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Two Approaches to Foundations in Greek Mathematics: Apollonius and Geminus

Published online by Cambridge University Press:  04 May 2010

Fabio Acerbi
Fabio Acerbi*
Affiliation:
Centre National de la Recherche Scientifique, Unité Mixte de Recherche 8163 “Savoirs, textes, langage”, B.P. 60149 – 59653 Villeneuve d'Ascq Cedex, France
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Argument

This article is the sequel to an article published in the previous issue of Science in Context that dealt with homeomeric lines (Acerbi 2010). The present article deals with foundational issues in Greek mathematics. It considers two key characters in the study of mathematical homeomery, namely, Apollonius and Geminus, and analyzes in detail their approaches to foundational themes as they are attested in ancient sources. The main historiographical result of this paper is to show that there was a well-established mathematical field of discourse in “foundations of mathematics,” a fact that is by no means obvious. The paper argues that the authors involved in this field of discourse set up a variety of philosophical, scholarly, and mathematical tools that they used in developing their investigations.

Type
Research Article
Information
Science in Context , Volume 23 , Issue 2 , June 2010 , pp. 151 - 186
Copyright
Copyright © Cambridge University Press 2010

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References

Other literature cited

Acerbi, Fabio. 2007. “Una scuola matematica alessandrina?” In La Matematica, Vol. 1. I luoghi e i tempi, edited by Bartocci, Claudio, Odifreddi, Piergiorgio, 65–89. Turin: Einaudi.Google Scholar
Acerbi, Fabio. 2008. “Hero of Alexandria.” In New Dictionary of Scientific Biography, edited by Koertge, Noretta III:283–286. Detroit: Ch. Scribner's Sons.Google Scholar
Acerbi, Fabio. 2010. “Homeomeric Lines in Greek Mathematics.” Science in Context 23 (1):1–37.CrossRefGoogle Scholar
Acerbi, Fabio. Forthcoming. “Perché una dimostrazione geometrica greca è generale.” In Atti degli incontri di Gargnano, edited by Repellini, Ferruccio Franco and Nenci, Elio. Milan: LED.Google Scholar
Anaritius' Commentary on Euclid. The Latin Translation I-IV. 1994. Edited by Tummers, Paul M. J. E.. Nijmegen: Ingenium Publishers.Google Scholar
Barnes, Jonathan. 1990. “Logical Form and Logical Matter.” In Logica, mente e persona, edited by Alberti, Antonina, 7–119. Florence: Leo S. Olschki.Google Scholar
Barnes, Jonathan. 1993. “‘Third Sort of Syllogism’: Galen and the Logic of Relations.” In Modern Thinkers and Ancient Thinkers, edited by Sharples, Robert W., 172–194. London: UCL Press.Google Scholar
Barnes, Jonathan. 2007. Truth, etc: Six lectures in ancient logic. Oxford: Clarendon Press.Google Scholar
Blass, Friedrich. 1883. Dissertatio de Gemino et Posidonio. Kiel: C. F. Mohr.Google Scholar
Bowen, Alan C. and Todd, Robert B.. 2004. Cleomedes’ Lectures on Astronomy: a Translation of The Heavens with an Introduction and Commentary. Berkeley and Los Angeles: University of California Press.CrossRefGoogle Scholar
Caston, Victor. 1999. “Something and Nothing: The Stoics on Concepts and Universals.” Oxford Studies in Ancient Philosophy 17:145–213.Google Scholar
Euclide. 1990–2001. Les Éléments. Translated with commentary by Bernard Vitrac. 4 vols. Paris: Presses Universitaires de France.Google Scholar
Evans, James and Berggren, J. Lennart. 2006. Geminos's Introduction to the Phenomena. Princeton: Princeton University Press.Google Scholar
The Fihrist of al-Nadīm. A Tenth-century Survey of Muslim Culture. 1970. Translated by Dodge, Bayard. New York: Columbia University Press.Google Scholar
Fortenbaugh, William W., Huby, Pamela M., Sharples, Robert W., and Gutas, Dimitri, eds. 1993. Theophrastus of Eresus. Sources for His Life, Writings, Thought and Influence. 2 vols. Leiden: New York; Cologne: Brill.Google Scholar
Fried, Michael N. and Unguru, Sabetai. 2001. Apollonius of Perga's Conica. Text, Context, Subtext. Leiden, Boston, Cologne: Brill.CrossRefGoogle Scholar
von Fritz, Kurt. 1955. “Die APXAI in der griechischen Mathematik.” Archiv für Begriffsgeschichte 1:13–103.Google Scholar
Hankinson, Robert James. 1994. “Galen and the Logic of Relations.” In Aristotle in Late Antiquity, edited by Schrenk, Lawrence P., 57–75. Washington: Catholic University of America Press.Google Scholar
Heiberg, Johan Ludwig. 1884. “Griechische und römische Mathematik.” Philologus 43:321–346, 467–522.CrossRefGoogle Scholar
Hogendijk, Jan P. 2000. “Traces of the Lost Geometrical Elements of Menelaus in Two Texts of al-Sijzī.” Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 13:129–164.Google Scholar
Jaouiche, Khalil. 1986. La théorie des parallèles en pays d'Islam. Paris: Vrin.Google Scholar
Knorr, Wilbur Richard. 1986. The Ancient Tradition of Geometric Problems. Boston, Basel, Berlin: Birkhäuser.Google Scholar
Knorr, Wilbur Richard. 1989. Textual Studies in Ancient and Medieval Geometry. Boston, Basel, Berlin: Birkhäuser.CrossRefGoogle Scholar
Kouremenos, Theokritos. 1994. “Poseidonius and Geminus on the Foundations of Mathematics.” Hermes 122:437–450.Google Scholar
Lloyd, Geoffrey Ernest Richard. 1978. “Saving the Appearances.” Classical Quarterly 28:202–222.CrossRefGoogle Scholar
Mueller, Ian. 1981. Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge, MA and London: MIT Press.Google Scholar
Pappus of Alexandria. 1986. Book 7 of the Collection. Edited with Translation and Commentary by Jones, Alexander, 2 vols. New York, Berlin, Heidelberg, Tokyo: Springer-Verlag.Google Scholar
Poseidonius. 1982. Die Fragmente. Edited by Herausgegeben von Willy Theiler. Berlin, New York: De Gruyter.Google Scholar
Posidonius. 1989. I. The Fragments. Edited by Edelstein, Ludwig and Kidd, Ian Gray. 2nd ed.Cambridge: Cambridge University Press.Google Scholar
Rabouin, David. 2009. Mathesis universalis. L'idée de « mathématique universelle » d'Aristote à Descartes. Paris: Presses Universitaires de France.CrossRefGoogle Scholar
Rashed, Roshdi, and Houzel, Christian. 2005. “Thâbit ibn Qurra et la théorie des parallèles.” Arabic Science and Philosophy 15:9–55.CrossRefGoogle Scholar
Reinhardt, Karl. 1921. Poseidonios. Munich: C. H. Beck.Google Scholar
Sabra, Abdelhamid I. 1968. “Thâbit ibn Qurra on Euclid's Parallels Postulate.” Journal of the Warburg and Courtauld Institutes 31:12–32.CrossRefGoogle Scholar
Schmidt, Max C. P. 1886. “Philologische beiträge zu griechischen mathematikern. III. Was schrieb Geminos?” Philologus 45:63–81.CrossRefGoogle Scholar
Tannery, Paul. [1881] 1912–1950. “Quelques fragments d'Apollonius de Perge.” Bulletin des Sciences mathématiques 2nd series, V:124–136. Reprinted in idem, Mémoires Scientifiques, edited by Heiberg, Johan Ludwig & Zeuthen, Hieronymus Georg, 17 vols., I:124–138. Toulouse: Privat; Paris: Gauthier-Villars.Google Scholar
Tannery, Paul. 1882a. “Sur les fragments d'Eudème de Rhodes.” Annales de la Faculté des Lettres de Bordeaux, IV:70–76. Reprinted in idem, Mémoires Scientifiques I:168–177.Google Scholar
Tannery, Paul. 1882b. “Sur Sporos de Nicée.” Annales de la Faculté des Lettres de Bordeaux, IV:257–261. Reprinted in idem, Mémoires Scientifiques I:178–184.Google Scholar
Tannery, Paul. 1883–4. “Pour l'histoire des lignes et surfaces courbes dans l'antiquité.” Bulletin des Sciences mathématiques 2nd series, VII:278–291 and VIII:19–30, 101–112. Reprinted in idem, Mémoires Scientifiques II:1–47.Google Scholar
Tannery, Paul. 1901. “Le philosophe Aganis est-il identique à Géminus?” Bibliotheca Mathematica 3rd series, II:9–11. Reprinted in idem Mémoires Scientifiques III:37–41.Google Scholar
Tannery, Paul. 1887. La géométrie grecque. Paris: Gauthier-Villars.Google Scholar
Thabit ibn Qurra. 2009. Science and Philosophy in Nineteenth-Century Bagdad. Edited by Rashed, Roshdi. Berlin and New York: De Gruyter.Google Scholar
Tittel, Karl. 1895. De Gemini Stoici studiis mathematicis quaestiones philologae. Leipzig: Typis M. Hoffmanni.Google Scholar
Tittel, Karl. 1910. “Geminos (1).” In Paulys Real-Encyclopädie der classischen Altertumswissenschaft, edited by Wissowa, Georg et al. , VII.1:1026–1050. Stuttgart: J.B. Metzler 1894–1963.Google Scholar
Todd, Robert B. 2000. “Géminos.” In Dictionnaire des Philosophes Antiques, edited by Goulet, Richard III:472–477. Paris: CNRS Editions.Google Scholar
Vitrac, Bernard. 2005. “Les classifications des sciences mathématiques en Grèce ancienne.” Archives de Philosophie 68:269–301.CrossRefGoogle Scholar
Vitrac, Bernard. Forthcoming. “Faut-il réhabiliter Héron?” In L'homme et la science, Actes du XVIe Congrès de l'Association Guillaume Budé, Montpellier, Septembre 1–4, 2008. Paris: Les Belles Lettres.Google Scholar