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)

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.

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

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)

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.

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 MacFarlane explores how we might make sense of the idea that truth is relative. He provides new, satisfying accounts of parts of our thought and talk that have resisted traditional methods of analysis, including what we mean when we talk about what is tasty, what we know, what will happen, what might be the case, and what we ought to do.

For this second edition, the editors have returned to Austin's original lecture notes, amending the printed text where it seemed necessary.

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

What psychological and philosophical significance should we attach to recent efforts at computer simulations of human cognitive capacities? In answering this question, I find it useful to distinguish what I will call "strong" AI from "weak" or "cautious" AI. According to weak AI, the principal value of the computer in the study of the mind is that it gives us a very powerful tool. For example, it enables us to formulate and test hypotheses in a more rigorous and precise fashion. (...) But according to strong AI, the computer is not merely a tool in the study of the mind; rather, the appropriately programmed computer really is a mind, in the sense that computers given the right programs can be literally said to. (shrink)

Normative requirements are often overlooked, but they are central features of the normative world. Rationality is often thought to consist in acting for reasons, but following normative requirements is also a major part of rationality. In particular, correct reasoning – both theoretical and practical – is governed by normative requirements rather than by reasons. This article explains the nature of normative requirements, and gives examples of their importance. It also describes mistakes that philosophers have made as a result of confusing (...) normative requirements with reasons. (shrink)

This book is the one to put into the hands of those who have been over-impressed by Austin 's critics....[Warnock's] brilliant editing puts everybody who is concerned with philosophical problems in his debt.

This new edition of the classic Contemporary Theories of Knowledge has been significantly updated to include analyses of the recent literature in epistemology.

John Hyman explores central problems in philosophy of action and the theory of knowledge, and connects these areas of enquiry in a new way. His approach to the dimensions of human action culminates in an original analysis of the relation between knowledge and rational behaviour, which provides the foundation for a new theory of knowledge itself.

This study provides a comprehensive reinterpretation of the meaning of Locke's political thought. John Dunn restores Locke's ideas to their exact context, and so stresses the historical question of what Locke in the Two Treatises of Government was intending to claim. By adopting this approach, he reveals the predominantly theological character of all Locke's thinking about politics and provides a convincing analysis of the development of Locke's thought. In a polemical concluding section, John Dunn argues that liberal and (...) Marxist interpretations of Locke's politics have failed to grasp his meaning. Locke emerges as not merely a contributor to the development of English constitutional thought, or as a reflector of socio-economic change in seventeenth-century England, but as essentially a Calvinist natural theologian. (shrink)

This is a selection of essays on moral responsibility that represent the major components of John Martin Fischer's overall approach to freedom of the will and moral responsibility. The collection exhibits the overall structure of Fischer's view and shows how the various elements fit together to form a comprehensive framework for analyzing free will and moral responsibility. The topics include deliberation and practical reasoning, freedom of the will, freedom of action, various notions of control, and moral accountability. The essays (...) seek to provide a foundation for our practices of holding each other (and ourselves) morally and legally accountable for our behavior. A crucial move is the distinction between two kinds of control. According to Fischer, "regulative control" involves freedom to choose and do otherwise ("alternative possibilities"), whereas "guidance control" does not. Fischer contends that guidance control is all the freedom we need to be morally responsible agents. Further, he contends that such control is fully compatible with causal determinism. Additionally, Fischer argues that we do not need genuine access to alternative possibilities in order for there to be a legitimate point to practical reasoning. Fischer's overall framework contains an argument for the contention that guidance control, and not regulative control, is associated with moral responsibility, a sketch of a comprehensive theory of moral responsibility (that ties together responsibility for actions, omissions, consequences, and character), and an account of the value of moral responsibility. On this account, the value of exhibiting freedom (of the relevant sort) and thus being morally responsible for one's behavior is a species of the value of artistic self-expression. (shrink)

John Stuart Mill's Utilitarianism is one of the most important, controversial, and suggestive works of moral philosophy ever written. Mill defends the view that all human action should produce the greatest happiness overall, and that happiness itself is to be understood as consisting in "higher" and "lower" pleasures. This volume uses the 1871 edition of the text, the last to be published in Mill's lifetime. The text is preceded by a comprehensive introduction assessing Mill's philosophy and the alternatives to (...) utilitarianism, and discussing some of the specific issues Mill raises in Utilitarianism. (shrink)

This anthology is a thorough introduction to classic literature for those who have not yet experienced these literary masterworks. For those who have known and loved these works in the past, this is an invitation to reunite with old friends in a fresh new format. From Shakespeare's finesse to Oscar Wilde's wit, this unique collection brings together works as diverse and influential as The Pilgrim's Progress and Othello. As an anthology that invites readers to immerse themselves in the masterpieces of (...) the literary giants, it is must-have addition to any library. (shrink)

"A modern classic. Dewey's lectures have lost none of their vigor...The historical approach, which underlay the central argument, is beautifully exemplified in his treatments of the origin of philosophy."-- Philosophy and Phenomenological Research "It was with this book that Dewey fully launched his campaign for experimental philosophy."-- The New Republic Written by an eminent philosopher shortly after the shattering effects of World War I, this volume offers an insightful introduction to the concept of pragmatic humanism. Dewey presents persuasive arguments against (...) traditional philosophical constructs, suggesting their basis in self-justification. He proposes, instead, an examination of core values in terms of their ultimate effects on the self and others. This experimental philosophy was received with both outrage and acclaim for daring to mingle ethics and science. Delivered in 1919 as a series of lectures, Dewey's landmark work appears here in an enlarged edition that features an informative introduction by the author, written more than 25 years after the book's initial publication. (shrink)

I argue against the orthodox view of the relationship between propositional and doxastic justification. The view under criticism is: if p is propositionally justified for S in virtue of S's having reason R, and S believes p on the basis of R, then S's belief that p is doxastically justified. I then propose and evaluate alternative accounts of the relationship between propositional and doxastic justification, and conclude that we should explain propositional justification in terms of doxastic justification. If correct, this (...) proposal would constitute a significant advance in our understanding of the sources of epistemic justification. (shrink)

This was scanned from the 1909 edition and mechanically checked against a commercial copy of the text from CDROM. Differences were corrected against the paper edition. The text itself is thus a highly accurate rendition. The footnotes were entered manually.

John Broome; V*—Fairness, Proceedings of the Aristotelian Society, Volume 91, Issue 1, 1 June 1991, Pages 87–102, https://doi.org/10.1093/aristotelian/91.1.87.

Intuition is sometimes derided as an abstruse or esoteric phenomenon akin to crystal-ball gazing. Such derision appears to be fuelled primarily by the suggestion, evidently endorsed by traditional rationalists such as Plato and Descartes, that intuition is a kind of direct, immediate apprehension akin to perception. This paper suggests that although the perceptual analogy has often been dismissed as encouraging a theoretically useless metaphor, a quasi-perceptualist view of intuition may enable rationalists to begin to meet the challenge of supplying a (...) theoretically satisfying treatment of their favoured epistemic source. It is argued, first, that intuitions and perceptual experiences are at a certain level of abstraction the same type of mental state, presentations, which are distinct from beliefs, hunches, inclinations, attractions, and seemings. The notion of a presentation is given a positive explication, which identifies its characteristic features, accounts for several of its substantive psychological roles, and systematically locates it in a threefold division among types of contentful states. Subsequently, it is argued that presentations, intuitive no less than sensory, are by their nature poised to play a distinctive epistemic role. Specifically, in the case of intuition, we encounter an intellectual state that is so structured as to provide justification without requiring justification in turn—something which may, thus, be thought of as ‘given’. (shrink)

Creative Intelligence: Essays in the Pragmatic Attitude represents an attempt at intellectual cooperation. No effort has been made, however, to attain unanimity of belief nor to proffer a platform of "planks" on which there is agreement. The consensus represented lies primarily in outlook, in conviction of what is most likely to be fruitful in method of approach. As the title page suggests, the volume presents a unity in attitude rather than a uniformity in results. Consequently each writer is definitively responsible (...) only for his own essay. The reader will note that the Essays endeavor to embody the common attitude in application to specific fields of inquiry which have been historically associated with philosophy rather than as a thing by itself. Beginning with philosophy itself, subsequent contributions discuss its application to logic, to mathematics, to physical science, to psychology, to ethics, to economics, and then again to philosophy itself in conjunction with esthetics and religion. The reader will probably find that the significant points of agreement have to do with the ideas of the genuineness of the future, of intelligence as the organ for determining the quality of that future so far as it can come within human control, and of a courageously inventive individual as the bearer of a creatively employed mind. (shrink)

Reissued here in its corrected second edition of 1864, this essay by John Stuart Mill argues for a utilitarian theory of morality. Originally printed as a series of three articles in Fraser's Magazine in 1861, the work sought to refine the 'greatest happiness' principle that had been championed by Jeremy Bentham, defending it from common criticisms, and offering a justification of its validity. Following Bentham, Mill holds that actions can be judged as right or wrong depending on whether they (...) promote happiness or 'the reverse of happiness'. Although attracted by Bentham's consequentialist framework based on empirical evidence rather than intuition, Mill separates happiness into 'higher' and 'lower' pleasures, arguing for a weighted system of measurement when making and judging decisions. Dissected and debated since its first appearance, the essay is Mill's key discussion on the topic and remains a fundamental text in the study of ethics. (shrink)

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 ...

John Henry Newman was a theologian and vicar at the university church in Oxford who became a leading thinker in the Oxford Movement, which sought to return Anglicanism to its Catholic roots. Newman converted to Catholicism in 1845 and became a cardinal in 1879. He published widely during his lifetime; his work included novels, poetry and the famous hymn 'Lead, Kindly Light', but he is most esteemed for his sermons and works of religious thought. This volume, first published in (...) 1870, is an ambitious examination of the logical processes that underpin religious faith. Newman discusses how it is possible to believe what cannot be proven empirically, and postulates that the mind has the facility to bridge the logic gap to allow for humans to believe in things that they do not fully comprehend. A lucid and masterful work which remains relevant to contemporary discussions of faith. (shrink)

The religious autobiography of John Henry Newman (1801-1890), in which he discusses his conversion to Roman Catholicism.

Reissued in its revised 1866 second edition, this work by John Stuart Mill discusses the positivist views of the French philosopher and social scientist Auguste Comte. Comte is regarded as the founder of positivism, the doctrine that all knowledge must derive from sensory experience. The two-part text was originally printed as two articles in the Westminster Review in 1865. Part 1 offers an analysis of Comte's earlier works on positivism in the natural and social sciences, while Part 2 considers (...) its application in areas such as religion and ethics. Mill states that Comte is the first philosopher who has attempted to extend positivism 'to all objects of human knowledge'. Despite being critical of a number of Comte's views, such as the exclusion of psychology from positivist science, Mill acknowledges his fellow philosopher's influence in the face of common negative perceptions of the positivist movement. (shrink)

Conceptualism is the thesis that, for any perceptual experience E, (i) E has a Fregean proposition as its content and (ii) a subject of E must possess a concept for each item represented by E. We advance a framework within which conceptualism may be defended against its most serious objections (e.g., Richard Heck's argument from nonveridical experience). The framework is of independent interest for the philosophy of mind and epistemology given its implications for debates regarding transparency, relationalism and representationalism, demonstrative (...) thought, phenomenal character, and the speckled hen objection to modest foundationalism. (shrink)

This book, a reevaluation of a major issue in modern philosophy, explores the controversy that grew out of John Locke's suggestion, in the Essay Concerning Human Understanding (1690), that God could give to matter the power of thought.

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

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.

With an inside view from an expert in the field and a clear and engaging writing style, Asian Philosophies, Seventh Edition invites students and professors to think along with the great minds of the Asian traditions. Eminent scholar and teacher John M. Koller has devoted his life to understanding and explaining Asian thought and practice. He wrote this text to give students access to the rich philosophical and religious ideas of both South and East Asia. New to this seventh (...) edition: Added material on Confucianism, including focused coverage of (1) the Analects and society and (2) ren and nature; Additional information on Theravada Buddhism, Vajrayana Buddhism, and Zen Buddhism as well as new in-depth coverage of ecological attitudes in Buddhism; Expanded coverage of ecological attitudes in all of the Asian traditions; Brief excerpts from primary sources to help better explain the key concepts; Added timelines for essential texts in each tradition; Improved Glossary and Pronunciation Guide; Additional text boxes, to help students quickly understand key ideas, texts, and concepts; Updated Further Reading sections. (shrink)