Complex systems are made orderly through strange attractors. These strange attractors are the rules by which complex systems are ordered. Each system, to be a system, must have governing rules – though for any particular system the number and kind of rules changes. We get new, emergent kinds of rules with each new, emergent system. A small number of rules create all the atoms, which combine to generate the rules of chemistry. The rules of quantum physics still hold at the quantum atomic level, but the interaction of atoms with each other generates new rules to emerge. One does not get ions in quantum physics – one gets them only from chemical interactions between atoms, whose interactions are able to make use of unstable electron orbits and stabilize them through the chemical interaction, transferring the single unstable electron from the donor (often metal) atom to the electron shell of the acceptor (nonmetal) atom, whose outer electron shell is stabilized by having the optimal number of electrons. A sodium atom is stabilized through its chemical interaction with chlorine by donating its outer electron to the chlorine atom, stabilizing it. If one merely knew of quantum physics, and not chemistry, could one predict such an interaction? Quantum physics predicts the creation of neutral atoms through the combination of electrons, protons, and neutrons. But it also predicts, in conflict with this rule, the increased stability of a full electron shell – with decreasing stability the fewer the electrons in the shell. But how could an atom missing an electron, giving it a positive charge, be more stable? Obviously it cannot, unless it is combined with chlorine, to create a neutral chemical combination, each atom oppositely charged, each with more stable outer electron shells. Chlorine is so much more stable as an ion that chlorine ions are smaller than chlorine atoms, as the addition of the extra electron to the outer shell to complete it makes the electrons orbit closer to the nucleus, in a more stable state. One gets an emergent chemical rule as chemistry makes use of this agonal conflict between maintaining charge neutrality and electron shell stability.
We find the same situation as we move from chemistry into a certain arrangement of organic chemical systems – life. The creation of ions, hydrogen bonds, chemical bonds, and van der Waals forces in chemical interactions generate, in certain kinds of organic chemicals, the ability to self-replicate. Stuart Kauffman goes into great detail in The Origins of Order, particularly Part II of that book, and I recommend this book for those readers who wish to go into the emergence of life from prebiotic chemicals in much greater detail than is allowed by the scope of this work. I, however, shall skip ahead considerably, and only say that while Kauffman is generally correct, considerable work on RNA has been done since 1993 that strengthens the case for RNA or an RNA-like precursor, in combination with small polypeptides, since it has since been shown that it is the rRNA of the ribosome which catalyzes the peptide bonds, not the proteins. So we can talk more strongly about an RNA world as at least a precursor to life as we now know it. In chemical polymers such as self-replicative RNA, the ability to self-replicate is determined by the order of the constituent parts (in this case, ribonucleotides). Any alteration of the sequence pattern affects the polymers’ ability to reproduce themselves. Most were dead ends, but a few could reproduce themselves better. This self-replicative ability was an emergent property of a particular polymer sequence – and abided by its own particular rules, rules which emerged from, yet were still a part of, the rules of chemistry.
Genes provide a template for the rules that give rise to complete cellular systems. Gene combinations and alternative forms of regulation give rise to different cell types (S. Kauffman identifies different cell types as different strange attractors (202) – identifying each and every cell as a fractal). The combination of cell types gives rise to different kinds of organisms, which are structured using rules different from those which organize the cells themselves. Neurons are a kind of cell which are capable, due to their complex structures, of complex interactions with other neurons. In sufficient numbers and concentrations (i.e., in a brain), these interacting neurons lead to complex behaviors (more emergent rules), including the ability to language and be self-aware. We get the accumulation of more and more rules, from more and more complex interactions from more and more constituent parts. The remarkable thing is that though the number of parts increases dramatically, the number of rules increases at a much slower rate. S. Kauffman gives us a set of equations that help us see the relationship between the number of types of parts of a system and the number of rules (strange attractors) generated by those parts, as well as the number of possible expressions a system can generate from those rules. Using these very simple equations, we can see how we can get the level of complexity we find in the universe, starting from perfect symmetry (nothing), and what this suggests for art and literature.
Kauffman shows that for any system with a certain number of components (N), that system will have 2n/2 possible states within the system, but only N/e number of cycles, or possible basins of attraction, where e is the inverse natural logarithm (e=2.718281828449...)
Thus a system containing 200 elements would have only about 74 alternative asymptotic patterns of behavior. More strikingly, a system containing 10,000 elements and chaotic attractors with median lengths on the order of 25000 would harbor only about 3700 alternative attractors. This is already an interesting intimation of order even in extremely complex disordered systems. (S. Kauffman, 194)
Kauffman then shows that such systems are even more organized, since for a complex system:
The expected median state cycle length is about N. That is, the number of states on an attractor scales as the square root of the number of elements. A Boolean network with 10,000 elements which was utterly random within the constraint that each element is regulated by only two elements would therefore have a state space of 210,000 = 103000 but would settle down and cycle recurrently among a mere 10,000 = 100 states. . . . A system of 10,000 elements which localizes its dynamical behavior to 100 states has restricted itself to 10-2998 parts of its entire state space. Here is spontaneous order indeed. . . . The number of state cycle attractors is also about N. Therefore, a random Boolean network with 10,000 elements would be expected to have on the order of 100 alternative attractors. A system with 100,000 elements, comparable to the human genome, would have about 317 alternative asymptotic attractors (201).
And this is about how many kinds of cells one finds in the human body. But more importantly, systems with very large numbers of elements can and do have a very small number of ways of organizing themselves, though the number of ways of expressing those rules may be astronomical. For a system with N=200, the median cycle length, or possible states per system, is 2100 1030, “At a microsecond per state transition, it would require about a billion times the age of the universe to traverse the attractor” (Kauffman, 194). And that is for a tiny system with only 200 elements. Yet the actual different ways such a system would be expressed would be only 47. There would be 47 general forms, with 1030 specific forms. One could see strange attractors (though not these specific numbers I have used as examples, of course) as the different species of animals the “zoological system” could create, and the median cycle length as the number of particular individuals that could be generated.
But let us now use these equations as promised. If I am correct in identifying the universe and everything in the universe as complex fractal systems of these sorts, then Kauffman’s equations should be able to give us the complexity found in the universe, starting with nothing.
N = dimensions = elements of a system
2N/2 = median cycle length (MCL) = possible states per system
N/e = number of attractors
N = median state cycle (MSC) = local dynamic behavior
And, for each new emergent system, constituting all the elements of the previous system:
Nnext = MCL + number of attractors, as MCL and attractors constitute the combination of elements, both the physical components and the rules that made that system.
For those systems that do not use all of the elements from a previous system, such as biology, which only uses certain kinds of chemicals (thought admittedly at least trace amount of most), and emergent human intelligence, which does not use all organisms, but only uses its own cells (and not all of them; though, like all organisms, it needs a full body in which to function, and whose body need a full ecosystem in which to live), N would necessarily be smaller than suggested above. Nnext would work starting from the big bang, up through the creation of strings, while N would have to be derived in other ways for life, human intelligence, and the arts and humanities. But let us see if we can get to either 10 or 11 dimensional strings from N = 0, at the big bang.
For N = 0,
MCL = 20/2 = 1 = singularity of the big bang (so far so good)
# attractors = 0/e = 0
MCL = 21/2 = 2 1.41
# attractors = 1/e 0.37 (a fraction, which we would expect in a fractal)
MSC = 1 = 1
N = MCL + # attractors = 1.4 + 0.37 = 1.78
MCL = 21.77/2 1.85
# attractors = 1.78/e 0.65
MSC = 1.77 1.3
N = 1.85 + 0.65 2.50
MCL = 22.5/2 2.38
# attractors = 2.5/e 0.92
MSC = 2.5 1.58
MCL = 23.3/2 = 3.14
# attractors = 3.3/e 1.21
MSC = 3.3 1.82
MCL = 24.35/2 = 4.52
# attractors = 4.35/e 1.61
MSC = 4.35 2.09
N 6.12 = 4-D space, time, matter-energy
MCL 26.12/2 8.34
# attractors 6/e 2.25 = creation of gravity and GUT
MSC 6 2.47
N 10.59 = fractal strings between 10 and 11 dimensions, containing 4-D space, time, matter-energy, gravity, strong nuclear, weak nuclear, electromagnetic, (speed of light?)
MCL 210.59/2 39.26 number of potential string combinations (strange quarks, etc.)
# attractors 10/e 3.9 actual string types (quarks, electron, photon)
MSC 10 3.16
(the means "about" or "about equal to")
By this point, not all possible states are realized, as they become increasingly unstable at increasing distance from the stabilizing attractors. Starting with only these very simple equations, we get emergence all the way to strings having between ten and eleven dimensions. We can reconcile the 10-D and the 11-D theories, since we see using these calculations that strings have fractal dimensions – which we would expect in a fractal universe. Theories that see dimensions as whole numbers would naturally give us either ten or eleven dimensions. This latter aspect of strings has given people a great deal of trouble. But here we see those dimensions arising naturally from these calculations, once one sees a dimension as being an interactive element of a system. A system with 100 different elements is a system with 100 dimensions. This suggests that one could see quantum strings as systems containing around eleven elements – with these elements being such things as length, width, height, time, and, as I have suggested, various constants, including the speed of light (c). And one might also include other constants, including Planck’s constant (h=6.6 x 10-34 joule second = a constant action, making it a good candidate) bringing us up to ten, and, if I may be so bold as to go further out on this limb, perhaps pi, since in quantum physics h-cross = h/2, or the Golden Mean = 1.618, since we see this function expressed at all scales, including the Fibonacci spirals of spiral galaxies, to bring it up to eleven. But determining what constitutes the dimensions of strings goes far beyond the bounds of this work. I propose these hoping someone more able than I am with the mathematics of quantum physics will investigate the dimensions of strings along these lines, as elements in a system.
J.T. Fraser, in Time, Conflict, and Human Values, proposes that there have been 101000 number of organisms through the history of life on earth (he also suggests we would get a complexity of 10 at the quantum level, which we have shown to be the case in the above calculations, suggesting his numbers may have at least some rough validity). This would mean the MCL for biology would be (to use these very rough numbers) 101000 23000. N 6000, which would be about the number of kinds of generic genes, giving rise to 6000/e patterns of behavior, or over 2000 different kinds of organism, which would itself be contained within MSC 6000 80 different types. Naturally, at this point, we are being highly approximate. However, if we further use Fraser’s numbers, where the MCL for humans = 1010,000, for the number of possible brain states, we get N 60,000, # of strange attractors 2200, which would include all the elements that constitute human behavior, including the number of emotions, universals of human behavior, etc., and MSC 250, which appears to be approximately the number of human cultural universals. These numbers perhaps would not surprise many scientists, but there is still some controversy among – primarily postmodernist – scholars in the humanities. For this reason, we should take a closer look at this step, at the issue of cultural universals, or human instincts. Only if humans have instincts can there be such a thing as an evolutionary approach to art and literature applicable to humans. The existence of such instincts could also explain the very content of our art and literature.