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THE CONCEPTS OF GENERALIZED FRACTALITY AND CHAOS

THE SCIENTIFIC GENERALIZATION OF FRACTALITY. THE NON-EUCLIDEAN IDEA AND ITS THEORETICAL SENSE

A general method for the description of phenomena grounded in a theory of Identities is an idea that has to be philosophically limited and scientifically generalized. Differences, Catastrophes, Games, Multiplicities, Disseminations…also form such a method, but these concepts express the generality in an ultimately restricted and transcendent philosophical mode. They are indeterminate generalities that manifest themselves in half-singularities, and these half-singularities testify only to philosophy’s ability to survive and not to its “adaptation” to the description of real phenomena. By contrast, we are seeking from the outset a form of generalized fractality (= GF) or chaos that is a real limit or critique of every possible philosophy; singularities that can be described in the spirit of science rather than as remainders of ontology’s autodissolution. Before the “application” of the theory of Identities to philosophy, and after the expulsion of sciences’ epistemological image, the intermediary task is to draw the last meaning of this new experience of Identities: to pass from Identities to the form of real fractality they make possible, to establish a scientific thought of fractals capable of generalizing their concept. Just as we previously described the adequation (of-the-last-instance) of science’s essence and of nondecisional Identity (of) self, so we will have to describe on this basis the adequation of the Identities we will call fractal and of scientific representation, of the scientific instance of knowing. At the same time that we will reformulate a non-philosophical and nongeometric concept of fractals, which is generalized under the conditions of Identity and of thought-science, we will manage to liberate sciences from their philosophical image.

This theory of Identity, then of fractals and of chaos, can be generalized beyond every philosophical position only if it is drawn from the essence of science, from the scientific posture, rather than from the geometric “Mandelbrotian” concept of fractals. These fractals will serve us as materials, as the indication of problems to be resolved and as properties to be generalized; but they will not be the principle of generalization. This generalization is the Theory of nondecisional Identities (of) self. A direct generalization of geometrico-physical fractals, direct but external to their scientific sense, would produce a “philosophy of fractals,” a “fractal vision of the World,” which would be added to the philosophies of Multiplicities, Differences, and Catastrophes. How can we elaborate a truly universal concept of fractals if not by eliminating the geometrico-philosophical mixture, which would transcendently combine their limited geometric experience with a philosophical generalization of the “metaphor” of fractals? The authentic generalization of scientific givens must remain internal to science. It must not take the form of a hermeneutic and circular appropriation through a transcendent Tradition or a Reserve of thought. But it must also cease to transpose, metaphorically and continually, the geometric and physical givens of fractals into the philosophical space. As soon as its principle is absolutely internal (the Identity-of-the-last-instance) and as soon as it is exposed in the framework of “science itself” and not under the philosophical horizon, fractality is absolutely universal, in principle and as much as possible (more universal than its philosophical generalization-totalization). This does not mean that it is not subject to corrections. Quite the contrary.

How is this new generalization of already elaborated regional or “ontic” knowledges possible?

1/ Here the method of description of phenomena is general only insofar as it considers them from the perspective of their Identity-of-the-last-instance (identity-without-identification), rather than from the perspective of their philosophical identification to a transcendent instance (totality, mixture, dyad, doublet, etc.) as the Foundation, Being, the Concept, Differe(a)nce, the inconsistent Multiple…would be. It is a matter of relating them to what is less their “in itself” than a position of thought, the only one, that renders intelligible the fact that their meaning is scientific, that they are products of science rather than objects of philosophy, and that describes them in terms of its own internal requirements and not in terms of alleged properties, “objective in themselves.” Phenomena’s only real “in itself” is the cause of science and, in a sense, the very science of these phenomena. The Theory of Identities does not replace existing sciences as philosophy does (if only partially). Quite the contrary, it adds itself to them as the special science, which makes clear that the described phenomena have meaning only through and for science. This theory’s “objective” is not to discover Identity, and the fractality that derives from it, as just one property alongside others (the “ontic” properties). It is indeed a question of a new property, but since it is neither ontic (the object of already existing sciences, like the geometry of fractals, for example) nor ontological and philosophical, it is a “unary” or specific property of “first science.” In other words: it is the specifically “scientific” property of phenomena; their capacity in-the-last-instance to fall under a science and what protects them from falling absolutely under the philosophical Authorities. This special property—absolutely supernumerary and, as we will see, “fractal” vis-à-vis philosophy—is typical of the scientific representation of things and allows science to think itself as the science of these phenomena. Instead of an ontico-material or ontologico-philosophical property, Identity-of-the-last-instance (of) this or that phenomenon = x is what makes it possible to say that the alleged “properties” of X are, in-reality-of-the-last-instance, acquired scientific knowledges, produced and apparently reified in the form of data “in themselves.”

2/ The second condition for a scientific generalization is apparently opposed to the first. But it is only its complement and, in some sense, its consequence. As to their mode of donation, the phenomena to be known must be considered as given at first under the reason of their philosophical sense or form, in any case as virtually philosophizable. The Theory of Identities (and thus the theory of fractals) can be a science of materials-being, of object-of-science-being of these phenomena, only if it is also practiced as a lifting of the resistance that philosophy spontaneously exerts against it. First science, in the form of Theory of Identities, is the real critique of philosophical singularities and multiplicities, which are forms of resistance against their scientific species and mimic science against its own spirit.

If it is possible to realize an equivalent to the theory of fractals in the nonmathematical sphere of pure thought, of “representation” in its universal form, a theory valid for natural language and philosophy and not only for some dynamic or physical geometric forms, this new discipline, far from being opposed in any way (less than philosophy in any case) to its mathematical form, will bring it instead a kind of confirmation—not “empirical,” i.e., ontic, but transcendental or founded strictly on the internal requirements of scientific thought itself. But this amounts to saying that the theory does not depend in its existence on the mathematical and Mandelbrotian form. If the Mandelbrotian theory should by chance be invalidated, our theory of fractal would survive this improbable disaster, because the mathematical theory will have served as its material and indication and will not have directly served to constitute it in its essence (the generalized theory transposes fractality on a real, other-than-mathematical-or-physical basis, as we will elaborate it).

The Identity “of-the-last-instance” is the one that remains in its own immanence, regardless of the region of the real (the World, for example) in which it acts qua cause. Thanks to this radically “transcendental” property—to this essence—that is its own, it opens the possibility of a new type of ultraphilosophical generalization, which can be applied to philosophy but that has a scientific and no longer philosophical origin. Nevertheless, as science “of” philosophy (i.e., of the data or phenomena that philosophy is and uses as its materials in view of a science the One), it is not only that: it is the science of the essence of sciences, of scientific representation. It is not a simple discipline alongside others, but the one that thinks the essence of-the-last-instance of sciences and safeguards their sense as sciences from philosophical expropriation. This is why it represents, in two distinct modes, the equivalent of a non-Euclidean mutation in the experience of thought.

1/ This mutation does not take place at the interior of a science, but at first from philosophy (eventually a “philosophy of fractals”) to a science. Clearly, this is possible only if every science is by its essence also a thought and can thus be compared on equal terms to a philosophy. The Theory of Identities does not represent an ontic or transcendent, but a radically transcendental mutation. Here we give a new sense to the obvious non-Euclidean “metaphor”: nothing-but-transcendental and not empirically conditioned, scientific yet valid for first science or for the essence of every science.

2/ This mutation does not stay at the interior of an (“ontic”) science. It equally takes place from an ontic science (fractal geometry) to a first science (neither ontic nor ontological, but unary). The generalized theory of fractals can be ultimately called non-Mandelbrotian, but in a slightly different sense from the previous sense of the non. It is not possible here to critique Mandelbrot’s oeuvre, nor do we intend to do so; we do, however, intend to genuinely critique philosophy.

Since this mutation is not interior to science, since it does not go from a science to philosophy, but from an (ontic) science to another science (that of the essence of sciences), we will say that it is strictly transcendental, i.e., internal to the scientific practice, the only practice that is immanent and thereby distinguished from philosophy. It is less the fractals as supposedly “natural properties” that are generalized (extended, transferred to other regions of nature or to society, art, etc.) than their theory, which is also, strictly speaking, a generalized theory (of fractals). This is possible only if the essence of theory itself proves to be fractal in an other-than-geometric mode.

Because we are elaborating a radical transcendental concept of science (and thus non-philosophical, nonempirico-transcendental concept), we have to treat this procedure of non-Euclidean generalization in the same way. It undergoes a mutation that frees it from its geometric “Euclidean”/“non-Euclidean” division. In every respect, this new non-Euclidean use seems to be less metaphorical than philosophy’s own use. It consists—if the analysis is pushed further—in liberating the non-Euclidean Idea from its restrained ontic forms and its philosophical metaphors and in restoring to it its identity, which can only be the identity of science, of the essence of science. On this condition, the non-Euclidean Idea becomes a universal theoretical tool.

It should be noted that the transcendental in question here is science’s and not philosophy’s transcendental. So it is equivalent to an instance of radical, undivided immanence, which is not split in a circle, doublet or empirico-transcendental mixture. The real cause of-the-last-instance is the only transcendental; and only Identity thus conceived restores to the transcendental its identity, which philosophy had lost.

It is science, science alone but as first, that is the internal cause and “beneficiary” of this operation of generalization. This benefit is not a capture, a slicing off, since everything takes place between sciences. Above all, we do not generalize Mandelbrot here the way philosophy has already done in order extract from this theory (which it did not itself produce) a surplus value of sense and of energy opposed to science. Vis-à-vis Newton, Boltzmann, Cantor, Einstein, or Mandelbrot…, philosophy proceeds

•    by detaching and isolating a local theory cut off from its process—while we reinscribe it in the process of an (other) science;

•    by its reposition as factum, autoposition, and rational transcendental fact that gives the sense or representation of Being—whereas fractal knowledges remain simple knowledges and are not identified by philosophical autoposition with the real itself, with “Being”;

•    by philosophical extension or generalization, i.e., by a divided (generality/totality) and thus limited universality—whereas the universalization of knowledges or theories is no longer affected here by a decision that would divide it, but rediscovers its identity—of-the-last-instance, of course. This is enough to open a theoretical “space” of absolutely unlimited rectification to the theory of fractals, a space that is not cleaved or closed by philosophical teleologies.

As Benoit Mandelbrot indicates with implacable theoretical rigor, the fractal is a “response to a question that does not even seem worth asking.”¹ Obviously the question will be elaborated even better—this will be the fractal theory itself—if the response is already given. Nothing theoretically rigorous is carried out that does not start from a given. That fractality is given does not mean, however, that it has to be given in nature, according to the mode of existence of various irregular figures in the vital environment. If some knowledges were not also given, if fractality were not an emergent theoretical object “before” becoming a property of perceived objects, there would be no theory of fractals. The whole problem is to adequately identify and describe this precession and immanence of an object of knowledge, like GF in particular. The theoretical ordering of the problem (the theory can represent an object of-the-last-instance without modifying it, provided it is determined by this object, which is not alienated in it) allows us to resolve this problem by abandoning the specular philosophical circle of the supposedly given fractal object and its supposedly given model. It allows us to escape the drawbacks associated with this conception: the idealist forcing of the object by the fractal model that abusively generalizes; the empiricist reduction of theory to an idealist abstraction of the object. GF’s theory is not an idealizing extrapolation or extension of the supposedly given object (not a shred of philosophy is in any case given with this type of fractality). It is not an idealist attempt to intervene in philosophy and to fractalize it under its own conditions; GF is a relatively autonomous order of reality. Neither is philosophy a region of GF, nor is GF an intensification or else a deconstruction of philosophy (in both cases, its reaffirmation). There is an a priori fractal intuition; this is what must be described. We will not say that it is deduced from the One, but, more exactly, that it is the space of every possible theoretical deduction, starting from axioms that bear on Identity, axioms that operate with philosophical information, but that are no longer reduced to this information.

We hope that we have not produced a scientific ideology or a scientist’s spontaneous philosophy or a philosopher’s spontaneous science (as the philosophical or epistemological uses of sciences in general are). We hope that we have produced the theory of fractals, more radical or universal than the geometric fractals, but not “superior” to them in the sense in which philosophy presents itself as the “superior use” of existing things. We pass from ontic fractality to a unary fractality, whose concept is broader and applies to language itself (philosophy, poetry, literature, etc.) and no longer only to geometric phenomena. And thus it is not a question of a poor extension of the geometric to language, such as the one scientists sometimes practice, scientists who are seized by the most uncontrolled philosophical spontaneity.

SKETCH OF A DIMENSIONAL DESCRIPTION OF THE PHILOSOPHICAL DECISION: FIRST APPEARANCE OF A PHILOSOPHICAL FRACTALITY

Fractality is a problem of dimension and presupposes a prior dimensional description (adapted to these nongeometric disciplines) of science itself and of philosophy. Just as philosophy knows half-singularities and unfinished multiplicities, so it will know a half-fractality. This half-fractality will appear only if we can characterize the dimension of the philosophical decision. We will take this characterization of philosophy as our guide in the search for a more radical fractality adapted to the instances of science itself.

In general, dimension designates the minimum number of relatively autonomous coordinates or parameters that are necessary for representing an object (philosophy in this case). If the problem of philosophy’s dimension can still receive an onto-philosophical sense, it can also already point toward another problem: the specifically fractal dimension, the dimension of the most irregular manifold, a dimension that philosophy cannot thematize vis-à-vis itself. What we are looking for is the dimension-of-fractality of philosophy itself. A dimension set in play by a generalized fractal relation to philosophy. The concept of dimension can be itself generalized and applied to philosophy through an adequate transformation, which is consistent with philosophy’s theoretical possibility.

The dimensional description of philosophy was initiated in the affiliated form of transcendental arithmetic, which the Ancients in the Pythagorean and Platonist context employed to describe philosophy. This is the meaning of the great principles of oral and esoteric Platonism: the One and the Two, the Dyad—principles that can be easily transformed into the description of every philosophical decision. This decision is the most general invariant by which a philosophy is recognized and assumed. It contains first a Dyad, a coupling of contraries—precisely a “decision,” in the narrow sense of the term (this decision gives its name to the whole, since it is decisive for it); then a Unity that reaffirms the coupling, that reposits as such the dyad-unity of contraries. This arithmetic is not mathematical, but transcendental; it claims to determine or constitute the real itself.

In this form, it suffers from a lack of true empiricism: it does not know, properly speaking, the 0 that serves to designate, in the dimensional description, the term or point—which it miscognizes, rejects as too empirical, or divides into a couple. It begins straightaway with a couple of terms, with the relation of the Dyad or with the 2. As a result, it suffers in a complementary way from an excess of philosophical or transcendent…empiricism, since what it calls the Two, by immediately tracing the existence of two points or two terms, the dimensional description would instead call the 1. In this case, the symbol of the dimension is not traced from the number of empirical or constitutive “dimensions” (what, for reasons of terminological clarity, we will call the moments, and not the dimensions, of the philosophical structure). As to the One, to the role it plays in the Decision, we will say that it is a third term or a second principle, or the “first principle”; but we will nevertheless call it One and will use the arithmetical 1 to designate it.

A dimensional description of the philosophical Decision cannot confine itself to saying that the Decision comprises two terms + one. To be sure, it is already an important characterization to say that it contains two and/or three terms: the One is already included in a latent state in the Dyad and added to it as a supplement. Philosophy is, without doubt, a figure of thought whose dimension is comprised “between”—we will discover the sense of this “between” later on—1 and 2 (a line that tends to fill a surface, a half-line half-surface…). However, a more precise description would complicate the problem. For the Dyad of contraries does not have a dimension 1 as a line would, but a dimension 2…In fact, the terms of the Dyad are already unified or intersected in a latent way by the One, which is not only transcendent to the Dyad but also immanent; external and internal to “experience” (the Dyad). So much so the Dyad has instead the figure of a surface, a planar space with two coordinates starting from a point-One, which makes them relative to one another. The One itself, in its transcendence to the Dyad, adds a supplementary dimension. Philosophy has an intermediary dimension between 2 and 3. Not an integer, but a fraction whose sense is obviously empirico-transcendental and not simply empirical or arithmetic. We should compare this elementary description to the one Heidegger gives of onto-theology and its dimensions of generality (the plane of Being, ontology) and of totality (the vertical or the theological summit).

This ever supernumerary nature of philosophy, with respect to the couplings of terms with which it works, is significant and suffices in a sense to characterize its originality as an excessive, dehiscent thought, simultaneously internal and external to the “givens” or to “experience.” Nevertheless, an arithmetic that is simply projected onto a structure does not amount to a dimensional analysis. This analysis, even when it is applied to philosophy and becomes in some sense transcendental, will instead say the following: an isolated term has a dimension 0 (philosophy represses or does not know it); the relation between two terms or the Dyad has a dimension 1; the structure Dyad + One has a dimension 2. The philosophical Decision is an object with dimension 2, a surface or a plane rather than a volume. But it is a matter of a mixture, of a surface or of a plane that is equally transcendental and not purely geometric. And no arithmetic or geometry exhausts its ontological content.

This surface- or plane-nature appears, strictly speaking, only via the dimensional description. It is however clear that this surface is not simple. Already in its arithmetic characterization—let’s start again from it for a moment—the philosophical Decision shows a fractionary nature in all its dimensions. Not only is the Dyad already fractioned (2/1 or excess of the 2 over the 1, its transcendence); but it globally corresponds to the fraction 3/2, which expresses the excess of the One over its own identity with one of the two terms of the Dyad—in both cases, the nature of philosophy’s “transcendence.” But in its dimensional characterization as well, its transcendental and not simply geometric nature means that philosophy is an “intermediary” being—not only between the dimensions 1 and 2, but in each of its moments, because the Dyad with a dimension 1 is an integer only in appearance and must already be represented as a fraction. In philosophy, the fractionary character does not lie in the relation between two structural moments but in the very nature of these moments.

This is to say that if it has the dimension 2, which is in fact a surface and is represented by an integer, philosophy is fractionary in relation to geometry, not only through the play, the gap between two consecutive moments, but intrinsically and in the very nature of its moments. It is so qualitatively and thoroughly. Not only in the sum of its moments, but in each of them where this sum is reflected: apparently, the philosophical surface or plane is intrinsically “fractal.” The consequence is that philosophy already deals with a certain fractality, but is not an accidentally fractal object; that this special and no doubt limited fractality is not one of its properties, but its most internal essence. Within it the fractionary style is not “simply” arithmetic; it remains in the empirico-transcendental doublet, in the fraction formed from the transcendental term and its irreducibility to the empirical term, or in the dehiscence of the Other to Logos, etc., a qualitative alterity that distinguishes the moments among themselves, but also each moment from itself. This is a sign: if we must seek a nongeometric (nonontic) and non-philosophical (nonontological) fractality, we can take as a “signpost” of its problem, if not of its solution, its philosophical concept and the dehiscence of the transcendental to the empirical or the “irregularity” of the Other to Logos, etc.

So it is possible to conserve the concept of dimension for philosophy (and for science), but subject to its change in the direction of the transcendental. It then takes in principle a fractal or quasi-fractal value, and every dimension of the “philosophical” object is fractal as much as “topological.” This explains a special property of philosophy: it contains at one and the same time more objects than the Dyad, which serves as its basis or its given, and no more, since each object of a philosophy can be placed in correspondence with a point of the Dyad. Philosophy is an intermediate being, a monster or a demon, at once identical and superior to the Cosmos. We can say that every philosophy contains no more things than the dyad of the Heavens and the Earth, but also that it contains at least one more: philosophy itself.

Another meaningful change that philosophy introduces into the geometric concept of fractals and that could serve as an indicator is another understanding of what an “intermediate” being is. In geometry, the intermediate between two dimensions (between the surface and the volume, for example) has no autonomous conceptual existence. The fraction is expressed quantitatively (by a number like 1.627, for example), and the reality or identity of this “fractal” property is referred to without being exhibited, if not in the form of another property, the internal homothety, which is another geometric knowledge. On the other hand, philosophy exhibits (no doubt in a still transcendent mode) the identity or reality of this gap, of this fraction, which is qualitative at least as much as quantitative: not only between 1 and 2, but at once 1 and 2. As an intermediary being, it possesses a certain identity and does not interrogate its origin, donation, and functions, as we saw earlier. It proceeds, nevertheless, by identifying the distinct dimensions, which end up converging at infinity: the objects with dimension 2 end up corresponding to those of dimension 1 and fill this dimension with their excess. There is a specific reality of the fraction as transcendental; and a first real reason, a first cause of fractality is provided in the form of a transcendental identity rather than a simple “local” property of homothety. This is how Nietzsche, for instance, seeing the acme of contemplation in the coincidence of being and becoming, creates a fractal object, but in the philosophical mode; or Heidegger with the In-between, the Fold of Being and being; or Deleuze with the Fold of the desiring machine-flow/partial object.

THE PHILOSOPHICAL SEMIFRACTALITY AND THE CONDITIONS OF ITS SURPASSING

If the philosophical interiorization of fractality to the Decision itself as identity-of-the-Intermediate, as In-between or Difference, has the advantage of starting to exhibit the root of a transcendental fractality, it only surpasses ontic (geometric) fractality in the mode of its ontological appropriation, where Identity, instead of “founding” and guaranteeing the irreducibility of the fractal to the continuum, does its utmost to efface it. In philosophy it is Identity itself that is fractal, that is affected by the fraction and its division. This reversibility of intermediate-being and its identity produces a genuine circular disaster in which all parties are losers. Fractality itself undermines Identity, which is then no longer capable of sustaining it, falls back on it, and inserts it in a process of identification. Fractality injures the identity that exploits it. This game of diminishing returns does not lead to a nihilist, but to an overnhilist leveling, to a situation of equilibrium or hesitation: half-nihilism, half-counternihilism, a vacillation that is the way in which philosophy drowns fractalities in Differences, Disseminations, Multiplicities…. The real of fractality is no longer and was never Identity, but the circle of the Same or Differe(a)nce: restrained fractality, effaced as much as produced, “static” in the broad sense of the Same; tautological, if you like.

Philosophy will have merely realized a half-fractality, incapable of elaborating its most universal concept “once and for all.” The most sure sign of its concept’s “average” character is obviously the multiplicity of philosophical decisions themselves: each philosophy leaves aside, implicitly or not, a real it deems too irregular or particular to be rationalizable and masterable; a real that gives place to another, supplementary philosophy, which takes hold of it against the prior philosophy and is expressly constituted to rationalize it, but proceeds in such a way that its “decision” makes visible a new residue, and so on. Philosophical fractality, the fractality of each system, is limited by the concepts of fractality that other philosophies implicitly propose. The only unique and universal concept of fractality is therefore conflated with the hesitations and conflicts that compose the continuum of the philosophical Tradition.

Obviously the concept of an absolutely irreducible or non-negotiable “fractionary relation” should be elaborated. Fractionary: not only in the arithmetic-quantitative sense, or in arithmetic-transcendental sense of philosophy (one of whose avatars is clearly the “relation-without-relation” that Heidegger and Derrida speak of); rather, in the sense in which the “relation,” the “fractionary” nature, which always contains an ultimate possibility of correspondence and continuity, would be itself ultimately abandoned. To detach fractality (at least by its reality or its identity) from the arithmetic fraction, and even from the philosophical “relation-without-relation,” from relation in general, i.e., from the Dyad; to stop inscribing it in the form-dyad. To be more precise: “classical,” geometric, or philosophical fractality is always said of figures whose irregularity derives from their being, i.e., from their property as intermediate natures, a property inscribed in the element of relations or relationships. These “in-betweens” were precisely coextensive with the Identities, syntheses, or connections that were transformed with them. So if we now seek a generalized fractality, we have to, if not eliminate every relation, at least no longer allow the “fractionary” relation and its identity—which are the two essential components of every fractality—to limit or impede each other and have to discover another “relation” between them than the relation of the Dyad (which is not only the initial given of philosophy, but its determinant essence). Those are the givens of the problem; what is its solution?

This program can be realized only if we manage, without any philosophical contradiction, to consider and treat as given (given beforehand and nothing more) Identity as an instance that is de jure absolutely irreducible to every “fraction” and even to every philosophical “relation.” Identity is not or, more precisely, is no longer fractality. But it must constitute the condition of reality, the cause of fractality, which will maintain it outside its reductive inscription in a philosophical dyad or in a geometric “intermediary.” As we have understood it, Identity is no longer even a dimension that can be connected to others. It is the reality of a nondimensional thought, but it is all the more fractal, because fractality is detached from its geometrico-arithmetic dimensional references which could only efface it. This identity no longer entertains any relation with (codetermination, reciprocity) something else, with its representation for example. It is the finally positive essence—experienced as such—of the “without-relation.” Moreover, it will be able to afford fractality a new existence: a “relation” that no longer has the form of an identity and of a division—the division of this identity, precisely. Such a fractality will probably have no continuous relation to the philosophical Decision and its dimensions, to the half-fractality this decision is capable of.

A thought by dimensions (even by intrinsically fractionary dimensions) like philosophy gives place to a fuzzy, unstable fractality, deprived in any case of radical conceptual universality. The most irreducible and stable fractality is not intradimensional, but ultradimensional: it is fractality “in” dimensionality itself, at least in its geometric-philosophical forms. One/Dimension(s): this is the “gap” that cannot be reduced to any ever negotiable as much as non-negotiable gap. Alterity—if this concept can still be used here—no longer takes the form of a mediation, of an intermediary or of a mid-place [mi-lieu], the form of the universality of a milieu; it takes the form of a distance without return, of a unilaterality that is irreducible in proportion to its Identity-of-the-last-instance. To the fractality of philosophy, to the fractality of a circular dynamic system, we will oppose not a “linear,” but a “noncircular” fractality—a fractality that is identical to a certain order, which cannot be reduced to what it puts in order.

The elaboration of a fractal style in the scientific thought of philosophy (a style rather than fractal objects) assumes that we have passed from a particular science, a regional theory, to a more universal discipline. But this discipline always remains a science and never becomes a philosophy. The only science that is more universal than regional (more precisely, ontic) sciences is the science of the One, which we call “first” for reasons already examined. This amounts to going from the plane of ontic objects or data to the plane of the “object” we term Identities. It is a very particular object because, giving itself in the mode of a radical or in-the-final-instance immanence, it enjoys two properties foreign to every object: 1. it remains what it is, without being transformed or alienated wherever it acts as cause (this is the main sense of cause or determination “in-the-final-instance”); 2. it is thus never given in the mode of the object, of presence, of representation, of Being that is transformed or alienated with its donation; it is not discernible in the horizon of philosophy’s most general Greco-ontological presuppositions. In this sense generalized fractality, which will be a mode or a sequel of this Identity, will consecrate the break outlined in contemporary thought, not simply with “metaphysics,” presence or representation, with “logocentrism” and its modes, but with philosophy itself. It is the logic of continuity and the half-singularities it tolerates that is discarded; the entire thematic of mixtures or blends, syntheses and co-belongings, of reversibility and of topological neighborhoods at best, of circular hierarchies at worst. It will be a question of a genuine fractal opening “beyond” the simply conveyed philosophical closures and teleologies.

FIRST SKETCH OF GENERALIZED FRACTALITY’S (GF’S) CONDITIONS

Four invariant conditions have to be united for fractality “in general” to exist. They form a system in pairs:

•    an irregularity, a fragmentation, an interruption, etc.;

•    not arbitrary, but definable by a certain degree (to be fixed) of irreducibility to a continuity whose nature must itself be fixed;

•    furthermore, a principle of constancy (“homothety”) of this irregularity, a principle of self-similarity of fragmentation;

•    which is itself defined in correlation with scale variations (magnification, degree of resolution, etc.).

These are indeterminate generalities, which serve as our guiding thread or as indications to be elaborated.

Generalized fractality (GF) unites those four conditions according to particular modalities, which distinguish it from its Mandelbrotian form. We will describe them progressively. Here is, first of all, a schematic chart of conditions that have to be fulfilled before we can move to a GF.

1. The condition of constancy or identity: to discover and identify, not so much the greatest possible inadequation between two terms, but the reason or cause of this inadequation, greater than any “gap,” “differe(a)nce,” “inconsistency,” “dehiscence,” “dissemination,” “Other”; thus of a nature distinct from every form of first alterity. Since GF is “unequal” to the philosophical Decision, it must be unequal to all the forms of inequality, fragmentation, and partialization that philosophy can tolerate. In a sense, philosophy is virtually the most powerful logic. At least, it presents itself as the most autoenveloping machine, as our average or statistical intelligibility, as the common sense proper to thought (Principle of Sufficient Philosophy). Identity-of-the-last-instance is the cause most inadequate to philosophy, because it is itself the cause of inadequation.

2. The condition of irregularity or of inadequation itself, the “Other” proper: to locate and identify the instance of inequality to the continuum (in this case, the philosophical continuum); this instance no doubt cannot be appropriated by philosophy but, unlike its cause, will necessarily entertain some “relation” to philosophy. It has to obtain the essence or reality of its inadequation from this Identity. This is what we will call Unilaterality.

3. These two conditions together eliminate the solution of contemporary philosophies, which consists in employing a first Other as an immediate solution. What is at stake is a vicious circle, typical of philosophy: seeking the inequality of two terms, the philosopher confines himself to positing, by simple petitio principii, this inequality in itself in the form of an Other, which already necessarily has all the traits of philosophical “logic,” which is already doublet or fold, mixture of transcendence and the transcendent, “autoposition” in its own way. The “Other” puts the philosophers to work, but it is an argument as lazy as philosophy itself. The sole theoretically rigorous, noncircular solution consists in “locating” or “discovering” an absolutely and not relatively first term. This term implicates the greatest inequality to philosophy without being itself this inequality. Rather, this inequality will be deduced from it as the strongest relation of inequality to philosophy. Finding its cause or its reality in this term, it no longer forms a circle with philosophy. As a fractality that interrupts continuity without forming a circle with it, without being conditioned by it; and that is generalized in this way.

4. Geometric fractality is the property of objects that lack uniform, continuous properties, but whose irregularity, however extreme it might be, becomes remarkable, formulable, and quantitatively identifiable. GF is not a property of this type. It is a property of knowing rather than a property of natural objects. And this requires it to be purely qualitative and defined vis-à-vis the essentially continuous model that philosophy is. Its concept and its essence have to change. If it is no longer a question of measuring the degree of irregularity of a statement vis-à-vis the average philosophical norm, this is because GF’s conditions—its requisites—have an entirely different style and appeal to sense and natural language rather than to quantification. In particular, we will not say, in the style of philosophies, that the measurable or calculable aspect of irregularity is inessential or belongs to the thing’s “phenomenality.” Quite simply, it has no meaning, not even a secondary one, in a qualitative and natural-language science (“non-philosophy”). More than qualitative even: because it is thoroughly transcendental by its essence and because, as Identity, it remains entirely outside the quantity-quality couple that is for it one material among others of its representation.

To be sure, knowledge’s fractality does not signify that it is approximate or that it approximates the real. It is, on the contrary, because knowing is not itself real in the strict sense (Identity) that it is fractal. In general, the idea of approximation must not be conflated with the idea of measuring the degree of an object’s fractality. For the approximation may concern the measure of fractality itself. Fractality, in its irreducibility, corresponds instead to the duality Identity-Unilaterality/Material (in this case to philosophy, which is reduced in its claims). The ultraphilosophical generalization of natural fractality rests on all these conditions (Identity and nonresemblance, Unilaterality and not first Other, etc.).

THE CONDITION OF CONSTANCY OR OF SIMILITUDE: IDENTITY-OF-THE-LAST-INSTANCE AS CAUSE OF GF

What is the principle of fractality’s generalization, both the cause that produces it and the instance that reproduces it or endows it with its “self-similarity”? A geometric fractal is defined by a certain equality between the condition of irregularity and the condition of constancy or similarity—the equality of two knowledges. A fractal philosophy in the style of Difference is defined by a certain (invertible) hierarchy of irregularity and its constancy or identity. Finally, a generalized, unary, or real fractal (neither ontic nor ontological) is defined by the irreversible priority (neither simple equality of knowledge nor primacy-hierarchy, but determination-in-the-last-instance) of Identity over irregularity. Subject, of course, to the reformulation (already carried out) of the experience of Identity and its “relation” (Identity-of…) to fragmentation. Identity, the condition of constancy or internal similitude, must receive its full sense if it wants to be the cause of fractality and no longer simply one of its given conditions. No longer ontic or ontological Identity (transcendent in both cases), but radically immanent or transcendental. Philosophy can explain on its own a certain internal similitude as well as a certain fractality. Here we are dealing with something else: “internal similitude” as essence or cause (of the absolutely nonmetaphysical type) and no longer as simple property of an imposed inequality or irregularity, which is already given in the World.

The first condition of the fractals of knowing is Identity: not any identity whatsoever, not the Identity that would be alienated in fractality itself while effacing it, but the one that remains “in” itself, as it is, at the very heart of its effectivity. Thus: a cause that cannot be exchanged with its effect, but a cause that can be located as such at the very heart of this effect. Identity is “lived” and conserved as such, unobjectivated, at the heart of a couple of contraries for example, without being itself affected by the play, the exchanges of this couple. A cause is indeed at issue: it produces a single type of effect, its own identity, but an effect that is differentiated according to the material in the midst of which it acts, an effect that constitutes fractality. But it is also a transcendental cause: both by its radical immanence and by the relation of conditioning (a very special relation, since it is “of-the-last-instance”) it entertains with what serves it as “experience” or “manifold” (in this case, mainly the philosophical Decision) and in which it continues to be experienced without any change or becoming of its essence.

Only Identity as it is of-the-last-instance is reproduced as rigorously identical through…in or directly in… as well as despite… the permanent variations of content. It is not “reproduced” the Same, but subsists as it is in what is reproduced, without thereby giving place to a simple resemblance or similitude, to an analogy or a univocity, or to a variance of variance—that is, to philosophy’s representative generalities. Not only is it the sole Identity that knows itself as this Identity, and is not modified in its essence by its own knowing or involved in a becoming, but even as cause in the midst of its effect of unilaterality or of fractality it is not undermined by this fractality and can be manifested (a second time) as what it is (the first time or in itself).

GF is more than the self-similarity of any system drawn from nature. The system’s traits of irregularity and of interruption will not be simply similar or alike, give or take differences in scale or content. It is not simply a matter of a “family resemblance” or the objective appearance of the identity of Difference. There is a genuine identity-of-the-last-instance of reproduction, but it does not sink in this reproduction, and a “total” instance, “Being” or “Full Body,” does not fall back on irregularity, absorbing or even marginalizing it. The fractal must be opposed to the “different” and to the “molecular”—and not only to them, of course (to the “language game” for example…). What occurs as self-similar is not the simple identity, since this identity is the One. It is not the Other (of philosophy), the Irregularity or the Inequality itself. What occurs “in” Identity is the Inequality-to-philosophy, and it occurs as identical, for Identity is the essence of the Unequal (which cannot be violated by the Unequal). The greatest irregularity, the greatest gap, occurs as identical and no longer as the Same. This is a crucial distinction in relation to…“Difference” and to other neighboring concepts. The product is not the Same in which Identity would be mixed with the Unequal; it is Identity itself, which occurs—rather than returns—as Identity (of) Inequality. Quasi reproduction, but without return; noncircular dynamic, in which “linearity” does not give place to any “curve.”

This cause of GF, which is unknown to both geometry and philosophy, has a decisive effect on fractality itself. GF is an irregularity, an interruption, vis-à-vis philosophical logic, which is apparently the most extended logic. It is a “relation” of inadequation to philosophical intelligibility and to its procedures of continuity. But this definition is vague. What is it that is irreducible to the philosophical Decision? The Theory of Identity teaches us that Identity—and not some instance of the Other à la contemporary philosophies—is the instance that is most inadequate to philosophy; that it is the authentic “Other,” precisely because it is in no way an Other that is autoposited or supposed in the vicinity of philosophy; it is an instance that has already radically suspended and undifferentiated philosophy (the PSP). GF rests on a downgrading of the Other in relation to Identity’s anteriority or precession of-the-last-instance. Identity is not first in relation to a second; it determines every other instance to be strictly second or unilateralized in relation to it.

This point is decisive in the struggle against philosophical illusions. In effect, the absolute Indivision, without blending (but without blending because it knows itself to be undivided and knows this in its own mode of Indivision), is inadequate in principle (and even more than inadequate) to every division or decision, synthesis or coupling. Simply because it is immanent (to) itself, it cannot by definition be located in the World or in Being, where it nevertheless acts. This inadequation-to… is thus no longer an essential property, a predicate of the essence of Identity, one that would retransform it into a species of the Other. It is a simple effect of its essence and does not codetermine it. Identity is not of the nature of the Other, and it is for this reason that it can unilateralize or determine the instance of unilateralization for every other given; it is for this reason that it can give to this instance the figure of the Other. The dual’s “non”-relation, the “dual” of Identity and philosophy whose authority is already suspended, does not fall in principle in any philosophical distribution or economy of knowledges (geometric knowledges, for example). More than the Other, which always requires a form of coupling, it is “inadequate” to Inadequation itself. The “last-instance” is more “Other” than every Other; it is only “the principle” or the cause of the most radical fractality. Before describing this fractality in its condition of irregularity, it is important to mark the mutation that this type of Identity introduces into fractality.

There are two different types of conservation of fractality in terms of Identity, which serves as its principle. If Identity is given with fractality, convertible or reversible with it (Identity is itself fractal or affected by irregularity), fractality will only be conserved as effaced, in the form of a continuous yet superior curve (for example: the philosophical Tradition, Destiny, etc.) that necessarily accompanies it and in which it obtains its sense and value. A conservative or reproductive conservation. If Identity is, on the contrary, de jure inherent (to) self without decision or transcendence, it does not risk falling back on fractality, forming a whole or a mixture with it, a tradition or a continuous curve. Fractality occurs or emerges as new everywhere: not new in a preexisting element, in an indeterminate generality or transcendence that would attenuate it, but new because it occurs or emerges “each time” for the first and only time. And it emerges from Identity, with which it is not mixed, to which it does not return, rather than from a background, a tradition, or a reserve that would reappropriate it—rather, also, than as a form against a background. It is intrinsically unique each time as well as solitary—and received or lived as such by Identity itself, or “in” Identity, which is not a subject behind its act and its product, which is not alienated in them.

THE CONDITION OF IRREGULARITY OR OF INTERRUPTION: UNILATERALITY

Identity-of-the-last-instance is not only what conserves or “reproduces” fractality as identical (to) itself. It is, in the first place, what “produces” fractality; it is its cause. But Identity, as we said, is not itself fractal. We have to add to it an instance of the Other (fractality proper), which is here secondary and no longer primary and determinant as it is in philosophy. We will thus distinguish, on one hand, the “relation,” the nonrelation, what we call the “dual” (or the greatest gap, which is not a first gap) of Identity and the philosophical Decision (or of coupling in general, including the fraction). And, on the other hand, the solution to this “inadequation,” which is greater than every inadequation, a solution that takes the form of Unilaterality. The dual is the structure that is expressed or exists through Unilaterality, the Other’s true scientific concept and GF’s concept. The dual is the cause whose effect-of-unilaterality announces it as the cause that it is.

Qualitatively, what does this structure of Unilaterality consist of? With Unilaterality, we pass from vague concepts of irregularity, interruption, and fragmentation—which only become exact when quantified—to a qualitatively precise figure that is, moreover, the kernel or germ of every fractality. Unilaterality is a radical and oriented asymmetry, the pure irreversibility or the Uniface, the nonsystem of a radically open relation, not teleologically closed by an adverse term, because every supposedly adverse or reciprocal term is in reality absolutely pervaded by contingency. Geometric fractals are characterized by an “irregularity” of form, rhythm, figure, structure, i.e., transcendent and reversible properties, an irregularity that is reproduced and repeated, as if the internal similitude could only appear at the end of extreme variation. On the other hand, this irregularity loses here its transcendent figure. It is in its turn “interiorized” and “autonomized” into an instance or a “fractal order.”

The other concepts of fractality are in every respect more complex, more derived and transcendent. They confine themselves to interiorizing common sense’s and perception’s experience of irregularity outside itself (i.e., in the concept) or to geometrizing it: as a nevertheless symmetric structure in the asymmetry; a bilateral or bifacial, reciprocal, even reversible structure, which presupposes that irregularity is perceived, looked at, watched over by the observer, inscribed in the transcendence of the World. In contrast, Identity-of-the-last-instance does not leave itself and institutes with the World an absolutely tapered, stretched, infinitely distant or distancing relation. This relation must be effectively traversed rather than surveyed and thus “philosophized.” That is why we distinguish Unilaterality and the fractionary, always bilateral relation and thus the types of fractality that each of them respectively engender. Identity is more or much less than a homothety with symmetry and displacement. Likewise, Unilaterality absolutely excludes coincidence, intersection, or double points. GF’s definition is precisely that it contains no reversibility, but rather a pure distancing or a distance without loop. We will describe these phenomenal traits in detail later on.

What is now this fractality’s relation to its cause, the fractality that assumes the mode of Unilaterality? It is not “produced” by its cause in the way in which causes in the World or in Being are alienated so as to continually produce their effect. It is a relatively autonomous structure in its species or its quality, distinct from the immanence of Identity. It does not codetermine its cause in return; it is at last manifested, i.e., produced in a radical phenomenal mode, as it is, through Identity-of-the-last-instance. All these traits are implicated in the “determination-in-the-last-instance,” which signifies not only that the cause is not lost in its effect, but that it communicates its autonomy to this effect. Accordingly, Identity as cause makes it possible to dispossess the Other (fractality) not only of its traditional bilaterality, but, what amounts to the same thing, of the false illusions that result from its philosophical or first autoposition, and to constitute it into a relatively autonomous, yet secondary order of existence. Fractality ceases to be a “property” of certain transcendent objects, to be projected metaphysically, in order to become a sphere or an order of reality with relative autonomy, the order of knowledge or theory.

A CHANGE IN THE THEORETICAL TERRAIN: UNIFRACTALS AND BIFRACTALS

Let’s return to the condition of identity and its efficacy on fractality proper. This is the occasion to dissolve a few philosophical appearances that GF can provoke.

Consider the “geometric” definitions of “self-similarity” formulated here and there by Mandelbrot. “It is the property of a geometric form in which each part is a reduced image of the whole.” Scaling or the property of internal homothety “is said of a geometric image or a natural object whose parts have the same form or structure as the Whole, except that they are at a different scale.” All the variants of a fractal construction have a “scaling” character: “not requiring a new rule at each stage of construction,” but “copying the details of the prior stage, which one will have reduced beforehand to a smaller scale.” Every part or fragment has the same form as the whole or is a “reduced image” of the whole. This property is thus self-similarity. And when it is not a question of the Whole, it is the argument of statistics that takes its place: “When every piece of the coast is, statistically speaking, homothetic to the whole—save for a few details that we choose not to concern ourselves with—the coast will be said to possess an internal homothety…[a notion that] leads to measuring the degree of irregularity of curves that satisfy it, through the relative intensity of large and small details, and ultimately through a dimension of homothety.”²

It is quite clear that, theoretically, these definitions are not very certain, although their coherence is remarkable and they form a system with the idea that the fractal objects constitute hierarchized clusters and overclusters—but only apparently hierarchized, specifies Mandelbrot. Here there are perhaps only appearances that have a philosophical origin. And the conceptualization in terms of Whole/part can only feed a hermeneutic interpretation that would efface fractality once again. The idea of scale variation is more rigorous and allows us to abandon the circle Whole/part to the illusions of immediate perception, of common sense and of their philosophical extension. The Whole has no proper reality (like the part as part, the partial object for example, the fragment…that is opposed to it) except for philosophy. This explains why philosophy’s fractality—since such a fractality does exist—is fractionary in a transcendental (transcendental/empirical fraction) and not only empirical or arithmetic way. It is a fuzzy fraction because of the rejection of the Identity between terms and because of amphibology or the “Same.” The fractality of parts pertains to the superior law of the Whole, even when it is the fractality of the part as such and not the fractality of the relation of (dialectical, hermeneutic, etc.) subordination of the part to the Whole. Philosophy is the operation of drowning fractality in the Same, in Difference, if not in the Whole—in any case, in the unity-of-contraries or the Dyad. We oppose to this fractality of synthesis or of totality—obtained through philosophical synthesis and presupposing the operations of a third agent, an ex machina philosopher who uses it for his own benefit—a fractality of identity that will produce a philosophy of synthesis, which is called “non-philosophy,” instead of a philosophical synthesis.

Assuming its geometric form can be effectively conceptualized in terms of Whole/parts, self-similarity has to change its terrain in order to become an absolutely universal theory; it has to pass from the philosophical terrain of the Identity-Whole to the properly scientific terrain of Identity-of-the-last-instance; from the possible, where it is dissolved in its own reaffirmations, to the real where it gains life, movement, and being. Fractality can only be generalized if it abandons the Whole/part teleology and even the last closure of the partial as such, the fragmented, the disseminated… ; it has to abandon the spirit of hierarchy that its concepts carry, explicitly or as a last vestige from which they cannot be emancipated. It has to stop cofounding itself in the fractalized material and therefore in the mixture it forms with the material; it has to discover its cause in Identity-of-the-last-instance, the sole “foundation” that cannot efface it. If no Whole or hierarchy exists any longer, no perceptively detectable clusters with identifiable contours (whether curved or fuzzy, continuous or angular), no parts will exist; not even partial objects, which are always also associated with flows—and flows are the last avatars of the Whole. The effect of the suspension of the Whole and of the philosophical forms in general is that there are now only absolutely dispersed Unilateralities, whose chaos is not limited by philosophical teleologies. The unifractals of science or of theory must be distinguished in this way from the bilateral or divided fractals, the philosophical bifractals obtained by first division or distance and that pertain to the law of the Whole or of the Same (of Difference, etc.).

Unlike geometric fractality, GF—because it is immediately chaos—does not support the continuous interpolation, division, and insertion of new irregularities up to the infinitely small through scale variation. This is also what distinguishes it from philosophical multiplicities. These multiplicities are founded on a division, a generally positive division, of course, on a positive distance instead of the simple division of an identity. But this distance continues to integrate with an identity; in any case, it presupposes an identity in a relation of reciprocity, so much so that this distance and this multiple are primary and, paradoxically, are effaced in their primacy or autoposition. By contrast, GF is “generalized” only because it is no longer primary or autopositing, because it flows from an Identity-of-the-last-instance or comes after it. This Inequality “in-Identity” or “in-chaos” excludes every process of division or interpolation. It is a structural, static distance, already given in the wake of the Identity with which it is not contemporary and with which does not form a system. Fractal Inequality is inequality-to-the-World or to-philosophy, but this time it is itself “unequal” (to) Identity as well as (to) its cause.

Scientific unifractals are no doubt poorer or more elementary than the philosophical bi or difractals. But they are not poorer as fractals. Poorer in philosophy, they are more acute in science. They are fractals without curvature, beyond curvature, not susceptible to an infinitesimal analysis. We will oppose them from this standpoint to a geometric or scientific-ontic form that can always more easily than they can—although not without resistance—let itself be reinterpreted philosophically as a “miraculous line” grounded in the originary confusion (the similitude of the Whole and parts) and can give place to phantasms of the “living line.” Geometric fractality easily drifts toward the “serpentine line” of grace (Ravaisson) or toward the most continuous philosophical curve (Eternal Return of the Same, the Möbius strip), whereas GF and chaos are the most insubordinate critique of continuity.

If one is nevertheless committed to “saving” the Whole and the parts, it is ultimately necessary to admit that the terms image, similitude, and resemblance take on a radically objective sense; that such phenomena exist, but are not grounded-effaced in a relation of continuity, belonging, or reflection of the Whole and parts; that these latter are therefore flattened outside every relation (in particular any relation of hierarchy, of circle, etc.). To be sure, a Whole would exist, but it would not be first and autopositing. It would be nothing more than parts, which no longer have any relation “between” one another (internal and external relations, little matters now); each has only a relation with the fractal structure that determines them as generalized-fractal object. We will see further on that terms isolated in this way—deprived of their reciprocal relation, of their synthesis or of their opposition, and of the philosophical mode of these latter, and now having a relation as individuals or terms only to the fractal structure—form what we called a chaos and even a generalized chaos. And chaos is the only way to suspend the principle of internal relations as well as the principle of external relations and, furthermore, to suspend their amphibological conflict.

Another philosophical appearance must still be dissolved. GF means, of course, that, regardless of the degree or scale of description of a scientific knowledge and thereby of a philosophy, thought never reaches homogeneous and equal parts, but ceaselessly discovers new inequalities. It is, however, not enough to say that differentiation, far from diminishing, on the contrary increases with “magnification” and the emergence of new unsuspected details, that there is a power of differentiation of thought that prevents us from reaching the simple or the homogeneous. Philosophy itself can maintain this (for example, when it combats the nihilist leveling of differences and reaffirms them). We are dealing with something other than a continuous differentiation: the impossibility of drowning the Irregularity “in-Identity” within Difference, the most fractal singularity within the continuity of a process, or again the real within the “impossible” or the “incompossible.”

In a general way, in other works already, we have not stopped insisting on a new requirement that is unknown to the contemporary philosophers of “multiplicities” or the “inconsistent multiple”: the multiple will have no purpose, will have no real critical force if it is not accompanied by a duality, a de jure inequality that ends up breaking a priori not only the “metaphysical” style, but the generally unitary and philosophical style. The multiple through difference (Nietzsche) or through inconsistency (Cantor), but also dissemination through the Other or through Difference (Derrida) represent, from our point of view, an ultimate unitary normalization of the greatest fractality, which resides in inequality through unilaterality and which, far from crushing the multiple, emancipates it or gives it the space of its efficacy. The philosophies of multiplicities have, as always, only crossed the easy half of the path, and assumed as a limited target “metaphysics” alone, representation alone, and not philosophy itself.

The science of the One is no longer fractionary—neither in the arithmetic nor the philosophical sense. In general, it no longer inserts any relation of inequality between simultaneous terms (the One, Being, being); it places inequalities between nonsimultaneous terms—we can, in fact, call simultaneity the identity that is given with each of the terms in the mode of a last instance. Every fraction is a reciprocal relation. It even becomes reversible, as we see in philosophy, which is reduced to its essence of decision and rid of its “rational” objects. There is nothing of this kind, no relation—of fraction or otherwise—between the One, Being, and being as structures of science. What we call Determination-in-the-last-instance excludes all relations for the benefit of relatively autonomous orders, defined precisely by their identity. The only quasi-fractionary moment would lie between Being and being—again: it implicates no reciprocal relation between them.

The weakening of the fractionary relation is not a weakening of fractality. On the contrary: it is its liberation, its letting-be as autonomous order. Its generalization passes through the “destruction” of the fractionary, even the destruction of its philosophical or “superior” form. It is a mutation from the Fraction’s terrain to Identity’s. Instead of an interiorization to the concept, fractality changes its basis and its principle.

THE PHENOMENAL TRAITS OF FRACTAL OR NON-PHILOSOPHICAL OBJECTIVITY: THE FRACTAL A PRIORI

We have not yet elucidated the undeniably original relation of the fractal structure to its manifold—in this case: the philosophy it unilateralizes or “fractalizes.” Before taking up this task, we should examine the internal nature of the structure of Unilaterality itself, fractality’s peculiar constitution as an original “ontological” region or domain, as science’s sphere of existence.

Sometimes the geometric fractals can be obtained by recursive construction, by indefinite repetition of the same procedure or the same figure. We reproduce a certain number of these elements in a homothetic way, but at ever smaller scales. This procedure leads to an artificial fractal, ultimately to a fractal representation of a natural object that does not perhaps have the same irregularity. This constructivist and operative aspect (interpolation/extrapolation of the structure) presupposes an intuition in which the “concept” of fractality is constructed by an idealizing repetition. But even the “natural” fractals that can be identified “with the naked eye” presuppose such an intuition. They are simple figures, and their regularity or their “concept” is indivisible and can be grasped at once. And we know how much Mandelbrot insists on this aspect, with the complementary rejection of formalism and the axiomatic. So the intuition in all its forms (perceptive or a priori) seems to be a de jure condition of every fractality. What does it become in GF? Is the structure of Unilaterality—the most elementary fractality—intuitive, and what sort of intuition is at issue?

Clearly, we are no longer dealing with a geometric and/or perceptive intuition. The description of GF transferred it from this “ontic” (if not empirical) element onto the terrain of nondecisional-(of)-self Identity-of-the-last-instance, whose radical immanence excludes every transcendence. As a result, if GF is still “intuitive,” its cause rules out that it can be a matter of a sensible intuition, but also of an equally transcendent and autopositional intellectual intuition. In the structure of Unilaterality that GF conditions, not every transcendence nor even every intuition is excluded (since this structure is the very transcendence of scientific representation); what is excluded is the philosophical form of transcendence, its divided/folded form, its doublet or mixture form. Scientific representation, rethought in its essence and no longer simply induced [induite] from this or that local knowledge, is at any rate the dimension of the objectivity, of the theory and of the objects of knowledge it contains. And this objectivity is at once “sensible” and intellectual. But it is no longer by any means certain—quite the contrary—that we are still dealing with the philosophical form of objectivity, of intuition, of sensibility, and of the concept. The cause (of) science manifests or renders accessible the field of theoretical representation, its phenomenon; but it only manifests it under its own conditions. These conditions exclude the most general philosophical form of theory and suspend the doublet form (the form of autopositional faktum) that philosophy spontaneously gives to scientific knowledge. What is the residue, the remainder of this suspension, a remainder by which the essence of theory can be safely characterized? What is this real kernel that Identity-of-the-last-instance extracts in some sense from philosophy—the material of its efficacy—while purifying it of its philosophical form? Here are a few of its phenomenal traits.

• It has to be a question of a transcendence, an opening or an exteriority typical of every representation, whatever it may be.

• Like every representation, it has to be considered objective; it is the very dimension of objectivity, in this case in its form as theory or knowledge. “Scientific” fractality is thus related—we do not yet know how—to a manifold, to a material; and in this case it will only be the very form of (theoretical) objectivity of this material. GF is not only an “objective property” of theory—almost in the sense of the property of a “natural” object. It is also the property of objectivity, which is that of science and which, having a fractal nature, is thereby absolutely distinguished from the philosophical objectivity with which epistemologies generally conflate it.

• What does the fractality of objectivity itself signify? Obviously it does not mean that the form of any object, its contour, is a rugged, angular, or irregular curve. We are in the transcendental dimension of the essence and of the essence of theory “in general,” not in the dimension of perception and of its geometric givens…But it means that knowledges’ very objectivity, their status as objects-of-knowledge, is intrinsically fractal or uni-lateral, that this status therefore excludes the continuity of the doublet-form or the fold-form. An elaborated and validated knowledge—which does not mean metaphysically certain and unrectifiable—is a knowledge only insofar as it is affected by an elementary yet de jure irregularity, by a distancing or a trait of the “faraway,” whereby it does not re-turn to the real of which it is the knowledge, a distance at the end of which it occurs without ever returning to itself, without looping back to itself and thus giving the illusory impression of the real. This irregularity-to-philosophy, which is typical of scientific knowing, can ultimately be said to exist in relation-to-philosophy, to its continuity (always looped or drowned), although we do not yet know its exact relation to this material. But it cannot be said to exist in relation to the real-of-the-last-instance of which it is the knowledge. It can be this unilaterality of knowledge only because it does not measure itself against it; otherwise, it would conflate the real with a supposedly present object that is given in its turn in transcendence. GF is lived and received by the “man of science” as a state of affairs “in itself”: there is no science without this affect of theoretical existence as an existence that is not “in the distance” or “distancing” and “neutralizing,” but in-the-Faraway. The theoretical view is fractal because it is a view “within” the Distant, rather than an activity of distancing and thus of return, as is the philosophical objectivation. The dimension of the Faraway-without-return is a static property, a structural state of affairs of knowledges and does not result from an operation on the real, on Identity. Such is the phenomenal or real kernel of what we call here and there, without understanding their origin and sense, scientific “neutrality” and “objectivity.”

• What is the content in reality of this intimately fractal representation, and what reality is at stake, if it is neither a matter of Identity itself nor of philosophy’s mixed reality (real possibility, actual virtuality, etc.)? The antiepistemological interest of this description is to make clear the existence of an a priori theoretical intuition as constitutive of knowledge. The fractal dimension, previously described as immanent view “in”-the-Faraway-without-return, “in” the Distance-without-loop (no doubt the scientific form of infinity, science’s infinity rather than one of its objects, a mathematical or cosmological object for example), can no longer be reduced to its philosophical form, i.e., to the doublet, the (variously proportioned) mixture of a representation divided into intuition and concept. We know that we must think it in terms of its real essence or its cause and that this cause is “in-the-last-instance.” Hence two fundamental consequences.

On one hand, scientific representation (“theory” in the broad and full sense, neither empiricist nor idealist, as we understand it) is a priori. It is universal and necessary (this is the only necessity internal to knowledges; it is not an external necessity, but the “fractal” system of these knowledges, which we will later on redescribe as chaos). More universal and necessary than philosophy itself, since its traits lie in Identity and are no longer philosophically divided (necessity divided as a coherence of rules or formal a prioris and as necessity of a given manifold; universality divided as generality and as totality and thus at once complex or mixed, simultaneously impoverished and reaffirmed…).

From where does the theoretical or fractal a priori obtain this absolute and undivided necessity and universality, which render it simpler and more powerful than the philosophical a prioris? From its cause, no doubt, which absolutely precedes the a priori and no longer forms a circle of reciprocal conditioning with it, as is the case in every philosophy. The last-instance, having here at the same time a transcendental nature—i.e., it is radically immanent but also conditions the object of knowledge—finally determines the a priori without a vicious circle. The conditions of a rigorous, circleless “transcendental deduction” of theory are finally united: the theoretical a priori is “applied” to the manifold of the philosophical material and produces knowledge under the efficacy of its “transcendental cause” (the “relations” of this “application” will be described later on). Thus are liquidated the aporias—the circles—of relations between the a priori and the transcendental, which Kant and neo-Kantians were unable to resolve because they still thought these relations inside the doublet-or-mixture form, as apriorico-transcendental fold. There is here a veritable transcendental deduction of the fractal a priori (of theory) whose validity or pertinence for the philosophical given is determined under a double condition: first transcendental, the condition of the real or of Identity; then empirical, the condition of its material, namely philosophy, which is fractalizable to the extent that it is first reduced, suspended in its claims by Identity itself. Instead of a supplementary neo-Kantianism, we have a non-Kantian conception of the fractal at the same time as of “non”-Kantianism as fractalization of Kantianism.

• On the other hand, knowledge’s specific reality, the existence of scientific representation, is neither an intuition nor a concept nor their philosophical synthesis or difference; it is their identity—a whole other thing than the old “intellectual intuition,” which is a philosophical mixture. Instead of “concept,” we prefer to speak of “theory” in this case. This theoretical intuition prevents theory from becoming an abstract and transcendent construction: it has a properly theoretical content (this fractal objectivation, which is itself complex or specified by several moments described earlier in this book and that constitute the form of objectivity), but it also has an a priori content from the manifold, which is furnished to it at the beginning by its philosophical data and that we described as chôra. So much so that theory is never a simple indeterminate generality, induced or abstracted from a manifold of given objects, nor a simple empty form of the logico-symbolic type. GF is a thought (an objectivity), but it is an intuitive thought (it has a priori contents and not only forms). Their Identity-of-the-last-instance renders absolutely indiscernible—although without turning them into an amphibology, quite the contrary—the form of objectivity (the “intellectual” side) and the content or the manifold (the “intuitive” side) under the species of which theory is concretely given each time. No line of partition, no philosophical decision, can still pass between those that are thus grasped and received “in-One.”

We should thus recognize that there is not only a fractality of sensible intuition, of “perceptive” and natural geometry, but more profoundly a fractal a priori intuition that is no longer opposed to the theory or is its simple base, that is the very essence of every theory. There is what we could call a real fractal a priori (neither formal nor material), which exerts itself in the form of theory and guides the scientific labor of experimentation. It is a question of theory as we have described it and not as epistemologies isolate, limit, and restrict it to an activity of theorization. For theory is obviously not one operation among others, terminal and “superior.” It is the very essence of science and what distinguishes science from philosophical “thought.” It is thus by means of their scientific treatment that philosophical objects will acquire a fractal nature and a reality rather than a real possibility and thus become “non-philosophical.”

• Let’s come back to the problem of intuition. The extraordinary primacy of seeing, of the eye, both of the geometric and of ordinary perception, in the theory of natural fractals is the sign of a problem that GF resolves in its own way. What we called fractal objectivity or objectivity-through-unilaterality is quite unusual: if philosophy distinguishes and unites transcendence and the transcendent, the distance of objectivation… and the object, in a mixture, science identifies them absolutely. On one hand, the object-(of-knowledge) is directly the theoretical transcendence; it is identical to this transcendence. The object-(of-knowledge) does not float within theoretical transcendence as in a preexisting element, in an abhorrent universal vacuum. It does not add its own particular transcendence to a milieu of universal transcendence that is destined to receive it. In a sense, there is in theory nothing more than this pure movement of infinite, unlimited transcendence with which the objects of knowledge are conflated. But on the other hand, the distance of objectivation—and it is always somewhat distinct from the object according to philosophy, which assumes that this object is partially hidden, in withdrawal or in reserve within a subject or an agent situated behind it—is in science (where precisely the “last-instance” is not a back-world or a back-act) integrally “objectivated” in its turn, i.e., thoroughly visible like the object itself. Hence the hyperobjectivism that science gives the impression of, but that is in no way the sense philosophers give to it—the philosophical sense of an exacerbated objectivation. It is rather the destruction of every objectivation/alienation. The dimension of fractal objectivity is more “objective,” if possible, than its philosophical equivalent. It is, as it were, thoroughly objective, without an act or agent in withdrawal. There is an identity (but of-the-last-instance) of the object and its objectivation. It is a phenomenal state of affairs: that of a static or structural objectivity, without objective things that add their ontic transcendence to transcendence itself, to ontological objectivity, but also without objectivation to divide and double it, as a verso divides and doubles a recto. The “field” of objectivity, or rather the theoretical opening, is static or unconstitutable by its description and can only be described as it is. It is an entity that can resemble a line (for the line itself), but without being a line; a surface (for the surface itself), but without being a surface; a recto (for the recto itself), but without being a verso; a curve without being a curve, etc. This is precisely the more-than-fractionary sense of theory, its generalized-fractal sense, by which it does not coincide with any dimension of geometric figures, nor with the ontological dimensions of the philosophical decision. We have to meditate on these fractal “images” themselves, to penetrate their unilateral structure, in order to enter into the “spirit” of science and not to conflate it with the spirit of philosophy.

• Lastly, since scientific representation is absolutely stretched out without horizon, an opening without ekstatic-horizontal transcendence, it is rigorously unreflected or “opaque.” It is indeed a matter of an opening, of a theoretical and intuitive a priori, of a thought. But in its intimate nature, it is deprived of every structure of reflection, of fold and refold. It is impossible for scientific representation to be a metatheory (in the philosophical and meta-physical sense of the meta-, not in its scientific sense) or an overtheory; it is impossible for it to be separated from itself, while remaining in continuity with itself, in order to survey itself. Science puts an absolute end to the philosopher’s position of overhanging, mastery, or coextension, the philosopher who has already surveyed/anticipated theory (as the subject that must be able to accompany its representations) because he aims from the outset for a horizon of teleological closure of theory and loses the patience of knowledge. The exercise of theoretical knowledge is absolutely immanent, even when it is related to its objects or its material, and this immanence of its theorico-experimental criteria forbids it the hubris of anticipation as well as the nostalgia of retention. It makes the process (not continuous, but indivisible or fractal) of knowledge a contemplation—rather than an operation or a practice—even within operations and scientific practice. But this contemplation is nocturnal, completely internal, without the light of reason, but not without opening. Precisely because the opening is in-the-last-instance in the mode of intrinsically finite Identity, it is also radically open, without double point or loop, without end or telos; it is even “infinite.” This opening is not illuminated in an anticipative way from a horizon that would hang over and close it. It is necessary, not to traverse it (it does not exist prior to thought as an object would, even if it is static and already-open: thought does not exist prior to itself), but to effectuate it each time from “objective givens” and “phenomena,” provided in this case by philosophy.

Thought’s fractality does not thus correspond to any fold or doublet, to any reflection, which are instead processes with continuous curvature. It neither reflects nor posits itself; it remains “suspended” to its cause. Universality is a nonangular fractality, without the—always bilateral—angle or fold of reflection. This hyperobjectivity of representation, which follows from Identity-of-the-last-instance, allows us to give a new sense to the affinity that generally exists between the fractal and the artificially produced synthetic images, the computer-generated images, or computer graphics, etc. For, more profoundly, if there is an objectivity without objectivation, without object or subject, without poles of identification, without transcendent rules of organization, not only is every theory from the outset a fractal image by nature (although nonperceptive, nonworldly), but its fractality suggests that it is absolutely objective (and not semiobjective, semiobjectivating); that it is “flat” or “stretched out,” without any thickness or profundity (save for a “surface” one) and can be produced or reproduced at will without requiring reflexive or philosophical conditions, which would separate it from what it can do, which would destroy its Identity-of-the-last-instance and would give it the “depth” of mixture or doublet, the “flesh” of perception. Computer simulation, with the help of mathematical algorithms, of some natural forms that have already been decoded as fractals, cannot be of direct and practical interest to us here. But in the application of the science of Identities to the figures of philosophy, we can clearly admit that if nondecisional Identity (of) self, the “last instance,” is a sort of transcendental automaton, then scientific theory is for its part a sort of transcendental computer, which thinks, but in an absolutely unreflected way (and which does not content itself for that reason with calculating); which constantly realizes, through the knowledges it produces, artificial or synthetic images of philosophy and its statements, which are its own “nature,” the “natural forms” it can simulate. Here scientific theory contains a priori the conditions of philosophy’s fractal decryption and does not need to search for them in an external organon (for example, in a mathematization or a logicization of philosophical statements and operations). GF does not have a mathematical, but rather a transcendental origin, an origin internal to every scientific practice. It applies, as a result, to natural and philosophical language. This problem will be taken up again in connection with “Artificial Philosophy.”

THE CONDITION OF CONTINUITY: PHILOSOPHY

Before describing the fractal structure more precisely, we have to reintroduce the fractalized object itself: philosophy.

In effect, what is traditionally called “thinking”?

What is called “thinking a phenomenon” can be reduced to a few invariant operations that define what is called “philosophy” or the “philosophical style.” Whether it is a matter of a “regional” (economic, aesthetic, sociological, semiological, ethical, etc.) or “fundamental” and expressly philosophical interpretation, the minimum gesture could be the following: to any phenomenon = X that we desire to interpret, we associate a continuous plane or space—a universal—of diverse operations, rules, and objects. These latter can be specified as economic, aesthetic, etc.—little matters. What counts is the association of a universal space to the singular phenomenon, a space that can be partially controlled and that serves to interpret and control the singularity under conditions of continuity.

What does “to associate” mean here? In what way is it a matter of a superior “associationism” proper to the philosophical style? We suspect that it cannot be a matter of an indifferent juxtaposition and that, moreover, a dialectic would be only one possible version of the matter. Association means that the singularity and the universal space of control intimately co-belong and cannot be defined outside each other, that there is between them no neutral zone or indifferent field of reality. Philosophy is the theory and practice of singularity as mixture or blending of contraries, nothing else. This is an ultimate invariant to which every philosophy can be brought through a supplementary interpretation.

To interpret, to comment, to dialectize, but also to deconstruct, all these operations—their reciprocal difference matters little, it is now irrelevant—have something in common: they respond to the invariants of the philosophical operation, of which they are modes. And this operation is always an idealized and transcendental, nonscientific geometry. More exactly: at once—since it is a double or divided operation—a transcendental decision and a transcendental topology. There is no event = X to which a universal space of presentation, of resonance, or of interpretation is not connected; no individual without an “associated milieu” that outlines series of possible interpretations. Thought rigorously, the relation would thus be that of a “difference,” whatever the way in which this difference is then varied. “Association” thus implies several invariant phenomena:

• The singularity belongs to the plane; it is a point of universal space, which can be defined by its coordinates or its properties. At the same time, however, it does not belong to it or not entirely; by a more empirical side, it is given as falling outside the plane. Singularity is, by definition, divided by the plane itself into an empirico-multiple, empirico-singular side and an ideal-universal side. This division of the phenomenon, its splitting, is not an inevitable accident or evil; it is the first operation of every thought that obeys the philosophical paradigm, a practical operation of dyadic scission or decision. Singularity is then distributed on two sides, empirical and ideal. Since its unitary concept is divided in this way, it involves it in a becoming of synthesis, of reconciliation, of production, of conciliation, etc. Singularity is never given as such, but, on one hand, it is supposed to be given, affected by transcendence or divided and, on the other, it is the end of a process, the outcome of a “concrete becoming.” These are the real’s philosophical avatars…

• The second trait, after division, is precisely the trait of identity or identification. By its ideal side, the one inscribed in the space of control, the singularity is affected by an already mapped-out future: to finish its identification with the totality of the plane; to become adequate to its expanse; to be completely interpreted and assimilated by the system of parameters (economic, aesthetic, semiological, etc.). Thought rigorously, this operation of interpretation or of reading of the phenomenon amounts to refolding the space onto the singularity and to stretching out the singularity to unlimited, though always equally finite dimensions of the plane. After the singularity’s division (a nonempirical division but in the dimension of the transcendental), there comes its doublage, doubling, and redoubling in the form of a simultaneous becoming-universal-and-concrete. The duality of singularity was “in itself”; now it is “for itself” or manifests itself as such.

The singularity is thus given twice—this is its philosophical regime. Once as divided from itself or underdetermined and a second time, but it is the “same,” as doubled on itself or overdetermined. Determination is knowingly weakened so that it can be reinforced. To philosophize is to multiply the doublings and the envelopes, to accumulate the representations and the control over singularity. All these interpretations, which “have to be able to accompany” the singularity, begin by dissolving its reality in order to present themselves in their turn not only as an overdetermination, but as a constitutive codetermination of its new reality. Semiological, sociological, psychoanalytic…spaces—and this doubling of all doublings, the ontological space or philosophy—conceal singularity and reality by claiming to manifest and constitute them.

Reduced to its minimal “eidetic content,” this is what here and there we unwittingly call thinking. This “continuous” paradigm belongs to philosophy. Here nothing apparently allows us to speak of a philosophical massacre of singularity or of determination in the name of mixtures. And yet it is in this sense or this sentiment that we are engaged. The complexity of philosophical operations and the very great simplicity of this schema cannot be raised as objections in this case. The complexity is simply the development of the simplicity and thus proves nothing. On the other hand, in the apprehension of singularity as mixture by the philosophical paradigm, this forced labor, this interior and exterior decision, this operative (even if transcendental) side of philosophy—its “technology” side—leaves us scientifically unsatisfied. Transcendental technology to be sure, in that it bears on the whole or the essence of the real and not only on a part of the object. But if philosophy is a transcendental technology of continuity, in what sense does still respect singularities?

The problem, as we have said, is to know whether the real, in order to be what it is, needs to be associated with a possible, to be divided in itself and stretched in the form of a universal space or a continuous curve; or else, whether it not diminished by this amphibology with the possible. Philosophers, who are often all too hasty (some even integrated the haste into the essence of being), will admit without much hesitancy that the multiplication of fundamental principles (the One, Being and the Other; the Same and the Other; Identity and Difference), that the dyad of principles is a source of enrichment, of complexity and of rigor. This is a tragic illusion: the unitary multiplication of the principles of Being engenders their interinhibition, as if the possibilization of the real and the realization of the possible were equivalent, through their chiasmus and their accumulation, to a real that would have already been determined in itself. This philosophical procedure of division and of doubling is, in the same measure, the scarcity of interimpediments between principles, the war at the heart of the real. Within philosophy’s framework, sufficient determination never meant anything other than scarcity and war. The sufficiency of reason is another name for the blending of principles. And the underside of this blending, the ground on which complexity rests, can only be the impossible partition, the transcendental rarity that sows violence throughout the entire real. By definition, the real is enough; it is not rare, this problem does not arise for it. On the condition, however, that we first dissolve its amphibology with the false “sufficiency” of reason. If philosophy is a practice of weakening the real (it divides the real, refolds it on itself; it stretches, doubles, and envelops the singularity, represses or resists it), the extreme tip of this resistance to the Determined is precisely what is presented in the history of thought as the Principle of Sufficient Determination. A reevaluation of the reality and materiality of singularities should pass through the unconditional suspension of this principle, which condenses all the philosophical equivocations.

GF’S AUTONOMY: ITS CONSTITUTIVE ORDERS

We can now begin to describe the relations between the fractal structure and the fractalized object. We progressively make our way toward the scientific concept of “generalized chaos.”

We have defined GF by internal and specific criteria: by its essence (Identity) and by its specific structure (Unilaterality). It is remarkable that we do not need to define it as “relative-to…,” to the material it fractalizes. Here the Other, fractality, unlike what happens in philosophy, is relatively autonomous with regard to Identity, but it is sui generis, independent in its essence from the continuous curve it breaks. This is the essential step in the conquest of a GF. And yet this Other is clearly not without relation-to philosophy, even if it is without relation-to…Identity. This is the problem of the condition of continuity, which has become secondary here and which we have had the occasion to examine vis-à-vis philosophy.

In a certain way, GF is situated “between” Identity (of) self and the given of the philosophical Decision. It is thus true that in a first, very external, approximation it also has the nature of an intermediate being: between the continuous, more or less hesitant curve of philosophy and the Identity that is so “Other” it precedes philosophy absolutely and without alterity. If fractality exists, it is situated “between” nondecisional Identity (of) self and the philosophical Decision. But if the exact meaning of this “in-between” can no longer, by definition, be philosophical or associated with a philosophy, if it depends for its reality on Identity alone, then it must be particularly elucidated.

If philosophy’s fundamental space has a fractionary dimension 2/1 (subject to the aforementioned reservations), the fractal at issue will not confine itself to filling this space once again, nor even to overflowing it. For philosophy is already this line of thought, which, by dint of folds, points of catastrophe or transmutation, retrogressions and turns, can exceed this space and approach the surface. GF, moreover, cannot be posited in relation to the philosophical space taken as unit of measure. It does not even extend beyond this space in the manner of an Other, deconstructions’ Other. It comes from elsewhere—from the Identity that is not an “elsewhere”—and it comes “in front” of philosophy, interrupting it radically and without negotiation. It is GF that defines a new space of thought, a quasi-space, that includes in an unequal mode the philosophical 2/1 space and its restrained fractality. The meaning of fractal inequality changes. It is not the fractal that is unequal and relative to the norm. On one hand, it is real or “identical” as inequality. It becomes a fractality de jure—more than de jure and other than a “property.” It is “in self” [en soi] insofar as it is identical-in-the-last-instance, i.e., in itself [en elle-même] (where Identity acts) rather than far from itself (the “last-instance” is not the transcendence of a distant cause, quite the contrary). On the other hand—here is the consequence—it is instead the norm of philosophical continuity that becomes unequal to it, absolutely and irrevocably unequal because it is unilateralized by GF. Fractality is not itself fractal (not doublet or fold); rather, it fractalizes philosophy. Thus this inequality in philosophy is not a property of philosophy itself. It is the Inequality/Unilaterality born from the Identity-of-the-last-instance that is imposed on philosophy and “renders” or “makes” it unequal, radically fragments or interrupts it. It is the norm that is pervaded by fractality, rather than fractality that is decided in-relation-to the norm—but this is because fractality exists in some sense “in itself” and forms an autonomous sphere of existence. It is clearly the determinant-of-the-last-instance cause that requires this new distribution of unilaterality. The term fractality should be maintained because this apparently intermediate sphere is the Other. GF is no longer the Other-of…Identity, but the Other-of…philosophical continuity; it affects this continuity without becoming relative to it in its turn.

In relation to GF, philosophy’s fractionary space in a transcendental mode appears as a “whole,” for example, as a circle or a continuous curve. Even if philosophy’s line is not simply or only continuous, if it is semicontinuous, semidiscrete, it remains necessarily a curve, whether it is topologically drowned or not, and it is “in relation” to this state of thought that GF appears as something other than a curve or a fraction (or a doublet or a mixture). If we are committed to keeping the term fraction, not only should we call it transcendental, as we do for philosophy, but we should modify the sense and scope of the fractionary bar and stop positing it as primary. Instead, we should posit as primary the Identity (of) Inequality and assume that there is really no continuous bar that unifies the present parts, that Identity-of-the-last-instance is not a “part(y)” in a game or a division.

The “fractal dimension” thus continues to exist, even as generalized. It defines a degree or, more exactly, a type of irregularity:

• unquantifiable…In ontic sciences, the degree of irregularity or fragmentation of a curve can be quantified. Already in philosophy this is no longer possible or it is a matter of an ideal quantification. In non-philosophy there is no longer any possible quantification. The greatest fractality is qualitative and lived immanentally. And yet, as the science of the One that is a science of philosophy, of sense and of language-in-philosophy, it is indeed scientific. As we understand it, first science, which is homogeneous to other sciences, is a transcendental mathesis;

• without autonomy vis-à-vis Identity itself (not “first” like geometric or geometrico-philosophical fractality). Identities in themselves are not fractal and will never become so on pain of falling back in their turn (as a continuous Whole or a continuous Tradition) on fractality. But they are the condition of reality, the cause of fractal structures that form the tissue of the theoretical representation (of) these Identities or (of) the real. But even if fractality needs Identity, it is a richer concept than this Identity; it contains a supplementary “dimension,” the fractal instance proper that enjoys a relative autonomy. If philosophy makes no irreversible and stable distinction between Identities and fractality, which affects them in return, science is founded on this distinction and, furthermore, on the cause that renders it necessary and grants fractality a relative autonomy;

• autonomous on the other hand, in its essence at least, vis-à-vis the philosophical manifold it fractalizes;

• partially dependent, from the standpoint of its existence, of its object or of its material; it represents in itself a strict and infinite—“chaotic”—fractality (albeit without a procedure of interpolation or division), an essentially perfect fractality. It obtains this relative autonomy from its cause and it is with this autonomy that it occurs as radically self-similar.

GF is not an operation or the outcome of an operation on a prior or related identity: division, distanciation or differentiation, recursive interpolation, etc. Identity-of-the-last-instance is not prior—given in a presence—or associated with such an operation. It is given “in” itself outside every presence, in such a way that if it is followed by an effect, this will not be an operation or an act, but a phenomenal and static state of affairs to be described rather than produced. This state of affairs is fractality as unilaterality. And fractality, unable to form a system with Identity, which lets it be, is abandoned to itself, to the distance without closure that it is; opening without horizon or project, without loop or node. It is impossible to interpolate or extrapolate voids and additions, lacks and supplements, from an Identity that is not inscribed in Being or Presence, in Transcendence, an Identity that remains only “in” itself. Unlike geometric fractality, Identity is no longer the fractalized natural support, it is the cause of fractality; it frees it to itself, to its more pure state without limiting or inhibiting it: fractality in itself or as such. Conversely, the fractalized support or material (philosophy) is less an identity than the mixture of identity and of difference (infrafractal or semifractal mixture) that constitutes philosophy’s essence. Restrained fractality belongs to the great sphere of technological objects, of objects obtained by division. Generalized fractality belongs to the immanent or indivisible phenomenal sphere and can at most be described without being realized. The first has to do with properties (of objects givens in transcendence), the second with essences that, by definition, can never be detected or identified in the World, but belong to the most immanent experience of thought.

THE FRACTALIZED MATERIAL: OCCASIONALIST CONCEPTION OF THE FRACTAL OBJECT

What occurs as irregularity identical (to) self in-the-last-instance is Unilaterality’s qualitative or specific order, its structure insofar as it is manifested in the mode of Identity. It occurs, as radically identical, let’s say directly and immediately in the philosophical manifold that accompanies it as well as against it. As fractality of-the-last-instance, its relation to the manifold is complex. It cannot be identified as a “natural” property of this manifold (philosophy), it occurs elsewhere, and yet it forms a radical irregularity that does not work against its material except directly on this material, if we may phrase it this way. We will not therefore look for a “contradiction” to mediatize or resolve between the manifold of philosophy and the fractality that structures it according to this strange relation of the-last-instance. There is nothing to mediatize; there are no longer any “contraries.” Fractality is indeed the fractality of philosophy, but insofar—at any rate—as the of no longer signifies a belonging to philosophy, a “natural” and “originary” property of philosophy, but a coming (from) of fractality itself as an autonomous order to and toward philosophy, the advent of fractality “in front” of philosophy, which is really affected by it, but only from the perspective of this fractal order itself or of the Identity that subtends it.

GF is a causality that, like Identity’s causality but perhaps less radically, affects or transforms an object without this object conditioning the action in its essence, without its own action surpassing the action of an occasion. We have to be more precise. We have not set out in detail—this was done in the previous chapters—the exact role of this occasion. But GF, the GF of science over philosophy or of philosophy when it is “seen-in-science,” presupposes an “occasionalist” conception of philosophy’s role and thus of the fractal “object.” Philosophy, its manifold at least (PSP is suspended)—not only its invariant representational content (its statements), but its structure-of-decision—is what covers or recovers the fractal structure, what incarnates it: its role is limited to that. GF is no longer the reciprocal system of an external operation or figure and of a transcendent manifold that codetermines it in an equal way (the fractality of a figure, of a curvature of object, a fractality specific to them). It is a relatively autonomous order, specified by this manifold, but not specific to it; it no longer presupposes the action of this manifold for the essence of fractality itself, but only for its material.

If the “fractal” object—qua fractalized material or support—does not act on the essence of the fractal structure, if it is only a cause in the mode of an occasion, the fractal structure on the other hand acts on it and determines through it a fractal manifold whose concept will be crucial for the definition of chaos.

If Identity is not alienated in fractality, fractality will not be alienated in its material. It remains universal each time it occurs as unique—thus “always.” But, or because of this, it affects its materials in each of their points or sites, since it determines them as materials. This is not the case of geometric fractality, sandwiched between two generalities: its homothetic identity, which is not absolute or absolutely determined on the plane of the real, because it is merely one property or one knowledge among others; and its geometric or physical materials, which it is incapable of cutting through from end to end and remain partially outside fractality. In such a way that some play or some indifference, some nonfractality, envelops fractality and attenuates its vigor. Identity, whatever its place or the object at issue, must be seen “in” itself (what we elsewhere call the vision-in-One or the One-in-One) and must thus serve as an unbreakable guiding thread, so that fractality can—without deteriorating—penetrate its material and determine it precisely as fractal manifold. Determination-in-the-last-instance is neither a totalization nor the smoothing of a curve. It safeguards the fractal manifold from every reappropriation by philosophy’s continuous procedures. Only the most solitary, the least alienated Identity can give the fractal order its identity-of-instance and assert it with respect to philosophy and against it, against its autointerpretation.

THE CONDITION OF PHILOSOPHICAL SCALE-AND-DECISION VARIATION: THE CONSERVATION OF FRACTALITY

Complete fractality has a double aspect. It is not reduced to Identity even if it manifests Identity as it is; but it contains the side of Unilaterality as a radical irregularity. Unilaterality in its turn has two aspects: by one of its sides, it is this Inequality in principle, which does not cease to reproduce itself as identical-in-the-last-instance without erasing itself as a result; by its other side, it does not stop varying its philosophical content. This irregularity-to-philosophy, this distance-from-the-World, reappears identically through variations, those of the types of the philosophical decision, those of the scales at which the content or the details of this or that decision is apprehended. The condition of variation will thus change in relation to geometry.

On one hand, it is complexified in decision-and-scale variations proper. The manifold of natural language, which incarnates every philosophical decision, corresponds to the details of figures that can be accessed through rescaling or variations in magnitude. But the change in the philosophical decision corresponds to a variation in the region or domain of “natural” fractality. Consider a vague definition of the fractal as formed of similar structures whose elements are ever smaller. Just as the internal similitude is insufficient and must go up to identity, so on the side of dimension the scale-and-magnitude variations must go, in the objects of knowing, up to qualitative variations of the philosophical decision and not only of the order of magnitude in the “detailed” analysis of a philosophy. More precisely, the manifold to be fractalized is not so much objects or categories as three qualitative kinds of manifolds: 1. natural language or the invariant representational content of a particular philosophy, 2. the historico-systematic manifold of decisions, 3. finally, the manifold of structural moments of a philosophical decision as such. It is first and foremost in relation to these constitutive manifolds of the philosophical, and above all in relation to the last one, that fractality asserts at once its irregularity and this irregularity’s order or constancy.

On the other hand, the condition of variation no longer plays the same role as in fractality’s geometric or philosophical forms in which it coconstitutes and codetermines this fractality. It becomes contingent. This does not mean that it is not necessary, but only that it is not necessary to the structure and essence of fractality. The structure of Identity-Unilaterality is a sort of homothety, but it is absolutely internal. It is no longer legible in the World; it can only be read in thought insofar as its cause is absolutely immanent: a transcendental homothety. It is indeed an invariant, but, unlike philosophy’s semifractal invariants, it is entirely immanent by its cause and not given at once in the two supposedly equal modes of Immanence and of Transcendence. Far from having a statistical side like the philosophical, or at least an objective appearance of identity or “sameness,” far from being codetermined in its invariant nature by the manifold, it is a strict self-similarity that is not itself modified by the emergence of a new manifold, that does not give place to the appearance of a Same, but that on the other hand ceaselessly gives a new figure to this manifold. The change in philosophical decisions or in scales within the description of a decision does not modify fractality itself. It does not produce a simple “self-similarity,” a “family resemblance” or a continuous “curve,” an “allure” or a “style,” in which case the manifold would “fuse” with the fractal structure, as in philosophy where the very essence of fractality is continually modified and thus effaced or limited by its companionship with Identity.

GF is a constant and a specific constant for each text or statement produced as “fractal.” One constant is distinct from another inasmuch as it is specified by the manifold of text-materials. This manifold is in fact included in the fractal structure as what is fractalized: it does not determine the essence of fractality. On the other hand, it intervenes in fractality for some precise functions that are variation factors of fractality itself. It is not a matter of invariants or of variance-of-variance, where identity would be certainly invariant, yet mixed or combined with the philosophical forms or contents, and would be deformed with them. There is fractality because Identity remains what it is and is not transformed with its contents. This nonconservative “conservation” of radical Identity, which is everywhere the same in different philosophical milieus, produces fractal structures of a new type. The variations in philosophical decisions or contents, increasingly analyzed and diversified contents, reveal, or allow to be manifested, not the “same” structure, but rather the identity (of) a structure. If singularities and differences are founded on the Same, on the conservation of variations, gaps, or differences, generalized fractality is founded on the conservation of strict Identity. The details vary according to the chosen philosophical scale or decision (a given philosophy does not reveal the same content from the perspective of this or that other philosophy), but each has an identical structure or internal form. The main property—in sciences or “natural” fractal objects—is internal “resemblance” or “similitude,” but in the domain of the objects of science, then of philosophy, the property must go up to Identity. For if it were a question of a simple resemblance or similitude, philosophy would be powerful enough to explain it on its own. But we are searching for a fractality that cannot be reduced to philosophy’s laws of continuity.

Furthermore, this contingency of decisions is necessary for the exercise of fractality. We will show that it has to aim—but precisely without objectivating it, thus without letting itself be codetermined by it—for a certain manifold, supplied in this case by philosophy itself. The variations in magnitude or in position-and-decision are no longer necessary for fractality itself. They are relevant only for the construction or manifestation of a concrete fractal object in which they intervene as factors of transcendent specification and incarnation, as quasi-philosophical modes of existence of irregularity’s a priori structure. What is modified is the manifold itself, not only through its external decision-and-scale variations, but more profoundly through the fractal structure that is imposed on it and that is identical and not only similar, analogous, etc. As to this structure it does not vary insofar as it is thought and received from its cause, from its identity-of-the-last-instance, directly from the manifold, but only inasmuch as this manifold specifies and overdetermines it.

The introduction of radical contingency—more than the “aleatory”—signifies philosophies’ indifference. What we have elsewhere called the principle of equivalence of philosophical decisions receives its full sense here. Its very function is to introduce an element of chance, even of “chaos,” into the collection of objects or to randomize it. The material is randomized in the form of an equivalence—which is required by fractality itself and is not arbitrary or imposed from the outside—of decisions that organize the language that incarnates them. In this way the initial idea of fractals is radicalized, according to which there is no difference of nature between extreme variations in time and space, but rather the continuity of an irregularity. It is now the variations themselves, and not their difference in time and in region, that are contingent in relation to the fractal structure. We said that fractal Inequality emerges as Identity without returning as Same; that is to say: this type of invariance finds its reason in Identity as cause rather than in variations in the degree of resolution or in scale expansions.

This change in fractality’s terrain forces us now to distinguish the rigorously “self-similar” fractals, those of science itself, from philosophy’s, whose self-similarity is merely “statistical” or “average.” For philosophy itself, its own fractality is unstable. It become constant and forms the specific order of philosophical disorder only for a science of philosophy. However, the constancy of the fractal mechanism no longer means, as we saw, an analogy, a similitude, a resemblance, or a univocity— philosophy’s nihilist, be it “superior” and counternihilist, boredom. From one decision to the other and from one region to the other of the same decision, fractality is identical only in-the-last-instance.

Thus fractality can be truly generalized only if its cause, the Identity that remains in itself, is not alienated in it; and if, correlatively, the condition-of-variation becomes more contingent than it is in philosophy, but also in geometry where “chance” and the “aleatory” fulfill this function. Identity remains what it is without moving to the Same, without drowning in knowing or in nature. And the manifold of variations becomes for its part absolutely contingent and plays its role in the form of this very contingency. The two sides of fractality are emancipated from their ongoing identification, from their philosophical becoming.

FRACTAL INTENTIONALITY: INTERFACE AND UNIFACE

A last element—not the mixed form itself as a philosophical relationship or relation between two given terms, but the relation of this mixed form to its manifold—also undergoes the effect of Identity and enters into fractality. What does it become in order to be the new relation “of” these new terms?

On one hand, the mixed form (we analyzed it in previous chapters) lets itself be decomposed into a certain number of a priori functions or structures, those of the philosophical Decision or the procedures of its continuity. These structures lose, under the effect of Identity, the mixed form itself, their autoreflection (in this way reflection can no longer be said of itself). And they become non-philosophical a prioris, nondecisional and nonpositional (of) self.

On the other hand, these a prioris, which are now proper to the fractal structure, are, as such, endowed with an intentionality (a priori of…for… ), an aiming for experience or for the fractalized manifold. But this intentionality can, in fact, no longer be that of consciousness or Being, which has the mixture/doublet form and objectivates its “object.” This is the solution to the previous problem: the relation of GF—of Unilaterality—to the object “philosophy,” a relation of which we said that it was an “immediate presence to…but against…” this object, is nothing other than this nonmixed, nonfolded intentionality, this simple “aim” and this simple “of” that no longer objectivate that to which they are related.

It was a question of resolving the problem of geometry’s application to physics or of the validity of the former for the latter. This problem is generalized here in the following form: how can GF be the fractality “of” the object “philosophy”? How can first science become the science “of” philosophy? The solution has been established. Unilaterality “in-Identity” is indeed lived as close as possible to this manifold; it is oriented on it without being essentially conditioned by it. It has a transcendental function. It is not a transcendent and abstract form imposed on the manifold; it is not the “autoposited” or “supposed” Other, which alters this manifold of philosophy from the outside and arbitrarily. It is the One-Other, the Other “in-One,” and this explains the otherwise inexplicable precession of the Other in relation to this manifold.

At bottom, what we have elaborated, at least in principle, is a fractal a priori—an a priori fractality. It is thus also the existence of a genuine fractal intentionality in the following double sense:

1/ GF is not a simple transcendent property, inert and given with the World, whether this property is that of a “natural” object or a prior knowledge that is simply reified in “nature.” It is endowed with a genuine immanent intentionality of a special type, which does not objectivate its manifold. Having a radical cause that determines its nature, it is destined by this cause to fractalize its object. We must thus distinguish not only the cause of fractality, but fractality itself and the object it fractalizes.

2/ Conversely, the intentionality of fractality is itself fractal and its concept is obtained in this case, on the theoretical plane, through the fractalization of intentionality’s traditional phenomenological form.

What can this fractal intentionality mean for the neighboring concept of “interface”?

Science is a fractal thought whose essence of radical immanence nevertheless rules out that it take the form—even the metaphorical form—of a figure (line, surface, volume) in general. In contrast, philosophy is a curve or a line that tends toward a surface, as we saw, and that touches in each of its points on an extraphilosophical manifold. Science is so fractal that it does not take the form even of a semicontinuous curve and does not touch in each of its points on the philosophical as on a supposedly “nonscientific” manifold. In its essence, it is not an interface—at least in the traditional epistemological sense of the term—and does not have an interface with philosophy. But, as science of Identities, it nevertheless uses philosophy primordially. In thought’s domain only philosophy and technology are interfaces and semifractalities. And it is science that introduces the most radical fractality at the same time that fractality affects the philosophical curve, limits its power of interface as such, an action for which fractality uses no supplementary interface. In the particular relation it entertains with its “exterior,” with the philosophical, it suspends straightaway the philosophical’s claims over the real, the functions that philosophy exercises through its structures of interfacing; it reduces the philosophical to the state of materials stripped of every claim over itself and over the real. There are no lines and no surfaces of separation common to science and philosophy, which have essentially different functions. Scientific fractality is not codetermined in its essence by philosophy.

The description could, nevertheless, be more precise. Science is not an interface in the techno-philosophical sense of the term, but we can assume that it represents the real, radical kernel of every possible interface. This is what we call the scientific a priori representation (GF’s order or instance) as nonpositional reflection (of) the real. The ensemble of a priori structures of scientific representation, and thus of Unilaterality, functions as an absolute interface, a “surface” of monstration or manifestation of scientific knowing, which is infinitely open and is thus not destined to be received, captured by a third person, by a supplementary observer like the philosopher. Knowledge objectively “shows,” without any external subject to receive and redistribute it. Unilateral interface, if you want, which is in reality a Uniface, a structure without bilaterality, not a common frontier or limit of more or less interrupted or inhibited exchanges, but a “surface” without verso and without loop, without borders or folds. In this sense science is the experience of an a priori interface. But this interface does not form a system with the philosophical materials it treats.

Conceived in this sense, the interface is entirely fractal. For the fractal-generalized object par excellence is scientific representation. It is not obtained through division—interpolation and extrapolation—of a transcendent irregularity composed of folds or of angles, but through the Identity-of-the-last-instance (of) Unilaterality. So much so that the interface and the most irreducible fractality are identified without remainder in Unilaterality—we can see why fractality no longer takes in this case the ready-made form of the peaks and angles, the triangles, the promontories and bays that made it “natural.”

FRACTALITY OF THEORETICAL SPACE-TIME

Identity-of-the-last-instance is indeed an identity “of” contraries, but there is here no aiming for, no intentionality of these contraries by Identity, which does not emerge from itself and is not alienated in what is seen “in” it. If theory itself, the fractal or unilateral objectivity, as we have examined it, aims through an intentionality of a particular type for the materials it fractalizes, this structure is ruled out for the cause (of) theory, which does not entertain for its part any relation (of transcendence) with this structure. It is the manifestation, the phenomenalization of contraries, but in its own condition of nondecisional immanence (of) self. So much so that a new “economy” of these contraries flows from it, a distribution that is not mixed or unitary-hierarchized. It immanently individuates or “identifies” each term outside its relation to the other (hence the quasi-phenomenological layers, strata, spheres, instances, etc.)—for example, the theoretical and the experimental. But it does so each time by radically identifying the old “contraries” outside every simple relation of co-belonging or synthesis. Identity of each term liberated from the other? More exactly: liberated from its philosophical relation (internal, substantial, dialectical, external, differential, etc.) to this other term, but not from this term. Identity is certainly “present” in each point of the materials or of other instances. But it cannot be reduced to these instances. It is seen “in” itself only, “identified” in an absolute way and not as “X or Y,” as identity of X or Y. It implies that the philosophical couples of X and of Y cease to be couples, that their terms are adequately recovered, but once as X and another time as Y.

The dual, the greatest inadequation, proceeds as a strict or immediate identification of contraries, but under the sole law of Identity… This identification is immediate, nondialectical for example. It does not take place in the One itself—which remains “in” itself without being affected by this identification—but in an element that is produced as secondary and that renders this immediate identification possible. Yet even if it takes place outside the One, it nonetheless takes place through the One and with the contraries of the philosophy-or-mixture form. Even if it is not a matter of a dialectical identification, there is a production of a middle term, of a relatively autonomous instance: the a priori theoretical representation or science. It is in Transcendence, with it, in its element, not in the One itself, that this identification of contraries takes place and can be called absolute, antiamphibological, the identification that is given at the origin as a material in philosophical transcendence.

What form does this process of determination-in-the-last-instance take? Just as Difference is not 1 = 2 but the becoming 1 ↔ 2, and just as becoming is mediation become universal, so Identity—which is immediately the contraries (certainly in-the-last-instance)—nevertheless needs…or produces a third instance (also a pure becoming): theoretical representation. With this difference—since there has to be one: this becoming-(of)-knowledge is not the becoming of the real, of Identity itself, and does not involve it in any way. It is thus an absolutely straight or unlimited Becoming, without loop or topology; without returning to itself—without the Eternal Return of the same. Nothing returns, not even difference, and there is even no “difference” to return. Becoming through and through, already the Open or the Faraway, the always-already-Faraway as such; Unilateralization and never the bilaterality of Difference. Every survey-return, every reflexivity or semireflexivity of the type “Difference” is excluded.

The two contraries must “be” 1, must be-identical strictly, 1–1 and not 1–2. They must undergo the 1 that is nothing but 1, which is not at the same time 2, and which alone will determine them as 1. Here it is not the 2 of Difference that is autoposited and reflected as 1 (thus always still as 2) in itself. No: the 2 must be immediately 1 that is nothing but 1; it must undergo the form of Identity. This is possible only if it engaged in a becoming (every becoming is always a “synthesis” of contraries), but absolutely stretched out = 1, with dimension-1, deprived of every autopositional, carnal, reflexive thickness. A few nuances have to be introduced into the preceding description.

On one hand, it is tricky to speak of identification of contraries without adding anything more. This process would be that of Difference, in which A is identified not with B, but also with the becoming-of-B. It is thus a matter of partial identification, of becomings rather than of objects. Here, in contrast, Identity remains in itself and excludes every identification (even partial identifications). The contraries are not grasped through their reciprocal or reversible identification, but through their identity (of-the-last-instance), which is not engaged in their affairs. It cannot thus be a question of a simple identification of contraries among themselves, but of each of the terms as noncontrary, without contrariety. The Identity “of” the passage from one contrary to another, which remains itself without being alienated in this passage, or which is nonpositional (of) self, is not reflected in becoming and forces this becoming—or lets it be—to remain “pure” becoming, absolutely stretched out or flat, without horizon or loop, without cusp point or point of transmutation—a static becoming, a becoming without passage. Philosophy is the looping of time in space, for example, and vice versa; theoretical knowledge emancipates them “from each other.” This means: from their reciprocal relation rather than one (from) the other.

Can we then speak of “becoming”? Or else, once the philosophical con-traction of contraries, their syntax, is removed (their fold, coupling or doublet, their dialectic, etc.), must we speak of a contiguity of terms, a contiguity that excludes a priori every synthesis and every transcendent bond (dialectical or otherwise) that would claim to survey them? We have to speak once again of chaos; the sole organizing law, the only truly immanent economy of the terms individuated in this way is the fractal structure of Unilaterality, which alone determines them as the ingredients that form chaos. Contiguity is a last form of continuity; the juxtaposition of relatively indifferent terms is a last “logic.” For chaos to exist, indifference must not be relative, reciprocal, “between” terms; it must come, after the One-Multiple of-the-last-instance, from the fractal structure that determines them (precisely as terms) by being indifferent to them. The variously articulated relations that compose the Multiplicities, the Differences, the Catastrophes, the Folds…are lifted here for the benefit of each term’s Identity as 1. Even Transcendence’s content is seen or received “in-One” and receives an Identity that holds “for” it without being dependent on it.

GF is not Identity that is subjected to logic, to its positional transcendence, and that has become Principle of Identity—or subjected to ontology. On the contrary, it is as it were the logic of Identity, the law that imposes this logic on thought.

On the other hand, it is tempting to speak of a fractal time, of an absolutely originary fractality of time. In reality, this generalized fractality holds for space-time; it is even their point of radical or of-the-last-instance identity. No doubt it is a matter of a time without ontological temporality or of a temporality without transcendent temporal things that add their own time to an indeterminate temporality-generality. Yet the same is true of space: a space without ontological spatiality or a spatiality without spatial things. Transcendence’s most primitive essence, its phenomenal kernel, is the fractality we have described. It rests on a profoundly antiamphibological Identity, but it holds for the “contraries” that are space and time just as it holds for all the others. We thus eliminate the old hierarchies, antinomies, and aporias of space and time that philosophy produced in its history. This is why the Greek obsession with becoming and its return in modern and contemporary philosophy as the primacy of a temporalizing, originary, and open temporality of space itself must be now dismissed. We have described the phenomenal traits of theoretical representation: as nonhorizontal sprawl, as recto-without-verso, as nonlooped-opening, as unlimited becoming in the sense of each-time-one-time, etc. The categories of becoming, of passage, of continuity, of discontinuity, of temporalization, of temporal ekstasis…are philosophical categories charged with dominating Multiplicities, Catastrophes, and Singularities in the philosophical sense of these terms. And those categories are marked by a certain primacy of time over space. From this perspective, we will prefer more “fractal” descriptive categories like identity, unilaterality, spread(ness), flatness, “without” (horizontality-without-horizon, etc.). Fractality is “identically”—but in-the-last-instance—space (and) time. This prohibits every hierarchization that would be merely a way of reintroducing continuity where only the law of unilaterality reigns. The variously balanced amphibologies of time and of space are dissolved or suspended in the real theoretical labor. It is true that GF also consists in moving from a time blended with space to a purer time—as Henri Bergson and Martin Heidegger sought, although they did not succeed to the same extent. But if the “pure” or the “original” still forms a system with their mixture, on the basis of which it is extracted and which it conserves as a constraint that is impossible to absolutely lift, then it merely hierarchizes in another form what should no longer be hierarchized. Time and Space gain their identity as “terms” only if what is at stake is not their identification, but a “last-instance” that lets them be as such and protects them from their reciprocal determination, which would only inhibit one through the other. Here again, it is perhaps in terms of fractal chaos that the reciprocal “nonrelation” of a fractalized time (and) of a fractalized space must be thought every time, in a unique and uncommon way. This is fundamental for the description of the new fractality: the fractality of theoretical space-time and, in this way, that of language. This fractality is no longer manifested in the World as geometric fractality or in philosophy as temporal fractality. It is manifested as the purest form of transcendence itself; as the spatiotemporal opening, but without “ekstasis,” without topological, catastrophist, revolutionary encystment. It is a matter of a delooping (in all the senses of the term), a deteleologization of time and space as well as of thought and language.

GENERALIZED CHAOS; PHILOSOPHICAL ORDER AND DISORDER; FRACTAL ORDER AND CHAOS

The problem of GF’s “intermediate” or nonintermediate character contains yet another aspect and has to be reexamined. GF is not only an original “intermediate” being, asymmetric between Identity and philosophy. This time it is intrinsically intermediate inasmuch as its essence is the strict Identity that remains (in) itself. How, by what effect, can an Identity remain intact, unalienated, at the heart of a coupling, of a milieu formed at the origin of two terms? Identity is manifested by its effect, Unilaterality as GF, but what is GF’s internal concrete content?

The material on which Identity acts is what we called philosophy’s mixture or doublet—its semifractality. Identity annuls or invalidates philosophy’s spontaneous self-reflection: if there is a fractality “of” philosophy, it cannot be discerned unless philosophy stops thinking itself. But the residue that subsists is still the mixture, the doublet-form, though it is “fractalized.” It thus acts positively, as we saw, on a coupling of opposites, which has simply become sterile. There is no reason to choose one of the two terms rather than the other. Identity does not choose, and GF is not a decision, an irregularity of this type; it is, rather, a finished stricture, an absolute constriction, in short an Identity. Each of the two terms is purified of the other, of its blending with the other, more exactly: of the mixed form itself. It is identical (to) self in-the-last-instance and without passing through the mediation of this form, without drawing itself on the horizon of this curve. It is a term in the full sense of the word, which has no need to be mirrored and alienated in the other because it obtains its being from Identity itself. It is not reduced to Identity, but enjoys the plenitude of its quality or its nature. It is a relatively autonomous order, instance, or sphere. The rule is to describe, without any blending, what each term of the fundamental couple becomes according to Identity’s guiding thread, without nevertheless forgetting that the fractalized object is always composed of two terms and that each of them is therefore found in fractality and has to be described (for example, what is the “scientific experience” that is thought under the condition of Identity of-the-last-instance? And what does “theory” become under the same condition?).

It now suffices to combine the identical-in-the-last-instance terms and the structure of Unilaterality, of the GF that is related to them in the intentional, nonobjectivating mode and that functions as their fractal “law.” GF is the reason that distributes philosophy’s basic terms outside philosophy’s ultimate law and now according to the law of the greatest irregularity. This is why the terms supplied by philosophy are henceforth independent of the law of the mixed form, of every philosophizable syntax (this law is not arbitrary, but applies to every syntax given with the World and Transcendence). However, this disorganization of the philosophical is not a bad nihilist chaos, because this manifold of determinations is distributed according to GF’s now immanent structure, which is related identically to all these determinations. We call chaos, in a more complete sense than the initial and always scientific sense, the fractal and immanent distribution of a manifold of determinations. Here chaos is no longer only the One-Multiple but also Unilaterality insofar as it is exerted in toto within the manifold of philosophy. Philosophy, with its manifold of decisions, dimensions, and so forth, no longer intervenes as such in this case; it is reduced. It is indeed the philosophy-form that is GF’s material, but the philosophy-form that is reduced to a manifold without philosophical relevance. Chaos is then GF’s relation to its internal material, to the reduced manifold of scales, forms, and dimensions of the philosophical. Quite simply, it is “the whole” of science that is fractal, what we elsewhere called the Universe rather than the Cosmos. In philosophy, chaos is said of the World; in science, it is said of the Uni-verse.

There are evidently two concepts of chaos as of every thing:

1/ A chaos through autodissolution, or else through autoaffirmation of amphibologies, mixtures, or blendings—amphibologies that are constitutive of philosophy and that are simply proliferated, accelerated, and aggravated by philosophy itself as the factor of disorder. Hence the neighboring concepts or modes of this bad philosophical chaos: Differe(a)nces, Chaosmos, Multiplicities, Language Games, etc. This unfinished chaos corresponds to a limited destruction of the philosophical order, of its most representative and superficial forms, and to a reaffirmed respect, a reaffirmation of this order. Philosophy produces neither a radical, absolute chaos anterior to it nor some strict, unilateral, or noncircular order, but a simple dis-order in view of a normalization of the real and science. A semiorder, semichaos; a tendential yet limited dis-location, barely touched on, in which philosophy finds the reasons and means of its survival and its permanence as a tradition.

2/ A non-philosophical chaos whose cause-of-the-last-instance is the nondecisional Identity (of) self; whose specific element is the irreversible, absolutely fractal gap of Unilaterality; whose effect is to be the mode of existence, under its conditions of philosophical origin, of generalized fractality. It attests to the philosophically unintelligible or inadmissible fact that the purest order, the simplest, is the germ of the most acute fractality and the root of chaos. The real chaos of science—not of knowledges in general, but of knowledges from the perspective of their theoretical relation to the real—is more than the opposite, it is the real critique of the philosophical dis-order. This disorder corresponds to the fact that an order that is transcendent (and thus woven from disorder) is itself posited and presented as real and unintelligible by itself. Philosophical dis-order is merely one particular order, which is posited as universal and real and then lets itself be corrupted, dislocated, and dissolved by what it engenders, but certainly not destroyed. Science’s chaos no longer forms a system with philosophy’s bad nihilist or merely counternihilist dis-order; instead, it forms its system with the immanent One-Multiple and the primitive order that fractality introduces.

So we will not conflate fractality with the aleatory, a passably negative and reactive concept that presupposes a rational norm. The fractal is and describes an essential, if not “normal,” state of affairs; it integrates with science. It does not suppose as given or valid an order that it would perturb—in the manner of the aleatory. The fractal is itself the order: chaos has its essence—but in-the-last-instance alone—in the most primitive order. If it were not “in-the-last-instance alone,” then thought would return to philosophical errancies.

The Identity of Irregularity, as the specific qualitative structure of scientific knowing, is thus chaos-without-logos, the fractalization of every chaology, of chao-logical Difference itself. Of course it is hardly close—since it is its real critique—to the philosophical chaosmos, the doublet of the old philosophical Cosmos and its contemporary, but always philosophical rejection. Let’s call GF under its philosophical conditions of existence “generalized chaos.” GF registers the destruction of the autoposition of the Other for the benefit of Identity-of-the-last-instance-of-the-Other, just as generalized chaos registers the destruction of the philosophical semichaos that exists only through the autoposition of the metaphysical autodissolution.

Measured against the irreversible fractality of scientific knowing, philosophy appears as an enterprise of normalization of science and, perhaps, of false liberation, false or grounded in an ignorance. The most creative irregularity belongs to science; it is not the irregularity of the philosophical decision. Science’s chaos is ultradeconstructive. Thought’s fluctuations, productions, and creations (for example, the new theoretical discoveries or objects) should not be absorbed. They are not themselves the exception; they are the rule or the essence. The “exception” to the rule is not an exception and is more than the rule: the cause that turns transcendent regulation into a normalization…The opposition between the variable and the stationary state, the fear of variation, of shock, of aggression, the rejection or the negotiation of novelty, which is then treated as margins, differences…all this is surmounted. The philosophy of Multiplicities, of Catastrophes and Differences, of Inconsistency, of Fuzziness amounts to under-/derealizing irregularity for the benefit of the philosophical form that remains the game’s master…This game is still that of philosophy as the autoposition of Transcendence (of the Other in this case). In contrast, the power of the Other must be itself determined in-the-last-instance by Identity, the key that unlocks the kingdom of generalized fractality.

Science’s mechanism is intrinsically chaotic; not only infinitely open-without-teleology, but fractal in-the-last-instance, i.e., through and through. Scientific knowledge can fluctuate, vary, and be renewed for apparently external reasons; they stem from the emergence of new phenomena (experiences and theories) that are from the start virtually philosophizable. And science has always struggled against this, whether expressly or not. But these reasons become internal to the suspension of their philosophical sense; this suspension constitutes them as the data of an experience in the scientific sense of this term. No external—for example, techno-political—stimulus can enter into competition (with respect to science’s essence) with its fractality, which is identical (to) self and fractalizes these stimuli in their turn. This advent of scientific fractality takes place without returning after each variation—or each new knowledge validated by any means whatever—to a stationary or normal state. The paradigm of crisis and of normal science, like falsifiability, and in general the metaphysical constructions of Karl Popper, Thomas Kuhn, and so forth, are unitary, transcendent, and foreign to science’s real labor. If science is a chaotic process, even in theory, then all the transcendent constructions of epistemologies are useless and uncertain—if not as normalizations of science.

We have summarily described a semifractal economy and a semifractal anatomy of philosophy. Invariant forms in rare numbers regularly emerge from the philosophical Tradition: the circle, the abyss, the fold, the doublet, the (double) band, the arborescent network, or the rhizome…But they are hardly fractal and not quite unilateral or irreversible, though they are already transcendental rather than natural. But here again they are seminatural, semitranscendental. Philosophy is an idealized double of nature and entirely lacks spirituality and interiority on one hand and scientific immanence on the other. The functions of philosophy’s semifractal lines and surfaces involve the capture of a maximum of information, resistance to external disturbances, struggle in general, and concentration of information (fold, double, continuum, etc.) more than they involve order, distance, economy, Occam’s razor, and the destruction of every teleology—these are science’s functions. Philosophical fractality partially imitates nature’s. It is ordered to teleologies; it proceeds by redundancy as much as by strict or “identical” self-similarity, by Sameness rather than by Identity; by tautology and accumulation of the reserve of knowing and of culture rather than by poverty, order, and simplicity.

THE GENERALIZED CHAOTIC DISTRIBUTIONS OR NON-PHILOSOPHY

Perfect yet artificial fractal curves, constructed through recursive procedures of infinite interpolation, through the introduction of new ever smaller irregularities and thus conserving to infinity the property of homothety, can exist in geometry. Yet they only exist in a very improbable way in nature where the scale changes are punctuated by changes in order, reign, or domain, with the same phenomenon moving from the macroscopic to the microscopic, from the meteorological to the chemical, from the molecular to the quantum…The “passage to infinity” is prohibited in nature. This perhaps helps explain the usual presentation of fractality in the form of clusters or regroupings of the “same” fractal object, which are in general strongly lacunar but also regularly dispersed.

From the perspective not only of its support but of its principle as well, fractality presupposes some intermediate or transition zones and would annul itself if, for example, an irregular line really and adequately filled a whole area and became continuous again in its own way. Natural objects combine several fractal ensembles (“hierarchized clusters”); this prohibits or renders difficult the fixing of a scale on which an adequation for a unique object would take place. We cannot compare in a direct, intuitive way a line’s length and the surface it partially occupies or fills. There is a phenomenon of “clustering,” of grouping of objects in distinct clusters, which appear as the change in the scale of magnitude. It seems that the same must be true for a GF where the “non-philosophical” statements thus produced are not only distinct but must form fractal aggregates, cluster-identities. Regardless of what happens between geometry and physics, our problem is to describe what happens in thought between science and philosophy and how the GF is distributed.

On one hand, the object “philosophy” also knows scale-changes (we can examine or survey it while participating in it—“meta”—with different degrees of resolution) and successive appearances that are increasingly complete or increasingly fuzzy and incomplete, dimensions that constitute its space (a theme, then the Dyad in which it is included, then the philosophical Decision that structures it, etc.). Philosophy also knows—provided we take it as a global reign, as a tradition—changes in decision, a “same” object, “critique” for example, going from the Kantian to the Nietzschean critique…. From the perspective of its materials, it does not seem possible that GF would lead to an infinite unhindered dispersion and seems that it too has to be distributed or grouped in more or less hierarchized lacunar clusters whose last reason would be this specific heterogeneity of philosophy. A reason that is different from fractal irregularity and that limits it in some way.

On the other hand, GF exists in the form of terms that are strictly identical each time in their specific qualitative, autonomous nature; they are clustered in strictly nonhierarchized forms by the absolutely dispersive a priori fractal structure. These determinations no longer float in a universal element; they are, so to speak, the manifold of the Other itself, the one it aims for in an in-objective way. Not an identity-of-the-Other, a Same, but an absolutely and thoroughly fractal Other, which is only identical in the last-instance and exists in the form of this radical chaos. Along with Identity, fractality is chaos’s determinant reason, and chaos is fractality’s concrete existence. Inside chaos, which is the chaos “of” philosophical determinations, philosophy stops reigning. The continuous topological relations of connection or of vicinity, the singularities, the catastrophes, the games, the partial objects and flows, the differences and differends are invalidated. The old terms of philosophy’s representational content are no longer linked or connected, topologically or otherwise, by some relation of co-belonging that would be proper and coextensive to them. They have in common only the fact that they are subjected to an identical fractality, which emancipates them from philosophy’s continuous curve. Fractality alone, the most simple or the most irreducible to every philosophical curve, distributes these determinations. Just as GF eliminated for its part the geometric transcendent and symmetric figures of fractality, so the radically thought chaos excludes for its part the distribution into clusters and lacunae.

More exactly, and in order to take this double constraint into account, the clusters, lacunae, and hierarchies, which can always be philosophically regrasped, are now only terms or determinations distributed by a chaos so fractal that it suspends their operative character. And so we will not imagine—this would be one more philosophical imagination—that thought has an interest in adding the clusters and the lacunae, the promontories and the bays to the already old philosophical imagery of holes, sheetings, Möbius strips, valleys and rivers, ruins…and to the even older imagery of circles and vortices. It is a matter of thinking chaos and of ceasing to imagine it, for generalized fractal chaos is more complex than the chiliagon and defies philosophical understanding itself. Science’s “dimension” is traced by the axis that goes from Identity-of-the-last-instance to its correlate of chaos. Chaos is what we see “of” philosophy when we situate ourselves in this wholly “internal” fractality.

CHAOS AND THE CRITIQUE OF PHILOSOPHIES AS “THEORIES OF KNOWLEDGE”

If scientific representation is essentially chaotic (and not chaological), then it is the real critique not only of philosophies that are, above all, “theories of knowledge,” but of all philosophies insofar as they always, and in any case, have a certain conception of knowledge and of science.

Fractality does not only change its nature when it passes from the geometric and/or philosophical Same or Whole to Identity-of-the-last-instance. It also changes its field of exercise or its object: it passes from natural objects to thought objects, but above all from objects to their representations. From this point of view, we can detect some hesitation in fractality’s geometric or restrained theory: fractal is said at times of the natural object to be modeled, but at others also of its mathematical representation, which is deemed to be more adequate in this case. From there to the conclusion that the object itself is fractal, one only has to take the philosophical plunge of circularity in the mirror of the object and of its representation. For instance, we may call “fractal object” any “natural object that can be reasonably and usefully represented in a mathematical way by a fractal set” (Mandelbrot). As always, this term representation receives a very “representative” or specular use. It is conflated with the supposition that fractality is ultimately a form “common” to the object and its representation, which is its more or less wrought “tableau.” This is the heart of the theoretical intuitionism of geometric fractality. It cannot be up to us to decide in the present case whether fractality is the fractality of the object or of its representation, even though the conception of science at work is here quite clearly that of a transcendent realism of perception and even though science is understood as a more or less specular double of its object. This approach to science is possible only if the object and its properties are in some sense reified and posited as objects “in themselves,” as common sense wants to do. Theory is then no longer intrinsically and qualitatively distinct from its object.

By contrast, the theory of science we are defending is a realism in-the-last-instance alone and not a theory of perception or of transcendence. So much so that theory shares no common form with its object—which it does not modify. So much so that this object’s presumed properties “in themselves” are only older and more elementary knowledges that are reified and realized. The presumed object “in itself” must be dissolved in the real-of-the-last-instance and in objective knowledges. Science is the thought that remobilizes these presumed properties of the object as knowledges and makes them serve as materials for other, more universal knowledges. If these two states of knowledge share a “common form,” it is the form of theory or of scientific knowledge, which is no longer specular, bilateral, and transcendent, but unilateral. Instead of a third and transcendent form common to the object and its representation, effecting their synthesis for its own benefit, there is the intentional-nonobjectivating relation of theory—a kind of “common fractal form,” but absolutely immanent and not transcendent to these two knowledges, a relation that individuates them as terms, as knowledge-identity, at the same time that it fractalizes them. This amounts to saying that instead of this specularity, this third-mirror in which the object and its representation communicate and are exchanged, there is a chaotic or fractal dissemination of the “object” and of “knowledge,” i.e., of thoroughly universal knowledges. In this way, generalized fractal chaos destroys, to its root, the possibility of philosophies of science and not only explicit “theories of knowledge.” As such, we will say that, in becoming universal or generalized, fractality has passed from the object—and its intuitive images—to its representation alone. But this representation should be understood as an autonomous sphere or an identity of theory, absolutely foreign—precisely because it, and it alone, is intrinsically fractal—to its objects as data, to what serves it as materials. This fractality of theory, which is its “specific difference”—and more than its difference—from its object as it is presented from the start, is what fractalizes all the objects-of-knowledge and plugs them back into science’s living circuit: even these old reified knowledges, the natural or perceptive properties of objects—including the sensible or intuitive fractality of Britain’s coasts, of clouds, and of the Ocean.

Furthermore, a clear solution is provided to the similitude—from now on, the immanent fractal identity—of the object and of its representation. The geometrician’s hesitation on this point marked the problem in a symptomatic way: there must be something like a property of resemblance, at least the equivalent of a similitude between the object and knowledge. But instead of reifying this resemblance in the form of the mirror-agent presupposed by all philosophies, Identity-of-the-last-instance alone is enough to explain that fractality can be said (both) of the object (and) of its representation, of the thing and of knowledge, though it is only the fractality of knowledge. This is why by “GF” we mean as much the property of an object as the property of the knowing of this object. On one hand, the object is itself only an old fractal knowing whose fractality was effaced or denied. On the other, two knowledges are fractal not through a common fractal form that divides up two supposed givens between them while remaining itself one or undivided, but through the investment of a one-or-undivided-fractality in a material that it and it alone individuates or makes visible, not so much as 2 but as 1, 1, 1, etc. Chaos is not confusion. On the contrary, it is what individuates by means of fractality itself philosophy’s confused blendings or amphibologies—its “order.”

If philosophy is a statistically regular curve, despite its angularity—but it is always bilateral rather than unilateral—we will not say, on the other hand, that science too is a curve, that it is simply more interspersed with angles or points: the theoretical discoveries. In reality, science is a chaos in which even the local curves are fractal events or absolutely irreversible catastrophes. It is impossible to smooth out science in a becoming or a teleology as philosophy does (these attempts, nevertheless, are not lacking). Science does not have the nature of a reserve as philosophy is; it has the nature of a chaos.

The philosophical Tradition can be analyzed in an infinitesimal way. It subsists and resists and continues to accompany, with its more or less breached totality, the details of its decisions. At every point of its decisions’ curve, it is possible to extend these decisions with a straight line (a new philosophy) or a linear development. So much so that the Tradition is sprinkled with these altogether straight doctrinal lines that approach Tradition in each of its decisions and give to philosophy a simultaneously pointed and continuous character. We know, moreover, that the philosopher contrives to trace two lines from the same point: the one circular, the other straight (Plato, Leibniz, Heidegger, Nietzsche, Deleuze, Althusser, etc.).

Yet science is so profoundly irregular in its very essence that it is detached from the start from philosophical idealizations and makes use of idealization, of regularization, only as a local procedure. It is incapable of outlining a coherent, teleologically dominated future for its work—if not by entrusting itself to philosophy and the State, united in a prestigious and intimidating alliance. The angularity of knowledge in its relation to the real and, thereby, in its relation to another knowledge is one-sided, so to speak, and traces only irreversible paths. This does not mean that it is impossible to survey them, to recross them in reverse or to “philosophize” them. Science is not the development of possibilities to be realized or of a virtual to be actualized. It is ultimately—to use an ontic or regional metaphor—a Brownian, yet transcendental movement, a movement that concerns knowledges themselves in their universality while existing only under philosophical conditions of existence. So long as the description remains faithful to the essence of science and does not found itself on some objective knowing, which is philosophically objectivated a second time or reposited, fetishized, or factualized in the form of transcendent fakta, it can plunge into the details of knowledges without ever apperceiving these phenomena of totality, of systematicity, and of continuity that philosophy believes it detects in these details. The description does not follow any line and, as a result, a line that weakens its heterogeneity or its multiplicity. Each “term,” each knowledge, obtains its sufficiency from the Identity-of-the-last-instance, from the cause of science rather than from an ontological relation to others.

Whereas philosophy is thought-through-system as well as through the proliferation and alleviation of systems, science is thought-through-chaos. What distinguishes them is by no means the artificial couple thought/knowledge, but two heterogeneous types of essence or of cause of thought-knowledge. Scientific practice reveals a granular—yet transcendental rather than physical—structure of knowing; a subatomic identity that is no longer homogeneous with the atom, but is “utterly” distinguished from every ontic or ontological-transcendent identity. It is an identity-of-the-last-instance, the ingredient of chaos. It is enough to think a difference without a line of differentiation that accompanies it and to reserve the right to an analysis and a synthesis external to the very event of knowledge; without a possibilizing foundation that immerses this event in the possible. The continuum, but also difference, the inconsistent multiple, language games…are artifacts, crystallizations, and smoothings of certain phenomena that are too accentuated for the philosopher to overlook them; the operation of an arbitrary and fascinated vigilance, obsessed with life and especially with survival.

Left to its autointerpretation, philosophy envelops itself with a kind of unlimited skin (surface, plane, plateau, slippery or rugged ground, etc.), through which it simply slides on science’s “fractal” asperities and shelters itself with them. Granular skin, differentiated into organs and smooth zones of reception—but it is a skin, the interface with another philosophy; a skin that philosophy does not live so much from within as surveys, anticipates, and projects. Science is another experience of the real: not through interfacing, but through unifacing, as we said, through a fractality—the Other itself—that is directly felt on the philosophical skin.

GF’S FIELD OF APPLICATION AND PERTINENCE

We elaborated GF’s concept by making use of the “categories” of Identity and Difference and of categories affiliated with Difference (multiplicities, differends, différance, language games, etc.). They are types of “meta”-philosophical transcendentals that have a reflexive vocation and that allow us to clarify and thematize amphibologies in general (not only those of the understanding, the Kantian amphibologies). As a result, we can better understand that fractality itself, as we described it, participates in this metaphilosophical—now: metascientific—function or serves to describe knowing in its essence as well as to critique the great—philosophical—amphibology of science and of philosophy. It belongs to the theory of science and thereby to the science of philosophy. Without a doubt these universal fractals can only be the fractals of a knowing—it is now the object “theory,” rather than Britain’s coast, that is intrinsically fractal. It is a matter, as we said, of a theoretical mutation that touches on science’s essence, that saves it from its epistemological capture and in this way reaches philosophy itself.

Yet if GFs are objects of knowing rather than of “nature,” this does not mean that they are obtained through the autoreflection or autoposition of geometric knowledges, in the form of a faktum. They have no doubt the essential properties of self-knowing—they pertain to the theory of science—and, being related to knowing rather than to the object, they are transcendental. This is a generality that can also receive a philosophical sense, that can be understood precisely as a process of autoreflection, but it is in fact a matter of an unreflected knowing, nondecisional and nonpositional (of) self.

Identity’s fractals are open only to a pure description; they are immanent phenomenal givens that no philosophical decision of generalization affects. They are from the outset the “most” universal. For universality is not in this case a question of degree, but of a “qualitative” definition or of a definition according to the essence: it is not divided into generality/totality; it is undivided in-the-last-instance as a unilaterally open sphere of objectivity. We see each thing “in”-the-last-instance without each thing having the structure of a reflection (even an objective reflection); without each thing becoming productive or participating. It is a contemplation that only the static phenomenal givens can fill. Even “becoming,” time, and practice are pervaded by this static phenomenality that corresponds to their fractal structure. We would readily speak of a fractal vision or mysticism if these terms did not convey the worst philosophical confusions.

What does it mean to say that the fractal theory is generalized? We can always imagine a “superior” fractality, which is specific to Being or philosophy, obtained through idealization and interiorization of geometric fractals, “superior” insofar as it is a mixture of geometry and philosophy. But we showed that this concept, which subtends the contemporary philosophies of singularities, is in fact a semi-, at once an under- and an overfractality in which fractality is impoverished and effaced (qualitatively; it is not a question of degree). It is more rigorous and fertile to identify a full or finished fractality, absolutely universal de jure, a fractality that can apply-to philosophy itself from the start rather than remain caught under philosophy’s law and thus immersed in an ever indeterminate generality.

It does not result from a supplementary, philosophical idealization of geometric idealities; this idealization would, paradoxically, turn these idealities into faktums, into supposedly autonomous rational facts that can be extended by a philosophical reflection. For instance, we have treated the geometric property of internal homothety less as a natural property (which would only be a faktum produced by and for a philosophical decision) than as a simple scientific knowledge, a local theoretical tool. And so we considered it as the simple indication or material of an essentially theoretical problem to be resolved (rather than as a rational faktum): the problem of the fractal constancy of science itself. We have thus avoided extending it in the (ontico-ontological) mixture of a transcendental homothety. Its equivalent for science is Identity-of-the-last-instance; it is no longer a homothety, which would presuppose operations of displacement and inversion. We have ceased to capture science philosophically, and have began treating it as an unconstitutable phenomenal state of affairs that can only be described. By the same token, we have not interiorized fractality proper, the condition of irregularity, to the concept; we have not divided and sublimated it with a philosophical decision and thus produced a universal fractal operation. But we have treated it as an indivisible body or drive, as an undivided distance without reverting or returning in any way to Identity. Everything is here lived or received by the cause (of) science in its unalienable mode—this is the sense of “in-the-last-instance”—so much so that everything in science is a purely immanent = indivisible phenomenal state of affairs, even when, like theory, it proves to be complex.

The concepts of generalized fractal and of chaos are more powerful theoretical (as well as artistic) tools than “differences,” “inconsistent multiplicities,” “games,” “turns,” “disseminations”…because they are theoretical rather than philosophical and because they define a fractality in relation to philosophy itself, considered globally. On the basis of Identity-of-the-last-instance’s incommensurability to Being, it is the whole logic and ontology of mathematical multiplicities, the topology of differences, the philosophy of fractals and the philosophy of catastrophes—not their mathematical or geometric bedrock, of course—that appear as half-solutions and do not do justice to the scientific sense of purely geometric or set-theoretical givens.

In a correlative way, GF sees its relevance limited to philosophy, i.e., to a discipline that uses natural language, is not reduced to this language, and can even determine it. It is strictly “transcendental” in the sense of the flawless immanence of its cause. It is thus not empirically identifiable or locatable in the World, in History, in Power, in Sexuality, or in Language, because it is fractality-in-Transcendence itself. To fractalize every form of ontological transcendence, to fractalize Being itself rather than to defer or reserve it—this is GF’s effect. One particular mathematical or physical theory cannot therefore exemplify it. On the other hand, it is fractality for Transcendence, for the World, and, in particular, if it cannot be found within language, which does not provide it, it is “destined” to shape natural language, at least insofar as it is required and included in philosophy and ordered in the sense that a philosophical decision can be superimposed on it. It is not an ontic science. It is pertinent for natural, physical, or mathematical, but also social or unconscious phenomena only to the extent that they are prima facie given as clothed in sense, coated with the philosophy-form. More rigorously, it is not a matter of a regional scientific fractality that is applied to philosophy, but of the fractality of “science itself,” of theory as such, thus of its scientific sense. It is on this “transcendental” condition that it can apply-to philosophy. “Science itself” is not “fractalized” or represented by a fractal model; it is fractal in its very essence. On the other hand, it is philosophy that, not being spontaneously fractal or being so only in half, is fractalized. It is fractal theory alone that thus possesses an original and de jure critical dimension in its relation to the manifold or to the object, precisely because—unlike its geometric and philosophical forms—it is founded on the distinction between orders or spheres of reality.

GF’s status and its type of pertinence have aesthetic consequences. If GF has to have a more direct, particular affinity with certain phenomena, these phenomena will be literature and poetry more than painting. By right, all arts are equal before it, since everything must pass through philosophy’s mediation (they are virtually philosophizable). In fact, GF’s circuit of access to phenomena is evidently shorter when it is a matter of the arts or of disciplines that mainly use natural language (law, ethics, etc.).

But in all respects GF represents a theoretical tool (theoretical and not technological). It has at its disposal the same type of power as the geometric form: a theoretical and not philosophical power. Far from remaining narcissistically in itself and circularly autoapplying to itself in an indefinite, repetitive, and sterile way, it enjoys de jure a theoretical intentionality, an intentionality of knowledge. It has an object distinct from itself: philosophy. It enables the theoretical knowledge of philosophy, its transformation into a scientific continent. It produces new, emergent statements irreducible to philosophy’s data, to their spontaneous “ideological” representation (philosophical faith, the Principle of Sufficient Philosophy). It is thus not a matter of a simple transfer, like a technology-transfer, with the problems of inadequation that results from it, but of a theoretical mutation within the concept of fractality, on one hand and, on the other hand, of the recognition of a specific intentional aim directed at the object “philosophy,” an intention of knowledge that belongs de jure to this transcendental concept. We will not conflate this theoretical elaboration, this rupture in the theory of fractality, with the conceptions—stemming from the epistemological disaster—that reduce scientific labor to an activity of importing-exporting of concepts. We can invent or at least discover philosophy as a scientific continent on the condition that we abandon not only these practices, but the very ideal of philosophy-as-rigorous-science for the wholly other Idea of a science-of-philosophy, which requires a recasting of the theoretical tool: with the double goal of giving it a truly scientific and not philosophical pertinence and of rendering it adequate to its object, which is not arbitrary—no more here than elsewhere. GF’s concept, which condenses our previous research in the manner of a concentrate of non-philosophy, responds to this double objective.

GF’s concept has made us realize in hindsight that what we called non-philosophy was already a fractal type of practice of philosophy. The practice presupposed a remodeling of this concept, which is thus “spontaneously” extended to natural language and to philosophy. The produced statements (and, as a result, also those that form its theory) no longer respond to philosophical logic, to the norm of its statements’ production and admissibility, because they are produced as reproducing a structure characterized by the internal Identity of its inequality, despite multiple decision-and-scale variations. On the other hand, one of the major interests of this theory, alongside its immanent “application” to natural language (rather than to space and to ontic objects), is more clearly to mark the difference in nature between philosophy and science. As the criterion of space or of knowing, it distinguishes this space absolutely from the philosophical space, which is not fractal but coordinated or mixed.