Two Types of Multiplicities
by Beth Metcalf
Differentials of Thought
Deleuze tells us (Difference & Repetition 176-182) that problems are different in kind from their solutions. The problematic Ideas are the extra-propositional and sub-representative differentials of thought. When the sub-representative plane is left out, then we have only a plane of representative reference. We are caught in a closed circle of reference. We can break that circle only when we reach the sub-representative conditions from which conditioned solutions may be actualized. When the intensive problematic differentials are included, the virtual conditions no longer resemble the solutions that are actualized on the representational plane of reference.
In What is Philosophy?, Deleuze and Guattari tell us that the philosophical concept of consistency must not be confused with the scientific function of reference. The philosophical plane is that sub-representative plane of problematic Ideas. It is the plane of the differentials of thought. It must not be confused with the plane of scientific reference with its functions. When the concept is assumed to be on the plane of reference, the ‘concept’ has been turned into a function. Then the concept is caught in a circle of reference under the principle of identity and generality, and the conditions of the problem resemble the conditioned solutions. Conceptual identity within a circle of reference turns the concept into a function. But D&G show us that concepts are not functions on a plane of reference. Concepts are intensive problematic multiplicities (Ideas) on the sub-representative plane of consistency. The philosophical concepts are events on a plane of consistency. The intensity of the philosophical concept consists of layers of inseparable variations. The extensity of the scientific function is composed of independent variables. When the intensive differentials of thought are confused with extensive elements in variable relations, then the concept becomes caught up in a general circle of reference. The differentials of the intensive concept must not be confused with the extensive relations of sets of objects or parts of objects at the level of functions and their solutions. Intensive concepts, not extensive relations of elements, are the differentials of thought. The concept must not be confused with scientific functions if we are to break the circle of closed reference.
Gödel’s Incompleteness Theorem
Deleuze and Guattari say (What is Philosophy? 120-1, Example 10) “Set theory is the constitution of a plane of reference, which includes not only an endoreference (intrinsic determination of an infinite set) but also an exoreference (extrinsic determination). In spite of the explicit attempt by Cantor to unite philosophical concept and scientific function, the characteristic difference remains, since the former unfolds on a plane of immanence or consistency without reference, but the other on a plane of reference devoid of consistency (Gödel).” Cantor assumed that set theory could unite the intrinsic consistency of the concept with the extrinsic reference of the function. That is, he tried to unite the intrinsic form of the concept with extrinsic substance of reference. However, Gödel’s ‘Incompleteness Theorem’ shows that set theory is not adequate for the task. Deleuze and Guattari show us why. Set theory puts both consistency and reference on a plane of closed reference. Set theory presupposes one closed circle of reference. Then, it cannot give us both intrinsic consistency and extrinsic reference at the same time. The ‘concept’ on a closed plane of reference mistakes differentials of thought to be the relations of extensive elements under the principles of identity and generality. This turns the concept into a function of reference. Instead, we must reach the problematic differentials of intensive conceptual consistency that are different in kind from the solutions of extensive reference.
Set theory turns the concept into a function on a closed plane of reference. Intrinsic endoreference is mistaken to be the consistency of the concept. There is no endoconsistency or exoconsistency because the sub-representative plane of consistency is not included. Consistency and reference are both assumed to be on a plane of already formed matter of generalizing consistency and universalizing reference. Set theory puts both conceptual consistency and functional reference inside one extensive reference frame, because it leaves out the philosophical plane of sub-representative and extra-propositional differentials of problematic thought. Set theory leaves out the sub-representative plane. Then, without the sub-representative differentials of the concept, the representative plane closes into a circle of reference; turning the concept into a function. There must be two types of multiplicities that intersect, if we are to break the circle of reference to which set theory restricts us. There must be two types of multiplicity* from the outset.
Diagram of Two Types of Multiplicities
The diagram shows both the Philosophical Plane of Consistency (sub-representative) and the Scientific Plane of Reference (representative). This diagram adds a new plane of sub-representative problematic Ideas which is different in kind from the plane of representative solutions. The scientific function and the philosophical concept are two types of mutliplicities that intersect, each on its own plane. D&G believe these two types of multiplicities are in need of each other. We must reach a philosophical plane of incorporeal-intensive consistency that is not to be confused with the scientific plane of corporeal-extensive reference. Each plane has an extrinsic (exo-) dimension and an intrinsic (endo-) dimension.
Philosophical Consistency and Scientific Reference
What is Philosophy? p. 152, “There must be at least two multiplicities, two types, from the outset. This is not because dualism is better than unity but because the multiplicity is precisely what happens between the two.” The philosophical concept apprehends the event as inseparable variation on a plane of consistency. It must not be confused with functions of independent variables on a scientific plane of reference. The two types of multiplicities intersect, each on its own line. But when science constructs its objects with functions, its philosophical concept must still be discovered (WIP 117). The philosophical concept is not given with the scientific function.
So, there is the philosophical
multiplicity of conceptual consistency, and there is the
scientific multiplicity of functional reference. And, (WIP
155) "…. when we compare philosophical and scientific
multiplicities, conceptual and functional multiplicities, it may
be much too simple to define the latter by sets. Sets, as
we have seen, are of interest only as actualizations of the
limit;
D&G demonstrate (WIP 121-2) that Cantor’s set theory gives only the negative determination of a limit, because it is restricted to the scientific plane of reference. The intrinsic endoreference and extrinsic exoreference are negatively determined within a circle of reference. With set theory, extensive reference already mediates the determination. There is no immediate determination when the sub-representative differentials of thought are left out. Therefore, set theory is merely a negative determination of mediation that limits the infinite. It leaves out the sub-representative differentials of conceptual consistency. The differentials of set theory are merely extensive relations of negative determination. And, we know what Deleuze thinks of negative determinations --- (D&R 85-6) the determination of the concept cannot bear directly on the undetermined being. The philosophical concept of difference must reach the form of the determinable --- the empty form of time --- in order to find its internal singular difference that is a prior individuation of pre-individual singularity. We must reach the empty form of the determinable if we are to reach the immediate determination of a limit with the undetermined infinite. We must reach the philosophical multiplicity of incorporeal intensity if we are to reach conceptual consistency. Whenever the concept is turned into a function of conceptual identity on a plane of reference, then consistency is taken to be merely endoreference in a circle of reference. It never reaches endoconsistency or exoconsistency. But when the concept reaches its sub-representative consistency, it is not turned into a function of reference. Then it is the immediate determination that limits the infinite unformed Substance. It reaches singular internal difference of the intensive concept. It reaches pre-individual singularity from which things (and sets of things) will come.
(D&R 27) If we are to reach the
immediate determination of a limit --- if we are to avoid a
difference already mediated --- we must avoid that merely
generalizing difference that turns the concept into a function.
We must find intrinsic singular difference at the intensive level
of problematic Ideas that does not confuse the concept with a
function. We must find a repetition that is not a
repetition of objects (sets of objects or parts of objects)
presupposing conceptual identity mediated by variability of a
reference frame. When there is presupposed identity of a
concept --- when the concept is turned into a function --- then,
(DR 270) “Repetition…is represented outside the
concept, as though it were a difference without concept, but
always with the presupposition of an identical concept”.
In other words, although Cantor tried to find both the
consistency of the philosophical concept and the reference of the
scientific function, set theory is inadequate for the task.
Cantor's extrinsic is supposed to be without a concept. However,
it still presupposes the identity of a concept (one generalizing
consistency within totalizing circle of reference). It
still does not define space from the perspective of an observer
tied to that space. It still defines space from a
presupposed universalizing reference frame extrinsic to an
observer. Set theory can give us neither endoconsistency
nor exoconsistency, because it is restricted to a plane of
reference. To avoid this dilemma, we must include the
problematic multiplicities of the sub-representative concept.
We must find the intersection of two types of multiplicity.
Then, we can reach singular difference that may, each time, be
actualized as reference with consistency --- but can never be
generalized consistency or totalizable reference.
Logical Propositions
What is Philosophy? Chapter 6,
D&G tell us that the logical proposition turns the
philosophical concept into a scientific function. Then
(136) "...the concept itself is the function for the set of
objects that constitute its extension." So long as we
mistake the concept to be a function "....[t]here is no
qualitative difference between intension and
extension....intension being simply the condition of reference
and constituting an endoreference of the proposition, extension
constituting the exoreference. Reference is not left behind
by ascending to its condition; we remain within
extensionality.” Therefore, the logical proposition
sees no qualitative difference between extension and intension.
The conditions of reference constitute the concept’s
intension (logical description of a possible world). Conditioned
reference and the conceptual conditions of reference have no
qualitative distinction --- no real distinction. And since
the propositional function still confuses the condition
(intension) with the conditioned (extension), the intension of
the proposition never escapes the circle of extensive reference.
It never reaches singular difference, because it leaves out the
conceptual consistency of the philosophical plane with its
problematic and intensive differentials of thought.
Logical Propositions turn the concept into a
function. (WIP 135-6) Propositional and logical functions
presuppose an endoreference of the concept and the exoreference
of the function. But, as we have seen, they are mediated by
the circle of reference. It turns the concept into a
function of reference. The concept (and its consistency) is
still in the form of prior possibilities of extensive reference.
It is in the form of the generality of the function and its
conceptual identity. It does not reach singular difference
because it leaves out the sub-representative plane. The
propositional function is merely circular in its extensive
reference. How can this ever reach the singular
comprehension of a concept --- the concept’s endoconsistency
or exoconsistency? The virtual condition resembles that
which it conditions. But Deleuze says, (Logic of Sense
p.68) “.…it is futile to go from the conditioned to the
condition in order to think the condition in the image of the
conditioned as the simple form of possibility. The
condition cannot have with its negative the same kind of relation
that the conditioned has with its negative.”
(WIP 137-8) D&G explain Gödel’s Incompleteness Theorem by saying, “In short, in becoming propositional [i.e. in becoming a propositional function], the concept loses all the characteristics it possessed as a philosophical concept: its self-reference, its endoconsistency and its exoconsistency. This is because a regime of independence [variables]….has replaced that of inseparability [intensive zones of indiscernible variations that, with division, change in nature].” Functions have only extensive reference. They never reach the intensive consistency of the concept. (140) When the concept is confused with a function, there is a presupposition that logic (the propositional function) can reach singularity of individuation. But the concept is not a function with discursive reference. (143) The concept has no reference. It is constituted by inseparable variations that pass through zones of indiscernibility. They change nature in changing the contour of the zones.
So long as we mistake the concept to be a
function, we do not reach the intensive transcendental condition
on the sub-representative plane of consistency. Therefore,
(138) "reduction of the concept to the function inevitably
deprives it [the concept] of all its specific characteristics
that referred back to another dimension [the intensive and
extra-propositional dimension of the philosophical
concept]." Set theory can't escape conceptual identity
of reference that turns the philosophical concept into a
scientific function. Set theory does not escape the
identity or generality of the concept. Therefore, D&G
are saying that in order to find that singular concept of
difference, we must reach that sub-representative and
extra-propositional plane of the philosophical concept --- the
intensive multiplicities of singular consistency.
Logic of Signification Versus Logic of
Sense
When we are talking about propositional
logic, we find that sense is never a correspondence between
propositions and things. Sense is the articulation of their
difference. (LOS 28) “I never state the sense of what
I am saying. But on the other hand, I can always take the
sense of what I say as the object of another proposition whose
sense, in turn, I cannot state.” Sense proliferates in
regressive series. We find that (LOS 36) every unique
series of homogeneous terms subsumes under it two heterogeneous
series --- a signifying series and a series of signifieds. And,
(66) a paradoxical element traverses these heterogeneous series
to make them converge and resonate. But Deleuze tells us
(LOS Eleventh Series of Nonsense) that there are two ways that a
paradoxical element of nonsense can interact with heterogeneous
series. On the one hand, there can be determinations of
signification according to a theory of “types”.
But the regressive law still relates names in degrees of
“type” to classes or properties. On the other
hand, nonsense may enact a donation of sense. Then,
(70) the regressive law distributes intensive degrees in a
heterogeneous series of events. These two series may still
be called ‘signifying’ and ‘signified’, but
they no longer have a prior relation of signification. They
(content and expression, each with their own form and substance)
are open to real sub-representative difference. (LOS 51)
“For sense is not to be confused with signification, it is
rather what is attributed in such a way that it determines both
the signifier [Corporeal Expression] and the signified [Corporeal
Content] as such”. (75) The paradoxical element of
signification is an abnormal set. But the paradoxical
element of sense reaches a new transcendental field of
incorporeal nomadic distribution. It is the univocal empty
form of time. When Deleuze is misunderstood to be at the
level of a determination of signification, his logic of sense is
being confused with logic of signification. A logic of
signification still puts the concept at the level of a function
of extensive reference. And (122) when we confuse sense
with signification, we define a problem by its condition of
“resolvability”; that is, “we conceive of the
condition only in the image of the conditioned.” Then,
there is no real difference between the conditioned and its
condition.
We must reach the domain of the
production of incorporeal sense. Therefore, in order to
avoid conflating the determination of signification with a
donation of sense, we must reach asignifying sense. We
must reach a sub-representative sense where ‘sets’ are
not already signified by properties or classes. We must
reach asignifying domain of the incorporeal sense prior to the
corporeal propositions that turn the concept into a propositional
function. With the determination of signification, series
still presuppose a prior relation of signification, and signs
themselves are signifying. Therefore, we must reach two
heterogeneous series of deterritorialized flows (content and
expression). (A-O 241) “These figures [content and
expression] do not derive from a signifier nor are they even
signs as minimal elements of the signifier; they are nonsigns, or
rather nonsignifying signs…” We must reach an
empty square that nomadically distributes asignifying sense.
Today’s task is to reach the univocal empty form of time
beneath the level of determination of signification. We
must reach the intersection of two types of multiplicities.
We must reach the philosophical donation of sense where the
concept is not turned into a function. The philosophical
multiplicity reaches sense (incorporeal thing) --- sense is
expressed (incorporeal proposition). (Expressionism in
Philosophy 335) “In short, what is expressed everywhere
intervenes as a third term that transforms dualities. Beyond real
causality, beyond ideal representation, what is expressed is
discovered as a third term that makes distinctions infinitely
more real and identity infinitely better thought. What is
expressed is sense: deeper that the relation of causality, deeper
than the relation of representation.”
Sets
Sets are collections of objects or things. Things (and sets of things) are given on the scientific plane of reference. But, as we have seen, "things" (WIP 120) depend upon how "the limit and the infinite [are] clasped together in an embrace from which things will come". So, we must inquire into that by which the given (objects of science) are given in an immediate embrace. We must reach the immediacy of the form of the determinable that is the philosophical multiplicity of intrinsic singular (pre-individual) difference. That is, we must inquire into the disparate intensity (DR 222) of the philosophical multiplicity. This philosophical multiplicity is the immediacy of that by which the given (scientific multiplicity) is given.
Deleuze says (Cinema 1, p. 10-11)
that the whole must not be confused with sets. Sets are
closed. But this closing is artificial. Sets are
artificially closed sets of parts. But a whole is not
closed. It is open. The duration of the whole changes
qualitatively, and divides into objects (and sets of objects).
And objects lose their contours and unite into qualitatively
changing duration. “Movement relates the objects
between which it is established to the changing whole which it
expresses, and vice versa.” That is why, before we can
reach a qualified whole, we must find the immediate limit
of the infinite from which things (or sets of things) will come.
The whole has no parts. It is the unformed infinite
that has not been qualified by a limit. Things or sets of
things cannot come from the unqualified infinite. The whole
must first be clasped together in an immediate embrace with a
limit (just as Spinoza’s infinite Substance must be
qualified in the Attributes). But, as we have seen,
qualification by a limit must not be a negative determination.
It must be an “immediate” embrace with the infinite.
That immediacy cannot happen on a closed plane of reference.
We must reach the sub-representative intensive immediacy that
qualifies a whole from which things will come.
A qualified whole is indivisible in the
sense that it cannot be divided without changing qualitatively.
The whole must not be thought to unify parts as closed sets of
objects. Relations belong to a whole, not to objects.
Objects of a set can change position in space. But relation
is not the property of objects in space. Relations belong
to the duration of the whole where relations are external to
their terms. Relation is an open whole. Therefore,
relations are not internal to a closed whole whose parts are
given in variable relations. Rather, the whole consists of
external relations that change the whole qualitatively. Sets
are in space, but the ‘whole’ is open in duration.
The open whole is the ‘Dividual’ that changes in nature
as its intensive individuation changes.
So, D&G are saying that sets depend
upon functions that intersect with an immediate
pre-individual singularity, each time. Set theory assumes
that functions depend upon sets. But then sets merely
mediate a limit as a negative determination. Sets must be open to
duration of the whole that changes in nature with every division
or augmentation on the sub-representative plane of intensity.
When the multiplicities of extensive reference are open to
intensive multiplicities, then functions (not sets) are the true
objects of science. Functions actualize a singular limit of
‘slowing down’.
Sets of the Movement-Image
Bergson introduced a change in philosophy
of movement. This changed the notion of sets as fixed
images or objects that change position in space. He
introduced the notion that movement has two aspects --- two types
of multiplicities that intersect. First, there is
movement in space between objects or sets of objects. This
is movement in space with variable relations. But Bergson
introduced a second aspect --- the expression of duration of the
whole. As duration changes qualitatively, it is divided
into objects. As objects change shapes, they are re-united
in duration. Movement relates objects to a changing whole,
and vice versa. Movement divides the whole into objects and
objects are reunited in a whole that changes nature. Prior
to Bergson, these internal relations among sets had been cut off
from the other aspect of movement, making object and image the
same and taking movement away. Sets were merely immobile
sections which change their relations in space. Everything
happened in a closed circle of reference. But Bergson
includes duration of a whole whose change is expressed in
movement. He includes a philosophical plane of virtual
multiplicities that gives science a better metaphysical
foundation.
Bergson changes the notion of
‘sets’. If the whole is mistaken to totalize sets
of parts in variable combinations, there can be no real change in
the nature of the whole --- there can be no movement in space.
We need to include heterogeneous duration in order to reach sets
that change contour in space as the whole changes nature in
duration. So, the whole is not a closed set of parts as set
theory still leads us to assume. The whole is open to
heterogeneous duration. The whole traverses all sets.
The ‘Dividual’ is an open whole qualified by the
immediacy of a limit. It is the immediate embrace of the
limit and the infinite from which things will come. The
infinite is qualified as real singular difference. The
Dividual is that open whole which, with division or augmentation,
necessarily changes the nature of the whole. It is the open
whole from which things, and sets of things, will come. And
it is also that which prevents sets from closing. As the
whole changes in nature, objects change contour in space. With
movement, objects are re-integrated into an open whole that
changes in nature as the whole is re-differentiated into objects
(and sets of object).
Therefore, Bergson’s is not a
homogeneous space-time of set theory. He introduced
heterogeneous duration that flows from articulations of
movement-images. However, for Bergson’s
movement-image, time is still subordinate to movement. There
is a void or interval that articulates sets and parts in space,
on the one hand; and the whole that changes in kind as it
divides, on the other. With the movement-image, sets are
thought to be parts of an open whole. Sets are rational
cuts. Action is linked as stimulus and response across an
interval.
I take the movement-image to be the
scientific multiplicity on the plane of reference. It
remains open to the philosophical plane of real conceptual
duration. But time (duration) is still subordinate to
movement through space. I take the movement-image to be the
intersection of two types of multiplicities. But with the
movement-image, the intensive multiplicities of duration are
subordinate to the extensive multiplicities of movement. Time
as duration of the open whole is subordinate to movement. Movement
is the measure of time. There is the intersection of the
two types of multiplicities that keeps the extensive plane of
reference open to the intensive duration of the whole. However,
time is still the chronological time of Chronos and its
actualizations of reference.
Intensity and Extensity
As we have seen, there is a tendency for the
philosophical concept to be turned into a scientific function.
There is the tendency to think ‘intensity’ as already
developed in extensity and subordinate to qualities that fill
extensity. Intensity is the difference that measures the
time of an equalization that cancels itself in extensity in which
it is distributed. However, Deleuze says (D&R 222)
“…each intensity is already a coupling….thereby
revealing the properly qualitative content of
quantity….Disparity….difference or intensity….is
the sufficient reason of all phenomena….” It is
important to understand that Deleuze’s ‘intensity’
is not used in the same sense science gives to that term. Within
science, (D&R 223) “….we know intensity only as
already developed within an extensity, and as covered over by
qualities. Whence our tendency to consider intensive
quantity as a badly grounded empirical concept….” Intensity
is the difference that tends to cancel itself in extensity and
underneath quality. With the scientific function, extensity
cancels intensive difference in the new order of reference that
it installs. Intensity is the difference that is equalized
in the extensity which it creates. There is, therefore, a
real difference between the intensive order of the philosophical
concept and the extensive order of scientific reference. Intensive
and extensive are two types of multiplicities that are not closed
into one circle of reference. The extensive actualization
does not resemble the intensive virtual that is actualized.
Deleuze's 'intensity' is the philosophical
concept when it has not been confused with a function. I
take the intensive multiplicities to be continuous. Any
intensity of content may be coupled with any intensity of
expression to produce any new continuous variation. Inseparable
variations of continuity may be produced that are never derived
from a prior notion of what is possible or impossible. All
real difference in continuous variation is possible at this level
of sub-representative and extra-propositional intensity. At
this intensive level of continuous variation, any continuity may
be created that was never imagined to be possible. But the
multiplicities of continuous variation at the level of intensity
must not be confused with discrete multiplicities on the
extensive plane of reference. Discrete multiplicities are
on the extensive plane of the variability of functions. A
function has a discrete continuity only within the limit that
mediates its variability.
Intrinsic and Extrinsic
Set theory uses a principle of negative
opposition. Both the intrinsic and the extrinsic are within
closed extensive reference. There is intrinsic
endoreference and extrinsic exoreference. However, set
theory cannot reach that plane of conceptual consistency. It
cannot reach endoconsistency or exoconsistency. It can only
give us a circle of generalizing reference. It cannot reach
the intensive consistency of the singular concept.
However, when the two types of
multiplicities intersect, propositions are expressions (both
corporeal and incorporeal). Things are contents (both
corporeal and incorporeal). The incorporeal opens the
forms. The corporeal is substance open to new forms of
actualization. Both the incorporeal and the corporeal have
their own intrinsic content (endoconsistency and endoreference).
And both the incorporeal and the corporeal have their own
extrinsic expression (exoconstituency and exoreference). (See
Diagram.) When both types of multiplicities are included,
then we are no longer closed into one circle of reference. (LOS
22) “Sense is both the expressible or the expressed of
the proposition, and the attribute of the state of affairs.
It turns one side toward things and one side toward propositions.
But it does not merge with the proposition which expresses it any
more than with the state of affairs or the quality which the
proposition denotes. It is exactly the boundary between
propositions and things…..” I take this to mean
that sense (incorporeal thing) is the endoconsistency that is the
expressed of the incorporeal proposition (exoconsistency). Sense
inheres in the proposition and is attributed to things (corporeal
thing). Sense is the boundary that articulates propositions
and things, but does not merge with them.
The relevant distinction is not between intrinsic endoreference and extrinsic exoreference. The relevant distinction is between the two types of multiplicities. Conceptual consistency and functional reference intersect as two types of multiplicities (the intensive-incorporeal and the extensive-corporeal). Their intersection of consistency and reference is a singular internal difference, each time. Sense is that which inheres in the proposition and is attributed to things. Sense articulates the boundary between propositions and things, with singular difference each time. This intersection is the singular concept of the thing (corporeal thing) and is actualized as the singular concept’s own discursive reference (corporeal proposition), each time.
Set theory on a closed plane of reference
can only express a relation of opposition/limitation. Its
opposition is negative mediation and its limitation is a
hierarchy of slowing-down. However, with Deleuze’s
sub-representative plane of philosophical univocality included,
‘intrinsic’ and ‘extrinsic’ do not have that
negative (opposition/limitation) form. With the
intersection of two types of multiplicity, we can reach the immediate
determination of a qualified infinite from which real
singular difference of things will come, disparately each time
--- actualized and realized in non-totalizable frames of
reference. Only when the function finds the singularity of its
own concept, can its reference frame be actualized.
Only then is its own endoconsistency and exoconsistency
actualized and realized in its own endoreference and
exoreference, each time. Only then can a set have its
own conceptual singularity to be actualized in the
singularity of its own reference frame, disparately each
time.
The Cardinal and the Ordinal
As Deleuze’s Bergsonism tells us, to
count a number of things, or sets of things, we must symbolically
represent them as a homogenous quantity. They must be
represented as occupying separate positions in space. But
if we conceive of time as a medium for this counting, we have
merely homogeneous spatialized-time of a ‘badly analyzed
composite’. It is only when we include the
heterogeneous plane of duration that both planes intersect and
open into two types of multiplicities. Perception is given
as homogeneous extensity in space. But that by which this
given is given is heterogeneous duration. These are the two
types of multiplicities that intersect.
Deleuze says (D&R 232-3) set theorists
assume that ordinals cannot be the origin of number, because
ordinality presupposes cardinality. But this assumption is
due to a misunderstanding. To say that ‘the cardinal
results from the ordinal’ does not mean that ordination
presupposes repetition of the same unit of homogeneous space to
be cardinalised every time the following ordinal number is
reached. A given type of number does not retain an
inequality without cancelling it within a new order of extension.
This type of number is what Deleuze means by
‘intensity’. When an intensive number cancels its
difference, it does so by explicating it within the extension it
installs. Cardinal numbers result from ordinal intensive
numbers on the sub-representative plane of heterogeneity where
time is not merely a medium of homogeneous space. Cardinals
are the explication of the intensive ordinate. That is,
extensity cancels the intensive difference in the new order of
reference that it installs. Intensive ordinates are the
real internal difference that is equalized in the extensive
reference which it creates. An intensive ordinate does not
retain an inequality without cancelling it within the new order
of cardinal extension. “Ordinal construction does not
imply a supposed same unit but only….an irreducible notion
of distance – the distances implicated in the depth of an
intensive spatium (ordered differences). Identical
unity is not presupposed by ordination; on the contrary, this
belongs to cardinal number and presupposes an extensive equality
among cardinal numbers, a relative equivalence of exteriorised
terms. We should not, therefore, believe that cardinal
number results analytically from ordinal….In fact, ordinal
number becomes cardinal only by extension, to the extent that the
distances enveloped in the spatium are explicated, or
developed and equalized in an extensity established by natural
number. We should therefore say that, from the outset, the
concept of number is synthetic.”
(WIP 118-19) The endoreference (relation
between limit and variable) constitutes a slowing down in the
chaos. This endoreference is the basis for a counting.
It extends over the conditioned (exoreference) of slowing-down.
The exoreference is a system of coordinates as external framing
so that speed of abscissas can be coordinated. But there is
not only a diversity of these limits. There is disparity of
heterogeneous systems. This leads D&G to feel entitled
to doubt the unifying aspirations of science. Since
coordinate systems are disparate and non-totalizable, we may feel
entitled to doubt a unified field. Perhaps
‘unified’ must take on a new ontological sense of
univocal being. But, of course, that is a question only
science can answer.
Sets of the Time-Image
I take both the movement-image and the
time-image to be the intersection of two types of multiplicities.
They are each an intersection of extensive multiplicities of
reference and intensive multiplicities of consistency. However,
the movement-image subordinates time to movement. The
extensive multiplicities of scientific reference are of
chronological time (Chronos) which is subordinate to movement of
corporeal things through space. Time is still on its
hinges. But it spatializes time with real difference of
reference. Time is measured by movement (sensory-motor
links) through space. In contrast, I take the time-image to
be the philosophical multiplicities of intensive forces. On
the plane of incorporeal Aion, time is constituted inside the
crystal. The crystal is the constitutive division of
present into present which is passing and the past which is
preserved. Time is out of joint. Now, movement is
sub-ordinate to the constitution of time. The philosophical
multiplicity is the empty form of time that nomadically
distributes things in an open space without reference. The
incorporeal time-image is a space of undecidability of bodies.
It is a space of overlapping indiscernibility. It
constitutes sets of things without reference. ‘Sets’
are no longer divided by a cut that forms part of one of the sets
it separates. Interstice no longer form part of either set
(one has no end, and the other has no beginning). With the
time-image, (C2 212) interiority of thought is not a whole, not
even an open whole.
The time-image brings about a reversal.
An Outside hollows itself out along with its own inside.
Images are no longer sensory-motor associations across a void.
There are no longer rational cuts between sets that determine
commensurable relations between images. Images are
unlinked. Interstices between “sets” are now the
non-correspondence between parallel series of content and
expression. Interstices between sets are not determined as
part of either series. There is undecidability of sets.
Sets as homogeneous sequences in space are now replaced by
heterogeneous parallel series. The heterogeneous series
form an irrational cut between sets of each series. There
is a series of visible-images (content). There is a series
of sound-images (expression). Content and expression are
not mediated by sensory-motor links. Sets are now re-linked
on irrational cuts. The philosophical concept is no longer
subordinate to movements in space-time. Now, movement is
subordinate to time constituted inside the crystal.
(C2 179-181) With the time-image,
‘sets’ are not sets of objects that change position in
space, as set theory would have it. The Time-image is also
distinct from the still classical ‘sets’ of
Bergson’s movement-image that assumed rational cuts between
sets across an interval --- where a cut forms part of one of two
sets it separates. With the time-image, (179-80)
“Given one image, another image has to be chosen which will
induce an interstice between the two. This is not an
operation of association, but of differentiation…..It is the
method of BETWEEN, ‘between two images’….It is the
method of AND, ‘this and then that’…” Now,
the time-image is the constitutive ‘and’ of things and
sets of things. It is the constitutive interstice between
two images. The Outside is no longer the sensory-motor
links of an external world. The Outside constitutes false
continuity. I take the interstice to be the
indiscernibility between images. It is the interstice
between heterogeneous parallel series of see-able content and
say-able expression. It constitutes interstices between
things. The time-image is the interstice between irrational
non-correspondences between two heterogeneous parallel series of
content and expression. Whereas the movement-image assumed
‘rational’ sensory-motor links across a void, the time
image constitutes interstices between ‘sets’ that have
no rational link. Sets are now assemblages without common
properties. Deleuze says of the interstice, (181)
“….it is irrational and does not form part of either
set, one of which has no more an end than the other has a
beginning: False continuity is such an irrational cut.”
The time-image is power of the Outside. The interstices
present the direct constitution of time where
“…..continuity….is reconciled with the sequences
of irrational points, according to non-chronological time
relationships.” The time-image is not subordinate to the
chronological relations of corporeal Chronos. Rather, it is
the non-chronological nomadic distributions of incorporeal Aion
that constitutes time to which movement is now subordinate.
There may be actualizations of a time-image. However, it
may also resist actualization on a plane of reference. There
is counter-actualization.
The Empty Form of the Intensive Ordinal
We have already explored sets in relation to
the open whole of the movement-image. Now, we must ask how
we reach the intensive order of the philosophical concept. How
do we reach an order of time that is not already subordinate to
the functions of extensive movements through space
(movement-image)? (WIP 157-9) We must reach the time of the
event where movement is subordinate to time (time-image). The
Event is the pure order of the empty form that nomadically
constitutes time by the distribution of incorporeal
singularities. Singularities are nomadically distributed in
the static empty form of time --- the past (what has just
happened?) and the future (what is going to happen?). The
empty form of time is this static order. First is BEFORE.
Second is AFTER. The order of time is constituted in this
empty incorporeal form. Time is no longer subordinate to
movement. It constitutes the irrational interstices between
images or sets. The order of time is constituted without
subordination to a form of movement through space. The
intensive multiplicity of ordinality intersects with an extensive
multiplicity of cardinality. This constitutes a new ordinal
conceptual consistency, each time --- to be actualized as a new
coordinate system of reference, each time. There now can be
both consistency and reference, each time. But there is no
system of generality that can accommodate unifying consistency
and universalizing reference. To find reference and its
own consistency, we must reach the intersection of two types
of multiplicities (scientific reference and philosophical
consistency) in disparate domains of singular difference, each
time.
Therefore, it is not that things, or sets of
things, have a prior givenness. It is not that a closed
space is distributed to things, assigning each thing a position
in relation to other things. Rather, instead of a sedentary
distribution of things into a closed space of already categorized
sets, there is nomadic distribution of things into the open space
of undistributed being. The non-metric intensive
multiplicities of philosophical consistency occupy space without
counting, and are prior to any actualized frame of reference**.
Univocity and Quantum Interpretation
Philosophy has become important for
interpretation of quantum effects. Might univocity point
toward a new interpretation of quantum physics? Einstein
would not accept anything but a deterministic interpretation for
quantum effects. He said, “God does not play
dice”. However, modern quantum interpretation says
there is no deterministic causality at the sub-atomic level.
God does play dice. But, if D&G are correct, it’s
even worse than that! To find the philosophical concept,
the dice game (univocity) is not at all like those
statistical-probabilistic games with which we are acquainted
(LOS, Tenth Series of the Ideal Game). And, it’s even
worse than that. There is no ‘God’ that would
represent generalized conceptual consistency or totalizable
universal reference. D&G feel entitled to question the
unifying aspirations of science. Perhaps interpretation
would take into account the intersection of scientific and
philosophical multiplicities of univocity. And isn’t
Deleuze’s univocal being consistent with a multi-verse
interpretation? All divergent disjunctions are ramified as
real. They may be actualized as multiplicities of singular
universes.
*See Time and Free Will by Henri
Bergson, chapter II, for his discussion of the two types of
multiplicities.
**In A Thousand Plateaus, Chapter 12
Proposition VI, D&G write about the “numbering
number”. It distributes something in open space
instead of distributing divided space. The number becomes
independent of a relation to space. The number has a
concrete relation to smooth space --- occupied without being
counted. (ATP 389) “The number is no longer a means of
counting or measuring, but of moving….The more independent
space is from a metrics, the more independent the number is from
space…. The number becomes a principle whenever it occupies
a smooth space, and is deployed within it as a subject, instead
of measuring a striated space.” And (ATP 391)
“…the numbering, nomadic or war, number is always
complex, that is, articulated….A complex of numbers every
time….it in no way implies large, homogeneous quantities,
like State numbers….but rather produces its effect of
immensity by its fine articulation, in other words by its
distribution of heterogeneity in free space.”