1. Scope

 

There are two arm-waving views often expressed about the relationship between “determinism/causality” on the one hand and “predetermination/predictability in principle” on the other. The first treats them as essentially interchangeable: what is causally determined from instant to instant is thereby predetermined over any period - the Laplacian view. The second view is that this is a confusion, and they are two quite distinct concepts. What I have never seen thoroughly explored (and therefore propose to make a start on here) is the range of different cases which give rise to different relationships between determinism and predetermination. I will attempt to illustrate that, indeed, determinism is neither a necessary nor a sufficient condition for predetermination in the most general case.

To make the main argument clear, I will relegate various pedantic qualifications, clarifications and comments to ^[footnotes].

Most of the argument relates to cases of a physically classical, pre-quantum world (which is not as straightforward as often assumed, and certainly not without relevance to the world we experience). The difference that quantum uncertainty makes will be considered briefly at the end.

 

2. Instantaneous determinism

To start with it is useful to define what exactly we mean by an (instantaneously) determinist system. In simple terms this means that how the system changes at any instant is fully determined by the state of the system at that instant ^[1]. This is how physical laws work in a Newtonian universe. The arm-waving argument says that if this is the case, we can derive the state of the system at any future instant by advancing through an infinite number of infinitesimal steps. Since each step is fully determined, the outcome must be as well. However, as it stands this is a mathematical over-simplification. It is well known that an infinite number of infinitesimals is indeterminate as such, and so we have to look at this process more carefully - and this is where there turn out to be significant differences between different cases.

 

3. Convergent and divergent behaviour

To illustrate the first difference that needs to be recognized, consider two simple cases - a snooker ball just about to collide with another snooker ball, and a snooker ball heading towards a pocket. In the first case, a small change in the starting position of the ball (assuming the direction of travel is unchanged) results in a steadily increasing change in the positions at successive instants after impact - that is, neighbouring trajectories diverge. In the second case, a small change in the starting position has no effect on the final position hanging in the pocket: neighbouring trajectories converge. So we can call these “convergent” and “divergent” cases respectively. ^[1.1]

Now consider what happens if we try to predict the state of some system (e.g. the position of the ball) after a finite time interval. Any attempt to find the starting position will involve a small error. The effect on the accuracy of prediction differs markedly in the two cases. In the convergent case, small initial errors will fade away with time. In the divergent case, by contrast, the error will grow and grow. Of course, if better instruments were available we could reduce the initial error and improve the prediction - but that would also increase the accuracy with which we could check the final error! So the notable fact about this case is that no matter how accurately we know the initial state, we can never predict the final state to the same level of accuracy - despite the perfect instantaneous determinism assumed, the last significant figure that we can measure remains as unpredictable as ever. ^[2]

One possible objection that might be raised to this conclusion is that with “perfect knowledge” of the initial state, we can predict any subsequent state perfectly. This is philosophically contentious - rather analagous to arguments about what happens when an irresistable force meets an immovable object. For example, philosophers who believe in “operational definitions” may doubt whether there is any operation that could be performed to obtain “the exact initial conditions”. I prefer to follow the mathematical convention that says that exact, perfect, or infinite entities are properly understood as the limiting cases of  more mundane entities. On this convention, if the last significant figure of the most accurate measure we can make of an outcome remains unpredictable for any finite degree of accuracy, then we must say that the same is true for “infinite accuracy”.

 

The conclusion that there is always something unknown about the predicted outcome places a “qualitative upper limit”, so to speak, on the strength of predictability in this case, but we must also recognize a “qualitative lower limit” that is just as important, since in the snooker impact example whatever the accuracy of prediction that is desired after whatever time period, we can always calculate an accuracy of initial measurement that would enable it. (However, as we shall shortly see ^[3], this does not apply in every case.)  The combination of predictability in principle to any degree, with necessary unpredictability to the precision of the best available measurement, might be termed “truncated predictability”.

 

4. More general cases

The two elemementary cases considered so far illustrate the importance of distinguishing convergent from divergent behaviour, and so provide a useful paradigm to be kept in mind, but of course, most real cases are more complicated than this.

To take some examples, a system can have both divergent parts and convergent parts at any instant - such as different balls on the same snooker table; an element whose trajectory is behaving divergently at one instant may behave convergently at another instant; convergent movement along one axis may be accompanied by divergent movement relative to another; and, significantly, divergent behaviour at one scale may be accompanied by convergent behaviour at a different scale. Zoom out from that snooker table, round positions to the nearest metre or so, and the trajectories of all the balls follow that of the adjacent surface of the earth.

There is also the possibility that a system can be potentially divergent at all times and places. A famous case of such behaviour is the chaotic behaviour of the atmosphere, first clearly understood by Edward Lorentz in 1961. This story comes in two parts, the second apparently much less well known than the first.

 

5. Chaotic case: discrete

The equations normally used to describe the physical behaviour of the atmosphere formally describe a continuum, an infinitely divisible fluid. As there is no algebraic “solution” to these equations, approximate solutions have to be found numerically, which in turn require the equations to be “discretised”, that is adapted to describe the behaviour at, or averaged around, a suitably large number of discrete points. 

 

The well-known part of Lorenz’s work ^[4] arose from an accidental observation, that a very small change in the rounding of the values at the start of a numerical simulation led in due course to an entirely different “forecast”. Thus this is a case of divergent trajectories from any starting point, or “sensitivity to initial conditions” as it has come to be known.

 

The part of “chaos theory” that grew out of this initial insight describes the convergent trajectories from any starting point: they diverge exponentially, with a time constant known as the Kolmogorov constant for the particular problem case ^[5]. Thus we can still say, as we said for the snooker ball, that whatever the accuracy of prediction that is desired after whatever time period, we can always calculate an accuracy of initial measurement that would enable it.

 

6. Chaotic case: continuum

Other researchers might have dismissed the initial discovery of sensitivity to initial conditions as an artefact of the computation, but Lorenz realised that even if the computation had been perfect, exactly the same consequences would flow from disturbances in the fluid in the gaps between the discrete points of the numerical model.  This is often called the “Butterfly Effect” because of a conference editor's colourful summary that “the beating of a butterfly’s wings in Brazil could cause a tornado in Texas”.

 

It is important to note that the Butterfly Effect is not strictly the same as “Sensitivity to Initial Conditions” as is often reported ^[6], although they are closely related. Sensitivity to Initial Conditions is an attribute of some discretised numerical models. The Butterfly Effect describes an attribute of the equations describing a continuous fluid, so is better described as “sensitivity to disturbances of minimal extent”, or in practice, sensitivity to what falls between the discrete points modelled.

 

Since, as noted above, there is no algebraic solution to the continuous equations, the only way to establish the divergent characteristics of the equations themselves is to repeatedly reduce the scale of discretisation (the typical distance between the points on the grid of measurements) and observe the trend. In fact, this was done for a very practical reason: to find out how much benefit would be obtained, in terms of the durability of the forecast ^[7], by providing more weather stations. The result was highly significant: each doubling of the number of stations increased the durability of the forecast by a smaller amount, so that (by extrapolation) as the number of imaginary weather stations was increased without limit, the forecast durability of the model converged to a finite value^[8]. Thus, beyond this time limit, the equations that we use to describe the atmosphere give indeterminate results, however much detail we have about the initial conditions. ^[9]

 

Readers will doubtless have noticed that this result does not strictly apply to the earth’s atmosphere, because that is not the infinitely divisible fluid that the equations assumed (and a butterfly is likewise finitely divisible). Nevertheless, the fact that there are perfectly well-formed, familiar equations which by their nature have unpredictable outcomes after a finite time interval vividly exposes the difference between determinism and predetermination.

 

With hindsight, the diminishing returns in forecast durability from refining the scale of discretisation is not too surprising: it is much quicker for a disturbance on a 1 km scale to have effects on a 2 km scale than for a disturbance on a 100 km scale to have effects on a 200 km scale.

 

7. Consequences of quantum uncertainty

It is often claimed that the the Uncertainty Principle of quantum mechanics ^[10] makes the future unpredictable ^[11], but in the terms of the above analysis this is far from the whole story.

 

The effect of quantum mechanics is that at the scale of fundamental particles ^[12] the laws of physical causality are probabilistic. As a consequence, there is certainly no basis, for example, to predict whether an unstable nucleus will disintegrate before or after the expiry of its half-life.

 

However, in the case of a convergent process at ordinary scales, the unpredictability at quantum scale is immaterial, and at the scale of interest predictability continues to hold sway. The snooker ball finishes up at the bottom of the pocket whatever the energy levels of its constituent electrons. ^[13]

 

It is in the case of divergent processes that quantum effects can make for unpredictability at large scales. In the case of the atmosphere, for example, the source of that tornado in Texas could be a cosmic ray in Colombia, and cosmic radiation is strictly non-deterministic. The atmosphere may not be the infinitely divisible fluid considered by Lorenz, but a molecular fluid subject to random quantum processes has just the same lack of predictability.

 

[EDIT] How does this look in terms of the LW-preferred Many Worlds interpretation of quantum mechanics?^[14] In this framework, exact "objective prediction" is possible in principle but the prediction is of an ever-growing array of equally real states. We can speak of the "probability" of a particular outcome in the sense of the probability of that outcome being present in any state chosen at random from the set. In a convergent process the cases become so similar that there appears to be only one outcome at the macro scale (despite continued differences on the micro scale); whereas in a divergent process the "density of probability" (in the above sense) becomes so vanishingly small for some states that at a macro scale the outcomes appear to split into separate branches. (They have become decoherent.) Any one such branch appears to an observer within that branch to be the only outcome, and so such an observer could not have known what to "expect" - only the probability distribution of what to expect. This can be described as a condition of subjective unpredictability, in the sense that there is no subjective expectation that can be formed before the divergent process which can be reliably expected to coincident with an observation made after the process. [END of EDIT]

 

8. Conclusions

What has emerged from this review of different cases, it seems to me, is that it is the convergent/divergent dichotomy that has the greatest effect on the predictability of a system’s behaviour, not the deterministic/quantised dichotomy at subatomic scales.

 

More particularly, in short-hand:-

Convergent + deterministic => full predictability

Convergent + quantised => predictability at all super-atomic scales

Divergent + deterministic + discrete => “truncated predictability”

Divergent + deterministic + continuous => unpredictability

[EDIT] Divergent + quantised => objective predictability of the multiverse but subjective unpredictability

 

Footnotes

1. The “state” may already include time derivatives of course, and in the case of a continuum, the state includes spatial gradients of all relevant properties.

1.1 For simplicity I have ignored the case between the two where neighbouring trajectories are parallel. It should be obvious how the argument applies to this case. Convergence/divergence is clearly related to (in)stability, and less directly to other properties such as (non)-linearity and (a)periodicity, but as convergence defines the characteristic that matters in the present context it seems better to focus on that.

2. In referring to a “significant figure” I am of course assuming that decimal notation is used, and that the initial error has diverged by at least a factor of 10.

3. In section 6.

4. For example, see Gleick, “Chaos”, "The Butterfly Effect" chapter.

5. My source for this statement is a contribution by Eric Kvaalen to the New Scientist comment pages.

6. E.G by Gleick or Wikipedia.

7. By durability I mean the period over which the required degree of accuracy is maintained.

8. This account is based on my recollection, and notes made at the time, of an article in New Scientist, volume 42, p290. If anybody has access to this or knows of an equivalent source available on-line, I would be interested to hear!

9. I am referring to predictions of the conditions at particular locations and times. It is, of course, possible to predict average conditions over an area on a probabilistic basis, whether based on seasonal data, or the position of the jetstream etc. These are further examples of how divergence at one scale can be accompanied by something nearer to convergence on another scale.

10. I am using “quantum mechanics” as a generic term to include its later derivatives such as quantum chromodynamics. As far as I understand it these later developments do not affect the points made here. However, this is certainly well outside my  professional expertise in aspects of Newtonian mechanics, so I will gladly stand corrected by more specialist contributors!

11. E.G. by Karl Popper in an appendix to The Poverty of Historicism.

12. To be pedantic, I’m aware that this also applies to greater scales, but to a vanishingly small extent.

13. In such cases we could perhaps say that predictability is effectively an “emergent property” that is not present in the reductionist laws of the ultimate ingredients but only appears in the solution space of large scale aggregates. 

14. Thanks to the contributors of the comments below as at 30 July 2013 which I have tried to take into account. The online preview of "The Emergent Multiverse: Quantum Theory According to the Everett Interpretation" by David Wallace has also been helpful to understanding the implications of Many Worlds.