♾️On Complex Numbers XVIII

"I consider it as one of the most important steps made by Analysis in the last period, that of not being bothered any more by imaginary quantities, and to be able to submit them to calculus, in the same way as the real ones." (Joseph-Louis de Lagrange, [letter to Antonio Lorgna] 1777)

"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages." (Carl F Gauss, [letter to Bessel] 1811)

"Without doubt one of the most characteristic features of mathematics in the last century is the systematic and universal use of the complex variable. Most of its great theories received invaluable aid from it, and many owe their very existence to it." (James Pierpont, "History of Mathematics in the Nineteenth Century", Congress of Arts and Sciences Vol. 1, 1905)

"There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology." (Paul A M Dirac, [Lecture delivered on presentation of the James Scott prize] 1939)

"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematican should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Algebraic geometry uses the geometric intuition which arises from looking at varieties over the complex and real case to deduce important results in arithmetic algebraic geometry where the complex number field is replaced by the field of rational numbers or various finite number fields." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"The very idea of raising a number to an imaginary power may well have seemed to most of the era’s mathematicians like asking the ghost of a late amphibian to jump up on a harpsichord and play a minuet." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Today it’s easy to see the beauty of i, thanks, among other things, to its prominence in mathematics’ most beautiful equation. Thus, it may seem strange that it was once regarded as akin to a small waddling gargoyle. Indeed, the simplicity of its definition suggests unpretentious elegance: i is just the square root of −1. But as with many definitions in mathematics, i’s is fraught with provocative implications, and the ones that made it a star in mathematics weren’t apparent until long after it first came on the scene." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

On Continuity XVI (The Beginnings)

"Since [...] nature is a principle of motion and mutation [...] it is necessary that we should not be ignorant of what motion is [...] But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. [..] frequently [...] those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous." (Aristotle, "Physics", cca. 350 BC)

"Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two. […] Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one."(Aristotle, "Physics", cca. 350 BC)

"When what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as a part in a whole. But when the thing is separate or in contact, it is immediately ‘in’ the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which touch are coincident." (Aristotle, "Physics", cca. 350 BC)

"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD) 

"But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent." (William of Ockham, cca. 1320)

"It is established that every continuum has further parts, and not so many parts finite in number that there are not further parts, and has all its parts actually and simultaneously, and therefore every continuum has simultaneously and actually infinitely many parts." (Gregory of Rimini [Gregorii Ariminensis], "Lectura super primum et secundum sententiarum", cca. 1350)

On Randomness XIX (Chaos II)

"The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results. Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical find randomness of the natural environment." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"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)

"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"Often, we use the word random loosely to describe something that is disordered, irregular, patternless, or unpredictable. We link it with chance, probability, luck, and coincidence. However, when we examine what we mean by random in various contexts, ambiguities and uncertainties inevitably arise. Tackling the subtleties of randomness allows us to go to the root of what we can understand of the universe we inhabit and helps us to define the limits of what we can know with certainty." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"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)

"There are only patterns, patterns on top of patterns, patterns that affect other patterns. Patterns hidden by patterns. Patterns within patterns. If you watch close, history does nothing but repeat itself. What we call chaos is just patterns we haven't recognized. What we call random is just patterns we can't decipher. what we can't understand we call nonsense. What we can't read we call gibberish. There is no free will. There are no variables." (Chuck Palahniuk, "Survivor", 1999)

"Heat is the energy of random chaotic motion, and entropy is the amount of hidden microscopic information." (Leonard Susskind, "The Black Hole War", 2008)

"Chaos is impatient. It's random. And above all it's selfish. It tears down everything just for the sake of change, feeding on itself in constant hunger. But Chaos can also be appealing. It tempts you to believe that nothing matters except what you want." (Rick Riordan, "The Throne of Fire", 2011)

"A system in which a few things interacting produce tremendously divergent behavior; deterministic chaos; it looks random but its not." (Chris Langton)

♾️On Complex Numbers XVI

"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power."  (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

 "√-1 is take for granted today. No serious mathematician would deny that it is a number. Yet it took centuries for √-1 to be officially admitted to the pantheon of numbers. For almost three centuries, it was controversial; mathematicians didn't know what to make of it; many of them worked with it successfully without admitting its existence. […] Primarily for cognitive reasons. Mathematicians simply could not make it fit their idea of what a number was supposed to be. A number was supposed to be a magnitude. √-1 is not a magnitude comparable to the magnitudes of real numbers. No tree can be √-1 units high. You cannot owe someone √-1 dollars. Numbers were supposed to be linearly ordered. √-1 is not linearly ordered with respect to other numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"From a formal perspective, much about complex numbers and arithmetic seems arbitrary. From a purely algebraic point of view, i arises as a solution to the equation x^2+1=0. There is nothing geometric about this - no complex plane at all. Yet in the complex plane, the i-axis is 90° from the x-axis. Why? Complex numbers in the complex plane add like vectors. Why? Complex numbers have a weird rule of multiplication […]" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"[…] i is not a real number-not ordered anywhere relative to the real numbers! In other words, it does not even have the central property of ‘numbers’, indicating a magnitude that can be linearly compared to all other magnitudes. You can see why i has been called imaginary. It has almost none of the properties of the small natural numbers-not subitizability, not groupability, and not even relative magnitude. If i is to be a number, it is a number only by virtue of closure and the laws of arithmetic." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The complex plane is just the 90° rotation plane-the rotation plane with the structure imposed by the 90° Rotation metaphor added to it. Multiplication by i is 'just' rotation by 90°. This is not arbitrary; it makes sense. Multiplication by-1 is rotation by 180°. A rotation of 180° is the result of two 90° rotations. Since i times i is -1, it makes sense that multiplication by i should be a rotation by 90°, since two of them yield a rotation by 180°, which is multiplication by -1." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Negative numbers posed some of the same quandaries that the imaginary numbers did to Renaissance mathematicians - they didn’t seem to correspond to quantities associated with physical objects or geometrical figures. But they proved less conceptually challenging than the imaginaries. For instance, negative numbers can be thought of as monetary debts, providing a readily grasped way to make sense of them." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the 'i times π' power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The fact that multiplying positive 4i times positive 4i yields negative 16 seems like saying that the friend of my friend is my enemy. Which in turn suggests that bad things would happen if i and its offspring were granted citizenship in the number world. Unlike real numbers, which always feel friendly toward the friends of their friends, the i-things would plainly be subject to insane fits of jealousy, causing them to treat numbers that cozy up to their friends as threats. That might cause a general breakdown of numerical civility." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Basis real and imaginary numbers have eternal and necessary reality. Complex numbers do not. They are temporal and contingent in the sense that for complex numbers to exist, we first have to carry out an operation: adding basis real and imaginary numbers together. Complex numbers therefore do not exist in their own right. They are constructed. They are derived. Symmetry breaking is exactly where constructed numbers come into existence. The very act of adding a sine wave to a cosine wave is the sufficient condition to create a broken symmetry: a complex number. The 'Big Bang', mathematically, is simply where a perfect array of basis sine and cosine waves start entering into linear combinations, creating a chain reaction, an 'explosion', of complex numbers - which corresponds to the 'physical' universe." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

♾️On Complex Numbers VII

“The remark which you make concerning roots that can not be extracted, and containing imaginary quantities which when added together give none the less a real quantity, is surprising and entirely new. One would never have believed that √(1 + √-3) + √(1 - √- 3) would make √6, and there is something hidden in this which is incomprehensible.” (Christaan Huygens, [letter to Gottfried W Leibniz] cca. 1670)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)

“Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality.” (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)

“For it ought to be considered that both –b   and –c  , as they stand alone, are, in some Sense, as much impossible Quantities as √(-b)  and √(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination.” (Thomas Simpson, “A Treatise of Algebra”, 1745) 

“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential."  (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748)

“Moreover, the whole method has the essential disadvantage that it occupies the mind with the distinction of a great number of cases that can be recognized only by inner intuition, and thus neutralizes an important part of that which algebra is supposed to accomplish, which is relieving the power of inner intuition. Finally, in such a treatment algebra loses a great part of the generality that it can obtain by the mutual connection of different problems, which becomes evident so easily when one uses isolated negative quantities. [...] Since imaginary quantities have to occur, science would certainly not win that much by avoiding negative quantities than it would lose in terms of clarity and generality.” (Johann P W Stein,  “Die Elemente der Algebra: Erster Cursus”, 1828) 

"Originally assuming the concept of the absolute integers, it extended its domain step by step; integers were supplemented by fractions, rational numbers by irrational numbers, positive numbers by negative numbers, and real numbers by imaginary numbers. This advance, however, occurred initially with a fearfully hesitant step. The first algebraists preferred to call negative roots of equations false roots, and it is precisely these where the problem to which they refer was always termed in such a way as to ensure that the nature of the quantity sought did not admit any opposite.” (Carl F Gauss, “Theoria residuorum biquadraticum. Commentatio secunda. [Selbstanzeige]”, Göttingische gelehrte Anzeigen 23 (4), 1831)

“[T]he notion of a negative magnitude has become quite a familiar one […] But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry. […] considering the prominent position which the notion occupies-say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it.” (Arthur Cayley, [address before the meeting of the British Association at Southport] 1870) 

“A satisfactory theory of the imaginary quantities of ordinary algebra, which is essentially a simple case of multiple algebra, with difficulty obtained recognition in the first third of this century. We must observe that this double algebra, as it has been called, was not sought for or invented; - it forced itself, unbidden, upon the attention of mathematicians, and with its rules already formed. [...] But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.” (Josiah W Gibbs, “On multiple Algebra”, Proceedings of the American Association for the Advancement of Science Vol. 35, 1886)

♾️On Complex Numbers VI

"We do not perceive any quantity such as that its square is negative!" (Bhāskara II, "Bijaganita", 12th century)

"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)

“Even zero and complex numbers are not excluded from the signification of a variable quantity.” (Leonhard Euler, “Introductio in Analysin Infinitorum” Vol. I, 1748)

“Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers” (Carl F Gauss, “Disquisitiones arithmeticae” [“Arithmetical Researches”], 1801)

"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question. but surely this is evident that since they lead to right conclusions they must have a logic! […] Till the doctrines of negative and imaginary quantities are better taught than they are at present taught in the University of Cambridge, I agree with you that they had better not be taught [...]" (Robert Woodhouse, [letter to Baron Meseres] 1801)

“At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial –or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications. “ (Carl F Gauss, [letter to Bessel,] 1811)

"Our general arithmetic, so far surpassing in extent the geometry of the ancients, is entirely the creation of modern times. Starting originally from the notion of absolute integers, it has gradually enlarged its domain. To integers have been added fractions, to rational quantities the irrational, to positive the negative .and to the real the imaginary. This advance, however, has always been made at first with timorous and hesitating step. The early algebraists called the negative roots of equations false roots, and these are indeed so when the problem to which they relate has been stated in such a form that the character of the quantity sought allows of no opposite. But just as in general arithmetic no one would hesitate to admit fractions, although there are so many countable things where a fraction has no meaning, so we ought not to deny to, negative numbers the rights accorded to positive simply because innumerable things allow no opposite. The reality of negative numbers is sufficiently justified since in innumerable other cases they find an adequate substratum. This has long been admitted, but the imaginary quantities - formerly and occasionally now, though improperly, called impossible-as opposed to real quantities are still rather tolerated than fully naturalized, and appear more like an empty play upon symbols to which a thinkable substratum is denied unhesitatingly by those who would not depreciate the rich contribution which this play upon symbols has made to the treasure of the relations of real quantities." (Carl F Gauss, "Theoria residuorum biquadraticorum, Commentatio secunda", Göttingische gelehrte Anzeigen, 1831)

“The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting.” (Bernhard Riemann, “Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse”, 1851)

“The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention […] And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning [...] It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression.” (William R Hamilton, “Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method… “, 1853) 

“The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.” (Bernhard Riemann, “Gesammelte Mathematische Werke”)

Bernhard Riemann - Collected Quotes

"The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting." (Bernhard Riemann, "Foundations of a general theory of functions of a variable complex magnitude", 1851)

"But even if it is of interest to grasp the possibility of this mode of treatment of geometry, the execution of the latter would be extremely fruitless, as by this means we would not find new theorems, and what seems simple and clear in the presentation in space, would thereby get involved and difficult." (Bernhard Riemann, "Nachlass", cca. 1852-1853) [on Euclid’s parallels axiom]

"Therefore, one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginery quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann, "Nachlass", cca. 1852-1853)

"If in [a continuous manifold] one passes from a certain [point] in a definite way to another, the [points] passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, than all the [points] so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forward or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"[…] let us take a continuous function of position within the given the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness. Every system of points where the function has a constant value, forms then a continuous manifoldness of fewer dimensions than the given one. These manifoldnesses pass over continuously into one another as the function changes; we may therefore assume that out of one of them the others proceed, and speaking generally this may occur in such a way that each point passes over into a definite point of the other; the cases of exception (the study of which is important) may here be left unconsidered. […] By repeating then this operation n times, the determination of position in an n-ply extended manifoldness is reduced to n determinations of quantity […]."  (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case we can only determine the more or less and not the how much." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics." (Bernhard Riemann, 1854)

"A theory of those functions [algebraic, circular or exponential, elliptical and Abelian] on the basis of the foundations here established would determine the configuration of the function (i.e., its value for each value of the argument) independently of any definition by means of operations [analytical expressions]. Therefore one would add, to the general notion of a function of a complex variable, only those characteristics that are necessary for determining the function, and only then would one go over to the different expressions that the function can be given." (Bernhard Riemann, "Theorie der Abel'schen Functionen", Journal für die reine und angewandte Mathematik 54, 1857)

"One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation." (Bernhard Riemann, "On the Number of Prime Numbers less than a Given Quantity" ["Ueber die Anzahl der Primzahlen unter einer gegebenen Gröosse"], Monatsberichte der Berliner Akademie, 1859) 

"By the concepts through which we conceive nature, not only are our perceptions complemented in each moment, but also future perceptions are singled out as necessary, or, insofar as the conceptual system is not complete enough for that purpose, determined as probable […] The conceptual systems which underlie them now [the exact sciences], have been formed by gradual change of older conceptual systems, and the reasons which resulted in new modes of explanation, can be reduced to contradictions or improbabilites, which turned up in the older modes of explanations." (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

"For an understanding of Nature, questions about the infinitely large are idle questions. It is different, however, with questions about the infinitely small. Our knowledge of their causal relations depends essentially on the precision with which we succeed in tracing phenomena on the infinitesimal level." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)

"The method applied by Newton to the grounding of the Infinitesimal Calculus, and which since the beginning of this century has been recognised by the best mathematicians as the only one that furnishes sure results, is the method of limits. The method consists in this, viz., instead of considering a continuous transition from one value of a quantity to another, from one position to another, or, speaking generally, from one determination of a concept to another, one considers in the first place a transition through a finite number of intervals and then allows the number of these intervals to increase so that the distances of two successive points of division all decrease infinitely." (Bernhard Riemann, "General Relation of the Concept System of Thesis and Antithesis", Gesammelte Mathematische Werke", 1876)

"The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)

“With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses.” (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

"Just give me the insights. I can always come up with the proofs!" (Bernhard Riemann)

The Infinite and Its Difficulties II

“The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.” (Thomas Aquinas, “Summa theologia”, 13th century) 

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this […] is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.” (Galileo Galilei, "Two New Sciences", 1638)

“Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; Imagine what they are when combined. In spite of this men cannot refrain from discussing them.” (Galileo Galilei, "Two New Sciences", 1638)

“Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power.” (Thomas Hobbes, "Of Man", 1658)

“Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” (Blaise Pascal, "Pensées", 1670)

“Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

 “A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined.” (George B Mathews, “Theory of Numbers”, 1892)

“Like children who are not permitted to do certain things, we are not permitted by nature to think in terms of infinity.” (Robert Tuttle Morris, “Microbes and Men”, 1916)

 "The infinite in mathematics is always unruly unless it is properly treated."  (Edward Kasner & James Newman, “Mathematics and the Imagination”, 1940)

“I am incapable of conceiving infinity, and yet I do not accept finity.” (Simone de Beauvoir, “La Vieillesse”, 1970)

On Measurement (Unsourced)

"An essential defect of previous presentations of geometry is that one usually returns to discrete numerical ratios in the treatment of similarity theory. This procedure, which at first seems simple, soon enough becomes entangled in complicated investigations concerning incommensurable magnitudes, as we have already hinted above; and the initial impression of simplicity is revenged upon problems of a purely geometrical procedure by the appearance of a set of difficult investigations of a completely heterogeneous type, which shed no light on the essence of spatial magnitudes. To be sure, one cannot eliminate the problem of measuring spatial magnitudes and expressing the results of these measurements numerically. But this problem cannot originate in geometry itself, but only arises when one, equipped on the one hand with the concept of number and on the other with spatial perceptions, applies them to that problem, and thus in a mixed branch that one can in a general sense call by the name 'theory of measurement' […] To relegate the theory of similarity, and even that of surface area, to this branch as has previously occurred (not to the form but to the substance) is to steal the essential content from what is called (pure) geometry." (Hermann G Grassmann)

"For although it is certainly true that quantitative measurements are of great importance, it is a grave error to suppose that the whole of experimental physics can be brought under this heading. We can start measuring only when we know what to measure: qualitative observation has to precede quantitative measurement, and by making experimental arrangements for quantitative measurements we may even eliminate the possibility of new phenomena appearing." (Heinrich B G Casimir)

"Science depends upon measurement, and things not measurable are therefore excluded, or tend to be excluded, from its attention." (Arthur J Balfour)

"Sometimes there are heated arguments at meetings about how to interpret data. When you have very few facts, fully interpreting them can give rise to three or four interpretations - within the error bars, the uncertainties in the measurements. You get people adhering to one or the other interpretation for a while, and that’s not based on fact because there are not enough facts. Eventually more facts are gathered and it becomes clear what the answer is, and everybody agrees. In the end you have a new result. That’s the wonderful thing about science, that you can only find in science. There is a point when there is no doubt anymore. There is usually a lot of emotional stress before you get rid of some former idea. There may be a few crackpots who fight it, but if the evidence is good, eventually all accept it. I think that’s wonderful. One of the best things about science is that there are some objective answers." (Kristina Katsaros)

"There are two possible outcomes: if the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery." (Enrico Fermi)

"Through measure to knowing is the motto I would like to write above every physics laboratory." (Heike K Onnes)

"[…] we must not measure the simplicity of the laws of nature by our facility of conception; but when those which appear to us the most simple, accord perfectly with observations of the phenomena, we are justified in supposing them rigorously exact." (Pierre-Simon Laplace)

🦋On 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)

"[...] the Rössler attractor stretches and folds trajectories in phase space. [...] Stretching is responsible for the butterfly effect; when a stretch occurs, nearby trajectories are pushed farther apart. Folding of some sort is necessary to keep orbits bounded. If there was not any folding, orbits would tend toward infinity. It is thus not surprising that we observe stretching and folding in a three-dimensional chaotic system such as the Rössler equations in addition to the one-dimensional logistic equation." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 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)