Thermodynamics IV

"It is impossible by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects. [Footnote: ] If this axiom be denied for all temperatures, it would have to be admitted that a self-acting machine might be set to work and produce mechanical effect by cooling the sea or earth, with no limit but the total loss of heat from the earth and sea, or in reality, from the whole material world." (William Thomson, "On the Dynamical Theory of Heat with Numerical Results Deduced from Mr Joule's Equivalent of a Thermal Unit and M. Regnault's Observations on Steam", Transactions of the Royal Society of Edinburgh, 1851)

"Though the ultimate state of the universe may be its vital and psychical extinction, there is nothing in physics to interfere with the hypothesis that the penultimate state might be the millennium - in other words a state in which a minimum of difference of energy - level might have its exchanges so skillfully canalises that a maximum of happy and virtuous consciousness would be the only result." (William James, [Letter to Henry Adams] 1910)" (William James, [Letter to Henry Adams] 1910)

"Organic evolution has its physical analogue in the universal law that the world tends, in all its parts and particles, to pass from certain less probable to certain more probable configurations or states. This is the second law of thermodynamics." (D'Arcy Wentworth Thompson, "On Growth and Form", 1917)

"In classical physics, most of the fundamental laws of nature were concerned either with the stability of certain configurations of bodies, e.g. the solar system, or else with the conservation of certain properties of matter, e.g. mass, energy, angular momentum or spin. The outstanding exception was the famous Second Law of Thermodynamics, discovered by Clausius in 1850. This law, as usually stated, refers to an abstract concept called entropy, which for any enclosed or thermally isolated system tends to increase continually with lapse of time. In practice, the most familiar example of this law occurs when two bodies are in contact: in general, heat tends to flow from the hotter body to the cooler. Thus, while the First Law of Thermodynamics, viz. the conservation of energy, is concerned only with time as mere duration, the Second Law involves the idea of trend." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The second law of thermodynamics provides a more modem (and a more discouraging) example of the maximum principle: the entropy (disorder) of the universe tends toward a maximum." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"[...] thermodynamics knows of no such notion as the 'entropy of a physical system'. Thermodynamics does have the concept of the entropy of a thermodynamic system; but a given physical system corresponds to many different thermodynamic systems." (Edwin T Jaynes, "Gibbs vs Boltzmann Entropies", 1964)

"'You cannot base a general mathematical theory on imprecisely defined concepts. You can make some progress that way; but sooner or later the theory is bound to dissolve in ambiguities which prevent you from extending it further.' Failure to recognize this fact has another unfortunate consequence which is, in a practical sense, even more disastrous: 'Unless the conceptual problems of a field have been clearly resolved, you cannot say which mathematical problems are the relevant ones worth working on; and your efforts are more than likely to be wasted.'" (Edwin T Jaynes, "Foundations of Probability Theory and Statistical Mechanics", 1967)

"There is no end to this search for the ultimate ‘true’ entropy until we have reached the point where we control the location of each atom independently. But just at that point the notion of entropy collapses, and we are no longer talking thermodynamics." (Edwin T Jaynes, "Papers on Probability, Statistics, and Statistical Physics", 1983)

"No one has yet succeeded in deriving the second law from any other law of nature. It stands on its own feet. It is the only law in our everyday world that gives a direction to time, which tells us that the universe is moving toward equilibrium and which gives us a criteria for that state, namely, the point of maximum entropy, of maximum probability. The second law involves no new forces. On the contrary, it says nothing about forces whatsoever." (Brian L Silver, "The Ascent of Science", 1998)

On Continuity XIV (Progress)

"Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity." (Gottfried W Leibniz, "New Essays Concerning Human Understanding with an Appealing", 1704)

"Harmonious order governing eternally continuous progress - the web and woof of matter and force interweaving by slow degrees, without a broken thread, that veil which lies between us and the Infinite - that universe which alone we know or can know; such is the picture which science draws of the world, and in proportion as any part of that picture is in unison with the rest, so may we feel sure that it is rightly painted." (Thomas H Huxley, "Darwiniana", 1893–94)

"Mathematics is not a world empire where one man can exert a dominating influence over work along all the lines, but it is continually splitting up into self-determining republics." (Geroge A Miller, "Felix Klein and the History of Modern Mathematics", 1927)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers." (David Hilbert, "Anschauliche Geometrie", 1932)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The Philosophy of Physics", 1936)

"And yet, on looking into the history of science, one is overwhelmed by evidence that all too often there is no regular procedure, no logical system of discovery, no simple, continuous development. The process of discovery has been as varied as the temperament of the scientist." (Gerald Holton, "Thematic Origins of Scientific Thought: Kepler to Einstein", 1973)

On Equilibrium (-1799)

"Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance." (Archimedes, "On the Equilibrium of Planes" Vol. I, 3rd century BC)

"Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes." (Archimedes, "On the Equilibrium of Planes" Vol. I, 3rd century BC)

"Inequality is the cause of all local movements. There is no rest without equality." (Leonardo da Vinci, Codex Atlanticus, 1478)

"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented.” (Leonhard Euler, “Specimen de usu observationum in mathesi pura", Novi Commentarii academiae scientiarum Petropolitanae 6, 1756/57)

"Statics is the science of the equilibrium of forces. In general, force or power is the cause, whatever it may be, which induces or tends to impart motion to the body to which it is applied. The force or power must be measured by the quantity of motion produced or to be produced. In the state of equilibrium, the force has no apparent action. It produces only a tendency for motion in the body it is applied to. But it must be measured by the effect it would produce if it were not impeded. By taking any force or its effect as unity, the relation of every other force is only a ratio, a mathematical quantity, which can be represented by some numbers or lines. It is in this fashion that forces must be treated in mechanics." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

On Equilibrium (1875-1889)

"Any system in stable chemical equilibrium, subjected to the influence of an external cause tends to change either its temperature or its condensation (pressure, concentration, number of molecules in unit volume), either as a whole or in some of its parts, can only undergo such internal modifications as would, if produced alone, bring about a change of temperature or of condensation of opposite sign to that resulting from the external cause." (Henri L Le Chatelier, "A General Statement of the Laws of Chemical Equilibrium", Comptes rendus Vol. 99, 1884)

"Every situation is an equilibrium of forces; every life is a struggle between opposing forces working within the limits of a certain equilibrium." (Henri-Frédéric Amiel, "Amiel's Journal", 1885)

"Plasticity, then, in the wide sense of the word, means the possession of a structure weak enough to yield to an influence, but strong enough not to yield all at once. Each relatively stable phase of equilibrium in such a structure is marked by what we may call a new set of habits." (William James, "The Laws of Habit", 1887) 

"Every change of one of the factors of an equilibrium occasions a rearrangement of the system in such a direction that the factor in question experiences a change in a sense opposite to the original change." (Henri L Le Chatelier, "Recherches Experimentales et Theoriques sur les Equilibres Chimiques" ["Experimental and Theoretical Research on Chemical Equilibria"], Annales des Mines 8, 1888)

"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)

"That branch of physics which is at once the oldest and the simplest and which is therefore treated as introductory to other departments of this science, is concerned with the motions and equilibrium of masses. It bears the name of mechanics." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"The whole aspect of the universe changes with this new conception. The idea of force governing the world, of pre-established law, preconceived harmony, disappears to make room for the harmony that [Charles] Fourier had caught a glimpse of: the one which results from the disorderly and incoherent movements of numberless hosts of matter, each of which goes its own way and all of which hold each other in equilibrium." (Peter Kropotkin, "Anarchism: Its Philosophy and Ideal", 1896)

On Equilibrium (1940-1949)

"One may generalize upon these processes in terms of group equilibrium. The group may be said to be in equilibrium when the interactions of its members fall into the customary pattern through which group activities are and have been organized. The pattern of interactions may undergo certain modifications without upsetting the group equilibrium, but abrupt and drastic changes destroy the equilibrium." (William F Whyte, "Street Corner Society", 1943)

"How would we express in terms of the statistical theory the marvelous faculty of a living organism, by which it delays the decay into thermodynamical equilibrium (death)? [...] the device by which an organism maintains itself stationary at a fairly high level of orderliness […] really consists in continually sucking orderliness from its environment." (Erwin Schrödinger, "What Is Life?", 1944)

"A well ordered society would be one where the State only had a negative action, comparable to that of a rudder: a light pressure at the right moment to counteract the first suggestion of any loss of equilibrium." (Simone Weil, "Gravity and Grace", 1947)

"The behavior of two individuals, consisting of effort which results in output, is considered to be determined by a satisfaction function which depends on remuneration (receiving part of the output) and on the effort expended. The total output of the two individuals is not additive, that is, together they produce in general more than separately. Each individual behaves in a way which he considers will maximize his satisfaction function. Conditions are deduced for a certain relative equilibrium and for the stability of this equilibrium, i.e., conditions under which it will not pay the individual to decrease his efforts. In the absence of such conditions ‘exploitation’ occurs which may or may not lead to total parasitism." (Anatol Rapoport, "Mathematical theory of motivation interactions of two individuals," The Bulletin of Mathematical Biophysics 9, 1947)

"The general method involved may be very simply stated. In cases where the equilibrium values of our variables can be regarded as the solutions of an extremum (maximum or minimum) problem, it is often possible regardless of the number of variables involved to determine unambiguously the qualitative behavior of our solution values in respect to changes of parameters." (Paul Samuelson, "Foundations of Economic Analysis", 1947)

"The study of the conditions for change begins appropriately with an analysis of the conditions for no change, that is, for the state of equilibrium." (Kurt Lewin, "Quasi-Stationary Social Equilibria and the Problem of Permanent Change", 1947)

Discovery in Physics (unsourced)

"In physical science the discovery of new facts is open to every blockhead with patience, manual dexterity, and acute senses; it is less effectually promoted by genius than by co-operation, and more frequently the result of accident than of design." (John Thomson)

"Man’s first glance at the universe discovers only variety, diversity, multiplicity of phenomena. Let that glance be illuminated by science - by the science which brings man closer to God, - and simplicity and unity shine on all sides." (Louis Pasteur)

"Natural Philosophy consists in discovering the frame and operations of Nature, and reducing them, as far as may be, to general Rules or Laws - establishing these rules by observations and experiments, and thence deducing the causes and effects of things." (Isaac Newton)

"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics." (Johannes Kepler) 

"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler)

On Metaphysics (until 1699)

“Metaphysics is universal and is exclusively concerned with primary substance. […] And here we will have the science to study that which is, both in its essence and in the properties which, just as a thing that is, it has. […] That among entities there must be some cause which moves and combines things. […] There must then be a principle of such a kind that its substance is activity.” (Aristotle, “Metaphysics”, 340 BC)

“[…] he who wishes to attain to human perfection, must therefore first study Logic, next the various branches of Mathematics in their proper order, then Physics, and lastly Metaphysics.” (Moses Maimonides, “The Guide for the Perplexed”, 1190)

“When you understand physics, you have entered the hall; and when, after completing the study of natural philosophy, you master metaphysics, you have entered the innermost court and are with the king in the same palace.” (Moses Maimonides, “The Guide for the Perplexed”, 1190)

"That is better and more valuable which requires fewer, other circumstances being equal. [...] For if one thing were demonstrated from many and another thing from fewer equally known premises, clearly that is better which is from fewer because it makes us know quickly, just as a universal demonstration is better than particular because it produces knowledge from fewer premises. Similarly in natural science, in moral science, and in metaphysics the best is that which needs no premises and the better that which needs the fewer, other circumstances being equal." (Robert Grosseteste," Commentarius in Posteriorum Analyticorum Libros", cca. 1217–1220)

“Physic […] is situate in a middle term or distance between natural history and metaphysic. For natural history describes the variety of things; physic the causes, but variable or respective causes; and metaphysic the fixed and constant causes.” (Sir Francis Bacon, “The Advancement of Learning”, 1605)

“Reality cannot be found except in One single source, because of the interconnection of all things with one another. […] It is a good thing to proceed in order and to establish propositions (principles). This is the way to gain ground and to progress with certainty.” (Gottfried Leibniz, 1670)

"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum's composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysics except he who has passed through this [labyrinth]." (Gottfried Wilhelm Leibniz, "Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae", 1676)

“Blind metaphysical necessity, which is certainly the same always and everywhere, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being, necessarily existing.” (Isaac Newton, “The Principia: Mathematical Principles of Natural Philosophy”, 1687)

On Problem Solving X: Mathematicians II

“More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of problems and the entities considered. Carried to an extreme, this procedure justifies the well-known joke that a mathematician is a scientist who knows neither what he is talking about nor whether whatever he is talking about exists or not.” (Élie Cartan) 

"Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems." (George Polya) 

"The science of physics does not only give us [mathematicians] an oportunity to solve problems, but helps us also to discover the means of solving them, and it does this in two ways: it leads us to anticipate the solution and suggests suitable lines of argument." (Henri Poincaré)
 
“Solving problems can be regarded as the most characteristically human activity.” (George Polya, 1981)

 “Solving problems is the specific achievement of intelligence.” (George Polya, 1957)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.” (Paul Lockhart, “A Mathematician's Lament”, 2009)
 
“To learn mathematics is to learn mathematical problem solving.” (Patrik W Thompson, 1985)

„[...] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.“ (Ivars Peterson, „Islands of Truth: A Mathematical Mystery Cruise“, 1990)

"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"For solving problems in the real world, not only a knowledge of mathematics is useful, but also the mathematician is meaningful: the mathematician uses specific ways of thinking such as abstraction, generalization, and extraction of truth from various realities to crystallize to a simple statement as a theorem. Moreover, in various applications, mathematicians do not merely apply knowledge in an existing specific area of mathematics, but can build up a new theoretical system of mathematics for solving concrete problems." (Masahiro Yamamoto, "Mathematics for Industry: Principle, Reality and Practice, from the Point of View of a Mathematician", [in "What Mathematics Can Do for You"] 2013)

"What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved.“ (Julia Robinson)

Previous <<||>> Next

On Numeracy I

“The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.” (Herbert G Wells, „Mankind in the Making”,  1903)

“[…] statistical literacy. That is, the ability to read diagrams and maps; a “consumer” understanding of common statistical terms, as average, per cent, dispersion, correlation, and index number.” (Douglas Scates, “Statistics: The Mathematics for Social Problems”, 1943)

„Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” (Samuel S Wilks, 1951 [paraphrasing Herber Wells] )

“Just as by ‘literacy’, in this context, we mean much more than its dictionary sense of the ability to read and write, so by ‘numeracy’ we mean more than mere ability to manipulate the rule of three. When we say that a scientist is ‘illiterate’, we mean that he is not well enough read to be able to communicate effectively with those who have had a literary education. When we say that a historian or a linguist is ‘innumerate’ we mean that he cannot even begin to understand what scientists and mathematicians are talking about.” (Sir Geoffrey Crowther, “A Report of the Central Advisory Committee for Education”, 1959)

“It is perhaps possible to distinguish two different aspects of numeracy […]. On the one hand is an understanding of the scientific approach to the study of phenomena - observation, hypothesis, experiment, verification. On the other hand, there is the need in the modern world to think quantitatively, to realise how far our problems are problems of degree even when they appear as problems of kind.” (Sir Geoffrey Crowther, “A Report of the Central Advisory Committee for Education”, 1959)

“Numeracy has two facets-reading and writing, or extracting numerical information and presenting it. The skills of data presentation may at first seem ad hoc and judgemental, a matter of style rather than of technology, but certain aspects can be formalized into explicit rules, the equivalent of elementary syntax.” (Andrew Ehrenberg, “Rudiments of Numeracy”, Journal of Royal Statistical Society, 1977)

See also:
Numeracy II & III

Chaos Theory: The Butterfly Effect - A Retrospective

"Parvus error in principiis, magnus in conclusionibus" 

"Parvus error in principio, magnus est in fine"

“A small error in the beginning (or in principles) leads to a big error in the end (or in conclusions).” (ancient axiom)

"The wise tell us that a nail keeps a shoe, a shoe a horse, a horse a man, a man a castle, that can fight." (Freidank, Bescheidenheit, cca. 1230)

"[…] 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)

"A little neglect may breed mischief [...] 

for want of a nail, the shoe was lost;

for want of a shoe the horse was lost;

and for want of a horse the rider was lost." (Benjamin Franklin, Poor Richard's Almanac, 1758) 

"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)

"Every existence above a certain rank has its singular points; the higher the rank the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being may produce results of the greatest importance." (James C Maxwell, [letter] 1865) 

"What we call little things are merely the causes of great things; they are the beginning, the embryo, and it is the point of departure which, generally speaking, decides the whole future of an existence. One single black speck may be the beginning of gangrene, of a storm, of a revolution." (Henri-Frédéric Amiel, [journal entry] 1868)

"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)

"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)

"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 appliquées 5e (4), 1898)

"An exceedingly small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation 'approximately'. If that enabled us to predict the succeeding situation with 'the same approximation', that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. (Jules H Poincaré, "Science and Method", 1908) 

 

"Throwing a small stone may have some influence on the movement of the sun." (Grigore C Moisil, "Determinism si inlantuire", 1940)

"The predictions of physical theories for the most part concern situations where initial conditions can be precisely specified. If such initial conditions are not found in nature, they can be arranged." (Anatol Rapoport, "The Search for Simplicity", 1956)

“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)

"Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?" (Edward N Lorenz, [talk] 1972)

"If a single flap of a butterfly's wing can be instrumental in generating a tornado, so all the previous and subsequent flaps of its wings, as can the flaps of the wings of the millions of other butterflies, not to mention the activities of innumerable more powerful creatures, including our own species." (Edward N Lorenz, [talk] 1972)

"If the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. More generally, I am proposing that over the years minuscule disturbances neither increase nor decrease the frequency of occurrence of various weather events such as tornadoes; the most that they may do is to modify the sequence in which these events occur." (Edward N Lorenz, [talk] 1972) 

"[...] the influence of a single butterfly is not only a fine detail-it is confined to a small volume. Some of the numerical methods which seem to be well adapted for examining the intensification of errors are not suitable for studying the dispersion of errors from restricted to unrestricted regions. One hypothesis, unconfirmed, is that the influence of a butterfly's wings will spread in turbulent air, but not in calm air." (Edward N Lorenz, [talk] 1972)

"Given an approximate knowledge of a system's initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system. This assumption lay at the philosophical heart of science." (James Gleick, Chaos: Making a New Science, 1987)

"A slight variation in the axioms at the foundation of a theory can result in huge changes at the frontier." (Stanley P Gudder, "Quantum Probability", 1988)

"The principle of maximum diversity operates both at the physical and at the mental level. It says that the laws of nature and the initial conditions are such as to make the universe as interesting as possible.  As a result, life is possible but not too easy. Always when things are dull, something new turns up to challenge us and to stop us from settling into a rut. Examples of things which make life difficult are all around us: comet impacts, ice ages, weapons, plagues, nuclear fission, computers, sex, sin and death.  Not all challenges can be overcome, and so we have tragedy. Maximum diversity often leads to maximum stress. In the end we survive, but only by the skin of our teeth." (Freeman J Dyson, "Infinite in All Directions", 1988)

"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)

"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)

"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)

"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)

"Everywhere […] in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since […] these laws are time symmetric- they permit […] time-reverse. […] The initial conditions play a decisive role in endowing the world with its sense of temporal direction. […] some prescription for initial conditions is crucial if we are to understand […]" (John D Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 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 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)

"[…] 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)

"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)

"Chaos is a parody of any metaphysics of destiny. It is not even an avatar of such a metaphysics. The poetry of initial conditions fascinates us today, now that we no longer possess a vision of final conditions, and Chaos stands in for us as a negative destiny. [...] Destiny is the ecstatic figure of necessity. Chaos is merely the metastatic figure of Chance. Chaotic processes are random and statistical in nature and, even if they culminate in the hidden order of strange attractors, that still has nothing to do with the fulgurating notion of destiny, the absence of which is cruelly felt." (Jean Baudrillard, "The Illusion of the End", 1992)

"In nonlinear systems - and the economy is most certainly nonlinear - chaos theory tells you that the slightest uncertainty in your knowledge of the initial conditions will often grow inexorably. After a while, your predictions are nonsense." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)

"In the everyday world of human affairs, no one is surprised to learn that a tiny event over here can have an enormous effect over there. For want of a nail, the shoe was lost, et cetera. But when the physicists started paying serious attention to nonlinear systems in their own domain, they began to realize just how profound a principle this really was. […] Tiny perturbations won't always remain tiny. Under the right circumstances, the slightest uncertainty can grow until the system's future becomes utterly unpredictable - or, in a word, chaotic." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)

"How can deterministic behavior look random? If truly identical states do occur on two or more occasions, it is unlikely that the identical states that will necessarily follow will be perceived as being appreciably different. What can readily happen instead is that almost, but not quite, identical states occurring on two occasions will appear to be just alike, while the states that follow, which need not be even nearly alike, will be observably different. In fact, in some dynamical systems it is normal for two almost identical states to be followed, after a sufficient time lapse, by two states bearing no more resemblance than two states chosen at random from a long sequence. Systems in which this is the case are said to be sensitively dependent on initial conditions. With a few more qualifications, to be considered presently, sensitive dependence can serve as an acceptable definition of chaos [...]" (Edward N Lorenz, "The Essence of Chaos", 1993)

"Symmetry breaking in psychology is governed by the nonlinear causality of complex systems (the 'butterfly effect'), which roughly means that a small cause can have a big effect. Tiny details of initial individual perspectives, but also cognitive prejudices, may 'enslave' the other modes and lead to one dominant view." (Klaus Mainzer, "Thinking in Complexity", 1994)

"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)

"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"What is chaos? Everyone has an impression of what the word means, but scientifically chaos is more than random behavior, lack of control, or complete disorder. [...] Scientifically, chaos is defined as extreme sensitivity to initial conditions. If a system is chaotic, when you change the initial state of the system by a tiny amount you change its future significantly." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"Small changes in the initial conditions in a chaotic system produce dramatically different evolutionary histories. It is because of this sensitivity to initial conditions that chaotic systems are inherently unpredictable. To predict a future state of a system, one has to be able to rely on numerical calculations and initial measurements of the state variables. Yet slight errors in measurement combined with extremely small computational errors (from roundoff or truncation) make prediction impossible from a practical perspective. Moreover, small initial errors in prediction grow exponentially in chaotic systems as the trajectories evolve. Thus, theoretically, prediction may be possible with some chaotic processes if one is interested only in the movement between two relatively close points on a trajectory. When longer time intervals are involved, the situation becomes hopeless."(Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Swarm systems generate novelty for three reasons: (1) They are 'sensitive to initial conditions' - a scientific shorthand for saying that the size of the effect is not proportional to the size of the cause - so they can make a surprising mountain out of a molehill. (2) They hide countless novel possibilities in the exponential combinations of many interlinked individuals. (3) They don’t reckon individuals, so therefore individual variation and imperfection can be allowed. In swarm systems with heritability, individual variation and imperfection will lead to perpetual novelty, or what we call evolution." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Unlike classical mathematics, net math exhibits nonintuitive traits. In general, small variations in input in an interacting swarm can produce huge variations in output. Effects are disproportional to causes - the butterfly effect." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"What is chaos? Everyone has an impression of what the word means, but scientifically chaos is more than random behavior, lack of control, or complete disorder. [...] Scientifically, chaos is defined as extreme sensitivity to initial conditions. If a system is chaotic, when you change the initial state of the system by a tiny amount you change its future significantly." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"Surveying the bewildering damage from some historical hurricanes, an outside observer might wonder whether builders were suffering under the illusion of a chaos-free environment: one governed by underestimated deterministic forces. Of course, in reality, dynamical chaos is intrinsic to the atmosphere, and contributes significantly to the aleatory uncertainty in wind loading. It may take more than the flap of a butterfly’s wings to change a hurricane forecast, but chaos imposes a fundamental practical limit to windstorm prediction capability." (Gordon Woo, "The mathematics of natural catastrophes", 1999)

"The classic example of chaos at work is in the weather. If you could measure the positions and motions of all the atoms in the air at once, you could predict the weather perfectly. But computer simulations show that tiny differences in starting conditions build up over about a week to give wildly different forecasts. So weather predicting will never be any good for forecasts more than a few days ahead, no matter how big (in terms of memory) and fast computers get to be in the future. The only computer that can simulate the weather is the weather; and the only computer that can simulate the Universe is the Universe." (John Gribbin, "The Little Book of Science", 1999)

"Chaos theory reconciles our intuitive sense of free will with the deterministic laws of nature. However, it has an even deeper philosophical ramification. Not only do we have freedom to control our actions, but also the sensitivity to initial conditions implies that even our smallest act can drastically alter the course of history, for better or for worse. Like the butterfly flapping its wings, the results of our behavior are amplified with each day that passes, eventually producing a completely different world than would have existed in our absence!" (Julien C Sprott, "Strange Attractors: Creating Patterns in Chaos", 2000)

"In chaology, the initial conditions are likely to be out of all proportion to the consequences; indeed, origins are much more random, unpredictable, and unknowable and seemingly much less directly causal than in orderly systems. The sensitive dependence upon initial conditions means that similar phenomena or systems will never be wholly identical and that the results of those small initial changes may be radically different. These unpredictable initial conditions may, for instance, lead to the so-called butterfly effect, in which an extremely minor and remote act causes disruptions of a huge magnitude."

(Gordon E Slethaug, "Beautiful Chaos: Chaos theory and metachaotics in recent American fiction", 2000)

"Scientists tell us that the world of nature is so small and interdependent that a butterfly flapping its wings in the Amazon rainforest can generate a violent storm on the other side of the earth. This principle is known as the 'Butterfly Effect'. Today, we realize, perhaps more than ever, that the world of human activity also has its own 'Butterfly Effect' - for better or for worse." (Kofi Annan, [Nobel lecture] 2001)

"In chaos theory this 'butterfly effect' highlights the extreme sensitivity of nonlinear systems at their bifurcation points. There the slightest perturbation can push them into chaos, or into some quite different form of ordered behavior. Because we can never have total information or work to an infinite number of decimal places, there will always be a tiny level of uncertainty that can magnify to the point where it begins to dominate the system. It is for this reason that chaos theory reminds us that uncertainty can always subvert our attempts to encompass the cosmos with our schemes and mathematical reasoning." (F David Peat, "From Certainty to Uncertainty", 2002)

"Incidentally, the butterfly effect also has a good side to it. Since a butterfly in Brazil can disturb the serene weather in Florida, the same butterfly could calm a hurricane in Texas by simply flapping its wings in a certain fashion. This process is called 'controlling chaos' and has been put to use with some success in dealing with heart fibrillation. By applying small shocks at precisely the right moment, an erratic heartbeat can be regularized and a heart attack avoided." (George Szpiro, "Kepler’s Conjecture", 2002)

"A depressing corollary of the butterfly effect (or so it was widely believed) was that two chaotic systems could never synchronize with each other. Even if you took great pains to start them the same way, there would always be some infinitesimal difference in their initial states. Normally that small discrepancy would remain small for a long time, but in a chaotic system, the error cascades and feeds on itself so swiftly that the systems diverge almost immediately, destroying the synchronization. Unfortunately, it seemed, two of the most vibrant branches of nonlinear science - chaos and sync - could never be married. They were fundamentally incompatible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"A sudden change in the evolutive dynamics of a system (a ‘surprise’) can emerge, apparently violating a symmetrical law that was formulated by making a reduction on some (or many) finite sequences of numerical data. This is the crucial point. As we have said on a number of occasions, complexity emerges as a breakdown of symmetry (a system that, by evolving with continuity, suddenly passes from one attractor to another) in laws which, expressed in mathematical form, are symmetrical. Nonetheless, this breakdown happens. It is the surprise, the paradox, a sort of butterfly effect that can highlight small differences between numbers that are very close to one another in the continuum of real numbers; differences that may evade the experimental interpretation of data, but that may increasingly amplify in the system’s dynamics." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems", 2003)

"At the basis of the impossibility of making reliable predictions for systems such as the atmosphere, there is a phenomenon known today as the butterfly effect. This deals with the progressive limitless magnification of the slightest imprecision (error) present in the measurement of the initial data (the incomplete knowledge of the current state of each molecule of air), which, although in principle negligible, will increasingly expand during the course of the model’s evolution, until it renders any prediction on future states (atmospheric weather conditions when the forecast refers to more than a few days ahead) completely insignificant, as these states appear completely different from the calculated ones." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems", 2003)

"The butterfly effect came to be the most familiar icon of the new science, and appropriately so, for it is the signature of chaos. […] The idea is that in a chaotic system, small disturbances grow exponentially fast, rendering long-term prediction impossible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"These, then, are the defining features of chaos: erratic, seemingly random behavior in an otherwise deterministic system; predictability in the short run, because of the deterministic laws; and unpredictability in the long run, because of the butterfly effect." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"An apparent paradox is that chaos is deterministic, generated by fixed rules which do not themselves involve any elements of change. We even speak of deterministic chaos. In principle, the future is completely determined by the past; but in practice small uncertainties, much like minute errors of measurement which enter into calculations, are amplified, with the effect that even though the behavior is predictable in the short term, it is unpredictable over the long term." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"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", 2004)

"Natural laws, and for that matter determinism, do not exclude the possibility of chaos. In other words, determinism and predictability are not equivalent. And what is an even more surprising rinding of recent chaos theory has been the discovery that these effects are observable in many systems which are much simpler than the weather. [...] Moreover, chaos and order (i.e., the causality principle) can be observed in juxtaposition within the same system. There may be a linear progression of errors characterizing a deterministic system which is governed by the causality principle, while (in the same system) there can also be an exponential progression of errors (i.e., the butterfly effect) indicating that the causality principle breaks down." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"[…] some systems (system is just a jargon for anything, like the swinging pendulum or the Solar System, or water dripping from a tap)  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)

"We often wonder why the more complex systems seem to indicate a preferred direction of time, or an arrow of time, whereas their elementary counterparts do not. […] This has to do with the if-then nature of physics questions. Anything we observe involves laws of motion but also particular initial conditions. […] The initial conditions are what make a situation look peculiar when we time reverse it." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"[…] we would like to observe that the butterfly effect lies at the root of many events which we call random. The final result of throwing a dice depends on the position of the hand throwing it, on the air resistance, on the base that the die falls on, and on many other factors. The result appears random because we are not able to take into account all of these factors with sufficient accuracy. Even the tiniest bump on the table and the most imperceptible move of the wrist affect the position in which the die finally lands. It would be reasonable to assume that chaos lies at the root of all random phenomena." (Iwo Białynicki-Birula & Iwona Białynicka-Birula, "Modeling Reality: How Computers Mirror Life", 2004)

"Much of chaos as a science is connected with the notion of ‘sensitive dependence on initial conditions.’ Technically, scientists term as ‘chaotic’ those nonrandom complicated motions that exhibit a very rapid growth of errors that, despite perfect determinism, inhibits any pragmatic ability to render accurate long-term prediction. […] The most important fact is that there is a discernibly precise ‘moment’, with a corresponding behavior, which is neither chaotic nor nonchaotic, at which this transition occurs. Yes, errors do grow, but only in a marginally predictable, rather than in an unpredictable, fashion. In this state of marginal predictability inheres embryonically all the seeds of the chaotic behavior to come. That is, this transitional point, the legitimate child of universality, without full-fledged sensitive dependence upon initial conditions, knows fully how to dictate to its progeny in turn how this latter phenomenon must unfold. For a certain range of possible behaviors of strongly nonlinear systems - specifically, this range surrounding the transition to chaos - the information obtained just at the transition point fully organizes the spectrum of behaviors that these chaotic systems can exhibit." (Ray Kurzweil, "The Singularity is Near", 2005)

"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)

"Chaos can leave statistical footprints that look like noise. This can arise from simple systems that are deterministic and not random. [...] The surprising mathematical fact is that most systems are chaotic. Change the starting value ever so slightly and soon the system wanders off on a new chaotic path no matter how close the starting point of the new path was to the starting point of the old path. Mathematicians call this sensitivity to initial conditions but many scientists just call it the butterfly effect. And what holds in math seems to hold in the real world - more and more systems appear to be chaotic." (Bart Kosko, "Noise", 2006)

"'Chaos' refers to systems that are very sensitive to small changes in their inputs. A minuscule change in a chaotic communication system can flip a 0 to a 1 or vice versa. This is the so-called butterfly effect: Small changes in the input of a chaotic system can produce large changes in the output. Suppose a butterfly flaps its wings in a slightly different way. can change its flight path. The change in flight path can in time change how a swarm of butterflies migrates." (Bart Kosko, "Noise", 2006)

"Linearity means that the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now. The mathematics of linear systems exhibits a simple geometry. The simplicity allows us to capture the essence of the problem. Nonlinear dynamics is concerned with the study of systems whose time evolution equations are nonlinear. If a parameter that describes a linear system is changed, the qualitative nature of the behavior remains the same. But for nonlinear systems, a small change in a parameter can lead to sudden and dramatic changes in both the quantitative and qualitative behavior of the system." (Wei-Bin Zhang, "Discrete Dynamical Systems, Bifurcations and Chaos in Economics", 2006)

"Physically, the stability of the dynamics is characterized by the sensitivity to initial conditions. This sensitivity can be determined for statistically stationary states, e.g. for the motion on an attractor. If this motion demonstrates sensitive dependence on initial conditions, then it is chaotic. In the popular literature this is often called the 'Butterfly Effect', after the famous 'gedankenexperiment' of Edward Lorenz: if a perturbation of the atmosphere due to a butterfly in Brazil induces a thunderstorm in Texas, then the dynamics of the atmosphere should be considered as an unpredictable and chaotic one. By contrast, stable dependence on initial conditions means that the dynamics is regular." (Ulrike Feudel et al, "Strange Nonchaotic Attractors", 2006)

"This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment." (Greg Rae, Chaos Theory: A Brief Introduction, 2006)

"Global stability of an equilibrium removes the restrictions on the initial conditions. In global asymptotic stability, solutions approach the equilibrium for all initial conditions. [...] In a study of local stability, first equilibrium solutions are identified, then linearization techniques are applied to determine the behavior of solutions near the equilibrium. If the equilibrium is stable for any set of initial conditions, then this type of stability is referred to as global stability." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)

"Sensitive dependence on initial conditions is one of the criteria necessary for showing a solution to a difference equation exhibits chaotic behavior." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)

"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)

"Thus, nonlinearity can be understood as the effect of a causal loop, where effects or outputs are fed back into the causes or inputs of the process. Complex systems are characterized by networks of such causal loops. In a complex, the interdependencies are such that a component A will affect a component B, but B will in general also affect A, directly or indirectly. A single feedback loop can be positive or negative. A positive feedback will amplify any variation in A, making it grow exponentially. The result is that the tiniest, microscopic difference between initial states can grow into macroscopically observable distinctions." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"Yet, with the discovery of the butterfly effect in chaos theory, it is now understood that there is some emergent order over time even in weather occurrence, so that weather prediction is not next to being impossible as was once thought, although the science of meteorology is far from the state of perfection." (Peter Baofu, "The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos", 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)

"Prior to the discovery of the butterfly effect it was generally believed that small differences averaged out and were of no real significance. The butterfly effect showed that small things do matter. This has major implications for our notions of predictability, as over time these small differences can lead to quite unpredictable outcomes. For example, first of all, can we be sure that we are aware of all the small things that affect any given system or situation? Second, how do we know how these will affect the long-term outcome of the system or situation under study? The butterfly effect demonstrates the near impossibility of determining with any real degree of accuracy the long term outcomes of a series of events." (Elizabeth McMillan, Complexity, "Management and the Dynamics of Change: Challenges for practice", 2008)

"The butterfly effect demonstrates that complex dynamical systems are highly responsive and interconnected webs of feedback loops. It reminds us that we live in a highly interconnected world. Thus our actions within an organization can lead to a range of unpredicted responses and unexpected outcomes. This seriously calls into doubt the wisdom of believing that a major organizational change intervention will necessarily achieve its pre-planned and highly desired outcomes. Small changes in the social, technological, political, ecological or economic conditions can have major implications over time for organizations, communities, societies and even nations." (Elizabeth McMillan, "Complexity, Management and the Dynamics of Change: Challenges for practice", 2008)

"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. […] Dynamical systems that exhibit sensitive dependence on initial conditions produce remarkably different solutions for two initial values that are close to each other. Sensitive dependence on initial conditions is one of the properties to exhibit chaotic behavior. In addition, at least one further implicit assumption is that the system is bounded in some finite region, i.e., the system cannot blow up. When one uses expanding dynamics, a way of pull-back of too much expanded phase volume to some finite domain is necessary to get chaos." (Péter Érdi, "Complexity Explained", 2008)

"Chaos has three primary features: unpredictability, boundedness, and sensitivity to initial conditions. Unpredictability means that a sequence of numbers that is generated from a chaotic function does not repeat. This principle is perhaps a matter of degree, because some of the numbers could look as though they are recurring only because they are rounded to a convenient number of decimal points. [...] Boundedness means that, for all the unpredictability of motion, all points remain within certain boundaries. The principle of sensitivity to initial conditions means that two points that start off as arbitrarily close together become exponentially farther away from each other as the iteration process proceeds. This is a clear case of small differences producing a huge effect." (Stephen J Guastello & Larry S Liebovitch, "Introduction to Nonlinear Dynamics and Complexity" [in "Chaos and Complexity in Psychology"], 2009)

"A system of equations is deemed most elegant if it contains no un- necessary terms or parameters and if the parameters that remain have a minimum of digits. [...] Just as one can find the most elegant set of parameters for a given system, it is possible to find the most elegant set of initial conditions within the basin of attraction or chaotic sea. However, it is usually more useful to have initial conditions that are close to the attractor to reduce the transients that would otherwise occur."  (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"Another property of bounded systems is that, unless the trajectory attracts to an equilibrium point where it stalls and remains forever, the points must continue moving forever with the flow. However, if we consider two initial conditions separated by a small distance along the direction of the flow, they will maintain their average separation forever since they are subject to the exact same flow but only delayed slightly in time. This fact implies that one of the Lyapunov exponents for a bounded continuous flow must be zero unless the flow attracts to a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"In a chaotic system, there must be stretching to cause the exponential separation of initial conditions but also folding to keep the trajectories from moving off to infinity. The folding requires that the equations of motion contain at least one nonlinearity, leading to the important principle that chaos is a property unique to nonlinear dynamical systems. If a system of equations has only linear terms, it cannot exhibit chaos no matter how complicated or high-dimensional it may be." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"Most systems in nature are inherently nonlinear and can only be described by nonlinear equations, which are difficult to solve in a closed form. Non-linear systems give rise to interesting phenomena such as chaos, complexity, emergence and self-organization. One of the characteristics of non-linear systems is that a small change in the initial conditions can give rise to complex and significant changes throughout the system. This property of a non-linear system such as the weather is known as the butterfly effect where it is purported that a butterfly flapping its wings in Japan can give rise to a tornado in Kansas. This unpredictable behaviour of nonlinear dynamical systems, i.e. its extreme sensitivity to initial conditions, seems to be random and is therefore referred to as chaos. This chaotic and seemingly random behaviour occurs for non-linear deterministic system in which effects can be linked to causes but cannot be predicted ahead of time." (Robert K Logan, "The Poetry of Physics and The Physics of Poetry", 2010)

"The main defining feature of chaos is the sensitive dependence on initial conditions. Two nearby initial conditions on the attractor or in the chaotic sea separate by a distance that grows exponentially in time when averaged along the trajectory, leading to long-term unpredictability. The Lyapunov exponent is the average rate of growth of this distance, with a positive value signifying sensitive dependence (chaos), a zero value signifying periodicity (or quasiperiodicity), and a negative value signifying a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"Complexity carries with it a lack of predictability different to that of chaotic systems, i.e. sensitivity to initial conditions. In the case of complexity, the lack of predictability is due to relevant interactions and novel information created by them." (Carlos Gershenson, "Understanding Complex Systems", 2011)

"The things that really change the world, according to Chaos theory, are the tiny things. A butterfly flaps its wings in the Amazonian jungle, and subsequently a storm ravages half of Europe." (Neil Gaiman, "Good Omens", 2011)

"The key characteristic of 'chaotic solutions' is their sensitivity to initial conditions: two sets of initial conditions close together can generate very different solution trajectories, which after a long time has elapsed will bear very little relation to each other. Twins growing up in the same household will have a similar life for the childhood years but their lives may diverge completely in the fullness of time. Another image used in conjunction with chaos is the so-called 'butterfly effect' – the metaphor that the difference between a butterfly flapping its wings in the southern hemisphere (or not) is the difference between fine weather and hurricanes in Europe." (Tony Crilly, "Fractals Meet Chaos" [in "Mathematics of Complexity and Dynamical Systems"], 2012)

"The most basic tenet of chaos theory is that a small change in initial conditions - a butterfly flapping its wings in Brazil - can produce a large and unexpected divergence in outcomes - a tornado in Texas. This does not mean that the behavior of the system is random, as the term 'chaos' might seem to imply. Nor is chaos theory some modern recitation of Murphy’s Law ('whatever can go wrong will go wrong'). It just means that certain types of systems are very hard to predict." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)

"History is often the tale of small moments - chance encounters or casual decisions or sheer coincidence - that seem of little consequence at the time, but somehow fuse with other small moments to produce something momentous, the proverbial flapping of a butterfly's wings that triggers a hurricane." (Scott Anderson, "Lawrence in Arabia: War, Deceit, Imperial Folly and the Making of the Modern Middle East", 2013)

"One of the remarkable features of these complex systems created by replicator dynamics is that infinitesimal differences in starting positions create vastly different patterns. This sensitive dependence on initial conditions is often called the butterfly-effect aspect of complex systems - small changes in the replicator dynamics or in the starting point can lead to enormous differences in outcome, and they change one’s view of how robust the current reality is. If it is complex, one small change could have led to a reality that is quite different." (David Colander & Roland Kupers, "Complexity and the art of public policy : solving society’s problems from the bottom up", 2014)

"The sensitivity of chaotic systems to initial conditions is particularly well known under the moniker of the 'butterfly effect', which is a metaphorical illustration of the chaotic nature of the weather system in which 'a flap of a butterfly’s wings in Brazil could set off a tornado in Texas'. The meaning of this expression is that, in a chaotic system, a small perturbation could eventually cause very large-scale difference in the long run." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A system governed by a deterministic theory can only evolve along a single trajectory - namely, that dictated by its laws and initial conditions; all other trajectories are excluded. Symmetry principles, on the other hand, fit the freedom-inducing model. Rather than distinguishing what is excluded from what is bound to happen, these principles distinguish what is excluded from what is possible. In other words, although they place restrictions on what is possible, they do not usually determine a single trajectory." (Yemima Ben-Menahem, "Causation in Science", 2018)

"Chaos theory is a branch of mathematics focusing on the study of chaos - dynamical systems whose random states of disorder and irregularities are governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of complex, chaotic systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is a sensitive dependence on initial conditions)." (Nima Norouzi, "Criminal Policy, Security, and Justice in the Time of COVID-19", 2022)

On Extrema II (Extrema in Context)

“As in mathematics, when there is no maximum nor minimum, in short nothing distinguished, everything is done equally, or when that is not nothing at all is done: so it may be said likewise in respect of perfect wisdom, which is no less orderly than mathematics, that if there were not the best (optimum) among all possible worlds, God would not have produced any.” (Gottfried W Leibniz, “Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil”, 1710)

“For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.” (Leonhard Euler, “De Curvis Elasticis”, 1744)

“Like a great poet, Nature produces the greatest results with the simplest means.” (Heinrich Heine)

“The triumph of a theory is to embrace the greatest number and the greatest variety of facts.” (Charles A Wurtz, “A History of Chemical Theory from the Age of Lavoisier to the Present Time”, 1869) 

“It is the constant aim of the mathematician to reduce all his expressions to their lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language.” (James J Sylvester, 1877)

“In pure mathematics the maximum of detachment appears to be reached: the mind moves in an infinitely complicated pattern, which is absolutely free from temporal considerations. Yet this very freedom – the essential condition of the mathematician’s activity – perhaps gives him an unfair advantage. He can only be wrong – he cannot cheat.” (Kytton Strachey, “Portraits in Miniature”, 1931)

“[…] science, properly interpreted, is not dependent on any sort of metaphysics. It merely attempts to cover a maximum of facts by a minimum of laws.” (Herbert Feigl, “Naturalism and Humanism”, American Quarterly, Vol. 1, No. 2, 1949)

“The concept of the ‘singleness of the superlative’ is simple: no problem in dynamics can be properly formulated in terms of more than one superlative, whether the superlative in question is stated as a minimum or as a maximum (e.g., a minimum expenditure of work can also be stated as a maximum economy of work). If the problem has more than one superlative, the problem itself becomes completely meaningless and indeterminate.” (George Kingsley Zipf, “Human Behavior and the Principle of Least Effort: An Introduction of Human Ecology”, 1949)

“To a considerable degree science consists in originating the maximum amount of information with the minimum expenditure of energy. Beauty is the cleanness of line in such formulations along with symmetry, surprise, and congruence with other prevailing beliefs.” (Edward O Wilson, “Biophilia”, 1984)

“The main goal of physics is to describe a maximum of phenomena with a minimum of variables.” (CERN Courier)

On Extrema I

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of the great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." (Herbert G Wells, "Mankind in the Making", 1903)

"According to Leibniz our world is the best possible. That is why its laws can be described by extremal principles." (Carl L Siegel)

"In Continuity, it is impossible to distinguish phenomena at their merging-points, so we look for them at their extremes." (Charles Fort, "The Book of the Damned", 1919)

"Change is most sluggish at the extremes precisely because the derivative is zero there." (Steven Strogatz, The Joy of X: A Guided Tour of Mathematics, from One to Infinity, 2012)

"Most practical questions can be reduced to problems of largest and smallest magnitudes […] and it is only by solving these problems that we can satisfy the requirements of practice which always seeks the best, the most convenient." (Pafnuty L Chebyshev)

"[…] nothing takes place in the world whose meaning is not that of some maximum or minimum." (Leonhard Euler)

"The world is not dialectical - it is sworn to extremes, not to equilibrium, sworn to radical antagonism, not to reconciliation or synthesis." (Jean Baudrillard)

"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler)

"We shall consider the simplest maximum and minimum problem that points to a natural transition from functions of a finite number of variables to magnitudes that depend on an infinite number of variables." (Vito Volterra)

"When a quantity is greatest or least, at that moment its flow neither increases nor decreases." (Isaac Newton)

Misquoted: Herbert G Wells on Mathematical Literacy

"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write."

The above quote on statistical literacy is often attributed to Herbert G Wells though it belongs to the statistician Samuel S Wilks, who in a 1951 presidential address was paraphrasing Wells:

"Perhaps H. G. Wells was right when he said ‘statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write’!" [4]

The original quote comes from “Mankind in the Making”, first published in 1903 (and not in 1911 as Wikipedia states):

"The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." [1]

Even if Wells mentions averages, maxima and minima, tools of statistics, the text refers to mathematical analysis and not statistics. Wilk’s paraphrasing makes sense in nowadays contexts, and seems somehow natural, even if statistical literacy is more about understanding and (critically) evaluating statements that involve rates and percentages.

Another paraphrasing of the same quote and probably closer to the essence of statistical literacy can be found in George A Lundberg paper published in 1940, however without giving credit to Wells:

"The time is perhaps at hand when it will be recognized that for intelligent living in modern society it is as necessary to be able to think in averages, percentages, and deviations as it is to be able to read and write." [2]

Previous Post <<||>> Next Post

References:
[1] “Mankind in the Making”, by Herbert G Wells, 1903 [Source]
[2] “Statistics in Modern Social Thought”, by George A Lundberg [in “Contemporary Social Theory”, Ed. by H. E. Barnes, H. Becker & F. Becker, 1940] [Source]
[3] “The H. G. Wells Quote on Statistics: A Question of Accuracy”, by James W Tankard Jr., Historia Mathematics 6, 1979 [Source]
[4] “Undergraduate Statistical Education”, by  Samuel S Wilks, Journal of the American Statistical Association, Vol. 46, 1951 [Source]

Infinite, Nature and Mathematics

“But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.” (Aristotle, “Generation of Animals”)

 “There is a single general space, a single vast immensity which we may freely call Void: in it are innumerable globes like this on which we live and grow; this space we declare to be infinite, since neither reason, convenience, sense-perception nor nature assign it a limit.” (Giordano Bruno)

"Just as the stone thrown into the water becomes the centre and cause of various circles, [so] the sound made in the air spreads out in circles and fills the surrounding parts with an infinite number of images of itself." (Leonardo da Vinci)

"I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author." (Gottfried W Leibniz)

“What is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything. Since he is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret; he is equally incapable of seeing the Nothing from which he was made, and the Infinite in which he is swallowed up.” (Blaise Pascal, "Pensées", 1670)  

"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)

“In an infinite number universe, every point can be regarded as the center, because every point has an infinite of stars on each side of it.” (Stephen Hawking, "A Brief History of Time", 1988)