Existence vs. Conceivability in Aristotle: Are Straight Lines Infinitely Extendible?
Aristotle is committed to finitism in mathematics. But there are certain uses of infinity in mathematics which are indispensable, even in the mathematics of his day. So we had better understand Aristotle’s finitism in a way that is compatible with the mathematics of his time (unless we are willing to ascribe complete naivete to him about these mathematics).
In particular, we have to address the issue of infinitely extendible lines, which are used in Greek mathematics – for example, in Euclid’s definition of parallel lines. Aristotle denies that such lines exist: like any other object which exceeds the fixed and finite size of the cosmos, infinitely extendible lines are definitely excluded from Aristotle’s physics. Moreover, due to Aristotle’s immanentism, they are excluded from his mathematics, too.
But how can Aristotle do mathematics without infinitely extendible lines? In the following I will suggest a possible solution, based on an analysis of the procedure of converse increasing, which Aristotle introduces and discusses in Ph. III 6. This procedure is both infinite – it is infinitely iterable – and does not require the existence of any infinite (or infinitely extendible) magnitudes.
The procedure is interesting, but it must be handled with care. On the one hand, if one fails to acknowledge its subtle mathematical content, one also risks compromising the philosophical interpretation, by ascribing to Aristotle gratuitous naiveties. On the other hand, if one overstates this mathematical content, one risks ascribing to him anachronistic, ‘non-Euclidean’ intents. At the end of the paper I will discuss two cases in point.
- Cleary, John J. 1995. Aristotle and mathematics: Aporetic method in cosmology and metaphysics. Leiden: Brill.Google Scholar
- De Risi, Vincenzo, ed. 2015. Mathematizing space: The objects of geometry from antiquity to the early modern age. Basel: Birkhäuser.Google Scholar
- Diels, H., ed. 1882. Simplicii in Aristotelis physicorum libros quattuor priores commentaria. In Commentaria in Aristotelem Graeca, vol. IX. Berlin: G. Reimer.Google Scholar
- Heath, Thomas L. 1921. A history of greek mathematics, vol. 2. Oxford: Oxford University Press (reprint: New York: Dover 1981).Google Scholar
- Hintikka, Jaakko. 1973. Time and necessity: Studies in Aristotle’s theory of modality. Oxford: Clarendon Press.Google Scholar
- Hussey, Edward. 1983. Aristotle, physics. Books III and IV. Oxford: Clarendon Press.Google Scholar
- Knorr, Wilbour R. 1982. Infinity and continuity: The interaction of mathematics and philosophy in antiquity. In Infinity and continuity in ancient and medieval thought, ed. N. Kretzmann, 112–145. London: Cornell University Press.Google Scholar
- Mendell, Henry. 2008. Aristotle and mathematics. In The stanford encyclopedia of philosophy. Available via DIALOG. http://plato.stanford.edu/archives/win2008/entries/aristotle-mathematics/
- Mendell, Henry. 2015. What’s location got to do with it? Place, space, and the infinite in classical greek mathematics. In Mathematizing space: The objects of geometry from antiquity to the early modern age, ed. V. De Risi, 15–63. Basel: Birkhauser.Google Scholar
- Miller, Fred D. 1982. Aristotle against the atomists. In Infinity and continuity in ancient and medieval thought, ed. N. Kretzmann, 112–145. London: Cornell University Press.Google Scholar
- Moiraghi, Francesco. 2017. Geometrie non-euclidee nella matematica greca. (Dissertation, Università degli studi di Milano a.a. 2016/2017).Google Scholar
- Mueller, Ian. 1990. Aristotle’s doctrine of abstraction in the commentators. In Aristotle transformed: The ancient commentators and their Influence, ed. R. Sorabji, 463–479. London: Duckworth.Google Scholar
- Toth, Imre. 1998. Aristotele e i fondamenti assiomatici della geometria. Prolegomeni alla comprensione dei frammenti non-euclidei nel “Corpus Aristotelicum” nel loro contesto matematico e filosofico. 2nd revised and corrected ed. (Italian) Milano: Vita e Pensiero.Google Scholar
- Ugaglia, Monica. 2009. Boundlessness and iteration: Some observation about the meaning of AEI in Aristotle. Rhizai 6: 193–213.Google Scholar
- Ugaglia, Monica. 2012. Aristotele, Fisica. Libro III. Roma: Carocci editore.Google Scholar
- Ugaglia, Monica. 2016. Is Aristotle’s cosmos hyperbolic? Educação e Filosofia 30: 1–21.Google Scholar
- Unguru, Shabetai. 2013. Greek geometry and its discontents: The failed search for non-Euclidean geometries in the greek philosophical and mathematical corpus. International Journal of History and Ethics of Natural Sciences Technology and Medicine 21: 299–311.Google Scholar
- Vitrac, Bernard. 1990. Euclide: Les Éléments. Traduction et commentaires par B. Vitrac. vol. I. Paris: P.U.F.Google Scholar
- Wieland, Wolfgang. 1970. Die aristotelische Physik. Göttingen: Vandenhoeck and Ruprecht.Google Scholar