Fractal Ontology

A language thus has this curious and striking feature. It has no immediately perceptible entities. And yet one cannot doubt that they exist, or that the interplay of these units is what constitutes linguistic structure. That is undoubtedly a characteristic which distinguishes languages from all other semiological institutions.

Ferdinand de Saussure (Course in General Linguistics, 105)

After having investigated the physiological mechanisms of speech, Saussure turns to consider the nature of linguistic signs. In his analysis we find a clear formulation of the first principle of his “new” linguistics, namely that the sign is arbitrary.

At first glance this statement seems to suggest an argument for an evolutionary linguistics, but Saussure draws our attention to the fact that “this very same factor tends to protect a language from any attempt to change it.” (73) As a system of arbitrary signs, there is no rational basis for language; paradoxically this makes language itself “inaccessible” to reason.

There is no logical reason we should prefer “sister” over “hermana,” and thus no grounds for argument about change. The great number of signs necessary to constitute a language (and the complex character of the linguistic system itself) also contribute to the force of collective inertia resisting linguistic innovation:

“At any time a language belongs to all its users. It is a facility unreservedly available throughout a whole community… [it] is something in which everyone participates all the time, and that is why it is constantly open to the influence of all. This key fact is by itself sufficient explain why a linguistic revolution is impossible.” (74)

A community naturally exerts a restrictive, conservative influence upon a language. Nonetheless, the passage of time also allows linguistic signs to be changed with “some rapidity” — hence both variability (diachrony) and invariability (synchrony) are characteristic of the linguistic sign. These two characteristics are “intimately connected” — a sign is only subject to change because it “continues through time.”
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Fractals are an example of a discovery from what claims to be a very new kind of science — a science of feedback, turbulence and emergence — which harnesses an old but ubiquitous species of chaos. A fractal can informally be defined as a recursively self-similar figure. They are not extrapolated from a geometric logic based on units. Rather fractals are constructed on the basis of an infinite program, such that the result ‘overflows’ any unitary dimension, and so cannot be said to (only) occupy ‘regular’ plane geometries (of two, three, four dimensions, or indeed any whole number of dimensions at all.)

Fractals are freaks, geometric monstrosities, pure chaos — and yet, a strange order persists behind their intricate complexity, a drunken infinity, a sublimely-tangled mess of dimensions. Fractals are a movement-image, but exactly what dynamism does a fractal capture? Not just a fracturing of the one, for fractals also construct their own plane of consistency and originate metrical regimes.

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I’ve been visiting the massthink blog (populated by Ryan and Aless) for a couple of months now, but, as a bad reader, I failed to leave comments. I was thrilled not only with the content, but more so with the way in which these two have organized their site. I was inspired to see their text page (which encouraged me to put up the bibliography, still under construction) and by the way in which they described their project.

They way that they describe themselves and their differing styles and personalities reminds me of the different forces at play between Joe’s writing and my own. He studies math, computer science, and philosophy, while I come from a training in comparative literature, literary theory and philosophy. Though that is one of the more obvious contrasts.

I first found their site looking at material on Nietzsche. Their post on Nietzsche focused on Danto’s reading, though at the same time it was evident that the authors were widely read beyond this one reading. I think it was here that I could tell that massthink had a great deal in common with the work Joe and I have been doing here at fractal.

I didn’t know at the time that our friendly exchange would be more than a coincidence. In a show of friendship and brotherhood, these two have extended a gesture of respect to the work that has been done here. The posts that they have discussed are important not only for their content, but also as a difference in tone: the posts they discuss are Joe’s from before he and I began collaborating together, and I think that it’s important to see the changes that his writing has undergone over the two years he’s had this blog.

In any case, one usually keeps reading because of an inspiration to think, an ‘affect’ to start. Reading through their site earlier this week, I found two extremely enlightening posts: one on Deleuze, DeLanda, and Darwninism; the other on the hylomorph and the monster (identity). I could not tell the author of the texts, but I can note a different style and orientation of writing. The first text is expository, and follows a line of thought that traces through different understandings of theories of evolution (I love the emphasis on a change in relation among parts and not of the parts themselves as primary for evolution). This text is clearer, more concise and and at least as important as anything I’ve read on the topic (but don’t trust me, see for yourself). On the other hand, their text on the hylomorph and the monster had a different approach to the reader, a different call, almost an assurance to the reader to allow difference to play itself out, to dissolve the images of thought that plague us through psychic/social repression.

Of course, this is only a small selection of their posts. These two know their stuff–none of the rhetorical foreplay, none of the abstract regurgitation of concepts. This is real philosophy, not diluted but distilled so as to capture its essence. It’s stronger that way, and definitely has a better kick!

“What does man actually know about himself? Is he, indeed, ever able to perceive himself completely, as if laid out in a lighted display case? Does nature not conceal most things from him — even concerning his own body — in order to confine and lock him within a proud, deceptive consciousness, aloof from the coils of the bowels, the rapid flow of the blood stream, and the intricate quivering of the fibers! She threw away the key.” (On Truth and Lying in an Extra-moral Sense, Part 1)

“To calm the imagination of the invalid, so that at least he should not, as hitherto, have to suffer more from thinking about his illness than from the illness itself! –that, I think, would be something! It would be a great deal! Do you now understand our task?” (Daybreak 54)

A Simple Theory of the Unconscious?

There is no simple theory of the unconscious in Nietzsche’s work. This is because the unconscious is complex, a site for transformation and not a singular ‘object’ of analysis. The unconscious would be everything which accounts for image-object-thought associations, and therefore that by which we could explain relations between thoughts and activities. However, Nietzsche clearly recognized that we cannot simply analyze the unconscious as a thing in-itself: it was very important for him that we should not be taken in by the idea that our explanations for things are adequate expressions of an underlying reality. Because in fact there is no necessary relation between human beings and reality; rather, we artistically create the mode in which we confront and understand the world. Thus there are no longer any laws of nature for Nietzsche: “…what is a law of nature as such for us? We are not acquainted with it in itself, but only with its effects, which means in its relation to other laws of nature — which, in turn, are known to us only as sums of relations. Therefore all these relations always refer again to others and are thoroughly incomprehensible to us in their essence.” (Truth and Lying, Part 1) We cannot reach the essence of a relation except by being deceived into thinking they are simple; like the will, which we are apt to conceive as a pure simple essence of participation, an ‘inclination,’ when it fact it may be the most complex phenomena in the entire world.
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Cellular automata

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty. D’arcy Thompson

All organisms are modular: life always consists of sub-organisms which are involved together in a biological network. The interrelations between organ and organism form a series of feedback loops, forming a cascading and complex surface. Each organ parasites off the next, but this segmentation is not spontaneous. Rather, it is development itself, the decoupling of non-communicating spaces for the organization of divergent series. Creative evolution, self-organization and modularity are the same idea.

The theory of the development of metabolic modularity is called morphogenesis. ‘Morphogenesis’ in its literal sense means the creation of shapes or forms. But in the (relatively) narrow sense we intend it here, morphogenesis is a self-symmetry of the biological structure (onto itself) which allows it to develop in such a way as to divide while remaining unseparated, that is: to ‘individuate,’ or split apart into fused symmetrical segments.
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Artificial intelligence stands in need of a fresh thought: a new thinking of complexity, of the virtual, and of machines. Instead of a virtual founded upon forms which remain forever the same, we need an idea of the virtual founded upon difference itself. We need a creative virtuality.

The task of building a robot demands a lucid and algorithmic way of grasping the frame problem. An adaptive principle of distinguishing problem spaces, some genetic evolution culminating in the capacity to mark a difference. So how do the sense organs evolve? Which is another way of asking: how does experience form?

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Recent theories of galaxy-formation and cosmic evolution are inching closer to embracing the radical geometry of general relativity. In the decades since the publication of the Schwarzchild solutions to the Einstein field equations as given in his theory of general relativity, many physicists have staked and made their professional careers on the seemingly abstruse mathematical issues involved at the heart of the debate.

Scholars have been fascinated and even sometimes ‘consumed’ by the study of black holes and the associated conceptual complexities, perhaps especially the novel interpretations of non-linear geometry relating to the prediction of an extreme curvature of space and time. Initially a seemingly bizarre consequence of the field equations, it was perhaps all the more so due to the very unique conditions under which this “affine” torsion, or gravity vortex, ought to occur.

Since the days of those first thought-provoking theoretical intuitions, the body of scientific theory regarding black holes has grown enormously. We now believe that black holes are cosmic anchors at the center of most galaxies, curving the geometry of the universe at large, perhaps even “fracturing” it into infinite geodesic ‘slices.’ We are fairly certain that there is an awful lot of something providing an enormous amount of gravitational radiation.

We also now have a deeper understanding using differential geometry of the Newtonian assertion that gravity is acceleration. We are even beginning to look for gravitational waves, or ‘graviton’ particles, which amounts to something like looking for an accelerating shift in the geometry of spacetime.

Most of us know, but perhaps have not fully recognized the importance of the geometric innovation within Einstein’s thought in shifting from special to general relativity. The explanation of this shift can be summarized as follows: in special relativity, Einstein is still using a more or less linear geometry of curved space, which is called a Minkowski space.

With the shift to general relativity, Einstein has gone beyond curved Minkowski spaces to a new kind of ’self-constructing’ space called a Lorentzian manifold, which possesses a radical non-linear geometry of momenta, whose curvature is defined by stress-energy tensors (or momentum.) Einstein had fully embraced an exotic auto-metric geometry where solutions of the system of equations are potential spatio-temporal geometries.

We began by considering black holes to be ‘marginal’ occurrences, but we are beginning towards a theory where they play a central role in the evolution of the cosmos. The lesson of the history of black holes is that autopoesis is an adequate paradigm at any scale, from the quantum to the cosmic to the social. Evolution is what the shift to ‘non-linear’ geometry in general relativity means: not only that spacetime has a curvature, but that this curvature has a curvature, that it can be fast or slow, self-destructing or evolving, divergent or harmonious, unified or fragmented, self-similar or infinitely differential, and arguably, even self-creating. Evolution is a cosmic principle that applies equally to tiny particles as to living creatures and galaxies.

But we must always remember that the relevance of evolution is political first. The history of the discovery of black holes reminds us of the importance of seemingly marginal occurrences. In other words, we ought to distinguish more clearly between the event of change and the process of change. Most will agree that the process of change and evolution is always in some sense amenable to observational modelling, because reality as such obeys only an internal and self-created measure, or law of motion. The event of change is, on the other hand, ontologically transversal — a new space embedded within an old space.

It may also be helpful to think of this evolutionary ‘event’ as the shift between different spaces, or spaces moving at different speeds relative to one another. After all, even separated spaces can be made to intersect, and we can immerse a space within another space. So this is the cosmic situation: all spaces are interconnected, but all relationships are in flux. Knowledge, like life, is a rhizome, spreading out in all directions at once, ever-shifting, evolving and involving, gradually or quickly adapting to always-changing conditions.

The question of knowledge is always which situation it is deployed into, the transversal path it travels between spaces, or into an outer space from an inner space. What is important to recognize is that this fractal shift is nothing transcendental: it is a purely mathematical function. Consider a mapping f from a space A to a space B immersed within it.

f: A -> B

What is it that we should take the ‘->’ to represent? The ‘->’ stands for an energy-transformation method, or transversal operation. In other words, ‘->’ is any pairwise-matching rule that establishes a ‘rhythm’ between inter-facing geometries. We can understand this transformation in terms of Nash’s work on immersion. Simply consider that for a given element x of an inner space, we are guaranteed some information-preserving mapping onto the outer space (into which the subspace is immersed)– regardless of the global topogical structure or local geometrical structure of either of the spaces.

Now, most people know that Godel proved that a large class of deductive systems cannot find themseles consistent or complete. But we think it ought to be more widely known than it is that, since the publication of Spencer Brown’s The Laws of Form, we can consider mathematics, logic, set theory and category theory, while essentially unfoundable, incomplete and inconsistent, they are such only insofar as they remain confined within their own non-self-referential containers and categorizations (even ‘meta-mathematical’ ones.) Yet Brown explains that this isn’t bad news, but the best news: it was a clue that we can make use of a more fundamental pre-logic to recursively found mathematics itself. In fact such a ‘calculus of self-reference’ has been constructed, whose simple but somewhat non-intuitive rules allow us to inferentially derive logic, and even arithmetic and calculus, as well as set and category theory, and so on. In short, we can inscribe and prove even the most complicated results of some of the most advanced fields of mathematics using Brown’s ‘primary algebra’ (or pa.)

In the pa, we begin with the void. It is an undistinguished space; we could argue there is no Universe because there has been no distinction made. Since all distinctions are made by an observer, the Universe is still a void while there is no observer present. Not a vacuum but vacuous: those things which may exist have not yet been told they may exist. Upon this void, let us suppose an observer arrives. Now we have a distinction: there are two spaces and thus a Universe, that is, we have made a mark, crossed a limit, traversed an interface via a singular point of local geometrical tension, which thereby accounts for the construction and subsequent immersion-into-itself of a complex topological space. We shall call such a separable or self-immersible space an observer space. Note the recursivity of this definition, for we are not concerned here with a projective geometry (though it is simple enough to derive it from the primary arithmetic) or even the seemingly critical question of vision; what we are rather interested in is the ‘original’ nomad transversal operator–the work of the observer– which illuminates the heart of cognition, with the fact that the whole of mathematics (set theory, logic, arithmetic) can be derived from the primary arithmetic merely being the most clear example. Observation is the primary transversal operation.

Tranversal operators thus allow us to describe the evolving ex-tension of space from the involving in-tension of observation, from the functionalizing of the enunciatory-apparatus, or assemblage of presignifying intensities. Such an operator is a worker, and as such circumscribes and resonantes with the tension, or differential movement, between two spaces’ rules of measuring force. We could write: A(x) -> B(x), the particle x being the object-flow or subject-group. This can also be read as presenting the differential momenta between two non-linear dynamic systems. We can further decode this as a mapping which embeds or immerses an open space within itself, as in the conventional notation f(A)=B which often gets shortened even further A(B).

But the real question is this: where does the difference originate? Where did the tension come from in the first place? Not of course the tension within each space which forces every movement onto an observable path — but rather the tension ‘between’ spaces observed in the transveral (function generating) operator ->. The transformation-operation encodes a transversal action across any spaces. Thus it describes the molecular evolution of a subject group, and not the position of a null transcendent absolute point of view, but precisely that of an engaged self-observing agency acting within time to fragment, tear apart, and finally dissociate a space entirely — what eventually gets called ‘working’ — in order to transform it into another.