Certain kinds of object, even rather large, intricate ones, will appear spontaneously under fairly general conditions. Crystals are the most obvious example of this. A sodium chloride crystal, for example, is a very specific arrangement of sodium and chlorine atoms. The chance of a large crystal forming by chance is too remote to consider. If, for example, you were to arrange black counters and white counters randomly on a large square grid, you would not produce the perfect pattern of alternation, like a chess board, that we see in sodium chloride. The appearance of sodium chloride is therefore a case of self-organisation.

Probably the most impressive examples of self organisation are fractals. The Mandelbrot set, for example, is enormously intricate, one might even say infinitely so, yet it is generated by a rather simple rule. Also, while the Mandelbrot set is a mathematical abstraction rather than a natural object, there are apparently natural fractals.

These examples, while impressive, point to a possible limitation of self organisation, however. For crystals and fractals share the feature of being highly self similar. If you move a crystal over by one atom, it maps exactly onto itself. It is "invariant", as they say, "under spatial translation". Fractals don't have this kind of self similarity. But they are self-similar, or invariant, under changes of scale. As you progressively "zoom in" on the Mandelbrot set, for example, you just see more and more of the same. So, despite being very intricate, crystals and fractals are kinda boring. This raises the following questions:

- Do self-organised objects have to be self similar?
- If they do, then why is this?

- Yes
- It results from symmetries in the underlying dynamics, specifically causal locality and invariance under spatial translation.

I'm not going to repeat the whole argument here, as it can be found in my paper. But I will reflect on the argument, from three different perspectives. The argument can be seen supporting a limitative principle, analogous to Gödel's incompleteness theorem, the Löwenheim-Skolem theorem, or Bell's theorem. The argument is a fairly typical symmetry argument. The argument also stands in a tradition of arguing that a cause can produce an effect only if the effect is somehow "present" in the cause.

I. Limitative Results: Gödel

There is a fairly honourable tradition of limitative, or "no go" theorems in science. One of Gödel's theorems, for example, entails that there is no complete system of rules of inference in second-order logic, or indeed in any formal language in which a categorical set of axioms of arithmetic exists. This result is disappointing to many, since it shows that rational inference cannot be "captured", as it were, in rules that manipulate symbols. (Whether human inference can be so captured is another matter. I like to say that Gödel's theorem shows that God isn't a machine.)

Another delicious result is Bell's theorem, concerning quantum theory, which says that no local, hidden-variable theory can be empirically equivalent to standard quantum mechanics.

Each of these theorems, like all such theorems, is negative. It says that, while you might keep trying to do a certain thing, it will never work. You might say they are logical Jeremiahs, or prophets of doom. This would be a little unfair to Jeremiah, however, since even that much-maligned prophet did urge an alternative to defeat and exile. These results, by contrast, make no positive suggestion at all. Bell's theorem gives no clue about how to interpret quantum mechanics, and Gödel's theorem doesn't suggest any better understanding of rational inference.

My theorem is also pure doom and gloom. Let's be honest: It offers no positive suggestion at all.

Can such negative claims be part of science? It is often said that a scientist must propose hypotheses that are empirically testable. That's not really true, however. While that's a big part of science, a lot of good scientific work is indeed negative. Much useful work is done by experimentalists who show that, while hypothesis H might predict empirical result E, E doesn't actually occur. Also, while most theorists are busily showing that H predicts E, other theorists very helpfully point out that H doesn't really predict E at all, even though we thought it did. A really negative scientist might even show that no hypothesis of a certain type will ever predict E.

I'm afraid I'm one of those really negative scientists. I've shown that no hypothesis in a very broad class predicts the existence of complex living organisms. More precisely, life cannot self organise in any dynamical system whose laws are local and invariant under spatial translation.

At this point a worrying possibility emerges. This no-go theorem is so broad that it rules out just about any naturalistic theory of the origin of life! It certainly seems to rule out all the naturalistic theories presently proposed. Yet, the whole business of science is to provide natural explanations for phenomena, so this result is unscientific after all. (Even if it is technically correct, take note.)

Well, this is an awkward business! What are we to do?

1. It is surely irrational to disregard results that attack the viability of one's project, simply on the grounds that they attack the viability of one's project!

2. While I regard my limitative result as scientific, I myself am a philosopher (of science) by trade, so I don't mind too much if it's considered philosophy. But it is science really.

3. This kind of thing has happened before. Maxwell found that his equations for the electric and magnetic fields were incompatible with the mechanical philosophy of his day. In those days, to give a scientific explanation of something was to give a mechanical model for it. Yet Maxwell, despite much effort, was unable to understand his equations as describing displacement fields in any kind of solid. He had, in effect, a limitative theorem that no such model was possible. Yet science didn't end in the late 19th century! Instead, they realised that non-mechanical explanations are better than none. And if the world isn't a giant machine, then those are going to be the true explanations.

4. So I think we have to broaden our horizons, and be open to new kinds of explanation. Perhaps it won't be that bad? And we have no other choice, if we want our explanations to be true.

[A mischievous aside. I like to say that Maxwell and Einstein refuted the mechanical philosophy over a century ago, but no one told the biologists! Now, one might reasonably object to this, saying that a model that's been shown to be inadequate in one context might work perfectly well elsewhere. I see. So mechanical models are too clumsy to handle the subtleties of the electromagnetic field, but are adequate for such simple phenomena as life, consciousness and rational thought?]

II. Symmetry Arguments

Symmetry arguments are common in logic, mathematics and physics. In logic, it is generally held that logical consequence preserves symmetry, so that if the premises of an argument (considered together) are symmetric in some respect, then the conclusion must also be symmetric in that respect. I will take this for granted. For example, suppose you're investigating a murder, and you have some evidence that Fred did it. On the other hand, you have exactly the same sort of evidence that Mike did it. (Perhaps you know that the murderer is tall, left-handed and male, and both Fred and Mike have these attributes. They have equal motives, and so on.) In this epistemic situation, you cannot draw a conclusion about Fred doing the murder without drawing a similar conclusion about Mike. In other words, you should assign the same probability to each.

In some physical experiments, the initial situation is symmetric in some respect. For example, the state might have reflective symmetry about some plane, so that the state maps to itself under reflection in that plane. What should we believe about the outcome of this experiment, given this initial symmetry?

Two answers have been considered here:

- The outcome will have reflective symmetry about that plane.
- The probability function over the outcomes will have reflective symmetry about that plane.

Nothing interesting can be proved without premises, of course. What premise(s) will allow me to prove (2)? The first premise is the one above, that logical consequence preserves symmetry. The second premise is what philosophers call the Principal Principle, as follows:

If the chance of a future event E is known, then the epistemic probability of E is equal to it.

So, for example, if you know that a coin is biased in such a way that its chance of heads is 0.331 on each toss, then the subjective utility of a gamble that pays $1 on heads (and nothing on tails) is 33.1 cents. The Principal Principle, called PP for short, is almost universally accepted.

The third premise is no less certain. It says that the chance of an outcome in an experiment depends only on the possible causes of that outcome, i.e. on the causally relevant physical conditions. In other words, if you repeat an experiment exactly, perfectly duplicating all the relevant physical circumstances, then the same chances obtain for the outcomes. This doesn't mean that the outcomes will be the same, except in the deterministic case. For example, suppose you do some experiment with the Stern-Gerlach apparatus, and the chance of the particle going UP is q. Then, in every exact repetition of this experiment, the chance of UP will be q as well.

Let's refer to this third premise as the claim that chances are physical.

With these three premises in place, we now consider some ideal physicist who has maximal knowledge of the initial state of some experiment, together with maximal knowledge of all the relevant dynamical laws. Since chances are physical, it follows that this physicist will know the chances of all possible outcomes of the experiment. Now we assume that the initial conditions are symmetric in some way, and consider what this physicist can infer about the outcome of the experiment. Any such conclusion can only be inferred from what the physicist knows, i.e. from his knowledge of the initial conditions. Now, since logical consequence preserves symmetry, any symmetric outcomes A and B must have equal epistemic probability for this physicist. Then, using the Principal Principle, the outcomes A and B must have equal chances as well. (If they did not, then PP entails that their epistemic probabilities would be different.)

This concludes our proof of (2). Now suppose that the system is deterministic, so that the chance of each outcome is either 0 or 1. We will argue by reductio ad absurdum. Suppose that the initial state of the experiment has symmetry with respect to transformation R, so that the initial state maps to itself under R, but the outcome of the experiment does not. In other words, if the actual outcome is E, then R(E) ≠ E. Now, assuming determinism, whatever occurs has chance 1, and whatever doesn't occur has chance 0. R(E) doesn't occur, and so has chance 0. E occurs, and so has chance 1. But, using (2), symmetric outcomes have equal chance. This is a contradiction. Thus the result is proven, i.e. the outcome must be R-symmetric as well in the deterministic case.

My proofs of (1) and (2) show that symmetry arguments are valid in physics as well as in logic and mathematics.

My limitative theorem is established, in part, by just such a symmetry argument. I assume that the dynamical laws operate locally, and are invariant under arbitrary spatial translation. These are rather strong symmetries to impose, but they are well accepted by physicists. I assume a random initial state, so that every possible initial state has the same chance. Such an initial chance distribution has just about every kind of symmetry you might name.

To be continued ...