Descartes's geometry as spiritual exercise

Critical Inquiry

Volumn 28, Issue 1, 2001, Pages 40-71

  Jones, Matthew L ^a  

^a Och Spine at New York Presbyterian Hospitals

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[No Author keywords available]

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EID: 14644422485     PISSN: 00931896     EISSN: None     Source Type: Journal    
DOI: 10.1086/449032     Document Type: Article

References (124)

1. see Susan Bordo and Mario Moussa, "Rehabilitating the 'I'," in Feminist Interpretations of René Descartes, ed. Bordo (University Park, Penn., 1999), pp. 280-304. One can easily multiply examples to include those from phenomenology (especially Husserl and Heidegger), cognitive science, and analytic philosophy. There are several important compendia of articles, often discussing and invoking Descartes.
2. See, for example, Who Comes after the Subject? ed. Eduardo Cadava, Peter Connor, and Jean-Luc Nancy (New York, 1991), and Penser le sujet aujourd'hui, ed. Elisabeth GuibertSledziewski and Jean-Louis Viellard-Baron (Paris, 1988).
3. Descartes used the term subject in the traditional Aristotelian manner. There are numerous other historical corrections to the fable. See Stephen Gaukroger's introduction to Descartes: An Intellectual Biography (Oxford, 1995)
4. and John Schuster, "Whatever Should We Do with Cartesian Method? Reclaiming Descartes for the History of Science," in Essays on the Philosophy and Science of René Descartes, ed. Stephen Voss (New York, 1993), pp. 195-223. A key approach has been to underline Descartes's continuity with late scholasticism; for two recent examples,
5. see Dennis Des Chene, Physiologia: Natural Philosophy in Late Aristotelian and Cartesian Thought (Ithaca, N.Y., 1996)
6. and Roger Ariew, Descartes and the Last Scholastics (Ithaca, N.Y., 1999). Another key movement has been to undermine the view that Descartes had no room for the senses.
7. See, for example, Desmond M. Clarke, Descartes's Philosophy of Science (University Park, Penn., 1982)
8. and Daniel Garber, Descartes Embodied: Reading Cartesian Philosophy through Cartesian Science (Cambridge, 2001).
9. see Michel Foucault, Le Souci de soi, vol. 3 of Histoire de la sexnalilé, 3 vols. (Paris, 1984), esp. pp. 51-85
10. and Arnold I. Davidson, "Ethics as Ascetics: Foucault, the History of Ethics, and Ancient Thought," in Foucault and the Writing of History, ed. Jan Goldstein (Oxford, 1994), pp. 63-80.
11. See Dennis L. Sepper, Descartes's Imagination: Proportion, Images, and the Activity of Thinking (Berkeley, 1996), p. 135 and his comments on the expansion of ingenium's power through exercise, p. 140.
12. Descartes, The Philosophical Writings of Descartes, trans. Robert Stooth off et al., 3 vols. (Cambridge, 1984), 1:36; hereafter abbreviated CSM.
13. For metamathematical concerns in the seventeenth century, see Hermann Schilling, Die Geschichte der Axiomatischen Methode in 16. und beginnenden 17. Jahrhundert (Hildesheim, 1969)
14. and Paolo Mancosu, The Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (New York, 1996).
15. see Michael S. Mahoney, "Algebraic vs. Geometric Techniques in Newton's Determinations of Planetary Orbits," in Action and Reaction: Proceedings of a Symposium to Commemorate the Tercentenary of Newton's "Principia," ed. Paul Theerman and Adele F. Seeff (Newark, 1993), pp. 183-205;
16. Domenico Bertoloni Meli, Equivalence and Priority: Newton versus Leibniz (Oxford, 1993);
17. Michel Blay, La Naissance de la mécanique analytique: La Science du mouvement au toumant des XVIIe et XVIIIe siècles (Paris, 1992);
18. and Douglas M. Jesseph, Squaring the Circle: The War between Hobbes and Wallis (Chicago, 1999), among others.
19. See, for example, Mario Biagioli, "The Social Status of Italian Mathematicians, 1450-1600," History of Science 27 (Mar. 1989): 41-95
20. and Galileo, Courtier: The Practice of Science in the Culture of Absolutism (Chicago, 1993);
21. Stephen Johnston, "Mathematical Practitioners and Instruments in Elizabethan England," Annals of Science 48 (July 1991): 314-44;
22. Robert S. Westman, "The Astronomer's Role in the Sixteenth Century: A Preliminary Study," History of Science 18 (June 1980): 105-47;
23. J. A. Bennett, "The Mechanics' Philosophy and the Mechanical Philosophy," History of Science 24 (Mar. 1986): 1-28;
24. Peter Dear, Discipline and Experience: The Mathematical Way in the Scientific Revolution (Chicago, 1995);
25. and Mahoney, The Mathematical Career of Pierre de Feninat, 1601-1665, 2d ed. (Princeton, N.J., 1994), pp. 1-14, 20-25.
26. David Lachterman rightly notes, "one will need to practice and apply it, not memorize or passively receive it" (David Rapport Lachterman, The Ethics of Geometry: A Genealogy of Modernity [New York, 1989], p. 134).
27. In a penetrating study of Descartes's Meditations, Foucault insisted that recognizing the intelligibility of Descartes's choices rested on a "double reading" of his text as both a system and an exercise (Foucault, "My Body, This Paper, This Fire," Aesthetics, Method and Epistemology, vol. 2 of Essential Works of Foucault, 1954-1984, ed. James D. Faubion [New York, 1998], p. 406);
28. compare the considerably more historically grounded Gary Hatfield, "The Senses and the Fleshless Eye: The Meditations as Cognitive Exercises," in Essays on Descartes's "Meditations," ed. Amelie Oksenberg Rorty (Berkeley, 1986), pp. 45-79; for a negative assessment of Hatfield,
29. see Bradley Rubidge, "Descartes's Meditations and Devotional Meditations," Journal of the History of Ideas 51 (Jan-Mar. 1990): 27-49.
30. and for positive ripostes, see Dear, "Mersenne's Suggestion: Cartesian Meditation and the Mathematical Model of Knowledge in the Seventeenth Century," in Descartes and His Contemporaries, ed. Roger Ariew and Marjorie Grene (Chicago, 1995), pp. 44-62;
31. and Sepper, "The Texture of Thought: Why Descartes's Meditiationes Is Meditational, and Why It Matters," in Descartes's Natural Philosophy, ed. Gaukroger, John Andrew Schuster, and John Sutton (New York, 2000), pp. 736-50.
32. see also Geneviève Rodis-Lewis, "Descartes et les mathématiques au collège," in "Le Discours" et sa méthode: Colloque pour le 350 anniversaire du "Discours de la méthode" ed. Nicolas Grimaldi and Marion (Paris, 1987), pp. 187-211.
33. See Jean-Louis Guez de Balzac, "Suite d'un entretien de vive voix, ou de la conversation des Romains," in Oeuvres diverses, ed. Roger Zuber (1644;
34. See Emmanuel Bury, "Le Sourire de Socrate ou, peut-on être à la fois philosophe et honnête homme?" in Le Loisir lettré à l'âge classique, ed. Marc Fumaroli, Philippe-Joseph Salazar, and Bury (Geneva, 1996), pp. 197-212.
35. Blaise Pascal, Pensées, in Oeuvres complètes de Blaise Pascal, ed. Louis Lafuma (Paris, 1963), no. 533, p. 578.
36. see also Marion, Sur l'onlologie grise de Descartes: Science cartésienne et savoir aristotélicien dans les "Regulae" (Paris, 1975), p. 151.
37. Cicero, De inventione, in De inventione; De optimum genere oralorum; Topica, trans. H. M. Hubbell (Cambridge, Mass., 1949), 2.51.156, p. 324;
38. see also Quintilian, The Institutio Oratoria of Quintilian, trans. H. E. Butler, 4 vols. (New York, 1921), 3.8.22, 1:490
39. and Montaigne "De l'utile et l'honneste," Essais, ed. Maurice Rat 2 vols. (Paris, 1962) 3.1, 2:205-21.
40. On honnêteté and Descartes's physical work, see Dear, "A Mechanical Microcosm: Bodily Passions, Good Manners, and Cartesian Mechanism," in Science Incarnate: Historical Embodiments of Natural Knowledge, ed. Christopher Lawrence and Steven Shapin (Chicago, 1998), pp. 51-82, esp. pp. 62-63.
41. see Zuber, "Die Theorie der Honnêteté," in Frankreich und Niederlände, vol. 2 of Die Philosophie des 17. Jahrhunderts, ed. Jean-Pierre Schöbinger (Basel, 1993), pp. 156-66;
42. Domna C. Stanton, The Aristocrat as Art: A Study of the "Honnête Homme" and the Dandy in Seventeenth-and Nineteenth-Century French Literature (New York, 1980);
43. Maurice Magendie, La Politesse mondaine et les théories de I'honnêteté, en Fiance au XVll' siècle, de 1600 à 1660, 2 vols. (Paris, 1925);
44. and Nannerl O. Keohane, Philosophy and the State in France: The Renaissance to the Enlightenment (Princeton, N.J., 1980), esp. pp. 283-88.
45. See Balzac, letter to Boisrobert, 28 Sept. 1623
46. quoted in Bernard Beugnot, "La Précellence du style moyen," in Histoire de la rhétorique dans I'Europe moderne, 1450-1950, ed. Fumaroli (Paris, 1999), p. 542.
47. See Zuber, "L'Urbanité française," Les Émerveillements de la raison: Classicismes littéraires du XVlI' siècle franςais (Paris, 1997), pp. 151-61.
48. See Henri-Jean Martin, "Les Formes de publications au milieu du XVII' siècle," Ordre et contestation au temps des classiques, ed. Roger Duchêne and Pierre Ronzeaud, 2 vols. (Paris, 1992), 2:209-24
49. and Jean-Pierre Cavaillé, "Descartes: Stratège de la destination," XVII' siècle, no. 177 (Oct.-Dec. 1992): 551-59.
50. As reported by Schooten to Christiaan Huygens. Compare Gaston Milhaud, Descartessavant (Paris, 1921), p. 160.
51. Compare the similar attack on Descartes's friend Guez de Balzac as a "Gascon," discussed in Jean Jehasse, Guez de Balzac et le génie romain, 1597-1654 (Saint-Étienne, 1977), p. 117.
52. See Henk J. M. Bos, "Arguments on Motivation in the Rise and Decline of a Mathematical Theory: The 'Construction of Equations,' 1637-ca. 1750," Archive for History of Exact Sciences 30 (Nov. 1984): 331-80, esp. p. 363.
53. See also Gaukroger, "The Nature of Abstract Reasoning: Philosophical Aspects of Descartes's Work in Algebra," in The Cambridge Companion to Descartes, ed. John Cottingham (Cambridge, Mass., 1992), pp. 91-114, esp. pp. 106-8.
54. Nicely noted in Morris Kline, Mathematical Thought from Ancient to Modern Times (New York, 1972), p. 308.
55. Note, for example, those of Girolamo Cardano, Lodovico Ferrari, and Niccolo Tartaglia. See Oystein Ore, Cardano, the Gambling Scholar (Princeton, N.J., 1953), esp. pp. 53-107;
56. Mahoney, The Mathematical Career of Pierre de Fenmat, 1601-1665, pp. 6-7.
57. See Bos, "The Structure of Descartes's Géométric" in Descartes: II metodo e i saggi, ed. Giulia Belgioioso et al. (Rome, 1990), pp. 349-69, esp. pp. 352-56.
58. See Bos, "The Structure of Descartes's Géométrie" and "On the Representation of Curves in Descartes's Géomé trie," Archive for History of Exact Sciences 24 (Oct. 1981): 295-338. He defined multiplication as follows: Let AB be unity. If one wants to multiply BD by BC, then one only needs to join C to A, and draw the parallel from C to E. Then BE is the desired product (since AB:BD::BC:BE and AB is unity, AB.BE = l.BE = BC.BD);
59. Not all commentators agree on this point, in large part because Descartes was well aware that his algebra could produce nongeometric solutions. See chapter 5 of Peter A. Schouls, Descartes and the Possibility of Science (Ithaca, N.Y., 2000).
60. see especially chapter 2 of Emily Grosholz, Cartesian Method and the Problem of Reduction (Oxford, 1991).
61. Compare Gaukroger, Descartes, p. 153.
62. Bury, "Le Sourire de Socrate ou, peut-on être à la fois philosophe et honnête homme?" p. 205.
63. Such an emphasis is fundamental in John Bossy, Christianity in the West, 1400-1700 (Oxford, 1985). For the notion of "social disciplining,"
64. see Gerhard Oestreich, Neostoicism and the Early Modern State, trans. David McLintock, ed. Brigitta Oestreich and H. G. Koenigsberger (Cambridge, Mass., 1982)
65. for the use of social disciplining and confessionalization in describing the Reformation and Counter Reformation, see the overview of German scholarship in R. Po-chia Hsia, Social Discipline in the Reformation: Central Europe, 1550-1750 (New York, 1989), esp. chap. 6.
66. For a survey of the recent social history of elites in this period, see Jean-Marie Constant, "Absolutisme et Modernité," in Histoire des elites en France du XVI' au XX' siècle, ed. Guy Chaussinand-Nogaret (Paris, 1991), pp. 145-216
67. and the useful but older study by John Hearsey McMillan Salmon, Society in Crisis: France in the Sixteenth Century (New York, 1975), pp. 92-113; for educated elites in the late sixteenth century,
68. see George Huppert, Les Bourgeois Gentilshommes: An Essay on the Definition of Elites in Renaissance France (Chicago, 1977); for elites and the transformation of manners,
69. see Norbert Elias, The Court Society, trans. Edmund Jephcott (New York, 1983).
70. Orest Ranum, "Courtesy, Absolutism, and the Rise of the French State, 1630-1660," Journal of Modern History 52 (Sept. 1980): 426-51;
71. and Ellery Schalk, "The Court as 'Civilizer' of the Nobility: Noble Attitudes and the Court in France in the Late Sixteenth and Early Seventeenth Centuries," in Princes, Patronage, and the Nobility: The Court at the Beginning of the Modern Age, c. 1450-1650, ed. Ronald G. Asch and Adolf M. Birke (Oxford, 1991), pp. 245-63.
72. esiècle," in Staatsräson: Studien zur Geschichte eines politischen Begriffs, ed. Roman Schur (Berlin, 1975), p. 101.
73. see Michel Hermans and Michel Klein, "Ces Exercices spirituels que Descartes aurait pratiqués," Archives de Philosophie 59 (Jul.-Sept. 1996): 427-40. For studies of such echoes,
74. see, for example, the study of textual similarities in Walter John Stohrer, "Descartes and Ignatius Loyola: La Flèche and Manresa Revisited," Journal of the History of Philosophy 17 (Jan. 1979): 11-27.
75. For intimations on this, see John Stephenson Spink, French Free-Thought from Gassendit ol Voltaire (London, 1960), esp. chap.8
76. and John Cottingham, Philosophy and the Good Life: Reason and the Passions in Greek, Cartesian, and Psychoanalytic Ethics (New York, 1998), chap. 3. For the uneasy coexistence of the ideals of pagan and Christian antiquities in the period,
77. see Zuber, "Guez de Balzac et les deux Antiquités," XVII' siècle, no. 131 (Apr.-Jun. 1981): 135-48. Hadot has examined how physical thought figured in Marcus Aurelius's general scheme of self-cultivation. Knowing the physical world allows one to concentrate fully on that which can be changed, to achieve the state of apatheia.
78. See Hadot, Exercices spirituels, "LΈxpérience de la méditation," Magazine litleraire, no. 342 (Apr. 1996): 73-76
79. and Hadot, Philosophy as a Way of Life: Spiritual Exercises from Socrates to Foucanlt, trans. Michael Chase, ed. Davidson (Oxford, 1995). Lisa Sarasohn recently has emphasized how Gassendi's physical thought similarly figured within his Epicurean ethic of self-cultivation.
80. See Lisa T. Sarasohn, Gassendi's Ethics: Freedom in a Mechanistic Universe (Ithaca, N.Y., 1996).
81. Compare, for Germany, Pamela H. Smith, The Business of Alchemy: Science and Culture in the Holy Roman Empire (Princeton, N.J., 1994), pp. 41-44, and for England,
82. see Julie Robin Solomon, Objectivity in the Making: Francis Bacon and the Politics of Inquiry (Baltimore, 1998), pp. 37-43. For the humanist tradition of history reading as moral instruction and the collapse of this in the late Renaissance,
83. see Timothy Hampton, Writing from History: The Rhetoric of Exemplarity in Renaissance Literature (Ithaca, N.Y., 1990). For differing accounts of the pedagogic role of logical treatises,
84. see Wilhelm Risse, Die Logik der Neuzeit, 2 vols. (Stuttgart-Bad Cannstatt, 1964), vol. 1, chap. 6. For a fine account of artificial versus natural logic and their relations to comtemplative versus active ideals,
85. see Nicholas Jardine, "Keeping Order in the School of Padua: Jacopo Zabarella and Francesco Piccolomini on the Offices of Philosophy," in Method and Order in Renaissance Philosophy of Nature: The Aristotle' Commentary Tradition, ed. Daniel A. Di Liscia, Eckhard Kessler, and Charlotte Methuen (Aldershot, 1997), pp. 183-210, esp. p. 201.
86. See Rodis-Lewis, Descartes: His Life and Thought, trans. Jane Marie Todd (Ithaca, N.Y., 1998), pp. 15-16.
87. For humanist commonplace methodology, see, for example, Ann Blair, "Humanist Methods in Natural Philosophy: The Commonplace Book," Journal of the History of Ideas 53 (Oct.-Dec. 1992): 541-51.
88. See Andre Robinet, Aux sources de l'esprit cartesien: LΆxe La Ramée-Descartes: De la "Dialectique" des 1555 aux "Regulae" (Paris, 1996), pp. 191-96.
89. See Marion's annotations in Descartes, Règies utiles el claires pour la direction de l'esprit en la recherche de la vérité, pp. 217-18.
90. For this notebook and the vicissitudes of its transmission, see Henri Gaston Gouhier, Les Premieres Pensées de Descartes: Contribution à I'histoire de l'anti-renaissance (Paris, 1958)
91. and Rodis-Lewis, "Le Premier Registre de Descartes," Archives de philosophie 54, no. 3-4 (1991): 353-77, 639-57.
92. For discussion, see Sepper, Descartes's Imagination, pp. 76-77.
93. see Kenneth Manders, "Descartes et Faulhaber," Bulletin Cartésien, Archives de Philosophie 58 (Jul.-Sept. 1995): 1-12.
94. see his "Descartes's Early Doctrine of Clear and Distinct Ideas," The Genealogy of Knowledge: Analytical Essays in the History of Philosophy and Science (Aldershot, 1997), pp. 131-52. For a fuller discussion of evidence in the philosophical, theological, and rhetorical traditions, see, for the foundational texts, Aristotle, Physics, 1.1 (184a 17-22) and Posterior analytics, 2.19 (99b15-100b17);
95. Wesley Trimpi, Muses of One Mind: The Literary Analysis of Experience and Its Continuity (Princeton, N.J., 1983), pp. 117-20;
96. and Antonio PérezRamos, Francis Bacon's Idea of Science and the Maker's Knowledge Tradition (Oxford, 1988), pp. 201-15. For a late scholastic account,
97. The best account of deduction in Descartes is Doren A. Recker, "Mathematical Demonstration and Deduction in Descartes's Early Methodological and Scientific Writings," Journal of the History of Philosophy 31 (Apr. 1993): 223-44;
98. see also Frederick Van De Pitte, "Intuition and Judgment in Descartes's Theory of Truth," Journal of the History of Philosophy 26 (July 1988): 453-70.
99. Gaukroger, Cartesian Logic: An Essay on Descartes's Conception of Influence (Oxford, 1989);
100. Yvon Belaval, Leibniz: Critique de Descartes (Paris, 1960).
101. and Desmond M. Clarke, "Descartes's Use of 'Demonstration' and 'Deduction,'" in René Descartes: Critical Assessments, ed. Georges J. D. Moyal, 4 vols. (New York, 1991), 1: 237-47.
102. I use aesthetic in the modern sense and not the meaning of sensory impression of the seventeenth century. On this, see Paul Oskar Kristeller, "The Modern System of the Arts," Renaissance Thought and the Arts: Collected Essays (Princeton, N.J., 1990), pp. 163-227.
103. see Martin Kemp, "From Mimesis to Fantasia: The Quattrocento Vocabulary of Creation, Inspiration, and Genius in the Visual Arts," Viator 8 (1977): 347-98.
104. see Schouls, Descartes and the Possibility of Science, and to some extent Mahoney, "The Beginnings of Algebraic Thought in the Seventeenth Century," in Descartes: Philosophy, Mathematics, and Physics, ed. Gaukroger (Sussex, 1980), pp. 141-68; for the latter,
105. see Gaukroger, Cartesian Logic, pp. 72-88.
106. Compare AT, 10:232-3 on a machine for making conic sections.
107. See Michel Serfati, "Les Compas cartésiens," Archives de philosophie 56 (Apr.-Jun. 1993): 197-230, esp. pp. 213-14.
108. see, for example, John A. Schuster, "Descartes and the Scientific Revolution, 1618-1634: An Interpretation" (Ph.D. diss., Princeton University, 1977);
109. Van De Pitte, "The Dating of Rule IV-B in Descartes's Regulae ad directionem ingenii," Journal of the History of Philosophy 29 (July 1991): 375-95, and many others.
110. A. G. Molland shows how Descartes misrepresented the ancients better to make his case; see Molland, "Shifting the Foundations: Descartes's Transformation of Ancient Geometry," Historia Mathematica 3 (1976): 21-49, esp. pp. 35-37.
111. Noted as a "parallel" in Bos, "On the Representation of Curves in Descartes's Géométrie," p. 310.
112. See, for example, Paolo Mancosu, "Descartes's Géomé trie and Revolutions in Mathematics," in Revolutions in Mathematics, ed. Donald Gilles (Oxford, 1992), pp. 93-95 for Clavius as Descartes's likely source.
113. See Molland, "Shifting the Foundations: Descartes's Transformation of Ancient Geometry," pp. 26, 36;
114. Bos, "On the Representation of Curves in Descartes's Géométrie," p. 314;
115. and Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge, 1974), pp. 101-3.
116. For intended audiences, see Jean-Pierre Cavaillé, "'Le Plus Éloquent Philosophe de derniers temps': Les Stratégies d'auteur de René Descartes," Annales: Histoire, Sciences sociales, no. 2 (Mar.-Apr. 1994): 349-67; for reception of the Discourse,
117. see Garber, "Descartes, the Aristotelians, and the Revolution That Did Not Happen in 1637," Monist 71 (Oct. 1988): 471-86
118. and the essays in Problématique et réception du "Discours de la méthode" et des "Essais," ed. Henry Méchoulan (Paris, 1988), pp. 199-212. For "Cartesian women,"
119. see Erica Harth, Cartesian Women: Version and Subversions of Rational Discourse in the Old Regime (Ithaca, N.Y., 1992)
120. and Eileen O'Neill, "Women Cartesians, 'Feminine Philosophy,' and Historical Exclusion," in Feminist Interpretations of René Descartes, pp. 232-57.
121. See especially Neil M. Ribe, "Cartesian Optics and the Mastery of Nature," his 88 (Mar. 1997): 42-61.
122. Geoffrey V. Sutton, Science for a Polite Society: Gender, Culture, and the Demonstration of Enlightenment (Boulder, Colo., 1995);
123. and Dear, "A Mechanical Microcosm."
124. Ironically, he located it in the development of a capacity inborn to all, "good sense." See the beginning of the Discours, AT, 6:1.