Abstract
By ‘logic’ I mean ‘the analysis of argument or proof in terms of form’. The two main examples of Greek logic are, then, Aristotle’s syllogistic developed in the first twenty-two chapters of the Prior Analytics and Stoic propositional logic as reconstructed in the twentieth century. The topic I shall consider in this paper is the relation between Greek logic in this sense and Greek mathematics. I have resolved the topic into two questions: (1) To what extent do the principles of Greek logic derive from the forms of proof characteristic of Greek mathematics? and (2) To what extent do the Greek mathematicians show an awareness of Greek logic?
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Notes
The translations of the Elements are by T. Heath, The Thirteen Books of Euclid’s Elements, 3 vols., Cambridge, England, 1925.
John Philoponus, In Aristotelis Analytica Priora Commentaria (ed. by M. Wallies), Berlin, 1905, 246.3–4, gives a similar illustration of the fifth anapodeiktos: ‘The side is either equal to or greater than or less than the side; but it is neither greater nor less; therefore it is equal’. For details on the anapodeiktoi and other aspects of Stoic logic, see B. Mates, Stoic Logic, Berkeley 1961.
These divisions and their names are taken from Proclus, In Primum Euclidis Elementorum Librum Commentarii (ed. by G. Friedlein), Leipzig, 1873, 203.1–210.16. The rigidity which they suggest is fully confirmed by Euclid’s Elements; and the terms them-selves, or forms of them, can all be found in third-century mathematical works. For references, see C. Mugler, Dictionnaire Historique de la terminologie geomitrique des grecs, Paris 1958.
As in D. Hilbert, Foundations of Geometry (transl. by L. Unger), La Salle, 111., 10th ed., 1971.
J. Łukasiewicz, Aristotle’s Syllogistic, Oxford, 2nd ed., 1957, p. 1, asserts that Aristotle does not allow singular terms in syllogisms. If Lukasiewicz is right, then no Euclidean argument would be an Aristotelian syllogism.
See, for example, H. Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter, Copenhagen 1896, p. 117.
See Alexander of Aphrodisias, In Aristotelis Analyticorum Priorum Librum I Commentarium (ed. by M. Wallies ), Berlin 1883, 344. 13–20.
For inadequate attempts to explain the move, see Proclus, In Primum Elementorum, 49.4–57.8; and J. S. Mill, A System of Logic, London, 9th ed., 1875, III. ii. 2.
See, for example, F. Klein, Elementary Mathematics from an Advanced Standpoint (transl. by E. R. Hedrick and C. A. Noble), New York 1939, II, pp. 196–202.
H. Hasse and H. Scholz, ‘Die Grundlagenkrisis in der griechischen Mathematik’, Kant-Studien XXXIH (1928), 17, call it a first attempt at an axiomatization in the modern sense.
See ‘Continuity and Irrational Numbers’ in R. Dedekind, Essays on the Theory of Numbers (transl. by W. Beman), Chicago 1924, pp. 15–17.
F. Beckmann, in ‘Neue Gesichtspunkte zum 5. Buch Euklids’, Archive for History of Exact Sciences IV (1967/8), 106–107, lists twenty-four ‘tacit assumptions’ of Book V.
Łukasiewicz, Aristotle’s Syllogistic, p. 6, denies Platonic influence. But see W. and M. Kneale, The Development of Logic, Oxford 1962, pp. 44, 67–68.
Translation by T. Heath, Mathematics in Aristotle, Oxford 1949, p. 26. This is an excellent work to consult for details of the mathematical aspects of the passages I am discussing.
W. D. Ross, Aristotle’s Prior and Posterior Analytics, Oxford 1949, pp. 412–414.
For the details, see Simplicius, In Aristotelis Physicorum Libros Quattuor Priores Commentaria (ed. by H. Diels), Berlin 1882, 60.22–68.32.
The evidence is collected in F. Wehrli, Die Schule des Aristoteles, Basel/Stuttgart, 2nd ed., 1969, VIII, 11–20.
Quoted by Simplicius, In Aristotelis Categorias Commentarium (ed. by C. Kalbfieisch), Berlin 1907, 394.13–395.31.
Institutio, V.5 (said of those about Chrysippus). Diogenes Laertius, Vitae Philosophorum (ed. by H. S. Long), Oxford 1964, VII. 190, lists among Chrysippus’s works ‘On a true diezeugmenon’ and ‘On a true sunemmenon’
Sextus Empiricus, Adversus Mathematicos, VIII.443, in Opera II (ed. by H. Mutsch- mann and J. Mau), Leipzig 1914. Alexander associates the one-premissed arguments with ‘those about Antipater’ (In Topicorum, 8.16–19). Other relevant passages are col¬lected in C. Piantl, Geschichte der Logik im Abendlande, Leipzig 1855, I, 477–478.
In Analyticorum Priorum, 164.30–31, 278.6–14. At 284.13–17 Alexander ascribes to hoi apo tes Stoas the second, third, and fourth themata. On the themata, see O. Becker, Vber die vier Themata des stoischen Logik, in Zwei Untersuchungen zur antiken Logik, Klassisch-philologische Studien XVII (1957), 27–49.
See the passages in H. Bonitz, ‘Index Aristotelicus’, in Aristotelis Opera (ed. by I. Bekker), Berlin 1831–70, V, 70b4–13.
G. Vlastos, ‘Zeno of Sidon as a Critic of Euclid’, in The Classical Tradition (ed. by L. Wallach ), Ithaca, N.Y., 1966, pp. 154–155.
W. Cronert, Kolotes und Menedemos, Studien zur Palaeographie und Papyruskunde VI (1906), 109.
O. Neugebauer, über eine Methode zur Distanzbestimmung Alexandria-Rom bei Heron, Det Kgl. Danske Videnskabernes Selskab XXVI (1938), 21–24.
See. K Tittel, De Gemini Stoici Studiis Mathematicis Quaestiones Philologae, Leipzig 1895. Brehier (‘Posidonius d’Apamee’, pp. 46–49) thinks that Geminus’s work on math¬ematics is derived entirely from Posidonius.
See E. Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung, Leipzig, 5th ed., 1923, pt. 3, sec. 2, pp. 642–645. The same material is found in E. Zeller, A History of Eclecticism in Greek Philosophy (transl. by S. F. Alleyne), London 1883, pp. 113–117. The importance of the reawakening of interest in Aristotle’s work for the history of logic is stressed by J. Mau, ‘Stoische Logik’, Hermes LXXXV (1957), 147–158. The historical reconstruction of the present paper seems to provide support for Mau’s views.
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© 1974 D. Reidel Publishing Company, Dordrecht-Holland
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Mueller, I. (1974). Greek Mathematics and Greek Logic. In: Corcoran, J. (eds) Ancient Logic and Its Modern Interpretations. Synthese Historical Library, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2130-2_4
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