On Prediction XV: Systems III

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"Although a system may exhibit sensitive dependence on initial conditions, this does not mean that everything is unpredictable about it. In fact, finding what is predictable in a background of chaos is a deep and important problem. (Which means that, regrettably, it is unsolved.) In dealing with this deep and important problem, and for want of a better approach, we shall use common sense." (David Ruelle, "Chance and Chaos", 1991)

"Although the detailed moment-to-moment behavior of a chaotic system cannot be predicted, the overall pattern of its 'random' fluctuations may be similar from scale to scale. Likewise, while the fine details of a chaotic system cannot be predicted one can know a little bit about the range of its 'random' fluctuation." (F David Peat, "From Certainty to Uncertainty", 2002)

"[…] while chaos theory deals in regions of randomness and chance, its equations are entirely deterministic. Plug in the relevant numbers and out comes the answer. In principle at least, dealing with a chaotic system is no different from predicting the fall of an apple or sending a rocket to the moon. In each case deterministic laws govern the system. This is where the chance of chaos differs from the chance that is inherent in quantum theory." (F David Peat, "From Certainty to Uncertainty", 2002)

"Chaos is a phenomenon encountered in science and mathematics wherein a deterministic (rule-based) system behaves unpredictably. That is, a system which is governed by fixed, precise rules, nevertheless behaves in a way which is, for all practical purposes, unpredictable in the long run. The mathematical use of the word 'chaos' does not align well with its more common usage to indicate lawlessness or the complete absence of order. On the contrary, mathematically chaotic systems are, in a sense, perfectly ordered, despite their apparent randomness. This seems like nonsense, but it is not." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)