multistable chaotic dynamical systems and ... - Semantic Scholar

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Letters International Journal of Bifurcation and Chaos, Vol. 18, No. 6 (2008) 1821–1824 c World Scientific Publishing Company

Int. J. Bifurcation Chaos 2008.18:1821-1824. Downloaded from www.worldscientific.com by "CENTRO DE INVESTIGACIONES EN OPTICA, A.C. LIBRARY" on 01/17/13. For personal use only.

MULTISTABLE CHAOTIC DYNAMICAL SYSTEMS AND PHILOSOPHY VICENTE ABOITES Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico Received June 21, 2007; Revised October 30, 2007

It is proposed that any dynamical system with coexisting chaotic attractors has an emergent property. This provides a nonreductive explanation of mental states and their high sensitivity to noise and initial conditions. If metaphysical terms result from the mental states and these are emergent properties of dynamical systems with coexisting attractors, such as the brain, it is suggested that this may provide a physical explanation of metaphysical concepts. Keywords: Multistability; mental states; emergent properties; metaphysics.

In the course of the history of philosophy, in particular, analytic philosophy, metaphysics has been rejected since it goes against empirical knowledge, is doubtful, going beyond the limits of human knowledge and, sterile, because independently of whether metaphysical queries can be answered or not it is useless to worry about them. Using very different arguments Carnap [1932] tried to show that metaphysical statements simply make no sense. The so-called “Hume’s fork” [Hume, 1748] concerning the content of any book consists of two questions — first, does it contain any abstract reasoning of the sort found in mathematics or geometry? If not, does it contain any factual statements of a kind that can be observed or tested? If neither; “commit it then to the flames, for it contains nothing but sophistry and illusion”. Hume’s empirism is based on the assumption that it is not possible to have an “idea” of something which has not been experienced first as an “impression”. Every simple impression has a simple idea corresponding to it and vice versa, but this is not the case with complex ideas, because these can be built — as when we use our powers of imagination — out of simple perceptions in ways that do not correspond to any one of them

[Grayling, 1998]. However from Hume’s point of view, since metaphysical terms are not the result of any impression they cannot refer to anything real. Do they? Could metaphysical terms be grounded on multistable chaotic states of our brain? Novel philosophic and scientific ideas presented next suggest so. In addition, these ideas may help to bridge the gap between what Snow [1959] once called the Two Cultures i.e. the breakdown of communication between the “two cultures” of modern society, the sciences and the humanities. The purpose of this Letter is to explain these ideas from the viewpoint of Nonlinear Dynamics. Reductionism is one solution to the problem of the relationship between different sciences. In its basic form reductionism states that the nature of complex things is reduced to the nature of sums of simpler or more fundamental things. For example, we teach that chemistry is based on physics, biology is based on chemistry, psychology is based on biology, sociology is based on psychology, and so on. Many see in scientific reductionism the unity of science. Even though the first two reductions are normally accepted, dissatisfaction with reductionism to describe mental phenomena has led to

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the analysis of nonreductive proposals. Emergence is one of them. According to Board [1929] “an emergent quality is roughly a quality which belongs to a complex, as a whole and not to its parts”. Therefore an emergent quality cannot be inferred even from a complete knowledge of the properties and performance of the parts of a system. It has been argued [Morton, 1988; Horgan & Tienson, 1992; Port & Gelder 1995; Combs, 1996; Newman, 1996, 2001, 2004] that there is a relation between mental phenomena and the chaos theory, in particular that mental properties are chaotic emergent properties of the brain. In this sense, it is possible to explain mental properties in physical terms. The trajectories of a dynamical dissipative system can converge to a region of the phase space called attractor. A fixed point, a limit circle and a quasiperiodic attractor are examples of nonchaotic attractors. However, there is a case when the trajectory converges to an attractor whose dimension is fractal, called strange attractor. A chaotic system will be understood as a system where its long term behavior can be described in terms of a strange attractor. Two examples of emergent chaotic phenomena are turbulent fluids and, of course, the weather. For instance, from a micro-scale point of view a turbulent fluid is made up of moving particles so the properties of a turbulent fluid supervene on the properties of these fluid particles. However the properties of the turbulent fluid cannot be defined in terms of the individual particles involved. Therefore the properties of a turbulent fluid are emergent properties that supervene on the properties of its particles. Mental properties of the brain supervene on chaotic properties of central nervous systems. It should be emphasized that emergent properties are not mysterious, only that we have a limited epistemic access to them (no mysterious entities are postulated by the emergence theory). According to Newman [1996] a more precise definition of emergence must accomplish four things: (i) it must make explicit the nature of the epistemic restriction on our knowledge of an emergent property, (ii) it must make explicit the rejection of reduction, (iii) it must make clear that an emergent property is supervenient on some set of physical properties and, (iv) it must elucidate how an emergent property can be explained without reducing it to other physical properties and without violating previous restrictions. To achieve these aims the following definition is proposed [Newman, 1996].

A property designated by a predicate P in an ideal theory T is emergent if and only if the following conditions are met: (1) T describes a class of systems SC which are structured aggregates of entities described by T (T being an ideal theory of those entities) and the entities described by T strongly supervene on those described by T . (2) Occurrences of the property designated by P are epistemically impossible to identify with occurrences of any property finitely describable in T . (3) Each occurrence of the property designated by P is an occurrence of a set of properties PC, which is modeled by T . Each member of PC is epistemically indistinguishable in T from some other member of PC. Fundamentally: Part one avoids the above definition from being applied to mysterious properties. Part two avoids the reduction of T to T . Part three guarantees that the property P is explicable only by reference to the class of lower-level properties that insatiate it. A simplified explanation of the argument leading to the conclusion that any dynamical system in the basin of attraction of a strange attractor has an emergent property follows (for details and a formalized presentation of Emergence see [Newman, 1996]). Since Chaos Theory supervenes on wellestablished physical theories, (1) is satisfied. On the other hand, a chaotic system has a sensitive dependence on initial conditions that introduces a fundamental epistemic limitation and unpredictability. Measurements can only specify the state space with limited experimental precision essentially making it impossible to predict how the system will evolve. It must be emphasized that this unpredictability is epistemic rather than metaphysic. Even knowing the system we are dealing with, the sensitive dependence prevents the prediction of any property. Also, the sensitive dependence and our limited precision in the measurement prevent us from identifying the chaotic system. Next, basins of attraction of coexisting attractors in a multistable system cannot be identified with specific physical properties of the system. Therefore (2) and (3) are also satisfied. As a result any dynamical system in the basin of attraction of a strange attractor has an emergent property. Even more, we may add that any dynamical system with coexisting chaotic attractors has an emergent property.

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Multistable Chaotic Dynamical Systems and Philosophy

Multistable systems are very complex due to the interaction between attractors. This introduces new characteristics to those already known of chaotic systems. In addition to a high sensitivity to the initial conditions, the basins of attraction in the space phase of multistable systems may be mixed in a very complex way. Also, a small change in a parameter may produce a change in the number of coexisting attractors, they may appear and disappear. Neurological research and electronic circuit simulations have shown evidence that the brain could be described as a complex multistable system. The first experimental evidence of multistability came from experiments on laser physics [Arecchi, 1982]. Multistability has been considered as a mechanism to explain memory storage and pattern recognition [Hertz et al., 1991; Canavier et al., 1993]. Chaotic attractors have been found in human EEG recordings [Basar, 1990], in the behavior of nerve membranes [Aihara, 1997], and mixed analog/digital circuit implementation of chaotic neuro-computer systems have been realized [Horio et al., 2002]. Multistability often appears in coupled systems [Kaneko et al., 1993] and in systems with delayed feedback [Martinez-Zerega et al., 2003; Balanov et al., 2005]. Since the brain may be simulated as a system of a very large number of coupled neurons with delayed feedback, multistability comes as no surprise [Foss & Milton, 2000]. The presence of noise greatly increases the complexity of dynamical systems. Noise may produce a complex jumping between coexisting attractors [de Souza et al., 2007] called noise-induced attractor hopping, this phenomenon is particularly important for low to intermediate noise strengths. There is enough experimental evidence showing the importance of noise in brain activity and this is a very promising area of research. Recently, Pisarchik and co-workers have shown that using noise or parameter perturbations multistability can be controlled [Pisarchik & Goswami, 2000; Pisarchik, 2001; Saucedo-Solorio et al., 2002; Jaimes-Reategui & Pisarchik, 2004]. The above research strongly suggests that there is plenty of room for exploration. For example, experiments in visual perception suggest that the concepts strongly depend on the initial conditions [Atteneave, 1971]. The analogy may go even further; it may be argued that the way in which each person views the world, her weltanschauung, also depends on her life initial conditions, i.e. the internal and

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external noise and initial conditions of our brain. If mental activity is an emergent property of the brain, a chaotic system, then brain states may result from internal or external signals or noise. Different brain states, coexisting chaotic attractors with their basins of attraction, would be linked, for instance, to language terms such as “cat”, “pain”, “red”, “grue”, “God”, or any other. It may be asked what would be the difference between a brain state for the word “cat” or “God”. Essentially none. Both would be the result of a physical neuronal brain process producing a brain state or strange attractor. In this way, philosophical enquiry like any other mental activity would be described in purely physical terms. Any physical or metaphysical term would be, from nonlinear dynamics point of view, nothing else but the result of an attractor, or sets of attractors generated by our brain. Metaphysical terms would be not any more ethereal than any other term referring to the physical world since they both are produced by our brain in the same way. Initial conditions and noise may be the clue to understand the generation and handling of different concepts by our brain. This may also be the “special process within the brain” [Greenfield, 2007] which may help to explain consciousness. The scientific explanation of mind and consciousness is perhaps the toughest scientific challenge of mankind. The author trusts that future research will eventually explain it even if this will imply a change in our reductionist paradigm of the world.

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