Deleuze, Leibniz and Projective Geometry in the Fold

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Notes

notes

I am grateful to the reviewer René Guitart for his constructive suggestions.

1 See, for example, Duffy, “Leibniz, Mathematics and the Monad.”

2 Panofsky 256.

3 Transcendental in this mathematical context refers to those curves that were not able to be studied using the algebraic methods introduced by Descartes.

4 A concept that was already in circulation in the work of Fermat and Descartes. Leibniz, Mathematische Schriften V: 126.

5 See ibid. 223.

6 Leibniz, Mathematische Schriften VII: 222–23.

7 Leibniz, Philosophical Papers and Letters 545.

8 The lettering has been changed to more directly reflect the isomorphism between this algebraic example and Leibniz's notation for the infinitesimal calculus.

9 This example presents a variation of the infinitesimal or “characteristic” triangle that Leibniz was familiar with from the work of Pascal. See Leibniz, “Letter to Tschirnhaus (1680)” in The Early Mathematical Manuscripts; and Pascal, “Traité des sinus du quart de cercle (1659)” in Œuvres Mathématiques.

10 Deleuze, Sur Leibniz, 22 Apr.

11 Leibniz, Mathematische Schriften V: 220–26.

12 Newton, Method of Fluxions and Infinite Series.

13 Newton's reasoning about geometrical limits is based more on physical insights rather than mathematical procedures. In “Geometria Curvilinea,” Newton develops the synthetic method of fluxions which involves visualizing the limit to which the ratio between vanishing geometrical quantities tends.

14 Lakoff and Núñez 224.

15 Bos 6.

16 Leibniz, Methodus tangentium inversa; see Katz 199.

17 See Bos 6.

18 See Boyer 287. While Leibniz had already envisaged the convergence of alternating series, and by the end of the seventeenth century the convergence of most useful concrete examples of series, which were of limited quantity, if not finite, was able to be shown, it was Cauchy who provided the first extensive and significant treatment of the convergence of series. See Kline 963.

19 For an account of this problem with limits in Cauchy, see Potter 85–86.

20 See Potter 85. While the epsilon-delta method is due to Weierstrass, the definition of limits that it enshrines was actually first proved by Bernard Bolzano (1741–1848) in 1817 using different terminology (Ewald 225–48); however, it remained unknown until 1881 when a number of his articles and manuscripts were rediscovered and published.

21 Boyer 287.

22 See Bell.

23 The infinitesimal is now considered to be a hyperreal number that exists in a cloud of other infinitesimals or hyperreals floating infinitesimally close to each real number on the hyperreal number line (Bell 262). The development of non-standard analysis, however, has not broken the stranglehold of classical analysis to any significant extent, but this seems to be more a matter of taste and practical utility rather than of necessity (Potter 85).

24 Robinson 2.

25 Non-standard analysis allows “interesting reformulations, more elegant proofs and new results in, for instance, differential geometry, topology, calculus of variations, in the theories of functions of a complex variable, of normed linear spaces, and of topological groups” (Bos 81).

26 For a more extensive discussion of this aspect of Deleuze's project, see Duffy, The Logic of Expression.

27 Deleuze, Sur Leibniz, 29 Apr.

28 Ibid.

29 The concept of neighbourhood, in mathematics, which is very different from contiguity, is a key concept in the whole domain of topology.

30 Deleuze, The Fold 15.

31 Which was actually known to the Babylonians one thousand years earlier, although Pythagoras is considered to be the first to have proved it.

32 Cache 34–41, 48–51, 70–71, 84–85.

33 See Lakhtakia et al. 3538.

34 Leibniz's distinction between the three kinds of points – physical, mathematical, and metaphysical – will be returned to in the following section.

35 Deleuze, Sur Leibniz, 15 Apr.

36 Bassler 870.

37 And that Deleuze characterizes as “vice-diction” (The Fold 59).

38 Deleuze, Sur Leibniz, 29 Apr.

39 It did not achieve prominence as a field of mathematics until the early nineteenth century through the work of Poncelet (1788–1867), Gergonne (1771–1859), Steiner (1796–1863), von Staudt (1798–1867) and Plücker (1801–68). One of the leading themes in Poncelet's work is the “principle of continuity” which he coined and in a broad philosophical sense goes back to the law that Leibniz used in connection with the calculus. However, Poncelet advanced it as an absolute truth and applied it to prove many new theorems of projective geometry. See Kline 843.

40 “Letter to Lady Masham (1704)” in Leibniz, Philosophical Essays 290.

41 Leibniz provides a mathematical representation of the metaphysical points in his ontological proof of God as ∞/1. If the infinite is the set of all possibilities, and if the set of all possibilities is possible, then there exists a singular individual who corresponds to it, and this singular individual is God represented mathematically by ∞/1. From God to the monad is to go from the infinite to the individual unit that includes an infinity of predicates. The metaphysical point that occupies the position of a monad's point of view is the inverse of the position occupied by God, and is represented mathematically by 1/∞. There is an infinity of 1/∞ (monads), and one all-inclusive ∞/1 (God). “For Leibniz the monad is… the inverse, reciprocal, harmonic number. It is the mirror of the world because it is the inverted image of God” (The Fold 129).

42 “Principles of Nature and Grace (1714)” in Leibniz, Philosophical Papers and Letters §13.

43 In the preface to New Essays on Human Understanding, Leibniz says that “noticeable perceptions arise by degrees from ones which are too minute to be noticed” (56).

44 Leibniz, Philosophical Essays 120.

45 “Letter to Simon Foucher (1693)” in Leibniz, Die philosophischen Schriften I: 415–16.

46 Pacidus Philalethi in Leibniz, Opuscules et fragments 614–15.

47 Panofsky 259. This method was systematized by Gaspard Monge (1746–1818) in what he called “descriptive geometry.”

48 Leibniz, Philosophical Essays 146. See Garber 34–40.

49 See Grene and Ravetz 141. Deleuze also poses the question of whether this topological account can be extended to Leibniz's concept of the vinculum (The Fold 111). If so, the topology of the vinculum would have to be isomorphic to that of matter; however, it would be so within each monad, and would be complicated by itself being a phenomenal projection. For further discussion of the vinculum in Leibniz see Look.