Il est tout à fait possible de soutenir que l’espace de Newton manque d’objectivité physique (ce qui est un corollaire de la théorie einsténienne) et néanmoins prendre tout à fait au sérieux la thèse de l’espace euclidien comme condition de possibilité de l’expérience. Condition de possibilité de l’émergence d’un sujet qui configure son monde en remettant tout point de son environnement à une métrique. Cette métrique ne serait autre que celle qui donne sens à la géométrie que l’on a appris (...) à l´école, c’est à dire, la métrique euclidienne. Tout en manquant de réalité physique, tout en n`étant pas fertile à l’heure de rendre compte de la forme de l’univers, la métrique euclidienne serait la seule modalité sous laquelle il y aurait pour l’homme une prise sur le monde. Qu’on l’appelle ou pas transcendantale, la reconnaissance de la métrique euclidienne comme condition de possibilité de l’expérience nous rapprocherait de Kant sans pour autant répudier Einstein ou Poincaré. (shrink)

What role does genius play in scientific invention and discovery? This was a question at the fore of many discussions in the seventeenth and eighteenth centuries, especially as the Romantic movement came to a head. It also serves as a frame for Idit Chikurel's focused and fascinating account of Salomon Maimon's views about mathematical invention and discovery. As with many of his positions, Maimon adopted a rather iconoclastic theory of genius and invention, and Chikurel does an excellent job of charting (...) the ways in which Maimon sought to categorize the means by which we can make inventions and discoveries in mathematics. The project is fairly narrow and assumes a good deal of familiarity with Maimon, so it is... (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map (...) onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

RésuméDans l'article qui précède, l'auteur s'efforce, à l'intention surtout de ceux qui enseignent les Eléments, de mettre en lumière la signification et l'importance de deux ouvrages concernant la géométrie. Le court écrit de M. G. Bouligand fait apparaǐtre la structure algébrique et logique de cette science et présente une ȧxiomatique introduisant les notions d'ensemble et de groupe de transformation. Ainsi s'élabore une classification progressive des problèmes selon le genre des solutions qui leur conviennent. Le livre beaucoup plus étendu de M. (...) F. Gonseth analyse les aspects intuitif, expérimental et théorique de ce qui constituait l'élémentaire dans les traités classiques, puis expose les recherches suscitées par le postulat d'Euclide qui ont amené au développement de la méthode axiomatique. Le triomphe de l'atomisme, découvrant un monde où les notions élémentaires de la géométrie ne s'applíquent plus intégralement, vient rendre plus aigu encore le problème du fondement de la vérité géométrique et celui de la nature de l'espace. Grǎce à l'introduction de notions telles que celle de correspondance schématique et celle de modèle, la géométrie apparaǐt comme une science rationnelle en évolution et à laquelle l'axiomatique sert de critère. Mais bien loin de présenter les caractères d'une pure rationalité, cette axiomatique retient certaines données intuitives, měme dans son application aux géométries non euclidiennes; elle organise un contrǒle de nos intuitions les plus élémentaires les unes par les autres, et assure par leur intermédiaire la portée expérimentale et la cohérence de la géométrie. On assiste ainsi à une véritable mutation de l'élémentaire; mais celle‐ci n'est pas due au hasard. La méthode qui a présidé à cette transformation, dont les étapes historiques sont décrites, est clairement dégagée et il est visible qu'elle régit également les formes nouvelles de la physique dont la constitution théorique ne diffère pas essentiellement de la géométrie considérée comme la science rationnelle de l'espace. Cette méthode est le statut d'une science ouverte. Ainsi se trouve pleinement légitimé un exposé comme celui de M. Bouligand. En s'illustrant l'un l'autre, les deux ouvrages offrent des horizons nouveaux à l'enseignement, měme élémentaire, de la géométrie. La portée philosophique de ces ouvrages n'est pas moindre que leur portée pédagogique.ZusammenfassungIn der voranstehenden Besprechung möchte der Rezensent insbesondere diejenigen unter den Lesern, die die Elemente unterrichten, auf zwei bedeutende Geometriewerke hinweisen. G. Bouligands knapp gehaltene Schrift stellt den algebraisch‐logischen Aufbau dieser Wissenschaft ins Licht und bietet eine Axiomatik, die die Begriffe der Menge und der Transformations‐gruppe einführt. Ausgehend von der passenden Lösungsart gelangt der Verfasser zu einer fortschreitenden Klassifizierung der Aufgaben. In seinem viel breiter angelegten Werk untersucht F. Gonseth das in den klassischen Lehrbüchern als das Elementare angesehene vom intuitiven, experimentellen und theoretischen Gesichtspunkte aus und macht daran anschliessend mit den von Euklids Postulat ausgehenden Nachforschungen, die zur Entwicklung der axiomatischen Methode geführt haben, bekannt. Der eigentliche Siegeszug des Atomismus und die mit dieser Wissenschaft verbundene Entdeckung einer Welt, in der die grundsätzlichen Begriffe der Geometrie ihre volle Bestätigung nicht mehr erfahren, macht die Frage nach den Grundlagen der Geometrie und die nach dem Wesen des Raums besonders brennend. Die Einführung von Begriffen wie Schema und Modell lässt die Geometrie als eine in Entwicklung begriffene Rationalwissenschaft erscheinen, welcher die Axiomatik als Prüfstein dient. Letztere trägt aber durchaus nicht die Züge reiner Vernunftmässigkeit, sie hält vielmehr, selbst in ihrer Anwendung auf die nichteuklidischen Geometrien, an gewissen intuitiven Voraussetzungen fest; sie ermöglicht eine wechselseitige Kritik unserer elementarsten intuitiven Erkenntnisse unter sich und gewährleistet so die experimentelle Tragweite und die Kohärenz der Geometrie. Es findet also eine eigentliche Mutation des Elementaren statt, doch bleibt dieser Wandel nicht dem Zufall überlassen. Die Planmässigkeit, mit der er sich vollzog, wird unter Beschreibung der geschichtlichen Entwicklungsstufen klar dargelegt, womit erkennbar wird, dass er auch die neueren Formen der Physik bestimmt, die sich ja nach ihrem theoretischen Grundgesetz von der Geometrie als der rationalen Wissenschaft vom Raume nicht wesentlich unterscheidet. Eine solche Methode erfüllt das Lebensgesetz einer offenen Wissenschaft. Durch sie finden Ausführungen wie die Bouligands ihre volle Rechtfertigung. Sich gegenseitig ergänzend und erläuternd eröffnen beide Werke selbst dem Elementarunterricht in der Geometrie neue Möglichkeiten. Der pädagogischen kommt ihre philosophische Bedeutung gleich.In the foregoing article the author endeavours—especially for all who are teaching the Elements—to set forth the significance and importance of two books about geometry. Mr. G. Bouligand's short writing brings out the algebraic and logical structure of that science and presents an »axiomatic« introducing both the notions of »whole« and »group of transformation«. A progressive classification of problems is thus elaborated according to the kinds of solutions that suit them. Mr. F. Gonseth, in his far more comprehensive work, analyses the intuitive, experimental and theoretical aspects of what used to constitute the elementary in classical treatises, then states the researches raised up by Euclid's postulate and leading to the development of the axiomatic method. The triumph of atomism, discovering a world to which the elementary notions of geometry do no longer apply fully, makes it still more difficult to solve the problems of the basing of both geometrical truth and the nature of space. Thanks to his introducing such ideas as »schematic correspondence« and »model«, geometry appears as a rational science in progress using the »axiomatic« as its criterion. But far from offering the characteristics of pure rationality, such an »axiomatic« retains some intuitive bases, even when applied to non‐Euclidian geometries; it organizes a control of our most elementary intuitions by setting them against one another, and secures through their medium the experimental significance and coherence of geometry. Thus a real mutation of the elementary takes place; but it is not owing to mere chance. The method which has presided over that transformation, the historical changes and stages of which are described, is clearly shown, and it is made evident that it also determines the new forms of physics, the theoretical constitution of which does not differ essentially from geometry considered as the rational science of space. That method is the statute of an open science. So Mr. Bouligand's account is fully justified. By elucidating each other, both books open new horizons to the teaching—even elementary—of geometry. The philosophical significance of those books is of no less importance than the pedagogical one. (shrink)

El problema fundamental para que la Teoría de la relatividad pueda ser acorde con la filosofía de Kant es el de la utilización de una geometría no euclídea. Que sus principios sean interpretados como juicios sintéticos a priori es, a nuestro entender, un problema secundario. Si queremos que los principios de una ciencia de la naturaleza sean universales y necesarios sin recurrir a dogmatismos, no queda otra posibilidad que entenderlos trascendentalmente. Como se observa en el principio de relatividad, Einstein también (...) pensaba que las leyes físicas son universales y necesarias. Sin embargo, su perspectiva fue más racionalista que crítica. Sea como fuere, si los principios de la Teoría de la relatividad son a priori y enlazan sintéticamente la experiencia posible, esta teoría nos servirá para entender que las geometrías no euclídeas no son ajenas a los planteamientos de Kant, si bien requieren dar un rodeo epistemológico. La doctrina que nos permite fundamentar las geometrías no euclidianas desde la filosofía de Kant la llamamos “Álgebra de la experiencia”.The main problem for the Theory of the relativity to be in agreement with the Kant´s philosophy is the use of a non-Euclidean geometry. The fact that their principles could be interpreted a priori as synthetic trials is, in our opinion, a secondary problem. If we want that the principles of a science of the nature would be universal and necessary ones without appealing to dogmatisms, then, he have to understand them transcendentally. As in the principle of relativity it is observed, Einstein also thought that the physical laws are universal and necessary ones. However, their perspective was more rationalistic than critic. In any case, if the Theory of Relativity principles are a priori and they synthetically connect the possible experience, then, this theory will allow us to understand that the non-Euclidean geometries are not far away from the Kant´s positions, although they require to made a epistemological turn around. The principle that allows us to base the geometries non Euclidian from the Kant´s philosophy is called “Algebra of the experience”. (shrink)

By a thorough study of the Posterior Analytics and related Aristotelian texts, Richard McKirahan reconstructs Aristotle's theory of episteme--science. The Posterior Analytics contains the first extensive treatment of the nature and structure of science in the history of philosophy, and McKirahan's aim is to interpret it sympathetically, following the lead of the text, rather than imposing contemporary frameworks on it. In addition to treating the theory as a whole, the author uses textual and philological as well as philosophical material to (...) interpret many important but difficult individual passages. A number of issues left obscure by the Aristotelian material are settled by reference to Euclid's geometrical practice in the Elements. To justify this use of Euclid, McKirahan makes a comparative analysis of fundamental features of Euclidian geometry with the corresponding elements of Aristotle's theory. Emerging from that discussion is a more precise and more complex picture of the relation between Aristotle's theory and Greek mathematics--a picture of mutual, rather than one-way, dependence. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Recent advances in physics render a reconsideration of the Place of Mathematics in the Interpretation of the Universe particularly timely. On the one hand, we have the introduction of non-euclidian geometry, which has given rise to much controversy informed or otherwise; on the other hand, we find mysterious forms of mathematics, invented to cope with the quantum difficulties, which so far have escaped metaphysical investigation or criticism. It would seem most desirable that these modes of interpreting reality should (...) be examined from the broader philosophical point of view. (shrink)

On the nature of mathematical reasoning -- Mathematical magnitude and experience -- Non-Euclidian geometries -- Space and geometry -- Experience and geometry -- Classical mechanics -- Relative and absolute motion -- Energy and thermodynamics -- Hypotheses in physics -- Theories of modern physics -- Probability calculus -- Optics and electricity -- Electrodynamics -- The end of matter.

Traditional philosophy of science regards theoretical reasoning, based on the example of Euclidian geometry, as the hallmark of a mature science. There is, however, a parallel tradition of practical reasoning based on specific cases that goes back to Aristotle. In this paper I argue that practical reasoning is an essential part of the practice of chemistry and should be understood and appreciated on its own merits rather than regarded as a symbol of the immaturity and inferiority of chemistry (...) as a science. (shrink)

In ?155 of his New Theory of Vision Berkeley explains that a hypothetical ?unbodied spirit? ?cannot comprehend the manner wherein geometers describe a right line or circle?.1The reason for this, Berkeley continues, is that ?the rule and compass with their use being things of which it is impossible he should have any notion.? This reference to geometrical tools has led virtually all commentators to conclude that at least one reason why the unbodied spirit cannot have knowledge of plane geometry (...) is because it cannot manipulate a ruler or a compass. In this article I will show that such an interpretation is flawed. I will instead argue that Berkeley's understanding of Euclidian geometry was based on Isaac Barrow's account of the foundations of geometry. On this view geometrical objects are conceived in terms of the idealized motion that generates the objects of geometry. Consequently, that what the unbodied spirit cannot do in this context is to form an idea of motion rather than being unable to handle geometrical tools. 1All references to Berkeley are from, A. A. Luce and T. E. Jessop (eds.), The Works of George Berkeley, Bishop of Cloyne (London: Thomas Nelson and Sons, Ltd., 1948) The following abbreviations are used: An Essay Towards A New Theory of Vision, section x = New Theory x; Philosophical Commentaries, entry x = Commentaries x; Part I of A Treatise concerning the Principles of Knowledge, section x = Principles x. All other references to Berkeley's works are of the form The Works of George Berkeley, Bishop of Cloyne, volume x, page y = Works, x, y. (shrink)

The dialogue develops arguments for and against a broad new world system - info-computationalist naturalism - that is supposed to overcome the traditional mechanistic view. It would make the older mechanistic view into a special case of the new general info-computationalist framework (rather like Euclidian geometry remains valid inside a broader notion of geometry). We primarily discuss what the info-computational paradigm would mean, especially its pancomputationalist component. This includes the requirements for a the new generalized notion of (...) computing that would include sub-symbolic information processing. We investigate whether pancomputationalism can provide the basic causal structure to the world and whether the overall research program of info-computationalist naturalism appears productive, especially when it comes to new approaches to the living world, including computationalism in the philosophy of mind. (shrink)

This authoritative new introduction draws on both Richard H. Popkin's unparalleled scholarship and a wealth of historical and philosophical sources to highlight the real influences behind Spinoza's thought. Popkin reconstructs Spinoza the man, and his theories, contrasting these findings with some of the popularity held misconceptions. Locating him within the context of his family and background, the author assesses the impact on Spinoza of everything from his infamous excommunication, to his affection for Euclidian geometry and the work of (...) Descartes. With a full account of Spinoza's groundbreaking Tractatus and Ethics, and an overview of his influence on both of his contemporaries and those who were to folow, this concise survey offers a variety of new perspectives, and will be warmly welcomed by students, scholars and interested readers alike. (shrink)

This article describes and discusses the ideas of Patrick Heelan about the nature of visual perception, which he argues as not being intrinsically Euclidian, but rather as being described both in Euclidian and not-Euclidian geometries. Another objective is to place Heelan’s ideas in the context of the attempts to grasp the nature of science from the hermeneutical standpoint, in agreement with the so called analytical philosophy of science, noticing that his philosophical perspective is determined by phenomenology, in (...) particular that inspired by Heidegger. Heelan’s phenomenology and philosophy of science are innovative and original deserving to be described and analysed. It is proposed that the investigation of Heelan’s project opens up new perspectives in both phenomenology and philosophy of science. (shrink)

The belief in the old type of a priori knowledge is gone. Cartesian innate ideas vanished with the severe criticism of Kant who held that only the form, not the content, of experience is known a priori. More recent criticism of Kant's a priori has taken place indirectly by way of the development of non-Euclidian geometries so that today there is no longer the belief that either the content or the form of experience is fixed and changeless or known (...) with certainty a priori. But has the a priori phase of experience passed altogether? If not, what in recent thought has taken its place? (shrink)

The title of this book was chosen "to encourage and stimulate the broadest attitude toward thought... If, in imagination, all of the people of the world were to stand upon one another's shoulders, they would make nine complete chains between the earth and the moon. If it is not so far to the moon, then it is not so far to the limits - whatever, whenever, or wherever they may be." The only limits to our thinking, then, should be the (...) limits of the universe itself. In Nine Chains to the Moon R. Buckminster Fuller takes up his own challenge and brilliantly displays the unique daring of his ideas. In the course of his analysis he makes us question our commonly-accepted notions of Euclidian geometry, "good" architecture, classical solutions to economic problems, and even our use of words. It is this kind of courageous thinking that led Fuller to his invention of the Geodesic dome and the formulation of his comprehensive anticipatory design science. (shrink)

The biocognitive theory presented in this paper offers an alternative to the attribution of cause perpetuated by the life sciences in our western culture. Historically, biology has based its epistemology on physics to understand life, whereas cognitive science has grounded its ontology in a convergence of biology, physics, and philosophy to provide models of self that range from a passive acceptance of an outside world to the active creation of an inner world. While Newtonian physics has served us well in (...) the physical sciences, the life sciences continue to embrace the limitations of its reductionism without advancing to the more inclusive concepts offered by complexity and quantum theories. As long as the biological and cognitive sciences remain married to Newtonian physics and Cartesian philosophy, mind will be relegated to an epiphenomenon of biology that will continue to separate cognitive processes from biological functions. Rather than choosing between upward causality that explains cause from the simplest level of the organism and downward causality that explains it from the most complex to the simplest, biocognitive theory offers contextual coemergence where the simultaneous resonance between fields of bioinformation is the genesis of cause. In this model of coemergent causality, cognition, biology, and cultural history are viewed as biocognitions that communicate within a bioinformational field that has both linear processes in Euclidian geometry and non-linear processes in fractal geometry. Because of the simultaneous and reciprocal nature of mind and body communication, it is argued that biology creates thought and thought creates biology. Just as mind and body cannot be separated, to attempt a separation of mind and world would create an artificial split between observer and observation that assumes we can “step out” of the world we are attempting to observe. (shrink)

Theoretical physics is a deductive discipline which presupposes the validity and applicability of certain other disciplines. Among these are logic, algebra, analysis, and geometry. Before relativity, Euclidean geometry was the only one thought to be important for physical space. These disciplines correlate well with experience, and, in the course of time, a priori validity came to be ascribed to them. To Kant, for example, the universe could not possibly be based on any geometry other than Euclid's. The (...) discovery of non-Euclidean geometries by Bolyai and Lobatchevsky, its systematic extension by Riemann, and the critical scrutiny of foundations by Mach and Poincaré shook the “self-evident truth” of Euclidean geometry as applied to the world of physics. The death knell of the a priori truth of any geometry was supplied by Einstein. He showed that a wider body of experience was coordinated by a Riemannian geometry than by Euclidean. He traced the success of Euclidian geometry back to the existence of so-called rigid bodies, in terms of which operations can be associated with numbers which correlate well with numbers associated in a purely mathematical way with mathematical constructs put in correspondence with these bodies. For the physicist, then, the geometry of the real world is now a physical theory whose “truth” like that of all physical theory, is measured by its degree of success in coordinating experience. (shrink)

The general editorial plan behind The Collected Papers of Bertrand Russell calls for two parallel series, one concerned with Russell's work on philosophy, logic and mathematics, the other with his less technical contributions in areas such as politics, practical ethics, history and education. Volume 1, sub-titled Cambridge Essays, 1888–99, is in a sense the ancestral volume of both series, for it comprises both technical and non-technical subjects. Russell appears here as diarist, public speaker, political commentator, as well as apprentice philosopher (...) and expert on non-Euclidian geometries. The Collected Papers as a whole will span the more than 80 years of Russell's writings devoted to a formidable range of topics from the personal to the highly abstract but, large as it is, this publishing project will make no attempt to bring together the totality of his writings, since many of these remain in print and are readily available. Instead, the project's distinct emphasis will be on Russell's shorter writings, such as essays, articles, reviews and speeches, which are not so easily located, even when previously published. Grouped together in convenient subdivisions, these papers will chronicle for us Russell's intellectual and social growth to an extent even greater than that attempted by Russell himself in his three-volume Autobiography or in My Philosophical Development, and they will also serve to acquaint us with the thoughts, feelings and attitudes of the venerable figure behind the well-known major books. The project's essential aim, then, is to mine and refine the vast Russell Archives in a way that will yield a permanent foundation of study for anyone, whether mere admirer or life-long scholar, who has any interest in Russell's life and work. (shrink)

This volume investigates the use of diagrams for the purposes of information. Using key scenes from Euclidian geometry and late medieval physics, it shows how diagrams interconnect space and rules, logic and epistemic surplus, double-meanings and precision, in such a way as to generate new insights. For the first time, the study integrates previously scattered notions into a systematic whole.

Geometry is an a priori science. However, its apriority is saddled with problems. The aim of this paper will be to show 1) how Kant understands that the contents of geometry are synthetic a priori judgments in the Critique of Pure Reason, and 2) if it’s still relevant to study Kant’s theory of geometry after the challenges posed by non-Euclidian theories of space. With respect to point 1: Kant understands geometry as the discipline that objectifies (...) the pure intuition of space. Every geometric concept is built upon the pure intuition of space through a synthetic ostensive process. Furthermore, the pure intuition of space is the form of external experiences. Thus, geometry and external phenomena share a common ground – pure space. This common ground is what provides an answer to the question of the possibility of mathematics as a universal and a priori science. With respect to point 2: the relevance of studying Kant’s theory of geometry lies not only in the fact that geometry can serve as an example to philosophy based on the fact that it establishes its propositions a priori, but also because the object-study of geometry – the pure intuition of space– forces the reader to review Kant’s thoughts about sensibility and its relation to space. The analysis of Kant’s theory of geometry then amounts to studying Kant’s theory of sensibility. (shrink)

Natural philosophers make mistakes. Descartes got the laws of inertia wrong, Kant misunderstood the primacy of Euclidian geometry, and almost everyone (except perhaps Aristarchus of Samos) prior to the discovery of the telescope mistakenly thought that the solar system was geocentric. That we find Aristotle mistaken on questions in the life sciences — questions which required advances such as the microscope to even articulate — should come as little surprise. There seems nothing remarkable in the fact that Aristotle (...) mistakenly thought that the constitutive elements of the world were four (earth, wind, air, fire) or that the “organ” of thought was not the brain, but the heart. But the matter is otherwise when scholars examine Aristotle’s remarks about the female in his biological writings. When Aristotle claims that the leader of a hive is the king bee (HA 8 [9].40.623b9-10), that woman have smaller brains than men (PA 2.7.653a28-9), and that the female of the species have fewer teeth than the male (HA 2.3.501b19-21), Aristotle’s critics have suspected something more nefarious than simply poor observation at work. When Aristotle goes on to claim that “the female is, as it were, a mutilated male” (GA 2.3.737a27-8) or that “the female is more dispirited and more despondent than the male, more shameless and more lying, readier to deceive and possessing a better memory for grudges” (HA 8 [9].1.608b10-12), critics have accused Aristotle of trying to pass off misogyny as science. (shrink)

This chapter contains sections titled: Classical Economics Neoclassical Economics Neoclassical Methodology Positivism, Popper and Beyond The New Institutional Economics Public Choice Rational Choice Sociology Summary and Conclusion Notes.

Academia is fascinated by the process of how Western science and mathematical knowledge converged with the Chinese mathematical tradition. This topic not only reflects East–West interaction, but also turns upon the philosophical question of whether the Chinese have the scientific or cognitive ability to comprehend critical, logical, and abstract thinking. Fundamental to mathematical knowledge is arithmetic, which deals with numbers and their operations. This paper re-evaluates the significance of Alexander Wylie’s (1815–1887) pioneering role in translating Western arithmetical knowledge in China, (...) through (1) unearthing a mathematician-missionary, Edward Moncrieff (1824–1857), who compiled a textbook of arithmetic and translated associated mathematical knowledge, in particular Euclidian geometry, one year earlier than Wylie, and (2) critically analyzing the concept of arithmetic in their textbooks by means of the method of Begriffsgeschichte. (shrink)

A slight modification of Helmholtz’s metrical approach to the foundations of geometry leads to the locally Euclidian character of space without restriction of the curvature. A bolder generalization involving time measurement leads to the locally Minkowskian character of spacetime. Some philosophical consequences of these results are drawn.Keywords: Hermann Helmholtz; Space; Time; Spacetime.

In the frame of the celebration of the century of the publication that gave way to the special theory of relativity, a semblance of the discussions generated from the philosophical implications that derive from the possibility of accepting non-euclidian frames for our space of representation. Poncaré’s argument is exposed in defense of geometrys principle of relativity and the criticism formulated by Hans Reichenbach is discussed.

This paper seeks to demonstrate that propositions 23–27 of the Euclidian Optics originated in the context of geometrical astronomy. These entries, which deal with the geometry of spheres and rays, present material that overlaps considerably with propositions 1–3 of Aristarchus of Samos’ On the Sizes and Distances of the Sun and the Moon. While all these theorems deal with material that could conceivably be native to celestial illumination, the proofs do not work for binocular vision. It therefore seems (...) probable that the proofs were borrowed from Aristarchus or, more likely, some common astronomical predecessor. As an extension of this observation, this paper argues that the Optics displays a far less unified purpose than has typically been assumed. Instead, it reflects a conglomerate of concerns, goals and dependencies, insofar as it presents multiple proofs that emerge from a set of geometrical concerns not directly related to vision as a physical phenomenon. These entries complicate the idea that the Optics is about explaining light, sight or false appearances in any straightforward sense. Instead, the text is oriented toward a somewhat more diffuse goal: simply to articulate the geometry of vision and rays. Optics as a discipline should should therefore be understood not as a purely mathematical explanation of vision, but as a target-field of explanation constituted by features and boundaries derived from extra-visual geometric concerns. (shrink)

This high-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether. Additional topics include Einstein's electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry, time and causality, and other subjects. Highlights include a rich exposition of the elements of the special and general theories of relativity.

Die Geometrie ist seit Euklids Elementen nicht nur Vorbild fur Theorieform und Wissenschaftlichkeit. Sie hat auch seit der Antike die erkenntnistheoretische Diskussion uber das Verhaltnis von Wirklichkeit und Erkenntnis beeinflusst. In ihrer formalaxiomatischen Auffassung der physikalischen Geometrie durch A. Einstein wird die Wissenschaftstheorie des 20. Jahrhunderts in ihren Mehrheitspositionen auf einen Logischen Empirismus festgelegt. Vollstandig ausgeklammert bleibt dabei das philosophische Problem der Gegenstandskonstitution. Wovon ist Geometrie eine Wissenschaft, und wodurch erhalt sie ihre Passung auf die Anwendungen in Handwerk, Technik und (...) messender Naturwissenschaft? (shrink)

Why does ‘ethnic cleansing’ occur? Why does the rise of nationalist feeling in Europe and of Black separatist movements in the United States often go hand in hand with an upsurge of anti-Semitism? Why do some mixings of distinct religious and ethnic groups succeed, where others (for example in Northern Ireland, or in Bosnia) fail so catastrophically? Why do phrases like ‘balkanisation’, ‘dismemberment’, ‘mutilation’, ‘violation of the motherland’ occur so often in warmongering rhetoric? All of these questions are, it will (...) turn out, connected. To understand how they are connected we we will need to examine how human beings acquire a relationship to specific chunks of land, a relationship that is emotionally so strong that they are prepared to die – or kill – to protect that land for themselves or to win it back from others. Territoriality, the biologically rooted predisposition to defend core areas of home ranges against intruders, is a near-universal phenomenon amongst animals of all species. But the ways in which defended territories are conceived and demarcated differ widely from species to species and from group to group. By coming to an understanding of the geometry of these differences we can come to understand also some of the factors which give rise to interethnic conflict. (shrink)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, (...) but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom, which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then (...) introduce two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion (...) that was first conceived by ancient Greek mathematicians. (shrink)

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn the (...) fundamental principles behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (shrink)

On an ordinary view of the relation of philosophy of science to science, science serves only as a topic for philosophical reflection, reflection that proceeds by its own methods and according to its own standards. This ordinary view suggests a way of writing a global history of philosophy of science that finds substantially the same philosophical projects being pursued across widely divergent scientific eras. While not denying that this view is of some use regarding certain themes of and particular time (...) periods, this essay argues that much of the epistemology and philosophy of science in the early twentieth century in a variety of projects looked to the then current context of the exact sciences, especially geometry and physics, not merely for its topics but also for its conceptual resources and technical tools. This suggests a more variable project of philosophy of science, a deeper connection between early twentieth-century philosophy of science and its contemporary science, and a more interesting and richer history of philosophy of science than is ordinarily offered. (shrink)

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The (...) spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (shrink)

This book reconstructs, both from the historical and theoretical points of view, Leibniz's geometrical studies, focusing in particular on the research Leibniz ...

The role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the “geometry of visibles”, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s “geometry of visibles” and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a (...) choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles. (shrink)