"Due to this sensitivity any uncertainty about seemingly insignificant digits in the sequence of numbers which defines an initial condition, spreads with time towards the significant digits, leading to chaotic behavior. Therefore there is a change in the information we have about the state of the system. This change can be thought of as a creation of information if we consider that two initial conditions that are different but indistinguishable (within a certain precision), evolve into distinguishable states after a finite time." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"In a real experiment the noise present in a signal is usually considered to be the result of the interplay of a large number of degrees of freedom over which one has no control. This type of noise can be reduced by improving the experimental apparatus. But we have seen that another type of noise, which is not removable by any refinement of technique, can be present. This is what we have called the deterministic noise. Despite its intractability it provides us with a way to describe noisy signals by simple mathematical models, making possible a dynamical system approach to the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"In fact, in all those cases in which the initial state is
given with limited precision (if we assume that the space-time is continuous
this is always the case because a generic point turns out to be completely
specified only by an infinite amount of information, for example by an infinite
string of numbers), we can observe a situation in which, when time becomes
large, two trajectories emerge from the 'same' initial point. So, even though
there is a deterministic situation from a mathematical point of view (the uniqueness
theorem for ordinary differential equations is not in question), nevertheless
the exponential growth of errors makes the time evolution self-independent from
its past history and then nondeterministic in any practical sense."
"Roughly speaking the dimension of a set is the amount of information needed to specify points in it accurately." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"Very often a strange attractor is a fractal object, whose geometric structure is invariant under the time evolution maps." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"And you should not think that the mathematical game is arbitrary and gratuitous. The diverse mathematical theories have many relations with each other: the objects of one theory may find an interpretation in another theory, and this will lead to new and fruitful viewpoints. Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole." (David Ruelle, "Chance and Chaos", 1991)
"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)"[…] if a system is sufficiently complicated, the time it takes to return near a state already visited is huge (think of the hundred fleas on the checkerboard). Therefore if you look at the system for a moderate amount of time, eternal return is irrelevant, and you had better choose another idealization." (David Ruelle, "Chance and Chaos", 1991)
"If we have several modes, oscillating independently, the motion is, as we saw, not chaotic. Suppose now that we put a coupling, or interaction, between the different modes. This means that the evolution of each mode, or oscillator, at a certain moment is determined not just by the state of this oscillator at that moment, but by the states of the other oscillators as well. When do we have chaos then? Well, for sensitive dependence on initial condition to occur, at least three oscillators are necessary. In addition, the more oscillators there are, and the more coupling there is between them, the more likely you are to see chaos." (David Ruelle, "Chance and Chaos", 1991)
"Because there are regularities in the structure of the universe, and because life can take advantage of it, a new feature of life, which we call intelligence, has slowly emerged." (David Ruelle, "Chance and Chaos", 1991)
"But natural selection does not explain how we came to understand the chemistry of stars, or subtle properties of prime numbers. Natural selection explains only that humans have acquired higher intellectual functions; it cannot explain why so much is understandable about the physical universe, or the abstract world of mathematics." (David Ruelle, "Chance and Chaos", 1991)
"In brief, an algorithm is a systematic way of performing a certain task. […] The algorithmic complexity of a problem depends therefore on the availability of efficient algorithms to handle the problem." (David Ruelle, "Chance and Chaos", 1991)
"In brief, the way we do mathematics is human, very much so. But mathematicians have no doubt that there is a mathematical reality beyond our puny existence. We discover mathematical truth, we do not create it. We ask ourselves what seems to be a natural question and start working on it, and not uncommonly we find the solution (or someone else does). And we know that the answer could not have been different." (David Ruelle, "Chance and Chaos", 1991)
"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you wouldn't do it without some strong urge." (David Ruelle, "Chance and Chaos", 1991)
"Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole. Mathematics is a great kingdom, and that kingdom belongs to those who see." (David Ruelle, "Chance and Chaos", 1991)"The definition of information was modeled after that of entropy, the latter measuring the amount of randomness present in a system. Why should information be measured by randomness? Simply because by choosing one message in a class of possible messages you dispel the randomness present in that class." (David Ruelle, "Chance and Chaos", 1991)
"The ideas of chaos apply most naturally to time evolutions with 'eternal return'. These are time evolutions of systems that come back again and again to near the same situations. In other words, if the system is in a certain state at a certain time, it will return arbitrarily near the same state at a later time." (David Ruelle, "Chance and Chaos", 1991)
"[...] the mean information of a message is defined as the amount of chance (or randomness) present in a set of possible messages. To see that this is a natural definition, note that by choosing a message, one destroys the randomness present in the variety of possible messages. Information theory is thus concerned, as is statistical mechanics, with measuring amounts of randomness. The two theories are therefore closely related." (David Ruelle, "Chance and Chaos", 1991)
"The problem of meaning is obviously deep and complex. It is tied among other things to the question of how our brain works, and we don't know too much about that. We should thus not wonder that today's science can tackle only some rather superficial aspects of the problem of meaning." (David Ruelle, "Chance and Chaos", 1991)
"[…] the standard theory of chaos deals with time evolutions that come back again and again close to where they were earlier. Systems that exhibit this "eternal return" are in general only moderately complex. The historical evolution of very complex systems, by contrast, is typically one way: history does not repeat itself. For these very complex systems with one-way evolution it is usually clear that sensitive dependence on initial condition is present. The question is then whether it is restricted by regulation mechanisms, or whether it leads to long-term important consequences." (David Ruelle, "Chance and Chaos", 1991)
"The starting point of a mathematical theory consists of a few basic assertions on a certain number of mathematical objects (instead of mathematical objects, we might speak of words or phrases, because in a sense that is what they are). Starting from the basic assumptions one tries, by pure logic, to deduce new assertions, called theorems." (David Ruelle, "Chance and Chaos", 1991)
"The unity of mathematics is due to the logical relation between different mathematical theories. The physical theories, by contrast, need not be logically coherent; they have unity because they describe the same physical reality." (David Ruelle, "Chance and Chaos", 1991)
"The universe has quite a bit of randomness in it, but also quite a bit of structure." (David Ruelle, "Chance and Chaos", 1991)
“This transition from uncertainty to near certainty when we observe long series of events, or large systems, is an essential theme in the study of chance.” (David Ruelle, "Chance and Chaos", 1991)"To avoid getting mired in mathematical questions beyond human capabilities, perhaps you should stay closer to physics." (David Ruelle, "Conversations on Nonequilibrium Physics With an Extraterrestrial", Physics Today, 2004)
“Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception.“ (David Ruelle, “The Mathematician's Brain”, 2007)
“Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures.” (David Ruelle, “The Mathematician's Brain”, 2007)
“The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes.” (David Ruelle, “The Mathematician's Brain”, 2007)
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