(Based on his book “At Home in the Universe”, Oxford University Press, New York, 1995,321 pp.)

We can have “order for free”, Kauffman argues. By “freelt he means sporitaneous, but of course paid for in the usual thermodynamic coinage. He presents a series of increasingly complex models to show this. From this he concludes that self-organization in complex systems plays a role in evolution alongside of natural selection. Selection alone, while powerful, would not be enough.

“Order for free” is already obvious when we try to connect buttons with strings. (This is ordinary graph network theory.) As complexity increases (more strings for the same number of buttons), there soon gets to be an inter-connected “giant” that we can lift all at one, with only a few small button groups still left isolated. Before this level is reached, there were mainly small isolated islands of button groups. There seems to be a “phase transition” from a subcritical to a supracritical region, such as is also seen in the subsequent more sophisticated models.

I think that we have all experienced such transitions, e.g. in rank-ordering nations on some indicator from an alphabetical list: at first the cross-outs of nations already accounted for are scattered randomly, but eventually blocks of adjacent cross-outs appear, until the whole list is exhausted. But the buttons-and-strings model is better because it is two-dimensional, while our list of nations has only one dimension.

In the buttons-and-strings model, the transition is plotted as an S-curve, such as I have discussed elsewhere for economic development, and transitions to peace, justice, and democracy. (See the “Pentagon of Values” (1990) and “Pax Democratica” (1994).)

His next model makes the self-organization of order a notch more concrete. He constructs Boolean networks, where the nodes (able to take on the binary values 0 or 1) react on each other by the Boolean functions OR and AND. (OR means that the node flips from 0 to 1 if anyone of its links says 1, AND means that the node flips from 0 to 1 if all its feed-ins say 1.) In these networks, if the interconnections are too numerous, the networks are in a state of chaos, i.e. never stop flipping; this mode is called “supracritical”. But if each node is linked to only two partners, order emerges (the system is “subcritical”): a large core is “frozen” without ever flipping, while a few chaotic islands remain outside. This can also be achieved, with more than two links per node, if most of the links are in the OR rather than the AND mode.

As we change either the number of links or the ratio of OR to AND junctions, we come to a definite phase transition. This time it goes from a few islands of order in a sea of chaos to a sea of order with a few islands of chaos. It is like the inversion of an oil and water emulsion, as in making mayonnaise. It is best for living systems to be near the. transition (“on the edge of chaos”) in order to preserve flexibility while maintaining stability. Kauffman’s claim is that sufficiently complex systems naturally navigate toward that critical edge, and therefore that life emerges naturally and is not an extremely improbable event, as is sometimes argued.

Then he takes a further step toward concreteness. Instead of these abstract Boolean networks, imagine mutually catalyzing chemical reactions. (The abstract Boolean networks are already models of this, but now he makes it explicit.) .When a set of interconnected chemical reactions, some of which catalyze some others, gets complex enough, we obtain collectively autocatalytic sets. This may be how life first began:

Note that life does not depend, in its definition, on particular chemical compounds (proteins and nucleic acids), but on reaction patterns. There may be other chemicals that would form such patterns, though we don’t know any on this planet. But we should be cautious in jumping to conclusions about other planets.

When take-off to sustainability is achieved, we call this autopoiesis, or self-organization. “Order for free.” Zero-order self-organization would be a self-catalyzed replicator, e.g. ribozyme RNA polymerase. First-order self-organization would be the collectively autocatalytic sets described above. These might be bacterial cells. Second-order self-organization might arise from links between these cells, through parasitism-predation to mutualism-synergy. An example might be the eukaryotic cell, or further on a lichen. A third-level might be multicellular organisms, a fourth level societies and eco-systems, a fifth level the whole biosphere. (Subsidiarity in action!) Here coevolution plays a pan, which Kauffman discusses in further models.

The cell needs to stabilize its order, through homeostasis (which is a network of negative feedbacks), but it also needs to preserve some flexibility (positive feedbacks, which it practices in mitotic cell division). However, Kauffman considers cells and organisms to be mainly in the subcritical regimes of order (with a few islands of chaos), while the biosphere as a whole is supracritical, because it is still growing and diversifying. (But it does have islands of order in the sea of chaos.) Local eco-systems may be just on the edge, as they maintain a rough equilibrium between speciations and extinctions.

In coevolution of different species in an eco-system, each species seeks fitness maxima in a changing fitness landscape. The landscape is shifting because of the very efforts of the different species, as they create or destroy niches for each other. It is in general impossible to find the universal fitness maximum. This is among the difficult computational problems, like that of a travelling salesman who wants to visit 27 cities by the shortest route. Even the fastest computer would take a time far greater than the age of the universe to find it. So in real life, we merely climb the highest fitness peaks we can find, not the very best – but this can be fairly good. In Kauffman’s words, we seek excellence, but not perfection (which is unattainable).

One very concrete example of collectively autocatalytic networks is the interplay between genes and the protein promoters and inhibitors that turn the genes on or off during embryonic development (ontogeny). Each cell derived from the divisions of the fertilized ovum has the full complement of genes, the same for all cells; yet the cells differentiate to some 256 different cell types (in humans), because different genes are turned on (become expressed. i.e. make proteins) in each cell type. About 70% of the genes are turned on in all cells; these are the “housekeeping genes” which every cell needs for its own functioning. These 70% are in the largest “frozen core” of order, as in the Boolean networks; they are invariable. The other 30% are in the smaller “attractorvalleys”, and are different in each cell type.

Kauffman explains the details. In Boolean networks there are “state cycles” that the network transits, but some of these are extremely long (it would take far more time than the age of the universe to transit them), so that they look chaotic. However, there are “attractors” and “valleys of attraction” to which the network eventually converges, and, surprisingly, in the midst of this “combinatorial explosion” to huge numbers, there are only a limited number of these valleys of attraction. He estimates them as 317, which is the same order of magnitude as the 256 different cell types. So perhaps each differentiated cell finds its own valley of attraction in the midst of all the chaos. This is perhaps the most spectacular example of spontaneous “order for free”. Kauffman emphasizes that there is nothing mystical (by which he means mysterious) about it, it is part of the mathematics of these model networks.

Kauffman goes on to discuss the evolution of artifacts, like the automobile, which has some similarities to biological evolution. The fact that cultural/technological evolution depends on human intentionality is not very relevant, because the consequences of our intended actions are often unpredictable, so that we do not get what we aimed for. Like biological evolution, cultural/technological evolution is still directed by “a blind watchmaker”, because we the watchmakers really don’t know what we are doing.

That relates to the question I asked when the unexpected events of 1989-91 occurred in world politics: who or what makes history? We not only did not consciously intend this, we did not even predict it. . Kauffman says: “History arises when the space of possibilities is too large by far for the actual to exhaust the possible.” History is the result of navigating blindly in the midst of vast combinatorial explosions. We do attain some peaks, but not the best ones, and the terrain keeps shifting. But tracking peaks on deforming landscapes is central to survival. We try to do our best, on the whole, some of us. So “who or what makes history?” You and I, and yet not you and I.

Since we cannot reliably predict the results of our actions, because we shift each other’s fitness landscapes like the competing/cooperating species in an eco-system, and since we can only move a step at a time and so cannot see where even the local peaks are, especially when the terrain is very rugged, we have to resort to certain tricks to navigate for survival. In biology, sexual reproduction was such a trick, because it allowed the species that practiced it to move more than one step at a time in the landscape (through recombination of genes) and thus perhaps find the local peaks – while risking falling into the local deep crevasses as well.

In cultural evolution, one trick is “simulated annealing”, going to an equivalent of “a higher temperature” to facilitate the search and then “cooling it” to stabilize whatever was found. (The analogy is the annealing of steel by heating and slow cooling, which freezes in a better crystal structure to give the steel a greater strength. But the simulation involves algorithms that I can’t quite follow.) Even this, Kauffman says, is too slow and laborious to apply in practical life, so what we usually do is “patching”.

In patching, we divide a big problem into a bunch of smaller problems, and optimize (or “satisfice”) each patch separately. This does not always give the best result (sometime ago, Kenneth Boulding warned against “sub-optimizing”), but at least it’s doable in the limited decision time that is sometimes available to us. The problem, of course, is deciding how big the patches should be. We have to attain the best balance between patching (subdividing) and “chunking” (combining problems).

Since big problems are usually “non-decomposable” into parts (i.e. strongly interlinked), there have to be compromises. All this is very relevant to the politics of subsidiarity. There we usually say “solve problems at the lowest level possible where there are no SIGNIFICANT external effects”. The questionable word is “significant”. In non-decomposable problems, how are we to judge this? There must be some arbitrary cut -off point. It is up to mere human judgment.

Another model Kauffman uses is “algorithmic chemistry” (which he playfully abbreviates to “alchemy”), in which strings of computer programs representing macromolecular chains are put into a “pot” and allowed to “react” with each other. Of course, the experiment is not done either “in vitro” or “in vivo’” but “in silico”. The “strings” can act both as “data” (being acted upon) and “programs” (acting on other strings. In the macromolecular analogy, the “strings” can be both “substrates” and “enzymes”. The whole model is a vast metaphor or analogy, from which insights may be gained.

All of Kauffman’s models are really such metaphors, as he himself emphasizes. The question in all such cases of using metaphors is really: are these similarities merely playful and superficial, or do they reveal deep underlying principles? Since many of Kauffman’s metaphoric models are mathematical in nature, I am betting on the deep principles.

The mathematics appropriate to complexity theory is combinatorial calculus and iteration, just as for classical Newtonian physics it is differential equations, for quantum mechanics Hamiltonian operators, for relativity Riemann geometry, and for thermodynamics/statistical mechanics factorials. The astronomer James Jeans once said that God is a mathematician. Perhaps this is not too far-fetched. At least it is one of God’s many attributes.