Mathematics as Interpretive Schema
Mathematics and art have enjoyed a tense but intriguing coupling dating as far back as the Pythagorean Brotherhood. It was here, in the undifferentiated infancy of science where spirituality and mathematics commingled in a garden of desire. The music of the spheres played symphonies in the mind of a foundational creator who was at once transcendent and immanent. It was the job of the mathematician then to recover this music and to make it available to those who could hear it. God spoke in riddles and proportions. God was a circle. God was a spiral. God was the energy of change moving over the face of immutability. In time, mathematics would become mathematics, void of the interpretive fancy of the sage which would work its way toward mysticism. Science became science, with no thought of gods or music; empiricism and measurement would reign there. And mysticism would persist in its mathematical form via the Free Masons and the cipher magick of angelic script born and propagated throughout the middle ages.
Nowadays it seems anachronistic or even childish to entertain these archaic systems of divination, or to spiritualize mathematics in any fashion. But the deep mystery, perhaps the only mystery left to the modern mind, is that of itself; and these methodologies of divination (among other things) are psychic maps of our own reckoning of this world, and without which would remain hidden to us. Indeed, abstract objects such as circles, and rectangles are more real to us than the shapes of atoms and molecules which have been modeled and remodeled by physicists for the past hundred years.
Rutherford’s first model of the atom was based on a configuration that borrowed its inspiration from our own solar system. As above, so below – er, not quite. The major pay off in working with mysticism and systems of divination is not whether or not they are “true” or “work” but what it means for the mind which engages them as true and working methods to understand reality. For my part and my learning, I consider them psychic maps of the unconscious replete with double entendres, condensation, projection, introjection, and retrojection. What does it mean to make sense out of the senseless? What does it mean that we always try to find sense in the perhaps random machinations of an impersonal universe?
As it turns out, mathematics has models for all the variant forms of “irrational art” which have persisted in one guise or another. And certainly no one would laud the “rationality” of a work of art. Rational art may be a contradiction in terms, and rationality proper may be less real than its counterpart “irrationality” which tends never to exclude any bits of data, regardless of how negligible or nonsensical they may be; it nonetheless is, and our irrational mind considers what the rational mind neglects.
It seems then impossible to talk about “rationality” without considering the mathematical form of the ratio, which has its linguistic counterpart in the form of an analogy. The analogy is a powerful but limited analytic tool used most often in the sciences to exemplify the relationship between some complex scientific phenomena and some obvious relationship which makes the complex scientific phenomena more accessible. The limit of the analogy is information loss, and by extension, because differences are not accounted for, but only similarities, reduction.
Likewise, analysis proper, or the breaking down of things into pieces, fails to consider the infinite divisibility of the parts, or that other such dissections are equally as valid. But the root of such techniques are always teleologically motivated. There is always a goal, always a desire, which drives the process, and yet is so ubiquitous, that the process can hardly be reducible to it. Desire itself is unanalyzable in the sense that it cannot be broken down into smaller pieces, but only aspected by the forms it takes. Desire is atomic, an energy or force, and it is one of the few pieces that AI gurus have yet to successfully emulate.
My desire in writing this piece is to make some connections between mathematical strategies as they relate to key issues in the humanities. The two maths whose consideration I believe deserves more critical attention are chaos theory and projection geometry. This text should be considered introductory and as existing as a persuasive essay which attempts to titillate your imagination into agreeing that mathematics offers the humanities some (really) exciting possibilities for inquiry.
Chaos Theory & Fractal Geometry
My work with chaos theory involves the usage of the fractal. Fractals are unique mathematical objects that are generated recursively and have infinite detail. Here, “generated recursively” refers to a process by which a function returns as its output a call to itself. For instance:
> function factorial ( byval n as integer ) as integer
> if n=1 then return n
> return n * factorial(n-1)
> end function
In this case, the factorial function is being solved recursively. If the value given for the function is 4 then it will take 3 loops through the function (or n-1) to return a value:
LOOP(1) = 4*3
LOOP(2) = 12*2
LOOP(3) = 24*1
When n=1 in the third iteration, the loop instruction to return n breaks the recursion and stops it from infinitely looping. Not only is there no clear distinction between the normally discreet categories of input and output, but the last output in fact becomes the very next input. There is no clear distinction, because the return value of the function contains a copy of itself, in a moment of self-reference. In terms of art-criticism, this phenomena is analogous to a mise en abyme, when an image contains a smaller copy of itself within the image. Such an effect can be reproduced by pointing the input feed of a video camera at the output screen. This creates a regress effect which appears to go on unto infinity.
Fractals share this property of mise en abyme. Take for instance the famous Mandelbrot set. You can zoom into the image of the Mandlebrot set and it will appear with the same infinite detail no matter how small the scale ad infinitum. The same holds true for the mise en abyme. If you could somehow travel into the screen whose input was pointed at its output, you experience the same infinite hall, with the same infinite detail.
The two properties of the fractal that this example accounts for are scale invariance, and self-similarity.
Self-similarity emerges out of the self-referential call of a function’s return value, or in the case of the videographer’s regress as an expression of the same vis a vis the input channel’s re-entry of the output channel. The self-reference ensures that a tiny copy is present at every point on the fractal. We can test this by zooming into the fractal, which will display self-similar “buds” which are smaller copies of a more evident larger copy, and this property is known as scale invariance.
Fractals are also interesting insofar as there is some debate as to whether they are two or three dimensional objects. Fractals have what are called a Hausdorff dimension which are actually allowed to be fractional. The Hausdorff dimension is analogous to a magnification factor, which is how far into the fractal you have zoomed. Fractal geometry requires this dimension to define a point in fractal space, since there is no smallest and there is no largest, a center must be defined which orients it UP/DOWN, LEFT/RIGHT, and IN/OUT, which is in fact the same as a three dimensional. However, in normal 3d space, a point will be assigned values to the x, y, and z axes whereas a fractal only requires values for x, y, with the Hausdorff dimension being nothing more than a scalar multiple which causes the interval (the space where data is plotted) to magnify or zoom out.
Chaos Theory and Attraction
An attractor, in the simplest possible terms, is a center around which data conglomerates. A variable passing through a dynamic system tends toward the attractor. The attractor can be understood to operate as a set, or a bundle, but further as having a defined shape. As Victoria Brockmeier notes:
Chaos theory was initially developed to model weather systems
– again, perhaps like myth. In such dynamic, evolving systems,
patterns often emerge; these loci mark the sites of what chaos theory
calls attractors. (Apostate Sing this World Forth, 13)
For example, if we wanted to talk about the Eden story, we could say that the “Fall” is an attractor. The story begins with Adam, and God. Adam gets to name some animals. His rib is extracted and Eve is built. The Serpent comes along, and tricks Eve into eating the apple. Thus the fall of man. The motif of “a fall” is thus the “variable” toward which the sequence of events unfold.
The motif of the “Fall” is not just one property of this myth, but rather the end state property, not necessarily “the point” in any conventionally rigid sense, but a telos which is not necessarily a “goal” or “intention.”
It would be slightly meaningless to say that the “point” of the Eden story is to “explain why mankind fell/there is suffering in the world”, because there are a myriad other possible ways this story could have unfolded and yet the motif set is unique to this story and those choices cannot be disregarded. Thus the notion of the attractor does not argue for the centrality of this or that motif, it merely sets a limit, an end point, the end point, toward which the myth evolves. Multiple myths can be mapped onto the “Fall” attractor such as the Golden Age of the Classics, or Plato’s Atlantis myth.
It bears mentioning here, that chaos theory developed based on the needs for solving non-linear differential equations such as fluid dynamics or whether patterns. The equations that describe such systems are generally broken down into discreet forces which interact in a dynamic system of inter-flowing energies.
In sketching out the pedagogical use of the attractor I hope to bring to bear the notion that older models which were based on archetypal methodologies like Jung and Lacan are easily translated over to the attractor without loss of information. It is worth noting to that the attractor describes a system of some sort, as opposed to the more abstracted notion of a “pattern” or a category. Mapping the attractor onto a bundle or a set however, would cause information loss. Here’s why:
We might have a curve along which myths featuring
kin-boundary transgression accumulated, and within
that, a section in which fathers kill daughters, a
section in which sons kill fathers, sections for different
incestuous relationships, and so on. A myth where a
particular event takes place multiple times would appear
close to one in which the same event occurs only once;
a myth in which event X happens and then event Y would
appear close to one in which Y happens and then X.
(Apostate Sing this World Forth, 15)
Thus unlike the bundle or the set, individual myths can be grouped by affinity within the set. In terms of computer science, this would afford the theorist a means of hashing or pre-organizing the data within the set, as opposed to the formless clump of bundle and set theory. Chaos theory’s attractors provide a systematic way to plot data onto shapes which allow us to see the distribution pattern with a god’s eye view.
Limit Cycles, Attractors, Temporal Spheres, & Projection Geometry
In the previous section, the sort of attractors we made reference to are fixed point attractors. The other sort of attractor I would like to draw your attention to is the limit cycle attractors. Without getting too overly technical about an extremely complex branch of mathematics, which was manufactured to handle some of the most difficult non-linear equations, a limit cycle implies a stable oscillator, or pulse which is operating somewhere within the system. The pulse could be a pendulum, a heartbeat, your breath, a rocking chair, anything which has an oscillating motion. As we will soon see, this oscillating motion is a temporal expression of the circle, moving about in a single dimension.
The typical expression of the circle is the familiar two dimensional variety. If we want to map this circle onto a line segment we will lose it’s expression as a circle unless we assume a temporal dimension. For instance, imagine walking from one end of a hall to another, but as soon as you get to the end, you realize you’re in the same place you began. Continuity between the extremes (as in a 2d projection map of the globe) is assumed and in fact the two end points (opposite poles to our eyes) are actually the same point. Such logic is familiar to us in terms of mysticism where phrases like I am the Alpha and Omega abound. It is our argument that this paradoxical logic operates on the behalf of a principle which transcends its own spatial geometry, as is the case with our circular line.
In zero dimensions, the circle maps onto two points, which represent the poles. Animating the image yields a vacillation between one pole and the other. Iambic pentameter is an example of such a 0d circle. It rocks back and forth like a pendulum swinging.
The Paradigm Example / Negation
This sentence is false.
If we are asked to evaluate a sentence whose “meaning” is its own negation we run into all of the motifs of chaos mentioned prior in single utterance, and introduce a new one.
“This sentence”, being a metanym for the entire sentence, is an example of self-reference. Since the entire sentence can be swapped out for the metanym, it exemplifies recursion.
When we carry out the substitution once, it yields:
(This sentence is false) is false.
After a second time it yields:
((This sentence is false is false) is false) is false.
After a third time it yields:
(((This sentence is false is false is false is false) is false is false) is false) is false.
In other words, $This sentence is false$ is mathematically identical to the function f(n) = 2^N or 2 raised to the power of N. This is called a binary logarithm and is used to generate all the binary numbers you associate with the amount of memory available on your hard disk. In essence, the variable N represents the amount of substitutions which have been made. The amount of is falses doubles for every iteration.
Firstly, the substitution works recursively because the last output of the sentence becomes its next input. It loops back into itself, it changes and morphs along an easily identifiable vector. The term this ensures its self-reference.
Logically speaking, $This sentence is false$ which is not strictly expressible in the finite space of a page, is true if an only if it is false.
For instance, if we assume this sentence is true, we are taking it at face value when it admits it is false. If we conclude then that it is false, we are denying the veracity of the statement being uttered which means it’s true. Whatever you assume, the negation flips it to the opposite pole (true to false) inverting the initial assumption. That assumption is then inverted upon re-entry, giving us the harmonics of the 0-sphere.
The reason why I like to employ methods such as mathematics to literary theory is based on a deep seeded desire to unify the knowledges. Specialist rhetoric and the impulse toward differentiation has created a veritable closed system of special sub-languages which are only accessible to the initiated. If this sounds a bit like Free Masonry well, I’d agree. It didn’t happen with any devious intentions at the heart, but nonetheless, breaking the terminological barrier from one for of discourse to the next is often difficult and seemingly not worth the effort it takes. The question then becomes to what extent can an artist or an art-critic invest their time in learning this foreign terminology. Well, now more than ever, artists whose medium is the internet combine technology, mathematics, and the sciences in specific ways, and critics in this field have been compelled to adapt. But the general impetus should not be pragmatism, but rather an openness to build a dialog between the disciplines; to build bridges between the knowledges. After all, we are all referring to the same reality, though are descriptions of it are infinite various. Perhaps the lesson of the fractal is that it should be this way, but too much compartmentalization and insularity causes confusion among thinkers who may in fact be describing the same event. I am not arguing for a streamlining of language into one prototype, but rather the exchangeability of the terms in their usage and the similarities between them allow one to be mapped onto the other. Perhaps one of the major successes of such an endeavor will be to create a prototype language which like the ontologists’ ontologies would need to be compressed for general discourse. But surely the foundation of such a prototype would not be anything as nebulous as being, but rather knowledge.
Victoria Brockmeier’s work can be found here: