Mathematics and the theory of multiplicities: Badiou and Deleuze revisited

Southern Journal of Philosophy

Volumn 41, Issue 3, 2003, Pages 411-449

  Smith, Daniel W ^a  

^a PURDUE UNIVERSITY   (United States)

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EID: 67650799904     PISSN: 00384283     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.2041-6962.2003.tb00960.x     Document Type: Article

References (116)

1. Gilles Deleuze, Foucault, trans. Sean Hand (Minneapolis: University of Minnesota Press, 1886), 2. Deleuze was speaking of Virilio's relation to Foucault.
2. Alain Badiou, L'être et l'événement (Paris: Editions du Seuil, 1988). An English translation of Badiou's magnum opus by Oliver Feltham is forthcoming from Continuum.
3. Gilles Deleuze, "A Philosophical Concept," in Who Comes After the Subject, ed. Eduardo Cadava, Peter Connor, Jean-Luc Nancy (New York: Routledge, 1991), 95.
4. Alain Badiou, Deleuze: The Clamor of Being, trans. Louise Burchill (Minneapolis: University of Minnesota Press, 2000), 4.
5. Badiou, Deleuze: The Clamor of Being, 1.
6. See Badiou, L'être et l'événement, 522: "the latent paradigm in Deleuze is 'natural'. . . . Mine is mathematical. " Similarly, in his review article of Deleuze's book on Leibniz, Badiou writes: "There have never been but two schemas or paradigms of the Multiple: the mathematical and the organicist. . . . This is the cross of metaphysics, and the greatness of Deleuze . . . is to choose without hesitation for the animal" (in Alain Badiou, "Gilles Deleuze, 'The Fold: Leibniz and the Baroque'," in Constantin V. Boundas and Dorothea Olkowski, ed. , Gilles Deleuze and the Theater of Philosophy [New York and London: Routledge, 1994], 55).
7. This same theme is continued in Badiou's article "De la Vie comme nom de l'Être," in Rue Descartes 20 (May 1998), 27-34
8. which is reprinted in revised form as "L'Ontologie vitaliste de Deleuze," in Alain Badiou, Court Traité d'ontologie transitoire (Paris: Seuil, 1998), 61-72.
9. See, for instance, the articles on Badiou's book by Eric Alliez, Arnaud Villani, and José Gil, collected in Future Antérieur 43.
10. Badiou, Court Traité, 72.
11. (see Badiou, L'être et l'événement, 20).
12. See Gilles Deleuze, Difference and Repetition, trans. Paul Patton (New York: Columbia University Press, 1994), 323, note 22: Given the irreducibility of 'problems' in his thought, Deleuze writes that "the use of the word 'problematic' as a substantive seems to us an indispensable neologism. "
13. Alain Badiou, "Une, multiple, multiplicité(s)," 4. This unpublished text is, to my knowledge, Badiou's only direct discussion of Deleuze's theory of multiplicities. I thank Peter Hallward for making the manuscript available to me.
14. This article has been published under the same title in Multitudes 1 (Mar 2000), 195-211; my page references are to the typescript.
15. See Gilles Deleuze and Félix Guattari, A Thousand Plateaus, trans. Brian Massumi (Minneapolis: University of Minnesota Press, 1987), 374: "Only royal science has at its disposal a metric power that can define a conceptual apparatus or an autonomy of science (including the autonomy of experimental science). " On page 486: "Major science has a perpetual need for the inspiration of the minor; but the minor would be nothing if it did not confront and conform to the highest scientific requirements. "
16. Deleuze and Guattari, A Thousand Plateaus, 362, 367 (emphases added).
17. Badiou, "Une, multiple, Multiplicité(s)," 4.
18. Badiou, Deleuze, 46: "I uphold that the forms of the multiple are, just like Ideas, always actual and that the virtual does not exist. " Deleuze agrees with this characterization of sets: "Everything is actual in a numerical multiplicity; everything is not 'realized,' but everything there is actual. There are no relationships other than those between actuals. "
19. Gilles Deleuze, Bergsonism, trans. Hugh Tomlinson and Barbara Habberjam (New York: Zone Books, 1988), 43.
20. Gilles Deleuze and Félix Guattari, Anti-Oedipus, trans. Robert Hurley, Mark Seem, and Helen R. Lane (New York: Viking, 1977), 371-2.
21. For Badiou's appeal to Lautréamont, see Court Traité, 72
22. and "De la Vie comme nom de l'Être," Rue Descartes, 34.
23. See Deleuze's well-known comments on his relation to the history of philosophy in "Letter to a Harsh Critic," in Negotiations, trans. Martin Joughin (New York: Columbia University Press, 1995), 5-6.
24. The best general works on the history of mathematics are Carl B. Boyer, History of Mathematics (Princeton: Princeton University Press, 1968)
25. and Morris Kline, Mathematical Thought From Ancient to Modern Times, 3 vols. (Oxford: Oxford University Press, 1972).
26. Proclus, Commentary of the First Book of Euclid's Elements, trans. Glenn R. Murrow (Princeton, N. J. : Princeton University Press, 1970), 63-7
27. as cited in Difference and Repetition, 163;
28. A Thousand Plateaus, 554, note 21;
29. and Logic of Sense, trans. Mark Lester, with Charles Stivale; ed. Constantin V. Boundas (New York: Columbia University Press, 1990), 54.
30. See also Deleuze's comments in The Time-Image, trans. Hugh Tomlinson and Barbara Habberjam (Minneapolis: University of Minnesota Press, 1989), 174: theorems and problems are are "two mathematical instances which constantly refer to each other, the one enveloping the second, the second sliding into the first, but both very different in spite of their union. " On the two types of deduction, see 185.
31. For instance, determining a triangle the sum of whose angles is 180 degrees is theorematic, since the angles of every triangle will total 180 degrees. Constructing an equilateral triangle on a given finite straight line, by contrast, is problematic, since we could also construct a non-equilateral triangle or a non-triangular figure on the line (moreover, the construction of an equilateral triangle must first pass through the construction of two circles). Classical geometers struggled for centuries with the three great "problems" of antiquity: - trisecting an angle, constructing a cube having double the volume of a given cube, and constructing a square equal to a circle - though it would turn out that none of these problems is solvable using only a straightedge and compass. See E. T. Bell's comments in Men of Mathematics (New York: Simon & Schuster, 1937), 31-2.
32. Deleuze, Logic of Sense, 54.
33. Edmund Husserl, Ideas: General Introduction to a Pure Phenomenology, trans. W. R. Boyce Gibson (New York: Macmillan, 1931), §74, 208.
34. See also Edmund Husserl's Origin of Geometry: An Introduction, ed. John P. Leavey, Jr. and David B. Allison (Stony Brook, N. Y. : H. Hayes, 1978), which includes Jacques Derrida's important commentary. Whereas Husserl saw problematics as "proto-geometry," Deleuze sees it as a fully autonomous dimension of geometry, but one he identifies as a "minor" science; it is a "proto"-geometry only from the viewpoint of the "major" or "royal" conception of geometry, which attempts to eliminate these dynamic events or variations by subjecting them to a theorematic treatment.
35. Deleuze, Difference and Repetition, 160 (emphasis added). Deleuze continues: "As a result [of using reductio ad absurdum proofs], however, the genetic point of view is forcibly relegated to an inferior rank: proof is given that something cannot not be, rather than that it is and why it is (hence the frequency in Euclid of negative, indirect and reductio arguments, which serve to keep geometry under the domination of the principle of identity and prevent it from becoming a geometry of sufficient reason). "
36. (citing Jules Houël, Essai critique sur les principes fondamentaux de la géométrie élémentaire [Paris: Gauthier-Villars, 1867], 3, 75).
37. Boyer makes a similar point in his History of Mathematics, 141: "Greek mathematics sometimes has been described as essentially static, with little regard for the notion of variability; but Archimedes, in his study of the spiral, seems to have found the tangent to the curve through kinematic considerations akin to the differential calculus. "
38. Boyer, History of Mathematics, 393.
39. transcripts of Deleuze's seminars, by Richard Pinhas, are available on-line at 〈http://www. webdeleuze. com/sommaire. html〉).
40. commenting on Léon Brunschvicg, Les étapes de la philosophie mathématique (Paris: PUF, 1947; new ed. : Paris: A. Blanchard, 1972).
41. Deleuze also appeals to a text by Michel Chasles, Aperçu historique sur l'origine et le développement de méthodes en géométrie (Brussels: M. Hayez, 1837)
42. which establishes a continuity between Desargues, Monge, and Poncelet as the "founders of a modern geometry" (A Thousand Plateaus, 554, note 28).
43. Gilles Deleuze and Félix Guattari, What is Philosophy? trans. Hugh Tomlinson and Graham Burchell (New York: Columbia University Press, 1994), 128, translation modified.
44. See Brunschvicg, Les étapes de la philosophie mathématique, 327-31.
45. See Carl B. Boyer, The History of the Calculus and Its Conceptual Development (New York: Dover, 1959), 267.
46. Deleuze praises Boyer's book as "the best study of the history of the differential calculus and its modern structural interpretation" (Logic of Sense, 339).
47. For a discussion of the various uses of the term "intuition" in mathematics, see the chapters on "Intuition" and "Four- Dimensional Intuition" in Philip J. Davis and Reuben Hersch, The Mathematical Experience (Boston, Basel, and Stuttgart: Birkhäuser, 1981), 391-405
48. as well as Hans Hahn's classic article "The Crisis in Intuition," in J. R. Newman, ed. , The World of Mathematics (New York: Simon and Schuster, 1956), 1956-76.
49. Boyer, The History of Mathematics, chapter 25, "The Arithmetization of Analysis," 598-619, at 598.
50. Guilio Giorello, "The 'Fine Structure' of Mathematical Revolutions: Metaphysics, Legitimacy, and Rigour," in Revolutions in Mathematics, ed. Donald Gilles (Oxford: Clarendon Press, 1992), 135. I thank Andrew Murphie for this reference.
51. See his article "A Requium for Postmodernism," which is available online at 〈http://mdcm. arts. unsw. edu. au/homepage/StaffPages/ Murphie〉.
52. See Penelope Maddy, Naturalism in Mathematics (Oxford: Oxford University Press, 1997), 51-2, for a discussion of Cantorian "finitism. "
53. For a useful discussion of Weierstrass's "discretization program" (albeit written from the viewpoint of cognitive science), see George Lakoff and Rafael E. Núñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being (New York: Basic Books, 2000), 257-324.
54. Maddy, Naturalism in Mathematics, 28.
55. Reuben Hersh, What is Mathematics, Really? (Oxford: Oxford University Press, 1997), 13.
56. Freeman Dyson, Infinite in All Directions (New York: Harper & Row, 1988), 52-3.
57. John Wheeler, in Frontiers of Time (Austin: Center for Theoretical Physics, University of Texas, 1978), has put forward the stronger thesis that the laws of physics are themselves "mutable" (13).
58. Kurt Gödel, cited in Hao Wang, From Mathematics to Philosophy (New York: Humanities Press, 1974), 86.
59. Hermann Weyl, The Continuum: A Critical Examination of the Foundations of Analysis [1918], trans. Stephen Pollard and Thomas Bole (New York: Dover, 1994), 23-4 (although Weyl still argues for a discrete interpretation of the continuous continuum).
60. Bertrand Russell makes the same point in his Principles of Mathematics (New York: Norton, 1938), 347, citing Poincaré: "The continuum thus conceived [i. e. , arithmetically or discretely] is nothing but a collection of individuals arranged in a certain order, infinite in number, it is true, but external to each other. This is not the ordinary [geometric or "natural"] conception, in which there is supposed to be, between the elements of the continuum, a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point. Of the famous formula, the continuum is a unity in multiplicity, the multiplicity alone subsists, the unity has disappeared" (347).
61. Deleuze, seminar of 29 April 1980.
62. Abraham Robinson, Non-Standard Analysis (Princeton: Princeton University Press, 1966), 83. See also 277: "With the spread of Weierstrass' ideas, arguments involving infinitesimal increments, which survived particularly in differential geometry and in several branches of applied mathematics, began to be taken automatically as a kind of shorthand for corresponding developments by means of the e, d approach. "
63. See Deleuze, Le Pli, 177: "Robinson suggested considering the Leibnizian monad as a infinite number very different from transfinites, as a unit surrounded by a zone of infinitely small [numbers] that reflect the converging series of the world. "
64. Hersch, What is Mathematics, Really? 289.
65. For discussions of Robinson's achievement, see Jim Holt's useful review, "Infinitesimally Yours," in The New York Review of Books, 20 May 1999
66. as well as the chapter on "Nonstandard Analysis" in Davis and Hersch, The Mathematical Experience, 237-54. The latter note that "Robinson has in a sense vindicated the reckless abandon of eighteenth-century mathematics against the straight-laced rigor of the nineteenth-century, adding a new chapter in the never ending war between the finite and the infinite, the continuous and the discrete" (238).
67. Jean Dieudonné, L'Axiomatique dans les mathématiques modernes, 47-8
68. as cited in Robert Blanché, L'Axiomatique, 91.
69. Nicholas Bourbaki, "The Architecture of Mathematics," in Great Currents of Mathematical Thought, ed. François Le Lionnais; trans. R. A. Hall and Howard G. Bergmann (New York: Dover, 1971), 31.
70. Henri Poincaré, "L'oeuvre mathématique de Weierstrass," Acta mathematica 22 (1898-1899), 1-18
71. See Deleuze, Le Pli: Leibniz et le baroque (Paris: Minuit, 1988), 64: "axioms concern problems, and escape demonstration. "
72. This section of the "Treatise on Nomadology" (361-74) develops in detail the distinction between "major" and "minor" science.
73. Deleuze is referring to the distinction between "problem" and "theory" in Georges Canguilhem, On the Normal and the Pathological, trans. Carolyn R. Fawcett (New York: Zone Books, 1978);
74. the distinction between the "problem-element" and the "global synthesis element" in Georges Bouligand, Le déclin des absolus methmatico-logiques (Paris: Éditions d'Enseignement supérieur, 1949);
75. On this score, Deleuze largely follows Lautman's thesis that mathematics participates in a dialectic that points beyond itself to a meta-mathematical power - that is, to a general theory of problems and their ideal synthesis - which accounts for the genesis of mathematics itself. See Albert Lautman, Nouvelles recherches sur la structure dialectique des mathematiques (Paris: Hermann, 1939), particularly the section entitled "The Genesis of Mathematics from the Dialectic": "The order implied by the notion of genesis is no longer of the order of logical reconstruction in mathematics, in the sense that from the initial axioms of a theory flow all the propositions of the theory, for the dialectic is not a part of mathematics, and its notions have no relation to the primitive notions of a theory" (13-14). Despite his occasional appeal to Lautman, Badiou is opposed to this Lautmanian appeal to a meta-mathematical dialectic.
76. Badiou, "Un, Multiple, Multiplicité(s)," 4.
77. Henri Bergson, The Creative Mind, trans. Mabelle L. Andison (Totowa, N. J. : Littlefield, Adams & Co. , 1975), 33. See also 191: "Metaphysics should adopt the generative idea of our mathematics [i. e. , change, or becoming] in order to extend it to all qualities, that is, to reality in general. "
78. See Deleuze, seminar of 29 April 1980.
79. For analyses of Deleuze's theory of multiplicities, see Robin Durie, "Immanence and Difference: Toward a Relational Ontology," in Southern Journal of Philosophy, 60 (2002), 1-29;
80. Keith Ansell-Pearson, Philosophy and the Adventure of the Virtual: Bergson and the Time of Life (London and New York: Routledge, 2002);
81. and Manuel Delanda, Intensive Science and Virtual Philosophy (London: Continuum, 2002).
82. Ian Stewart and Martin Golubitwky, Fearful Symmetry (Oxford: Blackwell, 1992), 42.
83. See Kline, Mathematical Thought, 759: "The group of an equation is a key to its solvability because the group expresses the degree of indistinguishability of the roots. It tells us what we do not know about the roots. "
84. citing C. Georges Verriest, "Evariste Galois et la théorie des équations algébraiques," in Oeuvres mathématiques de Galois (Paris: Gauthier-Villars, 1961), 41.
85. Gilles Deleuze, Essays Critical and Clinical, trans. Daniel W. Smith and Michael A Greco (Minneapolis: University of Minnesota Press, 1997), 149
86. citing a text by Galois in André Dalmas, Evariste Galois (Paris: Fasquelle, 1956), 132.
87. referring to Jules Vuillemin, La philosophie de l'algèbre (Paris: PUF, 1962). "Jules Vuillemin's book proposes a determination of structures [or multiplicities, in Deleuze's sense] in mathematics. In this regard, he insists on the importance of a theory of problems (following the mathematical Abel) and the principles of determination (reciprocal, complete, and progressive determination according to Galois). He shows how structures, in this sense, provide the only means for realizing the ambitions of a true genetic method. "
88. See Gilles Deleuze, "A quoi reconnait-on le structuralisme," in Histoire de la philosophie 8, ed. François Châtelet (Paris: Hachette, 1972-73), 315.
89. Albert Lautman, Essai sur les notions de structure et d'existence in mathématiques, vol. 1: Les schémas de structure; vol. 2: Les schémas de genèse (Paris: Hermann & Co. , 1938).
90. Although Badiou occasionally appeals to Lautman (see Deleuze, 98), his own ontology seems opposed to Lautman's; moreover, Badiou never considers Deleuze's own appropriation of Lautman's theory of differential equations, even though Deleuze cites it in almost every one of his books after 1968.
91. Lautman, Le Problème du temps (Paris: Hermann, 1946), 41-3; and Deleuze's seminar of 29 April 1980. Such singularities are now termed "attractors": using the language of physics, attractors govern "basins of attraction" that define the trajectories of the curves that fall within their "sphere of influence. "
92. For this reason, Deleuze's work has been seen to anticipate certain developments in complexity theory and chaos theory. Delanda in particular has emphasized this link in Intensive Science and Virtual Philosophy (see note 78).
93. For a presentation of the mathematics of chaos theory, see Ian Stewart, Does God Place Dice?: The Mathematics of Chaos (London: Blackwell, 1989), 95-144.
94. See Lautman, Essai, 43: "The constitution, by Gauss and Riemann, of a differential geometry that studies the intrinsic properties of a variety, independent of any space into which this variety would be plunged, eliminates any reference to a universal container or to a center of privileged coordinates. "
95. See Delanda, Intensive Science and Virtual Philosophy, 15 (on attractors), and chapters 2 and 3 (on symmetry-breaking cascades).
96. Delanda, Intensive Science and Virtual Philosophy, 102.
97. This conflation is stated most clearly in Badiou, Deleuze, 46:
98. "the univocal sovereignty of the One. " For discussions of Badiou's reading of the doctrine of univocity, see Nathan Widder, "The Rights of Simulacra: Deleuze and the Univocity of Being," in Continental Philosophy Review 34 (2001): 437-53
99. Deleuze and Guattari, What is Philosophy?, 35, 202-03.
100. See, for instance, Badiou, "Un, Multiple, Multiplicité(s), " 3: The One "can be called the Whole, Substance, Life, the Body without Organs, or Chaos. "
101. Ernst Mayr, "Is Biology an Autonomous Science?" in Toward a New Philosophy of Biology: Observations of an Evolutionist (Cambridge, Mass. : Harvard University Press, 1988), 8-23.
102. Deleuze, seminar of 22 April 1980. See also the seminar of 29 April 1980: "Everyone agrees on the irreducibility of differential signs to any mathematical reality, that is to say, to geometrical, arithmetical, and algebraic reality. The difference arises when some people think, as a consequence, that differential calculus is only a convention - a rather suspect one - and others, on the contrary, think that its artificial character in relation to mathematical reality allows it to be adequate to certain aspects of physical reality. "
103. Deleuze, Abécédaire, "H as in 'History of Philosophy" (overview by Charles J. Stivale available on-line at 〈http://www. langlab. wayne. edu/Romance/FreDeleuze. html〉. )
104. See Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth (New York: Hyperion, 1998), 17.
105. See Deleuze, Negotiations, 130: "Poincaré used to say that many mathematical theories are completely irrelevant, pointless. He didn't say they were wrong - that wouldn't have been so bad. "
106. Deleuze, seminars of 14 March 1978 and 21 March 1978. "The abstract is lived experience. I would almost say that once you have reached lived experience, you reach the most fully living core of the abstract. . . . You can live nothing but the abstract and nobody has lived anything else but the abstract. "
107. Deleuze, Bergsonism, 14.
108. Badiou's claim that Deleuze's methodology relies on intuition is discussed in Deleuze: The Clamor of Being, chapter 3, 31-40.
109. For the role of the scholia, see Gilles Deleuze, Expressionism in Philosophy: Spinoza, trans. Martin Joughin (New York: Zone Books, 1992), 342-50 (the appendix on the scholia);
110. for the uniqueness of the fifth book, see "Spinoza and the Three Ethics," in Essays Critical and Clinical, esp. 149-50.
111. See Badiou's essay on Spinoza, "L'ontologie fermée de Spinoza," in Court Traité, 73-93.
112. See also Hersh's comments on Descartes in What is Mathematics, Really? 112-13: "Euclidean certainty boldly advertised in the Method and shamelessly ditched in the Geometry. "
113. Deleuze, seminar of 22 February 1972: "The true axiomatic is social and not scientific. . . . The scientific axiomatic is only one of the means by which the fluxes of science, the fluxes of knowledge, are guarded and taken up by the capitalist machine. . . . All axiomatics are means of leading science to the capitalist market. All axiomatics are abstract Oedipal formations. "
114. And Anti-Oedipus, 255: "The theoretical opposition lies elsewhere: it is between, on the one hand the decoded flows that enter into a class axiomatic on the full body of capital, and on the other hand, the decoded flows that free themselves from this axiomatic. "
115. Deleuze and Guattari, What is Philosophy? 152.
116. Deleuze, Spinoza, Practical Philosophy, 128.