On Complexity I

"[Disorganized complexity] is a problem in which the number of variables is very large, and one in which each of the many variables has a behavior which is individually erratic, or perhaps totally unknown. However, in spite of this helter-skelter, or unknown, behavior of all the individual variables, the system as a whole possesses certain orderly and analyzable average properties. [...] [Organized complexity is] not problems of disorganized complexity, to which statistical methods hold the key. They are all problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole. They are all, in the language here proposed, problems of organized complexity." (Warren Weaver, "Science and Complexity", American Scientist Vol. 36, 1948)

"Crude complexity is ‘the length of the shortest message that will describe a system, at a given level of coarse graining, to someone at a distance, employing language, knowledge, and understanding that both parties share (and know they share) beforehand." (Murray Gell-Mann, "What is Complexity?" Complexity Vol. 1 (1), 1995)

"In contemplating natural phenomena, we frequently have to distinguish between effective complexity and logical depth. For example, the apparently complicated pattern of energy levels of atomic nuclei might easily be misattributed to some complex law at the fundamental level, but it is now believed to follow from a simple underlying theory of quarks, gluons, and photons, although lengthy calculations would be required to deduce the detailed pattern from the basic equations. Thus the pattern has a good deal of logical depth and very little effective complexity." (Murray Gell-Mann, "What is Complexity?", Complexity Vol. 1 (1), 1995)

"A dictionary definition of the word ‘complex’ is: ‘consisting of interconnected or interwoven parts’ […] Loosely speaking, the complexity of a system is the amount of information needed in order to describe it. The complexity depends on the level of detail required in the description. A more formal definition can be understood in a simple way. If we have a system that could have many possible states, but we would like to specify which state it is actually in, then the number of binary digits (bits) we need to specify this particular state is related to the number of states that are possible." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"When the behavior of the system depends on the behavior of the parts, the complexity of the whole must involve a description of the parts, thus it is large. The smaller the parts that must be described to describe the behavior of the whole, the larger the complexity of the entire system. […] A complex system is a system formed out of many components whose behavior is emergent, that is, the behavior of the system cannot be simply inferred from the behavior of its components." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"Complexity is that property of a model which makes it difficult to formulate its overall behaviour in a given language, even when given reasonably complete information about its atomic components and their inter-relations." (Bruce Edmonds, "Syntactic Measures of Complexity", 1999)

"[…] networks are the prerequisite for describing any complex system, indicating that complexity theory must inevitably stand on the shoulders of network theory. It is tempting to step in the footsteps of some of my predecessors and predict whether and when we will tame complexity. If nothing else, such a prediction could serve as a benchmark to be disproven. Looking back at the speed with which we disentangled the networks around us after the discovery of scale-free networks, one thing is sure: Once we stumble across the right vision of complexity, it will take little to bring it to fruition. When that will happen is one of the mysteries that keeps many of us going." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Complexity is the characteristic property of complicated systems we don’t understand immediately. It is the amount of difficulties we face while trying to understand it. In this sense, complexity resides largely in the eye of the beholder - someone who is familiar with s.th. often sees less complexity than someone who is less familiar with it. [...] A complex system is created by evolutionary processes. There are multiple pathways by which a system can evolve. Many complex systems are similar, but each instance of a system is unique." (Jochen Fromm, The Emergence of Complexity, 2004)

"The basic concept of complexity theory is that systems show patterns of organization without organizer (autonomous or self-organization). Simple local interactions of many mutually interacting parts can lead to emergence of complex global structures. […] Complexity originates from the tendency of large dynamical systems to organize themselves into a critical state, with avalanches or 'punctuations' of all sizes. In the critical state, events which would otherwise be uncoupled became correlated." (Jochen Fromm, "The Emergence of Complexity", 2004)

"Complexity arises when emergent system-level phenomena are characterized by patterns in time or a given state space that have neither too much nor too little form. Neither in stasis nor changing randomly, these emergent phenomena are interesting, due to the coupling of individual and global behaviours as well as the difficulties they pose for prediction. Broad patterns of system behaviour may be predictable, but the system's specific path through a space of possible states is not." (Steve Maguire et al, "Complexity Science and Organization Studies", 2006)

On Cybernetics (1990-1999)

"[Systems science] does not aim to find the one true representation for a given type of systems (e. g. physical, chemical or biological systems), but to formulate general principles about how different representations of different systems can be constructed so as to be effective in problem-solving." (Francis Heylighen, "Classical and non-classical representations in physics" I, Cybernetics and Systems 21, 1990)

"It seems that cybernetics is many different things to many different people. But this is because of the richness of its conceptual base; and I believe that this is very good, otherwise cybernetics would become a somewhat boring exercise. However, all of those perspectives arise from one central theme; that of circularity. When, perhaps a half century ago, the fecundity of this concept was seen, it was sheer euphoria to philosophize, epistemologize, and theorize about its unifying power and its consequences and ramification on various fields. While this was going on, something strange evolved among the philosophers, the epistemologists and, the theoreticians. They began to see themselves more and more as being included in a larger circularity; maybe within the circularity of their family; or that of their society and culture; or even being included in a circularity of cosmic proportions!" (Heinz von Foerster, "Ethics and Second-Order Cybernetics", 1991)

"The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence: the use of mathematical logic to understand the operation of the nervous system; the invention of information processing machines (as digital computers), thus laying the basis for artificial intelligence; the establishment of the metadiscipline of system theory, which has had an imprint in many branches of science, such as engineering (systems analysis, control theory), biology (regulatory physiology, ecology), social sciences (family therapy, structural anthropology, management, urban studies), and economics (game theory); information theory as a statistical theory of signal and communication channels; the first examples of self-organizing systems. This list is impressive: we tend to consider many of these notions and tools an integrative part of our life […]" (Francisco Varela, "The Embodied Mind", 1991)

"[...] that it views information as constructed and reconstructed by an individual interacting with the environment This provides an epistemological foundation of science, by viewing it as observer-dependent. Another characteristic of the new cybernetics is its contribution towards bridging the micro-macro gap. That is, it links the individual with the society." (Kenneth D Bailey, Sociology and the New Systems Theory: Toward a theoretical synthesis, 1994)

"At the very least (there is certainly more), cybernetics implies a new philosophy about (1) what we can know, (2) about what it means for something to exist, and (3) about how to get things done. Cybernetics implies that knowledge is to be built up through effective goal-seeking processes, and perhaps not necessarily in uncovering timeless, absolute, attributes of things, irrespective of our purposes and needs." (Jeff Dooley, "Thoughts on the Question: What is Cybernetics", 1995)

"Cybernetics is a science of purposeful behavior. It helps us explain behavior as the continuous action of someone (or thing) in the process, as we see it, of maintaining certain conditions near a goal state, or purpose." (Jeff Dooley, "Thoughts on the Question: What is Cybernetics", 1995)

"In sharp contrast (with the traditional social planning) the systems design approach seeks to understand a problem situation as a system of interconnected, interdependent, and interacting issues and to create a design as a system of interconnected, interdependent, interacting, and internally consistent solution ideas." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"With the subsequent strong support from cybernetics, the concepts of systems thinking and systems theory became integral parts of the established scientific language, and led to numerous new methodologies and applications - systems engineering, systems analysis, systems dynamics, and so on." (Fritjof Capra, "The Web of Life", 1996)

"Something of the previous state, however, survives every change. This is called in the language of cybernetics (which took it form the language of machines) feedback, the advantages of learning from experience and of having developed reflexes." (Guy Davenport, "The Geography of the Imagination: Forty Essays", 1997)

"Cybernetics is the science of effective organization, of control and communication in animals and machines. It is the art of steersmanship, of regulation and stability. The concern here is with function, not construction, in providing regular and reproducible behaviour in the presence of disturbances. Here the emphasis is on families of solutions, ways of arranging matters that can apply to all forms of systems, whatever the material or design employed. [...] This science concerns the effects of inputs on outputs, but in the sense that the output state is desired to be constant or predictable – we wish the system to maintain an equilibrium state. It is applicable mostly to complex systems and to coupled systems, and uses the concepts of feedback and transformations (mappings from input to output) to effect the desired invariance or stability in the result." (Chris Lucas, "Cybernetics and Stochastic Systems", 1999)

On Cybernetics (1950-1959)

"The concepts and methods of cybernetics are by no means restricted to the problems of servo-mechanisms, or even neural physiology, though the impetus came from these areas. […] Cybernetics analyzes all purposive behavior and provides an exact notion of communication and the transmital of information." (C West Churchman & Russel L Ackoff, "Purposive Behavior and Cybernetics", Social Forces, Volume 29 (1), 1950)

"The 'cybernetics' of Wiener […] is the science of organization of mechanical and electrical components for stability and purposeful actions. A distinguishing feature of this new science is the total absence of considerations of energy, heat, and efficiency, which are so important in other natural sciences. In fact, the primary concern of cybernetics is on the qualitative aspects of the interrelations among the various components of a system and the synthetic behavior of the complete mechanism." (Qian Xuesen, "Engineering Cybernetics", 1954) 

"The purpose of ‘Engineering Cybernetics’ is then to study those parts of the broad science of cybernetics which have direct engineering applications in designing controlled or guided systems. It certainly includes such topics usually treated in books on servomechanisms. But a wider range of topics is only one difference between engineering cybernetics and servomechanisms engineering. A deeper - and thus more important - difference lies in the fact that engineering cybernetics is an engineering science, while servomechanisms engineering is an engineering practice." (Qian Xuesen, "Engineering Cybernetics", 1954)

"Cybernetics is likely to reveal a great number of interesting and suggestive parallelisms between machine and brain and society. And it can provide the common language by which discoveries in one branch can readily be made use of in the others. [...] [There are] two peculiar scientific virtues of cybernetics that are worth explicit mention. One is that it offers a single vocabulary and a single set of concepts suitable for representing the most diverse types of system. [...] The second peculiar virtue of cybernetics is that it offers a method for the scientific treatment of the system in which complexity is outstanding and too important to be ignored. Such systems are, as we well know, only too common in the biological world!" (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Cybernetics treats not things but ways of behaving. It does not ask ‘what is this thing?’ but ‘what does it do?’ […] It is thus essentially functional and behaviouristic. Cybernetics deals with all forms of behavior in so far as they are regular, or determinate, or reproducible. The materiality is irrelevant. [...] The truths of cybernetics are not conditional on their being derived from some other branch of science. Cybernetics has its own foundations." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"The most fundamental concept in cybernetics is that of ‘difference’, either that two things are recognisably different or that one thing has changed with time. Its range of application need not be described now, for the subsequent chapters will illustrate the range abundantly. All the changes that may occur with time are naturally included, for when plants grow and planets age and machines move some change from one state to another is implicit. So our first task will be to develop this concept of 'change', not only making it more precise but making it richer, converting it to a form that experience has shown to be necessary if significant developments are to be made." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"There comes a stage, however, as the system becomes larger and larger, when the reception of all the information is impossible by reason of its sheer bulk. Either the recording channels cannot carry all the information, or the observer, presented with it all, is overwhelmed. When this occurs, what is he to do? The answer is clear: he must give up any ambition to know the whole system. His aim must be to achieve a partial knowledge that, though partial over the whole, is none the less complete within itself, and is sufficient for his ultimate practical purpose." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Cybernetics is not merely another branch of science. It is an intellectual revolution that rivals in importance the earlier Industrial Revolution. Is it possible that just as a machine can take over the routine functions of human muscle, another can take over the routine uses of human mind? Cybernetics answers, yes." (Isaac Asimov, [preface to Pierre de Latil’s "Thinking by Machine"] 1957)

"[Cybernetics is] the art of ensuring the efficacy of action." (Louis Couffignal, 1958)

"Cybernetics is the science of the process of transmission, processing and storage of information." (Sergei Sobolew, Woprosy Psychology, 1958)

On Cybernetics (1960-1969)

"Cybernetics is the general science of communication. But to refer to communication is consciously or otherwise to refer to distinguishable states of information inputs and outputs and /or to information being processed within some relatively isolated system." (Henryk Greniewski, "Cybernetics without Mathematics", 1960)

"Cybernetics offers a scientific approach to the cussedness of organisms, suggests how their behaviours can be catalysed and the mystique and rule of thumb banished." (Gordon Pask, "An Approach to Cybernetics", 1961)

"Cybernetics is concerned primarily with the construction of theories and models in science, without making a hard and fast distinction between the physical and the biological sciences. The theories and models occur both in symbols and in hardware, and by 'hardware’ we shall mean a machine or computer built in terms of physical or chemical, or indeed any handleable parts. Most usually we shall think of hardware as meaning electronic parts such as valves and relays. Cybernetics insists, also, on a further and rather special condition that distinguishes it from ordinary scientific theorizing: it demands a certain standard of effectiveness. In this respect it has acquired some of the same motive power that has driven research on modern logic, and this is especially true in the construction and application of artificial languages and the use of operational definitions. Always the search is for precision and effectiveness, and we must now discuss the question of effectiveness in some detail. It should be noted that when we talk in these terms we are giving pride of place to the theory of automata at the expense, at least to some extent, of feedback and information theory." (Frank H George, "The Brain As A Computer", 1962)

"The famous balance of nature is the most extraordinary of all cybernetic systems. Left to itself, it is always self-regulated." (Joseph W Krutch, Saturday Review, 1963)

"The famous balance of nature is the most extraordinary of all cybernetic systems. Left to itself, it is always self-regulated." (Joseph W Krutch, "If You Don't Mind My Saying So...: Essays on Man and Nature", 1964)

"Cybernetics is the science or the art of manipulating defensible metaphors; showing how they may be constructed and what can be inferred as a result of their existence." (Gordon Pask, "The Cybernetics of Human Performance and Learning", 1966)

"[…] cybernetics studies the flow of information round a system, and the way in which this information is used by the system as a means of controlling itself: it does this for animate and inanimate systems indifferently. For cybernetics is an interdisciplinary science, owing as much to biology as to physics, as much to the study of the brain as to the study of computers, and owing also a great deal to the formal languages of science for providing tools with which the behaviour of all these systems can be objectively described." (A Stafford Beer, 1966)

"In the language of cybernetics, maintaining reactions can be outlined as follows: the sensing material receives information about the external environment in the form of coded signals. This information is reprocessed and sent in the form of new signals through defined channels, or networks. This new information brings about an internal reorganization of the system which contributes to the preservation of its integrity. The mechanism which reprocesses the information is called the control system. It consists of a vast number of input and output elements, connected by channels through which the signals are transmitted. The information can be stored in a recall or memory system, which may consist of separate elements, each of which can be in one of several stable states. The particular state of the element varies, under the influence of the input signals. When a number of such elements are in certain specified states, information is, in effect, recorded in the form of a text of finite length, using an alphabet with a finite number of characters. These processes underlie contemporary electronic computing machines and are, in a number of respects, strongly analogous to biological memory systems." (Carl Sagan, "Intelligent Life in the Universe", 1966)

"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Cybernetics, based upon the principle of feedback or circular causal trains providing mechanisms for goal-seeking and self-controlling behavior." (Ludwig von Bertalanffy, "General System Theory", 1968)

"A more viable model, one much more faithful to the kind of system that society is more and more recognized to be, is in process of developing out of, or is in keeping with, the modern systems perspective (which we use loosely here to refer to general systems research, cybernetics, information and communication theory, and related fields). Society, or the sociocultural system, is not, then, principally an equilibrium system or a homeostatic system, but what we shall simply refer to as a complex adaptive system." (Walter F Buckley, "Society as a complex adaptive system", 1968)

"According to the science of cybernetics, which deals with the topic of control in every kind of system (mechanical, electronic, biological, human, economic, and so on), there is a natural law that governs the capacity of a control system to work. It says that the control must be capable of generating as much 'variety' as the situation to be controlled. (Anthony S Beer, "Management Science", 1968)

"Perhaps the most important single characteristic of modern organizational cybernetics is this: That in addition to concern with the deleterious impacts of rigidly-imposed notions of what constitutes the application of good 'principles of organization and management' the organization is viewed as a subsystem of a larger system(s), and as comprised itself of functionally interdependent subsystems." (Richard F Ericson, "Organizational cybernetics and human values", 1969)

On Paradox II

"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)

"When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"The paradox of reality is that no image is as compelling as the one which exists only in the mind's eye." (Shana Alexander, "Talking Woman", 1976)

 "The world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)

"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)

"The best reaction to a paradox is to invent a genuinely new and deep idea." (Ian Hacking, "An Introduction to Probability and Inductive Logic", 2001)

"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"Nature does weird things. It lives on the edge. But it is careful to bob and weave from the fatal punch of logical paradox." (Brian Greene, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", 2004)

"Paradoxes often arise because theory routinely refuses to be subordinate to reality." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

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On Paradox I

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

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

"The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself."  (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors." (Thomas H Huxley, "The Progress of Science", 1887)

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)

"The very name calculus of probabilities is a paradox. Probability opposed to certainty is what we do not know, and how can we calculate what we do not know?" (Henri Poincaré, "The Foundations of Science", 1913)

"Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man." (Bertrand Russell, "The Scientific Outlook", 1931)

"Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes." (James R Newman, "Mathematics and the Imagination", 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"A discovery in science, or a new theory, even when it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalysed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow; it takes a vast world unchallenged and for granted. This is one reason why, however great the novelty or scope of new discovery, we neither can, nor need, rebuild the house of the mind very rapidly. This is one reason why science, for all its revolutions, is conservative. This is why we will have to accept the fact that no one of us really will ever know very much. This is why we shall have to find comfort in the fact that, taken together, we know more and more." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

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On Statistics: Statistical Fallacies II

"A witty statesman said, you might prove anything by figures." (Thomas Carlyle, Chartism, 1840)

“Some of the common ways of producing a false statistical argument are to quote figures without their context, omitting the cautions as to their incompleteness, or to apply them to a group of phenomena quite different to that to which they in reality relate; to take these estimates referring to only part of a group as complete; to enumerate the events favorable to an argument, omitting the other side; and to argue hastily from effect to cause, this last error being the one most often fathered on to statistics. For all these elementary mistakes in logic, statistics is held responsible.” (Sir Arthur L Bowley, “Elements of Statistics”, 1901)

"Politicians use statistics in the same way that a drunk uses lamp-posts - for support rather than illumination." (Andrew Lang, [speech] 1910)

"Figures may not lie, but statistics compiled unscientifically and analyzed incompetently are almost sure to be misleading, and when this condition is unnecessarily chronic the so-called statisticians may be called liars." (Edwin B Wilson, "Bulletin of the American Mathematical Society", Vol 18, 1912)

"In earlier times they had no statistics and so they had to fall back on lies. Hence the huge exaggerations of primitive literature, giants, miracles, wonders! It's the size that counts. They did it with lies and we do it with statistics: but it's all the same." (Stephen Leacock, "Model memoirs and other sketches from simple to serious", 1939)

"It has long been recognized by public men of all kinds […] that statistics come under the head of lying, and that no lie is so false or inconclusive as that which is based on statistics." (Hilaire Belloc, "The Silence of the Sea", 1940)

“The enthusiastic use of statistics to prove one side of a case is not open to criticism providing the work is honestly and accurately done, and providing the conclusions are not broader than indicated by the data. This type of work must not be confused with the unfair and dishonest use of both accurate and inaccurate data, which too commonly occurs in business. Dishonest statistical work usually takes the form of: (1) deliberate misinterpretation of data; (2) intentional making of overestimates or underestimates; and (3) biasing results by using partial data, making biased surveys, or using wrong statistical methods.” (John R Riggleman & Ira N Frisbee, “Business Statistics”, 1951)

"Confidence in the omnicompetence of statistical reasoning grows by what it feeds on." (Harry Hopkins, "The Numbers Game: The Bland Totalitarianism", 1973)

"Fairy tales lie just as much as statistics do, but sometimes you can find a grain of truth in them." (Sergei Lukyanenko, "The Night Watch", 1998)

“Even properly done statistics can’t be trusted. The plethora of available statistical techniques and analyses grants researchers an enormous amount of freedom when analyzing their data, and it is trivially easy to ‘torture the data until it confesses’.” (Alex Reinhart, “Statistics Done Wrong: The Woefully Complete Guide”, 2015)

On Metaphors VIII

"The world is emblematic. Parts of speech are metaphors, because the whole of nature is a metaphor of the human mind." (Ralph W Emerson, "Nature", 1836)

"Everything which distinguishes man from the animals depends upon this ability to volatilize perceptual metaphors in a schema, and thus to dissolve an image into a concept. For something is possible in the realm of these schemata which could never be achieved with the vivid first impressions: the construction of a pyramidal order according to castes and degrees, the creation of a new world of laws, privileges, subordinations, and clearly marked boundaries - a new world, one which now confronts that other vivid world of first impressions as more solid, more universal, better known, and more human than the immediately perceived world, and thus as the regulative and imperative world." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"The various languages placed side by side show that with words it is never a question of truth, never a question of adequate expression; otherwise, there would not be so many languages. The ‘thing in itself’ (which is precisely what the pure truth, apart from any of its consequences, would be) is likewise something quite incomprehensible to the creator of language and something not in the least worth striving for. This creator only designates the relations of things to men, and for expressing these relations he lays hold of the boldest metaphors. To begin with, a nerve stimulus is transferred into an image: first metaphor. The image, in turn, is imitated in a sound: second metaphor. And each time there is a complete overleaping of one sphere, right into the middle of an entirely new and different one." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"One should employ a metaphor in science only when there is good evidence that an important similarity or analogy exists between its primary and secondary subjects. One should seek to discover more about the relevant similarities or analogies, always considering the possibility that there are no important similarities or analogies, or alternatively, that there are quite distinct similarities for which distinct terminology should be introduced. One should try to discover what the 'essential' features of the similarities or analogies are, and one should try to assimilate one’s account of them to other theoretical work in the same subject area - that is, one should attempt to explicate the metaphor." (Richard Boyd, "Metaphor and Theory Change: What Is ‘Metaphor’ a Metaphor For?", 1979)

"Myth is the system of basic metaphors, images, and stories that in-forms the perceptions, memories, and aspirations of a people; provides the rationale for its institutions, rituals and power structure; and gives a map of the purpose and stages of life." (Sam Keen, "The Passionate Life", 1983)

"Organizations are complex and paradoxical phenomena that can be understood in many different ways. Many of our taken-for-granted ideas about organizations are metaphorical, even though we may not recognize them as such. For example, we frequently talk about organizations as if they were machines designed to achieve predetermined goals and objectives, and which should operate smoothly and efficiently. And as a result of this kind of thinking, we often attempt to organize and manage them in a mechanistic way, forcing their human qualities into a background role. By using different metaphors to understand the complex and paradoxical character of organizational life, we are able to manage and design organizations in ways that we may not have thought possible before." (Gareth Morgan, "Images of Organization", 1986)

"We consider the notion of ‘system’ as an organising concept, before going on to look in detail at various systemic metaphors that may be used as a basis for structuring thinking about organisations and problem situations.“ (Mike Jackson, Creative Problem Solving: Total Systems Intervention, 1991)

"Perhaps our ultimate understanding of scientific topics is measured in terms of our ability to generate metaphoric pictures of what is going on. Maybe understanding is coming up with metaphoric pictures." (Per Bak, "How Nature Works: the science of self-organized criticality", 1996)

"We are a people captivated by the power and romance of metaphor, forever seeking the invisible through the image of the visible." (Lewis H Lapham, "Waiting For The Barbarians", 1997)

"Metaphor lives a secret life all around us. We utter about six metaphors a minute. Metaphorical thinking is essential to how we understand ourselves and others, how we communicate and learn, discover and invent." (James Geary, "I Is an Other: The Secret Life of Metaphor and How it Shapes the Way We See the World", 2011)

On Simplicity XIII (Complexity vs Simplicity V)

"Cultivate simplicity or rather should I say banish elaborateness, for simplicity springs spontaneous from the heart." (Charles Lamb, [Letter to Coleridge] 1790)

"In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance: - and […] the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth – and […] the better qualified does he become to free the exposition and application of principles from mathematical intricacy." (William J M Rankine, "On the Harmony of Theory and Practice in Mechanics", 1856) 

"The first obligation of Simplicity is that of using the simplest means to secure the fullest effect. But although the mind instinctlvely rejects all needless complexity, we shall greatly err if we fail to recognise the fact, that what the mind recoils from is not the complexity, but the needlessness." (George H Lewes, "The Principles of Success in Literature", 1865)

"Expansion means complexity, and complexity decay." (C Northcote Parkinson, "In-laws and Outlaws", 1962)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that." (Terry Eagleton, Against The Grain, 1986) 

"There is no over-arching theory of complexity that allows us to ignore the contingent aspects of complex systems. If something really is complex, it cannot by adequately described by means of a simple theory. Engaging with complexity entails engaging with specific complex systems. Despite this we can, at a very basic level, make general remarks concerning the conditions for complex behaviour and the dynamics of complex systems. Furthermore, I suggest that complex systems can be modelled." (Paul Cilliers," Complexity and Postmodernism", 1998) 

"People who pride themselves on their 'complexity' and deride others for being 'simplistic' should realize that the truth is often not very complicated. What gets complex is evading the truth." (Thomas Sowell, "Barbarians inside the Gates and Other Controversial Essays", 1999) 

"History, as well as life itself, is complicated; neither life nor history is an enterprise for those who seek simplicity and consistency." (Jared Diamond, "Collapse: How Societies Choose to Fail or Succeed", 2005)

"Simplicity in a system tends to increase that system’s efficiency. Because less can go wrong with fewer parts, less will. Complexity in a system tends to increase that system’s inefficiency; the greater the number of variables, the greater the probability of those variables clashing, and in turn, the greater the potential for conflict and disarray. Because more can go wrong, more will. That is why centralized systems are inclined to break down quickly and become enmeshed in greater unintended consequences." (Lawrence K Samuels,"Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013) 

On Simplicity XII (Complexity vs Simplicity IV)

"Simplicity of structure means organic unity, whether the organism be simple or complex; and hence in all times the emphasis which critics have laid upon Simplicity, though they have not unfrequently confounded it with narrowness of range." (George H Lewes, "The Principles of Success in Literature", 1865)

"The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, ‘Seek simplicity and distrust it’." (Alfred N Whitehead, "The Concept of Nature", 1919) 

"In products of the human mind, simplicity marks the end of a process of refining, while complexity marks a primitive stage." (Eric Hoffer, 1954)

"The machinery of the world is far too complex for the simplicity of men." (Jorge L Borges, "Dreamtigers", 1960)

"The ideas need not be complex. Most ideas that are successful are ludicrously simple. Successful ideas generally have the appearance of simplicity because they seem inevitable." (Sol LeWitt, "Paragraphs on Conceptual Art", 1967) 

"Simplicity does not precede complexity, but follows it." (Alan Perlis, "Epigrams on Programming", 1982)

"It is important to emphasize the value of simplicity and elegance, for complexity has a way of compounding difficulties and as we have seen, creating mistakes. My definition of elegance is the achievement of a given functionality with a minimum of mechanism and a maximum of clarity."  (Fernando J Corbató, "On Building Systems That Will Fail", 1991)

"When a musical piece is too simple we tend not to like it, finding it trivial. When it is too complex, we tend not to like it, finding it unpredictable - we don't perceive it to be grounded in anything familiar. Music, or any art form […] has to strike the right balance between simplicity and complexity […]" (Daniel Levitin, "This is Your Brain on Music", 2006) 

"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît Mandelbrot, "The Fractalist", 2012)

"I think there is a profound and enduring beauty in simplicity; in clarity, in efficiency. True simplicity is derived from so much more than just the absence of clutter and ornamentation. It's about bringing order to complexity." (Jonathan Ive, 2013) 

On Simplicity XI (Complexity vs Simplicity III)

"I would not give a fig for the simplicity this side of complexity, but I would give my life for the simplicity on the other side of complexity." (Oliver W Holmes Jr)

"If this seems complex, the reason is because Tao [nature] is both simple and complex. It is complex when we try to understand it, and simple when we allow ourselves to experience it." (Stanley Rosenthal)

"It is the last lesson of modern science that the highest simplicity of structure is produced, not by few elements, but by the highest complexity." (Ralph W Emerson)

"It would be simple enough, if only simplicity were not the most difficult of all things." (Carl G Jung)

"Out of intense complexities, intense simplicities emerge." (Winston S Churchill) 

"Progress is man's ability to complicate simplicity." (Thor Heyerdahl) 

"Simplicity is complexity resolved." (Constantin Brancusi)

"The art of simplicity is a puzzle of complexity." (Douglas Horton)

"The beauty of simplicity is the complexity it attracts." (Tom Robbins)

"The only simplicity for which I would give a straw is that which is on the other side of the complex - not that which never has divined it." (Oliver W Holmes Jr.)

"[...] the only simplicity to be trusted is the simplicity to be found on the far side of complexity." (Alfred N Whitehead)

"The world is a thing of utter inordinate complexity and richness and strangeness that is absolutely awesome. I mean the idea that such complexity can arise not only out of such simplicity, but probably absolutely out of nothing, is the most fabulous extraordinary idea. And once you get some kind of inkling of how that might have happened, it's just wonderful." (Douglas N Adams)

On Invention (1800-1849)

"Discoveries are not generally made in the order of their scientific arrangement: their connexions and relations are made out gradually; and it is only when the fermentation of invention has subsided that the whole clears into simplicity and order. " (William Whewell, "An Elementary Treatise on Mechanics" Vol. I, 1819)

"To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth." (David Hume, "Of the passions", 1826)

"For one person who is blessed with the power of invention, many will always be found who have the capacity of applying principles." (Charles Babbage, "Reflections on the Decline of Science in England, and on Some of Its Causes", 1830)

"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)

On Invention (-1799)

"[...] because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. Nor ought It to seem wonderful, that the invention of this as well as of other sciences, should receive its commencement from convenience and opportunity. Since whatever is carried in the circle of generation proceeds from the imperfect to the perfect." (Proclus Lycaeus, cca 5th century)

"Reflexion is careful and laborious thought, and watchful attention directed to the agreeable effect of one's plan. Invention, on the other hand, is the solving of intricate problems and the discovery of new principles by means of brilliancy and versatility." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture], cca. 15BC) 

"To invent is to discover that we know not, and not to recover or resummon that which we already know." (Francis Bacon, "Advancement of Learning", 1605)

"Moreover, the works already known are due to chance and experiment rather than to sciences; for the sciences we now possess are merely systems for the nice ordering and setting forth of things already invented; not methods of invention or directions for new works." (Sir Francis Bacon, "Novum Organum", 1620)

"To apply oneself to great inventions, starting from the smallest beginnings, is no task for ordinary minds; to divine that wonderful arts lie hid behind trivial and childish things is a conception for superhuman talents." (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632) 

"Invention, strictly speaking, is little more than a new combination of those images which have been previously gathered and deposited in the memory: nothing can come of nothing." (Joshua Reynolds, "Discourses on Art", 1769) 

"Nothing, in truth, has such a tendency to weaken not only the powers of invention, but the intellectual powers in general, as a habit of extensive and various reading without reflection." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792) 

On Invention (2000-2019)

"The passion and beauty and joy of science is that we humans have invented a process to understand the universe in a way that is true for everyone. We are finding universal truths." (Bill Nye, 2000) 

"The best reaction to a paradox is to invent a genuinely new and deep idea. (Ian Hacking, "An Introduction to Probability and Inductive Logic", 2001)

"The brain highlights what it imagines as patterns; it disregards contradictory information. Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it." (Benoît Mandelbrot, "The (Mis)Behavior of Markets", 2004)

"Metaphorizing is a manner of thinking, not a property of thinking. It is a capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects." (Ivan Mladenov, “Conceptualizing Metaphors: On Charles Peirce’s marginalia”, 2006)

"The strength of a theory is not what it allows, but what it prohibits; if you can invent an equally persuasive explanation for any outcome, you have zero knowledge." (Eliezer Yudkowsky, "An Alien God", 2007)

"We didn't invent nature. Nature invented us. Nature bats last, the saying goes, but, even more importantly, it's her playing field. We would be wise to learn the ground rules and play by them." (Kenny Ausubel, [speech at Bioneers Conference, 2003)

"Scientists often invent words to fill the holes in their understanding. These words are meant as conveniences until real understanding can be found. […] Words such as dimension and field and infinity […] are not descriptions of reality, yet we accept them as such because everyone is sure someone else knows what the words mean." (Scott Adams, "God's Debris: A Thought Experiment", 2004)

"The urge to discover, to invent, to know the unknown, seems so deeply human that we cannot imagine our history without it." (Alan Lightman, "The Discoveries: Great Breakthroughs in 20th-Century Science, Including the Original Papers", 2009)

On Invention (1975-1999)

"Mathematicians do not agree among themselves whether mathematics is invented or discovered, whether such a thing as mathematical reality exists or is illusory." (Albert L Hammond, "Mathematics - Our invisible culture", 1978)

"For the great majority of mathematicians, mathematics is […] a whole world of invention and discovery - an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor." (George F J Temple, "100 Years of Mathematics: a Personal Viewpoint", 1981)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)

"History is a constant race between invention and catastrophe." (Frank Herbert, "God Emperor of Dune", 1984)

"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) 

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"One cannot ‘invent’ the structure of an object. The most we can do is to patiently bring it to the light of day, with humility - in making it known, it is ‘discovered’. If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we are in no sense ‘making’ or ’building’ these ‘structures’. They have not waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we have been detecting or discovering, the reticent structure which we are trying to grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to ‘invent’ the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to ‘construct’ with the help of this language, bit by bit, those ‘theories’ which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them by a language in constant state improvement [...]. The sole thing that constitutes the true inventiveness and imagination of the researcher is the quality of his attention as he listens to the voices of things." (Alexander Grothendieck, "Récoltes et semailles –Rélexions et témoignage sur un passé de mathématicien", 1985)

"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)

"There is one qualitative aspect of reality that sticks out from all others in both profundity and mystery. It is the consistent success of mathematics as a description of the workings of reality and the ability of the human mind to discover and invent mathematical truths." (John D Barrow, "Theories of Everything", 1991)

"Ultimately, discovery and invention are both problems of classification, and classification is fundamentally a problem of finding sameness. When we classify, we seek to group things that have a common structure or exhibit a common behavior." (Grady Booch, "Object-oriented design: With Applications", 1991)

"As a result, surprisingly enough, scientific advance rarely comes solely through the accumulation of new facts. It comes most often through the construction of new theoretical frameworks. [..] To understand scientific development, it is not enough merely to chronicle new discoveries and inventions. We must also trace the succession of worldviews" (Nancy R Pearcey & Charles B Thaxton, "The Soul of Science: Christian Faith and Natural Philosophy", 1994)

"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)

"The controversy between those who think mathematics is discovered and those who think it is invented may run and run, like many perennial problems of philosophy. Controversies such as those between idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half thousand years. I do not expect to be able to convert those committed to the discovery view of mathematics to the inventionist view." (Paul Ernst, "Is Mathematics Discovered or Invented", 1996)

On Invention (1950-1974)

"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem." (Milton Friedman, "Essays in Positive Economics", 1953) 

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya,"Induction and Analogy in Mathematics", 1954)

"Invention is not the product of logical thought, even though the final product is tied to a logical structure." (Albert Einstein, "Autobiographische Skizze", 1955) 

"No part of the aim of normal science is to call forth new sorts of phenomena; indeed those that will not fit the box are often not seen at all. Nor do scientists normally aim to invent new theories, and they are often intolerant of those invented by others." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962) 

"We must now ask how changes of this sort can come about, considering first discoveries, or novelties of fact, and then inventions, or novelties of theory. That distinction between discovery and invention or between fact and theory will, however, immediately prove to be exceedingly artificial." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962) 

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963) 

"When the problems in physics become difficult we may often look to the mathematician who may already have studied such things and have prepared a line of reasoning for us to follow. On the other hand they may not have, in which case we have to invent our own line of reasoning, which we then pass back to the mathematician." (Richard Feynman,"The Character of Physical Law" , 1965) 

"The essential functions of the mind consist in understanding and in inventing, in other words, in building up structures by structuring reality." (Jean Piaget, 1971)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)

On Invention (1900-1949)

“With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society Vol. 8, 1902)

"The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us." (Henri Poincaré, "The Value of Science", 1905)

"It is the intuition of pure number, that of pure logical forms, which illumines and directs those we have called analysts. This it is which enables them not alone to demonstrate, but also to invent." (Henri Poincaré, "The Value of Science", 1905)

"Unadulterated, unsweetened observations are what the real nature-lover craves. No man can invent incidents and traits as interesting as the reality." (John Burroughs, "Ways of Nature", 1905)

"Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events. And the first thing to notice is this, that the only material we have at our disposal for making a picture of the whole world is supplied by the various portions of that world of which we have already had experience. We can invent no new forms of conception, applicable to the whole exclusively, and not suggested originally by the parts." (William James, "A Pluralistic Universe", 1908)

"The progressive development of man is vitally dependent on invention. It is the most important product of his creative brain. Its ultimate purpose is the complete mastery of mind over the material world, the harnessing of the forces of nature to human needs. This is the difficult task of the inventor who is often misunderstood and unrewarded. But he finds ample compensation in the pleasing exercises of his powers and in the knowledge of being one of that exceptionally privileged class without whom the race would have long ago perished in the bitter struggle against pitiless elements. Speaking for myself, I have already had more than my full measure of this exquisite enjoyment; so much, that for many years my life was little short of continuous rapture." (Nikola Tesla, "My Inventions", 1919) 

“We can invent as many theories we like, and any one of them can be made to fit the facts. But that theory is always preferred which makes the fewest number of assumptions.” (Albert Einstein [interview] 1929)

"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact." (Albert Einstein, 1930)

"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Levels of Infinity", cca. 1930)

"The discoveries in physical science, the triumphs in invention, attest the value of the process of trial and error. In large measure, these advances have been due to experimentation." (Louis Brandeis, "Judicial opinions", 1932)

"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya, “How to solve it”, 1944)

"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)

On Invention (1850-1899)

"Our inventions are wont to be pretty toys, which distract our attention from serious things. They are but improved means to an unimproved end." (Henry D Thoreau, "Walden ou la vie dans les bois", 1854) 

"They say that each generation inherits from those that have gone before; if this were so there would be no limit to man's improvements or to his power of reaching perfection. But he is very far from receiving intact that storehouse of knowledge which the centuries have piled up before him; he may perfect some inventions, but in others, he lags behind the originators, and a great many inventions have been lost entirely. What he gains on the one hand, he loses on the other." (Eugène Delacroix, 1854) 

"Nevertheless so profound is our ignorance, and so high our presumption, that we marvel when we hear of the extinction of an organic being; and as we do not see the cause, we invoke cataclysms to desolate the world, or invent laws on the duration of the forms of life!" (Charles Darwin, "On the Origin of Species by Means of Natural Selection", 1866)

"As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention." (James J Sylvester, "A plea for the mathematician", Nature Vol. 1, 1870) 

"The culture of the geometric imagination, tending to produce precision in remembrance and invention of visible forms will, therefore, tend directly to increase the appreciation of works of belles-letters." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32, 1875)

"Great inventions are never, and great discoveries are seldom, the work of any one mind. Every great invention is really an aggregation of minor inventions, or the final step of a progression. It is not usually a creation, but a growth, as truly so as is the growth of the trees in the forest." (Robert H Thurston, "The Growth of the Steam Engine", Popular Science, 1877) 

"A discoverer is a tester of scientific ideas; he must not only be able to imagine likely hypotheses, and to select suitable ones for investigation, but, as hypotheses may be true or untrue, he must also be competent to invent appropriate experiments for testing them, and to devise the requisite apparatus and arrangements." (George Gore, "The Art of Scientific Discovery", 1878) 

"Mathematics renders its best service through the immediate furthering of rigorous thought and the spirit of invention."  (Johann F Herbart, "Mathematischer Lehrplan fur Realschulen", 1890)

"The natural world has its laws, and no man must interfere with them in the way of presentment any more than in the way of use; but they themselves may suggest laws of other kinds, and man may, if he pleases, invent a little world of his own, with its own laws; for there is that in him which delights in calling up new forms - which is the nearest, perhaps, he can come to creation. When such forms are new embodiments of old truths, we call them products of the Imagination; when they are mere inventions, however lovely, I should call them the work of the Fancy: in either case, Law has been diligently at work." (George MacDonald, "The Fantastic Imagination", 1893)

"In scientific investigations, it is permitted to invent any hypothesis and, if it explains various large and independent classes of facts, it rises to the ranks of a well-grounded theory." (Charles Darwin, "The Variations of Animals and Plants Under Domestication" Vol. 1, 1896) 

On Complex Numbers X (Quantum Mechanics I)

"Meantime, there is no doubt a certain crudeness in the use of a complex wave function. If it were unavoidable in principle, and not merely a facilitation of the calculation, this would mean that there are in principle two wave functions, which must be used together in order to obtain information on the state of the system. [...] Our inability to give more accurate information about this is intimately connected with the fact that, in the pair of equations [considered], we have before us only the substitute - extraordinarily convenient for the calculation, to be sure - for a real wave equation of probably the fourth order, which, however, I have not succeeded in forming for the non-conservative case."(Edwin Schrödinger, "Quantisation as a Problem of Proper Values" , Annalen der Physik Vol. 81 (4), 1926)

"Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts. The usual method of getting the real part of a complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added." (Paul Dirac, "The Principles of Quantum Mechanics", 1930)

"In his desire to consider at any cost the propagation phenomenon of the waves ψ as something real in the classical sense of the word, the author had refused to acknowledge that the whole development of the theory increasingly tended to highlight the essential complex nature of the wave function." (Edwin Schrödinger. "Mémoires sur la mécanique ondulatoire", 1933) [author‘s comment in the French translation] 

"One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be alright in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another - it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the system and introduce an indeterminacy that will affect the second." (Ernst C K Stückelberg, "Quantum Theory in Real Hilbert Space", 1960) 

"It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex." (Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics" , Journal of Mathematical Physics Vol. 3, 1962)

"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C Baez, "Division Algebras and Quantum Theory", 2011)

"It is particularly helpful to use complex numbers to model periodic phenomena, especially to operate with phase differences. Mathematically, one can treat a physical quantity as being complex, but address physical meaning only to its real part. Another possibility is to treat the real and imaginary parts of a complex number as two related (real) physical quantities. In both cases, the structure of complex numbers is useful to make calculations more easily, but no physical meaning is actually attached to complex variables." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020)

"What is essentially different in quantum mechanics is that it deals with complex quantities (e.g. wave functions and quantum state vectors) of a special kind, which cannot be split up into pure real and imaginary parts that can be treated independently. Furthermore, physical meaning is not attached directly to the complex quantities themselves, but to some other operation that produces real numbers (e.g. the square modulus of the wave function or of the inner product between state vectors)." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020) 

On Complex Numbers IX

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." (Carl F Gauss, “Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)

"The employment of the uninterpretable symbol √-1 the intermediate processes of trigonometry furnishes an illustration of what has been said. I apprehend that there is no mode of explaining that application which does not covertly assume the very principle in question." (George Boole, "Laws of Thought", 1854)

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers." (Richard Dedekind, "Stetigkeit und irrationale Zahlen", 1872)

"Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers)." (Henry B Fine, "The Number-System of Algebra", 1890)

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900) 

 "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)
[French: "On a pu écrire depuis que la voie la plus courte et la meilleure entre deux vérités du domaine réel passe souvent par le domaine imaginaire." (Jacques Hadamard, Essai sur la psychologie de l'invention dans le domaine mathématique, 1945)]

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"Complex numbers seem to be fundamental for the description of the world proposed by quantum mechanics. In principle, this can be a source of puzzlement: Why do we need such abstract entities to describe real things? One way to refute this bewilderment is to stress that what we can measure is essentially real, so complex numbers are not directly related to observable quantities. A more philosophical argument is to say that real numbers are no less abstract than complex ones, the actual question is why mathematics is so effective for the description of the physical world." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020)

Mental Models XLIV (Spiritual & Religious Writings)

"When a soul has advanced so far on the spiritual road as to be lost to all the natural methods of communing with God; when it seeks Him no longer by meditation, images, impressions, nor by any other created ways, or representations of sense, but only by rising above them all, in the joyful communion with Him by faith and love, then it may be said to have found God of a truth, because it has truly lost itself as to all that is not God, and also as to its own self." (John of the Cross," Spiritual Canticle of The Soul and The Bridegroom", 1578) 

"Every man, as the Stoics used to say, is first and principally recommended to his own care; and every man is certainly, in every respect, fitter and abler to take care of himself than of any other person. Every man feels his own pleasures and his own pains more sensibly than those of other people. The former are the original sensations; the latter the reflected or sympathetic images of those sensations. The former may be said to be the substance; the latter the shadow. (Adam Smith, "The Theory of Moral Sentiments", 1759)

"Even Truth is of many types, like – Imaginative Truth, Practical Truth and Philosophical Truth. That which is in three times, that is called Truth and God itself is the first and the last truth.  But in practical life, truth takes many forms and as the practical truth I understand the sensible world's hard comprehensible truth. The one attained by the research of intellect, I call philosophical truth and imaginative that which illustrates through the subtle pictures of the mind." (Laxmi Prasad Devkota, "Art and Life", 1945)

"The eye projects and focuses the inner image (idea) onto the physical world in the same manner that a motion picture camera transfers an image onto a screen. The mouth creates words. The ears create sound. The difficulty in understanding this principle is due to the fact that we’ve taken it for granted that the image and sound already exist for the senses to interpret. Actually the senses are the channels of creation by which idea is projected into material expression." (Jane Roberts,"The Seth Material", 1970)

"[…] the intellect is incapable of knowing the supreme Truth; it can only range about seeking Truth, and catching fragmentary representations of it, not the thing itself, and trying to piece them together." (Ghose Aurobindo, "The Riddle of the World", 1973)

"Meditation is the emptying of the mind of the known. It cannot be done by thought or by the hidden prompting of thought, nor by desire in the form of prayer, nor through the self-effacing hypnotism of words, images, hopes, and vanities. All these have to come to an end, easily, without effort and choice, in the flame of awareness." (Jiddu Krishnamurti, "Meditations", 1979) 

"It is astonishingly beautiful and interesting, how thought is absent when you have an insight. Thought cannot have an insight. It is only when the mind is not operating mechanically in the structure of thought that you have an insight. Having had an insight, thought draws a conclusion from that insight. And then thought acts and thought is mechanical. So I have to find out whether having an insight into myself, which means into the world, and not drawing a conclusion from it is possible. If I draw a conclusion, I act on an idea, on an image, on a symbol, which is the structure of thought, and so I am constantly preventing myself from having insight, from understanding things as they are." (Jiddu Krishnamurti," On Mind and Thought", 1993)

"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)

Mental Models XLV

"Our knowledge springs from two fundamental sources of the mind; the first is the capacity of receiving representations (receptivity for impressions), the second is the power of knowing an object through these representations (spontaneity [in the production] of concepts)." (Immanuel Kant, "Critique of Pure Reason", 1781)

The traditional psychology talks like one who should say a river consists of nothing but pailsful, spoonsful, quartpotsful, barrelsful, and other moulded forms of water. Even were the pails and the pots all actually standing in the stream, still between them the free water would continue to flow. It is just this free water of consciousness that psychologists resolutely overlook. Every definite image in the mind is steeped and dyed in the free water that flows round it. With it goes the sense of its relations, near and remote, the dying echo of whence it came to us, the dawning sense of whither it is to lead." (William James, "On Some Omissions of Introspective Psychology", Mind, 1884) 

"The great basic thought that the world is not to be comprehended as a complex of ready-made things, but as a complex of processes, in which the things apparently stable no less than their mind-images in our heads, the concepts, go through an uninterrupted change of coming into being and passing away, in which, in spite of all seeming accidents and of all temporary retrogression, a progressive development asserts itself in the end - this great fundamental thought has, especially since the time of Hegel, so thoroughly permeated ordinary consciousness that in this generality it is scarcely ever contradicted." (Friedrich Engels," Ludwig Feuerbach and the Outcome of Classical German Philosophy", 1886)

"For a long time it has been known that the first systems of representations with which men have pictured to themselves the world and themselves were of religious origin. There is no religion that is not a cosmology at the same time that it is a speculation upon divine things. If philosophy and the sciences were born of religion, it is because religion began by taking the place of the sciences and philosophy." (Emile Durkheim, "The Elementary Forms of the Religious Life", 1912)

"If a fact is to be a picture, it must have something in common with what it depicts. […] What a picture must have in common with reality, in order to be able to depict it correctly or incorrectly - in the way it does, is its pictorial form. […] What any picture, of whatever form, must have in common with reality, in order to be able to depict it - correctly or incorrectly in any way at all, is logical form, i.e., the form of reality. […] Logical pictures can depict the world." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)

"A conceptual scheme is never discarded merely because of a few stubborn facts with which it cannot be reconciled; a conceptual scheme is either modified or replaced by a better one, never abandoned with nothing left to take its place."(James B Conant, "Science and Common Sense", 1951)

"As an empiricist I continue to think of the conceptual scheme of science as a tool, ultimately, for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries - not by definition in terms of experience, but simply as irreducible posits comparable, epistemologically, to the gods of Homer." (Willard v O Quine, "From a Logical Point of View", 1953) 

"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)

"The reason why new concepts in any branch of science are hard to grasp is always the same; contemporary scientists try to picture the new concept in terms of ideas which existed before." (Freeman J Dyson, "Innovation in Physics" , Scientific American, 1958)

Mental Models XLIII

"We both are, and know that we are, and delight in our being, and our knowledge of it. Moreover, in these three things no true-seeming illusion disturbs us; for we do not come into contact with these by some bodily sense, as we perceive the things outside of us of all which sensible objects it is the images resembling them, but not themselves which we perceive in the mind and hold in the memory, and which excite us to desire the objects. But, without any delusive representation of images or phantasms, I am most certain that I am, and that I know and delight in this." (Aurelius Augustinus, "The City of God", early 400s)

"The imagination is an eye where images remain forever." (Joseph Joubert, [Letter to Revd. Dr. Trusler] 1799) 

"There is a kind, I might almost say, of artistic satisfaction, when we are able to survey the enormous wealth of Nature as a regularly ordered whole - a kosmos, an image of the logical thought of our own mind." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1881) 

"I call the combination of a concept and a sound-image a sign, but in current usage the term generally designates only a sound-image, a word, for example (arbor, etc.). One tends to forget that arbor is called a sign only because it carries the concept ‘tree’, with the result that the idea of the sensory part implies the idea of the whole." (Ferdinand de Saussure, "Course in General Linguistics", 1915) 

"Thought is prior to language and consists in the simultaneous presentation to the mind of two different images." (Thomas E Hulme, "Notes on Language and Style", 1929) 

"In this way things, external objects, are assimilated to more or less ordered motor schemas, and in this continuous assimilation of objects the child's own activity is the starting point of play. Not only this, but when to pure movement are added language and imagination, the assimilation is strengthened, and wherever the mind feels no actual need for accommodating itself to reality, its natural tendency will be to distort the objects that surround it in accordance with its desires or its fantasy, in short to use them for its satisfaction. Such is the intellectual egocentrism that characterizes the earliest form of child thought." (Jean Piaget, "The Moral Judgment of the Child", 1932)

"What now is the answer to the question as to the bridge between the perception of the senses and the concepts, which is now reduced to the question as to the bridge between the outer perceptions and those inner image-like representations. It seems to me one has to postulate a cosmic order of nature - outside of our arbitrariness- to which the outer material objects are subjected as are the inner images […] The organizing and regulating has to be posited beyond the differentiation of physical and psychical […] I am all for it to call this ‘organizing and regulating’ ‘archetypes’. It would then be inadmissible to define these as psychic contents. Rather, the above-mentioned inner pictures (dominants of the collective unconscious, see Jung) are the psychic manifestations of the archetypes, but which would have to produce and condition all nature laws belonging to the world of matter. The nature laws of matter would then be the physical manifestation of the archetypes." (Wolfgang Pauli, [Letter to Markus Fierz], 1948) 

"My suggestion is that at each state the proper order of operation of the mind requires an overall grasp of what is generally known, not only in formal logical, mathematical terms, but also intuitively, in images, feelings, poetic usage of language, etc." (David Bohm,"Wholeness and the Implicate Order Wholeness and the Implicate Order", 1980) 

"Visual thinking is often driven more strongly by the conceptual knowledge we use to organize our images than by the contents of the images themselves. Chess masters are known for their remarkable memory for the pieces on a chessboard. But it's not because people with photographic memories become chess masters. The masters are no better than beginners when remembering a board of randomly arranged pieces. Their memory captures meaningful relations among the pieces, such as threats and defenses, not just their distribution in space."(Steven Pinker, "How the Mind Works", 1997)