About this topic
Summary (Under construction.) This category will index four overlapping topics: 1) Plato's philosophy of mathematics, in the sense of his remarks on mathematical reality and mathematical knowledge, 2) the presence and philosophical function of mathematics in the dialogues, 3) the role of mathematics and mathematicals in dialectic and the "theory of forms", and 4) the mathematical elements of Plato's late ontology, including the so-called "unwritten doctrines". For so-called "mathematical Platonism," see the category by that name (link below).
Key works (Under construction) Taylor 1926, Klein 1968 (of which Hopkins 2011 includes a detailed commentary),  Knorr 1975, Sayre 1983
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102 found
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  1. The Platonist Absurd Accumulation of Geometrical Objects: Metaphysics Μ.2.José Edgar González-Varela - 2020 - Phronesis 65 (1):76-115.
    In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.
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  2. The Theory of Ideas and Plato’s Philosophy of Mathematics.Bogdan Dembiński - 2019 - Philosophical Problems in Science 66:95-108.
    In this article I analyze the issue of many levels of reality that are studied by natural sciences. Particularly interesting is the level of mathematics and the question of the relationship between mathematics and the structure of the real world. The mathematical nature of the world has been considered since ancient times and is the subject of ongoing research for philosophers of science to this day. One of the viewpoints in this field is mathematical Platonism. In contemporary philosophy it is (...)
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  3. Measuring Humans Against Gods: On the Digression of Plato’s Theaetetus.Jens Kristian Larsen - 2019 - Archiv für Geschichte der Philosophie 101 (1):1-29.
    The digression of Plato’s Theaetetus (172c2–177c2) is as celebrated as it is controversial. A particularly knotty question has been what status we should ascribe to the ideal of philosophy it presents, an ideal centered on the conception that true virtue consists in assimilating oneself as much as possible to god. For the ideal may seem difficult to reconcile with a Socratic conception of philosophy, and several scholars have accordingly suggested that it should be read as ironic and directed only at (...)
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  4. Sugerencias sobre el modo de combinar las formas platónicas para superar las dificultades interpretativas del diálogo Parménides. La distinción entre la participación inmediata y la participación relacional.Gerardo Óscar Matía Cubillo - 2019 - Endoxa 43:41-66.
    Este trabajo pretende ser una referencia útil para los estudiosos de la filosofía de Platón. Aporta un enfoque original a la investigación de los procesos lógicos que condicionan que unas formas participen de otras. Con la introducción del concepto de participación relacional, abre una posible vía de solución a las distintas versiones del argumento del «tercer hombre». Puede resultar de interés asimismo el método de generación de los números a partir de lo par y lo impar, propuesto en la interpretación (...)
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  5. Mathematics, Mental Imagery, and Ontology: A New Interpretation of the Divided Line.Miriam Byrd - 2018 - International Journal of the Platonic Tradition 12 (2):111-131.
    This paper presents a new interpretation of the objects of dianoia in Plato’s divided line, contending that they are mental images of the Forms hypothesized by the dianoetic reasoner. The paper is divided into two parts. A survey of the contemporary debate over the identity of the objects of dianoia yields three criteria a successful interpretation should meet. Then, it is argued that the mental images interpretation, in addition to proving consistent with key passages in the middle books of the (...)
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  6. Univocity, Duality, and Ideal Genesis: Deleuze and Plato.John Bova & Paul M. Livingston - 2017 - In Contemporary Encounters with Ancient Metaphysics. Edinburgh University Press.
    In this essay, we consider the formal and ontological implications of one specific and intensely contested dialectical context from which Deleuze’s thinking about structural ideal genesis visibly arises. This is the formal/ontological dualism between the principles, ἀρχαί, of the One (ἕν) and the Indefinite/Unlimited Dyad (ἀόριστος δυάς), which is arguably the culminating achievement of the later Plato’s development of a mathematical dialectic.3 Following commentators including Lautman, Oskar Becker, and Kenneth M. Sayre, we argue that the duality of the One and (...)
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  7. One, Two, Three… A Discussion on the Generation of Numbers in Plato’s Parmenides.Florin George Calian - 2015 - New Europe College:49-78.
    One of the questions regarding the Parmenides is whether Plato was committed to any of the arguments developed in the second part of the dialogue. This paper argues for considering at least one of the arguments from the second part of the Parmenides, namely the argument of the generation of numbers, as being platonically genuine. I argue that the argument at 142b-144b, which discusses the generation of numbers, is not deployed for the sake of dialectical argumentation alone, but it rather (...)
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  8. Early Education in Plato's Republic.Michelle Jenkins - 2015 - British Journal for the History of Philosophy 23 (5):843-863.
    In this paper, I reconsider the commonly held position that the early moral education of the Republic is arational since the youths of the Kallipolis do not yet have the capacity for reason. I argue that, because they receive an extensive mathematical education alongside their moral education, the youths not only have a capacity for reason but that capacity is being developed in their early education. If this is so, though, then we must rethink why the early moral education is (...)
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  9. Is Plato a Coherentist? The Theory of Knowledge in Republic V–VII.Edith Gwendolyn Nally - 2015 - Apeiron 48 (2):149-175.
  10. Philosophy and Mathematics in the Teaching of Plato: The Development of Idea and Modernity.N. V. Mikhailova - 2014 - Liberal Arts in Russiaроссийский Гуманитарный Журналrossijskij Gumanitarnyj Žurnalrossijskij Gumanitaryj Zhurnalrossiiskii Gumanitarnyi Zhurnal 3 (6):468.
    It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects and theories really (...)
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  11. Plato's Problem: An Introduction to Mathematical Platonism.Marco Panza & Andrea Sereni - 2013 - Palgrave-Macmillan.
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  12. Kallikles i geometria. Przyczynek do Platońskiej koncepcji sprawiedliwości [Callicles and Geometry: On Plato’s Conception of Justice].Marek Piechowiak - 2013 - In Zbigniew Władek (ed.), Księga życia i twórczości. Księga pamiątkowa dedykowana Profesorowi Romanowi A. Tokarczykowi. Wydawnictwo Polihymnia. pp. vol. 5, 281-291.
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  13. A Case For The Utility Of The Mathematical Intermediates.H. S. Arsen - 2012 - Philosophia Mathematica 20 (2):200-223.
    Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined with Aristotle’s (...)
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  14. The Problem is Not Mathematics, but Mathematicians: Plato and the Mathematicians Again.H. H. Benson - 2012 - Philosophia Mathematica 20 (2):170-199.
    I argue against a formidable interpretation of Plato’s Divided Line image according to which dianoetic correctly applies the same method as dialectic. The difference between the dianoetic and dialectic sections of the Line is not methodological, but ontological. I maintain that while this interpretation correctly identifies the mathematical method with dialectic, ( i.e. , the method of philosophy), it incorrectly identifies the mathematical method with dianoetic. Rather, Plato takes dianoetic to be a misapplication of the mathematical method by a subset (...)
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  15. Mathematical Entities in the Divided Line.M. J. Cresswell - 2012 - Review of Metaphysics 66 (1):89-104.
    The second highest level of the divided line in Plato’s Republic appears to be about the entities of mathematics—entities such as particular triangles. It differs from the highest level in two respects. It involves reasoning from hypotheses, and it uses visible images. This article defends the traditional view that the passage is indeed about these mathematical ‘intermediates’; and tries to show how the apparently different features of the second level are related, by focussing on Plato’s need to give an account (...)
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  16. Inventing Intermediates: Mathematical Discourse and Its Objects in Republic VII.Lee Franklin - 2012 - Journal of the History of Philosophy 50 (4):483-506.
  17. Recollection and the Mathematician's Method in Plato's Meno.E. Landry - 2012 - Philosophia Mathematica 20 (2):143-169.
    I argue that recollection, in Plato's Meno , should not be taken as a method, and, if it is taken as a myth, it should not be taken as a mere myth. Neither should it be taken as a truth, a priori or metaphorical. In contrast to such views, I argue that recollection ought to be taken as an hypothesis for learning. Thus, the only methods demonstrated in the Meno are the elenchus and the hypothetical, or mathematical, method. What Plato's (...)
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  18. Introduction: Hypotheses and Progress.C. McLarty - 2012 - Philosophia Mathematica 20 (2):135-142.
    The unifying theme of this issue is Plato’s dialectical view of mathematical progress and hypotheses. Besides provisional propositions, he calls concepts and goals also hypotheses. He knew mathematicians create new concepts and goals as well as theorems, and abandon many along the way, and erase the creative process from their proofs. So the hypotheses of mathematics necessarily change through use — unless Benson is correct that Plato believed mathematics could reach the unhypothetical goals of dialectic. Landry discusses Plato on mathematical (...)
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  19. La Geometria Dell'anima: Riflessioni Su Matematica Ed Etica in Platone.Paolo Pagani - 2012 - Orthotes.
    Questo testo nasce da alcune indagini sul nesso tra matematica e filosofia in ambiente “accademico”. È interessante notare che l'esplorazione di tale nesso costituisce un felice tratto di continuità tra gli studi più classici e ...
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  20. A Likely Account of Necessity: Plato's Receptacle as a Physical and Metaphysical Foundation for Space.Barbara Sattler - 2012 - Journal of the History of Philosophy 50 (2):159-195.
    This paper aims to show that—and how—Plato’s notion of the receptacle in the Timaeus provides the conditions for developing a mathematical as well as a physical space without itself being space. In response to the debate whether Plato’s receptacle is a conception of space or of matter, I suggest employing criteria from topology and the theory of metric spaces as the most basic ones available. I show that the receptacle fulfils its main task–allowing the elements qua images of the Forms (...)
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  21. The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein.Burt C. Hopkins - 2011 - Indiana University Press.
    Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their (...)
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  22. "Quem não é geômetra não entre!" Geometria, Filosofia e Platonismo.Gabriele Cornelli & Maria Cecília Miranda N. Coelhdeo - 2007 - Kriterion: Journal of Philosophy 48 (116):417-435.
  23. Beginning the 'Longer Way'.Mitchell Miller - 2007 - In G. R. F. Ferrari (ed.), The Cambridge Companion to Plato's Republic. Cambridge University Press. pp. 310--344.
    At 435c-d and 504b ff., Socrates indicates that there is a "longer and fuller way" that one must take in order to get "the best possible view" of the soul and its virtues. But Plato does not have him take this "longer way." Instead Socrates restricts himself to an indirect indication of its goals by his images of sun, line, and cave and to a programmatic outline of its first phase, the five mathematical studies. Doesn't this pointed restraint function as (...)
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  24. Diagrams, Dialectic, and Mathematical Foundations in Plato.Richard Patterson - 2007 - Apeiron 40 (1):1 - 33.
  25. Can a Proof Compel Us?Cesare Cozzo - 2005 - In C. Cellucci D. Gillies (ed.), Mathematical Reasoning and Heuristics. King's College Publications. pp. 191-212.
    The compulsion of proofs is an ancient idea, which plays an important role in Plato’s dialogues. The reader perhaps recalls Socrates’ question to the slave boy in the Meno: “If the side of a square A is 2 feet, and the corresponding area is 4, how long is the side of a square whose area is double, i.e. 8?”. The slave answers: “Obviously, Socrates, it will be twice the length” (cf. Me 82-85). A straightforward analogy: if the area is double, (...)
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  26. ‘Mathematical Platonism’ Versus Gathering the Dead: What Socrates Teaches Glaucon &Dagger.Colin McLarty - 2005 - Philosophia Mathematica 13 (2):115-134.
    Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...)
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  27. He Shaping of Deduction in Greek Mathematics: A Study in Cognitive History; The Mathematics of Plato’s Academy: A New Reconstruction.J. Bergen - 2003 - Isis 94:134-136.
  28. Annotations to the Speech of the Muses (Plato Republic 546b-C).Michael Jacovides & Kathleen McNamee - 2003 - Zeitschrift für Papyrologie und Epigraphik 144:31-50.
    Annotations to the Speech of the Muses (Plato Republic 546b-c).
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  29. Plato’s Mathematical Construction.Reviel Netz - 2003 - Classical Quarterly 53 (2):500-509.
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  30. Plato’s Pythagoreanism.Constance Chu Meinwald - 2002 - Ancient Philosophy 22 (1):87-101.
  31. Platonic Number in the Parmenides and Metaphysics XIII.Dougal Blyth - 2000 - International Journal of Philosophical Studies 8 (1):23 – 45.
    I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...)
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  32. Plato on Why Mathematics is Good for the Soul.Myles Burnyeat - 2000 - In T. Smiley (ed.), Mathematics and Necessity: Essays in the History of Philosophy. pp. 1-81.
    Anyone who has read Plato’s Republic knows it has a lot to say about mathematics. But why? I shall not be satisfied with the answer that the future rulers of the ideal city are to be educated in mathematics, so Plato is bound to give some space to the subject. I want to know why the rulers are to be educated in mathematics. More pointedly, why are they required to study so much mathematics, for so long?
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  33. The Mathematical Turn in Late Plato.Patricia Curd - 1999 - Apeiron 32 (1):49 - 66.
  34. Figure, Ratio, Form: Plato's "Five Mathematical Studies".Mitchell Miller - 1999 - Apeiron 32 (4):73 - 88.
    A close reading of the five mathematical studies Socrates proposes for the philosopher-to-be in Republic VII, arguing that (1) each study proposes an object the thought of which turns the soul towards pure intelligibility and that (2) the sequence of studies involves both a departure from the sensible and a return to it in its intelligible structure.
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  35. Figure, Ratio, Form: Plato's Five Mathematical Studies.Mitchell Miller - 1999 - Apeiron 32 (4):73-88.
    A close reading of the five mathematical studies Socrates proposes for the philosopher-to-be in Republic VII, arguing that (1) each study proposes an object the thought of which turns the soul towards pure intelligibility and that (2) the sequence of studies involves both a departure from the sensible and a return to it in its intelligible structure.
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  36. Plato's Mathematics P. Pritchard: Plato's Philosophy of Mathematics. Pp. Vii + 191. Sankt Augustin: Academia Verlag, 1995. DM 58. ISBN: 3-88345-637-3. [REVIEW]Malcolm Schofield - 1998 - The Classical Review 48 (1):84-85.
  37. Plato's Mathematics - Pritchard P.: Plato's Philosophy of Mathematics. (International Plato Studies, 5.) Pp. Vii + 191. Sankt Augustin: Academia Verlag, 1995. DM 58. ISBN: 3-88345-637-3. [REVIEW]Malcolm Schofield - 1998 - The Classical Review 48 (01):84-85.
  38. Plato as "Architect of Science".Leonid Zhmud - 1998 - Phronesis 43 (3):211-244.
    The figure of the cordial host of the Academy, who invited the most gifted mathematicians and cultivated pure research, whose keen intellect was able if not to solve the particular problem then at least to show the method for its solution: this figure is quite familiar to students of Greek science. But was the Academy as such a center of scientific research, and did Plato really set for mathematicians and astronomers the problems they should study and methods they should use? (...)
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  39. Plato’s Philosophy of Mathematics. [REVIEW]Ian Mueller - 1997 - Ancient Philosophy 17 (2):458-461.
  40. Plato's Philosophy of Mathematics.Paul Pritchard - 1995 - Academia Verlag.
    Available from UMI in association with The British Library. ;Plato's philosophy of mathematics must be a philosophy of 4th century B.C. Greek mathematics, and cannot be understood if one is not aware that the notions involved in this mathematics differ radically from our own notions; particularly, the notion of arithmos is quite different from our notion of number. The development of the post-Renaissance notion of number brought with it a different conception of what mathematics is, and we must be able (...)
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  41. Mathematics and the Conversion of the Mind: Republic Vii 522c1-531e3.lan Robins - 1995 - Ancient Philosophy 15 (2):359-391.
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  42. Mathematics and the Conversion of the Mind: Republic VII 522c1-531e.lan Robins - 1995 - Ancient Philosophy 15 (2):359-391.
    An account of how the mathematical sciences turn the mind away from becoming and towards being. There are four main conclusions. 1. The study of numbers, when treated independently of the other sciences, uses a particular conception of the nature of numbers to detach the mind from the influence of perceptible objects. 2. The study of ratios and proportions, explicitly the core of Plato's harmonics, is fundamental also to plane and solid geometry and astronomy. 3. Ratios and proportion form the (...)
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  43. Ptolemy's Pythagoreans, Archytas, and Plato's Conception of Mathematics.Andrew Barker - 1994 - Phronesis 39 (2):113-135.
  44. Codici Nel Pentateuco E Matematica Egizio-Platonica.Gian Carlo Duranti - 1994 - Arcipelago.
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  45. Philosophy as Performed in Plato's "Theaetetus".Eugenio Benitez & Livia Guimaraes - 1993 - Review of Metaphysics 47 (2):297 - 328.
    We examine the "Theaetetus" in the light of its juxtaposition of philosophical, mathematical and sophistical approaches to knowledge, which we show to be a prominent feature of the drama. We suggest that clarifying the nature of philosophy supersedes the question of knowledge as the main ambition of the "Theaetetus". Socrates shows Theaetetus that philosophy is not a demonstrative science, like geometry, but it is also not mere word-play, like sophistry. The nature of philosophy is revealed in Socrates' activity of examination (...)
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  46. VIastos on Elenchus and Mathematics.Kenneth Seeskin - 1993 - Ancient Philosophy 13 (1):37-53.
  47. La génesis de las dimensiones en Platón.Juao de Dios Bares - 1992 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 7 (1-3):451-471.
    This paper deals with the ontological genesis of the series point-line-plane-solid in Plato’s philosophy. The texts of the Dialogues concerning this subject are presented, and passages of the Unwritten Doctrines that we know from Aristotle and other sources are specially considered. Certain problems within this context, such as the postulation of indivisible Iines, or the relation between each of the dimensions and the figures that can be placed in them, are considered in detail.
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  48. The Meno and the Mysteries of Mathematics. Lloyd - 1992 - Phronesis 37 (2):166-183.
  49. Platonism and Mathematics. [REVIEW]H. J. Blumenthal - 1991 - The Classical Review 41 (1):101-103.
  50. What Are the Topnoi in Philebus 51C?Todd Compton - 1990 - Classical Quarterly 40 (02):549-.
    In an interesting passage in the Philebus , Plato associates pure beauty with geometrical forms created by certain measuring tools used both by mathematicians and carpenters. The ‘beauty of figures’ is analysed as' something straight [εθ τι]… and round [περιφερς] and the two- and three-dimensional figures generated from these by [τρνοι] and ruler [κανσ7iota;] and set-squares [γωναι]' He continues: ‘For I maintain that these things are not beautiful in relation to something, as other things are, but they are always beautiful (...)
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