"The term ‘chaos’ currently has a variety of accepted meanings, but here we shall use it to mean deterministically, or nearly deterministically, governed behavior that nevertheless looks rather random. Upon closer inspection, chaotic behavior will generally appear more systematic, but not so much so that it will repeat itself at regular intervals, as do, for example, the oceanic tides." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)
"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)
"Chaos demonstrates that deterministic causes can have random effects […] There's a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. […] This paradox, that symmetry can get lost between cause and effect, is called symmetry-breaking. […] From the smallest scales to the largest, many of nature's patterns are a result of broken symmetry; […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)
"Is a random outcome completely determined, and random only by virtue of our ignorance of the most minute contributing factors? Or are the contributing factors unknowable, and therefore render as random an outcome that can never be determined? Are seemingly random events merely the result of fluctuations superimposed on a determinate system, masking its predictability, or is there some disorderliness built into the system itself?” (Deborah J Bennett, "Randomness", 1998)
"[...] an apparently random universe could be obeying every whim of a deterministic deity who chooses how the dice roll; a universe that has obeyed perfect mathematical laws for the last ten billion years could suddenly start to play truly random dice. So the distinction is about how we model the system, and what point of view seems most useful, rather than about any inherent feature of the system itself." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"Indeed a deterministic die behaves very much as if it has six attractors, the steady states corresponding to its six faces, all of whose basins are intertwined. For technical reasons that can't quite be true, but it is true that deterministic systems with intertwined basins are wonderful substitutes for dice; in fact they're super-dice, behaving even more ‘randomly’ - apparently - than ordinary dice. Super-dice are so chaotic that they are uncomputable. Even if you know the equations for the system perfectly, then given an initial state, you cannot calculate which attractor it will end up on. The tiniest error of approximation – and there will always be such an error - will change the answer completely." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"It should also be noted that the novel information generated by interactions in complex systems limits their predictability. Without randomness, complexity implies a particular non-determinism characterized by computational irreducibility. In other words, complex phenomena cannot be known a priori." (Carlos Gershenson, "Complexity", 2011)
"Just by looking at accelerating complexification of the Universe of which we are an integral part, we can conclude that we are not subjected to a random walk of evolution, nor are we subjected to a deterministic script of Nature, the truth lies somewhere in between – we are part of teleological evolution." (Alex M Vikoulov, "The Syntellect Hypothesis: Five Paradigms of the Mind's Evolution", 2019)
"A system in which a few things interacting produce tremendously divergent behavior; deterministic chaos; it looks random but its not." (Chris Langton)
No comments:
Post a Comment