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  1. Berkeley and Proof in Geometry.Richard J. Brook - 2012 - Dialogue 51 (3):419-435.
    Berkeley in his Introduction to the Principles of Human knowledge uses geometrical examples to illustrate a way of generating “universal ideas,” which allegedly account for the existence of general terms. In doing proofs we might, for example, selectively attend to the triangular shape of a diagram. Presumably what we prove using just that property applies to all triangles.I contend, rather, that given Berkeley’s view of extension, no Euclidean triangles exist to attend to. Rather proof, as Berkeley would normally assume, requires (...)
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  • Origins and Application of Geometry in the Thera Prehistoric Civilization Ca. 1650 BC.D. Fragoulis, A. Skembris, C. Papaodysseus, P. Rousopoulos, Th Panagopoulos, M. Panagopoulos, C. Triantafyllou, A. Vlachopoulos & C. Doumas - 2005 - Centaurus 47 (4):316-340.
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  • Kant’s Philosophy of Mathematics and the Greek Mathematical Tradition.Daniel Sutherland - 2004 - Philosophical Review 113 (2):157-201.
    The aggregate EIRP of an N-element antenna array is proportional to N 2. This observation illustrates an effective approach for providing deep space networks with very powerful uplinks. The increased aggregate EIRP can be employed in a number of ways, including improved emergency communications, reaching farther into deep space, increased uplink data rates, and the flexibility of simultaneously providing more than one uplink beam with the array. Furthermore, potential for cost savings also exists since the array can be formed using (...)
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  • The Beginnings of Formal Logic: Deduction in Aristotle’s Topics Vs. Prior Analytics.Marko Malink - 2015 - Phronesis 60 (3):267-309.
  • Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert†.John T. Baldwin - 2018 - Philosophia Mathematica 26 (3):346-374.
    We give a general account of the goals of axiomatization, introducing a variant on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and (...)
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  • La Définition V. 8 desElémentsd'Euclide.B. Vitrac - 1996 - Centaurus 38 (2-3):97-121.
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  • From Practical to Pure Geometry and Back.Mario Bacelar Valente - unknown
    The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically (...)
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  • Interpreting the Infinitesimal Mathematics of Leibniz and Euler.Jacques Bair, Piotr Błaszczyk, Robert Ely, Valérie Henry, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, Patrick Reeder, David M. Schaps, David Sherry & Steven Shnider - 2017 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 48 (2):195-238.
    We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...)
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  • A Methodology for Teaching Logic-Based Skills to Mathematics Students.Arnold Cusmariu - 2016 - Symposion: Theoretical and Applied Inquiries in Philosophy and Social Sciences 3 (3):259-292.
    Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to (...)
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  • The Relationship Between Hypotheses and Images in the Mathematical Subsection of the Divided Line of Plato’s Republic.Moon-Heum Yang - 2005 - Dialogue 44 (2):285-312.
    In explaining the relationship between hypotheses and images in the Line of Plato’s Republic VI, I first focus on Plato’s elucidation of the nature of mathematics as the mathematician himself understands it. I go on to criticize traditional interpretations of the relationship above based on the doubtful assumption that mathematics concerns Platonic Forms. To formulate my view of that relationship I exploit the notion of “structure.” I show how “hypotheses” as principles of proof can determine the structures of “images” by (...)
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  • Strategies for Conceptual Change: Ratio and Proportion in Classical Greek Mathematics.Paul Rusnock & Paul Thagard - 1995 - Studies in History and Philosophy of Science Part A 26 (1):107-131.
    …all men begin… by wondering that things are as they are…as they do about…the incommensurability of the diagonal of the square with the side; for it seems wonderful to all who have not yet seen the reason, that there is a thing which cannot be measured even by the smallest unit. But we must end in the contrary and, according to the proverb, the better state, as is the case in these instances too when men learn the cause; for there (...)
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  • Kant's Conception of Number.Daniel Sutherland - 2017 - Philosophical Review Current Issue 126 (2):147-190.
    Despite the importance of Kant's claims about mathematical cognition for his philosophy as a whole and for subsequent philosophy of mathematics, there is still no consensus on his philosophy of arithmetic, and in particular the role he assigns intuition in it. This inquiry sets aside the role of intuition for the nonce to investigate Kant's conception of natural number. Although Kant himself doesn't distinguish between a cardinal and an ordinal conception of number, some of the properties Kant attributes to number (...)
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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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  • Cognitive Artifacts for Geometric Reasoning.Mateusz Hohol & Marcin Miłkowski - forthcoming - Foundations of Science:1-24.
    In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...)
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  • For Some Histories of Greek Mathematics.Roy Wagner - 2009 - Science in Context 22 (4):535-565.
  • 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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  • Mathematical Concepts and Investigative Practice.Dirk Schlimm - 2012 - In Uljana Feest & Friedrich Steinle (eds.), Scientific Concepts and Investigative Practice. De Gruyter. pp. 3--127.
    In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts (...)
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  • Situating the Debate on “Geometrical Algebra” Within the Framework of Premodern Algebra.Michalis Sialaros & Jean Christianidis - 2016 - Science in Context 29 (2):129-150.
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  • Eudoxos and Dedekind: On the Ancient Greek Theory of Ratios and its Relation to Modern Mathematics.Howard Stein - 1990 - Synthese 84 (2):163 - 211.
  • On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought.Sören Stenlund - 2015 - Nordic Wittgenstein Review 4 (1):7-92.
    The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...)
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  • Didactical and Other Remarks on Some Theorems of Archimedes and Infinitesimals.A. Aaboe & J. L. Berggren - 1996 - Centaurus 38 (4):295-316.
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  • L’histoire des mathématiques de l’Antiquité.Maurice Caveing - 1998 - Revue de Synthèse 119 (4):485-510.
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  • The Debate Between H.G. Zeuthen and H. Vogt on the Historical Source of the Knowledge of Irrational Quantities.Maurice Caving - 1996 - Centaurus 38 (2-3):277-292.
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  • Definition in Mathematics.Carlo Cellucci - 2018 - European Journal for Philosophy of Science 8 (3):605-629.
    In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.
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  • The Wrong Text of Euclid: On Heiberg's Text and its Alternatives.Wilbur R. Knorr - 1996 - Centaurus 38 (2-3):208-276.
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