Infinity, continuity, and composition: The contribution of Gregory of Rimini

Medieval Philosophy and Theology

Volumn 7, Issue 1, 1998, Pages 89-110

  Cross, Richard ^a  

^a ORIEL COLLEGE   (United Kingdom)

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EID: 63149133879     PISSN: 10570608     EISSN: None     Source Type: Journal    
DOI: 10.5840/medievalpt1998716     Document Type: Article

References (21)

1. On this, see Anneliese Maier, Die Vorläufer Galileis im 14.Jahrhundert: Studien zur Naturphilosophie der Spätscholastik, Storia e Letteratura: Raccolta di Studi e Testi, 22 (Rome: Edizioni di Storia e Letteratura, 1949), pp. 172-73, 176-77
2. John E. Murdoch, "Infinity and Continuity," in The Cambridge History of Later Medieval Philosophy, ed. Norman Kretzmann, Anthony Kenny, and Jan Pinborg, (Cambrige: Cam-briige University Press, 1982) pp. 572-73
3. J. M. M. H. Thijssen, "Roger Bacon (1214-1292/1297): A Neglected Source in the Medieval Continuum Debate," Archives internationales d'histoire des sciences 34 (1984): 25-34 (p. 31)
4. Thijssen, "Het Continuum-Debat bij Gregorius van Rimini (1300-1358)," Algemeen Nederlands Tijdscrhift voor Wijsbegeerte 77 (1985): 109-19
5. A. W. Moore, The Infinite, The Problems of Philosophy: Their Past and Their Present (London and New York: Rout-ledge, 1990), pp. 51-54
6. On Harclay, see John E. Murdoch, "Henry of Harclay and the Infinite," in Studi sul xiv secolo in memoria di Anneliese Maier, ed. A. Maierù and A. Paravicini Bagliani, Raccolta di Studi e Testi, 151 (Rome: Storia e Letteratura, 1981)
7. On the immediate adjacence of Harclay's infinite indivisibles, see "Henry of Harclay and the Infinite," p. 230
8. Murdoch, "Infinity and Continuity," p. 577 n.39
9. quoted in Adam Wodeham, Tractatus de Indivisibilibus: A Critical Edition with Introduction, Translation, and Textual Notes, ed. Rega Wood, Synthese Historical Library, 31 (Dordrect, Boston, and London: Kluwer Academic Publishers, 1988), p. 289 n.2
10. S 1.35-6.1.1 ad 7 (3:223,11.27-29): "In one way an indivisible . . . lacks any actually divided parts, or [parts] potential to division or distinction, as many imagine a point to be." Of course, on this definition, the sort of indivisibles Gregory is talking about could be macro-indivisibles of the sort proposed by Adam Wodeham (on this, see Norman Kretzmann, "Adam Wodeham's Anti-Aristotelian Anti-Atomism," History of Philosophy Quarterly 1 [1984]: 381-98). But Gregory consistently ascribes to the view he is rejecting the claim that the relevant sort of indivisible is a point, and his own account of the real divisibility of a continuum excludes his accepting anything like Wodeham's position here. So I take it that Gregory regards divisibility as a necessary feature of any quantity
11. Curiously, Gregory does not believe the mathematical arguments to be effective against the postulation of imaginary indivisibles in geometry; perhaps, like Ockham, he regards mathematical theorems as conditionals, and some of them as counterfactual conditionals: for example, if indivisibles existed, then such and such a theorem would be factually true. For a discussion of Ockham, see Murdoch, "Ockham and the Logic of Infinity and Continuity," in Infinity and Continuity in Ancient and Medieval Thought, ed. Norman Kretzmann (Ithaca and London: Cornell University Press, 1982), p. 178
12. On the objection as found in Ockham, Wodeham, and Buridan, see the detailed discussion in Jack Zupko, "Nominalism Meets Indivisibilism," Medieval Philosophy and Theology 3 (1993): 158-85. Rega Wood describes the objection as the most interesting of the fourteenth-century objections to entitism: see her edition of Wodeham's DI, p. 15
13. On the reply in Wodeham, see Zupko, "Nominalism Meets Indivisibilism," p. 173
14. The only exception to this that I know of is Peter John Olivi: see his Quaestiones in Secundum Librum Sententiarum 2.31 (ed. Bernardus Jansen, 3 vols., Biblioteca Franciscana Medii Aevi, 4-6 [Quaracchi: Collegium S. Bonaventurae, 1922-1926], 1:554, 557)
15. S 2.2.2.1 (4:294, ll. 10-16). On the distinction between categorematic and syncategorematic words, see Norman Kretzmann, "Syncategoremata, Sophismata, Exponibilia," in The Cambridge History of Later Medieval Philosophy, pp. 211-45
16. On categorematic and syncategorematic uses of 'infinite,' see Kretzmann, William of Sherwood's Treatise on Syncategorematic Words (Minneapolis: University of Minnesota Press, 1968), pp. 41-43. Roughly, to use 'infinite' syncategorematically is to use it as a quantifier. Thus if we talk about - say - "infinitely many parts," we are referring to the parts, and noting that the parts are such that, for however many parts we have, we can always take more of them. To use 'infinite' categorematically is to refer to an infinite number: hence, a number "greater than any finite number."
17. see Murdoch, "Ockham and the Logic of Infinity and Continuity," p. 189
18. For "actual (conceptual) divisibility," see n.38 above
19. Modern commentators on Gregory tend to ascribe to him the view that a continuum is composed of infinitely many infinitesimal parts: see Maier, Die Vor-läufer Galileis, pp. 172-73
20. "Het Continuum-Debat," pp. 112-13
21. For a discussion of Alnwick's view, see John E. Murdoch, "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology," in Arts libéraux el philosophie au moyen âge: Actes du quatrième congrès international de philosophie médiévale (Montreal: Institut d'Etudes Médiévales, 1969), p. 220