Update (August 2022)
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This is an essay on philosophy of science and it close related to DINAMICA, and IDE for rapid construction and simulation of computational model of dynamic systems.
But being totally honest, it's not worth reading. Is based on an my first philosophical exercise on science where I tried to answer questions as when a model of the world can be considered a good model. What is the right number of dimensions to use, How complex it has to be. And now almost 10 years after I have theoretical answers to these set of questions. Answers that currently I'm coding in python and they looks very promising.
Any way, I'll leave this writing and its pdf essay only as prove that the knowledge has to evolve.
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Complexity, dynamic Systems and Cognitive Sciences
Science, in order to discover how universe works, has used models.
At the same time, models are the product of abstraction of reality. Such abstractions are constructed by means the most deep and highest mental capacities.
In this line of reasoning is easy to see how cognitive sciences can be used to understand the scientific method itself. And in a everlasting bucle, scientific method tries to understand how mind works when it constructs abstractions to understand reality. Then, science, mind and reality are part of the same think, think that try to understand to itself.
This research tries to put in clear how this happen.
Detailed document is attached as pdf
DYNAMICAL SYSTEMS + COGNITIVE SCIENCES
The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. Once the system can be solved, given an initial point it is possible to determine all its future points, a collection known as a trajectory or orbit.
In this way, then, dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space.
At any given time a dynamical system has a state given by a set of real numbers (a vector) which can be represented by a point in an appropriate state space (a geometrical manifold). Small changes in the state of the system correspond to small changes in the numbers. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.
In other hand, Cognitive science is the interdisciplinary study of how information is represented and transformed in the brain. It consists of multiple research disciplines, including psychology, artificial intelligence, philosophy, neuroscience, learning sciences, linguistics, anthropology, sociology, and education. It spans many levels of analysis, from low-level learning and decision mechanisms to high-level logic and planning; from neural circuitry to modular brain organization. The term cognitive science was coined by Christopher Longuet-Higgins in his 1973 commentary on the Lighthill report, which concerned the then-current state of Artificial Intelligence research. In the same decade, the journal Cognitive Science and the Cognitive Science Society were founded.
The term "cognitive" in "cognitive science" is "used for any kind of mental operation or structure that can be studied in precise terms" (Lakoff and Johnson, 1999). This conceptualization is very broad, and should not be confused with how "cognitive" is used in some traditions of analytic philosophy, where "cognitive" has to do only with formal rules and truth conditional semantics.
Recently has been developed a common point where Dynamic systems and Cognitive sciences are join. In this point mathematics have been the joining element between this two diciplines, this is a dynamical approch of the cognition.
The dynamical approach to cognition is closely related to ideas about the embodiment of mind and the environmental situatedness of human cognition, since it emphasizes commonalities between behavior in neural and cognitive processes on one hand with physiological and environmental events on the other. The most important commonality is the dimension of time that is common to all of these domains. This permits real-time coupling between them, where the dynamic of one system influences another. This embodiment or implementation of the congnition model could really be carried out basing on the Wolframs computation principle and his ideas about the simplest systems are capable to show complex behaviour.