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Proclus: A Commentary on the First Book of Euclid's Elements

Princeton University Press (1970)

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  1. Reductio ad absurdum from a dialogical perspective.Catarina Dutilh Novaes - 2016 - Philosophical Studies 173 (10):2605-2628.
    It is well known that reductio ad absurdum arguments raise a number of interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? What kind of absurdity do we obtain? Moreover, in the mathematics education literature number of studies have shown that students find it difficult to truly comprehend the idea of reductio proofs, which indicates the cognitive complexity of these constructions. In (...)
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  • History of Mathematics in Mathematics Education.Michael N. Fried - 2014 - In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. pp. 669-703.
    This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated to the ideas and (...)
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  • An Absurd Accumulation: Metaphysics M.2, 1076b11-36.Emily Katz - 2014 - Phronesis 59 (4):343-368.
    The opening argument in the Metaphysics M.2 series targeting separate mathematical objects has been dismissed as flawed and half-hearted. Yet it makes a strong case for a point that is central to Aristotle’s broader critique of Platonist views: if we posit distinct substances to explain the properties of sensible objects, we become committed to an embarrassingly prodigious ontology. There is also something to be learned from the argument about Aristotle’s own criteria for a theory of mathematical objects. I hope to (...)
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  • International Handbook of Research in History, Philosophy and Science Teaching.Michael R. Matthews (ed.) - 2014 - Springer.
    This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...)
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  • Colloquium 3: Why Beauty is Truth in All We Know: Aesthetics and Mimesis in Neoplatonic Science1.Marije Martijn - 2010 - Proceedings of the Boston Area Colloquium of Ancient Philosophy 25 (1):69-108.
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  • Colloquium 6: Physica More Geometrico Demonstrata: Natural Philosophy in Proclus and Aristotle.Dmitri Nikulin - 2003 - Proceedings of the Boston Area Colloquium of Ancient Philosophy 18 (1):183-221.
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  • Deleuze Challenges Kolmogorov on a Calculus of Problems.Jean-Claude Dumoncel - 2013 - Deleuze and Guatarri Studies 7 (2):169-193.
    In 1932 Kolmogorov created a calculus of problems. This calculus became known to Deleuze through a 1945 paper by Paulette Destouches-Février. In it, he ultimately recognised a deepening of mathematical intuitionism. However, from the beginning, he proceeded to show its limits through a return to the Leibnizian project of Calculemus taken in its metaphysical stance. In the carrying out of this project, which is illustrated through a paradigm borrowed from Spinoza, the formal parallelism between problems, Leibnizian themes and Peircean rhemes (...)
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  • PROCLUS ON PLATO'S TIMAEUS 89e3–90c7.Rüdiger Arnzen - 2013 - Arabic Sciences and Philosophy 23 (1):1-45.
    RésuméBien que l'existence d'une traduction arabe d'une section perdue en grec du commentaire de Proclus sur leTiméesoit connue depuis longtemps, ce texte n'avait fait jusqu'à présent l'objet d'aucune édition. Le présent article vise à remédier à ce manque, en proposant une édition critique du fragment arabe accompagnée d'une traduction anglaise annotée. L'étude qui l'accompagne, consacrée au contenu et à la structure du fragment transmis, montre qu'il présente, au plan formel, tous les éléments caractéristiques des commentaires de Proclus, quand bien même (...)
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  • What is it the Unbodied Spirit cannot do? Berkeley and Barrow on the Nature of Geometrical Construction.Stefan Storrie - 2012 - British Journal for the History of Philosophy 20 (2):249-268.
    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 (...)
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  • A formal system for euclid’s elements.Jeremy Avigad, Edward Dean & John Mumma - 2009 - Review of Symbolic Logic 2 (4):700--768.
    We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
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  • Proofs, pictures, and Euclid.John Mumma - 2010 - Synthese 175 (2):255 - 287.
    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 (...)
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  • Diagrams.Sun-Joo Shin - 2008 - Stanford Encyclopedia of Philosophy.
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  • On the status of proofs by contradiction in the seventeenth century.Paolo Mancosu - 1991 - Synthese 88 (1):15 - 41.
    In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations (...)
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  • Descartes and the tree of knowledge.Roger Ariew - 1992 - Synthese 92 (1):101 - 116.
    Descartes' image of the tree of knowledge from the preface to the French edition of the Principles of Philosophy is usually taken to represent Descartes' break with the past and with the fragmentation of knowledge of the schools. But if Descartes' tree of knowledge is analyzed in its proper context, another interpretation emerges. A series of contrasts with other classifications of knowledge from the seventeenth and eighteenth centuries raises some puzzles: claims of originality and radical break from the past do (...)
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  • Proclus and the neoplatonic syllogistic.John N. Martin - 2001 - Journal of Philosophical Logic 30 (3):187-240.
    An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypemegationlalpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the (...)
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  • Mathematical necessity and reality.James Franklin - 1989 - Australasian Journal of Philosophy 67 (3):286 – 294.
    Einstein, like most philosophers, thought that there cannot be mathematical truths which are both necessary and about reality. The article argues against this, starting with prima facie examples such as "It is impossible to tile my bathroom floor with regular pentagonal tiles." Replies are given to objections based on the supposedly purely logical or hypothetical nature of mathematics.
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  • The new sacred math.Ralph H. Abraham - 2006 - World Futures 62 (1 & 2):6 – 16.
    The individual soul is an ageless idea, attested in prehistoric times by the oral traditions of all cultures. But as far as we know, it enters history in ancient Egypt. I will begin with the individual soul in ancient Egypt, then recount the birth of the world soul in the Pythagorean community of ancient Greece, and trace it through the Western Esoteric Tradition until its demise in Kepler's writings, along with the rise of modern science, around 1600 CE. Then I (...)
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  • Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?Paolo Mancosu - 2009 - Review of Symbolic Logic 2 (4):612-646.
    Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...)
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  • On mathematical error.David Sherry - 1997 - Studies in History and Philosophy of Science Part A 28 (3):393-416.
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  • Geometrical First Principles in Proclus’ Commentary on the First Book of Euclid’s Elements.D. Gregory MacIsaac - 2014 - Phronesis 59 (1):44-98.
    In his commentary on Euclid, Proclus says both that the first principle of geometry are self-evident and that they are hypotheses received from the single, highest, unhypothetical science, which is probably dialectic. The implication of this seems to be that a geometer both does and does not know geometrical truths. This dilemma only exists if we assume that Proclus follows Aristotle in his understanding of these terms. This paper shows that this is not the case, and explains what Proclus himself (...)
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  • Newton and Proclus: Geometry, imagination, and knowing space.Mary Domski - 2012 - Southern Journal of Philosophy 50 (3):389-413.
    I aim to clarify the argument for space that Newton presents in De Gravitatione (composed prior to 1687) by putting Newton's remarks into conversation with the account of geometrical knowledge found in Proclus's Commentary on the First Book of Euclid's Elements (ca. 450). What I highlight is that both Newton and Proclus adopt an epistemic progression (or “order of knowing”) according to which geometrical knowledge necessarily precedes our knowledge of metaphysical truths concerning the ontological state of affairs. As I argue, (...)
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  • Determining the Determined State : The Sizing of Size From Aside/the Amassing of Mass by a Mass.Marvin Kirsh - 2013 - Philosophical Papers and Review 4 (4):49-65.
    A philosophical exploration is presented that considers entities such as atoms, electrons, protons, reasoned (in existing physics theories) by induction, to be other than universal building blocks, but artifacts of a sociological struggle that in elemental description is identical with that of all processes of matter and energy. In a universal context both men and materials, when stressed, struggle to accomplish/maintain the free state. The space occupied by cognition, inferred to be the result of the inequality of spaces, is an (...)
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  • The Readings of Apollonius' On the Cutting off of a Ratio.Ioannis M. Vandoulakis - 2012 - Arabic Sciences and Philosophy 22 (1):137-149.
    ExtractDuring the second half of the twentieth century an attention of historians of mathematics shifted to mathematics of the Late Antiquity and its subsequent development by mathematicians of the Arabic world. Many critical editions of works of mathematicians of the Hellenistic era have made their appearance, giving rise to a new, more detailed historical picture. Among these are the critical editions of the works of Diophantus, Apollonius, Archimedes, Pappus, Diocles, and others.Send article to KindleTo send this article to your Kindle, (...)
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  • Mathematical diagrams from manuscript to print: examples from the Arabic Euclidean transmission.Gregg De Young - 2012 - Synthese 186 (1):21-54.
    In this paper, I explore general features of the “architecture” (relations of white space, diagram, and text on the page) of medieval manuscripts and early printed editions of Euclidean geometry. My focus is primarily on diagrams in the Arabic transmission, although I use some examples from both Byzantine Greek and medieval Latin manuscripts as a foil to throw light on distinctive features of the Arabic transmission. My investigations suggest that the “architecture” often takes shape against the backdrop of an educational (...)
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  • Thales's sure path.David Sherry - 1999 - Studies in History and Philosophy of Science Part A 30 (4):621-650.
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  • Kepler's De quantitatibus.Giovanna Cifoletti - 1986 - Annals of Science 43 (3):213-238.
    The paper is an introduction to and an annotated translation of De quantitatibus, a mathematical manuscript by Johannes Kepler. Conceived as a philosophical treatise, the text collects, orders, and interprets the Aristotelian passages relevant to mathematics. Kepler thought of De quantitatibus as an introduction to Dasypodius's textbook, but by choosing the Aristotelian context, he distances himself from the tradition to which Dasypodius belonged. Dasypodius's works on mathematics, like Ramus's, were within the genre developed after the rediscovery of Proclus's commentary on (...)
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  • Proof-analysis and continuity.Michael Otte - 2004 - Foundations of Science 11 (1-2):121-155.
    During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. (...)
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  • The eclectic content and sources of Clavius’s Geometria Practica.John B. Little - 2022 - Archive for History of Exact Sciences 76 (4):391-424.
    We consider the Geometria Practica of Christopher Clavius, S.J., a surprisingly eclectic and comprehensive practical geometry text, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm (...)
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