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)

Are theories ‘underdetermined by the evidence’ in any way that should worry the scientific realist? I argue that no convincing reason has been given for thinking so. A crucial distinction is drawn between data equivalence and empirical equivalence. Duhem showed that it is always possible to produce a data equivalent rival to any accepted scientific theory. But there is no reason to regard such a rival as equally well empirically supported and hence no threat to realism. Two theories are empirically (...) equivalent if they share all consequences expressed in purely observational vocabulary. This is a much stronger requirement than has hitherto been recognised— two such ‘rival’ theories must in fact agree on many claims that are clearly theoretical in nature. Given this, it is unclear how much of an impact on realism a demonstration that there is always an empirically equivalent ‘rival’ to any accepted theory would have—even if such a demonstration could be produced. Certainly in the case of the version of realism that I defend—structural realism—such a demonstration would have precisely no impact: two empirically equivalent theories are, according to structural realism, cognitively indistinguishable. (shrink)

In his preface Mr Wilson writes 'I feel that a great many adults … would do better to spend less time in simply accepting the concepts of others uncritically, and more time in learning how to analyse concepts in general'. Mr Wilson starts by describing the techniques of conceptual analysis. He then gives examples of them in action by composing answers to specific questions and by criticism of quoted passages of argument. Chapter 3 sums up the importance of this kind (...) of mental activity. Chapter 4 presents selections for the reader to analyse, followed by questions of university entrance/scholarship type. This is a book to be worked through, in a sense a text-book. (shrink)

How do we acquire thoughts and beliefs about particulars by looking at pictures? One kind of reply essentially compares depiction to perception, holding that picture-perception is a form of remote object-perception. Lopes’s theory that pictures refer by demonstrative identification, and Walton’s transparency theory for photographs, constitute such remote acquaintance theories of depiction. The main purpose of this paper is to defend an alternative conception of pictures, on which they are not suitable for acquainting us with particulars but for acquainting us (...) with certain kinds of properties. This conception is outlined in §4, where it is argued that pictures are useful devices for what Heal has called indexical predication. In §2 and §3, I explain why I believe that remote acquaintance theories are false, and why picture-perception cannot function as a form of extended or remote object-perception. The main reason is that the contents of picture-perceptions do not themselves provide the kind of numerical and contextual information required for singular thought. Picture-reference is instead secured by independent beliefs or linguistic communication about the causal history of pictures as objects. In other words, it is beliefs about the numerical identity of pictures as objects that anchors the reference of the representational contents of pictures. (shrink)

Setting the scene -- Sentience and the sentient mind -- Special senses and their interpretation Survival strategies -- Social strategies -- Animals of the waters -- Animals of the air -- Animals of the savannah and plains -- Animals of the forests -- Close neighbours -- Our duty of care.

In this philosophically sophisticated and historically significant work, John H. Zammito reconstructs Kant's composition of The Critique of Judgment and reveals that it underwent three major transformations before publication. He shows that Kant not only made his "cognitive" turn, expanding the project from a "Critique of Taste" to a Critique of Judgment but he also made an "ethical" turn. This "ethical" turn was provoked by controversies in German philosophical and religious culture, in particular the writings of Johann Herder and (...) the Sturm und Drang movement in art and science, as well as the related pantheism controversy. Such topicality made the Third Critique pivotal in creating a "Kantian" movement in the 1790s, leading directly to German Idealism and Romanticism. The austerity and grandeur of Kant's philosophical writings sometimes make it hard to recognize them as the products of a historical individual situated in the particular constellation of his time and society. Here Kant emerges as a concrete historical figure struggling to preserve the achievements of cosmopolitan Aufkl-rung against challenges in natural science, religion, and politics in the late 1780s. More specifically Zammito suggests that Kant's Third Critique was animated throughout by a fierce personal rivalry with Herder and by a strong commitment to traditional Christian ideas of God and human moral freedom. "A work of extraordinary erudition. Zammito's study is both comprehensive and novel, connecting Kant's work with the aesthetic and religious controversies of the late eighteenth century. He seems to have read everything. I know of no comparable historical study of Kant's Third Critique."-Arnulf Zweig, translator and editor of Kant's ;IPhilosophical Correspondence, 1759-1799;X "An intricate, subtle, and exciting explanation of how Kant's thinking developed and adjusted to new challenges over the decade from the first edition of the Critique of Pure Reason to the appearance of the Critique of Judgment. "--John W. Burbidge, Review of Metaphysics "There has been for a long time a serious gap in English commentary on Kant's Critique of Judgment Zammito's book finally fills it. All students and scholars of Kant will want to consult it."--Frederick Beiser, Times Literary Supplement. (shrink)

Obviously medicine should be evidence-based. The issues lie in the details: what exactly counts as evidence? Do certain kinds of evidence carry more weight than others? (And if so why?) And how exactly should medicine be based on evidence? When it comes to these details, the evidence-based medicine (EBM) movement has got itself into a mess – or so it will be argued. In order to start to resolve this mess, we need to go 'back to basics'; and that means (...) turning to the philosophy of science. The theory of evidence, or rather the logic of the interrelations between theory and evidence, has always been central to the philosophy of science – sometimes under the alias of the 'theory of confirmation'. When taken together with a little philosophical commonsense, this logic can help us move towards a position on evidence in medicine that is more sophisticated and defensible than anything that EBM has been able so far to supply. (shrink)

This book investigates the central role of reason in Islamic intellectual life. Despite widespread characterization of Islam as a system of belief based only on revelation, John Walbridge argues that rational methods, not fundamentalism, have characterized Islamic law, philosophy and education since the medieval period. His research demonstrates that this medieval Islamic rational tradition was opposed by both modernists and fundamentalists, resulting in a general collapse of traditional Islamic intellectual life and its replacement by more modern but far shallower (...) forms of thought. However, the resources of this Islamic scholarly tradition remain an integral part of the Islamic intellectual tradition and will prove vital to its revival. The future of Islam, Walbridge argues, will be marked by a return to rationalism"--Provided by publisher. (shrink)

Man controls and dominates the habitat of most animals, both domestic and wild and there is a need for a pragmatic, workable approach to the problem of reconciling animal welfare with economic forces and the needs of man. It is the author's contention that much of the current philosophical discussion of animal welfare is misdirected now that it is possible to measure to some extent what animals think and feel and how much they can appreciate their quality of life. The (...) book deals with farm animals, pets, wild animals and laboratory animals and dicusses their environmental requirements, fear and stress, their response to pain, injury, disease and death, behaviour and aggression, and the implications of biotechnology and genetic engineering. Finally, the book tries to reconcile reverence for life with the inescapability of killing and reviews the prospects of preserving and enhancing quality of life for animals through legislations, education, economic and moral incentives. (shrink)

John Turri gives an example that he thinks refutes what he takes to be “G. E. Moore's view” that omissive assertions such as “It is raining but I do not believe that it is raining” are “inherently ‘absurd'”. This is that of Ellie, an eliminativist who makes such assertions. Turri thinks that these are perfectly reasonable and not even absurd. Nor does she seem irrational if the sincerity of her assertion requires her to believe its content. A commissive counterpart (...) of Ellie is Di, a dialetheist who asserts or believes that The Russell set includes itself but I believe that it is not the case that the Russell set includes itself. Since any adequate explanation of Moore's paradox must handle commissive assertions and beliefs as well as omissive ones, it must deal with Di as well as engage Ellie. I give such an explanation. I argue that neither Ellie's assertion nor her belief is irrational yet both are absurd. Likewise neither Di's assertion nor her belief is irrational yet in contrast neither is absurd. I conclude that not all Moore-paradoxical assertions or beliefs are irrational and that the syntax of Moore's examples is not sufficient for the absurdity found in them. (shrink)

A brief account of educational aims, focussing on preparation for a life of autonomous well-being.

I argue that incomprehensibility (what the ancient skeptics called acatalepsia) plays a central role in the skepticism of both Bayle and Hume. I challenge a commonly held view (recently argued by Todd Ryan) that Hume, unlike Bayle, does not present oppositions of reason--what Kant called antimonies.

Foreword I. Critical rationalism is a genuinely new philosophical perspective. It is not, however, one systematic view. The development of it by Popper and ...

1. 1735-1738 -- 2. 1738-1743 -- 3. 1743-1754 -- 4. 1755-1765 -- 5. 1765-1775 -- 6. 1776-1786 -- 7. 1787-91.

Whiskey Lao -- Fair warning -- Randomness at large -- We the addicted -- The best possible world -- The importance of being doomed -- Moral responsibility -- The upper limit to the value of possible worlds.

Could one give expression to a doctrine of essentialism without running afoul of semantical problems that are alleged to beggar systems of quantified modal logic? An affirmative answer is, I believe, called for at least in the case of individual essentialism. Individual essentialism is an ontological thesis concerning a kind of necessary connection between objects and their (essential) properties. It is not or anyhow not primarily a semantic thesis, a thesis about meanings, for example. And thus we are implicitly counselled (...) not to seek for the logical reconstruction of such necessities in theories in which necessity is a de ditto matter. A more natural-seeming target is an understanding of the structure of idioms of de re necessity. Of course, de re necessities may turn out to be special cases of necessities de ditto, with what consequences it is interesting to speculate upon, but it would be preferable if the theory of individual essences were to be prosecuted initially at least without prejudice to this other matter. A theory of individual essences therefore may find it a convenience to recognize a special category of open sentence by means of which to represent de re attributions of necessity. (shrink)