by Daniel Smith
20.
Edmund Husserl, Ideas: General Introduction to a Pure Phenomenology, trans. W. R. Boyce Gibson (New York: Macmillan, 1931), §74, 208. See also Edmund Husserl's Origin of Geometry: An Introduction, ed. John P. Leavey, Jr. and David B. Allison (Stony Brook, NY: H. Hayes, 1978), which includes Jacques Derrida's 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.
21.
DR 160. 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).
22.
The language of the rectilinear dominates ethics as well: to “rectify” a wrong, to “straighten” someone out, to make a situation “right”; the French term droit means both “straight,” in the geometric sense, and “right,” in the legal sense; an angle droit is a “right” angle; a moral person is someone who is “upright”; the wrong is a deviation from the “straight and narrow” (the line).
23.
See DR 174:
The mathematician Houël remarked that the shortest distance was not a Euclidean notion at all, but an Archimedean one, more physical than mathematical; that it was inseparable from a method of exhaustion; and that it served less to determine the straight line than to determine the length of a curve by means of a straight line—“integral calculus performed unknowingly” (citing Jules Houël, Essai critique sur les principes fondamentaux de la géométrie élémentaire [Paris: Gauthier-Villars, 1867], 3, 75)
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.
24.
Badiou, “Deleuze's Vitalist Ontology,” in Briefings on Existence, 70–1.
25.
Boyer, History of Mathematics, 393.
26.
TP 484. On the relation between Greek theorematics and seventeenth-century algebra and arithmetic as instances of “major” mathematics, see Deleuze, DR 160–1.
27.
Boyer, History of Mathematics, 394. Deleuze writes that “Cartesian coordinates appear to me to be an attempt of reterritorialization” (22 Feb 1972).
28.
TP 554 n23, commenting on Léon Brunschvicg, Les Étapes de la philosophie mathématique (Paris: PUF, 1947; new edn.: Paris: A. Blanchard, 1972). 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), which establishes a continuity between Desargues, Monge, and Poncelet as the “founders of a modern geometry” (TP 554 n28).
29.
See Brunschvicg, Les Étapes de la philosophie mathématique, 327–31.
30.
See Carl B. Boyer, The History of the Calculus and its Conceptual Development (New York: Dover, 1959), 267. Deleuze praises Boyer's book as “the best study of the history of the differential calculus and its modern structural interpretation” (LS 339).
31.
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 Hersh, The Mathematical Experience (Boston, Basel, and Stuttgart: Birkhäuser, 1981), 391–405, as well as Hans Hahn's classic article “The Crisis in Intuition,” in J. R. Newman, ed., The World of Mathematics (New York: Simon & Schuster, 1956), 1956–76.
32.
Boyer, The History of Mathematics, 598 (in the chapter on “The Arithmetization of Analysis”).
33.
Giulio Giorello, “The ‘Fine Structure’ of Mathematical Revolutions: Metaphysics, Legitimacy, and Rigour,” in Revolutions in Mathematics, ed. Donald Gilles (Oxford: Clarendon, 1992), 135. I thank Andrew Murphie for this reference.
34.
22 Feb 1972. See also DR 172: “The limit no longer presupposes the ideas of a continuous variable and infinite approximation. On the contrary, the notion of limit grounds a new, static and purely ideal definition of continuity, while its own definition implies no more than number.”
35.
See Penelope Maddy, Naturalism in Mathematics (Oxford: Oxford University Press, 1997), 51–2, for a discussion of Cantorian “finitism.”
36.
Deleuze provides a summary of these developments in DR 176:
The real frontier defining modern mathematics lies not in the calculus itself but in other discoveries such as set theory which, even though it requires, for its own part, an axiom of infinity, gives a no less strictly finite interpretation of the calculus. We know in effect that the notion of limit has lost its phoronomic character and involves only static considerations; that variability has ceased to represent a progression through all the values of an interval and come to mean only the disjunctive assumption of one value within that interval; that the derivative and the integral have become ordinal rather than quantitative concepts; and finally that the differential designates only a magnitude left undetermined so that it can be made smaller than a given number as required. The birth of structuralism at this point coincides with the death of any genetic or dynamic ambitions of the calculus.
37.
For a discussion of Weierstrass's “discretization program” (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, 2000), 257–324.
38.
Maddy, Naturalism in Mathematics, 28.
39.
Reuben Hersh, What is Mathematics, Really? (Oxford: Oxford University Press, 1997), 13.
40.
Badiou, Deleuze, 47.
41.
Freeman Dyson, Infinite in All Directions (New York: Harper & Row, 1988), 52–3. 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).
42.
Kurt Gödel, cited in Hao Wang, From Mathematics to Philosophy (New York: Humanities Press, 1974), 86.
43.
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). Bertrand Russell makes the same point in his Principles of Mathematics (New York: Norton, 1938), 347, citing Poincaré:
The continuum thus conceived [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.
44.
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 differenti
al 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.
45.
See FLB 129–30: “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.”
46.
Hersh, What is Mathematics, Really?, 289. For discussions of Robinson's achievement, see Jim Holt's useful review, “Infinitesimally Yours,” in The New York Review of Books, 20 May 1999, as well as the chapter on “Nonstandard Analysis” in Davis and Hersh, 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)
47.
Albert Lautman, Mathematics, Ideas, and the Physical Real, trans. Simon Duffy (London: Continuum, 2011), 88.
48.
Jean Dieudonné, L'Axiomatique dans les mathématiques modernes, 47–8, as cited in Robert Blanché, L'Axiomatique (Paris: PUF, 1955), 91.
49.
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. Bourbaki none the less insists—as do Deleuze and Guattari—that the analogy is not a precise one: mathematicians do not work mechanically as do workers on an assembly line, since “intuition” plays a fundamental role in their research.
This is not the intuition of common sense [explains Bourbaki], but rather a sort of direct divination (prior to all reasoning) of the normal behavior he has the right to expect from the mathematical entities which a long association has rendered as familiar to him as the object of the real world. (31)
Deleuze and Guattari make a similar point in AO 251.
50.
See 22 Feb 1972:
The idea of a scientific task that no longer passes through codes but rather through an axiomatic first took place in mathematics toward the end of the nineteenth century … One finds this well formed only in the capitalism of the nineteenth century.
Deleuze's political philosophy is itself based in part on the axiomatic-problematic distinction: “Our use of the word ‘axiomatic’ is far from a metaphor; we find literally the same theoretical problems that are posed by the models in an axiomatic repeated in relation to the State” (TP 455).
51.
TP 362. See also TP 141–2: “The phrase ‘politics of science’ is a good phrase for these currents, which are internal to science, and not simply circumstances and state factors that act upon it from the outside.”
52.
Henri Poincaré, “L’œuvre mathématique de Weierstrass,” Acta Mathematica 22 (1898–9), 1–18, as cited in Boyer, History of Mathematics, 601. Boyer notes that one finds in Riemann “a strongly intuitive and geometrical background in analysis that contrasts sharply with the arithmetizing tendencies of the Weierstrassian school” (601).
53.
See FLB 48: “axioms concern problems, and escape demonstration.”
54.
TP 361. This section of the “Treatise on Nomadology” (361–74) develops in detail the distinction between “major” and “minor” science.
55.
DR 323 n22. 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, 1978); the distinction between the “problem-element” and the “global synthesis element” in Georges Bouligand, Le Déclin des absolus mathématico-logiques (Paris: Éditions d'Enseignement Supérieur, 1949); and the distinction between “problem” and “solution” in Albert Lautman. All these thinkers insist on the double irreducibility of problems: problems should not be evaluated extrinsically in terms of their “solvability” (the philosophical illusion), nor should problems be envisioned merely as the conflict between two opposing or contradictory propositions (the natural illusion) (DR 161). 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 mathématiques (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)
Badiou frequently appeals to Lautman's name, but rarely (if ever) to his works, and is opposed to Lautman's appeal to a meta-mathematical dialectic.
56.
Badiou, “One, Multiple, Multiplicities,” in Theoretical Writings, 72.
57.
DR 161. See also DR 177–8: “If the differential disappears in the result, this is to the extent that the problem-instance differs in kind from the solution-instance.”
58.
Henri Bergson, The Creative Mind, trans. Mabelle L. Andison (Totowa, NJ: Littlefield, Adams, 1946), 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.”
59.
DR 179. See also D ix: “It seems to us that the highest objective of science, mathematics, and physics is multiplicity, and that both set theory and the theory of spaces is still in its infancy.”
60.
For analyses of Deleuze's theory of multiplicities, see Robin Durie, “Immanence and Difference: Toward a Relational Ontology,” in Southern Journal of Philosophy, Vol. 60 (2002), 1–29; Keith Ansell-Pearson, Philosophy and the Adventure of the Virtual: Bergson and the Time of Life (London and New York: Routledge, 2002); and Manuel De Landa, Intensive Science and Virtual Philosophy (London: Continuum, 2002).
61.
Ian Stewart and Martin Golubitwky, Fearful Symmetry (Oxford: Blackwell, 1992), 42.
62.
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.”
63.
DR 180, citing C. Georges Verriest, “Évariste Galois et la théorie des équations algébriques,” in Œuvres mathématiques de Galois (Paris: Gauthier-Villars, 1961), 41.
64.
ECC 149, citing a text by Galois in André Dalmas, Évariste Galois (Paris: Fasquelle, 1956), 132.
65.
DR 170, 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. (DI 306 n26)
66.
Albert Lautman, “Essay on the Notions of Structure and Existence in Mathematics,” in Albert Lautman, Mathematics, Ideas, and the Physical Real, trans. Simon Duffy (London : Continuum, 2011), 87–193. In this important volume, Duffy has made available to English-speaking readers almost the entirety of Lautman's work in the philosophy of mathematics. Although Badiou occasionally appeals to Lautman (see Deleuze, 98), his own ontology is largely oppo
sed 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.
67.
For discussions of Poincaré, see 29 Apr 1980, as well as Kline, Mathematical Thought, 732–8 and Lautman, Mathematics, Ideas, and the Physical Real, 259. 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.”
68.
For this reason, Deleuze's work has been seen to anticipate certain developments in complexity theory and chaos theory. De Landa in particular has emphasized this link in Intensive Science and Virtual Philosophy (London: Continuum, 2002). 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.
69.
See Lautman, Mathematics, Ideas, and the Physical Real, 112:
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.