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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