In my previous essay “Complexification and the Mind of God”, I referred to the book by Paul Davies “The Mind of God”, in which he defined two important terms:
Algorithmic Complexity (AC) is the length of the minimal program that would yield the observed output in a computer simulation. For sufficiently complex systems, the program is (almost) as long as the output. We say that the output is algorithmically incompressible.
Logical depth (LD) is the running time for the minimal program to generate the observed output. Simple systems are logically shallow, complex systems are logically deep; sometimes the running time is infinite, i.e. the program would never halt.
We can combine low and high AC and LD in 4 ways, as follows, along with typical examples as indicated:
Scheme A: Low AC, low LD – small crystal or a tiled bathroom floor. The repeating patterns do not extend indefinitely, hence we specify “small” crystal and “bathroom floor”.
Scheme B: Low AC, high LD – Koch snowflake and other fractal structures, even the entire Mandelbrot set. The program is quite short, but it runs forever.
Scheme C: High AC, low LD – embryo development. The program is very long (the entire genome), but the running time is limited (9 months for humans, much less for many other species).
Scheme D: High AC, high LD – bioloqical evolution. The very long program is still running, and will continue until the end of the Earth as a living planet.
Stuart Kauffman in “At Home in the Universe” considers embryonic development “a subcritical phase” (a sea of order with some islands of chaos, i.e. pretty well set with occasional errors), the biosphere as a whole “a supracritical phase” (a sea of chaos.with some islands of order, i.e. subject to Gould’s condition of “contingency”), as still growing. Ontogeny does recapitulate phylogeny, but only up to the point where that particular organism has evolved. Ecosystems are probably at the point of transition from subcritical to supracritical, “on the edge of chaos”.
Complexity theory, as it fits into my scheme of the expanded “windmill model” of the sciences (see “Unfinished Road of Penetration Into Truth” and its sequel, “Combining the Mandala of the Sciences and the Room with Many Doors), uses schemes B, C and D as defined above. Scheme A is the subject of crystallography and geometry. Scheme B deals with fractals, schemes C and D with life processes.
There are transitions: from A to B we still Use repetition (iteration mathematically speaking), but at progressively diminishing scales, as in the Koch snow~lake and same of Escher’s pictures. From (A or B) to C we st~ll have growth. but in C it is differentiating rather than repetitive growth. From C to D we go from predetermined growth to unpredictable growth.
A is the order of rocks (lithosphere), B is the order of seashores, lakeshores, mountains, and clouds, (hydrosphere and atmosphere), and C + D is’ the order of the biosphere. Going in the other direction, C + D is the biota, B + C + D is the biosphere in the extended sense (i.e. the biota plus where it resides), and A + B + C + D is Gaia — the whole surface of planet Earth. Is Gaia alive? Sphere A (the rocks) not too much, but perhaps it is like bone or shell or the wood in trees; sphere B (water and air) more so, ever changing in climate and weather; sphere C definitely, but stuck in species specificity and hence somewhat rigid; sphere D supremely so, with unfulfilled potential still beckoning. The unfairly devalued polytheistic nature religions saw spirits not only in animals and plants, but also in lakes, mountains, and rocks; all were holy. This is God (or Goddess) immanent, whatever the transcendent God may be.
What do we value? Diamonds are merely crystals, qold is a mere product of the rocks. King Midas found that the golden touch was a liability, since it transformed living beings into mere statues. An opposite process is indicated in the myths of Galatea and pygmalion, as well as Pinnochio. We value (or should value) entities with great algorithmic complexity and great logical depth.