equilavent but separate: the role of mathematics in deleuze and guattari

7/30/2019 Equilavent but Separate: The Role of Mathematics in Deleuze and Guattari

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Jody Collins

LIT255

4/28/2012

Equivalent but Separate: The Role of Mathematics in Deleuze and Guattari

This paper explores the ways that mathematics functions as the

organizing mechanism the three chaoids (philosophy, science, and art) as

presented in What is Philosophy? by Gilles Deleuze and Flix Guattari. Ipostulate that philosophy can be correlated with nomadic science, with

problematics, and thus creates planes of immanence out of chaos, while

science correlates with royal science, axiomatics, and creates planes of

reference. Art begins with axiomatics, then covers it with problematics,

creating a new dimensional space: the plane of composition.

Further, I propose that phenomenal reality is created out of the

incessant shuttle of translation between the three planes by the mind/brain,

either through a top-down or a bottom-up approach. Neither approach is

reciprocal, since the starting point modifies what and were the end point will

be. The three chaoids are equivalent, but not reducible one to another. This

paper will focus less on the specific characteristics of the three chaoids as it

will inquire into the connective links between them, illuminating that which

separates the chaoids into distinct parts and at the same time allows them to


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interact with one another in the arena of thought. The mechanism by which

the chaoids move and create order out of chaos is mathematics, not solely in

its common use on manipulating numbers, but also in its linguistic function.

The primary tension in What is Philosophy? is between philosophy and

science, between concept and function. The difference between the two can

be found as the difference between the two types of mathematics as

explained by Daniel W. Smith:

. . .the ontology of mathematics is not reducible to axiomatics, but

must be understood much more broadly in terms of the complex

tension between axiomatics and what he [Deleuze] calls

problematics. Deleuze assimilates axiomatics to major or royal

science, . . . which constantly attempts to effect a reduction or

repression of the problematic pole of mathematics, itself wedded to a

minor or nomadic conception of science. (412)

Thus, what Deleuze and Guattari refer to as science is not so much science

as used in common parlance, but rather royal science, axiomatics or

theorematics, which offsets a philosophy which corresponds with the

nomadic science, problematics. The two poles of mathematics can never

be reconciled, since they approach chaos from utterly opposite ways.

Axiomatics, or science, relinquishes the infinite, infinite speed, in order to

gain a reference able to actualize the virtual (Deleuze and Guattari 118,

italics in original). Problematics, or philosophy, on the other hand, retains the


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infinite and gives consistency to the virtual through concepts (ibid ). Smith

concurs, saying: In the minor geometry of problematics, figures are

inseparable from their inherent variations, affections, and events; the aim of

major theorematics, by contrast, is to uproot variables from their state of

continuous variation in order to extract from them fixed points and constant

relations (416). Problematics deals in change, while science desires to slow

change down so that points of reference may be calculated. Since the two

approaches have different methods and goals with regard to chaos, to

change, they can never parallel each other, though they are capable of

intersecting, interacting, and influencing each other.

The interaction between problematics and axiomatics is enacted upon

the field of mathematic logic, what Simon Duffy terms the logic of the

calculus of problems (565). It is not calculus as such, nor even

mathematical problems per se, which are at stake here, but rather theinteractions within calculus which can be imposed upon the interaction of the

chaoids in What is Philosophy? . Duffy continues to say that this logic is not

simply characteristic of the relative difference between Royal and nomadic

science . . . It is rather characteristic of the very logic of the generation of

each mathematical problematic itself (ibid). While it is clear that there are

inherent differences between philosophy and science, between problematics

and axiomatics, it is not their differences with which this calculus is primarily

concerned. It focuses, instead, with the creation of the actual out of the


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virtual, and on the inevitable reversion of the actual to the virtual. It is

concerned, in short, with creation from chaos.

A brief explanation of chaos as conceptualized by Deleuze and Guattari

here is useful, since the nature of chaos can by no means be assumed. In

What is Philosophy? they state:

Chaos is defined not so much by its disorder as by the infinite speed

with which every form taking shape in it vanishes. It is a void that is

not a nothingness but a virtual , containing all possible particles and

drawing out all possible forms, which spring up only to disappear

immediately, without consistency or reference, without consequence.

Chaos is an infinite speed of birth and disappearance. (118, italics in

original)

Hence, what characterizes chaos is infinite speed and infinite possibility,

pure change. According to Deleuze and Guattari, chaos has three

daughters, depending on the plane that cuts through it: these are the

Chaoids art, science, and philosophy as forms of thought or creation

(208, italics in original). The chaoids, for all that they provide us with ordered

reality, are not easily controlled: Philosophy, science, and art want us to

tear open the firmament and plunge into the chaos (202). Mathematics is

what keeps the chaoids from dissolving our thoughts in the chaos of reality,

by placing a system of checks and balances upon their operations and by

providing them a mechanism with which to move from chaos to order.


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This is why Deleuze and Guattari specifically give the term chaoids

to philosophy, science, and art, because the three desire chaos, are born of

chaos, and yet they enter chaos to bring back order, manipulate the fabric of

potential to create the actual: They are the three planes, the rafts on which

the brain plunges into and confronts chaos (ibid 210). All three chaoids

derive meaning and order from chaos in different ways, as mandated by the

mathematic system which models them. It is the virtual which the chaoids

seek to express: by keeping infinite speed in the form of absolute survey to

give the virtual consistency, by actualizing the virtual through limiting the

infinite, or by manipulating the limits of the actual to extract the sensations

of the virtual philosophy, science, and art.

When confronting chaos, the virtual which the chaoids seek to act

upon is called the event. The event holds within it both a problem and a

solution, and as all possibilities are contained within chaos, so too is themultiplicity of events drawn from chaotic possibility. As Bent Sorensen says:

The problem appears as a real multiplicity by being produced as a

problematic; the problem becomes an event (125-126). Thus, what first

approaches and captures the event is philosophy as problematics. Deleuze

and Guattari concur, saying: It is a concept that apprehends the event, its

becoming, its inseparable variations (158, italics in original). The concept, a

purely philosophical tool, first grasps the problem that is the event within

chaos, and with the problem it must grasp the solution, since the two are

twinned. Sorenson continues to say that at the same time as the solution is


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inscribed in the actual event of the problem, the relevant problem to which it

is a solution, must be counteractualized into its virtual phase, in a perpetual

state of becoming, that is, becoming a ctualized (126, italics in original). The

problem as event is always already virtual, and any solution to the problem,

be it in the form of concept, function, or aesthetic construction, necessarily

presupposes the actualized problem and re-inscribes the problem in the

virtual. This is why events are always becoming and never are.

An event cannot be actualized without retaining recourse to the

infinite, the virtual. Even in the actualization of an event through states of

affairs, the function in science, the event maintains a virtuality which eludes

capture by functions. As Deleuze and Guattari put it: No doubt, the event is

not only made up from inseparable variations, it is itself inseparable from the

state of affairs, bodies, and lived reality in which it is actualized or brought

about. But we can also say the converse: the state of affairs is no moreseparable from the event that nonetheless goes beyond its actualization in

every respect (159). Philosophy may speak the event, but science

actualizes the event through states of affairs, and both are necessary for the

event to be drawn from chaos (ibid 21). In its need to actualize the event,

however, science is actually attempting to destroy the event, to sever it from

its connection to the infinite, to grasp it whole and thus make it be.

Science cannot apprehend becomings.


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Here we must turn again to mathematics to see how philosophy and

science interact to effectuate the event. Smith argues that:

. . . mathematics is replete with events, to which he [Deleuze] grants

full ontological status, even if their status is ungrounded and

problematic; multiplicities in the Deleuzian sense are themselves

constituted by events. In turn, axiomatics, by its very nature,

necessarily selects against and eliminates events in its effort to

introduce rigor into mathematics and to establish its foundations.

(413)

We can see how philosophy, operating as problematics, can apprehend the

event, while axiomatic science struggles with the event. Problematics grasps

the event, the problem, and creates a solution. In creating a solution,

however, the problem is thrown back into virtuality, into chaos, and a new

event arises to fuel philosophys movement. Axiomatics, on the other hand,

attempts to wrest the event whole from chaos, to embody it within a state of

affairs, to reign it in with functions, and in so doing to remove from the event

that which makes it the event the virtual problem in the first place.

Axiomatics is primarily concerned with solutions, and problematics primarily

with the problems, though both encounter the event whole.

Smith goes on to say that problematics and axiomatics (minor and

major science) together constitute a single ontological field of interaction,

with the latter perpetually effecting a repression or more accurately, an


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arithmetic conversion of the former (414). The phenomenal reality of an

event necessitates both branches of mathematics, both philosophy and

science, since without embodiment the event cannot appear. But it is also

here that mathematics transcends mere numerical manipulation and enters

the linguistic register. It is the translation of the event from problematics to

axiomatics and back again which allows our apprehension of the event.

Deleuze and Guattari concur, saying: The task of philosophy when it creates

concepts, entities, is always to extract an event from things and beings . . .

space, time, matter, thought, the possible as events (33). Axiomatics

embodies the event in things and beings, and philosophy in turn

disembodies the event and places it back within the virtual. It is the

translation, the constant shuttle between science and philosophy, which

creates change, movement.

The path between embodiment and the virtual does not remain thesame coming up as it is going down. Here enters the difference between

differential and integral calculus within the logic of the differential calculus of

problems. Simon Duffy explains the mathematic difference as follows: The

differential calculus consists of two branches which are inverse operations:

differential calculus, which is concerned with calculating derivatives, or, in

Leibnizian terms, differential relations or quotients; and integral calculus,

which is concerned with integration, or the calculation of the infinite sum of

the differentials in the form of series (566). The differential calculus thus

contains two approaches: differential calculus, or axiomatics, and integral


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calculus, or problematics. Though both are contained within the calculus, the

difference is crucial it is the difference between a top-down or a bottom-up

approach, and the starting point determines the outcome.

Placing the difference between integration and differentiation back into

the context of What is Philosophy? , we see how the paths between the

virtual and the actual alter their shape depending on the approach, and how

science and philosophy modify each other:

Science passes from chaotic virtuality to the states of affairs and

bodies that actualize it. However, it is inspired less by the concern for

unification in an ordered actual system than by a desire not to distance

itself too much from chaos, to seek out potentials in order to seize and

carry off a part of that which haunts it, the secret of the chaos behind

it, the pressure of the virtual.

Now, if we go back up in the opposite direction, from states of

affairs to the virtual, the line is not the same because it is not the same

virtual (we can go down it as well without it merging with the previous

line). The virtual is no longer the chaotic virtual but rather virtuality

that has become consistent, that has become an entity formed on a

plane of immanence that sections the chaos. This is what we call the

event, or the part that eludes its own actualization in everything that

happens. (155-156, italics in original).


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The path of science is the path of differential calculus, of embodying the

virtual in states of affairs, of incessantly calculating derivatives in order to

impose reason on chaos. To move back from this place, from embodiment

back to the virtual, from science to philosophy, however, takes us back not

to the original chaotic virtuality from which science calculated the

differential, but rather to a new virtuality, a virtuality ordered by integral

calculus. The creation of the differential comes at the price of creating the

integral. The path is never the same twice chaotic virtuality is lost in the

mathematical operation as translation.

We must not forget, however, that even in the actualization of the

virtual through functions and bodies, there is always a part of the virtual

which resists embodiment. The event cannot be separated from its virtuality,

from its continual becomings. As Smith points out, in the calculus, the

differential is by nature problematic, it constitutes the internal character of the problem as such, which is precisely why it must disappear in the result

or solution. On the other hand, . . . the differential provides him [Deleuze]

with a mathematical symbolism of the problematic form of pure change

(426, italics in original). Even though the differential functions in science as a

way to embody the virtual, the form of the differential itself speaks to the

virtual which cannot be contained by it. The form of the differential, or, to

slip into the linguistic register, the sign of the differential, is problematic, and

as such points to the virtual, to the event which it cannot through its function

apprehend.


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The differential is the sign of the virtual, which is how Deleuze and

Guattari can say that science is paradigmatic , whereas philosophy is

syntagmatic (124, italics in original). Science looks to the meaning, the

solution within the differential, and so misses the event, the becoming, which

is immanent to the sign of the differential and which is acted upon by

philosophy. As Smith argues, one might say that while progress can be

made at the level of theorematics and axiomatics, all becoming occurs at

the level of problematics (424). Science can certainly lay claim to progress,

insofar as progress consists of creating solutions to more and more

problems. The problems themselves, however, their becomings, cannot be

truly addressed by science. It is the arena of problematics, of philosophy,

which is capable of dealing with events in their virtuality, in their constant

flux.

Science and philosophy clearly operate in different ways. They both,however, operate upon the same thing: both delve into chaos to create order

through mathematics. But as Deleuze and Guattari say, when an object a

geometrical space, for example is scientifically constructed by functions, its

philosophical concept, which is by no means given in the function, must still

be discovered (117, italics mine). The operations of a scientific function

cannot lead us to the philosophical concept which is operating in dialogue

with the function. The very fact that such a function exists, however, its very

form, points to a problematic behind its construction. This is why Smith

suggests that axiomatics is a foundational but secondary enterprise in


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mathematics, dependent for its very existence on problematics (422).

Although this line of reasoning is persuasive, we must nonetheless

remember that the event, the problem, cannot remain only a virtuality, but

must be embodied in states of affairs. Moving from science to philosophy

suggests that the philosophical concept appears prior to the scientific

function, but moving in the opposite direction suggests that the concept is

merely derived from the function. Deleuze and Guattari explain this tangle,

claiming that nonphilosophy is found where the plane [of immanence]

confronts chaos. Philosophy needs a nonphilosophy that comprehends it; it

needs a nonphilosophical comprehension just as art needs nonart and

science needs nonscience (218, italics in original). Each chaoid requires the

existence of the other two in order to function: The three planes, along with

their elements, are irreducible (ibid 216). The three inform each other,

though none can work in parallel with another. Each chaoid is equal to the

others; none can appear first or work better, since the three are equally

dependent on each other and equally distinct from each other.

The need of one chaoid for another refers us back to the necessity of

translation posited earlier. Smith acknowledges this need, stating that what

is crucial in the interaction between the two poles [axiomatics and

problematics] are thus the processes of translation that take place between

them (423). The translation between the chaoids occurs in the brain, at the

level of thought: The brain is the junction not the unity of the three

planes (Deleuze and Guattari 208, italics in original). It is crucial to


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recognize that the chaoids are never unified, not even when translated and

communicated within thought. Rather, they intersect, relate, and inform

each other within the space of thought, within the brain. As Sorensen claims:

Thinking itself is a practice that is able to produce the plane of immanence,

just as science and art are . . . The infinite movement of thought makes it

capable of traversing vast distances and multiple flows in a single flash, and

as a practice thinking draws a plane that maps these distances and these

flows (127). Although the planes which philosophy, science, and art draw

are different, all three planes are used by thought to cut the chaos and

create order.

Thus far my discussion has been mostly limited to the interaction

between science and philosophy, between axiomatics and problematics. The

third chaoid, art, has been intentionally placed to one side, because while

the mathematical interaction between science and philosophy is relativelyclear, the inclusion of art carries with it the potential to muddy the waters.

Smith, however, provides us with a rather oblique point of entry for art:

Even in mathematics, the movement from a problem to its solutions

constitutes a process of actualization; though formally distinct, there is no

ontological separation between these two instances (432). We have already

seen how philosophy and science deal with the problem event and its

solution(s). Art enters the scene between philosophy and science, in the

process of actualization. Since all three chaoids must confront chaos to

bring order, and since all three are by nature distinct, each has its own


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method for dealing with the virtual and the actual. Philosophy creates a

plane of immanence, science creates a plane of reference, and art creates a

plane of composition . This is how Deleuze and Guattari define art:

Composition, composition is the sole definition of art. Composition is

aesthetic, and what is not composed is not a work of art (191). Art draws

from axiomatics and problematics, but is not either it is the composition of

elements from the two poles.

Art as composition must be distinguished from the kind of composition

which science creates. Art is not the construction of functions or equations,

though it may borrow from or be built upon just such an axiomatic

foundation. Deleuze and Guattari claim emphatically that technical

composition, the work of the material that often calls on science . . . is not to

be confused with aesthetic composition, which is the work of sensation. Only

the latter fully deserves the name composition , and a work of art is neverproduced by or for the sake of technique (191-192, italics in original).

Composition includes technique, but is never simply reducible to technique.

Art also borrows from philosophy, in that it interacts with a becoming,

with an event. Deleuze and Guattari explain the difference between

conceptual becoming and aesthetic becoming by saying that there are

sensations of concepts and concepts of sensations. It is not the same

becoming. Sensory becoming is the action by which something or someone

is ceaselessly becoming-other (while continuing to be what they are) . . .


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whereas conceptual becoming is the action by which the common event

itself eludes what is (177). For the concept, the event escapes actualization,

even escapes itself. For art, on the other hand, the event is constantly

composed in such a way that it becomes other, that is, it is composed so that

it presents itself and presents an other, simultaneously.

Art deals with the event as self and other, as two contrasts welded into

one sensation, and as such, it uses both axiomatics and problematics in its

composition. This use of both branches of mathematics is directly addressed

by Deleuze and Guattari: Abstract art, and then conceptual art, directly

pose the question that haunts all painting that of its relation to the concept

and the function (183). Although art is neither a concept nor a function,

both are necessary starting points for art to be realized.

The primary role of art is to withdraw sensation from chaos, and

though art may use axiomatics or problematics as a starting point, it goes

beyond the capabilities of either branch of mathematics alone. This is clear

when Deleuze and Guattari say: Art enjoys a semblance of transcendence

that is expressed not in a thing to be represented but in the paradigmatic

character of projection and in the symbolic character of perspective (193).

Art borrows from the paradigmatic nature of science and the syntagmatic

nature of philosophy to compose something which is greater than the sum of

its parts, to give sensation to the event. Just as the event needs philosophy


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to give it meaning and science to give it form, so too does it need art to give

it sensation.

Also like philosophy and science, art must go through a series of

translations to produce sensation. Deleuze and Guattari explain this

aesthetic translation process, saying:

On this plane of composition, as on an abstract vectorial space,

geometrical figures are laid out cone, prism, dihedron, simple plane

which are no more than cosmic forces capable of merging, being

transformed, confronting each other, and alternating . . . The planes

must now be taken apart in order to relate them to their intervals

rather than to one another and in order to create new affects. (187)

The interaction between problematics and axiomatics, the form and meaning

of mathematical shapes and functions, is dismantled by art, rearranged, and

translated into an aesthetic form. Not only is art translating science and

philosophy, it is translating itself, changing the form and meaning of a cube

into the form and meaning of say, a room, which itself takes on the form and

meaning of something else within the context of the artwork.

Art, then, operates under the same guidelines as both science and

philosophy. It is still modeled upon mathematics, even if secondhand. If art is

to be found in mathematics pure, however, Smith may provide an avenue:

Transfinites and infinitesimals are two types of infinite number, which

characterize degrees of infinity in different fashions (422). Transfinites and


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infinitesimals are always becoming-others: they are both infinite and not-

infinite. To a person untrained in mathematics, they can even provide

sensation, if only the sensation of vertigo.

The relationship between art and infinity is as unique to art as the

respective relationships between philosophy and science are to infinity.

Deleuze and Guattari explain the respective relationships by stating that

philosophy wants to save the infinite by giving it consistency . . . Science,

on the other hand, relinquishes the infinite in order to gain reference (197).

Art, in its turn, wants to create the finite that restores the infinite (ibid).

Each chaoid acts upon infinity, upon virtuality, upon chaos, to shape

something orderly. Philosophy clings to the infinite, science creates the

finite, and art moves through both to create something finite which captures

an infinity: Even if the material lasts for only a few seconds it will give

sensation the power to exist and be preserved in itself in the eternity that coexists with this short duration (ibid 166, italics in original). There is a

degree of infinity inherent in every piece of art, even in a short series of

musical notes. For the space of the duration of the composed material, the

sensation extracted from chaos exists infinitely. Creating the infinite within

the finite is the work of art.

Creation drawn from the infinity of chaos is what characterizes thought

across all three planes of philosophy, science, and art. Deleuze and Guattari

sum the relationship between the three chaoids, saying:


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The three thoughts intersect and intertwine but without synthesis or

identification. With its concepts, philosophy brings forth events. Art

erects monuments with its sensations. Science constructs states of

affairs with its functions. A rich tissue of correspondences can be

established between the planes. But the network has its culminating

points, where sensation itself becomes sensation of concept or

function, where the concept becomes concept of function or of

sensation, and where the function becomes function of sensation or

concept. (198-199)

The threads which hold the three planes together are the threads of

mathematics, of mathematics as problematics, as axiomatics, as infinite

numbers. The suture points where the chaoids intersect one another are the

points where form, meaning, and sensation come together in the

mathematical narrative of the mind. Equivalent, but separate, philosophy,science, and art move us though chaos and into ordered, phenomenal,

reality.


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Works Cited

Deleuze, Gilles and Flix Guattari. What is Philosophy? . New York: Columbia

University Press,

1994. Print.

Duffy, Simon. The Role of Mathematics in Deleuzes Critical Engagement

with Hegel.

International Journal of Philosophical Studies 17.4 (2009): 563-582.

Print.

Smith, Daniel W. Mathematics and the Theory of Multiplicities: Badiou and

Deleuze

Revisited. The Southern Journal of Philosophy 41 (2003): 411-449.

Print.


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Sorensen, Bent Meier. Immaculate Defecation: Gilles Deleuze and Flix

Guattari in

Organization Theory. The Sociological Review 53 (2005): 120-133.

Print.

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