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|if the particle were at one of those locations instead, it would follow a phase trajectory which diverges from the one it would follow from the given location. (Note that this amounts to a sensitive dependence on initial conditions relative to a manifold of particular position where the values of orthogonal observables such as momentum could vary. This is a problem with trying to apply phase space descriptions to the quantum world, where we should properly be speaking terms of Hilbert space. The inaccuracy of the phase space description does not compromise the point I am making, however.) It would appear that this will remain true unless quantum mechanics should embrace a quantum picture of spacetime itself.|
LetÕs turn now to the stickier question of infinite detail in macroscopic abstractions such as centre of mass or fluid velocity. Smith uses an example from celestial mechanics in which we are interested in the motion of a planetary object in terms of its centre of mass. He correctly observes that the actual centre of mass is a vague point because at any given moment it is indeterminate what particles should be counted as part of the planetary body. This brand of indeterminacy might be very small, and we might rightly count it as irrelevant to the kinds of questions we would want to ask about the planetÕs motion, but as long as there is any vagueness at all, Smith is right in concluding that it cannot exhibit behaviour which is infinitely intricate. Essentially, if we want to paint an infinitely detailed picture we need an infinitesimal paintbrush, and unless we can shrink the centre of mass to a precisely defined point the paintbrush will always be too large.
The problem does not go away if we treat this kind of vagueness as a problem of knowing the location of a point which really is there but which we cannot locate. It is tempting just to say, for instance, well, this planetÕs centre of mass is here, plus or minus a possible error of 5cm in all directions. In that case we would be interested in tracing the time evolution of the centre point of our area of uncertainty. Without assuming any kind of hidden variables interpretation of quantum mechanics, we could draw a rough analogy with the case for quantum indeterminacy. To use again the example property of sensitive dependence of initial conditions, we might say that the centre of mass of a planetary body was somewhere within this cloud of indeterminacy, and wherever it might turn up for this particular observation, there is some other place arbitrarily close to it where it might have turned up instead and which would lead the planet along a different phase trajectory.
But this is inadequate for the reason that there just isnÕt in most cases a well-defined centre of mass. ItÕs not a problem of ignorance, of there actually being a centre of mass which is just difficult to find. It is a problem with the centre of mass abstraction itself: just as Smith observed, at any particular moment it is just indeterminate which particles we should or shouldnÕt include in calculating a bodyÕs centre of mass.
But how much can we make of this kind of vagueness? Smith would have us believe that since we canÕt even decide where a centre of mass should be, we have no business trying to understand the pointÕs behaviour through a mathematical model which pins it down not just very precisely but can pin it down with an infinitely intricate relationship between its initial location and its location at some future moment of time. IsnÕt it a bit like trying to track the evolution of a nationÕs GDP without a clear idea of what kinds of output should be included or whether we should include industries from the nationÕs territories and protectorates? I think there is something about applying infinitely intricate models to vague abstractions which so far has escaped scrutiny.
Specifically, infinitely intricate models might still apply to any appropriateÑperhaps even arbitraryÑsharpening (pace Dummett and others) of vague abstractions such as centre of mass. In other words, we might say that if there were a well-defined centre of mass precisely at such and such a point, then that point would evolve through time to this other location in phase space. This is akin to saying that if a hill of sand must contain at least 4 million of grains of sand entirely within and not touching the border of a 15 cm radius, then after such and such a wind has blown on this hill of sand for a particular precise length of time the hill will have blown away. In practice, our definition of Ôhill of sandÕ is even more vague than our definition of centre of mass, but that doesnÕt mean it would be useless to have a mathematical model which let us know how long it took for a hill of a given size to disappear under a given wind condition.
For macroscopic objects on the scale in which weÕre interested, the vagueness of abstractions such as centre of mass or hill of sand lies entirely in the abstraction and not in the real world. If we want to calculate something about the behaviour of such an abstraction with dynamical models, we must give the model something real and precise to chew on. When the model has digested the real input and given us back a description of behaviour, we cannot say the output is useless just because it was gotten from a sharpened real input which doesnÕt match our normal vague use of the abstraction; we must simply apply our vague abstraction again and recognise that the centre of mass or whatever is, as we will use it, more blurry than the answer weÕve got. The model is like the monster who chews up nails and spits out tacks, even if all weÕre wanting to know about is bumpy blobs of unfinished steel. Essentially we consider a vague quantity such as the location of a centre of mass, sharpen it up and feed it to the mathematical model, and then keep in mind that the output itself remains too sharp for our normal use and must be fuzzified to bring it in line with what we might actually observe. (Note that this is exactly analogous to our earlier conclusion drawn from the thermodynamics of black holes that we are applying continuous models to physical systems with only a finite number of distinguishable access states.)
The reason we must resort to such an inelegant interpretation of mathematical models of things like centres of mass is that we really shouldnÕt expect there to be any natural laws governing the behaviour of quasi-classical centres of mass. We should expect there to be laws governing gravity and the strong and electroweak forces which influence all the particles that go into making a body with a centre of mass. But looking for a natural law governing an abstraction like centre of mass for a macroscopic object is like looking for a natural law governing the behaviour of boomerangs. We might derive approximate models of boomerang behaviour by appeal to fluid dynamics, but boomerangs are as ill-defined as centres of mass. To apply a precise model of boomerang behaviour to the real world, we would have to give the model a well-defined boomerang, the likes of which would never be observed in the real world, and interpret what the model had to say about our abstract boomerang according to the kinds of vague boomerangs we might actually observe. That this is necessary isnÕt a fact which should be blamed on the precision of our model, it should be blamed on the messiness of our boomerang abstraction.
Now someone might object along the lines of SmithÕs approach that even if we actually could use an infinitely intricate model to describe the time evolution of vague abstractions just by appropriately sharpening inputs and fuzzifying outputs, there would remain a whole class of models without infinite intricacy which would offer at least as good or probably a better isomorphism with what we observe in the real world and abstract from it. That is, why should we bother with an infinitely detailed model when weÕre just going to throw away all that detail when we actually apply the model? Our answer here will be something that Smith has already dismissed as not of paramount importance, but I believe his dismissal was too quick.
We have already noted SmithÕs observation that models are what really get applied to the world, that equations are merely a way of specifying the model. But I suggest that what we are really after are the equations which somehow really do govern the motion of dynamical systems. If there are no equations really governing boomerangs as abstractions, we want the equations governing sharpened boomerangs as well-defined constructions of particles for which there are governing equations. This begins to sound like a comment on realism. Smith mentions as an aside in several places that his view might be taken to imply some brand of anti-realism, but he does not take up the issue fully. In order to get a handle on whether apparently gratuitous infinite detail in chaotic models makes them inferior models of the behaviour of vague abstractions lacking infinite detail, we must briefly engage questions of realism edge on.
I propose that there are equations which precisely describe the motion of at least some kinds of bodies in the world. Whether or not the equations of quantum mechanics as it presently stands are a representative sample of such equations, I believe that there is some body of equations more or less like quantum mechanics which would precisely describe the dynamics of some kinds of bodies (but not ill-defined boomerangs or vague centres of mass). This is little more than saying I believe the world operates according to particular laws of Nature. So far this is not a particularly stunning comment on the realism vs. anti-realism debate.
The implication of such a view is that there could be equations governing the behaviour of at least some bodies which are in fact sets of coupled nonlinear differential equations which can exhibit chaotic behaviour. This is independent of whether we have ever actually observed real systems governed by the equations exhibiting something like sensitive dependence on initial conditions. (Indeed, it would be impossible to observe sensitive dependence on initial conditions either in the real world or in mathematical models since we would need to try experiments over an infinite number of phase space points.) This is not as strong a realist position as it might at first sound; there is nothing startling about saying that even if all humans were born without any way of directly observing light MaxwellÕs equations might still be true. MaxwellÕs equations might still explain many things we could observe. In the same vein, although we cannot observe something like sensitive dependence on initial conditions, chaotic equations of motion might explain some of the behaviour we do observe through vague abstractions such as centre of mass.
This still does not answer the objection, however, that some other non-chaotic equations might just as accurately for our purposes describe the phenomena we are able to observe. But surely we are concerned not just with whether two candidate models, perhaps with appropriate sharpening or fuzzifying or both, can each describe what we see. We are also concerned with the simplicity of the explanations the two models offer. Suppose for instance that someone offered a theory of planetary motion in terms of superpositions of various geometric constructions which, while highly complex, yielded a picture of planetary motion quite close to the kinds of motion we could actually observe. Suppose that someone else offered a simpler theory of planetary motion which didnÕt require such complex constructions but which yielded a picture of planetary behaviour similarly in keeping with what we could actually observe. When physicists finally finally gave up trying to make complex epicycles work, it was both because they couldnÕt make the discrepancies with observed planetary motion disappear and because of the vastly greater simplicity, or parsimony, of the alternative. I would argue that even if there were no conflicts with observed planetary motion, perhaps because astronomical instruments werenÕt sufficiently capable, or perhaps even because there was some limit in principle to what they could measure, we should still count epicycles out in favour of simpler alternatives, whether NewtonÕs or EinsteinÕs. Likewise, even if there are other models which describe, for instance, the time evolution of a vague abstraction such as centre of mass, without involving infinite intricacy and the attendant sharpening and fuzzifying, if these two models are in practice indistinguishable, we should opt for the one with simpler equations.
The comment on realism then is simply this: that one set of equations or natural laws is correct as applied to appropriately sharpened abstractions and that we are then justified in applying standards such as parsimony in choosing one model over another experimentally indistinguishable model. And as far as I can see, the simplicity and elegance of the equations of chaotic models is unlikely to be surpassed by another model which gives up the power of coupled nonlinear differential equations for the sake of avoiding infinite detail. If anyone should produce a model of a physical system which does do this job, then we would be well advised to opt for it over a chaotic alternative. Shortly we will discuss the suitability of (often computationally simpler) stochastic models for applications where we are not interested so much in what actual equations governing physical systems but only want a model which displays behaviour similar to that of the real physical system.
All this so far is well and good, but there still is one more twist in the story of intricacy in models of vague abstractions such as centre of mass. Recall the earlier discussion about tracing time evolution of neural networks at different levels of description. There we noted that dynamical behaviour is underdetermined at the w and y levels, despite deterministic l dynamics. The rationale behind this point was that several points distinct in l space might be included in the same point in w or y space and that these points might lie on phase trajectories which diverge not only in l space but also in w or y space. There is an analogous problem with applying mathematical models of limited intricacy to abstractions like centre of mass. The problem can be understood either as a difficulty with using deterministic mathematical constructions to model physical phenomena described at a level that might be underdetermined or as a difficulty with unduly reducing the dimensionality of the model system to below that of the real system being modelled.
Returning to the example of centre of mass as it relates to the first way of understanding the problem, we can note that an infinite number of possible distinct physical arrangements yield the same centre of mass for a macroscopic object of a given mass, approximate shape, and density. Thus, if it should happen that any characteristics of the actual position/momentum distributions of the particles in the macroscopic body (beyond the rough information given by the centre of mass) are ever relevant to that bodyÕs dynamics as understood through the centre of mass abstraction, then any model which doesnÕt track these characteristics will be inherently subject to error. Indeed, if the actual equations governing the dynamics of the particles in the macroscopic body (whether we know these equations or not) are chaotic, these errors may ultimately become quite large. Thus even if we cannot in practice observe fine details of the structure of a macroscopic body, a mathematical model which lacks the (in practice, unobservable) intricacy of the real physical system may not describe the behaviour of what can be observed precisely enough to be useful. Without exploiting the fine details of a system at a level of description