On Mathematical Thinking

"It should therewith be remembered that as mathematics studies neutral complexes, mathematical thinking is an organizational process and hence its methods, as well as the methods of all other sciences and those of any practice, fall within the province of a general tektology. Tektology is a unique science which must not only work out its own methods by itself but must study them as well; therefore it is the completion of the cycle of sciences." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

“Philosophy in its old form could exist only in the absence of engineering, but with engineering in existence and daily more active and far reaching, the old verbalistic philosophy and metaphysics have lost their reason to exist. They were no more able to understand the ‘production’ of the universe and life than they are now able to understand or grapple with 'production' as a means to provide a happier existence for humanity. They failed because their venerated method of ‘speculation’ can not produce, and its place must be taken by mathematical thinking. Mathematical reasoning is displacing metaphysical reasoning. Engineering is driving verbalistic philosophy out of existence and humanity gains decidedly thereby.” (Alfred Korzybski,  “Manhood of Humanity”, 1921)

"We can now return to the distinction between language and symbolism. A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking." (Robin G Collingwood, "The Principles of Art", 1938)

"Figures and symbols are closely connected with mathematical thinking, their use assists the mind. […] At any rate, the use of mathematical symbols is similar to the use of words. Mathematical notation appears as a sort of language, une langue bien faite, a language well adapted to its purpose, concise and precise, with rules which, unlike the rules of ordinary grammar, suffer no exception." (George Pólya, "How to solve it", 1945)

"Geometrical intuition plays an essential role in contemporary algebro-topological and geometric studies. Many profound scientific mathematical papers devoted to multi-dimensional geometry use intensively the 'visual slang' such as, say, 'cut the surface', 'glue together the strips', 'glue the cylinder', 'evert the sphere' , etc., typical of the studies of two and three-dimensional images. Such a terminology is not a caprice of mathematicians, but rather a 'practical necessity' since its employment and the mathematical thinking in these terms appear to be quite necessary for the proof of technically very sophisticated results."(Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed [...] were made explicit when logic was formalized early in the this century [...] These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a [...] conjecture. [...] Heuristic arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. [...] Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Topology is a child of twentieth century mathematical thinking. It allows us to consider the shape and structure of an object without being wedded to its size or to the distances between its component parts. Knot theory, homotopy theory, homology theory, and shape theory are all part of basic topology. It is often quipped that a topologist does not know the difference between his coffee cup and his donut - because each has the same abstract 'shape' without looking at all alike." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"The chief feature of mathematical thinking is that it is logical. Certainly there is room for intuition in mathematics, and even room for guessing. But, in the end, we understand a mathematical situation and/or solve a problem by being very logical. Logic makes the process dependable and reproducible. It shows that what we are producing is a verifiable truth." (Steven G Krantz," Essentials of Mathematical Thinking", 2018)

"Musical form is close to mathematics - not perhaps to mathematics itself, but certainly to something like mathematical thinking and relationships." (Igor Stravinsky)

Edmund C Berkeley - Collected Quotes

"A machine can handle information; it can calculate, conclude, and choose; it can perform reasonable operations with information. A machine. therefore, can think." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"Another scientific problem to which new machinery for handling information applies is the problem of understanding human beings and their behavior. This increased understanding may lead to much wiser dealing with human behavior." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"As everyone knows, it is not always easy to think. By thinking, we mean computing, reasoning, and other handling of information. By information we mean collections of ideas - physically, collections of marks that have meaning. By handling information, we mean proceeding logically from some ideas to other ideas - physically, changing from some marks to other marks in ways that have meaning." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"For at least two centuries, solving differential equations to answer physical problems has been a main job for mathematicians. Mathematics is supposed to be logical, and perhaps you would think this would be easy. But mathematicians have been unable to solve a great many differential equations; only here and there, as if by accident, could they solve one. So they often wished for better methods in order to make the job easier." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"From a narrow point of view, a machine that only thinks produces only information. It takes in information in one state, and it puts out information in another state. From this viewpoint, information in itself is harmless; it is just an arrangement of marks; and accordingly, a machine that thinks is harmless, and no control is necessary." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"Now when we speak of a machine that thinks, or a mechanical  brain, what do we mean? Essentially, a mechanical brain is a machine that handles information, transfers information automatically from one part of the machine to another, and has a flexible control over the sequence of its operations. No human being is needed around such a machine to pick up a physical piece of information produced in one part of the machine, personally move it to another part of the machine, and there put it in again. Nor is any human being needed to give the machine instructions from minute to minute. Instead, we can write out the whole program to solve a problem, translate the program into machine language, and put the program into the machine." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"One of the operations of algebra that is important for a mechanical brain is approximation, the problem of getting close to the right value of a number. [...] Another important operation of algebra is interpolation, the problem of putting values smoothly in between other values."  (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"Probably the foremost problem which machines that think can solve is automatic control over all sorts of other machines. This involves controlling a machine that is running so that it will do the right thing at the right time in response to information." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"Programming - the way to give instructions to machines [...] (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"The amount of human effort needed to handle information correctly depends very much on the properties of the physical equipment expressing the information, although the laws of correct reasoning are independent of the equipment." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"These machines are similar to what a brain would be if it were made of hardware and wire instead of flesh and nerves. It is therefore natural to call these machines mechanical brains. Also, since their powers are like those of a giant, we may call them giant brains." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"Understanding an idea is basically a standard process. First, we find the name of the idea, a word or phrase that identifies it. Then, we collect true statements about the idea. Finally, we practice using them. The more true statements we have gathered, and the more practice we have had in applying them, the more we understand the idea." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"We can even imagine what new machinery for handling information may some day become: a small pocket instrument that we carry around with us, talking to it whenever we need to, and either storing information in it or receiving information from it." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"A computer is a person or machine that is able to take in information (problems and data), perform reasonable operations on the iformation, and put out answers. A computer is identified by the fact that it (or he) handles information reasonably." (Edmund C Berkeley & Lawrence Wainwright, Computers: Their Operation and Applications", 1956)

"Information is a set of marks that have meaning. Physically, the set of marks is a set of physical objects or a set of arrangements of some physical equipment. Then, out of this set, a selection is made in order to communicate, to convey meaning. For meaning to exist, there has to be a society of at least two persons or machines, a society that requires communication, that desires to convey meaning. By convention, the society establishes the meaning of the marks." (Edmund C Berkeley & Lawrence Wainwright, Computers: Their Operation and Applications", 1956)

"The precision of a number is the degree of exactness with which it is stated, while the accuracy of a number is the degree of exactness with which it is known or observed. The precision of a quantity is reported by the number of significant figures in it." (Edmund C Berkeley & Lawrence Wainwright, Computers: Their Operation and Applications", 1956)

"The moment you have worked out an answer, start checking it - it probably isn't right." (Edmund C Berkeley, "Right Answers: A Short Guide for Obtaining Them", Computers and Automation, Vol. 18 (10), 1969)

"The World is more complicated than most of our theories make it out to be." (Edmund C Berkeley, "Right Answers: A Short Guide for Obtaining Them", Computers and Automation, Vol. 18 (10), 1969)

"There is no substitute for honest, thorough, scientific effort to get correct data (no matter how much it clashes with preconceived ideas). There is no substitute for actually reaching a correct chain of reasoning. Poor data and good reasoning give poor results. Good data and poor reasoning give poor results. Poor data and poor reasoning give rotten results." (Edmund C Berkeley, Computers and Automation, 1969)

"Most problems have either many answers or no answer. Only a few problems have one answer." (Edmund Berkeley, Computers and Automation, 1970)

On Knowledge (-1699)

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)

"Thinking is different from perceiving and is held to be in part imagination, in part judgement: we must therefore first mark off the sphere of imagination and then speak of judgement. If then imagination is that in virtue of which an image arises for us, excluding metaphorical uses of the term, is it a single faculty or disposition relative to images, in virtue of which we discriminate and are either in error or not? The faculties in virtue of which we do this are sense, opinion, knowledge, thought." (Aristotle, "De Anima", cca. 350 BC)

"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)

"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)

"We can get some idea of a whole from a part, but never knowledge or exact opinion. Special histories therefore contribute very little to the knowledge of the whole and conviction of its truth. It is only indeed by study of the interconnexion of all the particulars, their resemblances and differences, that we are enabled at least to make a general survey, and thus derive both benefit and pleasure from history." (Polybius, "The Histories", cca. 150 BC)

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular." (Proclus Lycaeus, cca 5th century)

"All knowledge or cognition possessed by creatures is limited. Infinite knowledge belongs solely to God, because of His infinite nature." (John of Salisbury, "Metalogicon", 1159)

"All things have a way of adding up together, so that one will become more proficient in any proposed branch of learning to the extent that he has mastered neighboring and related departments of knowledge." (John of Salisbury, "Metalogicon", 1159)

"In our acquisition of [scientific] knowledge, investigation is the first step, and comes before comprehension, analysis, and retention. Innate ability, although it proceeds from nature, is fostered by study and exercise. What is difficult when we first try it, becomes easier after assiduous practice, and once the rules for doing it are mastered, very easy, unless languor creeps in, through lapse of use or carelessness, and impedes our efficiency. This, in short, is how all the arts have originated: Nature, the first fundamental, begets the habit and practice of study, which proceeds to provide an art, and the latter, in turn, finally furnishes the faculty whereof we speak. Natural ability is accordingly effective. So, too, is exercise. And memory likewise, is effective, when employed by the two aforesaid. With the help of the foregoing, reason waxes strong, and produces the arts, which are proportionate to [man’s] natural talents." (John of Salisbury, "Metalogicon", 1159)

"There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured […] Of these sciences the gate and key is mathematics […] He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world." (Roger Bacon, "Opus Majus", 1267)

"There are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth unless the mind discovers it by the path of experience." (Roger Bacon, "Opus Majus", 1267)

"That faculty which perceives and recognizes the noble proportions in what is given to the senses, and in other things situated outside itself, must be ascribed to the soul. It lies very close to the faculty which supplies formal schemata to the senses, or deeper still, and thus adjacent to the purely vital power of the soul, which does not think discursively […] Now it might be asked how this faculty of the soul, which does not engage in conceptual thinking, and can therefore have no proper knowledge of harmonic relations, should be capable of recognizing what is given in the outside world. For to recognize is to compare the sense perception outside with the original pictures inside, and to judge that it conforms to them." (Johannes Kepler, "Harmonices Mundi" ["Harmony of the World"] , 1619)

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"[…] the highest probability amounts not to certainty, without which there can be no true knowledge." (John Locke, "An Essay Concerning Human Understanding", 1689)

On Equilibrium (1900-1919)

"When we study the structure of the atom, we shall arrive at the conclusion that it is an immense reservoir of energy solely constituted by a system of imponderable elements maintained in equilibrium by the rotations, attractions and repulsions of its component parts." (Gustave Le Bon, "The Evolution of Matter", 1907)

"All thinking is of disturbance, dynamical, a state of unrest tending towards equilibrium. It is all a mode of classifying and of criticising with a view of knowing whether it gives us, or is likely to give us, pleasure or no." (Samuel Butler, "Thinking - The Note-Books of Samuel Butler", 1912)

"The network of ideas remains and forms as it were a moving cobweb in which repose wriggles and tosses, incapable of finding a stable equilibrium." (Jean H Fabre, "The Life of the Fly", 1913)

"We rise from the conception of form to an understanding of the forces which gave rise to it [...] in the representation of form we see a diagram of forces in equilibrium, and in the comparison of kindred forms we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other." (D'Arcy Wentworth Thompson, "On Growth and Form" Vol. 2, 1917)

"A society in stable equilibrium is - by definition - one that has no history and wants no historians." (Henry Adams, "The Degradation of the Democratic Dogma", 1919)

"All biologic phenomena act to adjust: there are no biologic actions other than adjustments. Adjustment is another name for Equilibrium. Equilibrium is the Universal, or that which has nothing external to derange it." (Charles Fort, The Book of the Damned, 1919)

On Patterns (1990-1999)

"Mathematics is an exploratory science that seeks to understand every kind of pattern - patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns." (Lynn A Steen, "The Future of Mathematics Education", 1990)

"Phenomena having uncertain individual outcomes but a regular pattern of outcomes in many repetitions are called random. 'Random' is not a synonym for 'haphazard' but a description of a kind of order different from the deterministic one that is popularly associated with science and mathematics. Probability is the branch of mathematics that describes randomness." (David S Moore, "Uncertainty", 1990)

"Systems thinking is a framework for seeing interrelationships rather than things, for seeing patterns rather than static snapshots. It is a set of general principles spanning fields as diverse as physical and social sciences, engineering and management." (Peter Senge, "The Fifth Discipline", 1990)

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 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)

"In everyday language, the words 'pattern' and 'symmetry' are used almost interchangeably, to indicate a property possessed by a regular arrangement of more-or-less identical units […]” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

"Scientists have discovered many peculiar things, and many beautiful things. But perhaps the most beautiful and the most peculiar thing that they have discovered is the pattern of science itself. Our scientific discoveries are not independent isolated facts; one scientific generalization finds its explanation in another, which is itself explained by yet another. By tracing these arrows of explanation back toward their source we have discovered a striking convergent pattern - perhaps the deepest thing we have yet learned about the universe." (Steven Weinberg, "Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature", 1992)

"Searching for patterns is a way of thinking that is essential for making generalizations, seeing relationships, and understanding the logic and order of mathematics. Functions evolve from the investigation of patterns and unify the various aspects of mathematics." (Marilyn Burns, "About Teaching Mathematics: A K–8 Resource", 1992)

"Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"World view, a concept borrowed from cultural anthropology, refers to the culturally dependent, generally subconscious, fundamental organization of the mind. This conceptual organization manifests itself as a set of presuppositions that predispose one to feel, think, and act in predictable patterns." (Kenneth G Tobin, "The practice of constructivism in science education", 1993)

"[For] us to be able to speak and understand novel sentences, we have to store in our heads not just the words of our language but also the patterns of sentences possible in our language. These patterns, in turn, describe not just patterns of words but also patterns of patterns. Linguists refer to these patterns as the rules of language stored in memory; they refer to the complete collection of rules as the mental grammar of the language, or grammar for short." (Ray Jackendoff, "Patterns in the Mind", 1994)

"A neural network is characterized by A) its pattern of connections between the neurons (called its architecture), B) its method of determining the weights on the connections (called its training, or learning, algorithm), and C) its activation function." (Laurene Fausett, "Fundamentals of Neural Networks", 1994)

"At the other far extreme, we find many systems ordered as a patchwork of parallel operations, very much as in the neural network of a brain or in a colony of ants. Action in these systems proceeds in a messy cascade of interdependent events. Instead of the discrete ticks of cause and effect that run a clock, a thousand clock springs try to simultaneously run a parallel system. Since there is no chain of command, the particular action of any single spring diffuses into the whole, making it easier for the sum of the whole to overwhelm the parts of the whole. What emerges from the collective is not a series of critical individual actions but a multitude of simultaneous actions whose collective pattern is far more important. This is the swarm model." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Self-organization refers to the spontaneous formation of patterns and pattern change in open, nonequilibrium systems. […] Self-organization provides a paradigm for behavior and cognition, as well as the structure and function of the nervous system. In contrast to a computer, which requires particular programs to produce particular results, the tendency for self-organization is intrinsic to natural systems under certain conditions." (J A Scott Kelso, "Dynamic Patterns : The Self-organization of Brain and Behavior", 1995)

"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & ?K Nishijima , "Broken Symmetry: Selected Papers Of Y Nambu", 1995)

"Whatever the reasons, mathematics definitely is a useful way to think about nature. What do we want it to tell us about the patterns we observe? There are many answers. We want to understand how they happen; to understand why they happen, which is different; to organize the underlying patterns and regularities in the most satisfying way; to predict how nature will behave; to control nature for our own ends; and to make practical use of what we have learned about our world. Mathematics helps us to do all these things, and often it is indispensable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"If we are to have meaningful, connected experiences; ones that we can comprehend and reason about; we must be able to discern patterns to our actions, perceptions, and conceptions. Underlying our vast network of interrelated literal meanings (all of those words about objects and actions) are those imaginative structures of understanding such as schema and metaphor, such as the mental imagery that allows us to extrapolate a path, or zoom in on one part of the whole, or zoom out until the trees merge into a forest." (William H Calvin, "The Cerebral Code", 1996)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"The more complex the network is, the more complex its pattern of interconnections, the more resilient it will be." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"Mathematics can function as a telescope, a microscope, a sieve for sorting out the signal from the noise, a template for pattern perception, a way of seeking and validating truth. […] A knowledge of the mathematics behind our ideas can help us to fool ourselves a little less often, with less drastic consequences." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"Mathematics is a way of thinking that can help make muddy relationships clear. It is a language that allows us to translate the complexity of the world into manageable patterns. In a sense, it works like turning off the houselights in a theater the better to see a movie. Certainly, something is lost when the lights go down; you can no longer see the faces of those around you or the inlaid patterns on the ceiling. But you gain a far better view of the subject at hand." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Mathematics, in the common lay view, is a static discipline based on formulas taught in the school subjects of arithmetic, geometry, algebra, and calculus. But outside public view, mathematics continues to grow at a rapid rate, spreading into new fields and spawning new applications. The guide to this growth is not calculation and formulas but an open-ended search for pattern." (Lynn A Steen, "The Future of Mathematics Education", 1998)

"A neural network consists of large numbers of simple neurons that are richly interconnected. The weights associated with the connections between neurons determine the characteristics of the network. During a training period, the network adjusts the values of the interconnecting weights. The value of any specific weight has no significance; it is the patterns of weight values in the whole system that bear information. Since these patterns are complex, and are generated by the network itself (by means of a general learning strategy applicable to the whole network), there is no abstract procedure available to describe the process used by the network to solve the problem. There are only complex patterns of relationships." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Mathematics has traditionally been described as the science of number and shape. […] When viewed in this broader context, we see that mathematics is not just about number and shape but about pattern and order of all sorts. Number and shape - arithmetic and geometry - are but two of many media in which mathematicians work. Active mathematicians seek patterns wherever they arise." (Lynn A Steen, "The Future of Mathematics Education", 1998)

"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)

"Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world."  (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Complexity is looking at interacting elements and asking how they form patterns and how the patterns unfold. It’s important to point out that the patterns may never be finished. They’re open-ended. In standard science this hit some things that most scientists have a negative reaction to. Science doesn’t like perpetual novelty." (W Brian Arthur, 1999)

"Randomness is the very stuff of life, looming large in our everyday experience. […] The fascination of randomness is that it is pervasive, providing the surprising coincidences, bizarre luck, and unexpected twists that color our perception of everyday events." (Edward Beltrami, "Chaos and Order in Mathematics and Life", 1999)

"The first view of randomness is of clutter bred by complicated entanglements. Even though we know there are rules, the outcome is uncertain. Lotteries and card games are generally perceived to belong to this category. More troublesome is that nature's design itself is known imperfectly, and worse, the rules may be hidden from us, and therefore we cannot specify a cause or discern any pattern of order. When, for instance, an outcome takes place as the confluence of totally unrelated events, it may appear to be so surprising and bizarre that we say that it is due to blind chance." (Edward Beltrami. "What is Random?: Chance and Order in Mathematics and Life", 1999)

On Continuity XI (Thought II)

"The function of man’s highest faculty, his reason, consists precisely of the continuous limitation of infinity, the breaking up of infinity into convenient, easily digestible portions - differentials. This is precisely what lends my field, mathematics, its divine beauty." (Yevgeny Zamiatin, "We", 1924)

"Rationality consists [of] the continuous adaptation of our language to our continually expanding world, and metaphor is one of the chief means by which this is accomplished." (Mary B Hesse, "Models and Analogies in Science", 1966)

"Truth is a totality, the sum of many overlapping partial images. History, on the other hand, sacrifices totality in the interest of continuity." (Edmund Leach, "Brain-Twister", 1967)

"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent 'forward' movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission." (Carl Eckart, "Our Modern Idol: Mathematical Science", 1984)

"To form a mental picture of the event, the knowledge developer attempts to integrate his or her perception of the situation with the expert’s perception. That mental picture is then recorded. What happens is a continuous shuttle process; the knowledge developer mentally moves back and forth from the initial impression of the event to the later evaluation of the event. What is finally recorded is the evaluation made during this retrospective period. Because a time lapse can make details of a situation less clear, the information is not always valid." (Elias M Awad, "Knowledge Management", 2003)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Continuity XIV (Progress)

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

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

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

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

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

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

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

On Imagination (2000-2024)

"One of the most fundamental notions in mathematics is that of number. Although the idea of number is basic, the numbers themselves possess both nuance and complexity that spark the imagination." (Edward B Burger, "Exploring the Number Jungle", 2000)

"To say that a thing is imaginary is not to dispose of it in the realm of mind, for the imagination, or the image making faculty, is a very important part of our mental functioning. An image formed by the imagination is a reality from the point of view of psychology; it is quite true that it has no physical existence, but are we going to limit reality to that which is material? We shall be far out of our reckoning if we do, for mental images are potent things, and although they do not actually exist on the physical plane, they influence it far more than most people suspect." (Dion Fortune," Spiritualism and Occultism", 2000)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music." (Northrop Frye, "The Educated Imagination", 2002)

"[…] because observations are all we have, we take them seriously. We choose hard data and the framework of mathematics as our guides, not unrestrained imagination or unrelenting skepticism, and seek the simplest yet most wide-reaching theories capable of explaining and predicting the outcome of today’s and future experiments." (Brian Greene, "The Fabric of the Cosmos", 2004)

"There is a strong parallel between mountain climbing and mathematics research. When first attempts on a summit are made, the struggle is to find any route. Once on the top, other possible routes up may be discerned and sometimes a safer or shorter route can be chosen for the descent or for subsequent ascents. In mathematics the challenge is finding a proof in the first place. Once found, almost any competent mathematician can usually find an alternative often much better and shorter proof. At least in mountaineering we know that the mountain is there and that, if we can find a way up and reach the summit, we shall triumph. In mathematics we do not always know that there is a result, or if the proposition is only a figment of the imagination, let alone whether a proof can be found." (Kathleen Ollerenshaw, "To talk of many things: An autobiography", 2004)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"To have the courage to think outside the square, we need to be intrigued by a problem. This intrigue will encourage us to use our imaginations to find solutions which are beyond our current view of the world. This was the challenge that faced mathematicians as they searched for a solution to the problem of finding meaning for the square root of a negative number, in particular v-1." (Les Evans, "Complex Numbers and Vectors", 2006)

"Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’" (Les Evans, "Complex Numbers and Vectors", 2006)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"If worldviews or metanarratives can be compared to lenses, which of them brings things into the sharpest focus? This is not an irrational retreat from reason. Rather, it is about grasping a deeper order of things which is more easily accessed by the imagination than by reason." (Alister McGrath, "If I Had Lunch with C. S. Lewis: Exploring the Ideas of C. S. Lewis on the Meaning of Life", 2014)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"The mental model is the arena where imagination takes place. It enables us to experiment with different scenarios by making local alterations to the model. […] To speak of causality, we must have a mental model of the real world. […] Our shared mental models bind us together into communities." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

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On Creativity (Mathematics I)

"Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing." (Robert C Buck, "Teaching Machines and Mathematics Programs", American Mathematical Monthly 69, 1962)

"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory." (Mark Kac, "Enigmas Of Chance", 1985)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful!" (James Tanton, "Thinking Mathematics")

"Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there." (Meir Shalev)

"No discovery has been made in mathematics, or anywhere else for that matter, by an effort of deductive logic; it results from the work of creative imagination which builds what seems to be truth, guided sometimes by analogies, sometimes by an esthetic ideal, but which does not hold at all on solid logical bases. Once a discovery is made, logic intervenes to act as a control; it is logic that ultimately decides whether the discovery is really true or is illusory; its role therefore, though considerable, is only secondary." (Henri Lebesgue)

"The essential feature of mathematical creativity is the exploration, under the pressure of powerful implosive forces, of difficult problems for whose validity and importance the explorer is eventually held accountable by reality." (Alfred Adler)

Daniel Kahneman - Collected Quotes

"Theories of choice are at best approximate and incomplete. One reason for this pessimistic assessment is that choice is a constructive and contingent process. When faced with a complex problem, people employ a variety of heuristic procedures in order to simplify the representation and the evaluation of prospects. These procedures include computational shortcuts and editing operations, such as eliminating common components and discarding nonessential differences. The heuristics of choice do not readily lend themselves to formal analysis because their application depends on the formulation of the problem, the method of elicitation, and the context of choice." (Amos Tversky & Daniel Kahneman, "Advances in Prospect Theory: Cumulative Representation of Uncertainty" [in "Choices, Values, and Frames"], 2000)

"A general law of least effort applies to cognitive as well as physical exertion. The law asserts that if there are several ways of achieving the same goal, people will eventually gravitate to the least demanding course of action. In the economy of action, effort is a cost, and the acquisition of skill is driven by the balance of benefits and costs. Laziness is built deep into our nature." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"A general limitation of the human mind is its imperfect ability to reconstruct past states of knowledge, or beliefs that have changed. Once you adopt a new view of the world (or any part of it), you immediately lose much of your ability to recall what you used to believe before your mind changed." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Knowing the importance of luck, you should be particularly suspicious when highly consistent patterns emerge from the comparison of successful and less successful firms. In the presence of randomness, regular patterns can only be mirages." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Intelligence is not only the ability to reason; it is also the ability to find relevant material in memory and to deploy attention when needed." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"It is the consistency of the information that matters for a good story, not its completeness. Indeed, you will often find that knowing little makes it easier to fit everything you know into a coherent pattern." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"The confidence that individuals have in their beliefs depends mostly on the quality of the story they can tell about what they see, even if they see little." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"The confidence we experience as we make a judgment is not a reasoned evaluation of the probability that it is right. Confidence is a feeling, one determined mostly by the coherence of the story and by the ease with which it comes to mind, even when the evidence for the story is sparse and unreliable. The bias toward coherence favors overconfidence. An individual who expresses high confidence probably has a good story, which may or may not be true." (Daniel Kahneman, "Don't Blink! The Hazards of Confidence", 2011)

"The idea that the future is unpredictable is undermined every day by the ease with which the past is explained." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"The illusion that we understand the past fosters overconfidence in our ability to predict the future." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"This is the essence of intuitive heuristics: when faced with a difficult question, we often answer an easier one instead, usually without noticing the substitution." (Daniel Kahneman, "Thinking, Fast and Slow", 2011) 

"We are far too willing to reject the belief that much of what we see in life is random."(Daniel Kahneman, "Thinking, Fast and Slow", 2011) 

"We are often confident even when we are wrong, and an objective observer is more likely to detect our errors than we are." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"We are prone to overestimate how much we understand about the world and to underestimate the role of chance in events." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"When people believe a conclusion is true, they are also very likely to believe arguments that appear to support it, even when these arguments are unsound." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"You know you have made a theoretical advance when you can no longer reconstruct why you failed for so long to see the obvious." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"A defining feature of system noise is that it is unwanted, and we should stress right here that variability in judgments is not always unwanted." (Daniel Kahneman, "Noise: A Flaw in Human Judgment", 2021)

"A general property of noise is that you can recognize and measure it while knowing nothing about the target or bias." (Daniel Kahneman, "Noise: A Flaw in Human Judgment", 2021) 

"Bias and noise - systematic deviation and random scatter - are different components of error. […] To understand error in judgment, we must understand both bias and noise. Sometimes, as we will see, noise is the more important problem. But in public conversations about human error and in organizations all over the world, noise is rarely recognized. Bias is the star of the show. Noise is a bit player, usually offstage. […] Wherever you look at human judgments, you are likely to find noise. To improve the quality of our judgments, we need to overcome noise as well as bias." (Daniel Kahneman, "Noise: A Flaw in Human Judgment", 2021)

On Generalization (1800-1849)

"An idea is always a generalization, and generalization is a property of thinking. To generalize means to think." (Georg W F Hegel, "The Philosophy of Right", 1820)

"To minds of a certain cast there is nothing so captivating as simplification and generalization." (Thomas R Malthus, "Principles of Political Economy", 1820)

"General assertions, like general truths, are not always applicable to individual cases; though Fortune's wheel is generally on the turn, sometimes when it gets into the mud, it sticks there." (Letitia E Landon, "Romance and Reality", 1831)

"It is not easy to anatomize the constitution and the operations of a mind which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)

"Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "System of Logic", 1843)

"The mere accumulation of unconnected observations of details, devoid of generalization of ideas, may doubtlessly have tended to create and foster the deeply rooted prejudice, that the study of the exact sciences must necessarily chill the feelings, and diminish the nobler enjoyments attendant upon a contemplation of nature." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe", 1845)

On Technology IV

"If you think technology can solve your security problems, then you don't understand the problems and you don't understand the technology." (Bruce Schneier, "Secrets and Lies: Digital Security in a Networked World", 2000)

"Ultimately, progress in applications is not deterministic, but opportunistic, exploiting for new applications whatever new science and technology happen to be coming along." (Herbert Kroemer, "Quasi-Electric Fields and Band Offsets: Teaching Electrons New Tricks", [Nobel Lecture], 2000)

"Technology can relieve the symptoms of a problem without affecting the underlying causes. Faith in technology as the ultimate solution to all problems can thus divert our attention from the most fundamental problem - the problem of growth in a finite system - and prevent us from taking effective action to solve it." (Donella H Meadows & Dennis L Meadows, "The Limits to Growth: The 30 Year Update", 2004)

"Although the Singularity has many faces, its most important implication is this: our technology will match and then vastly exceed the refinement and suppleness of what we regard as the best of human traits."  (Ray Kurzweil, "The Singularity is Near", 2005)

"The Singularity will represent the culmination of the merger of our biological thinking and existence with our technology, resulting in a world that is still human but that transcends our biological roots. There will be no distinction, post-Singularity, between human and machine or between physical and virtual reality. If you wonder what will remain unequivocally human in such a world, it’s simply this quality: ours is the species that inherently seeks to extend its physical and mental reach beyond current limitations." (Ray Kurzweil, "The Singularity is Near", 2005)

"Chance is just as real as causation; both are modes of becoming.  The way to model a random process is to enrich the mathematical theory of probability with a model of a random mechanism. In the sciences, probabilities are never made up or 'elicited' by observing the choices people make, or the bets they are willing to place.  The reason is that, in science and technology, interpreted probability exactifies objective chance, not gut feeling or intuition. No randomness, no probability." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Synergy occurs when organizational parts interact to produce a joint effect that is greater than the sum of the parts acting alone. As a result the organization may attain a special advantage with respect to cost, market power, technology, or employee." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

"What’s next for technology and design? A lot less thinking about technology for technology’s sake, and a lot more thinking about design. Art humanizes technology and makes it understandable. Design is needed to make sense of information overload. It is why art and design will rise in importance during this century as we try to make sense of all the possibilities that digital technology now affords." (John Maeda, "Why Apple Leads the Way in Design", 2010) 

"Today, technology has lowered the barrier for others to share their opinion about what we should be focusing on. It is not just information overload; it is opinion overload." (Greg McKeown, "Essentialism: The Disciplined Pursuit of Less", 2014)

"For a successful technology, reality must take precedence over public relations, for nature cannot be fooled." (Richard Feynman)

Richard E Nisbett - Collected Quotes

"Multiple regression, like all statistical techniques based on correlation, has a severe limitation due to the fact that correlation doesn't prove causation. And no amount of measuring of 'control' variables can untangle the web of causality. What nature hath joined together, multiple regression cannot put asunder." (Richard Nisbett, "2014: What scientific idea is ready for retirement?", 2013)

"What nature hath joined together, multiple regression cannot put asunder."  (Richard Nisbett, "2014: What scientific idea is ready for retirement?", 2013)

"A basic problem with MRA is that it typically assumes that the independent variables can be regarded as building blocks, with each variable taken by itself being logically independent of all the others. This is usually not the case, at least for behavioral data. […] Just as correlation doesn’t prove causation, absence of correlation fails to prove absence of causation. False-negative findings can occur using MRA just as false-positive findings do - because of the hidden web of causation that we’ve failed to identify." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Deductive and inductive reasoning schemas essentially regulate inferences. They tell us what kinds of inferences are valid and what kinds are invalid. […] Dialectical reasoning isn’t formal or deductive and usually doesn’t deal in abstractions. It’s concerned with reaching true and useful conclusions rather than valid conclusions. In fact, conclusions based on dialectical reasoning can actually be opposed to those based on formal logic." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Dialectical thinking opposes formalism because of its separation of form from content. We make errors by abstracting the elements of a problem into a formal model and ignoring facts and contexts crucial to correct analysis. Overemphasis on logical approaches leads to distortion, error, and rigidity." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Multiple regression analysis (MRA) examines the association between an independent variable and a dependent variable, controlling for the association between the independent variable and other variables, as well as the association of those other variables with the dependent variable. The method can tell us about causality only if all possible causal influences have been identified and measured reliably and validly. In practice, these conditions are rarely met." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"One technique employing correlational analysis is multiple regression analysis (MRA), in which a number of independent variables are correlated simultaneously (or sometimes sequentially, but we won’t talk about that variant of MRA) with some dependent variable. The predictor variable of interest is examined along with other independent variables that are referred to as control variables. The goal is to show that variable A influences variable B 'net of' the effects of all the other variables. That is to say, the relationship holds even when the effects of the control variables on the dependent variable are taken into account." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Science is often described as a 'seamless web'. What’s meant by that is that the facts, methods, theories, and rules of inference discovered in one field can be helpful for other fields. And philosophy and logic can affect reasoning in literally every field of science."(Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"The closer that sample-selection procedures approach the gold standard of random selection - for which the definition is that every individual in the population has an equal chance of appearing in the sample - the more we should trust them. If we don’t know whether a sample is random, any statistical measure we conduct may be biased in some unknown way." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"The correlational technique known as multiple regression is used frequently in medical and social science research. This technique essentially correlates many independent (or predictor) variables simultaneously with a given dependent variable (outcome or output). It asks, 'Net of the effects of all the other variables, what is the effect of variable A on the dependent variable?' Despite its popularity, the technique is inherently weak and often yields misleading results. The problem is due to self-selection. If we don’t assign cases to a particular treatment, the cases may differ in any number of ways that could be causing them to differ along some dimension related to the dependent variable. We can know that the answer given by a multiple regression analysis is wrong because randomized control experiments, frequently referred to as the gold standard of research techniques, may give answers that are quite different from those obtained by multiple regression analysis." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"The fundamental problem with MRA, as with all correlational methods, is self-selection. The investigator doesn’t choose the value for the independent variable for each subject (or case). This means that any number of variables correlated with the independent variable of interest have been dragged along with it. In most cases, we will fail to identify all these variables. In the case of behavioral research, it’s normally certain that we can’t be confident that we’ve identified all the plausibly relevant variables." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"The theory behind multiple regression analysis is that if you control for everything that is related to the independent variable and the dependent variable by pulling their correlations out of the mix, you can get at the true causal relation between the predictor variable and the outcome variable. That’s the theory. In practice, many things prevent this ideal case from being the norm." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"We are superb causal-hypothesis generators. Given an effect, we are rarely at a loss for an explanation. Seeing a difference in observations over time, we readily come up with a causal interpretation. Much of the time, no causality at all is going on—just random variation. The compulsion to explain is particularly strong when we habitually see that one event typically occurs in conjunction with another event. Seeing such a correlation almost automatically provokes a causal explanation. It’s tremendously useful to be on our toes looking for causal relationships that explain our world. But there are two problems: (1) The explanations come too easily. If we recognized how facile our causal hypotheses were, we’d place less confidence in them. (2) Much of the time, no causal interpretation at all is appropriate and wouldn’t even be made if we had a better understanding of randomness." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"We don’t recognize how easy it is to generate hypotheses about the world. If we did, we’d generate fewer of them, or at least hold them more tentatively. We sprout causal theories in abundance when we learn of a correlation, and we readily find causal explanations for the failure of the world to confirm our hypotheses. We don’t realize how easy it is for us to explain away evidence that would seem on the surface to contradict our hypotheses. And we fail to generate tests of a hypothesis that could falsify the hypothesis if in fact the hypothesis is wrong. This is one type of confirmation bias." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

On Coherence IV

"Man's general way of thinking of the totality, i.e. his general world view, is crucial for overall order of the human mind itself. If he thinks of the totality as constituted of independent fragments, then that is how his mind will tend to operate, but if he can include everything coherently and harmoniously in an overall whole that is undivided, unbroken and without border (for every border is a division or break) then his mind will tend to move in a similar way, and from this will flow an orderly action within the whole." (David Bohm, "Wholeness and the Implicate Order", 1980)

"Nature is disordered, powerful and chaotic, and through fear of the chaos we impose system on it. We abhor complexity, and seek to simplify things whenever we can by whatever means we have at hand. We need to have an overall explanation of what the universe is and how it functions. In order to achieve this overall view we develop explanatory theories which will give structure to natural phenomena: we classify nature into a coherent system which appears to do what we say it does." (James Burke, "The Day the Universe Changed", 1985)

"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding. " (Mark Johnson, "The Body in the Mind", 1987)

"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend, "Farewell to Reason", 1987)

"When loops are present, the network is no longer singly connected and local propagation schemes will invariably run into trouble. [...] If we ignore the existence of loops and permit the nodes to continue communicating with each other as if the network were singly connected, messages may circulate indefinitely around the loops and process may not converges to a stable equilibrium. […] Such oscillations do not normally occur in probabilistic networks […] which tend to bring all messages to some stable equilibrium as time goes on. However, this asymptotic equilibrium is not coherent, in the sense that it does not represent the posterior probabilities of all nodes of the network." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference", 1988)

"A world view is a system of co-ordinates or a frame of reference in which everything presented to us by our diverse experiences can be placed. It is a symbolic system of representation that allows us to integrate everything we know about the world and ourselves into a global picture, one that illuminates reality as it is presented to us within a certain culture. […] A world view is a coherent collection of concepts and theorems that must allow us to construct a global image of the world, and in this way to understand as many elements of our experience as possible." (Diederick Aerts et al, "World views: From Fragmentation to Integration", 1994)

"There are a variety of swarm topologies, but the only organization that holds a genuine plurality of shapes is the grand mesh. In fact, a plurality of truly divergent components can only remain coherent in a network. No other arrangement-chain, pyramid, tree, circle, hub-can contain true diversity working as a whole. This is why the network is nearly synonymous with democracy or the market." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"Falling between order and chaos, the moment of complexity is the point at which self-organizing systems emerge to create new patterns of coherence and structures of behaviour." (Mark C Taylor, "The Moment of Complexity: Emerging Network Culture", 2001)

Ralph D Stacey - Collected Quotes

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"In a linear world of equilibrium and predictability, the sparse research into an evidence base for management prescriptions and the confused findings it produces would be a sign of incompetence; it would not make much sense. Nevertheless, if organizations are actually patterns of nonlinear interaction between people; if small changes could produce widespread major consequences; if local interaction produces emergent global pattern; then it will not be possible to provide a reliable evidence base. In such a world, it makes no sense to conduct studies looking for simple causal relationships between an action and an outcome. I suggest that the story of the last few years strongly indicates that human action is nonlinear, that time and place matter a great deal, and that since this precludes simple evidence bases we do need to rethink the nature of organizations and the roles of managers and leaders in them." (Ralph D Stacey, "Complexity and Organizational Reality", 2000)

"The model [of reality] takes on a life of its own, in which its future is under perpetual construction through the micro interactions of the diverse entities comprising it. The "final" form toward which it moves is not given in the model itself, nor is it being chosen from outside the model. The forms continually emerge in an unpredictable way as the system moves into the unknown. However, there is nothing mysterious or esoteric about this. What emerges does so because of the transformative cause of the process of the micro interactions, the fluctuations themselves." (Ralph D Stacey et al, "Complexity and Management: Fad or Radical Challenge to Systems Thinking?", 2000)

"The central proposition in [realistic thinking] is that human actions and interactions are processes, not systems, and the coherent patterning of those processes becomes what it becomes because of their intrinsic capacity, the intrinsic capacity of interaction and relationship, to form coherence. That emergent form is radically unpredictable, but it emerges in a controlled or patterned way because of the characteristic of relationship itself, creation and destruction in conditions at the edge of chaos." (Ralph D Stacey et al, "Complexity and Management: Fad or Radical Challenge to Systems Thinking?", 2000)

"Organizations are not systems but the ongoing patterning of interactions between people. Patterns of human interaction produce further patterns of interaction, not some thing outside of the interaction. We call this perspective complex responsive processes of relating." (Ralph D Stacey, "Experiencing Emergence in Organizations", 2005)

"Systems are wholes consisting of parts interacting with each other in a self-generating, self-organising way and it is in this interaction that both parts and whole emerge without prior design." (Ralph D Stacey, "Strategic Management and Organisational Dynamics", 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)

"Organizations are not systems but the ongoing patterning of interactions between people. Patterns of human interaction produce further patterns of interaction, not some thing outside of the interaction. We call this perspective complex responsive processes of relating." (Ralph Stacey)

Systems Thinking V

"[Systems thinking is] A new way to view and mentally frame what we see in the world; a worldview and way of thinking whereby we see the entity or unit first as a whole, with its fit and relationship to its environment as primary concerns." (Stephen G Haines, "The Managers Pocket Guide to Systems Thinking & Learning", 1998)

"The beauty of this [systems thinking] mindset is that its mental models are based on natural laws, principles of interrelationship, and interdependence found in all living systems. They give us a new view of ourselves and our many systems, from the tiniest cell to the entire earth; and as our organizations are included in that great range, they help us define organizational problems as systems problems, so we can respond in more productive ways. The systems thinking mindset is a new orientation to life. In many ways it also operates as a worldview - an overall perspective on, and understanding of, the world." (Stephen G Haines, "The Managers Pocket Guide to Systems Thinking & Learning", 1998)

"As a meta-discipline, systems science will transfer its content from discipline to discipline and address problems beyond conventional reductionist boundaries. Generalists, qualified to manage today’s problem better than the specialist, could be fostered. With these intentions, systems thinking and systems science should not replace but add, complement and integrate those aspects that seem not to be adequately treated by traditional science." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"Systems thinking expands the focus of the observer, whereas analytical thinking reduces it. In other words, analysis looks into things, synthesis looks out of them. This attitude of systems thinking is often called expansionism, an alternative to classic reductionism. Whereas analytical thinking concentrates on static and structural properties, systems thinking concentrates on the function and behaviour of whole systems. Analysis gives description and knowledge; systems thinking gives explanation and understanding." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"System Thinking is a common concept for understanding how causal relationships and feedbacks work in an everyday problem. Understanding a cause and an effect enables us to analyse, sort out and explain how changes come about both temporarily and spatially in common problems. This is referred to as mental modelling, i.e. to explicitly map the understanding of the problem and making it transparent and visible for others through Causal Loop Diagrams (CLD)." (Hördur V Haraldsson, "Introduction to System Thinking and Causal Loop Diagrams", 2004)

"[systems thinking is]  thinking holistically and conscientiously about the world by focusing on the interaction of the parts and their influence within and over the system." (Kambiz E Maani, "Systems Thinking and the Internet from Independence to Interdependence", Encyclopedia of Information Science and Technology, Second Edition, 2009)

"Systems thinking, in contrast, focuses on how the thing being studied interacts with the other constituents of the system - a set of elements that interact to produce behaviour - of which it is a part. This means that instead of isolating smaller and smaller parts of the system being studied, systems thinking works by expanding its view to take into account larger and larger numbers of interactions as an issue is being studied. This results in sometimes strikingly different conclusions than those generated by traditional forms of analysis, especially when what is being studied is dynamically complex or has a great deal of feedback from other sources, internal or external. Systems thinking allows people to make their understanding of social systems explicit and improve them in the same way that people can use engineering principles to make explicit and improve their understanding of mechanical systems." (Raed M Al-Qirem & Saad G Yaseen, "Modelling a Small Firm in Jordan Using System Dynamics" [in "Handbook of Research on Discrete Event Simulation Environments: Technologies and Applications"], 2010)

"Understanding interdependency requires a way of thinking different from analysis. It requires systems thinking. And analytical thinking and systems thinking are quite distinct. [...] Systems thinking is the art of simplifying complexity. It is about seeing through chaos, managing interdependency, and understanding choice. We see the world as increasingly more complex and chaotic because we use inadequate concepts to explain it. When we understand something, we no longer see it as chaotic or complex." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture", 2011)

"Systems thinking focuses on optimizing for the whole, looking at the overall flow of work, identifying what the largest bottleneck is today, and eliminating it." (Matthew Skelton & Manuel Pais, "Team Topologies: Organizing Business and Technology Teams for Fast Flow", 2019)

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On Imagination (1950-1974)

"[…] observation is not enough, and it seems to me that in science, as in the arts, there is very little worth having that does not require the exercise of intuition as well as of intelligence, the use of imagination as well as of information." (Kathleen Lonsdale, "Facts About Crystals", American Scientist Vol. 39 (4), 1951)

"There is always an analogy between nature and the imagination, and possibly poetry is merely the strange rhetoric of that parallel." (Wallace Stevens, "The Necessary Angel", 1951)

"All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things." (Max Born, "Physical Reality", Philosophical Quarterly Vol. (11), 1953)

"The creative act owes little to logic or reason. In their accounts of the circumstances under which big ideas occurred to them, mathematicians have often mentioned that the inspiration had no relation to the work they happened to be doing. Sometimes it came while they were traveling, shaving or thinking about other matters. The creative process cannot be summoned at will or even cajoled by sacrificial offering. Indeed, it seems to occur most readily when the mind is relaxed and the imagination roaming freely." (Morris Kline, Scientific American, 1955)

"Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"The ultimate origin of the difficulty lies in the fact (or philosophical principle) that we are compelled to use the words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind; by analysing visible motions it has developed two ways of representing them by elementary processes; moving particles and waves. There is no other way of giving a pictorial description of motions - we have to apply it even in the region of atomic processes, where classical physics breaks down." (Max Born, "Atomic Physics", 1957)

"In imagination there exists the perfect mystery story. Such a story presents all the essential clews, and compels us to form our own theory of the case. If we follow the plot carefully we arrive at the complete solution for ourselves just before the author’s disclosure at the end of the book. The solution itself, contrary to those of inferior mysteries, does not disappoint us; moreover, it appears at the very moment we expect it." (Leopold Infeld, "The Evolution of Physics", 1961)

"The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1962)

"That perfected machines may one day succeed us is, I remember, an extremely commonplace notion on Earth. It prevails not only among poets and romantics but in all classes of society. Perhaps it is because it is so widespread, born spontaneously in popular imagination, that it irritates scientific minds. Perhaps it is also for this very reason that it contains a germ of truth. Only a germ: Machines will always be machines; the most perfected robot, always a robot." (Pierre Boulle, "Planet of the Apes", 1963)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience." (Northrop Frye, "The Educated Imagination", 1964)

"The imagination equips us to perceive reality when it is not fully materialized." (Mary C Richards, "Centering in Pottery, Poetry, and the Person", 1964)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"Fantasies are more than substitutes for unpleasant reality; they are also dress rehearsals, plans. All acts performed in the world begin in the imagination." (Barbara G Harrison, [Ms. Magazine] 1973)

"Equilibrium is a figment of the human imagination." (Kenneth Boulding, Toward a General Social Science, 1974)

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Richard L Daft - Collected Quotes

"A mental model can be thought of as an internal picture that affects a leader's actions and relationships with others. Mental models are theories people hold about specific systems in the world and their expected behavior." (Richard L Daft, "The Leadership Experience" , 2002)

"Systems thinking means the ability to see the synergy of the whole rather than just the separate elements of a system and to learn to reinforce or change whole system patterns. Many people have been trained to solve problems by breaking a complex system, such as an organization, into discrete parts and working to make each part perform as well as possible. However, the success of each piece does not add up to the success of the whole. to the success of the whole. In fact, sometimes changing one part to make it better actually makes the whole system function less effectively." (Richard L Daft, "The Leadership Experience", 2002)

"Systems thinking is a mental discipline and framework for seeing patterns and interrelationships. It is important to see organizational systems as a whole because of their complexity. Complexity can overwhelm managers, undermining confidence. When leaders can see the structures that underlie complex situations, they can facilitate improvement. But doing that requires a focus on the big picture." (Richard L Daft, "The Leadership Experience", 2002)

"A symbol is an object, act, or event that conveys meaning to others. Symbols can be considered a rich, non-verbal language that vibrantly conveys the organization’s important values concerning how people relate to one another and interact with the environment" (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"Data are raw facts and figures that by themselves may be useless. To be useful, data must be processed into finished information, that is, data converted into a meaningful and useful context for specific users. An increasing challenge for managers is being able to identify and access useful information." (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"The receiver decodes the symbols to interpret the meaning of the message. Encoding and decoding are potential sources for communication errors because knowledge, attitudes, and context act as filters and create noise when translating from symbols to meaning. Finally, feedback occurs when the receiver responds to the sender’s communication with a return message. Without feedback, the communication is one-way; with feedback, it is two-way. Feedback is a powerful aid to communication effectiveness because it enables the sender to determine whether the receiver correctly interpreted the message." (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"A paradigm is a shared mindset that represents a fundamental way of thinking about, perceiving, and understanding the world." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

"Synergy is the combined action that occurs when people work together to create new alternatives and solutions. In addition, the greatest opportunity for synergy occurs when people have different viewpoints, because the differences present new opportunities. The essence of synergy is to value and respect differences and take advantage of them to build on strengths and compensate for weaknesses." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

"Synergy occurs when organizational parts interact to produce a joint effect that is greater than the sum of the parts acting alone. As a result the organization may attain a special advantage with respect to cost, market power, technology, or employee." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

Jamshid Gharajedaghi - Collected Quotes

"A commonly accepted principle of systems dynamics is that a quantitative change, beyond a critical point, results in a qualitative change. Accordingly, a difference in degree may become a difference in kind. This doesn't mean that an increased quantity of a given variable will bring a qualitative change in the variable itself. However, when the state of a system depends on a set of variables, a quantitative change in one variable beyond the inflection point will result in a change of phase in the state of the system. This change is a qualitative one, representing a whole new set of relationships among the variables involved." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Analyzing the behavior of a nonlinear system is like walking through a maze whose walls rearrange themselves with each step you take (in other words, playing the game changes the game)." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"But one of the major findings of quantum theory is the recognition that the universe is an open system. Open systems are neg-entropic (-S) and exhibit a tendency toward order. This means that according to quantum theory, the universe is a self-organizing system, continuously expanding and moving toward increasing order and complexity. In this context even subatomic particles show open system behavior and self-referencing tendencies." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Complementary to the principle of multidimensionality and parallel to it is the concept of plurality. Plurality of function, structure, and process, as we will see later on, is at the core of systems theory of development. It makes the high/high a possibility and choice a reality. Plurality simply maintains that systems can have multiple structures and multiple functions and be governed by multiple processes; it denies the classical view of a single structure with a single function in a single cause-and-effect relationship."  (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Complexity is a relative term. It depends on the number and the nature of interactions among the variables involved. Open loop systems with linear, independent variables are considered simpler than interdependent variables forming nonlinear closed loops with a delayed response." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Despite seemingly contradictory requirements for pursuit of opposing ends, there are processes that would make the attainment of both ends feasible. For instance, both freedom and security are attainable by a process called participation, stability and change by adaptation, and order and complexity by organization." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"In the context of a zero-sum game, opposing tendencies are formulated in two distinct ways. First, conflicting tendencies are conceptualized as two mutually exclusive, discrete entities. The conflicts are treated as dichotomies that are usually expressed as X or NX. If X is right then NX has to be wrong. This represents an or relationship, a win/lose struggle with a moral obligation to win. The loser, usually declared wrong, is eliminated. Second, opposing tendencies are formulated in such a way that they can be represented by a continuum. Between black and white are a thousand shades of gray." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Laws of complexity hold universally across hierarchical scales (scalar, self-similarity) and are not influenced by the detailed behavior of constituent parts." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Natural science has discovered 'chaos'. Social science has encountered 'complexity'. But chaos and complexity are not characteristics of our new reality; they are features of our perceptions and understanding. We see the world as increasingly more complex and chaotic because we use inadequate concepts to explain it. When we understand something, we no longer see it as chaotic or complex. Maybe playing the new game requires learning a new language." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Nature's tendency for iteration, pattern formation, and creation of order out of chaos creates expectations of predictability. It seems, however, that nature, because of varying degrees of interaction between chance and choice, and the nonlinearity of systems, escapes the boredom of predictability." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Operational thinking is about mapping relationships. It is about capturing interactions, interconnections, the sequence and flow of activities, and the rules of the game. It is about how systems do what they do, or the dynamic process of using elements of the structure to produce the desired functions. In a nutshell, it is about unlocking the black box that lies between system input and system output." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Remember that to map the dynamic behavior of a system is to capture the interaction of positive and negative feedback loops. These interactions, in essence, define the interdependencies, which in turn are responsible for nonlinearity in the system. It is the interdependency that poses the major challenge to our cognitive abilities. It is this challenge that we need to overcome by using operational modeling. Pattern recognition is critical for understanding and changing undesirable behavior. This leads us to the need for development of interactive operational representation of the phenomenon under investigation." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Understanding interdependency requires a way of thinking different from analysis. It requires systems thinking. And analytical thinking and systems thinking are quite distinct. [...] Systems thinking is the art of simplifying complexity. It is about seeing through chaos, managing interdependency, and understanding choice. We see the world as increasingly more complex and chaotic because we use inadequate concepts to explain it. When we understand something, we no longer see it as chaotic or complex." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"The real challenge to building a viable social system is the ability to create unity in diversity, meeting the varying interests of independent members operating in an interdependent whole. […] In the long run, the society and the individual either stand together or fall separately. A win/win relationship is achieved not through zero-sum or even compromise. For both of them to win requires reconceptualization of the nature and the relationship of the whole and the parts." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"The second law states that a general tendency in the universe (as a closed system) is toward elimination of all differences. Thus, the ultimate state is sameness and randomness, a chaotic simplicity. Entropy (S), the measure of randomness, will therefore always increase. However, we know that living systems are neg-entropic (–S). They are able not only to negate this formidable process by differentiation, but also to move toward a predefined order, an organized complexity. Using the formula I = –S, which indicates that a neg-entropic system must have information, one might conclude that movement toward complexity and order is only possible if the system has a means of knowing and an internal image of what it wants to be." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)