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)

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

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)

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)

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.

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)

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)

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)

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)

The esteemed psychologist and thinker John Dewey headed for previously unexplored philosophical territory with this influential work. Written shortly after World War I, it embodies Dewey's system of pragmatic humanism and maintains that individuals can attain "a more ordered and intelligent happiness" by reconsidering the ultimate effects of their deepest beliefs and feelings. With its promise of achieving an understanding of the past and attaining a brighter future, Reconstruction in Philosophy remains ever relevant. "A modern classic." — Philosophy and (...) Phenomenological Research. (shrink)

With precision and profundity born of 30 years of devoted study, John Boyle offers an essential introduction to St. Thomas Aquinas on Scripture, shedding helpful light on the goals, methods, and commitments that animate the Angelic Doctor's engagement with the sacred page. Because the genius of St. Thomas's approach to the Bible lies not so much in its novelty but rather in the fidelity and clarity with which he recapitulates the riches of the preceding interpretive Tradition, this initiation into (...) St. Thomas's vision of Scripture is itself an orientation to the Church's vision of Scripture, from the Fathers through and beyond the Middle Ages. St. Thomas's embeddedness within the Church's Tradition and his own historical context is integral to his approach to Scripture, yet it sets him at some distance from modern readers, for whom his interpretive vision may seem perplexing or even impenetrable. In this Primer, Boyle first provides an acclimation to this medieval context through a survey and explanation of pertinent technical terminology used by St. Thomas and characteristic of the scholastic theology of the time. With an eye to the medieval practice of considering Scripture according to the fourfold division of causes, Boyle builds on this initial foundation by exploring in turn St. Thomas's accounts of the end or use of Scripture (final cause), its divine and human authorship (efficient cause), its order and division (material cause), and its literary styles or genres (formal cause). Drawing on writings from across St. Thomas's corpus, but especially his work On the Commendation and Division of Sacred Scripture and the prologues to his biblical commentaries, Boyle masterfully elucidates both the hermeneutical principles and deep wisdom of the Angelic Doctor's approach to Scripture, imparting invaluable guidance not only for reading and understanding St. Thomas and other great masters of the Tradition, but also-and ultimately-for understanding Scripture in light of this Tradition and reading it with greater benefit and joy. (shrink)

Despite recurring efforts, a gap exists across a variety of contexts between the protection of patients' safety in theory and in practice. This timely Research Handbook highlights these critical issues and suggests both legal and policy changes are necessary to better protect patients' safety. Multidisciplinary in nature, this Research Handbook features contributions from eminent academics, policy makers and medical practitioners from the Global North and South, discussing the essential facets concerning patient safety and the law. It highlights how the role (...) of legislation and case law has the potential to influence, both positively and negatively, medical practice and the quality of care. Chapters explore patient safety and the global health agenda; physiotherapy; 'non-therapeutic' clinical research with children; patient safety awareness in healthcare education; and the increasing use of robotics and artificial intelligence in healthcare. Outlining a wide range of international perspectives on patient safety and the law, this Research Handbook will appeal to academics and researchers specialising in health and medical law, human rights and healthcare regulation. It will also serve as a valuable resource for legal and medical practitioners alike, as well as clinicians and professionals working in healthcare governance. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The (...) first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

Theories of discourse bring to realism new ideas about how knowledge develops and how representations of reality are influenced. We gain an understanding of the conceptual aspect of social life and the processes by which meaning is produced. This collection reflects the growing interest realist critics have shown towards forms of discourse theory and deconstruction. The diverse range of contributions address such issues as the work of Derrida and deconstruction, discourse theory, Eurocentrism and poststructuralism. What unites all of the contributions (...) is a sense that it is essential to provide a realist alternative to the hitherto dominance of social constructionism, hermeneutics and postmodernism, over many of the issues discussed. By developing a realist perspective the different authors attempt to embed discourse within the structured nature of the reality of the world. Realism can situate language, discourse and ideology within context specific, or 'causally efficacious' circumstances. Realism can help to uncover issues of power, representation, and subjectivity and how discursive and other social practices produce real effects. This can help us understand the manner in which social structures are reproduced through various forms of ideology and discourse. And by knowing this, we can start to address questions concerning human emancipation and how the world is to be transformed. (shrink)

This vital study offers a new interpretation of Hume's famous "Of Miracles," which notoriously argues against the possibility of miracles. By situating Hume's popular argument in the context of the 18th century debate on miracles, Earman shows Hume's argument to be largely unoriginal and chiefly without merit where it is original. Yet Earman constructively conceives how progress can be made on the issues that Hume's essay so provocatively posed about the ability of eyewitness testimony to establish the credibility of marvelous (...) and miraculous events. (shrink)

Though the revised edition of A Theory of Justice, published in 1999, is the definitive statement of Rawls's view, so much of the extensive literature on Rawls's theory refers to the first edition. This reissue makes the first edition once again available for scholars and serious students of Rawls's work.

Previous edition, 1st, published in 1971.

Part One reviews some recent history of epistemology, focusing on ways in which the intellectual virtues have been invoked to solve specific epistemological problems. This part gives a sense of the contemporary landscape that has emerged and clarifies some of the disagreements among those who invoke the virtues in epistemology. Part Two explores some problems about knowledge in greater detail, and defends a externalist approach in virtue epistemology.

This book continues and revises the ideas of justice as fairness that John Rawls presented in _A Theory of Justice_ but changes its philosophical interpretation in a fundamental way. That previous work assumed what Rawls calls a "well-ordered society," one that is stable and relatively homogenous in its basic moral beliefs and in which there is broad agreement about what constitutes the good life. Yet in modern democratic society a plurality of incompatible and irreconcilable doctrines--religious, philosophical, and moral--coexist within (...) the framework of democratic institutions. Recognizing this as a permanent condition of democracy, Rawls asks how a stable and just society of free and equal citizens can live in concord when divided by reasonable but incompatible doctrines? This edition includes the essay "The Idea of Public Reason Revisited," which outlines Rawls ' plans to revise _Political Liberalism,_ which were cut short by his death. "An extraordinary well-reasoned commentary on _A Theory of Justice_...a decisive turn towards political philosophy." --_Times Literary Supplement_. (shrink)

A Norton Library edition of Locke's Second Treatise of Government, edited by A. John Simmons.

Much as we would like to conceive empirical thought as rationally grounded in experience, pitfalls await anyone who tries to articulate this position, and ...

John Dupre warns that our understanding of human nature is being distorted by two faulty and harmful forms of pseudo-scientific thinking. Not just in the academic world but in everyday life, we find one set of experts who seek to explain the ends at which humans aim in terms of evolutionary theory, while the other set uses economic models to give rules of how we act to achieve those ends. Dupre demonstrates that these theorists' explanations do not work and (...) that, if taken seriously, their theories tend to have dangerous social and political consequences. For these reasons, it is important to resist scientism: an exaggerated conception of what science can be expected to do for us. Dupre restores sanity to the study of human nature by pointing the way to a proper understanding of humans in the societies that are our natural and necessary environments. Anyone interested in science and human life will enjoy this book--unless they are its targets. (shrink)

"What is a military aircraft? John Law shows in his beautiful analysis that it is a constant oscillation between multiplicity and singularity.

John Searle's Speech Acts (1969) and Expression and Meaning (1979) developed a highly original and influential approach to the study of language. But behind both works lay the assumption that the philosophy of language is in the end a branch of the philosophy of the mind: speech acts are forms of human action and represent just one example of the mind's capacity to relate the human organism to the world. The present book is concerned with these biologically fundamental capacities, (...) and, though third in the sequence, in effect it provides the philosophical foundations for the other two. Intentionality is taken to be the crucial mental phenomenon, and its analysis involves wide-ranging discussions of perception, action, causation, meaning, and reference. In all these areas John Searle has original and stimulating views. He ends with a resolution of the 'mind-body' problem. (shrink)

Though the revised edition of A Theory of Justice, published in 1999, is the definitive statement of Rawls's view, so much of the extensive literature on Rawls's theory refers to the first edition.

John Searle's Speech Acts (1969) and Expression and Meaning (1979) developed a highly original and influential approach to the study of language. But behind both works lay the assumption that the philosophy of language is in the end a branch of the philosophy of the mind: speech acts are forms of human action and represent just one example of the mind's capacity to relate the human organism to the world. The present book is concerned with these biologically fundamental capacities, (...) and, though third in the sequence, in effect it provides the philosophical foundations for the other two. Intentionality is taken to be the crucial mental phenomenon, and its analysis involves wide-ranging discussions of perception, action, causation, meaning, and reference. In all these areas John Searle has original and stimulating views. He ends with a resolution of the 'mind-body' problem. (shrink)

The latest volume in the Oxford Readings in Philosophy series, this work brings together some of the best and most influential recent philosophical scholarship on Nietzsche. Opening with a substantial introduction by John Richardson, it covers: Nietzsche's views on truth and knowledge, his 'doctrines' of the eternal recurrence and will to power, his distinction between Apollinian and Dionysian art, his critique of morality, his conceptions of agency and self-creation, and his genealogical method. For each of these issues, the papers (...) show Nietzsche's continuing philosophical importance. Giving a clear and accessible overview, while retaining an analytical philosophical approach throughout, this volume is essential reading for all students of Nietzsche. (shrink)

This chapter addresses the issues that motivate representationalist accounts, and it describes the different versions of representationalism as responses to these issues. It argues that the representationalist views do not adequately respond to the epistemological problems that motivate them and that they engender some ontological problems. The chapter presents an alternative ‘presentationalist’ account that preserves the straightforward sense of the mind's openness to the world. While representationalism and presentationalism agree that the relation between mental events or states is direct but (...) mediated, they radically differ in their views of the nature of the mediation involved in the mind's intentional directedness to the world. This difference and the preferability of the presentationalist account are explored. (shrink)

I. Quaestiones in librum Porphyrii Isagoge ; et , Quaestiones super Praedicamenta Aristotelis -- II. Quaestiones in libros Perihermenias Aristotelis ; Quaestiones super librum Elenchorum Aristotelis ; Theoremata -- III. Quaestiones super libros Metaphysicorum Aristotelis, libri I-V -- IV. Quaestiones super libros Metaphysicorum Aristotelis, libri VI-IX -- V. Quaestiones super secundum et tertium De anima.

This new edition includes a substantial postscript by the author, in which he responds to thirty years of discussion, criticism and further work in the field to ...

First published in 1899, The School and Society describes John Dewey’s experiences with his own famous Laboratory School, started in 1896. Dewey’s experiments at the Labora­tory School reflected his original social and educational philosophy based on American experience and concepts of democracy, not on European education models then in vogue. This forerunner of the major works shows Dewey’s per­vasive concern with the need for a rich, dynamic, and viable society. In his introduction to this volume, Joe R. Burnett states (...) Dewey’s theme. Industrialization, urbanization, science, and technology have created a revolution the schools cannot ignore. Dewey carries this theme through eight chapters: The School and Social Progress; The School and the Life of the Child; Waste in Education; Three Years of the University Elementary School; The Psychology of Elementary Education; Froebel’s Educa­tional Principles; The Psychology of Occupations; and the Development of Attention. (shrink)

Dewey's book on Democracy and Education established his credentials in the field of education and once counted as his most important book. It has been re-published in many editions and continuously in print ever since the original publication in 1916.

We will next speak of Liberality. Now this is thought to be the mean state, having for its object-matter Wealth: I mean, the Liberal man is praised not in the circumstances of war, nor in those which constitute the character of perfected self-mastery, nor again in judicial decisions, but in respect of giving and receiving Wealth, chiefly the former. By the term Wealth I mean all those things whose worth is measured by money.

Natural Law and Natural Rights is widely recognised as a seminal contribution to the philosophy of law, and an essential reference point for all students of the subject. This new edition includes a substantial postscript by the author responding to thirty years of comment, criticism, and further work in the field.

Here, in a single volume, is a comprehensive and definitive account of pragmatism and classical American philosophy. Pragmatism and Classical American Philosophy, now revised and expanded in this second edition, presents the essential writings of the major philosophers of this tradition: Charles S. Peirce, William James, Josiah Royce, George Santayana, John Dewey, and George Herbert Mead. Illuminating introductory essays, written especially for this volume by distinguished scholars of American philosophy, provide biographical and cultural context as well as original critical (...) and interpretive perspectives. This edition also includes all new selections and interpretive essays that situate pragmatism and classical American philosophy in a wider American philosphical context, including: Ralph Waldo Emerson and transcendentalism; Jane Addams, feminism, and writings of American women; Borden Parker Bowne, personalism, and idealism; Alain Locke and Afro-American thought; and John Herman Randall, Jr., nationalism and realism. Up-to-date suggestions for further reading will benefit both introductory and advanced readers. This American intellectual tradition speaks insightfully, creatively, and critically to our contemporary global society and its pressing problems. In unmatched quality and quantity, Pragmatism and Classical American Philosophy provides the resources necessary to understand and act on these insights. (shrink)

This book continues and revises the ideas of justice as fairness that John Rawls presented in _A Theory of Justice_ but changes its philosophical interpretation in a fundamental way. That previous work assumed what Rawls calls a "well-ordered society," one that is stable and relatively homogenous in its basic moral beliefs and in which there is broad agreement about what constitutes the good life. Yet in modern democratic society a plurality of incompatible and irreconcilable doctrines -- religious, philosophical, and (...) moral -- coexist within the framework of democratic institutions. Recognizing this as a permanent condition of democracy, Rawls asks how a stable and just society of free and equal citizens can live in concord when divided by reasonable but incompatible doctrines? This edition includes the essay "The Idea of Public Reason Revisited," which outlines Rawls' plans to revise _Political Liberalism,_ which were cut short by his death. "An extraordinary well-reasoned commentary on _A Theory of Justice_...a decisive turn towards political philosophy." -- _Times Literary Supplement_. (shrink)

Whether the law should permit voluntary euthanasia or physician-assisted suicide is one of the most vital questions facing all modern societies. Internationally, the main obstacle to legalisation has proved to be the objection that, even if they were morally acceptable in certain 'hard cases', voluntary euthanasia and physician-assisted suicide could not be effectively controlled; society would slide down a 'slippery slope' to the killing of patients who did not make a free and informed request, or for whom palliative care would (...) have offered an alternative. How cogent is this objection? This book provides the general reader with a lucid introduction to this central question in the debate, not least by reviewing the Dutch euthanasia experience. It will interest all in any country whether currently for or against legalisation, who wish to ensure that their opinions are better informed. (shrink)

In this lively book, John Macnamara shows how a number of important thinkers through the ages have approached problems of mental representation and the ...