricity of the result deriving from a posteriori existential presuppositions. (shrink)

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received (...) view, this essay provides a contrary analysis by introducing a formal account of Euclid's proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid's arguments. It specifies what diagrams Euclid's diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored. (shrink)

Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice.

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu , a recently developed (...) formal system of proof (presented in Mumma (Synthese 175:255–287, 2010 )) within which Euclid’s diagrammatic proofs can be represented. (shrink)

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

In line with Ken Manders’s seminal account of Euclid’s diagrammatic method in the “The Euclidean Diagram,” two proof systems with a diagrammatic syntax have been advanced as formalizations of the method FG and Eu. In a paper examining Eu, Nathaniel Miller, the creator of FG, has identified a variety of technical problems with the formal details of Eu. This response shows how the problems are remedied.

It is commonplace to view the rigor of the mathematics in Euclid's Elements in the way an experienced teacher views the work of an earnest beginner: respectable relative to an early stage of development, but ultimately flawed. Given the close connection in content between Euclid's Elements and high-school geometry classes, this is understandable. Euclid, it seems, never realized what everyone who moves beyond elementary geometry into more advanced mathematics is now customarily taught: a fully rigorous proof cannot rely on geometric (...) intuition. In his arguments he seems to call illicitly upon our understanding of how objects like triangles and circles behave rather than grounding everything rigorously in axioms.Though widespread, the attitude is in a historical sense puzzling. For over two millenia, mathematicians of all levels studied the arguments in Elements and found nothing substantial missing. The book, on the contrary, represented the limit of mathematical explicitness. It served as the paradigm for careful and exact reasoning. How it could enjoy this reputation for so long is mysterious if careful and exact reasoning demands that all inferences be grounded in a modern axiomatic theory in the way Hilbert did in his famous Foundations of Geometry. By these standards, Euclid's work is deeply flawed. The holes in his arguments are not minor and excusable, but massive and cryptic.With his book Euclid and His Twentieth Century Rivals, Nathaniel Miller makes substantial progress in clearing this mystery up. The book is an explication of FG , a formal system of proof developed by Miller which reconstructs Euclid's deductions as essentially diagrammatic. The holes in Euclid's arguments are taken to appear precisely at those steps which are unintelligible without an accompanying geometric diagram. Interpreting the reasoning in the Elements in terms of a modern axiomatization , …. (shrink)

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an (...) inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object‐wise, could underlie diagram‐based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions. (shrink)

The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.

Rav et Leitgeb défendent la thèse de l’autonomie des preuves informelles par rapport aux systèmes formels de preuve. Azzouni, au contraire développe une explication qui réduit les preuves informelles à un réseau de systèmes formels sous-jacents. L’objectif principal de cet article est de démontrer la possibilité d’une position tierce médiane mettant en avant une explication quasi formelle de la méthode de preuve dans les Éléments. L’explication est quasi formelle, plutôt que formelle, en ce qu’elle donne au contenu géométrique un rôle (...) irréductible dans les preuves d’Euclide en ce que ce rôle est sujet à des contraintes formelles. Les inférences qui sont basées sur concepts géométriques ont une occurrence à l’intérieur d’un cadre formel précisément défini.Rav and Leitgeb argue for the autonomy of informal proofs from formal systems of proof. In contrast, Azzouni develops an account which reduces informal proofs to a network of underlying formal systems. The general aim of this paper is to demonstrate the possibility of a third, middle position with a quasi-formal account of Euclid’s proof method in the Elements. The account is quasi-formal, rather than simply formal, in that it gives geometric content an irreducible role in Euclid’s proofs. It is quasi-formal, rather than simply informal, in that this role is subject to formal constraints. Inferences which are based on geometric concepts occur within a precisely defined formal framework. (shrink)

Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully advanced (...) by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science. (shrink)

The achievements of the textile industry in Roman Britain are often underestimated as a result of the meagreness of our available evidence. The Edict on maximum prices issued by Diocletian in A.D. 301 shows that British capes commanded high prices on the markets of the Empire, and that in the late third century A.D. British rugs were the best in the world. In view of the competition from the traditional centres of rug manufacture in the East, this is an astonishing (...) achievement. (shrink)

The received translation and interpretation of many of the technical terms current in the textile industry of the Roman Empire are inaccurate, because lexicographers have either fought shy of being precise, or have thought that they recognized in the ancient world technical processes which originated at a much later date. The evidence is often equivocal or insufficient, but may still yield details that have been overlooked. The textile expression scutulatus, to take an example, deserves more attention than Blümner has devoted (...) to it. When it refers to the appearance of a garment, Lewis and Short, s.v. scutulatus, translate ‘diamond or lozenge-shaped, checkered’. It seems worth while to attempt a more precise translation, and, if possible, to estimate the significance in context of the goods so designated. (shrink)

Living out of silence, out of a fully functioning, lovingly attentive mind, and not just out of thought, out of a partially functioning mind, is requisite for depth or profundity in living or relating. A fully attentive, truly silent or meditative mind sees that there is real dualism of time and the timeless and that time and the timeless each has its own unique value. The timeless, or real silence, that which alone can make for depth in one's living and (...) relating, can be there only when, in one's awareness, time and all that is related to time, have come to an end. An examination of a wide variety of statements made by writers, both past and present, shows that these understandings are seldom there in contemporary writings, just as they were not there in writings that have come down to us from the past. 'Living beyond the One and the Many' invites readers to consider the possibility that what is presented in it can bring deep meaning to one's way of living in the world"--Publisher's description, back cover. (shrink)

If the world itself is metaphysically indeterminate in a specified respect, what follows? In this paper, we develop a theory of metaphysical indeterminacy answering this question.

Husserlian phenomenology, as the study of conscious experience, has often been accused of solipsism. Husserl’s method, it is argued, does not have the resources to provide an account of consciousness of other minds. This chapter will address this issue by providing a brief overview of the multiple angles from which Husserl approached the theme of intersubjectivity, with specific focus on the details of his account of the concrete interpersonal encounter – “empathy.” Husserl understood empathy as a direct, quasi-perceptual form of (...) intentionality through which the sense of the Other is constituted. Furthermore, his account of empathy is holistically integrated with his overall theory of intersubjectivity, including his discussions of the objectivity of nature, and the social, historical, and communal aspects of subjectivity. Husserl’s theory of empathy continues to cast a long shadow, influencing both the analytic and continental approaches to the problem of other minds, as well as contemporary account of social cognition in the cognitive sciences. (shrink)

An Essay on Human Action seeks to provide a comprehensive, detailed, enlightening, and (in its detail) original account of human action. This account presupposes a theory of events as abstract, proposition-like entities, a theory which is given in the first chapter of the book. The core-issues of action-theory are then treated: what acting in general is (a version of the traditional volitional theory is proposed and defended); how actions are to be individuated; how long actions last; what acting intentionally is; (...) what doing one thing by doing another is; what basic action is; and what omitting to do something is. Attention is also given to the concepts of causation, intention, volition, deciding, choosing, and trying. Finally, a libertarian account of free action is tentatively proposed and defended. (shrink)

We develop and apply a multi-dimensional conception of explanatory depth towards a comparative analysis of inflationary and bouncing paradigms in primordial cosmology. Our analysis builds on earlier work due to Azhar and Loeb (2021) that establishes initial condition fine-tuning as a dimension of explanatory depth relevant to debates in contemporary cosmology. We propose dynamical fine-tuning and autonomy as two further dimensions of depth in the context of problems with instability and trans-Planckian modes that afflict bouncing and inflationary approaches respectively. In (...) the context of the latter issue, we argue that the recently formulated trans-Planckian censorship conjecture leads to a trade-off for inflationary models between dynamical fine-tuning and autonomy. We conclude with the suggestion that explanatory preference with regard to the different dimensions of depth is best understood in terms of differing attitudes towards heuristics for future model building. (shrink)

Telling a patient that the outcome of his illness is not good, or even hopeless, requires sensitivity and the ability to communicate with him in the setting of a hospital which is an unnatural environment divorced from family and friends. It is a task which must be taught and learned by doctors and nurses.

Global Media Ethics is the first comprehensive cross-cultural exploration of the conceptual and practical issues facing media ethics in a global world. A team of leading journalism experts investigate the impact of major global trends on responsible journalism. The first full-length, truly global textbook on media ethics; Explores how current global changes in media promote and inhibit responsible journalism; Includes relevant and timely ethical discussions based on major trends in journalism and global media; Questions existing frameworks in media ethics in (...) light of the impact of global media; Contributors are leading experts in global journalism and communication. (shrink)

The existential analysis -- The death of dasein -- The timeliness of dasein -- The derivation of time -- The time of being.

We live in turbulent times, our world is changing at accelerating speed. Information is everywhere, but wisdom appears in short supply when trying to address key inter-related challenges of our time such as; runaway climate change, the loss of biodiversity, the depletion of natural resources, the on-going homogenization of culture, and rising inequity. Living in such times has implications for education and learning. This book explores the possibilities of designing and facilitating learning-based change and transitions towards sustainability. In 31 chapters (...) contributors from across the world discuss (re)emerging forms of learning that not only assist in breaking down unsustainable routines, forms of governance, production and consumption, but also can help create ones that are more sustainable. The book has been divided into three parts: re-orienting science and society, re-connecting people and planet and re-imagining education and learning. This is essential reading for educators, educational designers, change agents, researchers, students, policymakers and entrepreneurs alike, who are concerned about the well-being of the planet and convinced of our ability to do better. The content and related issues can be discussed on the blog by editor Arjen Wals: Transformative learning. 'We are living in times of incertitude, complexity, and contestation, but also of connectivity, responsibility, and new opportunities. This book analyses the consequences of these times for learning in formal, non-formal, and informal education. It explores the possibilities offered by the concept of sustainability as a central category of a holistic paradigm which harmonizes human beings with Earth. To change people and to change the world are interdependent processes - this book contributes to both.' Moacir Gadotti, Director of Paulo Freire Institute, São Paulo, Brazil 'I hope you share my excitement about the innovations for sustainability that this book catalogues and analyses. While the ecological news is grim, the human news is not. Even in a time of accelerating change, people are showing their enormous capacities to learn, adapt, restore and protect.' From the Foreword by Juliet Schor, author of ‘True Wealth: how and why millions of Americans are creating a time-rich, ecologically-light, small-scale high-satisfaction economy’ 'This book implies a ‘culture of critical commitment’ in educational thinking and practice - engaged enough to make a real difference to social-ecological resilience and sustainability but reflexively critical enough to learn constantly from experience and to keep options open in working for a sustainability transformation.' From the Afterword by Stephen Sterling, Professor of Sustainability Education, Centre for Sustainable Futures, Plymouth University, United Kingdom. (shrink)

What Is Political Theory? provides students with a comprehensive overview of the current state of the discipline. Ten substantive chapters address the most pressing topics in political theory today, including: - what resources do the classic texts still provide for political theorists? - what areas will political theorists focus on in the future? - can western political theory alone continue to provide a framework for responding to the challenges of modern political life? The authors assess the intellectual challenges to conventional (...) political theory, such as post-structuralism and the scientific study of politics that have revitalized the field in the last 30 years. They also broaden the perspective to take in non-western ideas and to reconceptualize political theory in the light of specifically global challenges. Students and teachers of political theory and political philosophy will find this book invaluable in understanding the factors that have shaped current political theory and which will guide its future development. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Cognitive Error and Contemplative Practices:The Cultivation of Discernment in Mind and HeartWesley J. WildmanBrains are amazing organs in all creatures with central nervous systems and especially in human beings. But they are not perfect. Without forgetting the larger success story of cognitive evolution, I want to explore the way that cognitive biases sometimes produce errors in both religious and secular social settings and how such errors can be diagnosed (...) and corrected when they occur. This will involve noticing that error diagnosis and correction is a process that certain social groups have a vested interest in resisting or neglecting, in some respects, while the very same social groups may furnish resources that support the detection of cognitive errors, in other respects.This presents a moral quandary for both secular and religious groups. Should we educate children to be fully aware of their cognitive vulnerability to advertising, thereby learning how to resist and eventually become immune to one of the fundamental power sources of modern market economies? Should religious groups explain to young people their cognitive tendencies to posit the action of supernatural beings whether or not any such action exists, even though this may disrupt the power of religious groups to forge bracing social togetherness that supports psychologically useful coping skills? While I do not seek to answer such complex moral questions in this paper, I do argue that knowledge of cognitive biases and the resulting tendencies to cognitive error, self-defeating behaviors, and self-deception should be made available to those individuals and groups who are interested in promoting a high degree of critical self-awareness in the analysis of beliefs and behaviors in both secular and religious contexts.The term "error" is a potentially problematic one in that it misleadingly suggests that there might be a uniquely correct way in which cognition should work. I do not wish to suggest such a binary opposition. After all, biases exist in the human cognitive system either because they have been selected in the evolutionary process for their survival benefits or because they are side effects of other traits selected for their usefulness. My concern is with the wondrous human discovery that we can analyze [End Page 61] our behaviors and beliefs with such precision that we can sometimes detect when our cognitive biases produce mistaken beliefs and self-defeating behaviors. This forms the basis for my contention that cognition can often work better than it does—more accurately, less self-destructively—when tendencies to cognitive error are diagnosed, corrected, and perhaps even systematically resisted.The profound irony here is that the very same social groups (secular and religious) with an obvious vested interest in resisting such enlightenment in some respects can also promote processes of discernment and insight in other respects. Existing literature in the cognitive psychology of religion rarely bothers with such subtleties. It is common to assume that simply to notice the operation of cognitive biases in religion—say, in supporting belief in supernatural entities who providentially interact in human affairs, or in establishing and reinforcing people's willingness to defer to certain kinds of religious authority—is at the same time to establish the presence of rampant cognitive error, massive resistance to diagnosing and correcting it, and thus the infliction of pernicious cruelties on young children who have no way of escaping the resulting irrational indoctrination.This moral reflex to condemn religious groups because of their characteristic reluctance to acknowledge the role of cognitive biases in their beliefs and practices is understandable given the irrationality, ignorance, or denial that such reluctance suggests. I frankly acknowledge that I share this moral concern. But the actual complexities of religious practice demonstrate that the situation demands a subtler evaluation. Resistance to awareness of cognitive biases exists both inside and outside religious groups; economic and political practices in all eras and of all types have every bit as much to gain from neglecting to enlighten people about their cognitive operations as religious groups do—just consider the techniques employed in commercial advertising and political campaigns. Moreover, religious groups also promote methods of discernment and self-awareness that have historically been, and continue to be, the dominant method by which... (shrink)

The legal environment -- Regulation of nursing practice -- Nurses in legal actions -- Standards of care -- Defenses to negligence or malpractice -- Prevention of malpractice -- Nurses as witnesses -- Professional liability insurance -- Accepting or refusing an assignment/patient abandonment -- Delegation to unlicensed assistive personnel -- Patients' rights and responsibilities -- Confidential communication -- Competency and guardianship -- Informed consent -- Refusal of treatment -- Pain control -- Patient teaching and health counseling -- Medication administration -- Clients (...) with AIDS and HIV testing -- Abusive situations -- Reproductive services -- Restraints -- Emergency psychiatric admissions -- Organ and tissue donation and transplantation -- Discharge against medical advice -- The medical record and documentation standards -- Electronic health information and communications -- Interdependent nursing functions: verbal orders and telenursing -- Event reporting and compliance -- Quality and safety in nursing practice -- Forensic issues -- Employer and employee rights -- Contracts -- Corporate liability -- Employment contracts and unionization -- Employment status liability -- Staffing issues and floating -- Americans with Disabilities Act -- Employees with AIDS/HIV infection and exposure to bloodborne pathogens -- Impaired nurses -- Sexual harassment in the workplace -- Violence in the workplace -- Intentional torts -- Environmental health and safety -- Ethical decision making -- Professional boundaries and patient relationships -- Moral courage and patient advocacy -- Social media and online professionalism -- Reporting illegal, unethical, or unsafe conduct -- Maternal versus fetal rights -- Futility of care -- Advance directives and end-of-life decisions -- Assisted suicide. (shrink)