Random dynamics and the research programme of classical mechanics

The modern mathematical theory of dynamical systems proposes a new model of mechanical motion. In this model the deterministic unstable systems can behave in a statistical manner. Both kinds of motion are inseparably connected, they depend on the point of view and researcher's approach to the system. This mathematical fact solves in a new way the old problem of statistical laws in the world which is essentially deterministic. The classical opposition: deterministic‐statistical, disappears in random dynamics. The main thesis of the paper is that the new theory of motion is a revolution in the research programme of classical mechanics. It is the revolution brought about by the development of mathematics.     

Tools for connectionist modeling: The dynamical systems methodology

Some understanding of dynamical systems is essential to achieving competency in connectionist models. This mathematical background can be acquired either through a rigorous set of upper undergraduate and/or graduate formal courses or via disciplined self-teaching. As part of developing a course in connectionism, we feel that although certain very basic mathematical tools are most appropriately learned in their “pure” form (i.e., from mathematics textbooks and courses), more advanced exposure to dynamical systems theory can be given in the context of an introduction to connectionism. Students thus learn to write connectionist simulations by first writing programs for simulating arbitrary dynamical systems, then using them to learn some aspects of dynamical systems in general by simulating some special cases, and finally applying this technique to connectionist models of increasing complexity.     

Information Processing and Dynamics in Minimally Cognitive Agents

There has been considerable debate in the literature about the relative merits of information processing versus dynamical approaches to understanding cognitive processes. In this article, we explore the relationship between these two styles of explanation using a model agent evolved to solve a relational categorization task. Specifically, we separately analyze the operation of this agent using the mathematical tools of information theory and dynamical systems theory. Information‐theoretic analysis reveals how task‐relevant information flows through the system to be combined into a categorization decision. Dynamical analysis reveals the key geometrical and temporal interrelationships underlying the categorization decision. Finally, we propose a framework for directly relating these two different styles of explanation and discuss the possible implications of our analysis for some of the ongoing debates in cognitive science.     

On the dynamics and adaptivity of mental processes: Relating adaptive dynamical systems and self-modeling network models by mathematical analysis

In this paper, it is addressed by mathematical analysis how network-oriented modeling relates to the dynamical systems perspective on mental processes. It has been mathematically proven that any dynamical system can be modeled as a temporal-causal network model and that any adaptive dynamical system (of any order) can be modeled by a self-modeling network (of the same order).     

Self-organisation in dynamical systems: a limiting result

There is presently considerable interest in the phenomenon of “self-organisation” in dynamical systems. The rough idea of self-organisation is that a structure appears “by itself” in a dynamical system, with reasonably high probability, in a reasonably short time, with no help from a special initial state, or interaction with an external system. What is often missed, however, is that the standard evolutionary account of the origin of multi-cellular life fits this definition, so that higher living organisms are also products of self-organisation. Very few kinds of object can self-organise, and the question of what such objects are like is a suitable mathematical problem. Extending the familiar notion of algorithmic complexity into the context of dynamical systems, we obtain a notion of “dynamical complexity”. A simple theorem then shows that only objects of very low dynamical complexity can self organise, so that living organisms must be of low dynamical complexity. On the other hand, symmetry considerations suggest that living organisms are highly complex, relative to the dynamical laws, due to their large size and high degree of irregularity. In particular, it is shown that since dynamical laws operate locally, and do not vary across space and time, they cannot produce any specific large and irregular structure with high probability in a short time. These arguments suggest that standard evolutionary theories of the origin of higher organisms are incomplete.     

An analytical solution to the nonlinear evolutionary equations for nucleation and growth of particles

ABSTRACT

In this paper, we consider a mathematical model for the growing crystals in supersaturated or supercooled systems. An integro-differential equation of the Fokker-Planck type is analytically solved using the saddle point method for the Laplace integral. The solution describes an intermediate stage of the phase transition process with allowance for the power law of the growth rate of spherical particles and the nucleation mechanisms known as the Meirs and Weber-Volmer-Frenkel-Zeldovich kinetics. A dynamical dependencies for the supersaturation/supercooling and the distribution function of crystals by their size are obtained. The novelty of the theory is the use of a power law for the growth rate of crystals, which leads to new analytical solutions of the problem.     

The emergence of brain and mind amid chaos through maximum‐power evolution

A new brain algorithm based Neurological Positivism (NP) is described that is reconcilable with emergent evolution. The maximum‐power evolution of brain and mind amid chaos is described. It is proposed that with the maximum‐power evolution of mind (a) a chaotic/fractal dynamical algorithmic isomorphy among world, brain, and mind is erected, and (b) we witness the origin of the mechanism of evolutionary epistemology—the origin of knowing energy. The maximum‐power evolution of symbols is described as resulting from features of chaos and fractal geometry. Finally, a neurological positivistic explanation of the workability of mathematics in the real world is proposed.     

Coping with stress through decisional control: Quantification of negotiating the environment

Coping with stress through ‘decisional control’ – positioning oneself in a multifaceted stressing situation so as to minimize the likelihood of an untoward event – is modelled within a tree‐structure scenario, whose architecture hierarchically nests elements of varying threat. Analytic and simulation platforms quantify the game‐like interplay of cognitive demands and threat reduction. When elements of uncertainty enter the theoretical structure, specifically at more subordinate levels of the hierarchy, the mathematical expectation of threat is particularly exacerbated. As quantified in this model, the exercise of decisional control is demonstrably related to reduction in expected threat (the minimum correlation across comprehensive parameter settings being .55). Disclosure of otherwise intractable stress‐coping subtleties, endowed by the quantitative translation of verbal premises, is underscored. Formalization of decisional stress control is seen to usher in linkages to augmenting formal developments from fields of cognitive science, preference and choice modelling, and nonlinear dynamical systems theory. Model‐prescribed empirical consequences are stipulated.     

Coordination dynamics of rhythmical and discrete prehension movements: Implications of the scanning procedure and individual differences

The aim of this study was to compare the coordination dynamics of discrete and rhythmical reaching and grasping movements from a dynamical systems perspective. Previous research from this theoretical perspective had focused on rhythmical actions and it is unclear to what extent discrete movements are amenable to a similar dynamical systems analysis. Six adult subjects performed prehension in two conditions: a discrete, non-continuous mode and a rhythmical, continuous mode. A `scanning procedure' was implemented between pre- and post-tests in which the required time of final relative hand closure (T_rfc) was systematically varied. It was shown that the error in the reaching and grasping pattern was least at an attractor region and systematically increased with deviation from the attractor. Results also indicated that there were no differences between condition or trial block for the group. However, there were several within-subject effects of interest. The validity of the scanning procedure was found to be questionable in the discrete condition, where four subjects showed differences in T_rfc between pre- or post-test and the predicted T_rfc of the scanning procedure. Four out of six subjects also had different preferred T_rfc values for discrete and rhythmical movement, indicating that individual specific models might need to be constructed for future dynamical modelling of discrete movement.PsycINFO classification: 2330     

Systems evolution,structures of long‐waves,and systems management

By classifying the evolution of systems into inter‐level evolution and intra‐level evolution and dividing the intra‐level evolution into four stages which are interrelated but quite different from each other, we discuss structures of several typical systems long‐waves, and strategies of systems management.     

Undecidability,incompleteness and Arnol'd problems

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.Partially supported by FAPESP and CNP_q (Brazil, Philosophy Section).     

Birth of an Abstraction: A Dynamical Systems Account of the Discovery of an Elsewhere Principle in a Category Learning Task

Human participants and recurrent (“connectionist”) neural networks were both trained on a categorization system abstractly similar to natural language systems involving irregular (“strong”) classes and a default class. Both the humans and the networks exhibited staged learning and a generalization pattern reminiscent of the Elsewhere Condition (Kiparsky, 1973). Previous connectionist accounts of related phenomena have often been vague about the nature of the networks’ encoding systems. We analyzed our network using dynamical systems theory, revealing topological and geometric properties that can be directly compared with the mechanisms of non‐connectionist, rule‐based accounts. The results reveal that the networks “contain” structures related to mechanisms posited by rule‐based models, partly vindicating the insights of these models. On the other hand, they support the one mechanism (OM), as opposed to the more than one mechanism (MOM), view of symbolic abstraction by showing how the appearance of MOM behavior can arise emergently from one underlying set of principles. The key new contribution of this study is to show that dynamical systems theory can allow us to explicitly characterize the relationship between the two perspectives in implemented models.     

Modeling the Dynamics of Risky Choice

Individuals make decisions under uncertainty every day. Decisions are based on incomplete information concerning the potential outcome or the predicted likelihood with which events occur. In addition, individuals' choices often deviate from the rational or mathematically objective solution. Accordingly, the dynamics of human decision making are difficult to capture using conventional, linear mathematical models. Here, we present data from a 2-choice task with variable risk between sure loss and risky loss to illustrate how a simple nonlinear dynamical system can be employed to capture the dynamics of human decision making under uncertainty (i.e., multistability, bifurcations). We test the feasibility of this model quantitatively and demonstrate how the model can account for up to 86% of the observed choice behavior. The implications of using dynamical models for explaining the nonlinear complexities of human decision making are discussed as well as the degree to which the theory of nonlinear dynamical systems might offer an alternative framework for understanding human decision making processes.     

Axiomatic world theory: An overview the general theory of evolution in brief

The world, its many subsystems and all their theories, starting with logic, can be reduced to two related functions: a combinatorial system generator and a hamiltonian system organizer. These can be derived, in turn, from an Axiom of Lawfulness, the expansion being guided by pseudo‐category and pseudo‐functor analysis to produce an axiomatic theory of the world or general theory of evolution. Specifically, world evolution is generated by a constrained combinatorial world generator, F:G_(X), deduced from two related axioms: I. The Axiom of World Lawfulness and II. The Axiom of World Constraint Constants, c = c₁, c₂, of primordial physical combinatee (substance), c₁, and physical combinator (motion), c₂.

Axiom I postulates a lawful analysis by an analyzer adhering to appropriate coordinate systems, CS, of a lawful analysand obeying a conservation law, X = X. The analysand consists of a base combinatee (the set and elements), X = {x₁, x₂,… x_n}, and a base combinator, namely, the universal Boolean operator, NOR = NOT + OR. Base combinatee and combinator both have attributes of quantity combinatorially generated by NOR operating on the universal number, 1, and of quality generated by NOR operating on the universal dimensions, MLT (mass, length, time), including the null sets.

Axiom II fixes the base constants, c, = c₁, c₂, thereby converting X to material substance using c₁ and NOR to material motion using c₂. This comprehensive, quality and quantity‐competent foundational science is called Universal Combinatorics. Its elements comprise the logical alphabet or metavector, A = {c, 1, MLT; X, NOR}, where c is obtained from the remaining terms. These give: (1) the attributive pseudo‐functor, F = P_(c,1,MLT), where P is the power set of the indicated attributes, and (2) the logic generator, G_(X), where G = NOR_(NOR). F then maps G_(X) into world evolution, F:G_(X) → world evolution, as follows:

Expanding the abstract generator, F₁:G,_(x), with world constants eliminated, i.e., c = 0, generates Universal Grammar consisting of (1) the substantive content of the abstract science chain running from linguistic grammar to mathematics and logic and (2) a comprehensive epistemology equivalent to an explicit theory of the strategic aspects of the scientific method, including a universal hamiltonian theory structure informally related to a mathematical category. The four epistemological theorems are:

1. I. The Combinatorial System Generator, F:G_(X), (read as “The attributive functor, F, maps the logic generator, G_(X), into world theory” or “The world is an attributive combinatorial function of logic").

2. II. The Hamiltonian System Operating Theorem, h (an abstract theory‐category structure).

3. III. The System Stability Theorem, PI?, where PI is the extremal Performance Index or controlling law.

4. IV. The Intersystem Abstraction Ranking Theorem given by the Attributive Functor/ Function, F.

F₂ admits the world constants, c > 0, to materialize the grammar generator, G_(X), to an homologous concrete Euler combinatorial physical wave generator, namely, the superstring equation of quantum theory, E_(NI) = A(σ,τ), where E is the permutational function, NI, is the set of nonintegers and the solution is the dual amplitude, σ,τ. Expanding generates the elementary particles of nonadaptive physics and, by inference, the substantive content of Universal Physics consisting of three additional primary systems comprising the world, where a primary system is defined as one having a distinct but derivative extremal controlling law:

1. I. Nonadaptive physics and chemistry (harmonic hamiltonian wave systems) : Minimize Action, subject to conservation constraints.

2. II. Adaptive physics or biology (membrane bound duplicating polymer‐copolymer hamiltonian systems) : Maximize Survival, subject to energy constraints.

3. III. Sentient physics or sociopsychology (neuromatrix hamiltonian systems) : Maximize subjective Happiness, subject to survival constraints and

4. IV. Representational physics or language (a symbolic combinatorial routine): Maximizes the Information Gain, subject to happiness constraints.

The world can then be viewed as a perpetual superfluid computer implicitly using the epistemology of Axiom I as a world program to process the physical data base, c > 0, of Axiom II into world evolution. After evolving through Systems I and II, mankind, i.e., System III, evolves as an internal metacomputer which makes the combinatorial program explicit and uses it to put all four primary systems in standard hamiltonian theory (pseudo‐category) format and terminology. This can be viewed as a generalization of the Darwinian variation‐and‐selection theme in which combinatorial‐variation is recursively hamiltonian‐selected thereby incrementing world logic and logic constraints on successive primary systems. Because Universal Physics and Universal Grammar are functor‐related homologous concrete and abstract combinatorial pseudocategories, related by a pseudo‐functor, thus, differing only in the presence and absence, respectively, of the World Constants, c ≥ 0, they constitute, ipso facto, Universal Science (Formal Philosophy, World Evolution, World Unification, Explicit Theory of Everything, ETOE, or Axiomatic World Theory).

QED: Because intricate verified predictions, ranging from particles to personality types, mental disorders, political parties and the abstract sciences, result from a system which is merely expanding to fill its possibility set, it is concluded that the world is lawful and that this means it is an object deterministic but not fully analytically determinable combinatorial system. In the object domain, the world is system‐number complete at four. Dually, in the analytical codomain, understanding of it is approximately complete, as measured by a world information gain function. Hence, the dualistic, analysand‐analyzer world program is finite and has dualistic completion criteria, as required of an involuted program.     

Iterating between lessons on concepts and procedures can improve mathematics knowledge

Background Knowledge of concepts and procedures seems to develop in an iterative fashion, with increases in one type of knowledge leading to increases in the other type of knowledge. This suggests that iterating between lessons on concepts and procedures may improve learning. Aims The purpose of the current study was to evaluate the instructional benefits of an iterative lesson sequence compared to a concepts‐before‐procedures sequence for students learning decimal place‐value concepts and arithmetic procedures. Samples In two classroom experiments, sixth‐grade students from two schools participated (N = 77 and 26). Method Students completed six decimal lessons on an intelligent‐tutoring systems. In the iterative condition, lessons cycled between concept and procedure lessons. In the concepts‐first condition, all concept lessons were presented before introducing the procedure lessons. Results In both experiments, students in the iterative condition gained more knowledge of arithmetic procedures, including ability to transfer the procedures to problems with novel features. Knowledge of concepts was fairly comparable across conditions. Finally, pre‐test knowledge of one type predicted gains in knowledge of the other type across experiments. Conclusions An iterative sequencing of lessons seems to facilitate learning and transfer, particularly of mathematical procedures. The findings support an iterative perspective for the development of knowledge of concepts and procedures.     

Cultural Nuances,Assumptions, and the Butterfly Effect: Addressing the Unpredictability Caused by Unconscious Values Structures in Cross‐Cultural Interactions

Cultural values, cross‐cultural interaction patterns that are produced by dynamical (chaotic) systems, have a significant impact on interaction, particularly among and between people from different cultures. The butterfly effect, which states that small differences in initial conditions may have severe consequences for patterns in the long run, serves as a creative way of drawing attention to a particularly challenging aspect of such chaotic systems. Forty‐six accounts of cross‐cultural situations involving the interface of Asian and Western cultures were examined for underlying nuances (using Kluckhohn values orientations; F. R. Kluckhohn & F. L. Strodtbeck, 1961) and their possible effects. The discrepant relational perspective contributed the most to interaction difficulties but was not independent of other spheres.     

Extending Dynamical Systems Theory to Model Embodied Cognition

We define a mathematical formalism based on the concept of an ‘‘open dynamical system” and show how it can be used to model embodied cognition. This formalism extends classical dynamical systems theory by distinguishing a ‘‘total system’’ (which models an agent in an environment) and an ‘‘agent system’’ (which models an agent by itself), and it includes tools for analyzing the collections of overlapping paths that occur in an embedded agent's state space. To illustrate the way this formalism can be applied, several neural network models are embedded in a simple model environment. Such phenomena as masking, perceptual ambiguity, and priming are then observed. We also use this formalism to reinterpret examples from the embodiment literature, arguing that it provides for a more thorough analysis of the relevant phenomena.