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The Limits of Self Organisation

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I’m writing to tell people about a paper of mine that was published in Synthese last month, titled:  “Self-organisation in dynamical systems: a limiting result”.  While the paper doesn’t address intelligent design as such, it indirectly establishes strict limits to what such evolutionary mechanisms as natural selection can accomplish.  In particular, it shows that physical laws, operating on an initially random arrangement of matter, cannot produce complex objects with any reasonable chance in any reasonable time.

The published version may be downloaded (payment or subscription needed) from Springer at:


Alternatively, a pre-published version is freely available at:


The argument is based on a number of original mathematical theorems that are proved in the paper.  A less technical presentation of the argument is however given below.

1.  What is self organization?

The basic idea of self organization is that some objects can appear “spontaneously”, i.e. without an external cause.  Snowflakes are an obvious example.  No factory is needed to assemble those lovely six-pointed stars – they just come together, by themselves, under the action of physical laws.  Now self organization isn’t magic, of course.  The objects that appear are caused by something, namely the basic properties of matter and the underlying dynamical laws.

One point I make in my paper is that standard evolutionary theories in biology are all self organization theories.  According to Darwin, for example, all life (except possibly the very first cells) arose by self organization.  This follows from the fact that the theory makes no appeal to an external guiding force, or to a special initial state, or to massive amounts of luck or time.

Biologists today talk of self organization as a possible supplement to the standard Darwinian mechanisms.  One might ask, for example, “Was this feature selected for, or did it arise by self organization?”  Nevertheless, according to the most straightforward definition of the term, Darwinian evolution is itself an example of self organization.

This is actually a substantial point, not a merely verbal one.  For it is easy to slip into thinking of natural selection as a sort of external guiding force, even though it is just a matter of interactions within the biosphere.  Recognising Darwinian evolution to be a case self organization helps us to realize that it is limited by the underlying dynamics. 

Self organized structures are caused by the dynamical laws of the system, operating on just about any initial state.  But now consider Michael Polanyi’s claim in “Life’s Irreducible Structure” (Science, vol. 160, June 1968) that the structure of a machine is independent of the laws that govern its material parts.  If Polanyi is right, it would seem to follow that machines (such as Paley’s watch!) cannot self organize.  And, as he notes, living organisms are like machines in this respect.

In my paper I investigate mathematically one way in which self organization in a system is constrained by the dynamics.  But I realize that I am not a professional mathematician (just a philosopher with a B.Sc. in math) and no doubt I’ve barely scratched the surface of this topic.  So one of my main hopes with this paper is that others will be encouraged to investigate further.

2.  Limitative results

The main theorem of my paper is the “Limitative Theorem”, stated as follows:

A specific large, maximally irregular object cannot appear by self organisation in any dynamical system whose laws are local and invariant.

Limitative results, or “no go” theorems, are not uncommon in science.  The most famous such results are found in mathematical logic, e.g. Gödel’s incompleteness theorems, the Skolem-Löwenheim theorem, etc.  But they are also present in quantum theory (Bell’s theorem, the Kochen-Specker theorem), where they establish that no theory of a certain kind is possible.  One can even see the laws of thermodynamics as establishing a “no go” theorem that perceptual motion machines are impossible, and special relativity as ruling out Captain Kirk’s warp drive.

ID theorists also make such limitative claims.  Michael Behe’s recent book is titled The Edge of Evolution, and here ‘edge’ means ‘limit’, or ‘limitation’.  He is arguing that certain evolutionary mechanisms are limited in what they can accomplish.  In a similar way William Dembski and Robert Marks have proposed a “law of conservation of information” that would rule out some kinds of physical behaviour.  In fact, while I haven’t argued for ID, my main theorem does generally seem to support the limitative claims of ID theorists.

On the topic of the logician Kurt Gödel, there is evidence that he anticipated my limitative theorem, judging by his claim below:

“The formation within geological time of a human body, by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field, is as unlikely as the separation by chance of the atmosphere into its components…The complexity of living bodies has to be present either in the material or in the laws.”

         (Kurt Gödel, in conversation with Hao Wang.  See Wang, ‘On “computabilism” and physicalism: Some problems’, in J. Cornwell, ed., Nature’s Imagination, 1995.)

3.  Symmetry arguments

My argument is an example of a symmetry argument, which is a kind of reasoning commonly used by physicists, as well as mathematicians and logicians.  In arguments where the premises exhibit some kind of symmetry, one can sometimes exploit that symmetry to derive a conclusion very easily, while ignoring a lot of tedious detail.

In physics, we know that if the laws and initial conditions are symmetric in a certain way, then the later behaviour must also be symmetric in that respect.  (This is true of a deterministic system.  In a stochastic system it’s the probability distribution rather than the behaviour that preserves symmetry.)

Symmetry arguments are relevant to the question of dynamical self organization because dynamical laws are highly symmetric.  And, I believe, it’s this very symmetry that rules out the self organization of human bodies.  (Indeed, I rather think that by “other laws of similar nature” in the quote above Gödel means other laws with such symmetries.)  If, as physicists believe, symmetry is preserved from causes to effects, then the products of self organization (caused by the dynamical laws) must also be symmetric.  But, on the contrary, living organisms are rather irregular, and hence (as Polanyi said) have a structure that is largely independent of these laws.

4.  “Conservation of information” theorem

Like Dembski and Marks, I argue for a kind of “conservation of information” law.  In my paper, this arises from needing a quantitative relationship for self organisation between the time available, the chance of success, and the degree of constraint on the initial state.  In order to derive this result (the “random equivalence” theorem) I define a measure of “dynamical complexity” that is based on the familiar notion of algorithmic complexity.  The rough idea is just that a complex object is one that the system cannot produce without a large “input”.

After defining complexity in this way, it’s not surprising that one can prove that the products of self-organisation must be dynamically simple.  After all, the “self” part means without external help.  So, in a way, the conservation of information is rather trivial.  But it’s not entirely trivial, as without this theorem one might imagine that a small input (say a small constraint on the initial state) could have a large effect on the chance of success.  The theorem says, by contrast, that each “bit” of constraint can (at best) only double the chance of success, or halve the time required, etc.

5.  Conclusion

My argument is not especially concerned with the creative powers of natural selection, since it covers self-organisation in general.  But the limitative theorem does entail that natural selection cannot have the powers that are often claimed.  In this respect my argument is similar to, for example, Michael Behe’s argument involving the notion of irreducible complexity (e.g. in Darwin’s Black Box).  An advantage of my symmetry argument however is that it is not vulnerable to the charge of being an argument from ignorance.  After all, even Behe admits that the production of an irreducibly complex object by selectionist mechanisms isn’t ruled out by his argument.  It’s just difficult to come up with a plausible account.  My argument does simply rule out the self organization of complex objects, and should not be as easy to answer as Behe’s (though only time will tell).

Neil Rickert: Thanks for your comment. I see that I could have made things clearer. The results are supposed to apply to closed dynamical systems, so in this case the "system" is the whole earth, or at least the biosphere. We could even consider the solar system, so that gravitational interactions and solar radiation are included, but that shouldn't be necessary. It seems clear enough that sunlight, for example, isn't a source of the particular irregularity needed to build living organisms. Richard Johns
Your second criterion for "self organization" is that "There is no need for interaction with an external system." It is difficult to see why this does not rule out the applicability of your results. In the case of biological systems, the main target of your argument, these systems need to interact with an external source of nutrition. In the case of your example of the vortex in a bathtub, there needs to be interaction with an external source of water, or else the vortex will be short lived. In the case of crystals, these are usually built from an external source of molecules that are being incorporated into the growing crystal. Neil Rickert
albertderoos: Technically you're right, that external actions on the system could in principle break the symmetry in the causes, and thus circumvent my results. But it seems pretty clear that not just any external action will do here. In particular, such things as regular cycles will make no difference I would guess. The difficult thing is to derive a (large and particular) irregular effect from a simple and regular cause, such as the laws of physics. Adding a second cause that is also simple and regular won't help I think, although really this is something to check mathematically. Richard Johns
Richard, sorry, wrong reference, this one I meant: http://www.biology-direct.com/content/pdf/1745-6150-2-12.pdf albertderoos
Richard, when talking about the origin of life as a self-organising structure, I have used the diurnal or seasonal cycle as an external system that could drive self-organization. Similar to the PCR reaction, individual nucleotides could organize themselves based on temperature cycles (see de Roos, Artificial Life 2006: http://albertderoos.nl/wp-content/uploads/2006/10/de-roos-artificial-life-2006.pdf ) That would counter your assumptions underlying your definition, would it not? albertderoos
Richard Johns: That seems really interesting. I will need some time to read it accurately. gpuccio
Welcome Richard Johns! Oramus

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