<<TRANSMATHOME

Forces and Deductions

‘Un espace dynamique doit être défini du point de vue d’un observateur lié à cet espace, et non d’une position extérieure.

(A dynamic space must be defined from the point of view of an observer tied to that space, not from an external position)’.[1]

In chapters one and two, Deleuze’s identification of the network of faculties as constitutive of the transcendental method was explored, together with his attack on the principles of recognition and the image of thought in Kant’s critique. His relation with Kant operates on (at least) two faces simultaneously. At the systematic level he explores connections, functions and operations amongst the faculties; questioning the repetition of the model of common sense as a mechanism for the stabilization of these relations, and its complementary, good sense, which is the common sense of teleology, Deleuze begins to expose his real relation with Kant. As the last chapter remarked, this is firstly and foremostly positive; Deleuze does not destroy without utilizing the components he has disarticulated to build new machines, and this is the second aspect of his employment of Kant. He occupies a space, and then re-distributes it, from the inside, not from the position of an external observer.

If one wanted to describe a method in this aspect of Deleuze’s engagement with critique, it would be one of selection and connection; intensity is connected with ideas, and dialectics is re-distributed as a problem of real differences of magnitude; thought is connected with sense, removing the former from the rule of concepts and identity, and relating it with the now objective problematic of ideas; the thing-in-itself is connected with difference, with that through which the given is given as diversity. What forces thought is discovered in sense, rather than in the illusory figures of possibility, recognition, generality and the image.

A question which emerges from the re-wiring of the system of faculties is that of forces. Kant doesn’t explain the relations of forces in the first Critique in the distribution of intensive magnitudes; the real moment of a cause, as has been mentioned before,  is simply described as gravity, allowing force to be conceptualized in relation to substance. Mille Plateaux, in a discussion of a difference between nomad and royal science, opposes two models; that of the Compars, whose primary distinction is a hylomorphic one, between matter and form, constructed through the selection of constants and law, and that of the Dispars, the relevant distinction of which is ‘matériau-forces (material-forces)’ which compose themselves by ‘mettre les variables elles-même en état de variation continue (placing the variables themselves in a state of continuous variation)’.[2] Each model is characterized by different distributions; the Compars by logos, which divides ‘un espace laminaire, strié, homogène et centré (a laminar, striated, homogeneous, and centred space)’[3] and presupposes gravity, and Dispars by nomos, a tactile space of contact and affects, which ‘n’a-t-il d’homogénéité qu’entre points infiniment voisins (has no homogeneity, except between infinitely proximate points)’[4] yet which is not differentiated by pre-formed relations and connections.

The most obvious model of science in Kant is that of the Compars: he is famously an admirer of Newton.  However, Deleuze’s relation to Kant and his deployment of critique as essentially economic suggest that the other model must also be implicated in Kantian critique. In Difference et Répetition, L’anti-oedipe and Mille Plateaux, a continual emphasis is placed on the co-existence of the two models: the material-forces distinction does not replace the matter-form distinction anymore than the State replaces the nomads. ‘L’histoire ne fait que traduire en succession une coexistence de devenirs (all history does is to translate a coexistence of becomings into a succession)’[5]. A purely historical perspective, leading to an evaluation of Kant’s work in terms of what came before and followed after him, and the division of his philosophy as a whole into pre-critical and critical (and, in some cases, as post-critical and senile) writings, will discover only this translation. Deleuze looks instead for consistencies in the system, and for the weightings and privileges attendant on certain structures which repress or cover becomings, and translate them into chronological movements. His interests lies in critique as a singular and economic problem, rather than in the successive attempts at solving this problem which run throughout Kant’s work and are continued by his successors.

For Deleuze, the machinic elements of critique are in its systematics, hidden in the theory of forces, in the problem of Ideas, and in the network of the faculties, and it is on these that he focuses. His critique does not progress from Kant, but rather abstracts out the various machines operating in his work, allowing forces and patterns hidden beneath and covered over by royal and state divisions of space and operations of power to be exposed. The régimes of molar and molecular (which for the moment can be taken to correspond roughly with the division of Compars and Dispars) are immanent to each other; what differs - as Kant always says - is not the ideas themselves, but the use to which they are put. Royal science deploys ideas reproductively: reduce something to a unit and make more of the same. Nomad science follows ideas, and an idea which Deleuze follows in Kant is that of repulsive and attractive force, in order to uncover further the conditions of real production.

In the Metaphysical Foundations of Natural Science, Kant discusses fundamental qualities of material forces - repulsion and attraction (Zurückstoßungskraft/repulsive Kraft and Anziehungskraft). The latter functions in empty extended space, as action at a distance, and is constructed through negation and limitation; it belongs to the Compars model. The former is a force which, like nomos, distributes a space of contact, intensive magnitudes filling space without determinate measure and belongs to the Dispars. In this chapter, Deleuze’s method of deduction is explored, which functions simultaneously with his selection of the system of faculties as the real elements which constitute the problem of critique as such. But first, forces.

I           Attraction and Repulsion

According to Kant, ‘the only two moving forces that can be thought’ [6], and which are fundamental to matter, are repulsion and attraction. These are differentiated in a variety of ways:

Repulsion

‘[B]y means of the sense of feeling [Sinnes des Gefühls]’ repulsive forces provide ‘the size and shape [Größe und Gestalt] of an extended thing’[7]. Their magnitude is aesthetic and intensive, and contact amongst repelling forces is physical and immediate; there is ‘no actual distance of parts, which always constitute a continuum as regards all expansion of the space of the whole’ [8]. That empty space could not be proved through experience was made clear in the first Critique, in relation to both the Anticipations of Perception and the infinite divisibility of intensive magnitudes, and at length in relation to regulative judgement, which is governed subjectively by three logical maxims asserting the continuity of nature; the absence of a vacuum - non datur vacuum formarum; the impossibility of leaps in nature, transitions between species comprehending ‘all the smaller degrees of difference that mediate between them’[9] - datur continuum formarum: and the law of their conjunction, continuum specierum, which  ‘recognise[s] a relationship of the different branches, as all springing from the same stem’ [10]. Homogeneity and specification are thus joined in an arborescent form, leading to a problem of roots, and what grounds them. This logics of continuity presupposes a transcendental law, ‘lex continui in natura’[11]. Kant is at pains to avoid the suggestion that attraction at a distance is across a real empty space, or that  variations amongst species correspond to real gaps.    

In the MFNS, Kant’s concern is with an intensive continuum of force, and with the possibilities of constructing a concept of  full space which will give material weight to the law of continuity in nature and support his claim that nature knows no vacuums. The distributions of bodies considered as magnitudes expressing intensive qualities are not determinate, in relation either to themselves or to a geometric boundary; prior to the construction of the concept of quantity the most one can say is that there are regions of density and patterns of flows. In the Aesthetic Kant says that removal from the representation of a body of  those aspects belonging to sensation leaves extension and figure; however, there is nothing  in this remark that determines the nature and configuration of such figures, since determination is a function of understanding, and mathematical objects exist through construction in pure intuition, which requires productive imagination. There is thus no weight to the claim that Kant can be refuted by the existence of non-Euclidean geometries, or by forms of non-rational mathematics, since there is no grammar, as it were, to the pure forms of space and time. They underlie the possibility of geometry and mathematics, but the process of numbering is prior to the concept of a number, just as the figuration of spatial bodies is prior to geometric axioms. Kant’s elucidation of intuition is confined to a three-dimensional space and the one-dimensional line of time is axiomatically extensive and producing through the successive addition of units. But the forms of intuition themselves are empty.  However, this is a complex issue and outside the scope of this thesis, so will not be pursued. It must, however, be kept in mind that pure intuition is vacuous and that the construction of spatio-temporal figures concerns the relation of imagination to intuition, so is implicated with the functioning of the former, a topic for the next chapter, and that intuition itself does not contain pre-given restraints on the potential for such constructions. Deleuze exploits this to the full.

As the above list notes, repulsive forces are not limited by space; they have no exhaustive extension, but become infinitely diffuse, until ‘no assignable quantity of matter would be found in any assignable space.’[12] Repulsive force alone, therefore, gives no concept of the dynamic magnitude of a body; no concept of quantity is constructible from the diffuse indeterminacy of intensive magnitude; space is full but not denumerable, occupied without measure.

In Kant, this distribution is subjected to negation; the multiplicitous difference of degree is ‘represented...through approximation to negation = 0’[13], and intensity becomes thought of as a unit, representable as a point, the real moment of cause. The actual intensive continuum filling space can then be mathematically conceived as a uniformly homogeneous field of points, all interconnected with each other, no part more distant than any other, because their relation is intensive, rather than extensive. Through the medium of the point, repulsive force can conceived of in relation to a uniform and undifferentiated mathematical continuum, extending to infinity. 

If, as Kant desires, repulsion is to become the basis of the movable, the topic of mechanics, then it cannot itself be thought of as mobile, just as, for time to be the form of everything which changes, it cannot itself change. The immanent dynamics of repulsive force have to be distributed uniformly, which means they have to be recorded in a manner different to their production, because the mechanical field into which they are to be folded is based on a principle of a unity of force, whilst intensity is immanently differentiated. The homogenization of force is the first move in this recording process, and forms the concept of substance.

In relation to substance, force is determinately defined, as a state of matter, rather than as an intensive vector. In the latter case, the given as diversity and that by which the given is given as diverse are immanently entangled, rather than subordinated to the principle of an ‘ultimate subject of existence’ and there is neither assignable origin nor end to the vector.[14] Empirical time and space are constructed through the movement of forces, rather than through reference to axiomatized quanta. Difference is thus virtual, or immanent to the actual continuum, and rather than being subjected to limitation, changes in nature as it changes in degree. It has been said that space cannot limit repulsive force. Nor, however, is the filling of space self-limiting: immanently repulsive, matter is ‘compelled...to continuously expand’ its occupation of space.[15] In relation to substance, this expansion is necessarily a relation with unity, and thus of determinate and measurable extension. In relation only to itself, intensive expansion is not defined or conceivable, and hence becomes a problematic, rather than a theoretical issue. Deleuze, by critiquing the formation and assumptions of common sense and setting a different model of science against that of universal gravitation, opens up this space, and focuses on the problem of how repulsion is first set up (a matter addressed later in this chapter).

Repulsive force is a force of surfaces: every part touches every part, and there is no empty space: ‘physical contact is the reciprocal action of repulsive forces at the common boundary of two matters’ [16]. But the boundary is only common in the sense that two matters are infinitely proximate, for it immanent to the field of each. It is not common in the sense that both matters share a law which their relation instantiates nor is it formed through the subordination by one matter of another. The boundary or limit is not governed a priori by any element implicated in its formation, but produced as an effect of the relation of forces at different intensive degrees: it is common only in the sense that it is a difference common to all distributions. Contact, Kant says, is differential, a problem of ‘infinitely small distances’ [17].

The model here is not hylomorphic, since the dynamic filling of space by repulsive forces is materially distributive, but there is as yet no differentiation of matter and form. For this, contact has to be referred to a limit, a point: it is the same problem as that of sensation, noted above, where the subjective indeterminacy of sensation appeared to void the possibility of objectively measuring intensive magnitude. ‘La géométrie et l’arithmétique prennent la puissance d’un scalpel (Geometry and arithmetic take on the power of the scalpel)’[18], Deleuze and Guattari say in Mille Plateaux, and it is this function which orders the homogeneous and dead space across which the true force of attraction drops bodies and draws lines.

Attraction

Unlike repulsive forces, which fill space by means of the sense of feeling [Gefühl], (and thus are in the sphere of aesthetic, rather than speculative or practical judgement) attractive force is characterized as ambivalent in relation to sensation [Empfindung]: either there is ‘no sensation at all’ or there is sensation, but no determinate object, and it is this that makes it appear at first problematic as a fundamental force, since no determinate quanta of intuition can be correlated with the spread or absence of sensation. There is either zero sensation of intensity, which as Kant says ‘would involve the representation of the instant as empty, therefore = 0’, and repulsive forces necessitate the impossibility of this.[19] Or there is sensation but no determinate intensive magnitude, or ‘degree of influence on the sense’ which would validate the objectivity of sensation, attributing to it an objective cause.[20] So attractive force becomes open to the accusation of having only subjective validity, and of functioning in a space with no real dynamic qualities.

No positive concept of real attractive force can be constructed: it is inferred, Kant says, but not derived, and on the basis of the possibility of a general concept of matter, so its positivity is not real but conceived. Independently of repulsion, attraction becomes purely mathematical: if there were only attractive forces, the parts of matter would ‘coalesce in a mathematical point’ in empty space.[21] As Kant says, mathematics ‘presents the most splendid example of the successful extension of pure reason, without the help of experience’, and it is through mathematics that the heterogeneous involution and division of repulsive forces becomes tied to points of attraction, becoming uniform and inert.[22] Synthetic, and thus productive, but a priori and thus merely possible in relation to the real, Kant’s mathematics grounds the royal description of space, as ‘strié par la chute des corps, les verticales de pesanteur (striated by the fall of bodies, the verticals of gravity’.[23]

Space is inverted through the negation of dynamic intensities, and the real is sucked through an impenetrable point, its sign inverted. Its depth becomes empty, voided of continuously differentiated degrees of intensity and re-distributed as homogeneous, parallel, Euclidean,  inertly receptive to the mechanical principles of order: the shift is from feeling to sight, from an intensive distribution to a determinate vision, from a real space unobservable from outside to an ideal space only observable from outside. There is a cancellation of indeterminate sensations in favour of a split sensibility according to a difference imposed from outside, by understanding, rather than one which emerges from intensive magnitudes.

The force of attraction is defined in terms of the action of points at a distance, ‘through every space as an empty space’[24], but only two bodies at a time defining, as Deleuze and Guattari say, ‘la forme d’intériorité de toute science (the form of interiority of all science)’.[25] In the construction of this form real magnitudes are assigned a negative value, and space is covered over with extensive lines, pillars of force, giving rise to a third form of compression, resulting from the relation of repulsive and attractive forces, which establishes a direction to flows of force, distributing a before and an after of time in relation to which a before and an after of intensive distribution can be determined. The point becomes a present, but a vacant one, which defines a direction of time ‘du passé au futur, comme du particulier au général (from past to future as though from particular to general)’, from the determinate point or state of matter, the moment of gravity, to the homogeneous chaos of uniformly dispersed intensity.[26] Kant’s dynamics are thus not commensurate only with mechanics, but will also support a thermodynamics of good sense. The ‘thèmes d’une réduction de la différence, d’une uniformisation du diverse, d’une égalisation de l’inégal (themes of a reduction of difference, a uniformisation of diversity, and an equalisation of inequality)’, fused in thermodynamics, established basic definitions satisfying, Deleuze writes, ‘tout le monde, y compris à un certain kantisme (everybody, including a certain Kantianism)’.[27] Deleuze, however, finds a third relation, which conjuncts the purpose-driven directionality of force proper to teleology and thermodynamics (the force of good sense) and the determinate conceptualization of force as the moment of gravity (the force of common sense). This conjunction drives critique across the thresholds of rational ends and towards machinic or auto-critique, which is not principled by unity but according to a principle of difference: given nothing but difference there is nothing in common but there is still difference.

Extension is the cancellation and covering up of intensities, their incorporation into an mechanical common sense and eschatological good sense, which organizes things ‘dans les conditions de l’étendue et dans l’ordre du temps (in the order of time and under the conditions of extensity)’ so that difference is encouraged to cancel itself, as time becomes subject to logic and material forces become hylomorphically arranged.[28] The final moment in constructing a dynamic concept of matter, a substance commensurate with mechanical expression, is limitation, which defines and confirms the degree of negation necessary to generate a universal and permanently uniform containment of repulsion by a point of attraction, and form a general concept of matter.

Attractive force splits into true and apparent. Attraction is apparent when the combined force of two bodies is not biunivocal, and their approach is not intensively symmetrical: one body ‘has been driven [getrieben] toward the first body from elsewhere by impact’.[29] But impact is an empirical and derivative concept of force, rather than a fundamental property, and so includes an admixture of elements, both empirical and a priori. Yet although it results from  physical contact rather than being a function of the relation across empty space of the bodies involved, the effects of impact can, given a generalized concept of matter and a science of forces, be anticipated a priori. Apparent attraction is the negative of repulsive force and ‘proper object of our external perception’[30]; in order to discover the true attraction at its basis, the mass of a body must be understood in terms of a point at its centre, and the relation of forces understood as constant for all variables. True attraction, Kant says, is estimated without the intervention of repulsive force or the need to accommodate intensive variations, and it is in this, its true and mathematical sense, that attraction is the ground of possibility of matter as matter in general.

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