Abstract
The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined in the Sophist and the Statesman. Against this framework, I shall move to a close examination of specific examples drawn from the Theaetetus (154c–55d), the Republic (523c–24d), and the Phaedo (96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify.
Acerbi, F. 2005. “A Reference to Perfect Numbers in Plato’s Theaetetus”. Archive for History of Exact Sciences 59, 319–48.Search in Google Scholar
Adam, J. 1963. The Republic of Plato. Edited with critical notes, commentary and appendices, 2nd edition with an introduction by D. A. Rees. 2 vols. Cambridge.Search in Google Scholar
Artmann, B. 1999. Euclid. The Creation of Mathematics. New York.Search in Google Scholar
Auffret, T. 2018. “L’angle de contingence chez Platon et Protagoras”. Les Études philosophiques 1, 139–62.Search in Google Scholar
Benson, H. H. 2010. “Plato’s Philosophical Method in the Republic. The Divided Line (510b–511d)”. In: Cambridge Critical Guide to Plato’s Republic. Ed. by M. L. McPherran. Cambridge, 188–208.Search in Google Scholar
–. 2012. “The Problem is not Mathematics, but Mathematicians. Plato and the Mathematicians Again”. Philosophia Mathematica 20, 170–99.Search in Google Scholar
Bluck, R. S. 1961. “The Puzzles of Size and Number in Plato’s Theaetetus”. Proceedings of the Cambridge Philological Society 7, 7–9.Search in Google Scholar
Brandwood, L. 1976. A Word Index to Plato. Leeds.Search in Google Scholar
Brown, M. S. 1969. “Theaetetus: Knowledge as Continued Learning”. Journal of the History of Philosophy 7, 359–79.Search in Google Scholar
Burnyeat, M. F. 1978. “The Philosophical Sense of Theaetetus’ Mathematics”. Isis 69, 489–513.Search in Google Scholar
–. 2000. “Plato on Why Mathematics is Good for the Soul”. Proceedings of the British Academy 103, 1–81.Search in Google Scholar
Burnyeat, M./Levett, M. J. (eds.) 1990. The Theaetetus of Plato. Indianapolis/Cambridge.Search in Google Scholar
Byrd, M. 2007. “Dialectic and Plato’s Method of Hypothesis”. Apeiron 40, 141–58.Search in Google Scholar
–. 2018. “Mathematics, Mental Imagery, and Ontology. A New Interpretation of the Divided Line”. The International Journal of the Platonic Tradition 12, 111–31.Search in Google Scholar
Cattanei, E. 2003. “Le matematiche al tempo di Platone e la loro riforma”. In: Platone, La Repubblica. Ed. by M. Vegetti, vol. V, libri VI–VII. Napoli, 473–539.Search in Google Scholar
Clarke, T. 2012. “The Argument from Relatives”. Oxford Studies in Ancient Philosophy 42, 151–77.Search in Google Scholar
Cleary, J. 2003. “Paideia in Plato’s Laws”. In: Plato’s Laws: From Theory into Practice, Proceedings of the VI Symposium Platonicum. Selected Papers. Ed. by S. Scolnicov/L. Brisson. Sankt Augustin, 165–73.Search in Google Scholar
Cooper, J. M./Hutchinson, D. S. (eds.) 1997. Plato. Complete Works. Indianapolis/Cambridge.Search in Google Scholar
Cornford, F. M. 1953. Plato’s Theory of Knowledge. The Theaetetus and the Sophist of Plato translated with a running Commentary. London.Search in Google Scholar
Cresswell, M. J. 2012. “Mathematical Entities in the Divided Line”. The Review of Metaphysics 66, 89–104.Search in Google Scholar
Crombie, I. M. 1962–1963. An Examination of Plato’s Doctrines, 2 vols. London/New York.Search in Google Scholar
Desjardins, R. 1990. The Rational Enterprise. Logos in Plato’s Theaetetus. Albany.Search in Google Scholar
Duncombe, M. 2012. The Nature and Purpose of Relative Terms in Plato. Doctoral thesis, University of Cambridge.Search in Google Scholar
Ebrey, D. 2013. “A New Philosophical Tool in the Meno: 86e–87c”. Ancient Philosophy 33, 75–96.Search in Google Scholar
El Murr, D. 2006. “Paradigm and Diairesis. A Response to M. L. Gill’s ‘Models in Plato’s Sophist and Statesman’”. Plato 6, 1–9.Search in Google Scholar
Ferrari, F. (trans.) 2011. Platone, Teeteto. Milano.Search in Google Scholar
Fowler, D. H. 1979. “Ratio in Early Greek Mathematics”. Bulletin of the American Mathematical Society 1, 807–46.Search in Google Scholar
Fowler, D. H. 1999. The Mathematics of Plato’s Academy. A New Reconstruction, 2nd edition. Oxford.Search in Google Scholar
Franklin, L. 2010. “Investigation from Hypothesis in Plato’s Meno. An Unorthodox Reading”. Apeiron 43, 87–115.Search in Google Scholar
Franklin, L. 2011. “Particular and Universal. Hypothesis in Plato’s Divided Line”. Apeiron 44, 335–58.Search in Google Scholar
Franklin, L. 2012. “Inventing Intermediates. Mathematical Discourse and its Objects in Republic VII”. Journal of the History of Philosophy 50, 483–506.Search in Google Scholar
Frede, D. 2006. Das mathematische Curriculum in Platons Staat. In: Wissen und Bildung in der antiken Philosophie. Ed. by C. Rapp/T. Wagner. Stuttgart/Weimar, 127–46.Search in Google Scholar
Frege, G. 1960. The Foundations of Arithmetic. A Logico-Mathematical Enquiry into the Concept of Number, 2nd edition. Translated by J. L. Austin. New York. Originally published as Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau, 1884.Search in Google Scholar
Gallop, D. 1975. Plato. Phaedo. Oxford.Search in Google Scholar
Geach, P. T. 1969. God and the Soul. London.Search in Google Scholar
Gerson, L. P. 2018. “What are the Objects of Dianoia?”. Plato 18, 45–53.Search in Google Scholar
Gill, M. L. 2006. “Models in Plato’s Sophist and Statesman”. Plato 6, 1–16.Search in Google Scholar
Goldschmidt, V. 1947. Le paradigme dans la dialectique platonicienne. Paris.Search in Google Scholar
Hackforth, R. 1955. Plato’s Phaedo. Cambridge.Search in Google Scholar
Haring, E. S. 1992. “Socratic Duplicity: Theaetetus 154b1–156a3”. The Review of Metaphysics 45, 525–42.Search in Google Scholar
Harte, V. 2002. Plato on Parts and Wholes. The Metaphysics of Structure. Oxford.Search in Google Scholar
Iwata, N. 2015. “Plato on Geometrical Hypothesis in the Meno”. Apeiron 48, 1–20.Search in Google Scholar
–. 2016. “Plato’s Hypothetical Inquiry in the Meno”. British Journal for the History of Philosophy 24, 194–214.Search in Google Scholar
Jirsa, J. 2013. “Letters and Models. On the Statesman 277D–278E”. In: Plato’s Statesman, Proceedings of the Eighth Symposium Platonicum Pragense. Ed. by A. Havlíček/J. Jirsa/K. Thein. Prague, 134–50.Search in Google Scholar
Kato, S. 1995. “The Role of paradeigma in the Statesman”. In: Reading the Statesman, Proceedings of the III Symposium Platonicum. Ed. by C. J. Rowe. Sankt Augustin, 162–72.Search in Google Scholar
Klein, J. 1992. Greek Mathematical Thought and the Origin of Algebra. Translated by E. Brann. New York. Originally published as “Die griechische Logistik und die Entstehung der Algebra”. Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B, Bd. 3, H. 1 (1934), 18–105; H. 2 (1936), 122–235.Search in Google Scholar
Knorr, W. R. 1975. The Evolution of the Euclidean Elements. Dordrecht.Search in Google Scholar
Kouremenos, T. 2015. The Unity of Mathematics in Plato’s Republic. Stuttgart.Search in Google Scholar
–. 2018. Plato’s Forms, Mathematics and Astronomy. Berlin.Search in Google Scholar
Lane, M. 1998. Method and Politics in Plato’s Statesman. Cambridge.Search in Google Scholar
Lloyd, G. E. R. 1966. Polarity and Analogy: Two Types of Argumentation in Early Greek Thought. Cambridge.Search in Google Scholar
McDowell, J. 1973. Plato. Theaetetus. Oxford.Search in Google Scholar
Menn, S. 2002. “Plato and the Method of Analysis”. Phronesis 47, 193–223.Search in Google Scholar
Morrow, G. R. 1960. Plato’s Cretan City. A Historical Interpretation of the Laws. New York.Search in Google Scholar
Mueller, I. 1991. “Mathematics and Education. Some Notes on the Platonic Program”. Apeiron 24, 85–104.Search in Google Scholar
Mugler, C. 1958. Dictionnaire historique de la terminologie géométrique des grecs. Paris.Search in Google Scholar
Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. Cambridge.Search in Google Scholar
Oberhammer, A. A. 2016. Buchstaben als Paradeigma in Platons Spätdialogen: Dialektik und Modell im Theaitetos, Sophistes, Politikos und Philebos. Berlin/Boston.Search in Google Scholar
Ofman, S. 2014. “Comprendre les mathématiques pour comprendre Platon-Téétète (147d–148b)”. Revue de la Société de Philosophie des Sciences 1, 71–80.Search in Google Scholar
Owen, G. E. L. 1957. “A Proof in the ΠΕΡΙ ΙΔΕΩΝ”. The Journal of Hellenic Studies 77(1), 103–11.Search in Google Scholar
Pender, E. E. 2003. “Plato on Metaphors and Models”. In: Metaphor, Allegory and the Classical Tradition. Ancient Thought and Modern Revisions. Ed. by G. R. Boys-Stones. Oxford, 55–81.Search in Google Scholar
Piazza, M. 2012. “Plato and the Dice. A Reassessment of Theaetetus 154a–155d”. The Cambridge Classical Journal 58, 231–56.Search in Google Scholar
Robins, I. 1995. “Mathematics and the Conversion of the Mind. Republic vii 522c1–531e3”. Ancient Philosophy 15, 359–91.Search in Google Scholar
Rowe, C. J. (trans.) 1993. Plato. Phaedo. Cambridge.Search in Google Scholar
Russell, B. 2004. History of Western Philosophy. London/New York. First published in London, 1946.Search in Google Scholar
Sayre, K. M. 2006. Metaphysics and Method in Plato’s Statesman. Cambridge.Search in Google Scholar
Scott, D. 2006. Plato’s Meno. Cambridge.Search in Google Scholar
Sedley, D. 1998. “Platonic Causes”. Phronesis 43(2), 114–32.Search in Google Scholar
Sedley, D./Long, A. (eds.) 2010. Plato. Meno and Phaedo. Cambridge.Search in Google Scholar
Smith, N. D. 1996. “Plato’s Divided Line”. Ancient Philosophy 16, 25–46.Search in Google Scholar
Smith, N. 2018. “Unclarity and the Intermediates in Plato’s Discussions of Clarity in the Republic”. Plato 18, 97–110.Search in Google Scholar
Szabó, Á. 1978. The Beginnings of Greek Mathematics. Translated by A. M. Ungar. Dordrecht/Boston. Originally published as Anfänge der griechischen Mathematik. Budapest, 1969.Search in Google Scholar
Toth, I. 1998a. Aristotele e i fondamenti assiomatici della geometria. Prolegomeni alla comprensione dei frammenti non-euclidei del ‘Corpus Aristotelicum’. Prefazione e introduzione di G. Reale, traduzione italiana di E. Cattanei, Milano.Search in Google Scholar
Toth, I. 1998b. Lo schiavo di Menone. Il lato del quadrato doppio, la sua misura non-misurabile, la sua ragione irrazionale. Commentario a Platone, Menone 82b–86c. Introduzione, traduzione, bibliografia e indici a cura di E. Cattanei. Milano.Search in Google Scholar
Tschemplik, A. 2008. Knowledge and Self-Knowledge in Plato’s Theaetetus. Lanham.Search in Google Scholar
van der Waerden, B. L. 1975. Science Awakening I., 4th edition. Dordrecht.Search in Google Scholar
Vitrac, B. 2005. “Les classifications des sciences mathématiques en Grèce ancienne”. Archives de Philosophie 68, 269–301.Search in Google Scholar
Vlastos, G. 1969. “Reasons and Causes in the Phaedo”. Philosophical Review 78, 291–325.Search in Google Scholar
Vlastos, G. 1991. “Elenchus and Mathematics. A Turning-Point in Plato’s Philosophical Development”. In: Socrates, Ironist and Moral Philosopher. Ed. by G. Vlastos. Cambridge, 107–31.Search in Google Scholar
Wolfsdorf, D. 2008. “The Method ἐξ ὑποθέσεως at Meno 86e1–87d8”. Phronesis 53, 35–64.Search in Google Scholar
Ziolkowski, J. 2014. “Plato’s Similes. A Compendium of 500 Similes in 35 Dialogues”. Washington D.C., Centre for Hellenic Studies, https://plato.chs.harvard.edu/.Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
