A New (and very Old) Model Model for Nonlinear Computation
by Katya Walter, Ph.D.
Summary for the Presentation in Tokyo - July 1995
at the Sixth International Conference on Human-Computer Interaction
__________________________________
Brief:
Both the ancient oracle called the Chinese
I Ching & modern DNA exhibit the same model of nonlinear computation. This model combines binary chunks & analog flow in an amazing hybrid format. These two functions of number together provide 64 nonlinear equations that make up the 64 hexagrams of the I Ching, & also the 64 DNA swatches of the genetic code. Their common root can be found in chaos theory. Number is the key to this meeting of East and West. It provides fractal "analinear" equations that combine analog & linear functions. Learning from this model can even provide insights in how to make new & better kinds of computers.
Abstract:
This paper presents an innovative model for computation. Present-day computers are usually binary and do not take nonlinear aspects into account. They certainly cannot do both binary and analog computation in the same process. But such things are possible. How?
Two models of this paradigm may be found in two very different places: in modern DNA and an ancient Chinese document called the I Ching. Each combines binary/digital unitizing along with analog ratios in flow to develop a peculiar hybrid mathematics that turns both DNA and the I Ching into a sturdy survival package.
Each model manifests the principles of the new science of patterned chaos. The core of it rests in the period 3 window of the periodic tree of bifurcating data. James Yorke and Tien-Yien in their famous article “Period Three Implies Chaos,” published in 1975, showed mathematically that when the period 3 window appears in a bifurcation tree, abruptly we no longer have random chaos but patterned chaos (i.e., an orderly but nonlinear structure that is able to sustain and replicate itself with variation).
Mathematically the mRNA codon and the I Ching trigram each hold a period 3 window of chaos patterning. This window is nonlinear, combining both analog and linear functions to synthesize a transcendent third dynamic whereby the system escapes to a higher order of organization.
The DNA swatch and I Ching hexagram can each be demonstrated to hold two polarized period 3 windows in a bond. These two windows are counterpoised against each other to form a fail-safe package of complementary chaos (or cochaos for short). There are eight possible triplets. These may be arranged in polarized pairs to form 64 possible configurations...which naturally gives us the 64 codons of the genetic code or the 64 hexagrams of the I Ching.
DNA is formed by two period 3 windows paired and bonded and strung in succession along the double spiral. Probably DNA took this form because it offers the sturdiest possible mathematical structure. It uses the cochaos paradigm whereby one chaos system (a period 3 window) is bonded with another chaos system (a second period 3 window) to provide a fail-safe supersystem of cochaos that provides both stability along with the possibility for evolutionary change within that stability.
Paper:
Most contemporary gene sequencing programs handle only binary aspects, avoiding nonlinear aspects altogether. But to quote "Hacking the Genome" in the April 1992 issue of Scientific American: "The clarity of the answers will depend on asking the right questions." Asking the right question for hacking DNA might mean looking at the I Ching and asking, "Why do both show the same nonlinear code that utilizes the principles of cochaos?"
Probing for that answer may reveal much about the basic patterns in life's physical and mental systems. It may even offer a new way to build computers that can imitate the basic number framework that is hidden within life itself—its mind and its matter—showing how they interact, perhaps even leading to the discovery of a master code in fundamental physics at the root of universal nature itself.
Number itself forms the root. To explore this deep root, we need not discard traditional scientific linearity, but instead just add something new - analogs. Analog plus linear gives us analinear number. It is not just linear. It combines the chunky lumps of unitized sums with the flowing proportions of analog ratios to birth a new third way. Some call this kind of processing nonlinear, but Stanislaw Ulam often pointed out that the word nonlinear is rather uselessly silly, since most of life's problems are nonlinear. He said calling something nonlinear is akin to saying that most of the animals in the zoo are non-elephants. It clarifies very little.
Thus I prefer to use the word analinear since it implies both number functions to develop a synergistic third state.
Binary number seeks a goal, a solution, the end target of quantified units in a string. It is discrete, end-stopped by that goal that determines the answer. But analog number does not emphasize a solution, a goal, a final summation.
Instead, it discusses the quality of relationships along the way. It brings up all kinds of resonant associations that open the door to further processing rather than closing down into a final answer. That's the trouble with analogs, from a traditional point of view. They proliferate rather than end-stop. They engender messy resonances that linearity does not want to deal with. Linearity does not want to trigger a network of related resonances that reinforce entrainment, not when it's trying to stay tidy and neat and hurry to a quick solution.
Entrainment is the main signature of analog number. It does not care about quantified goals, but rather, about the shifting analog qualities of relationship along the way. It compares the balances in ratios of resonance, not striving for an end but rather for the consummate trip, so that finally it never ends, because the end is no goal at all.
Instead it just keeps on traveling. When analog and linear functions of number combine into analinear number, the result can do both; it can determine straight-line solutions and also keep traveling in cycles. The result becomes the dynamic spiral of change.
The ancient Chinese I Ching provides an astoundingly complete model of this process in a clever shorthand that uses binary/digital sequencing plus analog flow. Its basic structure readily shows the existence of binary number, a fact evident to the West ever since the German scientist Gottfried Wilhelm Leibnitz at the turn into the1700s was sent a copy of the I Ching from a Jesuit missionary in China.
To his shock, Leibnitz realized that the symbols called hexagrams may be read as binary numbers counting from 0 through 63. This showed that the binary code Leibnitz thought he'd invented had preceded him by perhaps 3,000 years. Other scholars soon concurred in this observation.
More recently, Western scientists recognized that the I Ching's yang and yin can even be cross-coded in a binary way with the genetic code. Gunther Stent discusses this procedure in The Coming of the Golden Age, published in 1969. Martin Schoenberger explores it in The I Ching and the Genetic Code, published in 1973. Scientific American'sFunctions and Models of Modern Biochemistry in the I Ching appeared in 1978. In 1991 came Johnson Yan's book DNA and the I Ching. I discussed my view on this interface between the genetic code and the I Ching in Tao of Chaos, published in 1994. Each author considers various aspects of the genetic code/I Ching parallels.
How could the ancient East and the modern West, so far apart in space and time, come upon the same mathematical model? The ancient East saw it as an oracle that codes for the flow of universal mind, the way of the Tao. The modern West sees this same mathematical structure and calls it the DNA code that builds organic matter. Mind and matter—could this shared mathematical code revel a common taproot that plunges even deeper in nature?
To explain this, one must consider the I Ching's fractal aspects. My book Tao of Chaos discusses in detail the ability of both DNA and the I Ching to combine analog and binary functions of number into fractals. Both the genetic code and the I Ching code are based on chaos theory, which can predict a trend without specifying its exact details. Chaos patterning can predict an overall pattern, but it cannot specify any exact detail of its next manifestation. A mathematician can determine its general form but not the exact contents.
Patterned chaos has its own special signature:
* Order in the midst of apparent disorder.
* Cycling that repeats with continual slight variation.
* Scaling that fits one level into another like nesting boxes
* Universal applicability.
Chaos theory has enabled us to find pattern within apparently random events. With it, we can rise to a new level of vision to discover simplicity within complex flux. Long ago in China, this ability was called following the way of the Tao.
Patterned chaos first began to be explored mathematically in the West to any extent during the 1960s, often on makeshift analog computers that charted peculiar cyclic patterns. Chaos theory developed an odd vocabulary where fractals, the Julia and Mandelbrot sets, the butterfly effect and the strange attractor suddenly opened up a new "nonlinear" reality.
This fractal development can be found in the DNA structure which Watson and Crick discovered in the 1950s. Amazingly, this phenomenal use of number also exists in the I Ching, developed according to apocryphal Chinese history, in 3322 BCE.
Briefly, here is a synopsis of their parallel structures:
Each of the eight I Ching trigrams holds the signature of of chaos patterning as a period 3 window on a bifurcation tree. The first step is to polarize the forking on the bifurcation tree up through three levels. Then this window can be read horizontally across the branches of a bifurcation tree in typical Western linear fashion to provide a binary result, or it can be read vertically going up the forks to provide an analog result. In other words, the system can read by both methods simultaneously, resulting in an analinear reading that combines both functions. This is quite remarkable!
Furthermore, one period 3 window (a trigram) can be bonded with a second period 3 window (another trigram) to create the 64 hexagrams. These may be read as 64 cochaos patterns made by 64 pairs of period 3 windows combining binary/digital units and analog flow. Each of the 64 hexagrams describes its own unique dynamic process.
Likewise, the DNA double helix may be seen in this same way. Its pyrimidines and purines use the same organizational plan with polarized pairs of triplets bonding across the double spiral. Thus, any hexagram may be read to encode the message of its concomitant amino acid, thereby providing the 64 codons of RNA.
Furthermore, it can be shown that the I Ching and the genetic code not only use the same mathematical structure, but they also cross-correlate according to the basic messaging. For example, the Stop codon of the genetic code actually equates mathematically to Hexagram 12, Standstill. Somehow mind and matter merge in this master code. Its paradigm apparently builds our bodies and our minds. It is bone-deep in the species, archetypal in the mind.
Thus, each system—genetic code or I Ching—gives a microcosmic rendition of a larger principle of cochaos theory. Fortunately, these two models, ancient and modern, provide a means to observe a mathematical paradigm that is perhaps inherent in the fabric of the universe itself.
Numbers create the patterns of the universe. Analogs form the networks of qualitative resonance in the timing and spacing of matter and energy, while linears develop the discrete sums that quantify units of whatever is being spaced or timed. Together—as analinear number—they give a flowing, connective quality to the universe's discrete quantities. To merge the analog with the linear in cochaos patterning provides a truly universal computation method.
In this view, each hexagram or each DNA swatch becomes a nonlinear equation. Both versions are rooted in chaos theory—more particularly, in a specific variant I call cochaos. It bonds two period 3 windows of chaos patterning into this secure and fail-safe master combination as cochaos.
Notice—concentrating on only the binary or the analog aspects of number in either of these two systems would deprive us of the true key to their coding—the two counterpoised period 3 windows of complementary chaos that create 64 distinct and unique dynamic events. If we do not see that, we completely overlook the amazing combination of binary and analog processing that reveals the master code. I suspect that the I Ching and the genetic code offer to the human eye two variations on a deeper paradigm that is embedded in the universe itself, the master code that builds the universe.
To balance and harmonize the analog and linear functions of number is the special gift of analinear computation. It is evident in the ancient I Ching and in modern DNA. By combining unitized counting with flowing proportions, this paradigm creates nonlinear equations, or more appropriately, analinear equations. They may one day even provide us with a new kind of computer. I believe the universe uses it already.
© Katya Walter 1995
Posts
0 comments:
Post a Comment