About this topic
Summary Philosophical discussions about mathematics have a long history, which basically coincides with the history of philosophy. The main historiographic divisions are thus the same as for philosophy in general, i.e. there is philosophy of mathematics in Ancient Philosophy, in Medieval Philosophy, in Early Modern Philosophy (16th-18th centuries), and in Late Modern Philosophy (19th-20th centuries). For a general introduction to the topic, including source material, see R. Marcus and M. McEvoy, eds., A Historical Introduction to the Philosophy of Mathematics: A Reader (Bloomsbury, 2016). For excerpts and translations from crucial authors since Kant, compare W. Ewald, ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vols. I-II (Oxford University Press, 1996).  And for the late 19th and the first half of the 20th centuries, see P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics: Selected Readings (2nd ed., Cambridge University Press, 1984).
Key works Logicism, formalism, intuitionism, structuralism, foundations, logic, proof, truth, axioms, infinity.
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354 found
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1 — 50 / 354
  1. added 2018-12-29
    Is Geometry Analytic?David Mwakima - 2017 - Dianoia 1 (4):66 - 78.
    In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.
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  2. added 2018-12-24
    Frege's Conception of Numbers as Objects.C. Wright - 1984 - Critical Philosophy 1 (1):97.
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  3. added 2018-12-12
    Grössenrelationen und Zahlen, eine psychologische Studie.Christian von Ehrenfels & Carlo Ierna - 2017 - In Jutta Valent & Ulf Höfer (eds.), Christian von Ehrenfels: Philosophie – Gestalttheorie – Kunst: Österreichische Ideengeschichte Im Fin de Siècle. De Gruyter. pp. 185-234.
  4. added 2018-12-12
    Zeno's Paradoxes.Nicholas Huggett - 2002
    Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato's Parmenides. There we learn that Zeno was nearly 40 years old when Socrates was a young man, say 20. Since Socrates was born in 469 BC we can estimate a birth date for Zeno around 490 BC. Beyond this, really all we know is that he was close to Parmenides (Plato reports the gossip that they were lovers when Zeno was young), (...)
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  5. added 2018-12-03
    Uwagi o arytmetyce Grassmanna.Jerzy Hanusek - 2015 - Diametros 45:107-121.
    Hermann Grassmann’s 1861 work [2] was probably the first attempt at an axiomatic approach to arithmetic. The historical significance of this work is enormous, even though the set of axioms has proven to be incomplete. Basing on the interpretation of Grassmann’s theory provided by Hao Wang in [4], I present its detailed discussion, define the class of models of Grassmann’s arithmetic and discuss a certain axiom system for integers, modeled on Grassmann’s theory. At the end I propose to modify the (...)
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  6. added 2018-11-23
    Mathematical Progress: Ariadne's Thread.Michael Liston - 2000 - In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge. Kluwer Academic Publishers. pp. 257--268.
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  7. added 2018-11-09
    On the Intuitionistic Background of Gentzen's 1935 and 1936 Consistency Proofs and Their Philosophical Aspects.Yuta Takahashi - 2018 - Annals of the Japan Association for Philosophy of Science 27:1-26.
    Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...)
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  8. added 2018-09-29
    Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - 2019 - Noûs 53 (1):160-191.
    The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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  9. added 2018-09-28
    A Life in Science, Philosophy, and the Public Domain: Three Biographies of PoincaréJeremy J. Gray. Henri Poincaré: A Scientific Biography. Princeton, NJ: Princeton University Press, 2013. Pp. Xii+592. $35.00/£24.95 .Ferdinand Verhulst. Henri Poincaré: Impatient Genius. New York: Springer, 2012. Pp. Xi+260. $49.95 ; $39.95 .Jean-Marc Ginoux and Christian Gerini. 2012. Henri Poincaré: Une Biographie au Quotidien. Paris: Ellipses, 2012. Pp. Iv+298. €24.00 . [Henri Poincaré: A Biography Through the Daily Papers. Singapore: World Scientific, 2013. Pp. 260. $29.00 ; $22.00 .]. [REVIEW]David J. Stump - 2016 - Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (2):309-318.
  10. added 2018-09-24
    Michel Fichant; Sophie Roux . Louis Couturat : Mathématiques, langage, philosophie. 363 pp., apps., bibl., index. Paris: Classiques Garnier, 2017. €46. [REVIEW]David J. Stump - 2018 - Isis 109 (1):195-196.
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  11. added 2018-09-20
    Predicativity and Feferman.Laura Crosilla - 2017 - In Feferman on Foundations. Springer Verlag. pp. 423-447.
    Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell's type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predica-tivity have since taken a life of their own, (...)
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  12. added 2018-09-10
    Idealization in Mathematics.Thomas Mormann - 2012 - Discusiones Filosóficas 13 (20):147 - 167.
  13. added 2018-09-06
    Wilfried Sieg. Hilbert's Programs and Beyond. Xii + 440 Pp., Illus., Bibl., Index. Oxford: Oxford University Press, 2013. $85 .William Ewald;, Wilfried Sieg ., Michael Hallett . David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Xxv + 1,062 Pp., Tables, Bibl., Indexes. Berlin: Springer, 2013. $139. [REVIEW]Tom Archibald - 2015 - Isis 106 (2):481-483.
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  14. added 2018-07-30
    Review: Eckart Menzler-Trott’s — Logic’s Lost Genius: The Life of Gerhard Gentzen. [REVIEW]John N. Crossley - 2008 - Australasian Journal of Logic 6:83-86.
    Review of Eckart Menzler-Trott’s book, Logic’s Lost Genius: The Life of Gerhard Gentzen.
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  15. added 2018-07-29
    Hermann von Helmholtz, Philosophische Und Populärwissenschaftliche Schriften. 3 Bände.Gregor Schiemann, Michael Heidelberger & Helmut Pulte (eds.) - 2017 - Hamburg: Meiner.
    Aus dem vielfältigen Werk von Hermann von Helmholtz versammelt diese Ausgabe die im engeren Sinne philosophischen Abhandlungen, vor allem zur Wissenschaftsphilosophie und Erkenntnistheorie, sowie Vorträge und Reden, bei denen der Autor seine Ausnahmestellung im Wissenschaftsbetrieb nutzte, um die Wissenschaften und ihre Institutionen in der bestehenden Form zu repräsentieren und zu begründen. -/- Ein Philosoph wollte Helmholtz nicht sein, aber er legte der philosophischen Reflexion wissenschaftlicher Erkenntnis und wissenschaftlichen Handelns große Bedeutung bei. Vor allem bezog er, in der Regel ausgehend von (...)
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  16. added 2018-06-06
    Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...)
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  17. added 2018-05-01
    Salomon Maimons Philosophie der Mathematik.Hugo Bergmann - 1931 - Isis: A Journal of the History of Science 16 (2):220-232.
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  18. added 2018-04-29
    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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  19. added 2018-04-18
    Aristoteles’in Matematik Felsefesi ve Matematik Soyut­lama.Murat Kelikli - 2017 - Beytulhikme An International Journal of Philosophy 7 (2):33-49.
    Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this rea­ son, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the (...)
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  20. added 2018-03-20
    Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta.Maria Zack & Dirk Schlimm (eds.) - 2017 - New York: Birkhäuser.
    Proceedings of the Canadian Society for History and Philosophy of Mathematics.
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  21. added 2018-03-19
    On A.A. Markov’s Attitude Towards Brouwer’s Intuitionism.Ioannis M. Vandoulakis - 2015 - Philosophia Scientae 19:143-158.
    The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his endnotes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.
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  22. added 2018-03-19
    Proofs as Spatio-Temporal Processes.Petros Stefaneas & Ioannis M. Vandoulakis - 2014 - Philosophia Scientae 18:111-125.
    The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non-constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can be defined as (...)
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  23. added 2018-03-07
    il computer di Kant.Luigi Borzacchini (ed.) - 2015 - Bari, Italy: dedalo editore.
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  24. added 2018-03-07
    il computer di ockham.Luigi Borzacchini - 2010 - Bari, Italy: dedalo editore.
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  25. added 2018-03-07
    il computer di platone.Luigi Borzacchini (ed.) - 2005 - Bari, Italy: dedalo editore.
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  26. added 2018-03-06
    Gödel's Argument for Cantorian Cardinality.Matthew W. Parker - 2017 - Noûs.
    On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's (...)
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  27. added 2018-02-26
    Truth in Fiction: Origins and Consequences of Leibniz’s Doctrine of Infinitesimal Magnitudes.Douglas Jesseph - 2008 - In Douglas Jesseph & Ursula Goldenbaum (eds.), Infinitesimal Differences: Controversies Between Leibniz and His Contemporaries. Walter de Gruyter.
  28. added 2018-02-26
    Ueber Leibnizens Methode der Direkten Differentiation.Braanislav Petronievics - 1934 - Isis: A Journal of the History of Science 22 (1):69-76.
  29. added 2018-02-17
    Review of M. Baaz, C. H. Papadimitriou, H. W. Putnam, D. S. Scott, and C. L. Harper, Jr (Eds.), Kurt Godel and the Foundations of Mathematics: Horizons of Truth[REVIEW]Peter Smith - 2012 - Philosophia Mathematica 20 (2):260-266.
  30. added 2018-02-17
    Aristotle’s Prohibition Rule on Kind-Crossing and the Definition of Mathematics as a Science of Quantities.Paola Cantù - 2010 - Synthese 174 (2):225-235.
    The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...)
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  31. added 2018-02-17
    Purity as an Ideal of Proof.Michael Detlefsen - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 179-197.
    Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content.
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  32. added 2018-02-17
    Dedekind’s Analysis of Number: Systems and Axioms.Wilfried Sieg & Dirk Schlimm - 2005 - Synthese 147 (1):121-170.
    Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.
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  33. added 2018-02-17
    To Err is Humeant.Mark Wilson - 1999 - Philosophia Mathematica 7 (3):247-257.
    George Boolos, Crispin Wright, and others have demonstrated how most of Frege's treatment of arithmetic can be obtained from a second-order statement that Boolos dubbed ‘Hume's principle’. This note explores the historical evidence that Frege originally planned to develop a philosophical approach to numbers in which Hume's principle is central, but this strategy was abandoned midway through his Grundlagen.
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  34. added 2018-01-17
    Wittgenstein Und Die Philosophie der Mathematik.Bromand Joachim & Reichert Bastian (eds.) - forthcoming - Mentis Verlag.
    Ludwig Wittgenstein selbst hielt seine Überlegungen zur Mathematik für seinen bedeutendsten Beitrag zur Philosophie. So beabsichtigte er zunächst, dem Thema einen zentralen Teil seiner Philosophischen Untersuchungen zu widmen. Tatsächlich wird kaum irgendwo sonst in Wittgensteins Werk so deutlich, wie radikal die Konsequenzen seines Denkens eigentlich sind. Vermutlich deshalb haben Wittgensteins Bemerkungen zur Mathematik unter all seinen Schriften auch den größten Widerstand provoziert: Seine Bemerkungen zu den Gödel’schen Unvollständigkeitssätzen bezeichnete Gödel selbst als Nonsens, und Alan Turing warf Wittgenstein vor, dass aufgrund (...)
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  35. added 2018-01-16
    Dedekind’s Structuralism: Creating Concepts and Deriving Theorems.Wilfried Sieg & Rebecca Morris - 2018 - In Erich Reck (ed.), Logic, Philosophy of Mathematics, and their History: Essays in Honor W.W. Tait. College Publications.
    Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...)
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  36. added 2018-01-12
    Character and Object.Rebecca Morris & Jeremy Avigad - 2016 - Review of Symbolic Logic 9 (3):480-510.
    In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation. In this essay, we (...)
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  37. added 2017-12-27
    Models in Geometry and Logic: 1870-1920.Patricia Blanchette - 2017 - In Seppälä Niniiluoto (ed.), Logic, Methodology and Philosophy of Science - Proceedings of the 15th International Congress. College Publications. pp. 41-61.
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  38. added 2017-12-27
    Frege on Mathematical Progress.Patricia Blanchette - 2016 - In Sorin Costreie (ed.), Early Analytic Philosophy: New Perspectives on the Tradition. Springer. pp. 3 - 19.
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  39. added 2017-11-22
    Intuição e Conceito: A Transformação do Pensamento Matemático de Kant a Bolzano.Humberto de Assis Clímaco - 2014 - Dissertation, Universidade Federal de Goiás, Brazil
  40. added 2017-11-21
    Diálogo Sobre a Imutabilidade do céu: Aristóteles e Galileu.Arthur Feitosa de Bulhões - 2012 - Dissertation, UFPE, Brazil
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  41. added 2017-10-21
    Frege's Begriffsschrift is Indeed First-Order Complete.Yang Liu - 2017 - History and Philosophy of Logic 38 (4):342-344.
    It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is (...)
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  42. added 2017-10-10
    Kurt Gödel, Paper on the Incompleteness Theorems (1931).Richard Zach - 2004 - In Ivor Grattan-Guinness (ed.), Landmark Writings in Mathematics. Amsterdam: North-Holland. pp. 917-925.
    This chapter describes Kurt Gödel's paper on the incompleteness theorems. Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Gödel's first incompleteness theorem showed that this assumption was false: it states that there are sentences of number theory that are (...)
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  43. added 2017-09-22
    Philosophy and Mathematics, From Plato to the Present.Robert J. Baum - 1973 - San Francisco, Freeman, Cooper.
  44. added 2017-09-15
    Paul Erickson, The World the Game Theorists Made. [REVIEW]Philip Mirowski - 2017 - Hopos: The Journal of the International Society for the History of Philosophy of Science 7 (1):160-163.
  45. added 2017-09-08
    Wittgenstein’s ‘Notorious Paragraph’ About the Gödel Theorem.Timm Lampert - 2006 - In Contributions of the Austrian Wittgenstein Societ. pp. 168-171.
    In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...)
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  46. added 2017-09-04
    Reflections on Recollections: A Jewish Mathematician’s Life. [REVIEW]Mark Zelcer - 2017 - Metascience 26 (3):503-505.
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  47. added 2017-08-16
    NATUREZA E MOVIMENTO EM GALILEU: críticas à concepção aristotélica de movimento natural.Luiz Antonio Brandt - 2013 - XVI Semana Acadêmica de Filosofia da Unioeste.
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  48. added 2017-08-04
    Le Tracé Continu des Sections Coniques À la Renaissance: Applications Optico-Perspectives, Héritage de la Tradition Mathématique Arabe.Dominique Raynaud - 2007 - Arabic Sciences and Philosophy 17 (2):299-345.
    The perfect compass, used by al-Qūhī, al-Sijzī and his successors for the continuous drawing of conic sections, reappeared after a long eclipse in the works of Renaissance mathematicians like Francesco Barozzi in Venice. The resurgence of this instrument seems to have depended on its interest to solve new optico-perspective problems. Having reviewed the various instruments designed for the drawing of conic sections, the article is focused on the sole conic compass. Theoretical and empirical applications are detailed. Contrarily to the common (...)
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  49. added 2017-07-31
    Kant's Theory of Experience at the End of the War: Scholem and Benjamin Read Cohen.Julia Ng - 2012 - Modern Language Notes 127 (3):462-484.
    At the end of one side of a manuscript entitled “On Kant” and housedin the Scholem Archive in Jerusalem, one reads the following pro-nouncement: “it is impossible to understand Kant today.” 1 Whatever it might mean to “understand” Kant, or indeed, whatever “Kant” is heremeant to be understood, it is certain, according to the manuscript,that such understanding cannot come about by way of purporting tohave returned to or spoken in the name of “Kant.” For “[t]oday,” sothe document begins, “there are (...)
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  50. added 2017-07-28
    Acts of Time: Cohen and Benjamin on Mathematics and History.Julia Ng - 2017 - Paradigmi. Rivista di Critica Filosofica 2017 (1):41-60.
    This paper argues that the principle of continuity that underlies Benjamin’s understanding of what makes the reality of a thing thinkable, which in the Kantian context implies a process of “filling time” with an anticipatory structure oriented to the subject, is of a different order than that of infinitesimal calculus—and that a “discontinuity” constitutive of the continuity of experience and (merely) counterposed to the image of actuality as an infinite gradation of ultimately thetic acts cannot be the principle on which (...)
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