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 simple equations, we can get the level of complexity we find in the universe, starting from perfect symmetry (nothing).
Kauffman shows that for any system with a certain number of components (N), that system will have 2^(n/2) (or, 2 raised to the n/2 power) 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 pN. 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 p10,000 = 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 pN. 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).
This is about how many kinds of cells one finds in the human body. 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. These strange attractors (though not the specific numbers I have used as examples, of course) are the different species of animals the "zoological system" can create; the median cycle length is 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.
With:
N = dimensions = elements of a system
2^(N/2) = median cycle length (MCL) = possible states per system
N/e = number of attractors
N^(1/2) = 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 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 the body needs 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 = 2^(0/2) = 1 = singularity of the big bang (so far so good)
# attractors = 0/e = 0
N= 1
MCL = 2^(1/2) = p2 =1.41
# attractors = 1/e =0.37 (a fraction, which we would expect in a fractal)
MSC = 1^(1/2) = 1
N = MCL + # attractors = 1.4 + 0.37 = 1.78
MCL = 2^(1.77/2) =1.85
# attractors = 1.78/e =0.65
MSC = 1.77^(1/2) =1.3
N = 1.85 + 0.65 =2.50
MCL = 2^(2.5/2) =2.38
# attractors = 2.5/e =0.92
MSC = 2.5^(1/2) =1.58
N =3.30
MCL = 2^(3.3/2) = 3.14
# attractors = 3.3/e =1.21
MSC = 3.3^(1/2) =1.82
N =4.35
MCL = 2^(4.35/2) = 4.52
# attractors = 4.35/e –1.61
MSC = 4.35^(1/2) =2.09
N =6.12 = 4-D space, time
MCL = 2^(6.12/2) =8.34
# attractors = 6/e =2.25
MSC = 6^(1/2) = 2.47
N =10.59 = fractal dimensions 10 and 11 dimensions for quantum strings
MCL = 2^10.59/2 =39.26 =number of potential string combinations (strange quarks, etc.)
# attractors = 10/e =3.9 =Bosons, or forces (graviton, gluon, photon, bosons)
MSC = 10^1/2 =3.16
By this point, not all possible states are realized – they become increasingly unstable at increasing distance from the stabilizing attractors. So we should expect, as we in fact find, that only some of the MCL of the last set of equations are stable -- electrons, electron neurtrinos, up and down quarks, gravitons, photons (electromagnetism), gluons (strong nuclear force), and the bosons (weak nuclear force), versus the unstable muon, muon neutrino, tau, tau neutrino, and charm, strange, top, and bottom quarks. Using only these 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 these calculations give strings with fractal dimensions – which we would expect in a fractal universe. Theories that see dimensions as whole numbers would naturally give either ten or eleven dimensions. This latter aspect of strings created a great deal of trouble. But those dimensions arise 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 and creating around 43.5 elements – with these elements being such things as length, width, height, time, bosons, fermions, and various constants. And one might in fact include constants such as Planck’s constant (h=6.6 x 10-34 joule second = a constant action, making it a good candidate) and perhaps pi, since in quantum physics h-cross = h/2B. But these are details for the quantum physicists.
In Time, Conflict, and Human Values, J.T. Fraser proposes that there have been 10^1000 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). This would mean the MCL for biology would be (to use these very rough numbers) 10^1000 = 2^3000. 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^(1/2) =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 = 10^10,000, 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.
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