On Nonlinearity VI

"Up until now most economists have concerned themselves with linear systems, not because of any belief that the facts were so simple, but rather because of the mathematical difficulties involved in nonlinear systems [... Linear systems are] mathematically simple, and exact solutions are known. But a high price is paid for this simplicity in terms of special assumptions which must be made." (Paul A Samuelson, "Foundations of Economic Analysis", 1966)

"Linear relationships are easy to think about: the more the merrier. Linear equations are solvable, which makes them suitable for textbooks. Linear systems have an important modular virtue: you can take them apart and put them together again - the pieces add up. Nonlinear systems generally cannot be solved and cannot be added together. [...] Nonlinearity means that the act of playing the game has a way of changing the rules. [...] That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems." (James Gleick, "Chaos: Making a New Science", 1987)

"Never in the annals of science and engineering has there been a phenomenon so ubiquitous‚ a paradigm so universal‚ or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg‚ for beneath it lies a much finer structure of immense complexity‚ a geometric labyrinth of endless convolutions‚ and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore‚ thereby forfeiting their only chance to grapple with reality." (Leon O Chua, "Editorial", International Journal of Bifurcation and Chaos, Vol. l (1), 1991) 

"It remains an unhappy fact that there is no best method for finding the solution to general nonlinear optimization problems. About the best general procedure yet devised is one that relies upon imbedding the original problem within a family of problems, and then developing relations linking one member of the family to another. If this can be done adroitly so that one family member is easily solvable, then these relations can be used to step forward from the solution of the easy problem to that of the original problem. This is the key idea underlying dynamic programming, the most flexible and powerful of all optimization methods." (John L Casti, "Five Golden Rules", 1995)

"When it comes to modeling processes that are manifestly governed by nonlinear relationships among the system components, we can appeal to the same general idea. Calculus tells us that we should expect most systems to be 'locally' flat; that is, locally linear. So a conservative modeler would try to extend the word 'local' to hold for the region of interest and would take this extension seriously until it was shown to be no longer valid." (John L Casti, "Five Golden Rules", 1995)

"A system at a bifurcation point, when pushed slightly, may begin to oscillate. Or the system may flutter around for a time and then revert to its normal, stable behavior. Or, alternatively it may move into chaos. Knowing a system within one range of circumstances may offer no clue as to how it will react in others. Nonlinear systems always hold surprises." (F David Peat, "From Certainty to Uncertainty", 2002)

"In a linear system a tiny push produces a small effect, so that cause and effect are always proportional to each other. If one plotted on a graph the cause against the effect, the result would be a straight line. In nonlinear systems, however, a small push may produce a small effect, a slightly larger push produces a proportionately larger effect, but increase that push by a hair’s breadth and suddenly the system does something radically different." (F David Peat, "From Certainty to Uncertainty", 2002)

"Complex systems are full of interdependencies - hard to detect - and nonlinear responses." (Nassim N Taleb, "Antifragile: Things That Gain from Disorder", 2012)

On Syllogism I

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, The New Organon, 1620)

"[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words." (Charles Nordmann, "Einstein and the Universe", 1922)

"Knowledge is ours only if, at the moment of need, it offers itself to the mind without syllogisms or demonstrations for which there is no time." (André Maurois, "Un Art de Vivre" ["The Art of Living"], 1939)

"A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules and syllogisms, without motivation or goal." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

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

"Since mental models can take many forms and serve many purposes, their contents are very varied. They can contain nothing but tokens that represent individuals and identities between them, as in the sorts of models that are required for syllogistic reasoning. They can represent spatial relations between entities, and the temporal or causal relations between events. A rich imaginary model of the world can be used to compute the projective relations required for an image. Models have a content and form that fits them to their purpose, whether it be to explain, to predict, or to control." (Philip Johnson-Laird, "Mental models: Toward a cognitive science of language, inference, and consciousness", 1983)

"Whenever I have talked about mental models, audiences have readily grasped that a layout of concrete objects can be represented by an internal spatial array, that a syllogism can be represented by a model of individuals and identities between them, and that a physical process can be represented by a three-dimensional dynamic model. Many people, however, have been puzzled by the representation of abstract discourse; they cannot understand how terms denoting abstract entities, properties or relations can be similarly encoded, and therefore they argue that these terms can have only 'verbal' or propositional representations." (Philip Johnson-Laird, "Mental Models: Towards a Cognitive Science of Language, Inference and Consciousness", 1983)

"Formal logic and the logical syllogism encapsulate connectedness in reasoning." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"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)

On Axioms (2000-2009)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"We start from vague pictures or ideas […] which we encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialog between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs" (E Brian Davies, "Science in the Looking Glass", 2003)

"A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists (and the greatest possibility of using different consistent axiom systems)." (Paul Cohen, "Skolem and pessimism about proof in mathematics". Philosophical Transactions of the Royal Society A 363 (1835), 2005)

"An axiomatic theory starts out of some primitive (undefined) concepts and out of a set of primitive propositions, the theory’s axioms or postulates. Other concepts are obtained by definition from the primitive concepts and from defined concepts; theorems of the theory are derived by proof mechanisms out of the axioms." (Cristian S Calude, "Randomness & Complexity, from Leibniz to Chaitin", 2007)

"Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception." (David Ruelle, "The Mathematician's Brain", 2007)

"If you have the rules of deduction and some initial choice of statements as sumed to be true (called axioms), then you are ready to derive many more true statements (called theorems). The rules of deduc tion constitute the logical machinery of mathematics, and the axioms comprise the basic properties of the objects you are interested in (in geometry these may be points, line segments, angles, etc.). There is some flexibility in selecting the rules of deduction, and many choices of axioms are possible. Once these have been decided you have all you need to do mathematics." (David Ruelle, "The Mathematician's Brain", 2007)

"In mathematics, the first principles are called axioms, and the rules are referred to as deduction/inference rules. A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound." (Cristian S Calude et al, "Proving and Programming", 2007)

"Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures." (David Ruelle, "The Mathematician's Brain", 2007)

"Reducing theorems to a small number of axioms turns out to be deeply reminiscent of what scientists do. The mark of a good scientific theory, after all, is that it describes a large number of observations of the world while making only a small number of assumptions." (Marcus Chown, "God’s Number: Where Can We Find the Secret of the Universe? In a Single Number!", 2007)

"The fact that we have an efficient conceptualization of mathematics shows that this reflects a certain mathematical reality, even if this reality is quite invisible in the formal listing of the axioms of set theory." (David Ruelle, "The Mathematician's Brain", 2007)

"We axiomatize a theory not only to better understand its inner workings but also in order to obtain metatheorems about that theory. We will therefore be interested in, say, proving that a given axiomatic treatment for some physical theory is incomplete (that is, the system exhibits the incompleteness phenomenon), among other things." (Cristian S Calude, "Randomness & Complexity, from Leibniz to Chaitin", 2007)

"When we introduce the concept of a group, we do this by imposing certain properties that should hold: these properties are called axioms. The axioms defining a group are, however, of a somewhat different nature from the ZFC axioms of set theory. Basically, whenever we do mathematics, we accept ZFC: a current mathematical paper systematically uses well-known consequences of ZFC (and normally does not mention ZFC). The axioms of a group by contrast are used only when appropriate." (David Ruelle, "The Mathematician's Brain", 2007)

"Why are proofs so important? Suppose our task were to construct a building. We would start with the foundations. In our case these are the axioms or definitions - everything else is built upon them. Each theorem or proposition represents a new level of knowledge and must be firmly anchored to the previous level. We attach the new level to the previous one using a proof. So the theorems and propositions are the new heights of knowledge we achieve, while the proofs are essential as they are the mortar which attaches them to the level below. Without proofs the structure would collapse." (Sidney A Morris, "Topology without Tears", 2007)

Mathematical Models III

"Mathematical model making is an art. If the model is too small, a great deal of analysis and numerical solution can be done, but the results, in general, can be meaningless. If the model is too large, neither analysis nor numerical solution can be carried out, the interpretation of the results is in any case very difficult, and there is great difficulty in obtaining the numerical values of the parameters needed for numerical results." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Symmetries abound in nature, in technology, and - especially - in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the 'generic' behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity." (Ian Stewart, "Bifurcation with symmetry", 1988)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics" 2nd Ed., 1992)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"Modeling, in a general sense, refers to the establishment of a description of a system (a plant, a process, etc.) in mathematical terms, which characterizes the input-output behavior of the underlying system. To describe a physical system […] we have to use a mathematical formula or equation that can represent the system both qualitatively and quantitatively. Such a formulation is a mathematical representation, called a mathematical model, of the physical system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"An important aspect of the global theory of dynamical systems is the stability of the orbit structure as a whole. The motivation for the corresponding theory comes from applied mathematics. Mathematical models always contain simplifying assumptions. Dominant features are modeled; supposed small disturbing forces are ignored. Thus, it is natural to ask if the qualitative structure of the set of solutions - the phase portrait - of a model would remain the same if small perturbations were included in the model. The corresponding mathematical theory is called structural stability." (Carmen Chicone, "Stability Theory of Ordinary Differential Equations" [Mathematics of Complexity and Dynamical Systems, 2012])

"Models do not and need not match reality in all of its aspects and details to be adequate. A mathematical model is usually developed for a specific class of target systems, and its validity is determined relative to its intended applications. A model is considered valid within its intended domain of applicability provided that its predictions in that domain fall within an acceptable range of error, specified prior to the model’s development or identification." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Simplified description of a real world system in mathematical terms, e. g., by means of differential equations or other suitable mathematical structures." (Benedetto Piccoli, Andrea Tosin, "Vehicular Traffic: A Review of Continuum Mathematical Models" [Mathematics of Complexity and Dynamical Systems, 2012])

"Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set - called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

Joseph Weizenbaum - Collected Quotes

"A higher-level formal language is an abstract machine." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation", 1976)

"A theory is, of course, not merely any grammatically correct text that uses a set of terms somehow symbolically related to reality. It is a systematic aggregate of statements of laws. Its content, its very value as theory, lies at least as much in the structure of the interconnections that relate its laws to one another, as in the laws themselves." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"Computers make possible an entirely new relationship between theories and models. I have already said that theories are texts. Texts are written in a language. Computer languages are languages too, and theories may be written in them. Indeed, for the present purpose we need not restrict our attention to machine languages or even to the kinds of 'higher-level' languages we have discussed. We may include all languages, specifically also natural languages, that computers may be able to interpret. The point is precisely that computers do interpret texts given to them, in other words, that texts determine computers' behavior. Theories written in the form of computer programs are ordinary theories as seen from one point of view." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"Machines, when they operate properly, are not merely law abiding; they are embodiments of law. To say that a specific machine is 'operating properly' is to assert that it is an embodiment of a law we know and wish to apply. We expect an ordinary desk calculator, for example, to be an embodiment of the laws of arithmetic we all know." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"Man is not a machine, [...] although man most certainly processes information, he does not necessarily process it in the way computers do. Computers and men are not species of the same genus. [...] No other organism, and certainly no computer, can be made to confront genuine human problems in human terms. [...] However much intelligence computers may attain, now or in the future, theirs must always be an intelligence alien to genuine human problems and concerns." (Joesph Weizenbaum, Computer Power and Human Reason: From Judgment to Calculation, 1976)

"Programming systems can, of course, be built without plan and without knowledge, let alone understanding, of the deep structural issues involved, just as houses, cities, systems of dams, and national economic policies can be similarly hacked together. As a system so constructed begins to get large, however, it also becomes increasingly unstable. When one of its subfunctions fails in an unanticipated way, it may be patched until the manifest trouble disappears. But since there is no general theory of the whole system, the system itself can be only a more or less chaotic aggregate of subsystems whose influence on one another's behavior is discoverable only piecemeal and by experiment. The hacker spends part of his time at the console piling new subsystems onto the structure he has already built - he calls them 'new features' - and the rest of his time in attempts to account for the way in which substructures already in place misbehave. That is what he and the computer converse about." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"The aim of the model is of course not to reproduce reality in all its complexity. It is rather to capture in a vivid, often formal, way what is essential to understanding some aspect of its structure or behavior." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"The computer programmer is a creator of universes for which he alone is the lawgiver. No playwright, no stage director, no emperor, however powerful, has ever exercised such absolute authority to arrange a stage or field of battle and to command such unswervingly dutiful actors or troops." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"The connection between a model and a theory is that a model satisfies a theory; that is, a model obeys those laws of behavior that a corresponding theory explicitly states or which may be derived from it. [...] Computers make possible an entirely new relationship between theories and models. [...] A theory written in the form of a computer program is [...] both a theory and, when placed on a computer and run, a model to which the theory applies." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"There is a distinction between physically embodied machines, whose ultimate function is to transduce energy or deliver power, and abstract machines. i.e., machines that exist only as ideas. The laws which the former embody must be a subset of the laws that govern the real world. The laws that govern the behavior of abstract machines are not necessarily so constrained. One may, for example, design an abstract machine whose internal signals are propagated among its components at speeds greater than the speed of light, in clear violation of physical law. The fact that such a machine cannot actually be built does not prohibit the exploration of its behavior." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

On Coherence (-1949)

"Principles taken upon trust, consequences lamely deduced from them, want of coherence in the parts, and of evidence in the whole, these are every where to be met with in the systems of the most eminent philosophers, and seem to have drawn disgrace upon philosophy itself." (David Hume, "A Treatise of Human Nature", 1739-40)

"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35 (904), 1912)

"A system is said to be coherent if every fact in the system is related every other fact in the system by relations that are not merely conjunctive. A deductive system affords a good example of a coherent system." (Lizzie S Stebbing, "A modern introduction to logic", 1930)

"Even these humble objects reveal that our reality is not a mere collocation of elemental facts, but consists of units in which no part exists by itself, where each part points beyond itself and implies a larger whole. Facts and significance cease to be two concepts belonging to different realms, since a fact is always a fact in an intrinsically coherent whole. We could solve no problem of organization by solving it for each point separately, one after the other; the solution had to come for the whole. Thus we see how the problem of significance is closely bound up with the problem of the relation between the whole and its parts. It has been said: The whole is more than the sum of its parts. It is more correct to say that the whole is something else than the sum of its parts, because summing is a meaningless procedure, whereas the whole-part relationship is meaningful." (Kurt Koffka, "Principles of Gestalt Psychology", 1935)

"Statistics is a scientific discipline concerned with collection, analysis, and interpretation of data obtained from observation or experiment. The subject has a coherent structure based on the theory of Probability and includes many different procedures which contribute to research and development throughout the whole of Science and Technology." (Egon Pearson, 1936)

"[…] reality is a system, completely ordered and fully intelligible, with which thought in its advance is more and more identifying itself. We may look at the growth of knowledge […] as an attempt by our mind to return to union with things as they are in their ordered wholeness. […] and if we take this view, our notion of truth is marked out for us. Truth is the approximation of thought to reality […] Its measure is the distance thought has travelled […] toward that intelligible system […] The degree of truth of a particular proposition is to be judged in the first instance by its coherence with experience as a whole, ultimately by its coherence with that further whole, all comprehensive and fully articulated, in which thought can come to rest." (Brand Blanshard, "The Nature of Thought" Vol. II, 1939)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 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)

"Induction is the process of discovering general laws by the observation and combination of particular instances. […] Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further special cases." (George Pólya, "How to solve it", 1945)

On Nonlinearity II

"Indeed, except for the very simplest physical systems, virtually everything and everybody in the world is caught up in a vast, nonlinear web of incentives and constraints and connections. The slightest change in one place causes tremors everywhere else. We can't help but disturb the universe, as T.S. Eliot almost said. The whole is almost always equal to a good deal more than the sum of its parts. And the mathematical expression of that property - to the extent that such systems can be described by mathematics at all - is a nonlinear equation: one whose graph is curvy." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)

"Today the network of relationships linking the human race to itself and to the rest of the biosphere is so complex that all aspects affect all others to an extraordinary degree. Someone should be studying the whole system, however crudely that has to be done, because no gluing together of partial studies of a complex nonlinear system can give a good idea of the behaviour of the whole." (Murray Gell-Mann, 1997)

"Much of the art of system dynamics modeling is discovering and representing the feedback processes, which, along with stock and flow structures, time delays, and nonlinearities, determine the dynamics of a system. […] the most complex behaviors usually arise from the interactions (feedbacks) among the components of the system, not from the complexity of the components themselves." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The mental models people use to guide their decisions are dynamically deficient. […] people generally adopt an event-based, open-loop view of causality, ignore feedback processes, fail to appreciate time delays between action and response and in the reporting of information, do not understand stocks and flows and are insensitive to nonlinearities that may alter the strengths of different feedback loops as a system evolves." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Most physical processes in the real world are nonlinear. It is our abstraction of the real world that leads us to the use of linear systems in modeling these processes. These linear systems are simple, understandable, and, in many situations, provide acceptable simulations of the actual processes. Unfortunately, only the simplest of linear processes and only a very small fraction of the nonlinear having verifiable solutions can be modeled with linear systems theory. The bulk of the physical processes that we must address are, unfortunately, too complex to reduce to algorithmic form - linear or nonlinear. Most observable processes have only a small amount of information available with which to develop an algorithmic understanding. The vast majority of information that we have on most processes tends to be nonnumeric and nonalgorithmic. Most of the information is fuzzy and linguistic in form." (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)

"Swarm intelligence can be effective when applied to highly complicated problems with many nonlinear factors, although it is often less effective than the genetic algorithm approach [...]. Swarm intelligence is related to swarm optimization […]. As with swarm intelligence, there is some evidence that at least some of the time swarm optimization can produce solutions that are more robust than genetic algorithms. Robustness here is defined as a solution’s resistance to performance degradation when the underlying variables are changed. (Michael J North & Charles M Macal, Managing Business Complexity: Discovering Strategic Solutions with Agent-Based Modeling and Simulation, 2007)

"[…] our mental models fail to take into account the complications of the real world - at least those ways that one can see from a systems perspective. It is a warning list. Here is where hidden snags lie. You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long-term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays. You are likely to mistreat, misdesign, or misread systems if you don’t respect their properties of resilience, self-organization, and hierarchy." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

"A network of many simple processors ('units' or 'neurons') that imitates a biological neural network. The units are connected by unidirectional communication channels, which carry numeric data. Neural networks can be trained to find nonlinear relationships in data, and are used in various applications such as robotics, speech recognition, signal processing, medical diagnosis, or power systems." (Adnan Khashman et al, "Voltage Instability Detection Using Neural Networks", 2009)

"Linearity is a reductionist’s dream, and nonlinearity can sometimes be a reductionist’s nightmare. Understanding the distinction between linearity and nonlinearity is very important and worthwhile." (Melanie Mitchell, "Complexity: A Guided Tour", 2009)

On Nonlinearity I

"In complex systems cause and effect are often not closely related in either time or space. The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by nonlinear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive-feedback loops describing growth processes as well as negative, goal-seeking loops. In the complex system the cause of a difficulty may lie far back in time from the symptoms, or in a completely different and remote part of the system. In fact, causes are usually found, not in prior events, but in the structure and policies of the system." (Jay Wright Forrester, "Urban dynamics", 1969)

"Self-organization can be defined as the spontaneous creation of a globally coherent pattern out of local interactions. Because of its distributed character, this organization tends to be robust, resisting perturbations. The dynamics of a self-organizing system is typically non-linear, because of circular or feedback relations between the components. Positive feedback leads to an explosive growth, which ends when all components have been absorbed into the new configuration, leaving the system in a stable, negative feedback state. Non-linear systems have in general several stable states, and this number tends to increase (bifurcate) as an increasing input of energy pushes the system farther from its thermodynamic equilibrium. " (Francis Heylighen, "The Science Of Self-Organization And Adaptivity", 1970)

"[The] system may evolve through a whole succession of transitions leading to a hierarchy of more and more complex and organized states. Such transitions can arise in nonlinear systems that are maintained far from equilibrium: that is, beyond a certain critical threshold the steady-state regime become unstable and the system evolves into a new configuration." (Ilya Prigogine, Gregoire Micolis & Agnes Babloyantz, "Thermodynamics of Evolution", Physics Today 25 (11), 1972)

"An artificial neural network is an information-processing system that has certain performance characteristics in common with biological neural networks. Artificial neural networks have been developed as generalizations of mathematical models of human cognition or neural biology, based on the assumptions that: 1. Information processing occurs at many simple elements called neurons. 2. Signals are passed between neurons over connection links. 3. Each connection link has an associated weight, which, in a typical neural net, multiplies the signal transmitted. 4. Each neuron applies an activation function (usually nonlinear) to its net input (sum of weighted input signals) to determine its output signal." (Laurene Fausett, "Fundamentals of Neural Networks", 1994)

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

"[…] nonlinear interactions almost always make the behavior of the aggregate more complicated than would be predicted by summing or averaging."  (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)

“[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” (Fritjof  Capra, “The web of life: a new scientific understanding of living  systems”, 1996)

"Bounded rationality simultaneously constrains the complexity of our cognitive maps and our ability to use them to anticipate the system dynamics. Mental models in which the world is seen as a sequence of events and in which feedback, nonlinearity, time delays, and multiple consequences are lacking lead to poor performance when these elements of dynamic complexity are present." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Even if our cognitive maps of causal structure were perfect, learning, especially double-loop learning, would still be difficult. To use a mental model to design a new strategy or organization we must make inferences about the consequences of decision rules that have never been tried and for which we have no data. To do so requires intuitive solution of high-order nonlinear differential equations, a task far exceeding human cognitive capabilities in all but the simplest systems."  (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"All forms of complex causation, and especially nonlinear transformations, admittedly stack the deck against prediction. Linear describes an outcome produced by one or more variables where the effect is additive. Any other interaction is nonlinear. This would include outcomes that involve step functions or phase transitions. The hard sciences routinely describe nonlinear phenomena. Making predictions about them becomes increasingly problematic when multiple variables are involved that have complex interactions. Some simple nonlinear systems can quickly become unpredictable when small variations in their inputs are introduced." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)

On Symbols (1870-1879)

"Ideas are substitutions which require a secondary process when what is symbolized by them is translated into the images and experiences it replaces; and this secondary process is frequently not performed at all, generally only performed to a very small extent. Let anyone closely examine what has passed in his mind when he has constructed a chain of reasoning, and he will be surprised at the fewness and faintness of the images which have accompanied the ideas." (George H Lewes "Problems of Life and Mind", 1873)

"Mathematicians may flatter themselves that they possess new ideas which mere human language is yet unable to express. Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations of their minds." (James C Maxwell Scottish, "Thomson & Tait's Natural Philosophy", Nature Vol. 7, 1873) 

"The invention of a new symbol is a step in the advancement of civilisation. Why were the Greeks, in spite of their penetrating intelligence and their passionate pursuit of Science, unable to carry Mathematics farther than they did? and why, having formed the conception of the Method of Exhaustions, did they stop short of that of the Differential Calculus? It was because they had not the requisite symbols as means of expression. They had no Algebra. Nor was the place of this supplied by any other symbolical language sufficiently general and flexible; so that they were without the logical instruments necessary to construct the great instrument of the Calculus." (George H Lewes "Problems of Life and Mind", 1873)

"The leading characteristic of algebra is that of operation on relations. This also is the leading characteristic of Thought. Algebra cannot exist without values, nor Thought without Feelings. The operations are so many blank forms till the values are assigned. Words are vacant sounds, ideas are blank forms, unless they symbolize images and sensations which are their values. Nevertheless it is rigorously true, and of the greatest importance, that analysts carry on very extensive operations with blank forms, never pausing to supply the symbols with values until the calculation is completed; and ordinary men, no less than philosophers, carry on long trains of thought without pausing to translate their ideas (words) into images." (George H Lewes "Problems of Life and Mind", 1873)

"The rules of Arithmetic operate in Algebra; the logical operations supposed to be peculiar to Ideation operate in Sensation, There is but one Calculus, but one Logic; though for convenience we divide the one into Arithmetic the calculus of values, and Algebra the calculus of relations; the other into the Logic of Feeling and the Logic of Signs." (George H Lewes "Problems of Life and Mind", 1873)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)

"With Algebra we enter a new sphere, that of symbolical quantities; here letters are symbols of any values we please; all we deal with in them is the relations of equality which the letters symbolise. Although the values are changeable, jet, once assigned, they must remain fixed throughout the operation. Illogical reasoning, in philosophic as in ordinary minds, is not due to any irregularity in the normal operation, but to a departure from the values assigned." (George H Lewes "Problems of Life and Mind", 1873)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpretation is a trammel to the use. Symbols are essential to comprehensive argument." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"The strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

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

On Self-Organization V

"Clearly, if the state of the system is coupled to parameters of an environment and the state of the environment is made to modify parameters of the system, a learning process will occur. Such an arrangement will be called a Finite Learning Machine, since it has a definite capacity. It is, of course, an active learning mechanism which trades with its surroundings. Indeed it is the limit case of a self-organizing system which will appear in the network if the currency supply is generalized." (Gordon Pask, "The Natural History of Networks", 1960)

"It is inherent in the logical character of the abstract self-organizing system that all available methods of organization are used, and that it cannot be realized in a single reference frame. Thus, any of the tricks which the physical model can perform, such as learning and remembering, may be performed by one or all of a variety of mechanisms, chemical or electrical or mechanical." (Gordon Pask, "The Natural History of Networks", 1960)

"An autopoietic system is organized (defined as a unity) as a network of processes of production (transformation and destruction) of components that produces the components that: (a) through their interactions and transformations continuously regenerate and realize the network of processes (relations) that produce them and, (b) constitute it (the machine) as a concrete unity in the space in which they exist by specifying the topological domain of its realization as such a network." (Francisco Varela, "Principles of Biological Autonomy", 1979)

"Self-organization is seen as the process by which systems of many components tend to reach a particular state, a set of cycling states, or a small volume of their state space (attractor basins), with no external interference." (Luis M Rocha, "Syntactic Autonomy", Proceedings of the Joint Conference on the Science and Technology of Intelligent Systems, 1998) 

"Autopoietic systems, then, are not only self-organizing systems, they not only produce and eventually change their own structures; their self-reference applies to the production of other components as well. This is the decisive conceptual innovation. […] Thus, everything that is used as a unit by the system is produced as a unit by the system itself. This applies to elements, processes, boundaries, and other structures and, last but not least, to the unity of the system itself." (Niklas Luhmann, "The Autopoiesis of Social Systems", 1990)

"The second law of thermodynamics, which requires average entropy (or disorder) to increase, does not in any way forbid local order from arising through various mechanisms of self-organization, which can turn accidents into frozen ones producing extensive regularities. Again, such mechanisms are not restricted to complex adaptive systems." (Murray Gell-Mann, "What is Complexity?", Complexity Vol 1 (1), 1995)

"I propose a new concept based on an interpretation of ecosystems: sympoietic systems. These are complex, self-organizing but collectively producing, boundaryless systems. A subsequent distinction between sympoietic and autopoietic systems is discussed. This distinction arises from defining a difference between three key system characteristics: 1) autopoietic systems have self-defined boundaries, sympoietic systems do not; 2) autopoietic systems are self-produced, sympoietic systems are collectively produced; and, 3) autopoietic systems are organizationally closed, sympoietic systems are organizationally ajar." (Beth Dempster, "Sympoietic and Autopoietic Systems: A New Distinction for Self-Organizing Systems". 2000)

"Emergent self-organization in multi-agent systems appears to contradict the second law of thermodynamics. This paradox has been explained in terms of a coupling between the macro level that hosts self-organization (and an apparent reduction in entropy), and the micro level (where random processes greatly increase entropy). Metaphorically, the micro level serves as an entropy 'sink', permitting overall system entropy to increase while sequestering this increase from the interactions where self-organization is desired." (H Van Dyke Parunak & Sven Brueckner, "Entropy and Self-Organization in Multi-Agent Systems", Proceedings of the International Conference on Autonomous Agents, 2001)

"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"A self–organizing system acts autonomously, as if the interconnecting components had a single mind. And as these components spontaneously march to the beat of their own drummer, they organize, adapt, and evolve toward a greater complexity than one would ever expect by just looking at the parts by themselves." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

On Graph Theory I

"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude." (Gottfried Leibniz, [letter to Christiaan Huygens] 1670)

"A problem was posed to me about an island in the city of Königsberg, surrounded by a river spanned by seven bridges, and I was asked whether someone could traverse the separate bridges in a connected walk in such a way that each bridge is crossed only once. I was informed that hitherto no-one had demonstrated the possibility of doing this, or shown that it is impossible. This question is so banal, but seemed to me worthy of attention in that not geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind, with any number of bridges in any arrangement, whether or not such a round trip is possible […]" (Leonard Euler, [letter to Giovanni Marinoni] 1736)

"Graph theory is the study of sets of points that are joined by lines." (Martin Gardner, "Aha! Insight", 1978)

"At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied." (Timothy Gowers, "The two cultures of mathematics", 2000)

"Euler's proof that in Königsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all the bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Königsberg graph had four such nodes, one could not find the desired path." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws [...]. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The important graphs are the ones where some things are not connected to some other things. When the unenlightened ones try to be profound, they draw endless verbal comparisons between this topic, and that topic, which is like this, which is like that; until their graph is fully connected and also totally useless." (Eliezer S Yudkowsky,  "Mysterious Answers to Mysterious Questions", 2007)

"The mathematical structure known as a graph has the valuable feature of helping us to visualize, to analyze, to generalize a situation or problem we may encounter and, in many cases, assisting us to understand it better and possibly find a solution." (Arthur Benjamin, "The fascinating world of graph theory", 2015)

"The theory of graphs is the fundamental study of relations in their purest, non-trivial form: binary connections between abstract points. And as so often in combinatorics, this simple assemblage of trivial objects results in a dazzlingly rich theory of seemingly endless depths." (Felix Reidl, "Structural Sparseness and Complex Networks", 2015)

"The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation." (James J Sylvester) 

Mental Models LV

"To imagine - to form an image - we must have the numerous relations of things present to the mind, and see the objects in their actual order. In this we are of course greatly aided by the mass of organised experience, which allows us rapidly to estimate the relations of gravity or affinity just as we remember that fire burns and that heated bodies expand. But be the aid great or small, and the result victorious or disastrous, the imaginative process is always the same." (George H Lewes, "The Principles of Success in Literature", 1865)

"The degree in which each mind habitually substitutes signs for images will be, CETERIS PARIBUS [with other conditions remaining the same], the degree in which it is liable to error. This is not contradicted by the fact that mathematical, astronomical, and physical reasonings may, when complex, be carried on more successfully by the employment of signs; because in these cases the signs themselves accurately represent the abstractness of the relations. Such sciences deal only with relations, and not with objects; hence greater simplification ensures greater accuracy. But no sooner do we quit this sphere of abstractions to enter that of concrete things, than the use of symbols becomes a source of weakness. Vigorous and effective minds habitually deal with concrete images." (George H Lewes, "The Principles of Success in Literature", 1865)

"The steps to scientific as well as other knowledge consist in a series of logical fictions which are as legitimate as they are indispensable in the operations of thought, but whose relations to the phenomena whereof they are the partial and not unfrequently merely symbolical representations must never be lost sight of." (John Stallo, "The Concepts and Theories of Modern Physics", 1884)

"Myths and science fulfil a similar function: they both provide human beings with a representation of the world and of the forces that are supposed to govern it. They both fix the limits of what is considered as possible." (François Jacob, "The Possible and the Actual", 1982)

"Many people would accept that we do not really have knowledge of the world; we have knowledge only of our representations of the world. Yet we seem condemned by our consitution to treat these representations as if they were the world, for our everyday experience feels as if it were of a given and immediate world." (Francisco Varela, "The Embodied Mind", 1991)

"The seemingly stable scene you normally see is really a mental model that you construct - the eyes are actually darting all around, producing a retinal image as jerky as an amateur video, and some of what you thought you saw was instead filled in from memory." (William H Calvin, "How Brains Think", 1996) 

"[…] the 'reality' that we perceive is based on mental models in which things don't usually change their shapes or disappear, despite their changing appearances. We mainly react to what we expect - and tend to represent the things that we see as though they remain the same as we move atomic." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)

"We all construct mental models that describe our various mental states, bodies of knowledge about our abilities, depictions of our acquaintances, and collections of stories about our pasts. Then, whenever we use our models of ourselves, we tend to use terms like conscious - when those reflections lead to choices we make, and we use unconscious or unintentional to describe those activities that we regard as beyond our control." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)

"We solve easy problems in routine ways, scarcely thinking about how we accomplish these - but when our usual methods don't work, we start to 'reflect' on what went wrong and find ourselves to be switching around in a network of 'models', each of which purports to represent some facet or aspect of ourselves, so that we end representing ourselves with a loosely connected collection of images, models, and anecdotes." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006) 

"Why must those models be simplifications? Each model must help us to focus on only those aspects that matter in some particular context; that's what makes a map more useful to us than seeing the entire landscape that it depicts. The same applies to what we store in our minds. Consider how messy our minds would become if we filled them up with descriptions of things whose details had too little significance. So instead, we spend large parts of our lives at trying to tidy up our minds - selecting the portions we want to keep, suppressing others we'd like to forget, and refining the ones we're dissatisfied with." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)

Systems Thinking IV

"One of the strongest benefits of the systems thinking perspective is that it can help you learn to ask the right questions. This is an important first step toward understanding a problem. […] Much of the value of systems thinking comes from the different framework that it gives us for looking at problems in new ways." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)

"[...] information feedback about the real world not only alters our decisions within the context of existing frames and decision rules but also feeds back to alter our mental models. As our mental models change we change the structure of our systems, creating different decision rules and new strategies. The same information, processed and interpreted by a different decision rule, now yields a different decision. Altering the structure of our systems then alters their patterns of behavior. The development of systems thinking is a double-loop learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view and then redesign our policies and institutions accordingly." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

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

"Deep change in mental models, or double-loop learning, arises when evidence not only alters our decisions within the context of existing frames, but also feeds back to alter our mental models. As our mental models change, we change the structure of our systems, creating different decision rules and new strategies. The same information, interpreted by a different model, now yields a different decision. Systems thinking is an iterative learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view, reinventing our policies and institutions accordingly." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"There exists an alternative to reductionism for studying systems. This alternative is known as holism. Holism considers systems to be more than the sum of their parts. It is of course interested in the parts and particularly the networks of relationships between the parts, but primarily in terms of how they give rise to and sustain in existence the new entity that is the whole whether it be a river system, an automobile, a philosophical system or a quality system." (Michael C. Jackson, "Systems Thinking: Creative Holism for Manager", 2003) 

"Systems thinking is only an epistemology, a particular way of describing the world. It does not tell us what the world is. Hence, strictly speaking, we should never say of something in the world: ‘It is a system’, only: ‘It may be described as a system.’" (John Mingers, Realising" Systems Thinking: Knowledge and Action in Management Science", 2006)

"At a time when the world is more messy, more crowded, more interconnected, more interdependent, and more rapidly changing than ever before, the more ways of seeing, the better. The systems-thinking lens allows us to reclaim our intuition about whole systems and hone our abilities to understand parts, see interconnections, ask 'what-if' questions about possible future behaviors, and be creative and courageous about system redesign. (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"In ecology, we are often interested in exploring the behavior of whole systems of species or ecosystem composed of individual components which interact through biological processes. We are interested not simply in the dynamics of each species or component in isolation, but the dynamics of each species or component in the context of all the others and how those coupled dynamics account for properties of the system as a whole, such as its persistence. This is what people seem to mean when they say that ecology is ‘holistic’, an otherwise rather vague term." (John Pastor, "Mathematical Ecology of Populations and Ecosystems", 2008)

"Systems thinking is a mental discipline and framework for seeing patterns and interrelationships. It is important to see organizational systems as a whole because of their complexity. Complexity can overwhelm managers, undermining confidence. When leaders can see the structures that underlie complex situations, they can facilitate improvement. But doing that requires a focus on the big picture." (Richard L Daft, "The Leadership Experience", 2008)  

"Holism [is] the art - in contrast with reductionism - of seeing a complex system as a whole. Holism knows the limits to its understanding; it acknowledges that the system has its wildness, its privacy, its own reasons, its defenses against invasive explanation." (David Fleming, "Lean Logic", 2016)

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Systems Thinking III

"Systems thinking is a discipline for seeing the 'structures' that underlie complex situations, and for discerning high from low leverage change. That is, by seeing wholes we learn how to foster health. To do so, systems thinking offers a language that begins by restructuring how we think." (Peter Senge, "The Fifth Discipline", 1990)

"Systems thinking is a discipline for seeing wholes. It is a framework for seeing interrelationships rather than things, for seeing patterns of change rather than static 'snapshots'. It is a set of general principles- distilled over the course of the twentieth century, spanning fields as diverse as the physical and social sciences, engineering, and management. [...] During the last thirty years, these tools have been applied to understand a wide range of corporate, urban, regional, economic, political, ecological, and even psychological systems. And systems thinking is a sensibility for the subtle interconnectedness that gives living systems their unique character." (Peter Senge, "The Fifth Discipline", 1990)

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

"Systems philosophy brings forth a reorganization of ways of thinking. It creates a new worldview, a new paradigm of perception and explanation, which is manifested in integration, holistic thinking, purpose-seeking, mutual causality, and process-focused inquiry.” (Béla H. Bánáthy, "Systems Design of Education”, 1991)

"The new paradigm may be called a holistic world view, seeing the world as an integrated whole rather than a dissociated collection of parts. It may also be called an ecological view, if the term 'ecological' is used in a much broader and deeper sense than usual. Deep ecological awareness recognizes the fundamental interdependence of all phenomena and the fact that, as individuals and societies we are all embedded in (and ultimately dependent on) the cyclical process of nature." (Fritjof Capra & Gunter A Pauli, "Steering business toward sustainability", 1995)

"In the new systems thinking, the metaphor of knowledge as a building is being replaced by that of the network. As we perceive reality as a network of relationships, our descriptions, too, form an interconnected network of concepts and models in which there are no foundations. For most scientists such a view of knowledge as a network with no firm foundations is extremely unsettling, and today it is by no means generally accepted. But as the network approach expands throughout the scientific community, the idea of knowledge as a network will undoubtedly find increasing acceptance." (Fritjof Capra," The Web of Life: a new scientific understanding of living systems", 1996)

"It [system dynamics] focuses on building system dynamics models with teams in order to enhance team learning, to foster consensus and to create commitment with a resulting decision […] System dynamics can be helpful to elicit and integrate mental models into a more holistic view of the problem and to explore the dynamics of this holistic view […] It must be understood that the ultimate goal of the intervention is not to build a system dynamics model. The system dynamics model is a means to achieve other ends […] putting people in a position to learn about a messy problem [...] create a shared social reality […] a shared understanding of the problem and potential solutions [...] to foster consensus within the team [..]" (Jac A M Vennix, "Group Model Building: Facilitating Team Learning Using System Dynamics", 1996)

"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking - from the parts to the whole, from objects to relationships, from contents to patterns. […] Nourishing the community means nourishing those relationships." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"Systems thinking practices the exact opposite of this analytic approach. Systems thinking studies the organization as a whole in its interaction with its environment. Then, it works backwards to understand how each part of that whole works in relation to, and support of, the entire system’s objectives. Only then can the core strategies be formulated." (Stephen G Haines, "The Systems Thinking Approach to Strategic Planning and Management", 2000)

"Systems, and organizations as systems, can only be understood holistically. Try to understand the system and its environment first. Organizations are open systems and, as such, are viable only in interaction with and adaptation to the changing environment." (Stephen G Haines, "The Systems Thinking Approach to Strategic Planning and Management", 2000)

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Mental Models XL

"When the Soul wants to experience something she throws out an image in front of her and then steps into it." (Eckhart von Hochheim [aka Meister Eckhart] cca. 14th century)

"Materialism is the recognition of 'objects in themselves' , or outside the mind; ideas and sensations are copies of images of those objects." (Vladimir Lenin, "Materialism and Empirio-Criticism", 1908) 

"Images apparently occupy a curious position somewhere between the statements of language, which are intended to convey a meaning, and the things of nature, to which we only can give a meaning." (Ernst Gombrich, "Symbolic Images", 1972) 

 "When one analyzes the pre-conscious step to concepts, one always finds ideas which consist of 'symbolic images'. The first step to thinking is a painted vision of these inner pictures whose origin cannot be reduced only and firstly to the sensual perception but which are produced by an 'instinct to imagining' and which are re-produced by different individuals independently, i. e. collectively [...] But the archaic image is also the necessary predisposition and the source of a scientific attitude. To a total recognition belong also those images out of which have grown the rational concepts." (Wolfgang Pauli, [Letter to Markus Fierz] 1948) 

" The symbol is the tool which gives man his power, and it is the same tool whether the symbols are images or words, mathematical signs or mesons." (Jacob Bronowski, "The Reach of Imagination", 1967) 

"To imagine means to make images and to move them about inside one's head in new arrangements." (Jacob Bronowski, "The Reach of Imagination", 1967)

"These organizational processes result in our perceptions being structured into units corresponding to objects and properties of objects. It is these larger units that may be stored and later assembled into images that are experienced as quasi-pictorial, spatial entities resembling those evoked during perception itself [...] It is erroneous to equate image representations with mental photographs, since this would overlook the fact that images are composed from highly processed perceptual encodings." (Stephen Kosslyn, "Image and Mind", 1980) 

"Vision is a process that produces from images of the external world a description that is useful to the viewer and not cluttered with irrelevant information." (David Marr, "Vision", 1982)

"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) 

"But because of the way in which depictions represent, there is a correspondence between parts and spatial relations of the representation and those of the object; this structural mapping, which confers a type of resemblance, underlies the way images convey specific content. In this respect images are like pictures. Unlike words and symbols, depictions are not arbitrarily paired with what they represent." (Stephen Kosslyn et al," The Case for Mental Imagery", 2006)

From Parts to Wholes (1800-1849)

"In every moment of her duration Nature is one connected whole; in every moment each individual part must be what it is, because all the others are what they are; and you could not remove a single grain of sand from its place, without thereby, although perhaps imperceptibly to you, altering something throughout all parts of the immeasurable whole." (Johann G Fichte, "The Vocation of Man”, 1800)

"It is the destiny of our race to become united into one great body, thoroughly connected in all its parts, and possessed of similar culture. Nature, and even the passions and vices of Man, have from the beginning tended towards this end. A great part of the way towards it is already passed, and we may surely calculate that it will in time be reached." (Johann G Fichte, "The Vocation of Man", 1800)

"It is probable that what we call thought is not an actual being, but no more than the relation between certain parts of that infinitely varied mass, of which the rest of the universe is composed, and which ceases to exist as soon as those parts change their position with regard to each other." (Percy B Shelley, "On a Future State", 1815)

"Each of the parts of philosophy is a philosophical whole, a circle rounded and complete in itself. In each of these parts, however, the philosophical Idea is found in a particular specificality or medium. The single circle, because it is a real totality, bursts through the limits imposed by its special medium, and gives rise to a wider circle. The whole of philosophy in this way resembles a circle of circles. The Idea appears in each single circle, but, at the same time, the whole Idea is constituted by the system of these peculiar phases, and each is a necessary member of the organisation." (Georg W F Hegel, "Encyclopedia of the Philosophical Sciences", 1816)

"When the whole and the parts are seen at once, as mutually producing and explaining each other, as unity in multeity, there results shapeliness." (Samuel T Coleridge, "Letters", 1836)

"Science is nothing but the finding of analogy, identity, in the most remote parts." (Ralph W Emerson, 1837)

"So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically." (Antoine A Cournot, "Researches into the Mathematical Principles of the Theory of Wealth", 1838)

"The component parts of a vegetable or animal substance do not lose their mechanical and chemical properties as separate agents, when, by a peculiar mode of juxtaposition, they, as an aggregate whole, acquire physiological or vital properties in addition. Those bodies continue, as before, to obey mechanical and chemical laws, in so far as the operation of those laws is not counteracted by the new laws which govern them as organized beings. […] Though there are laws which, like those of chemistry and physiology, owe their existence to a breach of the principle of Composition of Causes, it does not follow that these peculiar, or, as they might be termed, heteropathic laws, are not capable of composition with one another." (John S Mill, "A System of Logic: Ratioconative and Inductive", 1843) [the heteropathic laws is synonymous with emergence]

From Parts to Wholes (1900-1909)

"And as the ideal in the whole of Nature moves in an infinite process toward an Absolute Perfection, we may say that art is in strict truth the apotheosis of Nature. Art is thus at once the exaltation of the natural toward its destined supernatural perfection, and the investiture of the Absolute Beauty with the reality of natural existence. Its work is consequently not a means to some higher end, but is itself a final aim; or, as we may otherwise say, art is its own end. It is not a mere recreation for man, a piece of by-play in human life, but is an essential mode of spiritual activity, the lack of which would be a falling short of the destination of man. It is itself part and parcel of man's eternal vocation." (George H Howison, "The Limits of Evolution, and Other Essays, Illustrating the Metaphysical Theory of Personal Idealism", 1901) 

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments." (David Hilbert, "Mathematical Problems", Bulletin American Mathematical Society Vol. 8, 1901-1902)

"For if society lacks the unity that derives from the fact that the relationships between its parts are exactly regulated, that unity resulting from the harmonious articulation of its various functions assured by effective discipline and if, in addition, society lacks the unity based upon the commitment of men's wills to a common objective, then it is no more than a pile of sand that the least jolt or the slightest puff will suffice to scatter.“ (Émile Durkheim, 1903)

"From that time, the universe has steadily become more complex and less reducible to a central control. With as much obstinacy as though it were human, it has insisted on expanding its parts; with as much elusiveness as though it were feminine, it has evaded the attempt to impose on it a single will. Modern science, like modern art, tends, in practice, to drop the dogma of organic unity. Some of the mediaeval habit of mind survives, but even that is said to be yielding before the daily evidence of increasing and extending complexity. The fault, then, was not in man, if he no longer looked at science or art as an organic whole or as the expression of unity. Unity turned itself into complexity, multiplicity, variety, and even contradiction." (Henry Adams, "Mont Saint Michel and Chartres", 1904)

"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)

"A system is a whole which is composed of various parts. But it is not the same thing as an aggregate or heap. In an aggregate or heap, no essential relation exists between the units of which it is composed. In a heap of grain, or pile of stones, one may take away part without the other part being at all affected thereby. But in a system, each part has a fixed and necessary relation to the whole and to all the other parts. For this reason we may say that a building, or a peace of mechanisme, is a system. Each stone in the building, each wheel in the watch, plays a part, and is essential to the whole." (James E Creighton, "An Introductory Logic"‎, 1909)

From Parts to Wholes (1930-1939)

"Everything abstract is ultimately part of the concrete. Everything inanimate finally serves the living. That is why every activity dealing in abstraction stands in ultimate service to a living whole." (Edith Stein, "The Ethos of Woman's Professions", 1930)

"To apply the category of cause and effect means to find out which parts of nature stand in this relation. Similarly, to apply the gestalt category means to find out which parts of nature belong as parts to functional wholes, to discover their position in these wholes, their degree of relative independence, and the articulation of larger wholes into sub-wholes." (Kurt Koffka, 1931)

"Even these humble objects reveal that our reality is not a mere collocation of elemental facts, but consists of units in which no part exists by itself, where each part points beyond itself and implies a larger whole. Facts and significance cease to be two concepts belonging to different realms, since a fact is always a fact in an intrinsically coherent whole. We could solve no problem of organization by solving it for each point separately, one after the other; the solution had to come for the whole. Thus we see how the problem of significance is closely bound up with the problem of the relation between the whole and its parts. It has been said: The whole is more than the sum of its parts. It is more correct to say that the whole is something else than the sum of its parts, because summing is a meaningless procedure, whereas the whole-part relationship is meaningful." (Kurt Koffka, "Principles of Gestalt Psychology", 1935)

"Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. Thus it may be that some part of what one knows may pertain to relations […] between the two subsystems (we shall limit ourselves to two), as follows: if a particular measurement on the first system yields this result, then for a particular measurement on the second the valid expectation statistics are such and such; but if the measurement in question on the first system should have that result, then some other expectation holds for that one the second. […] In this way, any measurement process at all or, what amounts to the same, any variable at all of the second system can be tied to the not-yet-known value of any variable at all of the first, and of course vice versa also." (Erwin Schrödinger, "The Present Situation in Quantum Mechanics", 1935)

"The existence of finality within the organism is undeniable. Each part seems to know the present and future needs of the whole, and acts accordingly. The significance of time and space is not the same for our tissues as for our mind. The body perceives the remote as well as the near, the future as well as the present." (Alexis Carrel, 1935)

"[Gestalt:] a system whose parts are dynamically connected in such a way that a change of one part results in a change of all other parts." (Kurt Lewin, "Principles of topological psychology", 1936)

"The realist method starts with the whole in order to distinguish the parts." (Étienne Gilson, "Methodical Realism", 1936)

"An organization is a subordinate system of a specific larger system, the cooperative system, whose components are physical, biological, and personal systems. The relations with other organizations… are outside this specific cooperative system. Other organizations are a part of the social environment of the organization. It is for this reason I have used the phrase 'complex of organizations' rather than 'system'. Usually the most significant relationships of a unit organization are those with the specific cooperative system of which it is a part. It is this system which primarily and on the whole in most instances determines the chief conditions of the organization's existence." (Chester Barnard, "The Functions of the Executive The Functions of the Executive", 1938)

"I see the tasks of social sciences to discover what kinds of order actually do exist in the whole range of the behavior of human beings; what kind of functional relationships between different parts of culture exist in space and over time, and what functionally more useful kinds of order can be created." (Robert S Lynd, "Knowledge of What?", 1939)

"The failure of the social sciences to think through and to integrate their several responsibilities for the common problem of relating the analysis of parts to the analysis of the whole constitutes one of the major lags crippling their utility as human tools of knowledge." (Robert S Lynd, "Knowledge of What?", 1939)

"The whole, though larger than any of its parts, does not necessarily obscure their separate identities." (William O Douglas, "Judicial Opinions", 1939)

From Parts to Wholes (1950-1959)

"For scientific endeavor is a natural whole the parts of which mutually support one another in a way which, to be sure, no one can anticipate." (Albert Einstein, "Out of My Later Years", 1950)

"What the worker needs is to see the plant as if he were a manager. Only thus can he see his part, from his part he can reach the whole. This ‘seeing’ is not a matter of information, training courses, conducted plant tours, or similar devices. What is needed is the actual experience of the whole in and through the individual's work." (Peter F Drucker, "The New Society", 1950)

"Every organism represents a system, by which term we mean a complex of elements in mutual interaction. From this obvious statement the limitations of the analytical and summative conceptions must follow. First, it is impossible to resolve the phenomena of life completely into elementary units; for each individual part and each individual event depends not only on conditions within itself, but also to a greater or lesser extent on the conditions within the whole, or within superordinate units of which it is a part. Hence the behavior of an isolated part is, in general, different from its behavior within the context of the whole. [...] Secondly, the actual whole shows properties that are absent from its isolated parts." (Ludwig von Bertalanffy, "Problems of Life", 1952)

"Individualism is the self-affirmation of the individual self as individual self without regard to its participation in its world. As such it is the opposite of collectivism, the self affirmation of the self as part of a larger whole without regard to its character as an individual self." (Paul Tillich, "The Courage to Be The Courage to Be", 1952)

"Every part of the system is so related to every other part that any change in one aspect results in dynamic changes in all other parts of the total system." (Arthur D Hall & Robert E Fagen, "Definition of System", General Systems Vol. 1, 1956)

"In our definition of system we noted that all systems have interrelationships between objects and between their attributes. If every part of the system is so related to every other part that any change in one aspect results in dynamic changes in all other parts of the total system, the system is said to behave as a whole or coherently. At the other extreme is a set of parts that are completely unrelated: that is, a change in each part depends only on that part alone. The variation in the set is the physical sum of the variations of the parts. Such behavior is called independent or physical summativity." (Arthur D Hall & Robert E Fagen, "Definition of System", General Systems Vol. 1, 1956)

"In our definition of system we noted that all systems have interrelationships between objects and between their attributes. If every part of the system is so related to every other part that any change in one aspect results in dynamic changes in all other parts of the total system, the system is said to behave as a whole or coherently. At the other extreme is a set of parts that are completely unrelated: that is, a change in each part depends only on that part alone. The variation in the set is the physical sum of the variations of the parts. Such behavior is called independent or physical summativity." (Arthur D Hall & Robert E Fagen, "Definition of System", General Systems Vol. 1, 1956)

"[...] the concept of 'feedback', so simple and natural in certain elementary cases, becomes artificial and of little use when the interconnexions between the parts become more complex. When there are only two parts joined so that each affects the other, the properties of the feedback give important and useful information about the properties of the whole. But when the parts rise to even as few as four, if every one affects the other three, then twenty circuits can be traced through them; and knowing the properties of all the twenty circuits does not give complete information about the system. Such complex systems cannot be treated as an interlaced set of more or less independent feedback circuits, but only as a whole. For understanding the general principles of dynamic systems, therefore, the concept of feedback is inadequate in itself. What is important is that complex systems, richly cross-connected internally, have complex behaviours, and that these behaviours can be goal-seeking in complex patterns." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"It is clear to all that the animal organism is a highly complex system consisting of an almost infinite series of parts connected both with one another and, as a total complex, with the surrounding world, with which it is in a state of equilibrium." (Ivan P Pavlov, "Experimental psychology, and other essays", 1957)

"[…] the comprehension of a structure requires intuitive knowledge of the ethology of its resistance and of its constituent materials." (Eduardo Torroja, "Philosophy of Structures/;, 1958)