Abstract
The opposition between the Continuous and the Discrete played a significant role in ancient Greek philosophy. This probably derived from the still more fundamental opposition concerning the One and the Many, an antithesis lying at the heart of early Greek thought.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Stokes (1971).
- 2.
For a general account of Oppositions, see Bell (2016).
- 3.
Pythagoras himself is believed to have been active between 540 and 520 B.C.
- 4.
See, e,g., Bell (2016).
- 5.
- 6.
That this was the Eleatic position may be inferred from Plato’s Parmenides .
- 7.
Kirk et al. (1983), pp. 249–50.
- 8.
Ibid., p. 262.
- 9.
Ibid., p. 399.
- 10.
For an account of Zeno’s paradoxes, see, e.g., Bell (2016).
- 11.
Barnes (1986).
- 12.
Furley (1967).
- 13.
Lucretius (1955), p. 51.
- 14.
Kirk et al. (1983), p. 360.
- 15.
Ibid. p. 358.
- 16.
Ibid. p. 371.
- 17.
Ibid. p. 407.
- 18.
Ibid. p. 408.
- 19.
See, e.g. Kirk et al. (1983), p. 408.
- 20.
As presented, e.g. in Heath (1981).
- 21.
Furley (1967).
- 22.
For Epicurus’s views see below.
- 23.
Quoted in Sambursky (1963), p. 153.
- 24.
But not the rigorous proof, which in his treatise On the Sphere and Cylinder Archimedes ascribes to Eudoxus.
- 25.
See Chap. 2.
- 26.
Quoted in Heath (1981), vol. I, p. 222.
- 27.
Yet in Book III infinitesimals appear in the form of “horn” angles: angles between curved lines. Proposition 16 asserts that the angle between a circle and a tangent is less than any rectilineal angle.
- 28.
Arhcimedes’ axiom, or primciple is now usually stated in the following form: if 0 < a < b, then there is a natural number n such that b < n.a.
- 29.
Boyer (1959), p. 50.
- 30.
- 31.
Furley (1967), p. 108 f.
- 32.
In Book VI of the Categories. Quantity (ποσον) is introduced by Aristotle as the category associated with how much. In addition to exhibiting continuity and discreteness, quantities are, according to Aristotle, distinguished by the feature of being equal or unequal.
- 33.
Here it must be noted that for Aristotle, as for ancient Greek thinkers generally, the term “number” – arithmos – means just “plurality”.
- 34.
Aristotle points out that (spoken) words are analyzable into syllables or phonemes, linguistic “atoms” themselves irreducible to simpler linguistic elements.
- 35.
Aristotle (1980), V, 3.
- 36.
Aristotle (1996a), Categories, VI.
- 37.
Aristotle (1980) VI, 1.
- 38.
Ibid.
- 39.
Ibid.
- 40.
Miller (1982).
- 41.
This is a crucial point for Aristotle in his refutation of Zeno’s dichotomy paradox, since Aristotle concedes to Zeno only that in order to reach a goal a moving body must pass through a potential infinity of half-distances. If the body were to traverse an actual infinity of such distances, it would have to make an infinite number of stops and starts. In other words, only an impossibly discontinuous motion (in Aristotle’s sense) would convert this potential infinity into an actual one.
- 42.
Aristotle (1980) VI, 9.
- 43.
Ibid., 2.
- 44.
- 45.
Aristotle (1980) V, 4.
- 46.
Ibid. VI, 10.
- 47.
Ibid. I, 4; 187b 18–21.
- 48.
This having been established above.
- 49.
Van Melsen (1952), p. 42.
- 50.
Ibid., p. 47.
- 51.
Pyle (1997), pp. 216–217.
- 52.
Ibid., p. 217.
- 53.
Quoted in Furley (1967) p. 111.
- 54.
Quoted ibid., p. 7.
- 55.
Ibid., p. 8.
- 56.
Alexander (1956), p. 54.
- 57.
Furley (1967), pp. 8–10.
- 58.
Epicurus’ position in this respect is similar to that of Cantor (see Chap. 4 below). Both can be said to have accepted the thesis that any potential infinity presupposes an actual infinity. But the consequences of this acceptance were quite different for the two. Epicurus, a finitist who repudiated actual infinity, was led necessarily to reject the potential infinite as well. But Cantor’s whole world view reposed on the actual infinite, so for him the thesis served not to demonstrate the non-existence of the potential infinite, but rather to reveal it as a shadow cast by the substantial reality of the actual infinite.
- 59.
For a penetrating discussion of the problem of material indivisibility, see Pyle (1997), Appendix 1.
- 60.
Aristotle (2000a), On Coming-to-Be and Passing Away., I, 8., 326a 32–33.
- 61.
Pyle (1997), Appendix 1.
- 62.
In this respect Epicurean minimal parts may be said to resemble the quarks that are currently presumed to be the ultimate building-blocks of matter: just as Epicurean minimal parts have no separate existence, quarks appear only in groups of two, three, or five.
- 63.
These difficulties are similar to those encountered by the Islamic atomists in the ninth and tenth centuries: see below.
- 64.
Quoted in Sambursky (1971), p. 93.
- 65.
Ibid., p. 94 f.
- 66.
White (1992), Ch. 7.
- 67.
Another possibility is to formulate the problem within the framework of smooth infinitesimal analysis, where intuitionistic logic prevents infinitesimals from being in general equal or unequal to zero. See Chap. 10 below.
- 68.
Sambursky (1963), p. 151.
- 69.
Quoted in Sambursky (1963), p. 151.
- 70.
Ibid.
- 71.
Ibid., pp. 151–2. Smooth infinitesimal analysis suggests another way of interpreting the Stoic conception of time.
- 72.
Lucretius (1955), p. 45
- 73.
In the sense that the collections of parts of a line and its half can be put into one-one correspondence.
- 74.
Quoted in Sorabji (1982), pp. 74–5.
- 75.
Needham (1954–), vol. II, pp. 190–1.
- 76.
Needham, (1954-), 19(h).
- 77.
Needham (1954-), 26(b).
- 78.
Needham (1954-), 26(b).
- 79.
See Gangopadhyaya (1980).
- 80.
Sorabji (1982), pp. 37–87.
- 81.
Maimonides (1956), p. 120.
- 82.
Ibid., p. 121.
- 83.
Ibid.
- 84.
Ibid., p. 122.
- 85.
Van Melsen (1952), p. 59.
- 86.
Quoted in ibid., p. 59.
- 87.
Nicholas of Autrecourt (1971), p. 82.
- 88.
Pyle (1997), p. 208.
- 89.
Nicholas of Autrecourt (1971), p. 82.
- 90.
In Pyle’s words:
We might almost claim that Nicholas was one of the founders of the doctrine of the infinitesimal, that curious creature greater than nothing yet less than anything, an infinity of which make up a magnitude. However great its heuristic value in the history of mathematics this doctrine is quite incoherent and the infinitesimal—as was already apparent in Zeno’s day—an impossible entity.
- 91.
See Chap. 4 below.
- 92.
Nicholas of Autrecourt (1971), p. 80.
- 93.
In his Opus Oxoniense. My source here is Murdoch’s discussion of medieval atomism in §52 of Grant (1974).
- 94.
Grant (1974), p. 317.
- 95.
He seems to have refrained, however, from subjecting the continuum to his celebrated “razor”.
- 96.
See, e.g. the papers of Murdoch and Stump in Kretzmann (1982).
- 97.
Burns (1916), p. 506.
- 98.
Ibid., p. 510.
- 99.
Ibid., p. 507.
- 100.
Ibid., p. 507.
- 101.
Ibid. p. 507–9.
- 102.
Murdoch (1957), p. 54.
- 103.
Op. cit.
- 104.
Oresme (1968).
- 105.
Ibid. pp. 45–47.
- 106.
Aristotle (1980), 234a.
- 107.
Boyer (1959), p. 73.
- 108.
Boyer and Merzbach (1989), p. 264 f.
- 109.
Cusanus (1954), p. 36.
- 110.
Quoted in Stones (1928).
- 111.
Boyer (1959), p. 91. The argument may well be of Greek origin.
Bibliography
Alexander, H.G., ed. 1956. The Leibniz-Clarke Correspondence. Manchester: University of Manchester Press.
Aristotle. 1980. Physics, 2 vols. Trans. Cornford and Wickstead. Loeb Classical Library, Harvard University Press and Heinemann.
———. 1996a. Metaphysics, Oeconomica, Magna Moralia. Vols. 2. Trans. Cooke and Tredinnick. Loeb Classical Library, Harvard University Press.
———. 1996b. The Categories, On Interpretation, Prior Analytics. Trans. Cooke and Tredinnick. Loeb Classical Library, Harvard University Press.
———. 2000a. On the Heavens. Trans. Guthrie. Loeb Classical Library, Harvard University Press.
Barnes, J. 1986. The Presocratic Philosophers. London: Routledge.
Bell, J.L. 2016. Oppositions and Paradoxes: Philosophical Perplexities in Science and Mathematics. Peterborough: Broadview Press.
Boyer, C. 1959. The History of the Calculus and Its Conceptual Development. New York: Dover.
Boyer, C., and U. Merzbach. 1989. A History of Mathematics. 2nd ed. Hoboken: Wiley.
Burns, C.D. 1916. William of Ockham on continuity. Mind 25: 506–512.
Cusanus, N. 1954. Of Learned Ignorance. Trans. Heron. London: Routledge and Kegan Paul.
Furley, D. 1967. Two Studies in the Greek Atomists. Princeton: Princeton University Press.
Gangopadhyaya, M. 1980. Indian Atomism. Calcutta: Bagchi.
Grant, E., ed. 1974. A Source Book in Medieval Science. Cambridge, MA: Harvard University Press.
Heath, T. 1981. A History of Greek Mathematics. Vol. 2. New York: Dover.
Kirk, G.S., J.E. Raven, and M. Schofield. 1983. The Presocratic Philosophers. 2nd ed. Cambridge: Cambridge University Press.
Kretzmann, N., ed. 1982. Infinity and Continuity in Ancient and Medieval Thought. Ithaca: Cornell University Press.
Lucretius. 1955. The Nature of the Universe. Trans. Latham. Penguin Books.
Maimonides, M. 1956. The Guide for the Perplexed. Trans. Friedländer. Dover.
Miller, F. 1982. Aristotle against the atomists. In Kretzmann 1982: 87–111.
Murdoch, J. 1957. Geometry and the Continuum in the 14th Century: A Philosophical Analysis of Th. Bradwardinne’s Tractatus de Continuo. Ph.D. thesis, University of Wisconsin.
Needham, J. 1954. Science and Civilisation in China. Vol. 4. Cambridge: Cambridge University Press.
Nicholas of Autrecourt. 1971. The Universal Treatise. Trans. Kennedy et al. Milwaukee: Marquette University Press
Oresme, N. 1968. Le Livre du Ciel et du Monde. Trans. Menut. Madison: University of Wisconsin Press.
Pyle, A. 1997. Atomism and Its Critics. Bristol: Thoemmes Press.
Sambursky, S. 1963. The Physical World of the Greeks. London: Routledge and Kegan Paul.
———. 1971. Physics of the Stoics. London: Hutchinson.
Sorabji, R. 1982. Atoms and time atoms. In Kretzmann 1982: 37–86.
Stokes, M. 1971. One and Many in Presocratic Philosophy. Cambridge: Harvard University Press.
Stones, G.B. 1928. The atomic view of matter in the XV th , XVI th and XVII th centuries. Isis 10: 444–465.
Van Melsen, A. 1952. From Atomos to Atom. Trans. Koren. Pittsburgh: Duquesne University Press.
White, M.J. 1992. The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford: Clarendon Press.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bell, J.L. (2019). The Continuous and the Discrete in Ancient Greece, the Orient, and the European Middle Ages. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-18707-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-18706-4
Online ISBN: 978-3-030-18707-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)