Abstract
What made the Oxford Calculators famous, or infamous, in the fourteenth and fifteenth centuries was their subtlety. In this, although their distinctiveness lay in their use of mathematics within physics, their reputation merged with that of English logicians such as William of Ockham, their work being put along with terminist logic into the general category of ‘Anglican subtleties.’ In recent papers I have tried to show that this linking of mathematical and logical work arose in part because of the importance of certain disputations at Oxford, in particular the disputations de sophismatibus and the disputations that constituted the ‘determinations’ that occurred at the time of the student becoming a bachelor of arts.1 Within the context of these student disputations, the Calculators’ facility with mathematics was valued because it provided students with complex, often counterintuitive, results which they might use to defeat their opponents.
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Notes
Edith Sylla, ‘The Oxford Calculators,’ in Norman Kretzmann, Anthony Kenny, and Jan Pinborg (eds.), The Cambridge History of Later Medieval Philosophy (Cambridge: Cambridge University Press, 1982), pp. 540–563.
A. Rupert Hall, ‘On the Historical Singularity of the Scientific Revolution of the Seventeenth Century,’ in J. H. Elliott and H. G. Koenigsberger (eds.), The Diversity of History. Essays in honour of Sir Herbert Butterfleld (Ithaca, New York: Cornell University Press, 1970), p. 207.
A. Rupert Hall, ‘On the Historical Singularity of the Scientific Revolution of the Seventeenth Century,’ in J. H. Elliott and H. G. Koenigsberger (eds.), The Diversity of History. Essays in honour of Sir Herbert Butterfleld (Ithaca, New York: Cornell University Press, 1970), p. 207. The latter phrase is quoted from Edward Grant, ‘Late Medieval Thought, Copernicus, and the Scientific Revolution,’ Journal of the History of Ideas, 23 (1962), 197.
Edward Grant, ‘Aristotelianism and the Longevity of the Medieval World View,’ History of Science, 16 (1978), p. 98.
Edward Grant, ‘Aristotelianism and the Longevity of the Medieval World View,’ History of Science, 16, note 16, p. 106.
John North, ‘Kinematics — More Etherial than Elementary,’ in Madeleine Pelner Cosman and Bruce Chandler (eds.), Machaut's World: Science and Art in the Fourteenth Century, Annals of the New York Academy of Sciences, Vol. 314 (New York: New York Academy of Sciences, 1978), pp. 89–102.
MS Cambridge, Peterhouse 272, f. 1ra. The following study is based for the most part on a transcription of Parts II–VI of the Summa as contained in this manuscript, with occasional variant readings from other manuscripts.
For a different and longer outline, but one which, I think, may overemphasize the role of the ‘doubts,’ see James A. Weisheipl, ‘Developments in the Arts Curriculum at Oxford in the Early Fourteenth Century,’ Medieval Studies, 28 (1966), pp. 169–172.
The drop is sharper than might appear because a single mention in the table may refer to a brief mention or to discussion of the cited book repeatedly throughout the chapter.
In the preface to the Summa Dumbleton appears to say, if I have interpreted his somewhat obscure Latin correctly, that he has not only collected diverse bits or points into a Summa, but has in some way ordered and matured them: ‘Plurimorum scribentium grati laboris dignique memoria particeps ad mensuram mee facultatis doni ex logicali materia communi et physica quandam summam veluti spicarum dispersarum manipulum quoquomodo maturatum et in compositum recolegi.’ MS Cambridge, Peterhouse 272, f. 1 ra.
Dumbleton’s references to other parts of the Summa in MS Peterhouse 272 are on ff. 11 va, llvb, 12vb, 13rb, 13va and vb, 15va, 16va, 19ra, rb, and va, 20rb, 26rb, 27va and vb, 28vb, 29ra and rb, 30rb and vb, and, finally, 31 rb.
Cf. Edward Grant, ‘Late Medieval Thought, Copernicus, and the Scientific Revolution,’ (above note 3), p. 205 and note 32.
See Appendix, quotations from ff. 17va, 18va, 20rb, 29va.
For instance, when on 31 rb, line 3 of transcription, MS Peterhouse 272 says ‘isto modo intelligendo,’ MS Vatican Lat. 954 reads ‘isto modo ymaginando.’
See Appendix, f. 29va.
See Appendix, quotations from ff. 18vb (first quotation), 19va, 29va, 31 rb.
Latin in Appendix, quotation from f. 19va.
See Appendix, quotations from ff. 16vb, 17va, 18ra, 19rb, 26va, 28rb, 29vb, 33ra.
Curtis Wilson, William Heytesbury. Medieval Logic and the Rise of Mathematical Physics (Madison, Wisconsin: University of Wisconsin Press, 1956), p. 25, cited in Edward Grant, ‘Later Medieval Thought…’(above, note 3), p. 205, n. 32.
MS Peterhouse 272, f. 8va. Later in the Summa, f. 20va, Dumbleton, in good Ockhamist fashion, denies that surfaces, points, and lines exist. Cf. also Appendix, quotation from f. 34ra, where Dumbleton says that points are imaginary.
MS Peterhouse 272, f. 14va. The reading ‘Lincoln’ is uncertain.
MS Peterhouse 272, f. 16va: Tertio modo proprissime magis et minus suscipere dicitur quod natum est suscipere gradum intenciorem post remisciorem vel econtra. Et sic intensio et remissio solum competit qualitatibus in tertia specie qualitatum et earum subiectis propter ipsas competit intencio et remissio.
MS Peterhouse 272, f. 16vb. Cf. Latin in Appendix.
MS Peterhouse 272, f. 17rb.
MS Peterhouse 272, f. 17rb.
MS Peterhouse 272, f. 17va. Cf. Latin in Appendix, 17va, first quotation.
MS Peterhouse 272 Cf. partial Latin in Appendix, second quotation from this folio.
MS Peterhouse 272, f. 17vb.
MS Peterhouse 272, See Latin text in Appendix.
MS Peterhouse 272, f. l7va.
MS Peterhouse 272, f. l8vb.
MS Peterhouse 272, f. 18va ff.
MS Peterhouse 272, f. 19va.
MS Peterhouse 272, f. 20ra.
MS Peterhouse 272, f. 20rb.
MS Peterhouse 272, f. 20ra.
Nicole Oresme did assume the existence of a physical entity, the quantity of quality, corresponding to this product. Cf. Marshall Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions (Madison, Wisconsin: University of Wisconsin Press, 1968), pp. 172–177, 404–407.
For a survey of this history see Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, Wisconsin: The University of Wisconsin Press, 1959), pp. 430–440.
See H. Lamar Crosby, Jr., (ed.), Thomas of Bradwardine. His Tractatus de Proportionibus. Its Significance for the Development of Mathematical Physics (Madison, Wisc: University of Wisconsin Press, 1955), pp. 110–117. Also: A. George Molland, ‘The Geometrical Background to the Merton School,’ The British Journal for the History of Science, 4 (1968), 108–125; Edith Sylla, ‘Compounding Ratios: Bradwardine, Oresme, and the first edition of Newton’s Principia,’ in Everett Mendelsohn (ed.), Transformation and Tradition in the Sciences. Essays Presented to I. Bernard Cohen (Cambridge: Cambridge University Press, 1984), pp. 11–43.
H. Lamar Crosby, Jr., (ed.), Thomas of Bradwardine. His Tractatus de Proportionibus. Its Significance for the Development of Mathematical Physics (Madison, Wisc: University of Wisconsin Press, 1955), (above, note 39), pp. 96–99.
Edith Sylla, ‘Medieval Concepts of the Latitude of Forms: The Oxford Calculators,’ Archives d’Histoire Doctrinale et Litteraire du Moyen Age, 40 (1973), pp. 264–271. In an earlier section of this article, I discuss some of the material from Part II of the Summa.
MS Peterhouse 272, f. 23va.
MS Peterhouse 272, f 23vb.
MS Peterhouse 272, f. 25va.
MS Peterhouse 272, f. 25va.
MS Peterhouse 272, f. 24ra. This is Dumbleton’s second counter-argument.
MS Peterhouse 272, f. 24ra.
MS Peterhouse 272, f. 24rb.
MS Peterhouse 272, f. 24rb.
This is Dumbleton’s first argument against the position; cf. MS Peterhouse 272, f. 24ra.
This is Dumbleton’s third argument against the position (MS Peterhouse 272, f. 24ra).
Since 1984 I have published two other papers relevant to these considerations, namely, ‘Mathematical physics and imagination in the work of the Oxford Calculators: Roger Swineshead’s On Natural Motions’, in Edward Grant and John Murdoch (eds.), Mathematics and its applications to science and natural philosophy in the Middle Ages. Essays in honor of Marshall Clagett (Cambridge, London, etc.: Cambridge University Press, 1987), pp. 69–101; and ‘Alvarus Thomas and the Role of Logic and Calculations in Sixteenth Century Natural Philosophy,’ in Stefano Caroti (ed.), Studies in Medieval Natural Philosophy Biblioteca di Nuncius. Studi e Testi, Vol. 1, (Florence: Leo S. Olschki, 1989), pp. 257–98. I also gave a paper at the Rutgers University Colloquium on Late Medieval Science and Technology, 18–19 April 1986, entitled, ‘The Oxford Calculators and Aristotelianism: Tradition and Transformation in Late Medieval Science as Exemplified by Paul of Venice and Alvarus Thomas.’
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Sylla, E.D. (1991). The Oxford Calculators and Mathematical Physics: John Dumbleton’s Summa Logicae et Philosophiae Naturalis, Parts II and III. In: Unguru, S. (eds) Physics, Cosmology and Astronomy, 1300–1700: Tension and Accommodation. Boston Studies in the Philosophy of Science, vol 126. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3342-5_7
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