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.
Gödels 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, Gödels 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
. 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; they depend on functions and not the converse, and the
function is the true object of science." (WIP 120)
D&G tell us that the infinite must find a limit from which
things will come. It is a limit of the infinite that makes
the limited thing possible.
the limit and the
infinite [are] clasped together in an embrace from which things
[and sets of things] will come. Is the function,
then, this limit? D& G tell us that this
determination by a limit must not be negative mediation. It
must not be mediated by a reference plane of already formed
matter. Every limit is illusory and every
determination is negation, if determination is not in an
immediate relation with the undetermined. The
determination of a limit with the undetermined infinite must
relate immediately if there is not to be merely a negative
determination. So, the function cannot be the immediate
limit, because it mediates reference. For set theory, the
function mediates a reference frame of opposition between
intrinsic (endoreference) and extrinsic (exoreference) and
installs a hierarchy of slowing down. It is still a
negative mediation of opposition/limitation that Deleuze rejects
they depend on functions and not the converse, and the function is the true object of science." (WIP 120) D&G tell us that the infinite must find a limit from which things will come. It is a limit of the infinite that makes the limited thing possible. the limit and the infinite [are] clasped together in an embrace from which things [and sets of things] will come. Is the function, then, this limit? D& G tell us that this determination by a limit must not be negative mediation. It must not be mediated by a reference plane of already formed matter. Every limit is illusory and every determination is negation, if determination is not in an immediate relation with the undetermined. The determination of a limit with the undetermined infinite must relate immediately if there is not to be merely a negative determination. So, the function cannot be the immediate limit, because it mediates reference. For set theory, the function mediates a reference frame of opposition between intrinsic (endoreference) and extrinsic (exoreference) and installs a hierarchy of slowing down. It is still a negative mediation of opposition/limitation that Deleuze rejects (DR 202-4).
D&G demonstrate (WIP 121-2) that Cantors 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.
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 concepts 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 concepts 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 Gödels 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. Todays 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 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 Spinozas 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. Sets are not internally related to a closed whole. 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, Bergsons is not a homogeneous space-time of set theory. He introduced heterogeneous duration that flows from articulations of movement-images. However, for Bergsons 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 Deleuzes 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 concepts 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 Deleuzes 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
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 to be cardinalised every time the following ordinal number is reached. A given type of number (intensive) does not retain an inequality without cancelling it within a new order of extension. When a type of number (intensive) cancels its difference, it does so by explicating it within the extension it installs. But it maintains its difference of intensity in the implicated order by which it is grounded. Natural numbers are ordinal (originally intensive). Cardinal numbers result from ordinal intensive numbers. 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. 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 Bergsons 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, its 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, its 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 isnt Deleuzes 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.
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