Skip to main content
SpringerLink
Log in

‘Mathesis Universalis’ in the Seventeenth Century

  • Chapter
Book cover Descartes’s Mathematical Thought

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 237))

Abstract

How did Descartes grasp ‘mathesis universalis’? We have already discussed its mathematical nature in Chapter 4, § 3. Now we turn to his philosophy of mathematics and attempt to support an argument about how ‘mathesis universalis’ in his youthful work Regulae ad directionem ingenii should be understood philosophically. For this purpose, we consider retrospectively his theory of mathematics there, turning our attention to its philosophical aspect.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. AT, X, pp. 365–366.

    Google Scholar 

  2. AT, X, p. 378.

    Google Scholar 

  3. n Aristotle’s discussions on the first principles, see T. H. Irwin, Aristotle’s First Pinciples (Oxford, 1988). The relation of metaphysical principles to mathematical priciples are more subtle. See, among others, Ian Mueller, “Sur les principes des mathématiques chez Aristote et Euclide,” Mathématiques et philosphie de l’Antiquité à l’âge classique, éd. par R. Rashed (Paris, 1991), pp. 101–113, and Richard D. McKirahan, Jr., Principles and Proofs: Aristotle’s Theory of Demonstrative Science (Princeton, 1992).

    Google Scholar 

  4. AT, VI, p. 31.

    Google Scholar 

  5. “Descartes, à (Amsterdam), à Mersenne, à Paris. (mi-janvier 1630),” AT, I, p. 110; AM, I, p. 112; CM, II, p. 373.

    Google Scholar 

  6. “Descartes, à Amsterdam, à Mersenne, à Paris. 15 avril 1630,” AT, I, pp. 144–145; AM, I, pp. 135–136; CM, II, pp. 430–431.

    Google Scholar 

  7. See the following statements: “[P]erhaps I may some day complete a little treatise of Metaphysics, which I began when in Frisia.” “Descartes, à (Leiden ou Amsterdam), à Mersenne, à Paris. (25 novembre 1630),” AT, I, p. 182; AM, I, p. 173; CM, II, p. 564. “Eight years ago, however, I wrote in Latin the beginnings of a treatise of metaphysics […J.” “Descartes, à (Leyde), à Mersenne, à Paris. (vers le 20 avril 1637),” AT, I, p. 350; AM, I, p. 329; CM, VI, p. 234.

    Google Scholar 

  8. See “Descartes, à Amsterdam, à Mersenne, en voyage. (6 mai 1630),” AT, I, p. 149; AM, I, p. 139; CM, II, p. 481: “As for the eternal truths, I say once more that they are true or possible only because God knows them as true or possible. They are not known as true by God in any way which would imply that they are true independently of Him.” This statement seems to have been written in reply to Mersenne’s question concerning Descartes’s creation theory of ‘eternal truths’ stated in the letter on April 15, 1630.

    Google Scholar 

  9. ’Margaret Dauler Wilson, Descartes (London/Henley/Boston, 1978), p. 121. Cf. Harry Frankfurt, “Descartes on the Creation of the Eternal Truth,” The Philosophical Review, 86 (1977), pp. 36–57; Eternal Truths and the Cartesian Circle: A Collection of Studies, ed. by Willis Doney (New York/London, 1987) and Geneviève Rodis-Lewis, Idées et vérités éternelles chez Descartes et ses successeurs (Paris, 1985).

    Google Scholar 

  10. A. Koyré, Essai sur l’idée de Dieu et les preuves de son existence chez Descartes (Paris, 1922), pp. 19–22.

    Google Scholar 

  11. For almost exhaustive documents on this doctrine, consult Émile Boutroux, Des Vérités éternelles chez Descartes, tr. par M. Canguilhem (Paris, 1985), Introduction, “Textes de Descartes relatifs aux vérités éternelles,” pp. 41–48.

    Google Scholar 

  12. “Descartes, à (Santpoort), à Mersenne, à Paris. (17 et 27 mai 1638),” AT, II, p. 138; AM, II, p. 263; CM, VII, pp. 227–228.

    Google Scholar 

  13. “Descartes au P. [Mesland] [Leyde, 2 mai 1644?],” AT, IV, pp. 118–119; AM, VI, pp. 145—146.

    Google Scholar 

  14. See Margaret J. Osier, “Divine Will and Mathematical Truth: Gassendi and Descartes on the Status of the Eternal Truths,” in Roger Ariew and Marjorie Grene, eds., Descartes and His Contemporaries: Meditations, Objections, and Replies (Chicago/London, 1995), pp. 145–158.

    Google Scholar 

  15. Ibid., p. 24

    Google Scholar 

  16. See É. Gilson, La Liberté chez Descartes et la théologie (Paris, 1987), Première Partie, Ch. IV: “La doctrine cartésienne de la liberté divine et la théologie de l’Oratoire,” pp. 157–210. A more comprehensive study can be seen in Henri Gouhier, Cartésianisme et augustinisme an XVII’ siècle (Paris, 1978).

    Google Scholar 

  17. Ibid. (n. 6), AT, I, p. 144; AM, I, p. 135; CM, II, p. 430.

    Google Scholar 

  18. G. Rodis-Lewis, Descartes (Introduction, n. 4), pp. 146–147; The English translation, p. 101. See also the same author’s “From Metaphysics to Physics,” in Stephen Voss, ed., Essays on the Philosophy and Science of René Descartes (New York/Oxford, 1993), pp. 242258, at p. 247.

    Google Scholar 

  19. Michio Kobayashi, La Philosophie naturelle de Descartes (Paris, 1993), pp. 27–42.

    Google Scholar 

  20. AT, VIII-1, pp. 327–329 (Part IV, Arts. 205–207).

    Google Scholar 

  21. See E. M. Curley, “Descartes on the Creation of the Eternal Truths,” The Philosophical Review, 43 (1984), pp. 569–597.

    Google Scholar 

  22. See, for example, Martial Gueroult, Descartes’ Philosophy Interpreted according to the Order of Reasons, Vol. II: The Soul and the Body, translated by Roger Ariew (Minneapolis, 1985; The original French edition, 1953), p. 28; Jacques Bouveresse, “La Théorie du possible chez Descartes,” Revue internationale de philosophie, 37C année, N° 146 (1983), pp. 293318; Hidé Ishiguro, “Reply to Jacques Bouveresse,” Ibid., pp. 311–318; Eadem, “The Status of Necessity and Impossibility in Descartes,” in Amélio O. Rorty, ed., Essays on Descartes’ Meditations (Berkeley/Los Angeles/London, 1986), pp. 459–471. And A. Boyce Gibson invokes Emile Boutroux’s distinction between absolute and relative contradiction in his “The Eternal Verities and the Will of God in the Philosophy of Descartes,” Proceedings of the Aristotelian Society, N. S., 30 (1929–30), pp. 31–54.

    Google Scholar 

  23. M. Wilson, Descartes (n. 9), pp. 125–126; H. Ishiguro, “Reply to Jacques Bouveresse” (n. 30), p. 317; Eadem, “The Structure of Necessity and Impossibility in Descartes” (n. 30), p. 468.

    Google Scholar 

  24. E. M. Curley, “Analysis in the Meditations: The Quest from Clear and Distinct Ideas,” in Stanley Tweyman, ed., René Descartes: Meditaions on First Philosophy in Focus (London/New York, 1993), pp. 159–184. Cf. Daniel Garber and Lesley Cohen, “A Point of Order: Analysis, Synthesis, and Descartes’ Principles,” and S. Tweyman, “Professor Cottingham and Descartes’ Methods of Anaysis and Syntheis,” in Ibid., pp. 135–147 & 148–158, respectively. See also Stephen Gaukloger, Cartesian Logic: An Essay on Descartes’s Conception of Inference (Oxford, 1989).

    Google Scholar 

  25. M. Gueroult, Descartes’ Philosophy Interpreted according to the Order of Reasons, Vol. I: The Soul and God, tr. by Roger Ariew (Minneapolis, 1984; The French edition, 1952), p. 47. For the use of the terminology ‘psychologism’, Gueroult refers to Edmund Husserl’s Logische Untersuchungen, Bd. I (Halle, 1900), 7. Kap. “Der Psychologismus als skeptischer Relativismus.” Cf. Yvon Bélaval, “Intuitinnisme cartésien et psychologisme,” Revue internationale de philosophie, 37e année, N° 146 (1983), pp. 319–325.

    Google Scholar 

  26. AT, VII, p. 140: “Principiorum enim notitia non solet a dialecticis scientia appellari.”

    Google Scholar 

  27. “Descartes â Clerselier. [Egmond, juin ou juillet 1646],” AT, IV, pp. 443–444; AM, VII, pp. 84–85. Descartes maintained that the logical principle of contradiction is a principle which is valid in general and that our soul exists is the first principle. The latter case seems to be closely related to the cogito. And he stated generally about the first principle: “I will also add that one should not require the first principle to be such that all other propositions can be reduced to it and proved by it. It is enough if it is useful for the discovery of many, and if there is no other proposition on which it depends, and none which is easier to discover. It may be that there is no principle at all to which alone all things can be reduced. They do indeed reduce other propositions to the principle that the same thing cannot both be and not be at the same time, but their procedure is superfluous and useless.” This discussion is certainly related to mathematical principles.

    Google Scholar 

  28. O. A. Kubitz, “Scepticism and Intuition in the Philosophy of Descartes,” The Philosophical Review, 48 (1939), pp. 472–491.

    Google Scholar 

  29. Peter Dear, “Marin Mersenne and the Probabilistic Roots of ‘Mitigated Scepticism’,” Journal of the History of Philosophy, 22 (1984), p. 202. See also Idem, Mersenne and the Learning of the Schools (Ithaca/London, 1988) and “Mersenne’s Suggestion: Cartesian Meditaion and the Mathematical Model of Knowledge in the Seventeenth Century” in Descartes and His Contemporaries: Meditations, Objections, and Replies (n. 14), pp. 44–62.

    Google Scholar 

  30. For a discussion on this discontinuity from the point of view of general philosophy, see M. Gueroult, Op. cit. (n. 33), pp. 30–32.

    Google Scholar 

  31. Richard H. Popkin, The History of Scepticism from Erasmus to Descartes (Assen, 1964; The First Edition, 1960), p. 184. The revised and expanded edition of the former was published under the new title of The History of Scepticism from Erasmus to Spinoza (Berkeley/Los Angeles/London, 1979). The cited passage is on p. 180.

    Google Scholar 

  32. R. H. Popkin, “Intellectual Autobiography: wars and all,” in Richard A. Watson and James E. Force, eds., The Sceptical Mode in Modern Philosophy: Essays in Honor of Richard H. Popkin (Dordrecht/Boston/Lancaster, 1988), pp. 103–149.

    Google Scholar 

  33. E. M. Curley, Descartes against the Skeptics (Cambridge, Mass., 1978).

    Google Scholar 

  34. Jean-Marie Beyssade, La Philosophie première de Descartes: Le temps et la cohérence de la métaphysique (Paris, 1979), pp. 105–106.

    Google Scholar 

  35. Popkin, “Secpticism, Theology and the Scientific Revolution,” in Imre Lakatos and Alan Musgrave, eds., Problems in the Philosophy of Science (Amsterdam, 1968), pp. 1–28; See also the discussion which follows Popkin’s paper, pp. 29–39.

    Google Scholar 

  36. Phillip R. Sloan, “Descartes, the Sceptics, and the Rejection of Vitalism in Seventeenth Century Physiology,” Studies in History and Philosophy of Science, 8 (1977), pp. 1–28. 45Ibid., pp. 11–12, footnote 29.

    Google Scholar 

  37. G. Rodis-Lewis, “Création des vérités éternelles, doute suprême et limites de l’impossible chez Descartes,” Idées et vérités éternelles chez Descartes et ses successeurs (n. 9), p. 120.

    Google Scholar 

  38. AT, VII, p. 130: “[…] libros ea de re complures ab Academicis & Scepticis scriptos dudum videssem […]. The translation of this sentence by J. Cottingham et al. reads: ”I had seen many ancient writings by the Academics and Sceptics on this subject.“ This is a serious mistranslation. ”Dudum“ should be traslated into ”sometime ago.“ In this point, the translation by Elizabeth S. Haldane and G. R. T. Ross in The Philosophical Works of Descartes, Vol. II (Cambridge, 21931) is basically correct: ”I had long ago seen several books written by the Academics and Sceptics about this subject.“ (p. 31.)

    Google Scholar 

  39. Popkin, The History of Scepticism from Erasmus to Spinoza (n. 39), Chapter II “The Revival of Greek Scepticism in the Sixteenth Century,” pp. 18–41.

    Google Scholar 

  40. C. B. Schmitt, “The Rediscovery of Ancient Skepticism in Modern Times,” in Myles Burnyeat, ed., The Skeptical Tradition (Berkeley/Los Angeles/London, 1983), pp. 225–251. An earlier version with the title “The Recovery and Assimilation of Ancient Scepticism in the Renaissance” appeared in Rivista critica di storia della fiosofia, 27 (1972), pp. 363–384.

    Google Scholar 

  41. On the writings which are supposed to have influenced Descartes, see Roger Ariew, John Cottingham, and Tom Sore11, eds., Descartes’ Meditations: Background Source Materials (Cambridge/New York, 1998).

    Google Scholar 

  42. Sextus Empiricus, Outlines of Pyrrhonism, translated by R. G. Bury (London/Cambridge, Mass., 1933), Book I, Chaps., X—XI, pp. 14–17. I have also referred to Descartes’s contemporary Greek and Latin version Sexti Empirici Opera quae extant (Paris, 1621), which has the same chapter divisions as Bury’s edition.

    Google Scholar 

  43. Ibid., Book I, Chap. III, pp. 4–7.

    Google Scholar 

  44. Ibid., Book I, Chap. XV, §164, pp. 94–95. See also Julia Annas and Jonathan Barnes, The Modes of Scepticism: Ancient Texts and Modern Interpretations (Cambridge, 1985), p. 182, and J. Barnes, The Toils of Scepticism (Cambridge, 1990).

    Google Scholar 

  45. Heath, The Thirteenth Books of Euclid’s Elements, Vol. 1 (New York, 1956), pp. 202–220. 55Sextus, Ibid., pp. 94–95.

    Google Scholar 

  46. Diogenes Laertius, Lives of Eminent Philosophers, Vol. II, translated by R. D. Hicks (London/Cambridge, Mass., 1925), pp. 500–501 (§§88–89).

    Google Scholar 

  47. Sextus Empiricus, Against the Professors, Vol. IV, translated by R. G. Bury (London/Cambridge, Mass., 1949), pp. 246–247 (§§7–8).

    Google Scholar 

  48. Ibid., pp. 248–249 (§ 8).

    Google Scholar 

  49. Heath, Op. cit. (n. 54), p. 153.

    Google Scholar 

  50. Sextus, Ibid., pp. 262–263 (§ 37).

    Google Scholar 

  51. Ibid., pp. 290–291 (§ 92).

    Google Scholar 

  52. Gassendi criticizes Descartes as follows: “I have no objection to what you say at the beginning of the Sixth Meditation, namely that material things are ‘capable of existing in so far as they are the subject matter of pure mathematics.’ In fact, however, material things are the subject matter of mixed, not pure, mathematics (objectum mixtae, non purae, matheseos), and the subject matter of pure mathematics including the point, the line, the surface, and the indivisible figures which are composed of these elements and yet remain invisible cannot exist in reality.” AT, VII, pp. 328–329; Gassendi, Disquistio metaphysica: seu Dubitationes et instantiae adversus Renati Cartesii metaphysicarn et responsa, Texte établi, traduit et annoté par Bernard Rochot (Paris, 1962), pp. 518–519. For Gassendi’s view of qualitative physics in contrast with the Cartesian mathematical physics, see B. Rochot, “Gassendi et les mathématiques,” Revue d’histoire des sciences, X (1957), pp. 69–78.

    Google Scholar 

  53. For a discussion about the epistemological aspect of Sextus’s attack on geometry, see Ian Mueller, “Geometry and Scepticism,” in Jonathan Barnes, et al., eds., Science and Speculation: Studies in Hellenistic Theory and Practice (Cambridge/Paris, 1982), pp. 69–95. On the Epicurian’s view of geometry, see, especially, “Appendix: Epicureans and Geometers,” pp. 92–95. Study I “Indivisible Magnitudes” in David J. Furley’s Two Studies in the Greek Atomists (Princeton, 1967), pp. 1–158 also contains some interesting arguments on the Atomists’ criticism of geometry.

    Google Scholar 

  54. Cicero, Academica, Vol. XIX, With an English Translation by H. Rackham (London/ Cambridge, Mass., 1933), Book II, § 106, pp. 602–603.

    Google Scholar 

  55. Michel de Montaigne, An Apology of Raymond Sebond, translated and edited with an introduction and notes by M. A. Screech (London, 1987), p. 109; OEuvres complètes, textes établis par Albert Thiboudet et Maurice Rat (Paris, 1962), p. 516.

    Google Scholar 

  56. See Popkin’ and Schmitt’s writings quoted in notes 48 and 49.

    Google Scholar 

  57. Consult Charles B. Schmitt, Cicero Scepticus: A Study of the Influence of the Academica in the Renaissance (The Hague, 1972), esp. pp. 78–91.

    Google Scholar 

  58. ee Henri Gouhier, La Pensée métaphysique de Descartes (Paris, 1978; First edition, 1962), pp. 34–35.

    Google Scholar 

  59. E. g. Léon Brunschvicq, Descartes et Pascal lecteurs de Montaigne (Paris, 1995; First edition., 1942); Popkin, Op. cit. (n. 48), p. 172; Michael G. Paulson, The Possible Influence of Montaigne’s Essais on Descartes’ Treatise on the Passions (Lanham/New York/London, 1988).

    Google Scholar 

  60. “Descartes au [Marquis du Newcastle] [Egmond, 23 Novembre 1646],” AT, IV, pp. 573 & 575; AM, VII, pp. 225–226.

    Google Scholar 

  61. Apology (n. 65), p. 277; OEuvres, p. 555.

    Google Scholar 

  62. See footnote 359 of Apology and footnote 439 of OEuvres, p. 1574, n. 1.

    Google Scholar 

  63. Apology, p. 115; OEuvres, pp. 521–522. For a possible source of this passage, see Henry Cornerius Agrippa, De incertitudo et vanitate scientiarurn atque artium (Antwerp, 1530); Opera, t. II (Hildesheim, 1970), pp. 6–7; On the Vanitie and Vncertaintie of Artes and Sciences, ed. by Catherine M. Dunn (Northridge, CA, 1975), p. 17. In his “Introduction” to the 1970 Opera edition, Popkin writes on this work: “De Vanitate has often been regarded as one of the early sceptical treatises of the Renaissance, leading to the ”nouveau pyrrhonisme“ of Montaigne and Charron, and contributing to the sceptical crisis that was to engulf the late 16th century and early 17th cenury intellectuals. Aucually De Vanitate is more anti-intellectual than sceptical.” (Introduction to Opera, I (Hildesheim, 1970), p. XVI.) “De Vanitate at least probably played a significant role in preparing the climate of opinion in whcich the revival of Greek scepticism could take place as a serious movement.” (Ibid., p. XX.)

    Google Scholar 

  64. Apology, pp. 115–116; (Euvres, p. 522

    Google Scholar 

  65. Apology, p. 185; (Mimes, p. 585

    Google Scholar 

  66. This does not necessarily mean that Montaigne was a strict Pyrrhonist. The following article has pointed put that Montaigne was closer to Socrates, not Pyrrho or Sextus. See Craig Walton, “Montaigne on the Art of Judgment: The Trial of Montaigne,” in The Sceptical Mode in Modern Philosophy (n. 40), pp. 87–102.

    Google Scholar 

  67. Apology, p. 66; OEuvres, p. 479.

    Google Scholar 

  68. Apology, p. 67; OEuvres, p. 480.

    Google Scholar 

  69. AT, VII, p. 384.

    Google Scholar 

  70. AT, VII, p. 140.

    Google Scholar 

  71. E. M. Curley, Descartes against the Skeptics (n. 41), p. 86.

    Google Scholar 

  72. Benedictus de Spinoza, The Principles of Descartes’ Philosophy, translated by Halbert Hains Britan (La Salle, Illinois, 1905), p. 14.

    Google Scholar 

  73. Gueroult, Op. cit. (n. 33), p. 60.

    Google Scholar 

  74. AT, VII, p. 550.

    Google Scholar 

  75. St. Augustine’s statement in Book II, Chapter 3 of De Libero arbitrio which Arnauld cited is as follows: “First, if we are to take as our starting point what is most evident, I ask you to tell me whether you yourself exist. Or are you perhaps afraid of making a mistake in your answer, given that, if you did not exist, it would be quite impossible for you to make a mistake?” (AT, VII, p. 198.) Cf. Gareth B. Matthews, Thought’s Ego in Augustine and Descartes (Ithaca/London, 1992), p. 11; Léon Blanchet, Les Antécedents historiques du ((Je pense, donc je suis» (Paris, 1920), pp. 27–33.

    Google Scholar 

  76. Gassendi, Disquisitio rnetaphysica (n. 62), “In Meditationem IV, Dubitatio IV, Instantia 2,” p. 463: “Circulum commissum, probando, esse Deum, insumque veracem, quia clara, distinctaque est ejus notitia; et, Notitiam Dei esse claram, distinctamque, quia Deus est, ipseque verax.” For Mersenne’s objection, see AT, VII, pp. 124–125; and for Arnault’s see AT, VII, p. 214.

    Google Scholar 

  77. AT, X, pp. 489–527.

    Google Scholar 

  78. For a description of this problem, see, for example, John Etchemendy, “The Cartesian Circle: Circulus ex tempore,” Studia Cartesiana, 2 (1981), pp. 5–42. See also papers contained in Eternal Truths and the Cartesian Circle: A Collection of Studies (n. 9)

    Google Scholar 

  79. See Gouhier, La Pensée métaphysique de Descartes (n. 68), pp. 293–319.

    Google Scholar 

  80. Anthony Kenny, “The Cartesian Spiral,” Revue internationale de philosophie, 37e année, N° 146 (1983), pp. 247–256. On Kenny’s precedent understanding of the ‘Cartesian circle’, see A. Kenny, Descartes: A Study of His Philosophy (New York, 1968), pp. 188–196.

    Google Scholar 

  81. AT, VII, p. 71. See also similar passages in the Sixth Meditation, pp. 71, 74 and 80. For the expression “objectum purae matheseos,” see David R. Lichterman, “Objectum Purae Matheseos: Mathematical Construction and the Passage from Essence to Existence,” in A. O. Rorty, ed., Essays on Descartes’ Meditations (n. 30), pp. 435–458.

    Google Scholar 

  82. AT, VIII-1, p. 78 (Part II, Art. 64).

    Google Scholar 

  83. “Descartes, à Leyde, à Mersenne, à Paris. (28 janvier 1641),” AT, III, pp. 297–298; AM, IV, p. 269; CM, X, p. 440.

    Google Scholar 

  84. “Daniel Garber, ”Semel in vita: The Scientific Background to Descartes’ Meditations,“ in A. O. Rorty, Op. cit. (n. 30), p. 82; Idem, Descartes Embodied: Reading Cartesian Philosophy through Cartesian Science (Cambridge, 2001), p. 223.

    Google Scholar 

  85. Metaphysica, 995a14–16.

    Google Scholar 

  86. Blaise Pascal, “The Geometrial Mind and the Art of Persuation,” translated by R. H. Popkin, in Pascal: Selections, ed. by Popkin (New York/London, 1989), p. 192; OEuvres complètes, Édition établie et annotée par Jacques Chevalier (Paris, 1954), pp. 599–600.

    Google Scholar 

  87. For example, see Élie Denisoff, Descartes, premier théoreticien de la physique mathématique: Trois essais sur le ((Discours de la méthode» (Louvain/Paris, 1970).

    Google Scholar 

  88. See, among others, Garber, Descartes’ Metaphysical Physics (Ch. 4, n. 132). On various aspects of Cartesian physics, see S. Gaukroger, J. Schuster and J. Sutton, eds., Descartes’ Natural Philosophy (London/New York, 2000), and on the development of Cartesian natural philosophy which culminated in his Principia philosophiae of 1644, see Stephen Gaukroger, Descartes’ System of Natural Philosophy (Cambridge, 2002).

    Google Scholar 

  89. Pierre Boutroux, L’Imagination et les mathématiques selon Descartes (Paris, 1900), p. 42. Boutroux composed the work when he was barely nineteen years old! Ronald S. Calinger, “Boutroux, Pierre Léon,” DSB, II, p. 357.

    Google Scholar 

  90. Boutroux, L’Imagination (n. 99), p. 25.

    Google Scholar 

  91. See, for example, H. Poincaré, “Sur la nature du raisonnement mathématique,” Revue de métaphysique et de morale, 2 (1894): Reprinted as Chap. I of La Science et l’hypothèse (Paris, 1903); English translation, “On the Nature of Mathematical Reasoning,” in Paul Benacerraf and Hilary Putnum, eds, Philosophy of Mathematics: Selected Readings (Cambridge, 21983), pp. 394–402.

    Google Scholar 

  92. Gueroult, Op. cit. (n. 33), p. 273.

    Google Scholar 

  93. Jürgen Mittelstraß, “Die Idee einer Mathesis Universalis bei Descartes,” Perspektiven der Philosophie, 4 (1978), pp. 177–192; Idem, “The Philosopher’s Conception of Mathesis Universalis from Descartes to Leibniz,” Annals of Science, 33 (1979), pp. 593–610.

    Google Scholar 

  94. J. E. Hofmann, Frans van Schooten der Jüngere, Boethius, II (Wiesbaden, 1962), p. 33, note 52.

    Google Scholar 

  95. “Fr van Schooen à Christiaan Huygens. 3 Juin 1648,” OEuvres complètes de Christiaan Huygens, t. I (La Haye, 1888), p. 98. We do not know exactly whether this treatise is identical with “Calcul de Mons. de Cartes. [Introduction a sa Geometrie],” a short monograph on elementary symbolic algebra, contained in AT, X, pp. 659–680. But the Principia has characteristics similar to that French tract.

    Google Scholar 

  96. For the establishment of the Savilian professor of geometry at Oxford, see my “Scientific Studies at Oxford and Cambridge in the Seventeenth Century—A Research Review,” Historia Scientiarum, No. 20 (1981), pp. 57–75, esp. pp. 60–66.

    Google Scholar 

  97. John Wallis, Mathesis universalis: Operum mathematicorum pars prima (Oxford, 1657), “Dedicatio”; Opera mathematica, I (Oxford, 1695), p. 13.

    Google Scholar 

  98. Mathesis universalis, p. 13; Opera, I, p. 18.

    Google Scholar 

  99. Mathesis universalis, p. 73; Opera, I, p. 56.

    Google Scholar 

  100. See my “The Acceptance of the Theory of Proportion in the Sixteenth and Seventeenth Centuries,” (Ch. 5, n. 162), esp. pp. 91–95; J. F. Scott, The Mathematical Work of John Wallis (London, 1938; New York, 1981), pp. 66–71; Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, Translated by Eva Brann (Cambridge, Mass., 1968), pp. 211–224. For the mathematical thought of Wallis in general, see A. Prag, “John Wallis, 1616–1703. Zur Ideengeschichte der Mathematik im 17. Jahrhundert,” Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Apt. B, 1 (1931), pp. 381–412.

    Google Scholar 

  101. D. T. Whiteside, ed., The Mathematical Papers of Isaac Newton, Vol. V (1683–1684) (Cambridge, 1972), pp. 538–621.

    Google Scholar 

  102. For the situation in which Newton composed Matheseos universalis specimina, see D. T. Whiteside, “General Introduction,” pp. xix—xx and “Introduction,” pp. 414–419 of The Mathematical Papers of Isaac Newton, Vol. IV (1674–1684) (Cambridge, 1971).

    Google Scholar 

  103. Ibid., pp. 526–589.

    Google Scholar 

  104. G. I. ‘s Gravesande, Matheseos Universalis Elementa. Quibus accedunt, Specimen commentarii in Arithmeticam Universalem Newtoni: ut & De determinanda Forma Seriei infinitae adsumtae Regula [Methodus] Nova (Leiden, 1727), p. 4: “Ars haec, Algebra, etiam Analysis, à quibusdam Mathesis Universalis vocatur.” See A. Rupert Hall, “’s Gravesande, William Jacob,” DSB, V, p. 510.

    Google Scholar 

  105. Heinrich Scholz, Mathesis Universalis: Abhandlungen zur Philosophie als strenger Wissenschaft (Basel/Stuttgart, 21969), p. 108.

    Google Scholar 

  106. G. W. Leibniz, Dissertatio de arte cornbinatoria (Leipzig, 1666): Die philosophischen Schriften (=PS), ed. C. I. Gerhardt, Bd. 4 (Berlin, 1880), p. 35: Leibniz, Philosophical Papers and Letters: A Selection, translated and edited, with an Introduction by Leroy E. Loemker (Dordrecht/Boston, 21969), pp. 76–77.

    Google Scholar 

  107. This point of view was invoked by Issac Barrow invoked in his Lectiones mathematicae (delivered from 1664 to 1666 and published in 1683) when he attacked Wallis’s analytical standpoint in the latter’s Savilian lectures the Mathesis universalis. See my Op. cit. (n. 115), esp. pp. 86–88.

    Google Scholar 

  108. “Roshdi Rashed, ”Combinatorial Analysis, Numerical Analysis, Diophantine Analysis and Number Theory,“ in R. Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2: Mathematics and the Physical Sciences (London/New York, 1996), p. 378; Further in detail in Prof. Rashed’s graduate lecture in the summer semester of 1995 at the University of Tokyo.

    Google Scholar 

  109. See G. H. R. Parkinson, “Introduction” to Leibniz, Logical Papers (Oxford, 1966), p. xii; E. J. Aiton, Leibniz: A Biography (Bristol/Boston, 1985), p. 18

    Google Scholar 

  110. Eberhard Knobloch, “The Mathematical Studies of G. W. Leibniz on Combinatorics,” Historia Mathematica, 1 (1974), pp. 409–430, esp. pp. 411–416.

    Google Scholar 

  111. Leibniz, Op. cit. (n. 121), p. 41; Loemker, p. 75.

    Google Scholar 

  112. On the dates of Leibniz’s writings, consult Kurt Müller und Gisela Knönert, hrsg., Leben und Werk von Gottfried Wilhelm Leibniz: Eine Chronik (Frankfurt am Main, 1969). On that of the “Mathesis universalis,” see p. 136. Mittelstraß conjectures that the “Elementa nova matheseos univeralis” dated from about 1675. See J. Mittelstraß, “The Philosopher’s Conception of Mathesis Universalis etc.” (n. 108), p. 604.

    Google Scholar 

  113. Louis Couturat, Opuscules et fragments inédits de Leibniz: Extraits des manuscripts de la Bibliothèque royale de Hanovre (Paris, 1903), p. 348: “(Idea libri cui titulus est) ELEMENTA NOVA MATHESEOS UNIVERSALIS.”

    Google Scholar 

  114. Couturat, Loc. cit.: “{Elementa Matheseos (Universalis) talia esse debent ut prodesse possint etiam ad Cryptographemata interpretanda, ad ludum Schaccorum ludendum, et talia id genus.}”

    Google Scholar 

  115. Ibid.: “Haec Elementa Matheseos universalis multo plus differunt a Speciosa hactenus cogita, quam ipsa Speciosa Viet æ aut Cartesii differt a Symbolica veterum. Ostendetur his Methodus Calculum Geometricum ad illa quoque problemata porrigendi quae Albgebram (hactenus receptam) transcendunt. Tradetur et Synthesis et Analysis, sive tam Combinatiria quam Algebra.”

    Google Scholar 

  116. “Leibniz an Oldenburg, Paris, 28. Dezember 1675,” Leibniz, Sämtliche Schriften und Briefe, Dritte Reihe: Mathematischer Naturwissenschaftlicher und Technischer Briefwechsel, Bd. 1 (Berlin, 1976), p. 331: “Ego vero agnosco, quicquid in hoc genere praebat algebra, non nisi superioris scientiae beneficium esse, quam nunc combinatoriam, nunc characteristicam appellare soleo […].” Cf. Mathematische Schriften (= MS), ed. C. I Gerhardt, Bd. 1 (Berlin, 1849), pp. 85–86; Loemker, p. 166.

    Google Scholar 

  117. Couturat, Ibid., pp. 350–351: “Mathodus solvendi problema est sel synthetica vel analytica. […] Synthetica (vel combinatoria) est cum alia problemata percurrimus et incidimus tandem in nostrum (et huc pertinet Methodus eundi à simplicibus ad composita problemata.) Analytica est, cum à nostro inchoantes regredimur ut perveniamus ad conditiones quæ ad ipsum sovendum sufficiunt.” For Leibniz’s general account for the dual method, see his “De Synthesi et Analysi universale seu Arte inveniendi et judicandi,” PS, Bd. 7 (Berlin, 1890), pp. 292–298; Loemker, pp. 229–234.

    Google Scholar 

  118. Ibid., p. 348: “Mathesis universalis tradere cebet Methodum aliquid exacte determinandi per ea qua sub imaginationem cadunt, sive ut ita dicam Logicam imaginationis. Itaque hint excluduntur Metaphysica circa res pure intelligibiles, ut cogitationem, actionem. Excluditur et Mathesis specialis circa Numeros, Situm, Motum. Imaginatio generaliter circa duo versatur, Qualitatem et Quantitatem, sive magnitudinem et formam; secumdum qua res dicuntur similes aut dissimiles, aquales aut inaquales. Et vero similitudinis considerationem pertinere ad Mathesin generalem non minus quam aqualitatis, ex eo patet quod Mathesis specialis, qualis est Geometria, sape investigat figurarum similitudines.”

    Google Scholar 

  119. See Aiton’s description in his Leibniz: A Biography (n. 124), pp. 96–98.

    Google Scholar 

  120. “Leibniz à Christiaan Huygens. 8 septembre 1679, Appendice,” OEuvres complètes de Christiaan Huygens, t. 8 (La Haye, 1899), pp. 219–220; MS, Bd. 2 (Berlin, 1849), p. 20; Loemker, p. 250.

    Google Scholar 

  121. Müller u. Krönert, Op. cit. (n. 127), p. 126. MS, Bd. 5 (Berlin, 1858), pp. 178–183; Loemker, pp. 254–258.

    Google Scholar 

  122. Couturat, Ibid., pp. 349–350, especially the editor’s note 1 on p. 350.

    Google Scholar 

  123. Mathematische Schriften, Bd. 7 (Halle, 1863), p. 53: “Mathesis universalis est scientia de quantitate in universum, seu de ratione aestimandi, adeoque limites designandi, intra quos aliquid cadat. Et quoniam omnis creatura limites habet, hint dici potest, ut Metaphysica est scientia rerum generalis, ita Mathesin universalem esse scientiam creaturarum generalem. Duasque habet partes: scientiam finiti (quae Algebrae nomine venit priorque exponetur), et scientiam infiniti, ubi interventu infiniti finitum determinatur.” For an account of the relationship between metaphysics and universal mathematics, see Gilles-Gaston Granger, “Philosophie et mathématique leibniziennes,” Revue de métaphysique et de morale, 86e année (1981), pp. 1–37, esp. pp. 1–5.

    Google Scholar 

  124. Ibid., p. 68.

    Google Scholar 

  125. P 54: “Et quemadmodum multi Logicam illustrare tentaverunt similitudine computi ipseque Aristoteles in Analyticis Mathematico more locutus est, ita vicissim et multo quidem rectius Mathesis praesertim universalis, adeoque Arithmetica et Algebra tractari possunt per modum Logicae, tanquam si essent Logica Mathematica, ut ita in effectu coincidat Mathesis universalis sive Logistica et Logica Mathematicorum; unde et Logistica nostra nomine Analyseos Mathematicae passim venit.” As for the translation, we refer partly to Gottfried Martin, Leibniz: Logic and Metaphysics, translated by K. J. Northcott and P. G. Lucas (Manchester, 1964; The original German edition, 1960), p. 84.

    Google Scholar 

  126. Leibniz, Nouveax essais sur l’entendement humain in Sämtliche Schriften und Briefe, Sechste Reihe, Bd. 6 (Berlin, 1962), Livre IV, Chap. 17, § 4, p. 478. See the English translation by Peter Remnant and Jonathan Bennett (Cambridge, 1981), which has the same pagination as the above edition.

    Google Scholar 

  127. Op. cit. (n. 128), p. 68.

    Google Scholar 

  128. See, for example, Edmund Husserl, Logical Investigations, translated by J. N. Findlay, Vol. I (London/New York, 1970; The original German edition, 1900), “Prolegomena to Pure Logic,” § 60, “Links with Leibniz,” pp. 218–220. “Through his Combinatoria he [Leibniz] is also the intellectual father of the pure theory of manifolds (die reine Mannigfaltigkeitslehre), a discipline close to pure logic and in fact intimately one with it.” (p. 220). On the development of Leibniz’s concept of ‘mathesis universalis’, see Dietrich Mahnke, “Leibnizens Synthese von Universalmathematik und Individualmetaphysik,” Jahrbuch für Philosophie und phänomenologische Forschung, Bd. 7 (1925), pp. 305–611.

    Google Scholar 

  129. Mittelstraß, Op. cit. (n. 127), p. 610. Cf. Idem, Neuzeit und Aufklärung: Studien zur Entstehung der Neuzeitlichen Wissenschaft und Philosophie (Berlin/New York, 1970), § 12 “Kunstsprache und Logikkalkül,” pp. 413–453; Louis Couturat, La Logique de Leibniz d’après des documents inédits (Paris, 1901), Chap. VII: “La mathématique universelle,” pp. 283–322.

    Google Scholar 

  130. Müller u. Krönert, Op. cit. (n. 127), p. 24. Aiton, on the other hand, ascribes this fragment to the first mathematical product during Leibniz’s sojourn in Paris during the years 1672–1676. See Op. cit. (n. 124), p. 42.

    Google Scholar 

  131. “Demonstratio proportionum primarum,” Sämtliche Schriften und Briefe, V-2 (Berlin, 1966), p. 479: “Ego ita sentio, nullam propositionem accipiendam esse nisi probatam […].” 159/bid., p. 480.

    Google Scholar 

  132. E. De Olaso, “Leibniz and Scepticsm,” in Richard H. Popkin, Ezequiel De Olaso, and Giorgio Tonelli, eds., Scepticism in the Enlightenment (Dordrecht/Bonston/London, 1997), pp. 99–130, at p. 100.

    Google Scholar 

  133. Loc. cit.: “Totum esse maius parte, primus demonstravit Hobbius, fundamentum Scientiae de Quantitate.” Cf. Thomas Hobbes, Elementorum philosophiae sectio prima de corpore (London, 1655), Chap. 8, § 25, p. 72. F. J. E. Woodbridge, Hobbes: Selections (New York, 1958), pp. 92–93.

    Google Scholar 

  134. “Ut Totum esse maius parte in Anglo contactus negavit Gregorius a S. Vincentio et in infinito Cardinalis Pallavicius.” For a discussion on the angle of contact, see J. E. Hofmann, Leibniz in Paris 1672–1676 (Cambridge, 1974), pp. 12–14. Note that for Leibniz the angle of contact and the infinite are not magnitudes.

    Google Scholar 

  135. Ibid., p. 482: “Cuius pars alteri toti aequalis est, id est mains per def. Majoris. Pars totius cde (nempe de) est aequalis toti de (nempe sibi ipsi). Ergo cde est mains quam de; totum parte. Quod erat demonstrandum.”

    Google Scholar 

  136. PS, Bd. 7 (Berlin, 1890), p. 355; Loemker, p. 677.

    Google Scholar 

  137. PS, Bd. 4, p. 422; Loemker, p. 291; Philosophical Essays, translated by Roger Ariew and Daniel Garber (Indianapolis & Cambridge, 1989), p. 23.

    Google Scholar 

  138. PS, Bd. 4, p. 423; Loemker, p. 292; Philosophical Essays, p. 25.

    Google Scholar 

  139. PS, Bd. 4, p. 426; Loemker, p. 294; Philosophical Essays, p. 27.

    Google Scholar 

  140. Pascal: Selections (n. 96), pp. 173–194; OEuvres complètes (n. 96), pp. 575–604.

    Google Scholar 

  141. Charles Parsons, “Platonism and Mathematical Intuition in Kurt Gödel,” The Bulletin of Symbolic Logic, 1 (1995), p. 45, n. 3.

    Google Scholar 

  142. “Projets et Essays pour arriver à quelque certitude pour finir une bonne partie des disputes et pour avancer l’art d’inventer,” Couturat, Opuscules et fragments inédits de Leibniz (n. 127), pp. 181–182.

    Google Scholar 

  143. PS, Bd. 4, pp. 427–428; Loemker, p. 304.

    Google Scholar 

  144. PS, Bd. 6 (Berlin, 1885), pp. 218–231.

    Google Scholar 

  145. PS, Bd. 6, pp. 612–614; Loemker, pp. 646–647.

    Google Scholar 

  146. “Animadversiones,” PS, Bd. 4, p. 355; Loemker, p. 383. See also Nouveaux essais (n. 143), p. 107 and 406–407. “Leibniz and Tschirnhaus examined Roberval’s papers and decided that the Elements of Geometry was not worth publishing.” P. Remnant and J. Bennett’s translation, note “Roberval”, pp. lxix—lxx. Now Roberval’s treatise was published as Éléments de géométrie de Roberval, texte présentés par Vincent Jullien (Paris, 1996) & Kokiti Hara, “Quelques Œuvrages des Géométrie More Veterum de Roberval, (1) et (2),” Historia Scientiarum, Vol. 2, Nos. 1–2 (1992), pp. 13–44 & 109–117.

    Google Scholar 

  147. “Animadversiones,” PS, Bd. 4, p. 355; Loemker, p. 384.

    Google Scholar 

  148. Ibid.

    Google Scholar 

  149. PS, Bd. 4, p. 356; Loemker, p. 384.

    Google Scholar 

  150. Nouveaux essais, Bk I, Ch. i, § 18, p. 82

    Google Scholar 

  151. Frege, Die Grundlagen der Arithmetik (Breslau, 1884), § 88; M. S. Mahoney’s translation in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics (Cambridge, 1983), p. 154.

    Google Scholar 

  152. Ibid. Cf. W. V. Quine’s criticism of logicism in his “Carnap and Logical Truth,” in Benacerraf and Putnam, eds., Op. cit. (n. 169), p. 358. Quine puts it: “Kant’s readiness to see logic as analytic and arithmetic as synthetic, in particular, is not superseded by Frege’s work.”

    Google Scholar 

  153. Margaret D. Wilson, “On Leibniz’s Explanation of ‘Necessary Truth’,” in Harry G. Frankfurt, ed., Leibniz: A Collection of Critical Essays (Notre Dame/London, 1972), pp. 401–419.

    Google Scholar 

  154. I understand Margaret D. Wilson to be claiming this point in her splendid essay “Leibniz and Locke on ‘First Truths’,” Journal of the History of Ideas, 28 (1967), pp. 347–366; Also reprinted in Ch. 24 of her Ideas and Mechanism: Essays on Early Modern Philosophy (Princeton, 1999), pp. 353–372.

    Google Scholar 

  155. Brown, Leibniz (Brighton, Sussex, 1984), pp. 11, footnote 7: ”I take Rationalism to be a view not simply that a deductive metaphysics is possible but that a fully demonstrated metaphysics is possible. Leibniz’s later metaphysical system is, as I shall attempt to show, hypothetico-deductive. Its principles are assumptions rather than self-evident truths.“

    Google Scholar 

  156. S. Brown, “Leibniz’s Break with Cartesian ‘Rationalism’,” in A. J. Holland, ed., Philosophy, Its History and Historiography (Dordrecht/Boston, 1985), pp. 195–208, esp. p. 202. “One way of expressing this reform is to say that Leibniz rejected the Cartesian appeal to our ideas in favour of a pragmatic criterion of metaphysical truth. We do not have intuitive knowledge of the essence of substance, as Descartes had mistakenly supposed.”

    Google Scholar 

  157. “Leibniz an Foucher,” PS, Bd. 1 (Berlin, 1875), p. 382. G. W. Leibniz, Discourse on Metaphysics and Related Writings, edited and translated with an Introduction, Notes and Glossary by R. Niall, D. Martin, and Stuart Brown (Manchester/New York, 1988), pp. 130131; See also Brown, Leibniz (n. 175), p. 74.

    Google Scholar 

  158. Brown, Leibniz, p. 74.

    Google Scholar 

  159. Brown, Leibniz, p. 73.

    Google Scholar 

  160. Leibniz, Nouveaux essais (n. 143), p. 367 183/bid., p. 411.

    Google Scholar 

  161. Ibid., p. 446

    Google Scholar 

  162. Ibid., pp. 452–453.

    Google Scholar 

  163. popkin, “Leibniz and the French Sceptics” (n. 173), p. 247.

    Google Scholar 

  164. Lakatos, Proofs and Refutations (Cambridge, 1976), p. 49, footnote 1.

    Google Scholar 

  165. José Ortega y Gasset, The Idea of Principle in Leibnitz and the Evolution of Deductive Theory, translated by Mildred Adams (New York, 1971), p. 111.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Tokyo, Yokohama, Japan

    Chikara Sasaki

Authors
  1. Chikara Sasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Sasaki, C. (2003). ‘Mathesis Universalis’ in the Seventeenth Century. In: Descartes’s Mathematical Thought. Boston Studies in the Philosophy of Science, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1225-5_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1225-5_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6487-5

  • Online ISBN: 978-94-017-1225-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation