ultra-complex systems UltraComplexSystems 2008-07-12 02:22:59 2008-10-18 15:31:45 Topic extremely complex systems highest level of existence in Categorical Ontology % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym,color,enumerate} \usepackage{xypic} \xyoption{curve} \usepackage[mathscr]{eucal} \setlength{\textwidth}{7.1in} %\setlength{\textwidth}{16cm} 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\renewcommand{\leq}{{\leqslant}} \renewcommand{\geq}{{\geqslant}} \def\red{\textcolor{red}} \def\magenta{\textcolor{magenta}} \def\blue{\textcolor{blue}} \def\<{\langle} \def\>{\rangle} \subsection{Ultra-complex systems} An \emph{ultra-complex system} represents the human mind from the standpoint of a generalized Categorical Ontology Theory of Levels as the highest level of complexity that emerged through biological and social coevolution over the last $2.2$ million years on Earth. \subsubsection{Preliminary Data:} One can represent in square categorical diagrams the emergence of ultra-complex dynamics from the super-complex dynamics of human organisms coupled {\em via} social interactions in characteristic patterns represented by \PMlinkname{Rosetta biogroupoids}{RosettaGroupoids}, together with the complex--albeit inanimate--systems with `chaos'. With the emergence of the ultra-complex system of the human mind-- based on the super-complex human organism-- there is always an associated progression towards higher dimensional algebras from the lower dimensions of human neural network dynamics and the simple algebra of physical dynamics, as shown in the following, essentially \emph{non-commutative} categorical diagram. \begin{definition} An \emph{ultra-complex system, $U_{CS}$} is defined as an object representation in the following non-commutative diagram of systems and dynamic system morphisms or `dynamic transformations': $$ \xymatrix@C=5pc{[SUPER-COMPLEX] \ar [r] ^{(\textbf{Higher Dim})} \ar[d] _{\Lambda}& ~~~(U_{CS}= ULTRA-COMPLEX) \ar [d]^{onto}\\ COMPLEX& \ar [l] ^{(\textbf{Generic Map})}[SIMPLE]} $$ \end{definition} Note that the above diagram is indeed not `natural' (i.e. it is not commutative) for reasons related to the emergence of the higher dimensions of the super--complex (biological/organismic) and/or ultra--complex (psychological/neural network dynamic) levels in comparison with the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems.