Investigations

Stuart Kauffman's book on Emergence

In Stuart Kauffman's book on emergence, he offers the following reasons why Goedel does not explain why the configuration space of the biosphere is not finitely prestate-able. He has, in doing so, convinced me that my previous intuitions to the contrary are, in fact, correct.

In this chapter I have been trying to say, argue, articulate the possibility that a biosphere is a profoundly generative -- somehow fundamentally always creative. The cornerstone of this dawning near-conviction lies in the belief I now hold with some confidence that we cannot finitely prestate the configuration space of a biosphere. New variables...persistently emerge. New language games and living games emerge. What is the status of my claim that we cannot finitely prestate the configuration space of a biosphere? I do think my claim is true. But why? I am not sure. It is wise to explore some possible reasons.

A first possibility is that the biosphere, like a complex algorithm, unfolds in ways that cannot be foretold. Recall that for many algorithms the behavior of the algorithm cannot be prestated in any form more compressed than simply watching the program unfold. The famous "halting problem" is the classic example. For many algorithms that are to compute an answer, then halt, we cannot say ahead of time whether the computer will halt in finite time.

I do not think the biosphere is akin to this difficulty with many algorithms, the building blocks of the algorithms...are well-stated, crisp, mathematical primitives. Our uncertainty about the unfolding of an algorithm does not lie in uncertainty about the primitives, but about the consequences of the arrangements of these agreed-upon primitives in a given computer code. For example, will the algorithm based on those primitives halt or not halt in finite time?

--Stuart Kauffman

Nothing in non-computability precludes development of new primitives which would make it possible to predict particular halting. In fact it's pretty easy to extend Turing to show it is always possible to do so, but the primitives produce new potential algorithms which we cannot predict.

Since I wrote the preceeding paragraph, I have learned it is not precisely true. I was assuming that Turing was trying to extend Goedel. But, in fact, his goal was to come up with an algorithmic end-run on Goedel's conclusion.

Instead, he found a proof that such algorithm was impossible. So a better way to say what I said is the following:

Nothing in non-computability precludes development of new primitives under which previously unpredictable haltings would become predictable. In fact it's pretty easy to extend Goedel to show it is always possible to do so, but the Goedelian primitives produce new sets of algorithmic definitions which will include haltings we cannot predict.

Kauffman continues:

Let's consider the possibility that the incapacity to finitely prestate the configuration space of a biosphere is related to Goedel's theorem. Goedel demonstrated that for axiomatic systems as rich or richer than arithmetic, given a set of axioms, there were always statements that were true but not formally derivable from the axioms. In addition, Goedel showed that it was always possible to enrich the axiom set, and from that enriched axiom set, it would be possible to prove the formally true but unprovable statements in the formal system. On the other hand, he also showed that the new enriched axiom system would itself have still further formally true but unprovable statements.

--Stuart Kauffman

The relation which Kauffman seems to be suggesting here is an analogy, which I will represent as:

statements which are true:statements which are formally derivable::A:B

Read this formalization of the analogy as: "Statements which are true are to statements which are formally derivable just as A is to B."

I think Kauffman is a little bit unclear about what he is suggesting we consider as values for A and B.

Kauffman adds:

I am not persuaded that the uncertainty about the configuration space of a biosphere is analogous to true but formally undecidable statements in a formal system. I base this upon an analogy between formal proof and causal consequences....If we are to represent causal consequences by a formal system, then the concept of a proof derived by formal procedures from axioms...is the natural way to represent causal consequence. If this parallelism is taken seriously, then statements in a formal language that are true but unprovable in that formal language can have no causal pathway--that is, proof--from the axioms to the desired consequence. But this analogy seems to fail with respect to the evolution of the biosphere. There is a perfectly fine causal account of Gertrude and her maiden flight. We can reconstruct that account after the fact, even if we could not have predicted it. Thus, it does not seem that our difficulty in prestating all exaptations is the same as the mathematical fact of formally undecidable statements in an axiom system.

--Stuart Kauffman

It seems to me Kauffman is confusing two different analogies (both of which assume B="observed true"):

In one analogy A = "is predictable before the fact," while in the other A = "has a causal account." I believe both of these analogies can be shown to meet all the criteria for Goedel's theorem to apply to them. Depending on the exact definitions of each, they may or may not be identical.

But each is a modal predicate which can be represented as axiomatic systems (under the loose definition of axiomatic system employed by Goedel).

We can create an axiomatic system which constructs causal accounts, thus allowing us to ask the question, "Is it possible to derive a causal account of Gertrude's maiden flight from this system?" This is formally equivalent to the Goedelian predicate "possible to derive a particular theorem."

We can also create an axiomatic system which makes predictions, thus allowing us to ask the question, "Is it possible to derive a prediction of Gertrude's maiden flight from this system?" This is formally equivalent to the same Goedelian predicate.

But that does not mean the two systems are necessarily the same, as Kauffman seems to assume. They need not even be related, although I will propose a method for drawing a relationship below which is suggested by Kauffman's next paragraph:

On the other hand, there may be a parallel between the exaptations of which we have spoken and Goedel's theorem and the augmentation of the axiom set such that formerly unprovable statements become provable. That is, the emergence of novel exaptations in evolution do seem rather like the emergence of novel primitive objects and primitive control operations--hence, novel axioms. In the examples above, the emergence of the genetic code and the emergence of chromosomes that duplicate and partition daughter chromosomes into two daughter cells, the evolution of controlled recombination, seem to become instantiated as "biological laws," even though they are entirely historically contingent. Changing the biological laws in evolution seems rather like the generation of a novel axiom which new consequences can be derived.

--Stuart Kauffman

This suggests that we understand the appropriate analogy to be A = "is predictable before the fact." If the system does not predict Gertrude's maiden flight (and other exaptations) before the fact, we may be able (after the fact) to add more axioms to the system, which will enable us to predict these additional exaptations (after the fact) and perhaps to predict other exaptations as well.

But we know in this instance there will be additional exaptations enabled by this extension of the predictable-before-the-fact system which will occur without being themselves predictable before the fact under the system. Indeed, we could even describe this as the emergence of new axioms.

And then Kauffman writes:

On this interpretation, my claim that we cannot finitely prestate the configuration space of a biosphere becomes the claim that the biosphere keeps generating new "causal axioms" from which it generates novel forms. Then just as we do not know where the new axioms of a formal system come from, save as the free invention of the logician involved, so it would seem that we cannot prestate the new generative exaptations that allow evolution to drive in new directions. I am not entirely persuaded by this analogy to finding ever new axioms in Goedel's theorem, but it does have some coherence.

--Stuart Kauffman

Indeed it does. In fact, it resolves the confusion between the two different analogies discussed earlier.

While Kauffman may remain unconvinced, I believe it is possible to prove both analogies are powerful, but different. Goedel's conclusions can be shown to necessarily follow in both cases, but we must recognize that they are separate, even if related.

Stillflame is working on developing metamathematical arguments which demonstrate this conclusively.

The Gottman Hypothesis

Perhaps the arguments of Gottman et al. in <i>The Mathematics of Marriage</i> (or some stronger form associated with Strogatz' <i>Nonlinear Dynamics and Chaos</i>) will make such efforts unnecessary. Gottman argues that the inability of linear systems to produce a stable set point means that real-world systems will have to have a non-linear dynamic function (or change function) which will inevitably produce emergent systems.

See also Math and The Goedel Connection.

— Scotus - 20 Apr 2002