Unpredictable Does Not Mean Incomputable

Debates about whether living systems are “just computation”, or somehow obviously exceed it, often make the same mistake in opposite directions. One side says: if we can describe the behaviour of a system computationally, then the system itself must be computational. The other says: if the system is chaotic, open-ended, and impossible to predict in practice, then computation cannot possibly capture it.

I do not think either conclusion follows.

The pancomputationalist temptation

Let us start with the first argument.

Suppose a system admits a computable model that tracks its behaviour to arbitrary precision. This is already a substantial claim. The naive conclusion is tempting: if computation reproduces the system’s behaviour, then the system is a computer. Perhaps, even more strongly, computation is what the system fundamentally is.

But why should that follow?

Simulability, implementation, and constitution are different claims. No theorem in computability theory takes us automatically from the first to the other two.

Philosophers of mind ran into a version of this problem long ago. Putnam observed that, if we allow a sufficiently permissive mapping between physical and computational states, almost any physical process can be made to implement almost any finite automaton. If implementation is this cheap, saying that a system “implements a computation” tells us almost nothing. Chalmers later tried to formulate stronger constraints. But the general problem remains: a mathematical mapping is not yet an ontology.

The pancomputationalist would like to conclude:

We can compute a model of it, therefore it is computation.

The difficulty is not necessarily with the conclusion. The difficulty is that the mathematics, by itself, has not yet earned it.

The anti-computationalist temptation

Now consider the argument from the opposite side.

Turing had already isolated part of the relevant distinction in 1950. For a discrete state machine, if you know the exact state and the transition rule, the next state is fixed. With some continuous physical systems, however, the practical situation is very different: a tiny error in the initial conditions can become a huge difference later, if we wait long enough. The three-body problem is the classical example; the double pendulum is the textbook one. Change the starting angle by a hair and, after a few swings, the two trajectories may have almost nothing in common.

Two runs starting a hair apart, evolving under one identical deterministic rule.

The naive conclusion now runs in the other direction: if the future of the system is unpredictable, perhaps computation has reached its limit. And, if living systems display this kind of sensitivity, perhaps they must therefore exceed computation.

The important word here is unpredictable. It does not mean incomputable.

Take a simpler case. Suppose I give you an exact state, an effective update rule, and ask where the system will be after (t) steps. You apply the rule (t) times. The calculation may be enormous; two almost identical starting points may lead to completely different answers. But, in principle, there is no mystery about the procedure: start here, apply the rule, stop after (t) steps.

Incomputability is a different kind of obstacle. The point is not that the calculation is too long, too delicate, or useless in practice. The point is that no general algorithm can always give the answer. Chaos, by itself, gives us nothing of this sort. It tells us that prediction may collapse because tiny errors in our knowledge of the initial state are amplified. This is a problem of access and precision. It is not automatically a failure of computation.

This distinction is not even foreign to strongly anti-computationalists. In the more careful technical literature, unpredictability and incomputability are not simply treated as synonyms. The problem usually appears one step later, when broader philosophical conclusions are drawn too quickly.

One concerns the collapse of prediction. The other concerns the existence of an algorithm.

The anti-computationalist would like to conclude:

We cannot predict it, therefore computation cannot capture it.

But chaos does not hypercompute. Sensitivity to initial conditions does not, by itself, produce a non-computable function, solve an undecidable problem, or take us beyond effective procedure. It may destroy prediction. That is already important. It is simply not the same claim.

So, in the end, the two familiar arguments mirror each other.

One moves too quickly from a successful computational description to an ontology of computation. The other moves too quickly from a failed prediction to a limit of computation.

Both add a premise that the mathematics simply does not contain.

Perhaps living systems are computational in a deep constitutive sense. Perhaps they are not. I have no intention of settling that question here. The more modest point is that computability and chaos, by themselves, do not settle it either.

A final note on the word itself

Part of the trouble is that “computationalism”, and with it “anti-computationalism” is not one specific claim.

(i) There is computation as a mathematical theory: calculability, the λ, μ-calculus (etc.), Post machines, the Church-Turing thesis in its mathematical form. (ii) There is computation as a universal ontology: the thesis that the world, the brain, the genome are, at bottom, discrete machines running code. And there is (iii) computation as simulability: the claim that a given physical process admits a faithful computable model, to arbitrary precision.

These three come apart. One can treat calculability as one of the deepest achievements of the last century and still deny that the universe is a Turing machine. The second and third are different bets. The positions are not even close to interchangeable, no doubt about it.

Once the word is split into (i), (ii) and (iii), the two temptations above become easy to name. The pancomputationalist takes success at the level of “simulability as an ontology”: a computable model tracks the system; therefore, the system is computation. The anti-computationalist makes the mirror move: a rejection of the universal metaphor, which may be entirely justified, the world is not code, and the genome is not software, is spent as if it had shown that no faithful computable model could exist at all.

Neither is licensed. Rejecting the metaphor is not the same as refuting simulability, and simulating a process is not the same as reading off its nature. So the discipline this post asks for is, first of all, a discipline about the word: before asking whether something is computational, it is worth asking which of the three claims is actually on the table.