Volume Two: Symposia and Invited Papers || Determinism in Deterministic Chaos

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Determinism in Deterministic ChaosAuthor(s): Roger JonesSource: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association,Vol. 1990, Volume Two: Symposia and Invited Papers (1990), pp. 537-549Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/193096 .

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Determinism in Deterministic Chaos

Roger Jones

University of Kentucky

1. Introduction

In a paper fifteen years ago about the meaning and the possibility of the beginning and end of time, our redoubtable session chair, John Earman, ended up like this:

...[T]he answers to the questions posed at the outset lie somewhere in a thicket of problems growing out of the intersection of mathematics, physics, and metaphysics. This paper has only located the thicket and engaged in a little initial bush beating. This is not much progress, but knowing which bushes to beat is a necessary first step.

Some philosophers will be disappointed that the thicket is populated by so many problems of a technical and scientific nature. On the contrary, I am encouraged by this result because it shows that a long-standing philosophical problem has a non- trivial and, indeed, a surprisingly large content. Moreover, this result is a good illus- tration of the artificiality and danger of trying to separate philosophy from science. (Earman 1977, p. 131-2)

These remarks are a kind of manifesto of an approach to philosophical questions about space and time dating from the late 1960's. Instead of debating these questions endlessly in the sort of conceptual or abstract way in which they had been debated ever since (at least) Newton and Leibniz, largely independent of actual thinking in the physical sciences, this new approach began by taking advantage of the latest mathematical representations of space-time structure within physics. By posing philosophical questions as conjectures in this mathematical language of space-time structure, certain types of moderately precise answers could be given. These answers generally took the form of pointing out that the conjecture was true in certain models of space- time structure and false m others. There was a kind of "you pays your money; you takes your choice" atmosphere about these answers.

Now I don't want to give the impession that the answers to all philosophically in- teresting questions about space and time can simply be read off models of space-time structure, though there are some beautiful results. I know, for instance, that Larry Sklar would object strenuously, as he amply demonstrates in the volume of his col- lected essays on space-time (Sklar 1985). Often, the very interpretation of "the physics" one invokes in some philosophical discussion of space-time structure turns on the philosophical issues one is discussing. One musn't forget the metaphysical bushes in John Earman's thicket.

PSA 1990, Volume 2, pp. 537-549 Copyright ? 1991 by the Philosophy of Science Association

In any case, all I want to claim is that this historical course of philosophical discussions about space and time structure - a course moving from exclusively conceptual and metaphysical discussions to bush beating at this intersection of mathematics, physics, amd metaphysics, from searches for in-principle canonical clarity to cata- logues of moderately precise "results" associated with particular mathematical structures - is characteristic of philosophical discussions about determinism as well.

In fact, there has been substantial overlap in discussions of space-time structure and those of determinism. Much such overlap is in evidence in what must be regarded as the state-of-the-art philosophical discussion of determinism in this new vein, John Earman's Primer on Determinism (Earman 1986), which gave Jeremy Butterfield the idea of organizing this symposium.

2. Laplacian Determinism

The notion of determinism that Earman dedicates himself to, for the most part, is what he calls Laplacian determinism. Motivated by the famous passage from Laplace, but cleansing its spirit of epistemological references to a "predictor", Earman casts his condition in terms of physically possible worlds - "worlds that sat- isfy the natural laws obtaining in the actual world" (Earman 1986, p. 7). A physically possible world is Laplacian deterministic just in case, given any other physically possible world, when the two worlds agree on all relevant physical properties at a given time, then they agree for all times (Earman 1986, p. 13).

Obviously, the $64 question is whether our own world is Laplacian deterministic. But, in the best spirit of the sort of space-time discussions I have just described, one is not going to get an answer from John Earman - or of course anybody - but rather a heavily annotated catalog. That is, one is going to get a whole lot of candidates for "physically possible worlds", and tests of the definng condition for determinism for them. The annotations come in various discussions of what constitutes "natural laws" and how they are distinguished from, for instance, boundary conditions; they come in discussions of "relevant physical properties", of difficulties with the concept of "a given time", of specifying physical properties at particular times, and such.

In his Primer, Earman's first catalog entries are for "classical" worlds. After a careful analysis of the importance of spacetime structure itself for the basic casting of the question of determinism, he turs to traditional Newtonian worlds. And very quickly, as he says, "the Gestalt of determinism safely and smoothly at work in Newtonian worlds" is switched "to puzzlement about how Laplacian determinism could possibly be true" (Earman 1986, p. 33). Many of his considerations have to do with effects imploding from and exploding to spatial infinity. Some have to do with very cleverly contrived circumstances of point particles. Throughout the difficulty lies in the status of supplementary conditions that must be imposed on the basic differential equations of motion in the particular worlds to guarantee uniqueness of evolution.

Mark Wilson, in his review of Earman's Primer for Philosophy of Science (Wilson 1989), is a little impatient with Earman's emphasis on spacetime structure. He is also a little impatient about some of the clever system/worlds designed as prima facie af- fronts to determinism, chiding Earman for "cast[ing] his net widely enough to sample cheerfully some fairly dubious arguments for indeterminism" (Wilson 1989, p. 527). Wilson chooses instead to concentrate on more homely systems, "ordinary machines" - two wheels linked by a connecting rod, three intermeshed gear wheels, real, inelas- tic billiard balls. The problems with these systems are generally problems of instability: in certain configurations their large-scale future behavior is extremely sensitive to minute changes in their initial conditions, rather like a pencil, balanced precarious- ly on its point. Now the instability one finds in these systems exists only at a point, or perhaps a few points, in their entire space of states. And though even the presence of one such unstable configuration makes these homely systems, as Wilson says, ade- quate to "provide a tolerable first picture of how 'determinism' goes awry in many of

Earman's examples" (Wilson 1989, p. 510), both Earman and Wilson are aware of the existence of an important class of systems for which such instability is a ubiquitious aspect of their state spaces, at least asymptotically. These are the systems studied by what is popularly known as "chaos theory". And they are thought by some to intro- duce important and special challenges to determinism. At least they are a new class of "physically possible worlds" against which to test the defining conditions.

3. Dripping Faucets

In the best spirit of Wilson's homely machines, I want to consider a particularly homely example. I want to talk about dripping faucets. There are a couple of reasons for choosing this example to talk about determinism in chaos theory. First is its very homeliness: it's a simple phenomenon about which everyone has intuitions bor of personal experience. Second, it exhibits in a simple way some of the techniques used m chaos theory, and indeed, some of the feel of research in chaos theory, which unfor- tunately remains an area in which there seem to be only skeptics and impassioned evangelists. The world owes largely to Robert Shaw, then a graduate student at UC Santa Cruz, the analysis of the systematics of faucet dripping (Shaw 1984). And I owe my appreciation of his work to a recent paper by Steven Kellert, Mark Stone, and Arthur Fine (Kellert et al. 1990).1

Dripping faucets are a part of everyone's experience, particularly in the middle of the night. What one tends to notice about them is the interval between drips. It is the waiting for the next drip that tends to produce agony. And I don't know which pro- duces more agony, inexorably regular dripping, or subtly random dripping. In any case the drop interval is surely one of the phenomenologicaly significant aspects of a dripping faucet, and most folks know that as the flow rate of water through the pipe is adjusted upward by the processes of corruption and decay that degrade faucet wash- ers, the drop interval becomes shorter and more and more irregular until a turbulent trickle is produced. The particular region of dripping faucet phenomenology I want to focus on is that in which the drop interval is irregular.2 (Fig. 1)

Figure 1

1--1 ;-I "" I - . :..rr~

__.. - -- I, -.

To understand the dynamics of this phenomenon of fluid flow using traditional hy- drodynamic models would be a very imposing task. To understand the changing shapes of the water droplets as they form and detach in full detail would involve an analysis with many (even infinitely many) degress of freedom and several force laws. But Shaw and his colleagues did not go about the task of analyzing the dripping faucet in this way. Their first tool of analysis was a display of the time series of drop intervals on a two dimensional plot in which the nth drop interval is plotted on the x axis and the n-plus-first interval is plotted on the y axis. Now when dripping is pretty regular, the nth drop interval is just equal to the n-plus-first, and the graph of all such pairs is approximately a point. (Fig. 2a) What would the appearance of such a graph be when the water flow rate is such that the drop interval is irregular? Well, if it really is irregular, and there is no correlation at all between one drop interval and the drop interval that comes after it, then the graph would simply be a random scatter of points. (Fig. 2b)

"periodic"

"stochastic"

(d (b) r

r r ? ', ? ??

r ,C I C C _ r I. Icr ,

rl I r . i' r lr'5 ? r r ?

r II? ' 'L rA r r r? I r ?r i , rL r r r ??'

28 (msec) 33

Figure2

But that's not the kind of pattern Shaw got. With very good, minutely regulable faucets that dripped even in the daytime, and precise measurements of drop intervals, he and his buddies took data that graphed up like this. (Fig. 2b,c)

Shaw and Co. had seen such shapes before.3 In fact, of course, they were on the lookout for them. And they knew that such shapes were often produced from remark- ably simple, but non-linear differential equations, of the sort that characterize, for instance, forced and damped pendulums. So Shaw set out to find such an equation that might model the behavior of the drop intervals in this irregular region.

The equation he found was based on an analogy between a filling water drop and a body of increasing mass suspended from a spring. (Fig. 3) As the mass of the body increases, the spring stretches, and the body accelerates downwards. Now Shaw sup- posed that at some critical stretch, the body separated into two bodies, the one at- tached to the spring remaining with the same mass as at the start of the process and springing back up to its original position, and the other body falling away freely. This is a cute and wonderfully simple model, on the analogy that the viscosity of water acts like a spring coefficient, and the water flow rate as a rate of mass change.

xo l 1 I I

Figure 3

Well, Shaw wrote down the differential equations that characterize this spring model, and proceeded to play with them on an analog computer. Lo and behold, he discovered that, as he says, "physical faucet data can be found which closely resemble time vs. time maps obtained from the analog simulation..." (Shaw 1984, p. 16; quoted in Kellert et al. 1990, p. 6). (Fig. 4) The resemblance he regarded as good evidence for the success of the variable mass pendulum analogy.

data analog model

(a) (b)

t' ~~~~~f~t \g1

80 (msec) 90 Figure 4

I am not going to discuss here the status of the sense of "confirmation" at stake here. This is what Kellert, Stone, and Fine talk about in their paper. But it is clear that Shaw took quite seriously the functions describing the mass and spring as indicative of the underlying dynamics of the dripping faucet. And there is no doubt of the power of the mathematics of the mathematical treatment. For Shaw used it to discover an attractor in the state space of the variable mass pendulum system, an attractor he recog- nized as a two-dimensional projection of "a R6ssler attractor in its 'screw type' or 'fun- nel' parameter regime." (Fig. 5) And, as Shaw puts it, "The close correspondence of model and experiment ... argues that such a structure is embedded in the infinite-dimensional state space of the fluid system" (Shaw 1984, p. 17; Kellert et al., p. 7).

Now it is this notion of an attractor that I am particularly concerned with here. For it is the character of such attractors as Shaw identified for his pendulum model system, strange attractors, as they are called, that gives rise to much of the discussion regarding determinism in chaotic systems.

The notion of an attractor in a space describing states of a physical system is not really a very exotic one. The basic idea is simply that of a region in the state space into which nearby dynamical trajectories converge. Energy conserving Hamiltonian systems do not have attractors in their state spaces, but systems in which energy is dissipated almost always do. An ordinary real-world pendulum, for instance, will sooner or later come to rest at zero deflection from the vertical, and trajectories on a two dimensional state space coordinatized in values of its angular deflection and angular velocity will spiral in to the origin, a fixed point attractor. A stabilized real-life pendulum, such as one in a grandfather clock, will have a roughly circular "limit cycle" in state space, and trajectories will spiral out to the limit cycle from inside it, and in to the limit cycle from outside it, all inevitably "attracted".

The strange attractor Robert Shaw identified for the variable mass pendulum model is an attractor in just this way. All state trajectories (within some limits) for the system end up "on it." But its shape and structure are certainly strange. In the first place, though all nearby state trajectories converge onto it, once on it, nearby trajectories rapidly, exponentially in fact, diverge. (Fig. 6) They are able to do so, and still remain within a bounded region of state space, because of the peculiar geometric na-

position

velocity

< (up)

Figure 5

ture of the attractor. It is as though the state space is stretched strongly in one dimension - creating the rapid divergence of nearby state trajectories, and then folded back on itself in another dimension - creating a reconvergence of trajectories, of course in a different direction: the trajectories never cross.

It is this structure that gives rise to both the good news and the bad news about determinism in systems whose state spaces feature such attractors - chaotic systems, as I will call them from now on. Among model systems, these include oscillators such as Shaw's variable mass pendulum, compound pendulums, and various electronic analogs; there are also several easy-to-write-down one and two dimensional iterative mappings. Among modeled systems, real world systems, one finds the dripping faucet, but also controlled fluid turbulence cells, chemical reactions, biological populations, heart cells, economic structures, and, fans of such analysis insist, many, many more.

The good news is that apparently chaotic behavior, such as that of the dripping faucet at particular flow rates, can be modeled by simple, mathematically deterministic processes. These processes - like that of the variable mass pendulum - are simple in that they involve a small number of degrees of freedom. And they are deterministic, in that their evolution is described by differential equations - non-linear differential equations, to be sure - but still equations for which the existence and uniqueness of solutions can be guaranteed. So one is guaranteed that for every set of (allowable) initial conditions, the system evolves in a unique way for all time; trajectories in state space never cross. There is no need to model such systems using theories with stochastic elements; their behavior need not be seen as random, in that sense. Nor

Figure 6

need the complexity in their behavior be considered to arise from the action of many competing processes, many degrees of freedom. The appearance of chaos, or random behavior, simply comes from the fact that nearby trajectories diverge so dramatically on the attractor that time series behavior, operationally obtained by sampling the state of the system at times that are long compared to the rate of divergence of the trajectories on the attractor, and even continuously modeled, is just all over the place. (Fig. 7)

But that is an important part of the bad news about determinism in "deterministically chaotic" systems. Arbitrarily nearby trajectories on the attractor diverge exponentially. Unless the state of the system is known exactly (in the real number sense), the future state of the system on the attractor is essentially wholly unpredictable. Strange attractors display, in the fairly standard parlance, sensitive dependence on initial conditions. And I do mean sensitive. This sensitive dependence on initial conditions of deterministically chaotic systems throws down a kind of verificationist gaunt- let to some classical ideas of determinism.

For it is a fact of a very serious nature that the outcomes of measurements and the input to calculations are restricted to finitely statable numbers. And yet for every finitely statable set of initial conditions for a chaotic system there exists a multiplicity of state trajectories with those conditions, trajectories which diverge on the attractor so explosively that all predictability is lost, and one would do as well to regard the evolution of the system as random. In practical terms, such minute details as a computer's round-off algorithm will be crucial in what future states it calculates.

Figure 7

4. Responses

Now let me consider some responses one might make to this essential failure of predictivity. One is simply to say, "So what?" This point is persuasively made by G. M. K. Hunt (1987) and Mark Stone (1989) in short papers on just this topic, and by John Earman repeatedly in his Primer. In the best tradition that I mentioned at the beginning of the paper of answering questions about determinism only in the context of precise theoretical specifications, the theory of chaotic systems - non-linear dynamics - is certainly Laplacian deterministic. I mean, everybody just calls it "deterministic chaos". The mathematics of the theory guarantees the existence and uniqueness of dynamical trajectories for physical systems. Laplacian determinism requires only that two physically possible worlds that agree on the values - point values in this case - of their physically relevant properties agree at all times. That is, if they are located at identical points on identical state-space trajectories, their total histories will "trajectorially coincide". This is simply guaranteed by the mathematics of the theory of chaotic systems.

If one wishes to import this entirely straightforward discussion into this "real world", Earman, Hunt, and Stone chorus, then certain philosophical duties appropriate to such importations must be paid. One is to recognize the distinction between metaphysics and epistemology. As a metaphysical doctrine about the world, Earman says,

"whether [Laplacian determinism] is fulfilled or not depends only on the structure of the world, independently of what we could do or could know of it" (1986, p. 7). But such metaphysical Laplacian determinism is not threatened in any way by chaotic systems. In accord with all I have been saying, such systems, insofar as they are part of the world, evolve along unique dynamical trajectores, associated uniquely with a set of dynamical properties at each instant. Of course, we can't identify these unique dynamical properties and follow these trajectories, because our measuring and computa- tional abilities are ineliminably limited. And any lack of precision in specification, because of the structure of the state space of these systems, makes trajectory prediction impossible. But that's a problem of epistemology. The fact that we can never use the deterministic equations of chaos theory to find the single trajectory which the world/system follows certainly does not threaten the existence of a trajectory, which again, is guaranteed by the mathematics of the theory.

Now this seems to me to be a proper philosophical response. But I am always nervous when perfectly clear results about theoretical systems are imported into "the world", and distinctions between metaphysics and epistemology are made. For there is actually some discussion among people in this chaos business about just what kind of dynamical properties "the world deals in. What I'm talking about here is not just some kind of basic verificationism, which would distinguish between theories in for- mulation and theories in use, and simply come down on the side of "observable specifications" as against "unobservable idealizations". No indeed. The discussion swirls around the fundamental appropriateness of real number specifications of state in dynamics, and it has led to proposals for whole new approaches to dynamics. The loud- est voices in these discussions are those of Ilya Prigogine (e.g., Pngogine and Stengers 1984) and Joseph Ford (e.g., Ford 1983, 1986), neither of whom ever men- tions the other. I want to talk about their views very briefly, and then end with a general point.

Both Prigogine and Ford argue for the inappropriateness of real number characteri- zations of state, and they both explicitly wish to avoid the charge of naive verificationism by appealing to "prnciple". The sort of principles they both mention as analogies are those associated with the speed of light and with Heisenbergian uncertainty. They both find a kind of "logical incompatibility" in the theory of "deterministic chaos", and they both embrace the chaos part - indeterminism, fundamental randomness - as the genuine way of the world. They reject the doctrine of Laplacian determinism for our world, even metaphysically, and want to construct a dynamics to reflect this indeterminism they see as fundamentally "necessary". The spirt of their positions is thus very much the same. But the details are certainly different.

Prigogine's arguments against real valued state specifications in classical (non-linear) dynamics are sometimes just discouraging conflations of "impossible in principle in our world" with "theoretically inconsistent . But he also offers much more sugges- tive invocations of a kind of rigorous complementarity in chaotic systems between real-valued, external-time-ordered state descriptions and distributional descriptions into which a kind of internal "system time" is built (Prigogine and Stengers 1984, pp. 272-290; Misra, Courbage, and Prigogine 1979; Batterman 1991). At present his pro- gramme is mathematically promissory, but if he can define such novel state descriptions rigorously, he assures us that choosing them as fundamental, on the basis of ex- plicit second law considerations, holds out the prospect of making a seamless whole of "dynamics and thermodynamics, the physics of being and the physics of becom- ing" (Prigogine and Stengers 1984, p. 277).

Joe Ford's case against real valued state descriptions for non-linear systems is considerably more elaborate. He takes very seriously the information-based theory of algorithmic complexity due (independently) to Solomonov, Kolmogorov, and Chaitin, (see, e.g., Chaitin 1975). Variously, he points out that:

i) chaotic trajectories cannot be computed by any algorithmic rule simpler than a computer program that simply copies the state values at various times;

ii) computations of chaotic trajectories really don't deserve to be called computations at all (and certainly not predictions), because the input information for the computations must, rigorously, be equivalent to a copy of the output;

iii) the time evolution of chaotic systems is computationally incompressible: it takes just as long to compute the evolution of the system as for the system itself to evolve (e.g., Ford 1986, pp. 350-1).

For all these reasons, he finds something galling, a glaring conceptual mismatch, if not an out-and-out logical incompatibility, between the "determinism" and the "chaos" in the theory of deterministic chaos. And the charlatan is the "determinism". For what kind of "determinism" is it when the product of its action is a chaotic trajectory that is "random and incalculable; its information content ... both infinite and im- compressible" (Ford 1983, p. 46)?

Ford places much of the blame for this mismatch on the assumed input for such deterministic equations - the randomly digited, incalculable, informationally infinite and incompressible real numbers. Chaos comes out of many such deterministic calculations only because chaos is put into them. So it's kind of an artificial chaos, resulting from a bogus determinism based on treating such entities as perfectly well-defined.

Ford's suggestion is for a "humanly meaningful number system that does not involve the assumption of infinite precision" (Ford 1983, p. 47). He proposes one, iso- morphic to a finite set of integers. (See Winnie 1991 for a critique of this proposal.) Other researchers are making practical efforts to develop what are called "cellular automata models" for physical processes in general.4 Such models are defined only on discrete sets of numbers; thus a physics based on them would automatically involve a truncation of real numbers, and the kind of resultant coarse-graining would guarantee the genuine indeterminacy of chaotic systems, and with it an absolute irreversibility that would provide, as another fan has said, "a complete justification of classical sta- tistical mechanics" (Jensen, p. 180), the same result that Prigogine seeks.

There is one cloud in this rosy vision. For as Ford admits, there is nothing in algorithmic complexity theory, or any other similarly deep and abstract theory, which has anything to say about a "natural bound on observational precision" (Ford 1983, p. 47). Roderick Jensen spells this out:

[I]t is possible that the scale at which the trunction of real numbers occurs may be so small that no practical consequences of the distinction between continuum and discrete theories can be deduced or verified. In that case, the issue of the ultimate discretization of the real world will pass from the domain of physics to that of philosophy". (Jensen 1987, p. 180)

The real world giveth; and the real world taketh away. Still, the domain of philosophy is not such a bad place for a doctrine such as Laplacian determinism to reside in. As I have mentioned several times, the most productive philosophical tradition of discussion of such issues is one in which various theoretical systems are carefully described and the issue decided for them. In this way we learn valuable lessons about the notion of a scientific theory, of laws of nature, of boundary and initial conditions, of experimental data, and other issues of concern to philosophers. As long as we can keep our annotated catalogs growing, we are making, in philosophy, a kind of Kuhnian progress.

1I learned a lot also from Steven Kellert's recent (1990) dissertation, Philosophical Questions About Chaos Theory. This dissertation, as far as I know, is the best large-scale study of its topic.

2Figures 1-5 and figure 7 are taken from Shaw 1984. Kellert et al. (1990) repro- duce figures 2-5 as well. Figure 6 comes from Crutchfield et al. (1986).

3For a readable history of chaos theory, including the early study of attractors by a meteorologist and a population biologist, see Gleick 1987.

4See, for example, the articles in "Cellular Automata", Physica 10D (1984).

References

Batterman, R. (1991), "Randomness and Probability in Dynamical Theories: On the Proposals of the Prigogine School.", Philosophy of Science 58: 241-263.

Chaitin, G. (1975), "Randomness and Mathematical Proof', Scientific American 232 (May): 47-52.

Crutchfield, J., Farmer, D., Packard, N., and Shaw, R. (1986), "Chaos", Scientific American 255: 46-57.

Earman, J. (1977), "Till the End of Time", in Foundations of Space-Time Theories, J. Earman, C.Glymour, and J. Stachel (eds.),(Minnesota Studies in the Philosophy of Science, Vol. VIII). Minneapolis: University of Minnesota Press, pp. 109-134.

_ _ _ . (1986), A Primer on Determinism. Dordrecht: Reidel.

Ford, J. (1983), "How Random Is A Coin Toss?", Physics Today 36: (April) 40-47.

?___-_-_-. (1986), "What Is Chaos, That We Should Be Mindful of It?", in The New Physics, P. C. W. Davies (ed.). Cambridge: Cambridge University Press, pp. 348-371.

Hunt, G. M. K. (1989), "Determinism, Predictability, and Chaos", Analysis 47: 129-133.

Jensen, R. (1987), "Classical Chaos", American Scientist 75: 168-180.

Kellert, S. (1990), Philosophical Questions About Chaos Theory (dissertation, Northwestern University)

Kellert, S., Stone, M., and Fine, A. (1990), "Models, Chaos, and Goodness of Fit", forthcoming in Philosophical Topics.

Misra, B., Prigogine, I., and Courbage, M. (1979), "From Deterministic Dynamics to Probabilistic Description", Physica 98A: 1-26.

Prigogine, I. and Stengers, I. (1984), Order out of Chaos. Boulder: Shambhala Publications.

Shaw, R. (1984), The Dripping Faucet as a Model Chaotic System. Santa Cruz: Aerial Press.

Sklar, L. (1985), Philosophy and Spacetime Physics. Berkeley: University of California Press.

Stone, M. (1989), "Chaos, Prediction, and Laplacian Determinism", American Philosophical Quarterly 26: 123-131.

Wilson, M. (1989), "Critical Notice: John Earman's A Primer on Determinism", Philosophy of Science 56: 502-531.

Winnie, J. (1991), "Computable Chaos", unpublished manuscript.

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