Cet article étudie les principales classifications grecques anciennes des sciences mathématiques. Je souligne le rôle joué par Platon dans cette topique.

Experience in thesis and peer reviewing revealed that most authors have much difficulty in presenting the structural supports of their study. This becomes evident in both their oral and written scientific reports in which little congruency or lack of it is frequently observed between title of the study, general objective (aim), and the hypothesis. Likewise, confusion is usually present when distinguishing purpose from objective when presenting the research problem. In order to aid in approaching the mentioned difficulties an exercise termed (...) Trivium is proposed and explained. This term that designated the main subjects in scholar formation during medieval times, is used to designate the effective coupling of the above three structural elements (title, objective, hypothesis) that, properly used, confer epistemological basis to their scientific reports. Likewise, in order to reinforce this exercise a Quadrivium is recommended in which a fourth structural element is added, i.e., a premise, as the theoretical basis that precisely indicates the purpose of the study and provides immediate support to the hypothesis. Examples to illustrate these exercises are provided. Abiding by this logic may significantly aid science students and some open-minded researchers in adequately constructing their scientific reports, while avoiding frustrating experiences due to uncomfortable observations by peers on their manuscripts. (shrink)

This study, regarding the problems surrounding 'time' during the 9th to 11th centuries, offers new insights into the scientific interests and rationality of the quadrivium, and argues for the existence of a 'discovery of nature' prior to the 12th century.

While classical schools today typically exhibit a carefully considered approach to the linguistic arts of the trivium, the equally important mathematical arts of the quadrivium have received relatively little consideration. This being so, mathematics is often approached in ways that are not distinctly classical. This article seeks to establish the importance of the quadrivial arts as a means of ascending from lower to higher things. Though most know Plato’s comparison of a lack of education to being imprisoned in a (...) cave, relatively few are familiar with the role the quadrivial arts play in ascending from the cave. Because the mathematical arts cultivate and direct the imagination, they enable students to move beyond sensible particulars to the formation of forms and figures by the mind. Thus, the mathematical arts help free us from an undue preoccupation with lower things and direct us toward the pursuit of knowledge and what truly is. (shrink)

Il faut aborder avec scepticisme les quelques références à Pythagore et aux Pythagoriciens contenues dans les dialogues de Platon, car il est difficile de prendre la mesure effective d'une influence pythagoricienne sur l'Athénien. En fait, les seules références explicites à Pythagore (Rép. X 600a-b) et aux Pythagoriciens (Rép. VII 530c-531a) ne nous disent pas grand-chose et, concernant les personnages traditionnellement considérés comme pythagoriciens, nos informations sont peu fiables, incertaines et temps contradictoires. Une telle prudence est particulièrement utile dans le cas (...) de Timée et du caractère homonyme de ce dialogue, dont le pythagorisme, bien que largement accepté par les commentateurs, semble hautement discutable. Ainsi, seule l'application d'une méthode historique prudente et lucide peut éviter cette dérive qui identifie comme pythagoricienne toute référence à la transmigration de l'âme et aux mathématiques au sens large. (shrink)

Today’s conflicts between the views that the humanities hold of science and engineering and the views science and engineering hold of the humanities weaken the very core of our culture. Their cause is lack of integration in today’s education among subjects that hark back to the medieval trivium and quadrivium. A new trivium is needed to provide every educated person with a basic understanding of the endeavors and instruments that help us address our world and shape a new morality—the (...) humanities, in the noblest sense of the word, to civilize, science to understand nature, and engineering, broadly defined, to encompass the kindred activities that modify nature. Integration of these endeavors is urgent. It involves, in turn, an intimate interaction (the “biosoma”) of biological organisms, society, and machines—a new quadrivium. No domain can any longer be considered and learned in isolation. (shrink)

The _Turba Philosophorum_ is at the confluence of many Greek traditions, and bears testimony both to the history of the transmission of Greek knowledge, and of its reception in Egypt in the ninth century.

Cet ouvrage est une réédition numérique d’un livre paru au XXe siècle, désormais indisponible dans son format d’origine.

In this paper, we examine Sextus Empiricus' treatise Against the geometers . We first set this treatise in the overall context of the sceptic's polemics against the liberal arts. After a discussion of Sextus' attitude to the quadrivium , we discuss the structure, the sources and the target of the Against the geometers . It appears that Euclid is not Sextus' source, and neither he, nor the professional geometers, seem to be Sextus' main targets. Of course, Sextus never really (...) makes clear his precise target, but his attacks are rather directed against geometry as a means of modelling the physical world, thus ruining the support geometry was intended to bring to the physical part of dogmatic philosophy. (shrink)

Dépositaire d’une partie des textes écrits par S. Weil à propos de la pensée grecque,son ami le P. J.-M. Perrin les publia sous le titre « Intuitions pré-chrétiennes ». Ce titre inscrit l’approche de la pensée grecque par S. Weil dans une perspective téléologique conformeà une apologétique héritée des Pères de l’Église, mais étrangère à la pensée de S. Weil.Pour elle, bien loin que la révélation chrétienne vienne accomplir ce qui n’était présentqu’en germe chez les Grecs, la vérité en était (...) déjà pleinement connue de ces derniers, commede bien d’autres civilisations disparues. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Role of the Heavens in the Thought of Philip MelanchthonCharlotte MethuenPhilip Melanchthon has long been recognized as one of the central figures in the German Lutheran Reformation. His theological contribution to the Reformation may be found in his codifying of Lutheran theology in the Confessio Augustana and in the Loci Communes, the first major Lutheran theological textbook, which long remained a central text for the teaching of theology (...) in Lutheran schools and universities. A teacher at the University of Wittenberg from 1518 until his death, Melanchthon was also involved in the reform of universities and the establishment of school systems throughout Lutheran Germany. These reforms facilitated the spread of the Reformation by encouraging the development of Lutheran educational principles and the establishing of Humanist education as the basis for the schools and universities of Lutheran Germany, and Melanchthon’s role in this process has led to his being accorded the title Praeceptor Germaniae. 1 More recently, Melanchthon has also been hailed as the initiator of a new “Lutheran” approach to natural philosophy. 2 It is my contention in this article that Melanchthon’s interest in natural philosophy is integrally connected to his educational interest and that these hang upon a particular theological and cosmological [End Page 385] understanding of the universe, particularly the heavens, as becomes clear from his defense of astronomy.I. Despite his theological interests and his commitment to the dissemination of Lutheran theology, Melanchthon was reluctant to teach theology and remained primarily concerned with the teaching of the subjects which made up the curriculum of the arts faculty in a sixteenth-century university. It was as Professor of Greek that he was originally appointed to the University of Wittenberg, and he had a deep interest in the teaching of ancient languages, especially Greek, and in the study of ancient texts in their original form rather than through later commentators. His promotion of the trivium (grammar, logic, and rhetoric) led him to write a textbook in each of these areas; and these works continued to be used as prescribed works in many German universities into the seventeenth century. 3 As an educator Melanchthon was thus firmly in the humanist mold. However, Melanchthon’s interests did not only take in the teaching of languages; he also promoted the teaching of both moral and natural philosophy. 4 Moreover, he was a strong advocate of the teaching of the mathematical sciences, particularly arithmetic, geometry, and astronomy. These, together with music, made up the quadrivium, which complemented the linguistic arts.In Melanchthon’s day the mathematical sciences had long been accepted as part of the arts curriculum, 5 even if in practice minimal attention had been paid them at many universities. 6 Their ambiguous status is reflected by the attitude taken by many humanist reformers, who did not regard arithmetic, geometry, astronomy, and music as an essential part of the curriculum. Thus Erasmus believed that music, arithmetic, and geography should be taught only to pupils who had a particular interest in these subjects 7 and that knowledge of nature, astrology, architecture, and other related subjects would come from reading the ancient authors. 8 In a similar vein Juan Luis [End Page 386] Vives argued that mathematics tended to divorce the mind from the practical concerns of life and that its study was not, therefore, to be recommended. 9 In contrast Melanchthon seems to have recognized the value of the study of the mathematical sciences from the beginning of his scholarly career. Perhaps he was inspired by his professor of mathematics at Tübingen, Johannes Stöffler, a gifted mathematician who wrote his own Arithmetic and was also fluent in the praise of his subject. 10 In an early oration, De artibus liberalis, delivered in Tübingen in 1517, Melanchthon praises arithmetic as a principal part of philosophy; geometry for its role in producing the wonderful machines of war and the beautiful ornaments of cities; music for its elegance and harmony; and astronomy (very briefly) for its applications in medicine. 11This oration should almost certainly be read as part of the tradition of orations and odes which extolled each of the liberal arts for its practical (and often also... (shrink)

Die Hauptthese von Sinn im Leben (SIL) besagt, dass die individuelle Orientierung am Quadrivium des Wahren, Schönen, Guten und Moralischen den Kern dessen ausmacht, was wir als sinnvolles Leben begreifen. Das, was unserem Leben Sinn verleiht, sind unter anderem Handlungen, die darauf abzielen, zum wissenschaftlichen Fortschritt beizutragen, künstlerische Projekte zu verwirklichen, anderen Menschen zu helfen oder die Welt gerechter zu gestalten. In diesem Kontext ist es nicht überraschend, dass ich die wissenschaftliche Suche nach einer philosophisch gerechtfertigten Sinntheorie selbst als (...) sinnstiftend betrachte. Daher bin ich Beatrix Himmelmann und Christoph Horn dankbar dafür, dass sie durch ihre kritischen Nachfragen meine Suche angeregt und vertieft haben, um hoffentlich dadurch zum Diskurs über das sinnvolle Leben beizutragen. Um auf die unterschiedlichen Fragen einzugehen, habe ich meine Antworten in drei Themenbereiche unterteilt. Ich hoffe, damit die meisten strittigen Diskussionspunkte zu adressieren und meine eigene Sichtweise klarer darzustellen. Es sollte selbstverständlich sein, dass dies nur der Beginn einer Diskussion und nicht ihr Ende ist. (shrink)

Dans la pensée antique, la perception intellectuelle et sensible constitue une réalité homogène, par exemple pour les Pythagoriciens, si la perception intellectuelle perçoit la structure mathématique du monde manifesté, la perception sensible est en adéquation avec le premier ordre perceptif. D’une façon très générale, nous dirons que dans la pensée antique, l’être se manifeste de deux façons, dans l’intellect et dans le sensible, seulement la réalité reste homogène. Avec Platon débute une an...

Ce livre est centré sur les rapports entre religion et philosophie en Grèce antique. Il s'agit en particulier de dégager les fondements philosophiques du rituel féminin des bacchantes et de montrer comment celui-ci trouve un écho dans la vie quotidienne des Grecs, théâtre compris, et un prolongement dans les doctrines philosophiques, notamment dans les textes des philosophes pythagoriciens, dans le platonisme par la voix de Diotime et dans le stoïcisme.

El recorrido de algunos momentos de la biografía de Gerberto de Aurillac nos muestra claramente al hombre que supo hermanar el pensamiento y la acción; el bien vivir con el bien decir, el trivium con el quadrivium; la enseñanza y la política; lo divino y lo humano. Estudiante de gramática en el monasterio de Aurillac y de matemáticas en la Marca Hispánica con el obispo Altón, como maestro, tenemos que asociarlo también a los emperadores Otón II y, sobre todo (...) a Otón III. En el filósofo de la disputa con Otrico tenemos que ver también al teólogo del De corpore et Sanguine Domini. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:BonaventureJay M. HammondChristopher M. Cullen, Bonaventure, Great Medieval Thinkers Series. Oxford: Oxford University Press, 2006. ISBN: 0-19-514925-4 (paperback); 0-19-514926-2 (hardback). Pages: xviii + 251.This volume makes a valuable contribution to the "great medieval thinkers" series from OUP by providing an accessible introduction to the philosophy and theology of the great Franciscan St. Bonaventure († 1274). The Preface presents the book's organizing principle: "to analyze Bonaventure's thought by following (...) his own division of the branches of philosophy and theology" (xi) as found in the Bonaventure's classic text On the Reduction of the Arts to Theology (text explained on 29-32; oddly the book never addresses the discrepancy in the dating of the text, c. 1254 or c. 1270). This text considers how the new core subjects of the arts, heavily influenced by Aristotle and expanding beyond the trivium and quadrivium, are illuminations that should lead back to theology, that is to Scripture, which "lies hidden in all knowledge and in all nature."Part I, contains two chapters: the Introduction and another on the definition of Christian Wisdom, the ultimate goal of the Reduction of the Arts. Part II, "The Light of Philosophical Knowledge," contains three chapters on Bonaventure's physics, metaphysics, and moral philosophy. Part III, "The Light of Theological Knowledge," includes seven chapters organized according to Bonaventure's Breviloquium: the Trinity, Creation, Sin, the Incarnation, Grace, the Sacraments, and the Last Things.The preface sets out three claims that color the rest of the book. First, as mentioned, the book attempts an "introduction to the philosophy and theology" of Bonaventure, which the book calls "Christian Philosophy" (a term introduced by Étienne Gilson, Leon Veuthey and later adopted by John Quinn). Yet, I agree with the preface that "it is not easy to say precisely what this means in practice" (xi). Although Bonaventure did employ the phrase "Christian Wisdom" throughout his writings, he never used the phrase "Christian [End Page 541] Philosophy." More explanation of this key term would have been helpful because it seems to relate more to the neo-scholastic disputes of the early to mid twentieth century than to Bonaventure's thirteenth century thought. Second, the book claims that "his Commentary on the Sentences should have a priority of place in any accurate reading" of his thought. On this point the book delivers by citing more from the Sentence Commentary than any other of Bonaventure's works (139 times overall, 63 from Book II alone), but with the much shorter Breviloquium coming in a close second (128 times). Four works come in a distant third (Hexaemeron 38, Itinerarium 32, De mysterio Trinitatis 21, and De scientia Christi 21). Unfortunately little attention is given to Bonaventure's sermons, scripture commentaries, hagiographies and spiritual tracts. Third, whatever shortcomings the book has regarding the coverage of Bonaventure's many works, the book still fulfills its claim to provide "an overview of [Bonaventure's] synthesis," which is "an elaborate and sophisticated synthesis, created in a specific historical context" (xiv).The book begins with an introductory sketch of Bonaventure's life framed by three key historic movements of the thirteenth century: the rediscovery of Aristotle, the emergence of the "universities," and the rise of the mendicant orders. These overlapping contexts are indispensable to understanding Bonaventure's writings, which the introduction surveys in general chronological order and according to genre. After reviewing his life and works, the Introduction concludes with a consideration of Bonaventure's influence. The Introduction is helpful in orienting the modern reader to Bonaventure's thought, particularly his unique synthesis of Augustine and Aristotle. However, some of the dating is imprecise. For example, in October 1255 (not 1256) Bonaventure responded to William of Saint-Amour's attacks against the mendicants with Q.1 De humilitate of his De perfectione evangelica. Thus, Bonaventure entered the secular-mendicant controversy before Thomas Aquinas and Thomas of York (12, 17). Likewise, Bonaventure most likely incepted into the Parisian consortium of masters in April 1254 not October 1257 (11).While all of Part II provides good information on the intricacies of Bonaventure's philosophy, the sections explaining [End Page 542] his hylomorphic metaphysics (42... (shrink)

아우구스티누스의 『질서론』(De ordine)은 카펠라의 저작에 앞서 7자유교과를 체계적으로 다룬 최초의 작품이다. 아우구스티누스는 이 저작에서 3학(trivium)과 4과(quadrivium)의 기본 형태를 제시하고 7자유교과의 의미와 정당성을 철학과 기독교적 가치로 입증하는 데 천착한다. 그에게 있어 7자유교과는 총체적 지식, 즉 우주적 질서에 도달하기 위한 방법임과 동시에 우주적 질서(ordo universitatis)의 일부분이다. 아우구스티누스는 보편적 우주의 질서에 도달하기 위해서는 자유교과가 반드시 가르침의 올바른 순서(ordo eruditionis)에 의해 확장해야만 한다고 말한다. 그는 7자유교과를 언어 분과와 수학분과로 나누는데, 이렇게 나눌 수 있는 근거도 필연적 발달 단계에 따른 것이다. 또한 그는 7자유교과가 이성적 (...) 활동에 최적화된 교과로 구성되었기 때문에 개별 교과는 이성(ratio)의 활동 여부에 따라 우선순위를 갖는다고 강조한다. 아우구스티누스의 7자유교과는 철학과 기독교적 가치를 강조한다는 점에서 로마의 학자들과 구분되며, 자유교과를 표준화하려고 노력한다는 점에서 그리스 학자들과 구분된다. 또한 자유교과의 위상을 신적 질서의 일부분으로 상정한다는 점에서 이전 기독교 학자들과도 구분된다. 아우구스티누스의 7자유교과는 단순히 로마의 자유교과를 기독교적으로 변용한 것이 아니라, 전문화된 로마의 자유교과 체제에 신적 질서와 그리스의 철학적 정당화를 종합하여 독특한 자유교과 체제를 구성한 것이다. 중세 학자들은 카펠라의 문학적인 내용을 적극 수용하여 7자유교과를 발전시켰지만, 자유교과의 존재 이유를 밝혀주는 아우구스티누스가 없었다면 800년 동안 7자유교과에 천착하는 일은 없었을 것이다. (shrink)

Thierry fue canciller de la escuela catedralicia de Chartres de 1142-1150. Comentador de Boecio, su obra refleja sin embargo una síntesis personal de fuentes platónicas, neoplatónicas y aristotélicas. Está convencido del valor propedéutico de las artes liberales para la teología, dando amplia cabida también al quadrivium. Este estudio, centrado en su Tractatus de sex dierum operibus, pone de manifiesto que la causalidad es el eje de su comprensión ontológica de la realidad :desde el punto de vista filosófico, la causalidad (...) explica la existencia real de lo múltiple a partir del Uno, y en la perspectiva teológica muestra la creación como obra de Dios Uno y Trino. La causalidad implica, para Thierry, la participación y garantiza por tanto la capacidad causal de los seres naturales, con la consiguiente autonomía real en su orden, pero dependiente de la causa suprema, sin la cual nada existiría. (shrink)

In twelfth-century Europe schools flourished in many centres. There were schools in monasteries and cathedrals, primarily for the education of monks and priests but often open also to laymen. In Italian towns, especially, there were lay schools teaching law and commercial skills to fee-paying students. In France, especially, also in England and other countries, there were schools for feepaying students of the liberal arts. The traditional list of the liberal arts included seven: grammar, logic and rhetoric (the "trivium"), and arithmetic, (...) geometry, astronomy and music (the "quadrivium"): most of the schools we're talking about taught the trivium, grammar, logic and rhetoric. These three disciplines in one way or other taught language skills; they were sometimes called the artes sermocinales. Students who had completed these "trivial" studies sometimes moved on to theology, or sometimes set up as teachers themselves of grammar, logic and rhetoric. Some students travelled from country to country looking for a good school, and sometimes made their living for a while by teaching before becoming students again. (shrink)

Introduction -- Prelude: a paradox of the ineffable. Schopenhauer's deep copy ; The Platonic solutions ; Four dialectical responses (after Nietzsche) -- Bloch's tone. The tone ; The natural klang ; The expressive tone ; Bloch's magic rattle ; The tone's inner ineffability ; The event-forms ; A dialectical account of music history ; Utopian musical speech -- Adorno's musical fracture. Adorno's tone ; Adorno's conception of history ; The tendenz des materials ; Music's language-like ineffability ; The immanent critique (...) ; The paradox of Mahler's vernacular ; The curve of inconsistency -- Interlude: Wittgenstein's silence -- Jankelevitch's inconsistency. Bergson and the inconsistency of time ; The aporetic source of fidelity ; Charm ; Cosmic silence ; Unwoven dialectics -- Deleuze and Guattari's rhythm. Deleuze's rhythm ; The rhythm of sense ; A structuralist quadrivium ; The rhythm of life ; Sonorous co-extensions -- Conclusion: a paradox of the vernacular. (shrink)

This article concerns the exact sciences in Byzantium, namely the four traditional sciences that were usually designated in the Middle Ages by the term quadrivium : arithmetic, astonomy, geometry and music. It offers a review of the Byzantine texts and documents preserved in these domains from the sixth century to the Fall of Constantinople. The article presents the leading trends : continuation of Hellenistic Greek science, Islamic Jewish or Latin influences, in practical or scholarly directions, as well as the (...) great historical changes that mark the stages of scientific knowledge. The ancillary sciences, geography, optics and mechanics, as well as astrology, are briefly sketched. (shrink)

A review is undertaken of the contributions of 38 classical authors, from Pythagoras in the 6th century BCE to Isidore in the 6th century CE, to the classification of the sciences. Such classifications include some that are more theoretical in function, some that are more practical. The emergence of the quadrivium and trivium is charted; the Greek concept of “enkýklios paideía” and the Latin term “artēs liberales” are defined; and the ways in which the form, content, and function of (...) science classifications change during this period are assessed. (shrink)

La centralité du chapitre A6 dans la composition du livre A de la Métaphysique a été justement soulignée : Aristote ne reconstitue pas ici une philosophie du passé, comme il l’avait fait dans les chapitres précédents, depuis ses origines jusqu’aux Pythagoriciens, mais traite de la philosophie du présent, celle de Platon et de son école, l’Académie, dont il avait été un élève pendant vingt ans. Le texte peut être divisé en trois parties principales : la première (987a29-b14) est une « (...) généalogie » du platonisme classique, celui que l’on retrouve dans les dialogues platoniciens, c’est-à-dire un dualisme ontologique de Formes intelligibles et de corps sensibles liés par une relation, qu’Aristote considère comme obscure et inexpliquée, de participation. La troisième partie (987b18-988a17) expose plutôt une doctrine platonicienne absente des dialogues, c’est-à-dire les causes et principes des Formes intelligibles, l’Un et la Dyade Indéfinie. Mais le passage sur lequel je m’attarderai est plutôt une étape intermédiaire entre la première et la troisième partie, qui introduit, à côté du dualisme ontologique classique de la première partie, des objets intermédiaires (μεταξύ) entre les Formes intelligibles et les corps sensibles, c’est-à-dire les objets des mathématiques (τὰ μαθηματικά) : l’ontologie platonicienne est donc une ontologie triadique. La conclusion du chapitre (988a7-17) résume l’ontologie platonicienne, mais sans mentionner les objets mathématiques. Dans cet essai, je traiterai de deux questions concernant l’ontologie triadique du platonisme, à savoir l’existence intermédiaire des entités mathématiques et ce que signifie être un intermédiaire entre les αἰσθητά et les εἴδη, en particulier la relation entre les objets mathématiques et la sphère des sensibles. La méthode suivie sera celle d’une « lecture lente » du texte, selon la formule proposée par Nietzsche, c’est-à-dire une lecture attentive avant tout aux détails plutôt qu’à l’ensemble du texte. (shrink)