"[…] the least initial deviation from the truth is multiplied later a thousand-fold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter. The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end.” (St. Thomas Aquinas, "De Ente et Essentia”, cca. 1252)
"Systems in physical science […] are no more than appropriate instruments to aid the weakness of our organs: they are, properly speaking, approximate methods which put us on the path to the solution of the problem; these are the hypotheses which, successively modified, corrected, and changed in proportion as they are found false, should lead us infallibly one day, by a process of exclusion, to the knowledge of the true laws of nature.” (Antoine L Lavoisier, "Mémoires de l’Académie Royale des Sciences”, 1777)
"In every moment of her duration Nature is one connected whole; in every moment each individual part must be what it is, because all the others are what they are; and you could not remove a single grain of sand from its place, without thereby, although perhaps imperceptibly to you, altering something throughout all parts of the immeasurable whole." (Johann G Fichte, "The Vocation of Man”, 1800)
"Nature builds up by her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring.” (Sir John F W Herschel, "The Cabinet of Natural Philosophy”, 1831)
"It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe." (Thomas Carlyle, "Sartor Resartus", 1836)
"There is a maxim which is often quoted, that ‘The same causes will always produce the same effects.’ To make this maxim intelligible we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more that once, so that the causes and effects cannot be the same in all respects. [...] There is another maxim which must not be confounded with that quoted at the beginning of this article, which asserts ‘That like causes produce like effects’. This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a great change in the final state of the system, as when the displacement of the ‘points’ causes a railway train to run into another instead of keeping its proper course." (James C Maxwell, "Matter and Motion", 1876)
"[…] the simplicity of nature which we at present grasp is really the result of infinite complexity; and that below the uniformity there underlies a diversity whose depths we have not yet probed, and whose secret places are still beyond our reach.” (William Spottiswoode, 1879)
"A tenth of a degree more or less at any given point, and the cyclone will burst here and not there." (Henri Poincaré, "Sur le probleme des trios corps et les equations de la dynamique", Acta Mathematica Vol. 113, 1890)
"In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. […] One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus applied by symmetry.” (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development”, 1893)
"Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state, that is a perpetual motion machine." (Ludwig Boltzmann, "The Second Law of Thermodynamics", [Address to a Formal meeting of the Imperial Academy of Science], 1886)
“Certainly, if a system moves under the action of given forces and its initial conditions have given values in the mathematical sense, its future motion and behavior are exactly known. But, in astronomical problems, the situation is quite different: the constants defining the motion are only physically known, that is with some errors; their sizes get reduced along the progresses of our observing devices, but these errors can never completely vanish.” (Jacques Hadamard, “Les surfaces à courbures opposées et leurs lignes géodésiques”, Journal de mathématiques pures et appliqués 5e (4), 1898)
"Every deep thinker and observer of the Natural Laws is convinced that Nature is an orderly arrangement of matter and forces; that, in a word, Nature is not chaos, but cosmos.” (Frederick Hovenden, "What is Life?”, 1899)
"[…] there is a God precisely because Nature itself, even in chaos, cannot proceed except in an orderly and regular manner." (Immanuel Kant) "There is no such thing as chaos, it tacitly asserts, in the sidereal world or outside of it. For chaos is the negation of law, and law is the expression of the will of God.” (Agnes Mary Clerke, "Problems in Astrophysics”, 1903)
"[…] the lifeless symmetry of architecture, however beautiful the design and proportion, no man would be so mad as to put in competition with the animated charms of nature.” (Fanny Burney, "Evelina”, 1909)
"It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details." (Henri Poincaré, "The Future of Mathematics”, Monist Vol. 20, 1910)
"Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation." (Paul A M Dirac, "The Relation Between Mathematics And Physics", Proceedings of the Royal Society of Edinburgh”, 1938-1939)
"Rut seldom is asymmetry merely the absence of symmetry. Even in asymmetric designs one feels symmetry as the norm from which one deviates under the influence of forces of non-formal character.” (Hermann Weyl, "Symmetry”, 1952)
“Since the fundamental character of the living thing is its organization, the customary investigation of the single parts and processes cannot provide a complete explanation of the vital phenomena. This investigation gives us no information about the coordination of parts and processes. Thus, the chief task of biology must be to discover the laws of biological systems (at all levels of organization). We believe that the attempts to find a foundation for theoretical biology point at a fundamental change in the world picture. This view, considered as a method of investigation, we shall call ‘organismic biology’ and, as an attempt at an explanation, ‘the system theory of the organism’” (Ludwig von Bertalanffy, “Kritische Theorie der Formbildung”, 1928)
“Throwing a small stone may have some influence on the movement of the sun [...]" (Grigore C Moisil, “Determinism si inlantuire”, 1940)
"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals – such were its abstract characteristics.” (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness”, 1955)
"The essential vision of reality presents us not with fugitive appearances but with felt patterns of order which have coherence and meaning for the eye and for the mind. Symmetry, balance and rhythmic sequences express characteristics of natural phenomena: the connectedness of nature - the order, the logic, the living process. Here art and science meet on common ground.” (Gyorgy Kepes, "The New Landscape: In Art and Science”, 1956)
"Chaos is but unperceived order; it is a word indicating the limitations of the human mind and the paucity of observational facts. The words ‘chaos’, ‘accidental’, ‘chance’, ‘unpredictable’ are conveniences behind which we hide our ignorance.” (Harlow Shapley, "Of Stars and Men”, 1958)
"One of mankind’s earliest intellectual endeavors was the attempt to gather together the seemingly overwhelming variety presented by nature into an orderly pattern. The desire to classify - to impose order on chaos and then to form patterns out of this order on which to base ideas and conclusions - remains one of our strongest urges.” (Roger L Batten, 1959)
"One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.” (Edward N Lorenz, "The Predictability of Hydrodynamic Flow", Transactions of the New York Academy of Sciences 25 (4), 1963)
“[…] the task of general systems theory is to find the most general conceptual framework in which a scientific theory or a technological problem can be placed without losing the essential features of the theory or the problem. The proponents of general systems theory see in it the focal point of resynthesis of knowledge. There was a time when the man of knowledge was a generalist rather than a specialist, that is, he embodied the knowledge of principles rather than skills. He was the philosopher and the sage, and his epistemological creed was most clearly stated by Plato, who believed that all real knowledge comes from within rather than from without, that is, from the contemplation of what must be rather than what seems to be.” (Anatol Rapoport, “General system theory”, The International Encyclopedia of Social Sciences, 1968)
"It is necessary to study not only parts and processes in isolation, but also to solve the decisive problems found in organization and order unifying them, resulting from dynamic interaction of parts, and making the behavoir of the parts different when studied in isolation or within the whole." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)
"The properties and modes of action of higher levels are not explicable by the summation of the properties and modes of action of their components taken in isolation. If, however, we know the ensemble of the components and the relations existing between them, then the higher levels are derivable from the components." (Ludwig von Bertalanffy, "System Theory: Foundations, Development, Applications", 1968)
"Modern theoretical physics […] has put our thinking about the essence of matter in a different context. It has taken our gaze from the visible-the particles-to the underlying entity, the field. The presence of matter is merely a disturbance of the perfect state of the field at that place; something accidental, one could almost say, merely a ‘blemish’. Accordingly, there are no simple laws describing the forces between elementary particles […] Order and symmetry must be sought in the underlying field." (Walter Thirring "‘Urbausteine der Materie”, Almanach der bterreichischen Akademie der Wissenschaften Vol. 118, 1968)
"The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos.” (Herbert A Simon, "The Sciences of the Artificial”, 1969)
"Unfortunately, non-chaotic systems are very nearly as scarce as hen’s teeth, despite the fact that our physical understanding of nature is largely based upon their study." (Joseph Ford, "How Random Is a Coin Toss?" Physics Today Vol. 36 (4), 1983)
"Nature is disordered, powerful and chaotic, and through fear of the chaos we impose system on it. We abhor complexity, and seek to simplify things whenever we can by whatever means we have at hand. We need to have an overall explanation of what the universe is and how it functions. In order to achieve this overall view we develop explanatory theories which will give structure to natural phenomena: we classify nature into a coherent system which appears to do what we say it does.” (James Burke, "The Day the Universe Changed”, 1985)
"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)
"Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the fluctuations of the wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities." (James Gleick, "Chaos: Making a New Science", 1987)
"[…] nature, at the fundamental level, does not just prefer symmetry in a physical theory; nature demands it.” (Jennifer T Thompson, "Beyond Einstein: The Cosmic Quest for the Theory of the Universe”, 1987)
"The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done.” (Ian Stewart, "Does God Play Dice?”, 1989)
"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)
"Unfortunately, recognizing a system as chaotic will not tell us all that we might like to know. It will not provide us with a means of predicting the future course of the system. It will tell us that there is a limit to how far ahead we can predict, but it may not tell us what this limit is. Perhaps the best advice that chaos ‘theory’ can give us is not to jump at conclusions; unexpected occurrences may constitute perfectly normal behavior." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)
"Order wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful.” (Pierre Maurice Marie Duhem, "The Aim and Structure of Physical Theory”, 1991)
"An asymmetry in the present is understood as having originated from a past symmetry.” (Michael Leyton, "Symmetry, Causality, Mind”, 1992)
"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)
"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present” (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?”, 1992)
“General evolution theory, based on the integration of the relevant tenets of general system theory, cybernetics, information and communication theory, chaos theory, dynamical systems theory, and nonequilibrium thermodynamics, can convey a sound understanding of the laws and dynamics that govern the evolution of complex systems in the various realms of investigation. […] The basic notions of this new discipline can be developed to give an adequate account of the dynamical evolution of human societies as well. Such an account could furnish the basis of a system of knowledge better able to orient human beings and societies in their rapidly changing milieu. (Ervin Laszlo et al., “Evolution: The grand synthesis”, 1993)
"Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible.” (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?”, 1992)
"We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty.” (Ivars Peterson, "Newton’s Clock”, 1993)
"The voyage of discovery into our own solar system has taken us from clockwork precision into chaos and complexity. This still unfinished journey has not been easy, characterized as it is by twists, turns, and surprises that mirror the intricacies of the human mind at work on a profound puzzle. Much remains a mystery. We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty.” (Ivars Peterson, "Newton’s Clock”, 1993)
“What is a system? A system is a network of interdependent components that work together to try to accomplish the aim of the system. A system must have an aim. Without an aim, there is no system. The aim of the system must be clear to everyone in the system. The aim must include plans for the future. The aim is a value judgment.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
"How surprising it is that the laws of nature and the initial conditions of the universe should allow for the existence of beings who could observe it. Life as we know it would be impossible if any one of several physical quantities had slightly different values.” (Steven Weinberg, "Life in the Quantum Universe", Scientific American, 1995)
"Randomness, chaos, uncertainty, and chance are all a part of our lives. They reside at the ill-defined boundaries between what we know, what we can know, and what is beyond our knowing. They make life interesting.” (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari”, 1998)
""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)
"The self-similarity of fractal structures implies that there is some redundancy because of the repetition of details at all scales. Even though some of these structures may appear to teeter on the edge of randomness, they actually represent complex systems at the interface of order and disorder." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)
"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox.” (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos”, 2004)
"Group theory is a powerful tool for studying the symmetry of a physical system, especially the symmetry of a quantum system. Since the exact solution of the dynamic equation in the quantum theory is generally difficult to obtain, one has to find other methods to analyze the property of the system. Group theory provides an effective method by analyzing symmetry of the system to obtain some precise information of the system verifiable with observations." (Zhong-Qi Ma & Xiao-Yan Gu, "Problems and Solutions in Group Theory for Physicists", 2004)
"[…] some systems […] are very sensitive to their starting conditions, so that a tiny difference in the initial ‘push’ you give them causes a big difference in where they end up, and there is feedback, so that what a system does affects its own behavior." (John Gribbin, "Deep Simplicity", 2004)
"Of course, the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So, the direct impact of this phenomenon on weather prediction is often somewhat overstated." (James Annan & William Connolley, "Chaos and Climate”, 2005)
"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate.” (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity” 5th Ed., 2007)
"[…] it would seem that randomness and order are both inevitable parts of any description of reality. When we try to understand some particular phenomenon we are, in effect, banishing disorder. Before a piece of mathematics is understood it stands as a random collection of data. After it is understood, it is ordered, manageable. […] Both properties - the randomness and the order - are present simultaneously. This is what should be called complexity. Complexity is ordered randomness.” (William Byers, "How Mathematicians Think”, 2007)
"The ‘butterfly effect’ is at most a hypothesis, and it was certainly not Lorenz’s intention to change it to a metaphor for the importance of small event.” (Péter Érdi, "Complexity Explained”, 2008)
"A characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings.” (Hans J Korsch et al, „Chaos: A Program Collection for the PC”, 2008)
"Symmetry may have its appeal but it is inherently stale. Some kind of imbalance is behind every transformation.” (Marcelo Gleiser, "A Tear at the Edge of Creation: A Radical New Vision for Life in an Imperfect Universe”, 2010)
"[…] in all things that live there are certain irregularities and deficiencies which are not only signs of life, but sources of beauty. No human face is exactly the same in its lines on each side, no leaf perfect in its lobes, no branch in its symmetry. All admit irregularity as they imply change; […]” (John Ruskin, "The Stones of Venice: The Sea Stories”, 2013)
"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)
"God has put a secret art into the forces of Nature so as to enable it to fashion itself out of chaos into a perfect world system.” (Immanuel Kant)
"Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.” (Chen Ning Yang)
"Science, like art, music and poetry, tries to reduce chaos to the clarity and order of pure beauty.” (Detlev W Bronk)
"The most general law in nature is equity - the principle of balance and symmetry which guides the growth of forms along the lines of the greatest structural efficiency.” (Herbert Read)
"We find, therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind that is not possible to obtain in any other way.” (Johannes Kepler)
"Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the fluctuations of the wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities." (James Gleick, "Chaos: Making a New Science", 1987)
"[…] nature, at the fundamental level, does not just prefer symmetry in a physical theory; nature demands it.” (Jennifer T Thompson, "Beyond Einstein: The Cosmic Quest for the Theory of the Universe”, 1987)
"The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done.” (Ian Stewart, "Does God Play Dice?”, 1989)
"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)
"Unfortunately, recognizing a system as chaotic will not tell us all that we might like to know. It will not provide us with a means of predicting the future course of the system. It will tell us that there is a limit to how far ahead we can predict, but it may not tell us what this limit is. Perhaps the best advice that chaos ‘theory’ can give us is not to jump at conclusions; unexpected occurrences may constitute perfectly normal behavior." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)
"Order wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful.” (Pierre Maurice Marie Duhem, "The Aim and Structure of Physical Theory”, 1991)
"An asymmetry in the present is understood as having originated from a past symmetry.” (Michael Leyton, "Symmetry, Causality, Mind”, 1992)
"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)
"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present” (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?”, 1992)
“General evolution theory, based on the integration of the relevant tenets of general system theory, cybernetics, information and communication theory, chaos theory, dynamical systems theory, and nonequilibrium thermodynamics, can convey a sound understanding of the laws and dynamics that govern the evolution of complex systems in the various realms of investigation. […] The basic notions of this new discipline can be developed to give an adequate account of the dynamical evolution of human societies as well. Such an account could furnish the basis of a system of knowledge better able to orient human beings and societies in their rapidly changing milieu. (Ervin Laszlo et al., “Evolution: The grand synthesis”, 1993)
"Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible.” (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?”, 1992)
"We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty.” (Ivars Peterson, "Newton’s Clock”, 1993)
"The voyage of discovery into our own solar system has taken us from clockwork precision into chaos and complexity. This still unfinished journey has not been easy, characterized as it is by twists, turns, and surprises that mirror the intricacies of the human mind at work on a profound puzzle. Much remains a mystery. We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty.” (Ivars Peterson, "Newton’s Clock”, 1993)
“What is a system? A system is a network of interdependent components that work together to try to accomplish the aim of the system. A system must have an aim. Without an aim, there is no system. The aim of the system must be clear to everyone in the system. The aim must include plans for the future. The aim is a value judgment.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
"How surprising it is that the laws of nature and the initial conditions of the universe should allow for the existence of beings who could observe it. Life as we know it would be impossible if any one of several physical quantities had slightly different values.” (Steven Weinberg, "Life in the Quantum Universe", Scientific American, 1995)
"Randomness, chaos, uncertainty, and chance are all a part of our lives. They reside at the ill-defined boundaries between what we know, what we can know, and what is beyond our knowing. They make life interesting.” (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari”, 1998)
""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)
"The self-similarity of fractal structures implies that there is some redundancy because of the repetition of details at all scales. Even though some of these structures may appear to teeter on the edge of randomness, they actually represent complex systems at the interface of order and disorder." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)
"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox.” (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos”, 2004)
"Group theory is a powerful tool for studying the symmetry of a physical system, especially the symmetry of a quantum system. Since the exact solution of the dynamic equation in the quantum theory is generally difficult to obtain, one has to find other methods to analyze the property of the system. Group theory provides an effective method by analyzing symmetry of the system to obtain some precise information of the system verifiable with observations." (Zhong-Qi Ma & Xiao-Yan Gu, "Problems and Solutions in Group Theory for Physicists", 2004)
"[…] some systems […] are very sensitive to their starting conditions, so that a tiny difference in the initial ‘push’ you give them causes a big difference in where they end up, and there is feedback, so that what a system does affects its own behavior." (John Gribbin, "Deep Simplicity", 2004)
"Of course, the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So, the direct impact of this phenomenon on weather prediction is often somewhat overstated." (James Annan & William Connolley, "Chaos and Climate”, 2005)
"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate.” (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity” 5th Ed., 2007)
"[…] it would seem that randomness and order are both inevitable parts of any description of reality. When we try to understand some particular phenomenon we are, in effect, banishing disorder. Before a piece of mathematics is understood it stands as a random collection of data. After it is understood, it is ordered, manageable. […] Both properties - the randomness and the order - are present simultaneously. This is what should be called complexity. Complexity is ordered randomness.” (William Byers, "How Mathematicians Think”, 2007)
"The ‘butterfly effect’ is at most a hypothesis, and it was certainly not Lorenz’s intention to change it to a metaphor for the importance of small event.” (Péter Érdi, "Complexity Explained”, 2008)
"A characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings.” (Hans J Korsch et al, „Chaos: A Program Collection for the PC”, 2008)
"Symmetry may have its appeal but it is inherently stale. Some kind of imbalance is behind every transformation.” (Marcelo Gleiser, "A Tear at the Edge of Creation: A Radical New Vision for Life in an Imperfect Universe”, 2010)
"[…] in all things that live there are certain irregularities and deficiencies which are not only signs of life, but sources of beauty. No human face is exactly the same in its lines on each side, no leaf perfect in its lobes, no branch in its symmetry. All admit irregularity as they imply change; […]” (John Ruskin, "The Stones of Venice: The Sea Stories”, 2013)
"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)
"God has put a secret art into the forces of Nature so as to enable it to fashion itself out of chaos into a perfect world system.” (Immanuel Kant)
"Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.” (Chen Ning Yang)
"Science, like art, music and poetry, tries to reduce chaos to the clarity and order of pure beauty.” (Detlev W Bronk)
"The most general law in nature is equity - the principle of balance and symmetry which guides the growth of forms along the lines of the greatest structural efficiency.” (Herbert Read)
"We find, therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind that is not possible to obtain in any other way.” (Johannes Kepler)
No comments:
Post a Comment