Mathematics and
Sciences addressing the new dimensions
and levels of understanding the great web of life.

"One of the principal sciences of the next century will be the study of
complex, autocatalytic, self-organizing, non-linear, adaptive systems...

Systems Theory: An interdisciplinary field which studies systems as a whole. Systems theory was founded on principles from physics, biology and engineering and later grew into numerous fields including philosophy, sociology, organizational theory, management, psychotherapy (within family systems therapy) and economics among others. Cybernetics is a related field, sometimes considered as a part of systems theory.

Systemics: An emerging branch of science that studies holistic systems and tries to develop logical mathematical, engineering and philosophical frameworks. General Systems Theory and Systems science are related field.

Complexity Theory (Com*plex"i*ty): A new set of interdisciplinary sciences from which are emerging explanatory principles and models of complex dynamic systems composed of many inter-acting parts. It offers radical new views of accelerating global ecological, climatic, economic, social & political changes, evolution, health & healing, immunology, consciousness and more. Examples: Inter(connected)net(work), the brain and body, an ecosystem, and financial markets. Many leading scientists believe that that systems, perhaps even life itself, arise and thrive on the edge of chaos with just enough order to give them pattern, but not so much to slow their adaptation and learning.

Inclusive of;
Automota Theory - the mathematical study of abstract computing machines (especially Turing machines) and the analysis of algorithms used by such machines.
Autopoiesis - the fundamental process of living systems, essentially the mechanism by which living systems continually produce themselves as autonomous unities.

Cybernetics - communication and control in living beings and in the machines built by humans.

Emergence - indication of the arising patterns, structures, or properties, that do not seem adequately explained by referring to the system's pre-existing components and their interactions.
Feedback - a process whereby some proportion or in general, function, of the output signal of a system is passed (fed back) to the input. In biological systems such as organisms, ecosystems, or the biosphere, most parameters must stay under control within a narrow range around a certain optimal level under certain environmental conditions. The deviation of the optimal value of the controlled parameter can result from the changes in internal and external environments.
Network - systems of interconnected components
Non-Linear Dynamical Systems - exhibit a completely unpredictable behavior, which might seem to be random (in deterministic systems). This unpredictable behaviour has been called chaos.

Chaos Theory - deals with dynamical systems that, while in principle deterministic, have a high sensitivity to initial conditions, because their governing equations are nonlinear. Examples; the atmosphere, plate tectonics, economies, and population growth. 

Fractal Geometry - images or objects that are self-similar (Koch snowflake) at many or all scales. Fractals look like the whole, as do parts of the parts (Mandelbrot set) (Julia set) (Abstract Expressionism)
The Butterfly Effect - "Sensitive dependence on initial conditions" i.e. the flapping of a butterfly's wing will create a disturbance that will become amplified eventually to change the large scale atmospheric motion.

Non-Linearity - the solutions to the equations do not form a vector space and cannot be superposed (added together) to produce new solutions. This makes solving the equations much harder than in linear systems.

Gaia Theory (Geophysiology) - broadly inclusive name for a group of ideas that living organisms on a planet modify the nature of the planet to make it more suitable for life. This set of theories holds that all organisms on a planet regulate the biosphere to the benefit of the whole.
Self-Organization - the various mechanisms by which pattern, structure and order emerge spontaneously in complex systems. Examples; sand ripples, the coordinated movements of flocks of birds or schools of fish, nests of termites, seashells, fingerprints, and the galaxy.
Symbiogenesis - a interaction between two organisms living together in more or less intimate association or even the merging of two dissimilar organisms.
holography and heirarchy commonalities ?
reductionistic and holistic complements ?
mutualism, competition, parasiticism ?
statistical analysis ? deterministic ? random ?

chaos, order, control ? fractal commonalities ?
- Capra, Fritjof - physicist
- Complexity and Nonlinear Dynamics
- Divine Right of Capital
- Ecoliteracy foundation
- EcoSteps Training
- Emergence Journal
- Fractal Expressionism
- Fusion Anamoly
- Integrity Research Institute
- Kevin Kelly's Out of Control
- Pegasus Communications
- Pioneers of Change
- Prototista Educational Center
- Sante Fe Institute
- Systems Thinker Newsletter
- Universal Aesthetic of Fractals
- Wolfram, Stephen - scientist
- Wolfram Science

Computational Equivalence - a question of translating inputs and outputs from one system to another. Systems found in the natural world can perform computations up to a maximal ("universal") level of computational power, and that most systems do in fact attain this maximal level of computational power. Consequently, most systems are computationally equivalent. For example, the workings of the human brain or the evolution of weather systems can, in principle, compute the same things as a computer.

Computational Irreducibility - the only way to determine the answer to a computationally irreducible question is to perform, or simulate, the computation. Some irreducible computations can be sped up by performing them on faster hardware, as the principle refers only to computation time.

Catastrophe Theory • Evolution Theory • Scientific Complexity
Experimental Mathematics • General Systems Theory
A.I. ? ? ? • A.L. ? ? ? • Nanotech ? ? ? • Statistical Mechanics ? ? ?
Simplicity: The property, condition, or quality of being simple or un-combined. It often denotes beauty, purity or clarity. Simple things are usually easier to explain and understand than complicated ones. Also refered to as Occom's Razor, or simple living lifestyle.

"Our life is frittered away by detail.... Simplify, simplify."
~ Henry David Thoreau

"Simplicity is the ultimate sophistication."
~ Leonardo Da Vinci

"I would have written a shorter letter, but I did not have the time."
~ Blaise Pascal, Mark Twain, T.S. Eliot....