on the actual impact of deterministic chaos

THEODOR LEIBER

ON THE ACTUAL IMPACT OF DETERMINISTIC CHAOS�

ABSTRACT. The notion of (deterministic) chaos is frequently used in an increasing numberof scientific (as well as non-scientific) contexts, ranging from mathematics and the physicsof dynamical systems to all sorts of complicated time evolutions, e.g., in chemistry, biology,physiology, economy, sociology, and even psychology. Despite (or just because of) thesewidespread applications, however, there seem to fluctuate around several misunderstandingsabout the actual impact of deterministic chaos on several problems of philosophical interest,e.g., on matters of prediction and computability, and determinism and the free will. In orderto clarify these points a survey of the meaning variance of the concept(s) of deterministicchaos, or the various contexts in which it is applied, is given, and its actual epistemologicalimplications are extracted. In summary, it turns out that the various concepts of deterministicchaos do not constitute a “new science”, or a “revolutionary” change of the “scientific worldpicture”. Instead, chaos research provides a sort of toolbox of methods which are certainlyuseful for a more detailed analysis and understanding of such dynamical systems which are,roughly speaking, endowed with the property of exponential sensitivity on initial conditions.Such a property, then, implies merely one, but quantitatively strong type of limitation oflong-time computability and predictability, respectively.

1. INTRODUCTION

Deterministic chaos1 has beenen voguenow for more than one decade.Applications on problems for long assumed to be quite “regular” andpredictable, as well as on problems for long intuitively known to be“chaotic” have been proposed, and quite a number of them successful-ly so. Meanwhile, the explanatory ambitions and applications of chaosresearch, more specifically of nonlinear dynamical systems theory, cov-er many types of “dynamical evolutions” in empirical sciences such asphysics, chemistry, biology, economy, sociology, physiology, and psychol-ogy (Basar 1990; Duke and Pritchard 1991; Haken 1983; Kuppers 1996;Mainzer 1997a; Mainzer 1997b; Skarda and Freedman 1987). Basical-ly, the various applications of nonlinear systems theory try to utilize theresults known from mathematical, and especially physical chaos researchand nonlinear dynamics. Thereby, they provide interesting and sometimesilluminating quantitative and qualitative models treatable by establishedmathematical methods (even in traditionally non-mathematized sciences).While the chaos industry is still expanding the interpretational practice, ormeaning variance of “chaos”, varies a lot.2

Synthese113: 357–379, 1998.c 1998Kluwer Academic Publishers. Printed in the Netherlands.

358 THEODOR LEIBER

A general lesson to be learned from the various phenomena of deter-ministic chaos in mathematics and physics simply is that, as was wellknown before, mathematical determinism, especially if it is taken to imply(long-time) computability, is an idealization never achievable in the empir-ical world of actual modelling, measurements, and computations. Morespecifically, it follows from deterministic chaos research that any actualperturbation of a dynamical system is amplified exponentially in the courseof time, and thereby (long-time) computability is strongly limited.

While it is not even clear whether different authors agree on that point,beyond it there seems to be not much conceptual conformity (and oftenclarity) in chaos research and application, i.e., different concepts of chaos(and the like) are in use while their interrelations are not (always) suffi-ciently clear. This situation has left open questions, and it has furthereda multitude of proposals, conjectures, and speculations, like e.g.: Doeschaos render the physical cosmos creative? Does it free us from deter-minism, and thus open a novel universe of freedom? Does deterministicchaos imply the end of physicalism, or materialism, in the sciences? Doesit mark an ultimate developmental endpoint of the explanatory success ofthe mathematized sciences? Does chaos introduce fundamental aspects ofnon-comprehensibility into science which therefore might qualify as being“post-modern” in character? Do we need a novel philosophy of science,specifically one taking into account the epistemological implications ofdeterministic chaos?

In order to answer at least some of these questions this paper is orga-nized as follows: First, an overview on the various approaches to determin-istic chaos is given with special emphasis on the open questions of chaosresearch (Section 2); second, epistemological conclusions are drawn, espe-cially such concerning the meaning of the concepts of chaos, determinism,determinability, and freedom (Section 3); finally some concluding remarksare given (Section 4).

2. THE MEANING VARIANCE OF DETERMINISTIC CHAOS

2.1. Chaos, Complexity, Computability, and Predictability

2.1.1. On the Definition of Mathematical ChaosDifferent accounts on “definitions of chaos” have been around in the lit-erature of the last years. Not very much surprisingly (but often ignored),there is, however, no general definition of deterministic chaos applicableto the majority of interesting cases.

It is only in the special case of the iteration of a function [e.g., thelogistic functionx ! ax(1� x), x 2 [0;1], a � 4] that there is agree-

ON THE ACTUAL IMPACT OF DETERMINISTIC CHAOS 359

ment on the mathematical (actually topological) properties characteristicof deterministic chaos (Peitgen et al. 1994, Chap. 1):

(i) sensitive dependence on the initial conditions (SD);3

(ii) dynamical mixing in state space (MIX);4

(iii) periodic points lying dense in state space (DPP ).5

Note that the mathematical definition of chaos, (MIX ^DPP ^SD) :$DC, presupposes the existence of a state space whose states are preciselylocalisable in principal, and it applies to closed systems. Recently, it hasbeen shown that theSD-,MIX-, andDPP -properties are not independentof each other, and the theorem can be proven (Banks et al. 1992; Peitgenet al. 1994, 115 f.) that any real continuous functionf , which is definedon an arbitrary subset of the reals and has theDPP - andMIX-property,also exhibits theSD-property. In summary, for mathematical chaos thefollowing logical relations hold:

(i) (MIX ^DPP ^ SD) :$ DC;(ii) MIX ^DPP ! SD;(iii) :((MIX ^DPP )$ SD).

Since computer arithmetics always deals with finitely many digits, inchaotic systems any initial value errors and computational errors of com-puted (symbolic) trajectories rapidly grow because of theSD-property: thecomputed trajectories will deviate arbitrarily much from the “true” (the-oretical) trajectories after a finite number of (computational) iterations.6

However, because of the finitely truncated computer arithmetics neitherchaotic nor periodic behavior can be observed exactly in the long term ona computer. Thus, it seems to be extremely problematic to (reliably) cal-culate chaotic trajectories on a (digital) computing machine at all (Adamset al. 1993; Lorenz 1989).

Theoretically, this problem can be solved for simple systems (as, e.g.,tent maps) according to the so-called Shadowing Lemma from mathemat-ical number theory which says (Coven et al. 1988; Peitgen et al. 1994, 122ff.): For chaotic systems every computer calculated trajectory after a shorttime interval, or small number of computational steps, is “false” because oftheSD-property. Nevertheless, it has (reliable) explanatory power becauseit astonishingly well approximates a “true” trajectory of the same chaoticsystem for all times. That is to say that a computer calculated trajectorystarting within an�-neighborhood of a true trajectory for all times stays inthat neighborhood, similar as the shadow of a wanderer always follows thelatter.

The validity of the Shadowing Lemma does, however, not allow usto circumvent the implications of theSD-property, i.e., it does not allow

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for any quantitative statement about the long-time behavior of chaoticsystems. Indeed, the Shadowing Lemma confirms that statistical propertiesof very simple chaotic systems which are measured by numerical computerexperiments are of reliable explanatory value.

2.1.2. Hamiltonian and Dissipative Chaos in PhysicsIn physics, a deterministic dynamical system is defined on the basis ofthe concepts of a state,�(t) = fx1(t), x2(t), x3(t), : : : , xN (t)g, anda deterministic equation of motion,_�(t) = B(�(t); t), which, giventhe (initial) state at any time,�0 = �(t0), (theoretically, or formally)uniquely and completely (i.e., non-probabilistically) determines the state�(t) = t(�0; t) at any future timet > t0, or past timet < t0, wherethe possible explicit time-dependence of t is assumed to have a discreteFourier spectrum.7

The distinguishing property of dynamical deterministic chaos is thechaotic long-time behavior (Thomas and Leiber 1994): Deterministic equa-tions of bounded motion with few degrees of freedom give rise to compli-cated solution trajectories (i) which do not exhibit any (quasi-) periodici-ties (without any external disturbances), and (ii) which are extremely (i.e.,exponentially) sensitive against (small) deviations in the initial conditions.

In physics, the characteristics of dynamical flows, namely the degreeof stability of the characteristic directions of the flow, can be investigatedon the basis of an eigenvalue problem where the largest positive eigen-value determines the long-time behavior of the system. These eigenvaluescorrespond to the so-called Lyapunov exponents�i which provide a (com-putable) measure for estimating the dynamical instability (or dynamicaldivergence of neighboring trajectories) of a system. Lyapunov exponentsplay an important role for characterizing Hamiltonian and dissipative chaosin physics, which are distinguished by different mechanisms for the routesto chaos as well as for the specific topological structures of chaos.8 Numer-ically, the Lyapunov exponents of a dynamical system are calculated fromthe mean rate of (exponential) divergence of initially close trajectories.

The case of “regular”, i.e., long-time effectively computable, dynamicsis given if the distanced(t) of neighboring trajectories is constant or growsalgebraically in the course of time,9

d � const or d � t�;

with a system dependent constant�. This implies that the length of thecomputable time interval of the trajectories’ evolution grows algebraicallywith the precision of initial data, too. The reason is that theN binary digitsof the initial data are increasingly “lost” with the trajectories’ evolution


because of the algebraic amplification of initial value and/or computationalerrors, i.e., in the regular case in the order ofN=2N bits per computationalstep are lost.

In the case of chaotic, i.e., not effectively long-time computable, motionneighboring trajectories diverge exponentially (exponential sensitivity),

d � exp(�+t); �+ > 0;

where�+ denotes the largest (characteristic) Lyapunov exponent. Accord-ingly, for the chaotic case the computable time interval,tc � 1=�+, merelygrows logarithmically with increasing precision of initial data; per itera-tion, then, approximately one bit of initial data is lost.

Therefore, in principal the empirical distinction of regular and chaoticdynamics in numerical experiment is achievable by quantitative estimates:for the case ofN -bit computing precision, in regular systems any (com-putable) correlation with initial data is lost after approximately 2N itera-tions, while in chaotic systems the same is true afterN iterations already.At the same time it is to be noted, however, that dynamical chaos in thesense of limited long-time computability is a matter of degree.

Although the properties of mathematical chaos cannot be rigorous-ly proved for typical dynamical systems of interest, they can be clearlydemonstrated at least qualitatively: According to the definition of mathe-matical chaos theSD-property is necessary for deterministic chaos, but notsufficient (see Section 2.1.1). In dynamical systems’s theory (of physics)this is reflected by the fact that positive Lyapunov exponents are necessary,but also not sufficient for dynamical chaos to appear. Besides the expo-nential divergence of trajectories (“stretching” of phase space volume), forbounded motions a further mechanism of repeated turning back (“folding”of phase space volume) is required. Topologically, this realizes theMIX-property (Peitgen et al. 1994, 73 ff.). Moreover, for Hamiltonian chaosthe topologicalMIX- andDPP -property can be demonstrated at leastqualitatively on the basis of the topology of tori and Poincare sections,respectively (Leiber 1996a, 382–384).

Besides the mathematical definition of chaos (see Section 2.1.1), whichis applicable only to relatively simple mathematical systems (as, e.g., thelogistic function, the tent map, and the Bernoulli shift), and besides thecharacteristics of Hamiltonian and dissipative chaos, a number of methodshave been invoked, especially in the physics of non-dissipative nonlineardynamical systems, to characterize the degree of complexity, and, accord-ing to the increasing degree of dynamical instability, or non-predictability,a true hierarchy of abstract dynamical systems has been established, rough-ly (i.e., neglecting intermediate degrees) ranging from (i) ergodicity, to (ii)

362 THEODOR LEIBER

mixing, and (iii) K- and Bernoulli systems.10 [Note that, unfortunately,it is widespread (ab-)use to denote all of these types of dynamical (in-)stability by the same word, chaos.] Thereby,K-systems satisfy the zero orone law from probability theory, i.e., the only events that can be predict-ed with certainty are those which independently of the past history haveprobability zero or one; while Bernoulli systems exhibit even a strongerdegree of randomness thanK-systems, and they are characterized by theproperty that any two outputs are probabilistically independent (Ornstein1974; Shields 1973). Thus, Bernoulli systems exhibit behavior that is in awell-defined sense as “random as a coin toss”.11

Whereas rigorous proofs for theDPP - andMIX-property of math-ematical chaos can only be given for very simple nonlinear systems, theoverwhelming majority of dynamical systems, which are of interest inphysics, and which are assumed to exhibit chaotic behavior, does not allowfor comparable proofs (and also not for proving the Shadowing Lem-ma). Therefore, such systems are investigated by means of a number ofconceptually and empirically nonequivalent procedures, e.g., canonicalperturbation theory, linear stability analysis, Lyapunov exponents, dynam-ical entropies, strange attractors, diffusion models),12 where in most casesnumerical computer calculations play a decisive role. Of course, these pro-cedures do not provide mathematical existence proofs for deterministicchaos in dynamical systems. Obviously, the question arises how it maybe proved whether a dynamical system, which looks chaotic, is in factchaotic, and not just (quasi-)periodic with a very long period, or endowedwith an extensive and complicated transient behavior. An answer, thoughquite general, fundamental, and to the negative to this question (and relatedones) has been given for Hamiltonian systems on the basis of the proof thatclassical physics (if appropriately axiomatized in a language of first ordercontaining classical elementary analysis, e.g., via Suppes-predicates) con-tains algorithmically undecidable and incomplete “sentences” (Costa andDoria 1995). Among other things, the detailed results show that there exista number of chaotic properties which are algorithmically undecidable, andthat there is no general algorithmic procedure available for proving a givenHamiltonian dynamical system to be chaotic, or not.13

2.1.3. ComputabilityThe intrinsic limitations of what a computer can do computationally maybe divided computation-theoretically into three main groups:

(i) physical limitations of realizability of computing machines (e.g., quan-tum uncertainty relation, heat dissipation);


(ii) numerical untreatability and uncomputability (e.g., too high compu-tational problem complexity)14;

(iii) deterministically chaotic systems.

Now, several authors have put forward “definitions of chaos” on thebasis of merely intuitive notions of algorithmic complexity [which, in part,have been convincingly criticized by Batterman (1993, 49 ff.)]. Such anapproach, however, has been worked out more precisely in the frameworkof algorithmic complexity theory as developed by A. N. Kolmogorov, G.Chaitin, and others. In this framework, the conditional complexityK

of a finite sequenceS given some initial informationI aboutS is (Kolmo-gorov-) defined in terms of the length of the shortest programmeP requiredto computeS on a given computer or algorithm (Batterman 1993, 56–57):

K(SjI) = minfkjk = l(P );(P; I) = Sg:

The sequenceS, then, has infinite (algorithmic) complexity (i.e.,S isuncomputable) if no such programmeP exists. Since Kolmogorov provedthat there exists a universal Turing machine�, such thatK�(SjI) �K(SjI)+c, wherec is a constant which depends on and�, but not onS or I, the algorithmic complexity can be used as a machine independentquantityK. The complexity definition of algorithmic randomness of asequenceS is now given by:

DEFINITION 2.1.S is random ifK(Sjlength(S)) � length(S)�d, where-by d is some nonnegative constant.

For infinite sequences the randomness defined in terms of algorithmiccomplexity usually adopted reads:

DEFINITION 2.2.S is random ifK(S) = limn!1K(n)=n > 0, whereK(n) is the complexity of a finite subsequence of lengthn.

The basic advantages of this definition are its generality, and its concep-tual specificity. There is, however, a “disadvantage” connected therewith,namely, the definition of chaos via algorithmic randomness (or effectivecomputability) is too strong a requirement: according to the definitionnot only exponentially unstable, i.e., chaotic systems are classified to berandom, but also algebraically unstable systems [for whichK(n) � na

(a � 1), and which are untreatable fora > 1 and computable fora = 1](Leiber 1996b, 38–41). In the same sense Batterman (1993; 1996) has givendetailed evidence for the fact that, in general, from algorithmic complexity

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we cannot infer to dynamical instability in the sense of the (exponential)SD-property.15

Note also, that from the theory of (recursive) computability in classi-cal (linear) analysis (Blum et al. 1989; Pour-El and Richards 1989), todate, almost nothing is available about the computational complexity ofnonlinear (i.e., potentially chaotic) problems. It is clear, however, that theclass of (numerically) untreatable systems (which are not effectively algo-rithmically computable because of exponentially growing computationalcomplexity) and the class of chaotic systems are not identical: every chaoticsystem is untreatable (i.e., not long-time computable) whereas untreatableproblems are not necessarily chaotic [e.g., there exist a lot of linear, non-chaotic problems which are untreatable (Leiber 1996b, 36–40)], i.e., thedeterministically chaotic systems constitute a true subset of the set ofuntreatable problems. Moreover, unique connections between dynamicalsystems properties like, e.g., nonlinearity, non-integrability, and dynami-cal instability on the one hand, and (algorithmic) complexity and limitedlong-time computability on the other hand, cannot be established. Forexample, there are effectively treatable (i.e., algorithmically computable)systems which do not admit of a closed-form solution as a function of time(e.g., transcendental equations); there are nonlinear systems which are inte-grable (e.g., solitons); there are linear systems which are uncomputable, oruntreatable (Leiber 1996b, 38–40); also, exponential instability is compat-ible with well-posedness, i.e., with the existence of a closed-form solution.In summary, for the instability hierarchy of dynamical systems there is nocomprehensive characterization available in terms of computability con-cepts.

2.1.4. Degrees of Predictability: Regular Versus Chaotic MotionObviously, the question arises whether the mathematical and physical con-cepts of deterministic chaos can be reduced to a smallest common denomi-nator, which is not only meaningful theoretically, but first of all empirically.A positive answer can be straightforwardly given in the framework of acorrelation function concept of the predictability of dynamical systems.

In the problem of predictability of dynamical evolutions actually threeprocesses are involved, namely the observed onex(t), the model oney(t),and the hypothetically underlying real processz(t). Then, the mean-squareerrorh�2i = h(x�y)2i as taken from (finite) empirical averaging providesa universally adopted measure for prediction accuracy,

h�2i =1N

NXj=1

[x(t0j + �)� y(t0j + �)]2;(1)


wherej counts the different observations, andt0j and t0j + � denote thestarting instant and the time instant of measurements, respectively. [It isassumed that the greater the number of observations performed the morefaithful is the error estimate in Equation (1)].

The degree of predictability can now be measured by a coefficient ofcorrelation between the observed process and the model process at the timemoment� after observation has started:

D(�) =hxyiqhx2ihy2i

:

Since the initial value of predictiony0 is taken to be equal tox0, we haveD(� = 0) = 1; empirically, with increasing� the degree of predictabilityD reaches zero. Generally, the closerD(�) to unity (from below) themore satisfactory the forecast, and the closerD(�) to zero (from above)the larger the discrepancy between observation and prediction. The timespan of predictable behavior,�pred, is defined byD(�pred) = 0:5 whichcorresponds to the situation that the absolute errorh�2i is of the same orderas the observed process’s invariancehx2i: h�2i � (hx2i+ hy2i)=2� hx2i.

While in the case of “regular” motion the mean-square prediction errorgrows algebraically

h�2i = ��X

�2 withX

�2 = (�2� + �2

f + �2�M );

in the case of deterministic chaos it grows exponentially

h�2i = exp(2�+�)X

�2;

with �2� , �2

f , and�2�M representing the contributions of “measurement

noise” (i.e., finite errors in measurement and numerical precision of initialdata), other physical noises (e.g., stochastic forces), and the impact ofmodel inaccuracy�M = Mx �Mz, respectively. Empirically, none ofthese noises is negligible while for the case of “pure” deterministic chaoswe may omit noises other than perturbations of initial data due to finiteprecision in measurement and numerical computation.

Equating the mean-square errorh�2iwith the observation variancehx2i,we can estimate the time of predictable behavior in terms of the (specified)signal-to-noise ratioSNR = hx2i=�2

� for the regular case,

�regpred�

hx2i

�2�

!1=�

;(2)

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and for the chaotic one:

�chaospred �

12�+

ln

hx2i

�2�

!:(3)

The feature of (local) exponential instability, or sensitivity on initial condi-tions as given in deterministic chaos means, e.g., that to increase�chaos

pred byan order of magnitude the signal-to-noise ratio must increase by the factore10 � 20:000. Also, the positive Lyapunov exponents considerably lowerthe predictable time span, especially when�+ � 1,SNR� 1 (i.e., smallerrors), and�+ � SNR.

In summary, any deterministic or stochastic process has a limited pre-dictability range stemming from inaccuracyof the model equation, perturb-ing action of instrumentation or strength of measurement noise, fluctuationsin the system concerned, bifurcations in the course of evolution etc. (andconstraints due to costs). Deterministic chaos in the sense of exponentialinstability of dynamical systems is a quantitatively severe limitation of thelong-time predictability of deterministic systems, because any sort of error,deviation, or perturbation is amplified exponentially, see Equation (3).

3. ON THE EPISTEMOLOGICAL IMPLICATIONS OF DETERMINISTIC CHAOS

3.1. Determinism Versus Determinability

The features of deterministic chaos question a number of implicit assump-tions of classical mechanical physics, the “mechanical world picture”, anda positivistic philosophy of science (and thereby also significantly changethe traditional conceptual content of “chaos”:16

(i) Chaos can be conceived as a property of deterministic systems whichare not subjected to any (external) stochastic perturbations.

(ii) Chaotic behavior may appear already in low-dimensional systems(with more than two state space dimensions).17

(iii) Chaos is an essential property of most (nonlinear) dynamical systems18

which are not effectively treatable by means of linear perturbationtheory.19

(iv) In contradistinction to purely statistical models chaotic behavior canbe (partially) subjected to quantitatively precise analyses (other thanprobabilistic methods).

(v) Mathematical determinism is empirically rather meaningless, and theassumption that mathematical determinism should imply numericallong-time computability is simply misguided; deterministic chaos is,


however, just one additional aspect strengthening this argument. Thus,mathematical determinism (theoretical or formal determinateness; tra-ditionally sometimes called “absolute predictability”20) and predict-ability (determinability; effective computability) are clearly to be dis-tinguished (even if there were no deterministic chaos at all).

(vi) Besides, deterministic chaos research demonstrates that a correctdeterministic nomological (DN -) explanation (based on a determinis-tic law) does not necessarily constitute a potentially correct long-timeprediction; the corresponding equivalence is only theoretically valid,namely if the initial data are known with arbitrary precision; especial-ly, in the case of deterministic chaos the initial data have to be knownwith precision increasing at least exponentially with the time intervalto be predicted.21

(vii) Deterministic chaos puts limitations on the feasibility of any theoret-ically intended reductive explanation which should be carried out viaprecise numerical computations of the system state; e.g., in the caseof micro-reductions effective numerical untreatability may preventthe effective execution of an intended (partial) reduction because thereducing problem formulation may be numerically untreatable whilethe reduced problem formulation may not be.

The researches on deterministic chaos provide mathematical and numer-ical refinements with regard to the solution structure of nonlinear differen-tial (and difference) equations with respect to their dynamical stability (andwith some connection to questions of their algorithmic computability); andthus, deterministic chaos in physics constitutes a distinct methodologicalprogress. Physical chaos research does, however, not constitute a newresearch programme, or novel theory of physics: The theoretical core (ornegative heuristic) is still constituted by the axioms and theorems of classi-cal mechanics.22 In this context, e.g., the researches in Hamiltonian chaosare merely one argument for maintaining that dynamical systems are richerin solution structure than the integrable part of mechanics, the importanceof which has been overemphasized for long. Epistemologically, drawing adirect comparison between the findings of deterministic chaos and the fun-damental changes of physical theorizing in the 20th century is, therefore,exaggerating. In contradistinction to relativistic and quantum mechanics,physical chaos research in the framework of classical mechanics does notdevelop novel “fundamental” structures of the micro- or macro-cosmos,but first of all has led to a certain “renaissance” of classical mechanics byemphasizing the (general and possibly unifying) question of (algorithmic,effective) computability of dynamical systems.23 Surely, with the adventof deterministic chaos in the natural sciences the “dream of physicalism”,

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in the sense of the belief in the feasibility of “perfect predictability” basedon mathematical determinism, has got an additional and (publicly) veryeffective counterargument; but it should also be clear that the thesis of“perfect predictability” was never tenable.

Moreover, the dynamic phenomena of deterministic chaos demonstratethat there is obviously no fundamental dichotomy between determinismand randomness in mathematical modelling, because unstable determinis-tic, chaotic systems may model specific random phenomena. For a varietyof physical examples of ergodic systems it can be shown (Ornstein andWeiss 1991) that a deterministic process is indistinguishable from a (non-deterministic) Markov process up to deviations due to a finite partition ofthe state space. If this partition is chosen to be the finite limit of mea-surement accuracy, the deterministic and a Markov process model of aSinai billard are observationally indistinguishable. Thus, results from theinvestigation of the instability hierarchy of dynamical systems show that itis not always unambiguously possible to decide on the basis of empiricalsuccess whether the model adopted should be mathematically determinis-tic, or indeterministic (i.e., stochastic). This again shows that mathematicaldeterminism is not very meaningful empirically, and that the “determin-ism” or “indeterminism” of our dynamical models should be conceivedas a matter of degree, and that the decision between a deterministic oran indeterministic model of description can be a convention dependingon the choice of the precision of analysis. Or, in the words of GregoireNicolis: “The connection between statistical and deterministic descriptionis quite intricate indeed. For certain (perhaps even for most) types of ourpredictions statistical description is operationally more meaningful, sinceit reflects the finite precision of measurement process and bypasses the fun-damental limitations associated with the instability of [the hypotheticallyunderlying deterministic] motion” (Nicolis 1996, 38).

3.2. Determinism, Chaos and Freedom

First of all, the question whether the deterministic chaos as found in thedynamics (of dissipative systems) is relevant for an understanding of (ani-mal, or human) freedom intimately depends on which concepts of deter-minism and freedom we utilize. Here, a customary distinction is betweenthe points of view of the (metaphysical) determinist, the libertarian, and thecompatibilist.24 The determinist maintains that there is no liberty of actionat all because any action is fully (and perhaps nomologically) determinedby the preceding circumstances, and therefore the concept of responsibilitycannot be justified by the idea of human freedom. The libertarian contendsthat there are (at least some) gaps in the deterministic (or causal) evolu-


tions in the natural world; thus freedom appears in the sense of non-physicalcauses of decisions and actions, and it serves as the only way to justifymoral responsibility. Strictly speaking, the metaphysical determinism andthe libertarianism both are neither provable, nor falsifiable empirically.The compatibilist position, as we take it here, tries to analyse the problemof determinism versus freedom by looking at details which can be sub-jected to empirical decision; in effect such a compatibilist position is notconcerned with (dis-)proving metaphysical determinism or metaphysicalfreedom, and it is an approach which is interdisciplinary in character inthe sense that any proponent of a specific discipline may contribute onlymodestly to the whole subject.

Second, in order to avoid any sort of substance dualism in the realmof the mind-body problem(s) we will argue in the sense of the regula-tive idea (but not necessarily the effective feasibility) of epistemologicalreductionism, but not eliminative materialism. That is to say we will arguefor a compatibilist position which, as one of its aspects, may search foran answer to the question: “Which brain mechanisms realize those actionswhich we call free?” Here, one possible answer is that there is already quitean amount of evidence that at least some brain activities can be conceivedas chaotic dynamical systems, and in this sense deterministically chaoticdynamics may be understood as constituting a sort of neurophysiologicalcondition of “relatively free” human actions, i.e., of actions governed bycognitive functions of the brain which can be taken to be (hypothetically)based on deterministic physiological processes, but nevertheless appear tobe random, or free, to some extent on the observational level. (Clearly, thiscan be taken as just another instance of the effective, empirical irreversibili-ty even in a classically described universe, or of the general conventionality,or empirical undecidability, of the determinism or indeterminism of ourmathematical modelling in dynamics.)

Although the investigation of the brain by means of the methods of non-linear systems theory is a relatively new field, there is increasing evidencethat a good deal of the brain’s neurophysiological functions (e.g., motoric,visual, and auditory activities) can be conceived as deterministically chaot-ic systems (Bas¸ar 1990; Bas¸ar and Roth 1996; Duke and Pritchard 1991;Elbert et al. 1992; Skarda and Freedman 1987). Among the typical prop-erties of dissipative chaos identified in electrophysiological measurementsof the brain are, e.g., limit cycles, intermittent bursts, nonlinear resonancephenomena, nonlinear phase transitions, and strange attractors with fewdegrees of freedom. Three examples are in order: (i) In computer simula-tions of the neuro- and electrophysiological patterns of smell recognitionin rabbits strange attractors play an important role. During the perception

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of a learned smell the EEG exhibits a higher degree of order, namely astrange attractor of lower dimensionality, than in the state of relaxationof the olfactory centre of the rabbit’s brain. The learning of smells canbe modelled as dynamic pattern formation in the neural network of theolfactory centre, and a specific smell is realized by a strange attractor inthe corresponding phase space. The recognition of smells, then, is mod-elled as dynamic pattern recognition in the olfactory neural net (Skardaand Freedman 1987). (ii) Also, different electrophysiological activities inthe human brain have already been investigated experimentally; e.g., theneural excitations correlating to the closing of the eyes, to tactile senseimpressions, to visual perception, to the formation of mental ideas and toword processing, to sleeping and hypnosis, to noise in the auditory system,to epilepsia and schizophrenia (Elbert et al. 1992). Despite a lot of openquestions still, there is evidence that the (strange attractor) EEG of humansat rest has a higher fractal dimension, i.e., it is “more chaotic”, than dur-ing a cognitive activity (Kruse et al. 1992). This can be interpreted in thefollowing way: as long as a human brain is not actively engaged with cogni-tive tasks, with sense impressions from the environment, or with decisions(e.g., in a state of rest), or, as long as in an human brain only few strangeattractor patterns have been established (e.g., in children), the neural netof the brain remains in a state constituted by strange attractors with higherdimension and/or these attractors are fewer in number. Such a brain statebeing “more chaotic", among other things correlates to a lower degree ofconceptual-constructive predeterminateness of perceptions and of deter-minateness of reactions, as well as a lower degree of information-basedand learned competence for decisions or rational judgement. (iii) Also,the question of guiltiness in acts committed in the heat of passion may betreated in the general framework of a neurodynamic approach: Fabian andStadler (1992) tried to dynamically model the behavior of committers inpsychosocial stress situations (this should not be taken as a comprehensiveexplanation). The fact that acts committed in the heat of passion often aretriggered by tiny causes leading to exaggerating reactions may be (par-tially) explained in the following way: psychosocial stress may lead to adestabilisation of the array of motivations, and at points of instability orbifurcation natural fluctuations may effect a drifting to more stable attrac-tors (as, e.g., the phylogenetically gained attractors of flight and attack).Even phenomena like compulsion neuroses may find a partial explanation:the irresistible compulsion of acting in a certain way could be neurolog-ically instantiated by the formation of a very stable attractor dominatingand suppressing other brain activities. Such a dominant attractor may only


be modified decisively, or replaced by an alternative one, by unlearningthat neurally coded pattern of behavior (e.g., in behavioral therapy).

These approaches and others show that deterministic (dissipative) chaosin the brain may explain the following: (i) While on the theoretical level thebrain function which is investigated may be assumed to obey mathematicaldeterminism, on the observational level we usually have long-term unpre-dictability, with a quite strong form given by deterministic chaos. Thus,on the observational level we always have some effective indeterminate-ness, or randomness in the dynamics of the brain because tiny deviationsblow up (exponentially) in the course of time. (ii) This is completely com-patible with our subjective experiences of spontaneous or free decisionsto take place (e.g., when a body’s behavior is governed by the switchingbetween different attractors of brain functioning at bifurcational points).(iii) Also, basic concepts from chaos research provide preliminary meanswhich allow for the approximate dynamical modelling of some specificpatterns of behavior which are determined (genetically or) by learning,and which are empirically predictable and testable at least on a macro-level of description (e.g., when rules or principles of behavior seem tocorrespond to strange attractors in the electrophysiological dynamics ofneural nets). (iv) Therewith, chaotic dynamics in our brain processes givesus dynamical models and partial explanations for capturing aspects ofbehavioral features like spontaneity, flexibility, creativity, and adaptivity.In neural dynamics questions concerning these features can now be posedmore precisely, and can be gradually subjected to empirical investigation,e.g., it can be investigated which boundary conditions are decisive forcertain changes of brain states.

However, it is not the task of physicists alone to find out what the mean-ing of freedom may be. Psychologists, cognitive scientists, neuroscientists,and philosophers have to add to the dynamical modelling of brain func-tions their (partial) explanations of the meaning of human freedom andautonomy, and the mechanisms of their realization on different levels oforganization and description of humans’ behavior. In this interdisciplinaryeffort, the chaotic dynamics of brain processes supplies us with (empiri-cally interesting) aspects of such an explanation which demonstrates thatempirical brain functioning is to be conceived as a complex mixture andinteraction of growth processes which are predictable to different degrees(which constantly change during the different stages of growth). In thissense chaotic trajectories and bifurcational scenarios are dynamical modelsfor higher degrees of unpredictability and freedom, while strange attractorscorrespond to structural stability and a higher degree of neuronal predeter-minateness of cognition.

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In summary, we might convincingly argue for “chaos now finally prov-ing the cosmos creative”, or “chaos providing a new chance for recon-ciling (human) freedom with the mechanistic universe”, only in a weak,non-libertarian sense of the conceptual meaning of creativity, or freedom.What chaotic models really do, first of all, is that they free us from themisguided assumption and the implications of (the feasibility of) absolute,i.e., unrestricted computability and absolute predictability (an assumptionwhich, however, was always erroneously ascribed to empirical systemswhich are always subjected to small errors, or noise). And chaos researchgives us measurable models which can be used for improving aspectsof our scientific understanding of spontaneity, flexibility, creativity, andadaptivity in our behavior.

4. CONCLUDING REMARKS

To conclude, we would like to summarize the most general conclusionsthat can be drawn from contemporary chaos research: theoretically, deter-ministic chaos is conceived as a property of systems which are strictlydeterministic in the sense of mathematical determinism; empirically, themost prominent feature of such systems, namely the exponential sensi-tivity on initial conditions, leads to an amplification of any perturbation,noise, or error, which grow exponentially in the course of time; determin-istic chaos always implies effective uncomputability (but not necessarilyuntreatability), i.e., the precision of initial data required for gaining a givenprecision of final data (and thus the computational complexity) increasesexponentially fast; unambiguous definitions of deterministic chaos do existonly for relatively simple mathematical maps (but not for interesting casesof dynamical systems).

There is a hierarchy of degrees of computability of formal systems,from which only the “edges” are known, to date. Within this hierarchy wemay call those systems chaotic which, basically because of their nonlin-earity, exhibit (at least some close analogues of) theSD-property becauseof theMIX- andDPP -property, and which are therefore algorithmicallyuncomputable (or have at least algebraically growing computational com-plexity) in the long term. Thus, deterministic chaos constitutes merely oneproblem type in the wide, and partially still unexplored range of prob-lems of effective computability. In a more general sense chaos researchprovides us with intuitive examples and arguments which should be usedto further our (often underdeveloped) abilities of “nonlinear thinking”(Mainzer 1997a), namely conceptualization in terms of nonlinear causalnets instead of mono-causal chains. Such abilities are well trained by study-


ing the dynamics of nonlinear systems with their emphasis on the role ofinstabilities leading to exponential error amplification.

Another lesson to be learned from chaos research is that the mathe-matical and physical models of dynamical systems theory, which stressesthe importance of generic properties and structural stability (like, e.g.,strange attractors, bifurcation scenarios), provide invaluable guidance inthe study of specific problems (e.g., many numerical results previouslyqualified as “anomalous” are now used for identifying chaotic behavior).Different methods are provided for analyzing solutions with interestingglobal properties in specific nonlinear models. Most of these methods areheavily relying on numerical experiment and have led to a number of newmethods of data analysis (e.g., dimension computations, Lyapunov expo-nents, Kolmogorov–Sinai entropy, phase space reconstruction, spectra ofdimensions). It seems, however, that the range of applicability and validityof such methods has not yet been investigated comprehensively, e.g., noneof the famous scenarios of “bifurcation to chaos” (like period doubling,quasiperiodic) seem to be sufficiently general that we can entirely dispensewith analyses to determine what actually occurs in specific models.

It should also be clear that chaos research does not constitute a “newscience”, or “novel theory”. Physical chaos research doesn’t even existas a very coherent field (comparable, e.g., to the quantum and relativitytheories), and there is no comprehensive methodology available for, e.g.,mathematical, Hamiltonian, and dissipative chaos, i.e., in these three gener-al cases the properties of chaos are (or have to be) investigated by differentmethods (like those mentioned in Section 2.1). Thus, chaos research isconstituted by a rather loose collection of ideas and methods, many of thelatter inherited from “classical” applied mathematics, which can be addedto the scientist’s toolbox.

In summary, the different approaches to deterministic chaos tell us thatthere are severe quantitative limits to (long-time) computability, and thuscontrollability, already in deterministic systems with few degrees of free-dom. Thus, deterministic chaos constitutes one argument out of many thatattack the general (positivistic) belief in the complete computability ofnature, advocated at least since Galilei, Hobbes, and Descartes, as well asby the 19th century mechanists, the 20th century positivist philosophy ofscience, and many others. The (mathematized) natural sciences are stillclose enough to the technocratic ideal of progress, but draw attention to itslimits today, i.e., we observe epistemological limits of the insights possi-ble for us into the dynamics of the material world (which have, however,only become accessible to precise methodical analyses in recent times):The limitations of computability are limits of controllability and feasibil-

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ity. Therewith, it is not the developmental endpoint of the mathematizednatural sciences which is announced, but it may well be the insight into theindispensible value of sometimes “more qualitatively” arguing quantitativesciences, which may be promoted.

NOTES

� The author gratefully acknowledges financial support from the Deutsche Forschungsge-meinschaft (DFG) through a Habilitationsstipendium. He also wishes to thank two anony-mous referees for criticism which led to considerable improvements of the manuscript.1 We explicitly distinguish between metaphysical, epistemological, and mathematical deter-minism. Epistemological determinism amounts to the working hypothesis of law-likenessof processes (or the heuristical, theoretical, and empirical fruitfulness of law-like scientificmodels). Mathematical determinism is given by the fundamental existence theorem of ordi-nary differential equations, i.e., the existence and uniqueness of solution for any initial state.We take it that metaphysical determinism is a transcendent assumption neither provable norrefutable, e.g., because: epistemological determinism is not strictly confirmable (becauseof the problem of induction), and mathematical determinism is an idealization which isnot strictly confirmable empirically (because measurements are always given with finiteprecision and are endowed with noise).

Whenever the term “chaos” appears in the following, it is meant to denote “deterministicchaos” in the sense of mathematical determinism (if not explicitly stated otherwise).2 Some scientists are even unhappy about the very notion of deterministic chaos itself[which has been accidentally introduced by (Li and Yorke 1975)].3 SD-property (or dynamical instability): For any given pointx0 in state space (or thedefining interval of the chaotic map) there is at least one pointy0 arbitrarily close tox0, sothat finally the iterates ofx0 andy0 will differ by a fixed threshold value which is the samefor all pointsx0 (sensitivity constant). More intuitively, theSD-property, especially in thesense of exponential sensitivity, is known as the most popular “butterfly effect”.4 MIX-property (or topological transitivity): For any two open intervalsI andJ (whichmay be arbitrarily small, but with positive length) initial values, which reachJ by iteration,can be found inI.5 DPP -property: Arbitrarily close to anyx0 a pointz0 can be found which is periodicunder the dynamics. In the generic case, this means that initial data of dynamic systemscorresponding to regular (but unstable periodic, e.g., hyperbolic points) and chaotic solutiontrajectories, respectively, are intimately mixed in phase space (thereby constituting fractalstructures) (Thomas and Leiber 1994, 166 ff.). (Number theoretically, the density of periodicpoints implies that any irrational number can be approximated arbitrarily close by periodicexpansions.)6 For example, for the logistic map already after approximately 7 iterations of two initiallyneighbored iterates the deviation amplitude will be of the same order as the amplitude ofthe iterates.7 For some details on the distinction of “weak determinism” (existence and uniqueness ofsolution) and “strong determinism” (linear stability), and its relation to chaos, see (Leiber1996a, 371–374; Thomas and Leiber 1994).8 Hamiltonian chaos may be generally characterized as follows: the perturbation expansionof a non-integrable dynamics (carried out around the integrable dynamics which takes


place onn-tori) is not convergent, because of resonances occurring between the degreesof freedom (and therefore the structure ofn-tori breaks down). It is shown: that evenHamiltonian systems exhibit motions which cannot be represented by quasi-periodic ones(i.e., by rational or irrational oscillations); and that for sufficiently small Hamiltonianperturbations and sufficiently large distances from the resonances, nevertheless, (infinitelymany) invariant tori are preserved having non-vanishing measure in phase space [KAMtheorem (Bennetin et al. 1984)]. The epistemological relevance of the KAM theoremmainly rests on the fact that for the solution classes of dynamical systems mathematicalexistence theorems and stability proofs, respectively, are still required, because (reliable)numerical computations are always possible only for finite time intervals (e.g., numericallyor mathematically, a typical Poincare section of a chaotic system cannot be explained insufficient detail, while the KAM theorem gives at least a qualitative explanation why thereoccur stable and unstable orbits).

The characterization of dissipative dynamics is mainly effected by the topologicalstructure of attracting regions on phase space which are called attractors of the motion:dissipative chaos is then characterized by “strange attractors” (at least one Lyapunov expo-nent is positive, one vanishes, all others are negative). Strange attractors have a complicatedgeometric structure which often can be represented via Cantor sets of simple structures anda fractal dimension (Peitgen et al. 1992; Thomas and Leiber 1994).

For further details about these chaotic mechanisms in dynamical systems (and alsosome graphical illustrations) see, e.g., (Leiber 1996a, 380–397).9 In this sense integrable systems are regular, but regular systems are not necessarily inte-grable.10 For definitions and (technical) details, see (Arnold and Avez 1968, 32 ff.; Batterman1991a; Lichtenberg and Liebermann 1983, Chap. 5; Ornstein and Weiss 1991).11 An appropriate measure for distinguishingK- and Bernoulli systems, which have positive(nonzero) entropy, from systems with less random ergodic properties (such as ergodicity andmixing), is the metric or Kolmogorov–Sinai entropy. Unlike the algorithmic complexity,which refers only to a single “random” sequence, the entropy characterizes the dynamicalsystem in terms of its complete set of possible sequences, and measures their diversity(Batterman 1993, 61). For a detailed account of the mathematical formalism behind theconcepts ofK- and Bernoulli systems, see (Batterman 1991a, 246 ff.). Batterman (1991a)and Verstraeten (1991) have also given an illuminating critique of unsuccessful attempts(by Prigogine and coworkers) to utilize the instability properties ofK-systems (equippedwith the corresponding properties of indeterminacy) for an approach to the foundation ofirreversibility in physical theory.12 For detailed accounts, see, e.g., (Buzug 1994; Lichtenberg and Liebermann 1983; Peitgenet al. 1994, Chap. 3; Tabor 1989).13 Moreover, in the framework of coding abstract dynamical systems as “symbolic dynam-ics” (Alekseev and Yakobson 1981) direct relations have been established between themathematical (topological) definition of chaos and Godelian undecidability, and it has beenshown that chaos in symbolic dynamics (e.g., in Bernoulli shifts) may be equivalent toundecidability (Agnes and Rasetti 1991).

For dissipative systems the situation is even worse: e.g., there are neither existenceproofs for deterministic dissipative chaos in the Lorenz equations, nor undecidability results;thus, to date, strange attractors in dissipative chaotic systems are only “confirmed” by com-puter calculations.14 Untreatable problems are those which, depending on some system parameter, have expo-nentially growing algorithmic complexity; if the computational complexity is infinite for

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any problem formulation, conceivable to date, we call the system uncomputable. For adiscussion of points (i) and (ii), and also a short survey on information-based complexityand computability problems (in linear analysis), see (Leiber 1996b, 26–40).15 These results have been derived on the basis of some highly complicated theorems fromergodic theory (which have been given between 1977 and 1993 by Ya. B. Pesin, A. A.Brudno, and H. White), and they are valid under the preconditions of the Brudno-White-Pesin theorems, namely it is required that the dynamical system be a continuous, inverselycontinuous, twice differentiable and measure-preserving mapping of a compact metrizablespace onto itself (Batterman 1996).16 For an historic outlook to some traditional concepts of chaos, see (Leiber 1996a).17 Also, integrability in the sense of classical mechanics (Arnold and Avez 1968; Tabor1989) is not uniquely correlated with the number of mechanical degrees of freedom.18 Nonlinearity (i.e., non-validity of the superposition principle) is a necessary (but not suf-ficient) condition for deterministic chaos (in the sense of exponential sensitivity) to appear.19 Perturbation theoretic methods have, however, proved very useful in the intimate neigh-borhood of Hamiltonian chaos, especially for establishing the KAM theorem (Leiber 1996a,380–385, 390–395).20 Note that Pierre Simon de Laplace equated mathematical determinism with absolute pre-dictability, but only for the case of a super-intelligent being (later called Laplacean demon)which would be able to know the initial conditions and interacting forces of a mechanicalsystem with absolute precision.21 From theSD-property in chaotic systems, and the unavoidable (initial value) measure-ment errors and computational deviations inherent in any computational process (even ofnon-chaotic systems), it has been concluded that (neglecting Duhem–Quine holism for thecase of anAllsatz) deterministically chaotic laws of motion (e.g., difference or differentialequations) are not falsifiable (from computer experiments) in the strict sense (i.e., deter-ministically), but are only treatable by means of the usual statistical methods (Dusberg1995). This claim is, however, not strictly valid at least for two reasons: (i) In the caseswhere the Shadowing Lemma can be proved explicitly. (ii) Moreover, such “degree of non-falsifiability” is only larger for chaotic systems (because in the long-term they do allow onlyfor probabilistic predictions, i.e., the predictions are of the type that the dynamic systemafter some time will be found in some infinitesimal intervaldx of the state space withprobability p(x)dx, where the probability densityp depends on the equation of motionconsidered [for an explicit example for the logistic equation, see (Dusberg 1995, 17)],while non-falsifiability is nevertheless not completely negligible for non-chaotic systems,simply because mathematical determinism (including the reversibility and reproducibilityof formal states) is an idealization which is not attainable by empirical science. Clearly, thedifference between regular and chaotic systems is that for chaotic systems the reliabilityof predicted values decreases exponentially with increasing prediction time span, while inregular systems it decreases merely algebraically. In this sense, on the observational levelall dynamical systems exhibit the property of “effective irreversibility”, but to differentdegrees (which may be of considerable importance, however, for practical purposes).22 This is not to say that physical chaos research doesn’t have its novelties, namely, its pos-itive heuristic (investigate nonlinear systems with more complicated solution behavior!),its novel predictions (e.g., homo- and heteroclinic points and related complicated orbits inHamiltonian chaos; strange attractors of different types in dissipative chaos), and its novelapplications (i.e., successful predictions).23 For the sake of completeness we mention that, in addition to the “standard” interpreta-tional problems of quantum physics (i.e., the questions of the possibility of hidden variables


and the interpretation of the act of measurement), the existence of classical deterministicchaos seems to pose a further problem of its own for the quantum-classical border, because,prima facie, quantum systems (conceived in the sense of the linear Schrodinger equationand the Kopenhagen standard interpretation) do not allow for any kind of deterministicallychaotic behavior at all. Basically, for those who nevertheless take “quantum chaos” to bean interesting field of research there are two (opposing) ways in responding to this ques-tion. One arises from deterministic, unorthodox models of quantum physics, like Bohmianmechanics from which the classical limit, more precisely the classical equation of motion– incorporating all features of deterministic chaos – is straightforwardly recovered, and thesignature of deterministic chaos finds a quite direct application. In Bohmian mechanics,theSD-property typically shows up at “bifurcational points” for Bohmian trajectories ina measurement (e.g., at the slits in a double-slit experiment, or at the magnet in a Stern–Gerlach experiment) (Durr et al. 1992). The second type of answer arises from the nowwell-established statistical properties of eigenvalue spectra typical of quantum systemswhose classical counterpart is chaotic. Since the chaotic behavior of a classical system is along-time property, it shows up, according to Bohr’s Correspondence Principle, in quantumsystems with high quantum numbers in the short-range properties of the correspondingspectrum of the energy level distances: the transition to classical chaos, in the quantumsystem is related to the transition from a Poisson distribution to a Wigner distribution of theenergy level spectrum (Monteiro 1994). Thereby, the vanishing (at the observational level)of explicit chaos in quantum theory has added a new ingredient to the quantum-classicalborder (Batterman 1991b; O’Connor et al. 1992), because deterministic chaos does notdisappear without trace, and in “quantum chaology” phenomena such as Wigner statistics,quantum scars, spectral modulations, and dynamical localization are studied (Casati andChirikov 1995).24 As is well known, there is much ambiguity in and between the numerous authors’ “lin-guistics of freedom and determinism”. For a brief survey on the vast literature on the subjectof free will, see also (Moreh 1994).

REFERENCES

Adams, E., W. F. Adams, W. Kuhn, W. Rufeger, and H. Spreuer: 1993, ‘ComputationalChaos May Be Due to a Single Local Error’,Journal of Computational Physics104,241–50.

Agnes, C. and M. Rasetti: 1991, ‘Undecidability of the Word Problem and Chaos inSymbolic Dynamics’,Il Nuovo Cimento106B, 879–907.

Alekseev, V. M. and M. V. Yakobson: 1981, ‘Symbolic Dynamics and Hyperbolic DynamicSystems’,Physics Reports75, 287–325.

Arnold, V. I. and A. Avez: 1968,Ergodic Problems of Classical Mechanics, W. A. Benjamin,New York.

Banks, J., J. Brooks, G. Cairns, G. Davis, and P. Stacey: 1992, ‘On Devaney’s Definitionof Chaos’,American Mathematical Monthly99, 332–34.

Basar, E. (ed.): 1990,Chaos in Brain Function, Springer, Berlin.Basar, E. and G. Roth: 1996, ‘Ordnung aus dem Chaos: Kooperative Gehirnprozesse bei

kognitiven Leistungen’, in Kuppers, pp. 290–322.Batterman, R. W.: 1991a, ‘Randomness and Probability in Dynamical Theories: On the

Proposals of the Prigogine School’,Philosophy of Science58, 241–63.

378 THEODOR LEIBER

Batterman, R. W.: 1991b, ‘Chaos, Quantization, and the Correspondence Principle’,Syn-these89, 189–227.

Batterman, R. W.: 1993, ‘Defining Chaos’,Philosophy of Science60, 43–66.Batterman, R. W.: 1996, ‘Chaos: Algorithmic Complexity vs. Dynamical Instability’, in

Weingartner and Schurz, pp. 211–35.Bennetin, G., L. Galgani, A. Giorgilli, and J.-M. Strelcyn: 1984, ‘A Proof of Kolmogorov’s

Theorem on Invariant Tori Using Canonical Transformations Defined by the Lie Method’,Il Nuovo Cimento79, 201–23.

Blum, L., M. Shub, and S. Smale: 1989, ‘On a Theory of Computation and Complexity overthe Real Numbers: NP-Completeness, Recursive Functions and Universal Machines’,Bulletin of the American Mathematical Society21, 1–46.

Buzug, T.: 1994,Analyse chaotischer Systeme, B.I.-Wissenschaftsverlag, Mannheim.Casati, G. and B. Chirikov (eds.): 1995,Quantum Chaos. Between Order and Disorder,

Cambridge University Press, Cambridge.Costa, N. C. A. da and F. A. Doria: 1995, ‘Godel Incompleteness, Explicit Expressions for

Complete Arithmetic Degrees and Applications’,Complexity1, 1–16.Coven, E., I. Kan, and J. A. Yorke: 1988, ‘Pseudo-Orbit Shadowing in the Family of Tent

Maps’,Transactions of the American Mathematical Society308, 227–41.Durr, D., S. Goldstein, and N. Zanghi: 1992, ‘Quantum Chaos, Classical Randomness, and

Bohmian Mechanics’,Journal of Statistical Physics68, 259–70.Dusberg, K. J.: 1995, ‘Deterministisches Chaos: Emige wissenschaftstheoretisch interes-

sante Aspekte’,Journal for General Philosophy of Science26, 11–24.Duke, D. and W. Pritchard (eds.): 1991,Measuring Chaos in the Human Brain, World

Scientific, Singapore.Elbert, T., W. Lutzenberger, B. Rockstroh, P. Berg, and R. Cohen: 1992, ‘Physical Aspects

of the EEG in Schizophrenics’,Biological Psychiatry32, 595–606.Fabian, T. and M. Stadler: 1992, ‘Applying Chaos Theory to Delinquent Behavior on

Psychosocial Stress Situations’, in: F. Losel, D. Bender, and T. Bliesner (eds.),Psychologyand Law, de Gruyter, Berlin, pp. 55–61.

Haken, H.: 1983,Synergetics: An Introduction. Nonequilibrium Phase Transitions andSelf-Organization in Physics, Chemistry, and Biology, Springer, Berlin.

Kruse, P., M. Stadler, B. Pavlekovic, and V. Gheorghiu: 1992, ‘Instability and CognitiveOrder Formation: Self-Organization Principles, Psychological Experiments, and Psy-chotherapeutic Interventions’, in: W. Tschacher, G. Schiepek, and E. J. Brunner (eds.),Self-organization and Clinical Psychology, Springer, Berlin, pp. 102–17.

Kuppers, G. (ed.): 1996,Chaos und Ordnung. Formen der Selbstorganisation in Natur undGesellschaft, Reclam, Stuttgart.

Leiber, T.: 1996a,Kosmos, Kausalitat und Chaos. Naturphilosophische, erkenntnistheo-retische und wissenschaftstheoretische Perspektiven, Ergon Verlag, Wurzburg.

Leiber, T.: 1996b, ‘Chaos, Berechnungskomplexitat und Physik: Neue Grenzen wis-senschaftlicher Erkenntnis?’,Philosophia Naturalis33, 23–54.

Li, T.-Y. and J. Yorke: 1975, ‘Period Three Implies Chaos’,American MathematicalMonthly82, 985–92.

Lichtenberg, A. J. and M. A. Liebermann: 1983,Regular and Stochastic Motion, Springer,New York.

Lorenz, E. N.: 1989, ‘Computational Chaos – A Prelude to Computational Instability’,PhysicaD35, 299–317.

Mainzer, K.: 1997a,Thinking in Complexity. The Complex Dynamics of Matter, Mind andMankind, (3rd ed.) Springer, Berlin.


Mainzer, K.: 1997b,Gehirn, Computer, Komplexitat, Springer, Berlin.Monteiro, T. S.: 1994, ‘Chaos in Atomic Physics’,Contemporary Physics35, 311–27.Moreh, J.: 1994, ‘Randomness, Game Theory and Free Will’,Erkenntnis41, 49–64.Nicolis, G.: 1996, ‘Natural Laws and the Physics of Complex Systems’, in Weingartner

and Schurz, pp. 36–9.O’Connor, P., S. Tomsovic, and E. J. Heller: 1992, ‘Semiclassical Dynamics in the Strongly

Chaotic Regime: Breaking the Log Time Barrier’,PhysicaD55, 340–57.Ornstein, D. S.: 1974,Ergodic Theory, Randomness, and Dynamical Systems, Yale Univer-

sity Press, New Haven.Ornstein, D. S. and B. Weiss: 1991, ‘Statistical Properties of Chaotic Systems’,Bulletin of

the American Mathematical Society24, 11–116.Peitgen, H.-O., H. Jurgens, and D. Saupe: 1992,Bausteine des Chaos – Fraktale, Klett-Cotta

and Springer, Stuttgart and Berlin.Peitgen, H.-O., H. Jurgens, and D. Saupe: 1994,Chaos – Bausteine der Ordnung, Klett-

Cotta and Springer, Stuttgart and Berlin.Pour-El, M. B. and J. I. Richards: 1989,Computability in Analysis and Physics, Springer,

Berlin.Shields, P.: 1973,The Theory of Bernoulli Shifts, University of Chicago Press, Chicago.Skarda, C. and W. Freedman: 1987, ‘How the Brain Makes Chaos in Order to Make Sense

of the World’,Behavioral and Brain Sciences10, 161–95.Tabor, M.: 1989,Chaos and Integrability in Nonlinear Dynamics. An Introduction, John

Wiley, New York.Thomas, H. and T. Leiber: 1994, ‘Determinismus und Chaos in der Physik’, in: K. Mainzer,

W. Schirmacher (Hrsg.),Quanten, Chaos und Damonen. Erkenntnistheoretische Aspekteder modernen Physik, B.I.-Wissenschaftsverlag, Mannheim, pp. 147–207.

Verstraeten, G.: 1991, ‘Some Critical Remarks Concerning Prigogine’s Conception ofTemporal Irreversibility’,Philosophy of Science58, 639–54.

Weingartner, P. and G. Schurz (eds.): 1996,Law and Prediction in the Light of ChaosResearch, Lecture Notes in Physics, Vol. 273, Springer, Berlin.

Institut fur Philosophiec/o Lehrstuhl fur Philosophie und WissenschaftstheorieUniversitat AugsburgUniversitatsstraße 10D-86135 AugsburgGermanyTel/Fax (0821) 598-5584e-mail [email protected]

on the actual impact of deterministic chaos

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