"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." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"Now, the main problem with a quasiperiodic theory of turbulence (putting several oscillators together) is the following: when there is a nonlinear coupling between the oscillators, it very often happens that the time evolution does not remain quasiperiodic. As a matter of fact, in this latter situation, one can observe the appearance of a feature which makes the motion completely different from a quasiperiodic one. This feature is called sensitive dependence on initial conditions and turns out to be the conceptual key to reformulating the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"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)
"A meaningful physical discussion always requires an operational background. Either this is provided by an existing theory, or you have to give it yourself by the sufficiently explicit description of an experiment that can, at least in principle, be performed." (David Ruelle, "Chance and Chaos", 1991)
"A purely psychological approach to science would miss the importance of the comprehensibility of mathematics, and of 'the unreasonable effectiveness of mathematics in the natural sciences'. In fact, some scientists in the 'soft' sciences seem to miss this as well. But mathematicians and physicists know that they deal with a reality that has laws of its own, a reality above our little psychological problems, a reality that is strange, fascinating, and in some sense beautiful." (David Ruelle, "Chance and Chaos", 1991)
"Although a system may exhibit sensitive dependence on initial condition, 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)
"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)
"By gluing a mathematical theory on a piece of physical reality we obtain a physical theory. There exist many such theories, covering a great diversity of phenomena. And for a given phenomenon there are usually several different theories. In the better cases one passes from one theory to another one by an approximation (usually an uncontrolled approximation)." (David Ruelle, "Chance and Chaos", 1991)
"First, strange attractors look strange: they are not smooth curves or surfaces but have 'non-integer dimension' - or, as Benoit Mandelbrot puts it, they are fractal objects. Next, and more importantly, the motion on a strange attractor has sensitive dependence on initial condition. Finally, while strange attractors have only finite dimension, the time-frequency analysis reveals a continuum of frequencies." (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)
"Mathematics is not just a collection of formulas and theorems; it also contains ideas. One of the most pervasive ideas in mathematics is that of geometrization. This means, basically, visualization of all kinds of things as points of a space." (David Ruelle, "Chance and Chaos", 1991)
"Quantum mechanics, like other physical theories, consists of a mathematical part, and an operational part that tells you how a certain piece of physical reality is described by the mathematics. Both the mathematical and the operational aspects of quantum mechanics are straightforward and involve no logical paradoxes. Furthermore, the agreement between theory and experiment is as good as one can hope for. Nevertheless, the new mechanics has given rise to many controversies, which involve its probabilistic aspect, the relation of its operational concepts with those of classical mechanics […]" (David Ruelle, "Chance and Chaos", 1991)
"Sometimes the old philosophical problems are clarified by science; sometimes they subvert science. But the questions that are suggested by introspection often remain unanswered, and when the answers come they tend to be intellectually convincing rather than psychologically satisfying." (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)
"What causes difficulties is the apparent contradiction between determinism and our free will, introspectively characterized by the fact that several possibilities are open, and we engage our responsibility by choosing one. Introducing chance into the laws of physics does not help us in any way to resolve this contradiction. […] what allows our free will to be a meaningful notion is the complexity of the universe or, more precisely, our own complexity." (David Ruelle, "Chance and Chaos", 1991)
"What is an attractor? It is the set on which the point P, representing the system of interest, is moving at large times (i.e., after so-called transients have died out). For this definition to make sense it is important that the external forces acting on the system be time independent (otherwise we could get the point P to move in any way we like). It is also important that we consider dissipative systems (viscous fluids dissipate energy by self-friction). Dissipation is the reason why transients die out. Dissipation is the reason why, in the infinite-dimensional space representing the system, only a small set (the attractor) is really interesting." (David Ruelle, "Chance and Chaos", 1991)
"What we call intelligence is the activity of the mind and takes place in the brain. Intelligence guides our actions on the basis of what we perceive from the outside universe, and the interpretation of visual messages is therefore part of it." (David Ruelle, "Chance and Chaos", 1991)
"What we now call chaos is a time evolution with sensitive dependence on initial condition. The motion on a strange attractor is thus chaotic. One also speaks of deterministic noise when the irregular oscillations that are observed appear noisy, but the mechanism that produces them is deterministic." (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)
"A first important remark is that nature gives us mathematical hints. […] A second important remark is that mathematical physics deals with idealized systems. […] The third important remark is that nature may hint at a theorem but does not state clearly under which conditions is true." (David Ruelle, "The Mathematician's Brain", 2007)
"A good problem solver must also be a conceptual mathematician, with a good intuitive grasp of structures. But structures remain tools for the problem solver, instead of the main object of study." (David Ruelle, "The Mathematician's Brain", 2007)
"A simple mathematical argument, like a simple English sentence, often makes sense only against a huge contextual background." (David Ruelle, "The Mathematician's Brain", 2007)
"At the root of science and scientific research is the urge, the compulsion, to understand the nature of things" (David Ruelle, "The Mathematician's Brain", 2007)
"'Doing mathematics' is thus working on the construction of some mathematical object and resembles other creative enterprises of the mind in a scientific or artistic domain. But while the mental exercise of creating mathematics is somehow related to that of creating art, it should remain clear that mathematical objects are very different from the artistic objects that occur in literature, music, or the visual arts." (David Ruelle, "The Mathematician's Brain", 2007)
"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)
"I think that the beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes." (David Ruelle, "The Mathematician's Brain", 2007)
"If you have the rules of deduction and some initial choice of statements as sumed to be true (called axioms), then you are ready to derive many more true statements (called theorems). The rules of deduc tion constitute the logical machinery of mathematics, and the axioms comprise the basic properties of the objects you are interested in (in geometry these may be points, line segments, angles, etc.). There is some flexibility in selecting the rules of deduction, and many choices of axioms are possible. Once these have been decided you have all you need to do mathematics." (David Ruelle, "The Mathematician's Brain", 2007)
"[...] if two conics have five points in common, then they have infinitely many points in common. This geometric theorem is somewhat subtle but translates into a property of solutions of polynomial equations that makes more natural sense to a modern mathematician." (David Ruelle, "The Mathematician's Brain", 2007)
"[...] it is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illuminated, and some of the meaning hidden in the nature of things is finally revealed." (David Ruelle, "The Mathematician's Brain", 2007)
"It thus stands to reason that mathematical structures have a dual origin: in part human, in part purely logical. Human mathematics requires short formulations (because of our poor memory, etc.). But mathematical logic dictates that theorems with a short formulation may have very long proofs, as shown by Gödel. Clearly you don't want to go through the same long proof again and again. You will try instead to use repeatedly the short theorem that you have obtained. And an important tool to obtain short formulations is to give short names to mathematical objects that occur repeatedly. These short names describe new concepts. So we see how concept creation arises in the practice of mathematics as a consequence of the inherent logic of the sub ject and of the nature of human mathematicians." (David Ruelle, "The Mathematician's Brain", 2007)
"Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems. And the culture evolves because its key concepts and results change, slowly or brutally, to be replaced by new mathematical beacons." (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)
"Mathematics is useful. It is the language of physics, and some aspects of mathematics are important in all the sciences and their applications and also in finance. But my personal experience is that good mathematicians are rarely pushed by a high sense of duty and achievement that would urge them to do something useful. In fact, some mathematicians prefer to think that their work is absolutely useless." (David Ruelle, "The Mathematician's Brain", 2007)
"Putting together a sequence of mathematical ideas is like taking a walk in infinite dimension, going from one idea to the next. And the fact that the ideas have to fit together means that each stage in your walk presents you with a new variety of possibilities, among which you have to choose. You are in a labyrinth, an infinite-dimensional labyrinth." (David Ruelle, "The Mathematician's Brain", 2007)
"The absence of figures, therefore, does not mean that geometric intuition is shunned. On the contrary, geometrization is welcomed by mathematicians: this consists in giving a geometric interpretation of mathematical objects (in algebra or number theory) that are a priori nongeometric." (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)
"The consideration of the mind may be irrelevant when we discuss the formal aspects of mathematics but not when we discuss conceptual aspects. Mathematical concepts indeed are a production of the human mind and may reflect its idiosyncrasies." (David Ruelle, "The Mathematician's Brain", 2007)
"The fact that we have an efficient conceptualization of mathematics shows that this reflects a certain mathematical reality, even if this reality is quite invisible in the formal listing of the axioms of set theory." (David Ruelle, "The Mathematician's Brain", 2007)
"The panoply of technical tools of mathematics reflects the inside structure of mathematics and is basically all we know about this inside structure, so that building a new theory may change the way we understand the structural relations of different parts of mathematics." (David Ruelle, "The Mathematician's Brain", 2007)
"[...] the structure of human science is largely dependent on the special nature and organization of the human brain." (David Ruelle, "The Mathematician's Brain", 2007)
"To me, mathematical physics has a unique character: Nature herself takes you by the hand and shows you the outline of mathematical theories that an unaided pure mathematician would not have seen. But many details remain hidden, and it is our task to bring them to light." (David Ruelle, "The Mathematician's Brain", 2007)
"We must admit, however, that our knowledge of the logical structure of mathematics and of the workings of the human mind remain quite limited, so that we have only partial answers to some questions, while others remain quite open." (David Ruelle, "The Mathematician's Brain", 2007)
"We must be prepared to find that the perfection, purity, and simplicity that we love in mathematics is metaphorically related to a yearning for human perfection, purity, and simplicity. And this may explain why mathematicians often have a religious inclination. But we must also be prepared to find that our love of mathematics is not exempt from the usual human contradictions." (David Ruelle, "The Mathematician's Brain", 2007)
"We speak of mathematical reality as we speak of physical reality. They are different but both quite real. Mathematical reality is of logical nature, while physical reality is tied to the universe in which we live and which we perceive through our senses. This is not to say that we can readily define mathematical or physical reality, but we can relate to them by making mathematical proofs or physical experiments." (David Ruelle, "The Mathematician's Brain", 2007)
"What we get at the end is a mathematical theory: a human construct that, unavoidably, uses concepts introduced by definitions. And the concepts evolve in time because mathematical theories have a life of their own. Not only are theorems proved and new concepts named, but at the same time old concepts are reworked and redefined." (David Ruelle, "The Mathematician's Brain", 2007)
"When we introduce the concept of a group, we do this by imposing certain properties that should hold: these properties are called axioms. The axioms defining a group are, however, of a somewhat different nature from the ZFC axioms of set theory. Basically, whenever we do mathematics, we accept ZFC: a current mathematical paper systematically uses well-known consequences of ZFC (and normally does not mention ZFC). The axioms of a group by contrast are used only when appropriate." (David Ruelle, "The Mathematician's Brain", 2007)
"With mathematics and particularly mathematical logic, we come to grips with the most remote, the most nonhuman objects that the human mind has encountered. And this icy remoteness exerts on some people an irresistible fascination." (David Ruelle, "The Mathematician's Brain", 2007)
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