On Knowledge (1940-1949)

"It is by abstraction that one can separate movements, knowledge, and affectivity. And the analysis is, here, so far from being a real dismemberment that it is given only as probable. One can never effectively reduce an [mental] image to its elements, for the reason that an image, like all other psychic syntheses, is something more and different from the sum of its elements. […] We will always go from image to image. Comprehension is a movement which is never-ending, it is the reaction of the mind to an image by another image, to this one by another image and so on, in principle to infinity. "(Jean-Paul Sartre, "The Imaginary: A phenomenological psychology of the imagination", 1940)

"In perception, a knowledge forms itself slowly; in the [mental] image the knowledge is immediate. We see now that the image is a synthetic act which unites a concrete, nonimagined, knowledge to elements which are more actually representative. The image teaches nothing: it is organized exactly like the objects which do produce knowledge, but it is complete at the very moment of its appearance." (Jean-Paul Sartre, "The Psychology of Imagination", 1940)

"Science, in the broadest sense, is the entire body of the most accurately tested, critically established, systematized knowledge available about that part of the universe which has come under human observation. For the most part this knowledge concerns the forces impinging upon human beings in the serious business of living and thus affecting man’s adjustment to and of the physical and the social world. […] Pure science is more interested in understanding, and applied science is more interested in control […]" (Austin L Porterfield, "Creative Factors in Scientific Research", 1941)

“A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling.” (Robert Musil, “Man Without Qualities”, 1943)

"It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge."  (George Pólya, "How to solve it", 1945)

"The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach." (George Pólya, "How to solve it", 1945) 

"Our theory has some bleaker consequences. [...] What is knowledge, if we are but a part of the mechanical system of the world we seek to know? What becomes of our ceaseless effort to explain the universe we live in, if explanation is but a part of the mechanical process?" (Kenneth Craik, "The Nature of Explanation", 1943)

"Whenever a man increases the content of his mind he gains new knowledge, and this occurs each time a new relation is established between the worlds on the two sides of the sense-organs - the world of ideas in an individual mind, and the world of objects existing outside individual minds which is common to us all." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"Science usually advances by a succession of small steps, through a fog in which even the most keen-sighted explorer can seldom see more than a few paces ahead. Occasionally the fog lifts, an eminence is gained, and a wider stretch of territory can be surveyed - sometimes with startling results. A whole science may then seem to undergo a kaleidoscopic ‘rearrangement’, fragments of knowledge being found to fit together in a hitherto unsuspected manner. Sometimes the shock of readjustment may spread to other sciences; sometimes it may divert the whole current of human thought." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"The former distrust of specialization has been supplanted by its opposite, a distrust of generalization. Not only has man become a specialist in practice, he is being taught that special facts represent the highest form of knowledge." (Richard Weaver, "Ideas have Consequences", 1948)

"We all inherit a great deal of useless knowledge, and a great deal of misinformation and error (maps that were formerly thought to be accurate), so that there is always a portion of what we have been told that must be discarded. But the cultural heritage of our civilization that is transmitted to us - our socially pooled knowledge, both scientific and humane - has been valued principally because we have believed that it gives us accurate maps of experience. The analogy of verbal words to maps is an important one [...]. It should be noticed at this point, however, that there are two ways of getting false maps of the world into our heads: first, by having them given to us; second, by creating them ourselves when we misread the true maps given to us." (Samuel I Hayakawa, "Language in Thought and Action", 1949)

On Knowledge (1990-1999)

"[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge." (Howard Gardner, "The Unschooled Mind", 1991)

"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. [...] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)

"We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance." (John A Wheeler, Scientific American Vol. 267, 1992)

"Indeed, knowledge that one will be judged on some criterion of ‘creativeness’ or ‘originality’ tends to narrow the scope of what one can produce (leading to products that are then judged as relatively conventional); in contrast, the absence of an evaluations seems to liberate creativity." (Howard Gardner, "Creating Minds", 1993)

"Knowledge is theory. We should be thankful if action of management is based on theory. Knowledge has temporal spread. Information is not knowledge. The world is drowning in information but is slow in acquisition of knowledge. There is no substitute for knowledge." (William E Deming, "The New Economics for Industry, Government, Education", 1993)

"Worldviews are social constructions, and they channel the search for facts. But facts are found and knowledge progresses, however fitfully. Fact and theory are intertwined, and all great scientists understand the interaction." (Stephen J Gould, "Shields of Expectation - and Actuality", 1993)

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

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

"Humans may crave absolute certainty; they may aspire to it; they may pretend, as partisans of certain religions do, to have attained it. But the history of science - by far the most successful claim to knowledge accessible to humans - teaches that the most we can hope for is successive improvement in our understanding, learning from our mistakes, an asymptotic approach to the Universe, but with the proviso that absolute certainty will always elude us. We will always be mired in error. The most each generation can hope for is to reduce the error bars a little, and to add to the body of data to which error bars apply." (Carl Sagan, "The Demon-Haunted World: Science as a Candle in the Dark", 1995)

"The amount of understanding produced by a theory is determined by how well it meets the criteria of adequacy - testability, fruitfulness, scope, simplicity, conservatism - because these criteria indicate the extent to which a theory systematizes and unifies our knowledge." (Theodore Schick Jr., "How to Think about Weird Things: Critical Thinking for a New Age", 1995)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states." (Robert Hafner & Jim Stewart, "Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’", Science Education 79 (2), 1995) 

"Generalization is the process of matching new, unknown input data with the problem knowledge in order to obtain the best possible solution, or one close to it. Generalization means reacting properly to new situations, for example, recognizing new images, or classifying new objects and situations. Generalization can also be described as a transition from a particular object description to a general concept description. This is a major characteristic of all intelligent systems." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"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)

"Discourses are ways of referring to or constructing knowledge about a particular topic of practice: a cluster (or formation) of ideas, images and practices, which provide ways of talking about, forms of knowledge and conduct associated with, a particular topic, social activity or institutional site in society. These discursive formations, as they are known, define what is and is not appropriate in our formulation of, and our practices in relation to, a particular subject or site of social activity." (Stuart Hall, "Representation: Cultural Representations and Signifying Practices", 1997)

"Data is discrimination between physical states of things (black, white, etc.) that may convey or not convey information to an agent. Whether it does so or not depends on the agent's prior stock of knowledge." (Max Boisot, "Knowledge Assets", 1998)

"The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)

"Analysis of a system reveals its structure and how it works. It provides the knowledge required to make it work efficiently and to repair it when it stops working. Its product is know-how, knowledge, not understanding. To enable a system to perform effectively we must understand it - we must be able to explain its behavior—and this requires being aware of its functions in the larger systems of which it is a part." (Russell L Ackoff, "Re-Creating the Corporation", 1999)

On Knowledge (1980-1989)

"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)

"A schema, then is a data structure for representing the generic concepts stored in memory. There are schemata representing our knowledge about all concepts; those underlying objects, situations, events, sequences of events, actions and sequences of actions. A schema contains, as part of its specification, the network of interrelations that is believed to normally hold among the constituents of the concept in question. A schema theory embodies a prototype theory of meaning. That is, inasmuch as a schema underlying a concept stored in memory corresponds to the meaning of that concept, meanings are encoded in terms of the typical or normal situations or events that instantiate that concept." (David E Rumelhart, "Schemata: The building blocks of cognition", 1980)

“Analogies, metaphors, and emblems are the threads by which the mind holds on to the world even when, absentmindedly, it has lost direct contact with it, and they guarantee the unity of human experience. Moreover, in the thinking process itself they serve as models to give us our bearings lest we stagger blindly among experiences that our bodily senses with their relative certainty of knowledge cannot guide us through.” (Hannah Arendt, “The Life of the Mind”, 1981)

"Knowledge specialists may ascribe a degree of certainty to their models of the world that baffles and offends managers. Often the complexity of the world cannot be reduced to mathematical abstractions that make sense to a manager. Managers who expect complete, one-to-one correspondence between the real world and each element in a model are disappointed and skeptical." (Dale E Zand, "Information, Organization, and Power", 1981)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"Definitions are temporary verbalizations of concepts, and concepts - particularly difficult concepts - are usually revised repeatedly as our knowledge and understanding grows." (Ernst Mayr, "The Growth of Biological Thought", 1982)

"We are drowning in information but starved for knowledge." (John Naisbitt, "Megatrends: Ten New Directions Transforming Our Lives", 1982)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to motor mechanisms' of the knowledge." (John F Sowa, "Conceptual Structures", 1984)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"Knowledge is the appropriate collection of information, such that it's intent is to be useful. Knowledge is a deterministic process. When someone 'memorizes' information (as less-aspiring test-bound students often do), then they have amassed knowledge. This knowledge has useful meaning to them, but it does not provide for, in and of itself, an integration such as would infer further knowledge." (Russell L Ackoff, "Towards a Systems Theory of Organization", 1985)

"Although science literally means ‘knowledge’, the scientific attitude is concerned much more with rational perception through the mind and with testing such perceptions against actual fact, in the form of experiments and observations." (David Bohm & F David Peat, "Science, Order, and Creativity", 1987)

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

"We admit knowledge whenever we observe an effective (or adequate) behavior in a given context, i.e., in a realm or domain which we define by a question (explicit or implicit)." (Humberto Maturana & Francisco J Varela, "The Tree of Knowledge", 1987)

"In the Information Age, the first step to sanity is FILTERING. Filter the information: extract for knowledge. Filter first for substance. Filter second for significance. […] Filter third for reliability. […] Filter fourth for completeness." (Marc Stiegler, "David’s Sling", 1988)

"Probabilities are summaries of knowledge that is left behind when information is transferred to a higher level of abstraction." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference", 1988)

"Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"A discovery in science, or a new theory, even where it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalyzed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow: it takes a vast world unchallenged and for granted." (James R Oppenheimer, "Atom and Void", 1989)

On Knowledge (1950-1959)

"Every bit of knowledge we gain and every conclusion we draw about the universe or about any part or feature of it depends finally upon some observation or measurement. Mankind has had again and again the humiliating experience of trusting to intuitive, apparently logical conclusions without observations, and has seen Nature sail by in her radiant chariot of gold in an entirely different direction." (Oliver J Lee, "Measuring Our Universe: From the Inner Atom to Outer Space", 1950)

"The essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"[The information of a message can] be defined as the 'minimum number of binary decisions which enable the receiver to construct the message, on the basis of the data already available to him.' These data comprise both the convention regarding the symbols and the language used, and the knowledge available at the moment when the message started." (Dennis Gabor, "Optical transmission" in Information Theory : Papers Read at a Symposium on Information Theory, 1952)

"The world is not made up of empirical facts with the addition of the laws of nature: what we call the laws of nature are conceptual devices by which we organize our empirical knowledge and predict the future." (Richard B Braithwaite, "Scientific Explanation", 1953)

"Knowledge rests on knowledge; what is new is meaningful because it departs slightly from what was known before; this is a world of frontiers, where even the liveliest of actors or observers will be absent most of the time from most of them." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

"Scientific metaphors are called models. They are made with the full knowledge that the connection between the metaphor and the real thing is primarily in the mind of the scientist. And they are made with a clearly definable purpose - as starting points of a deductive process. […] Like every other aspect of scientific procedure, the scientific metaphor is a pragmatic device, to be used freely as long as it serves its purpose, to be discarded without regrets when it fails to do so." (Anatol Rapoport, "Operational Philosophy", 1954)

"The laws of science are the permanent contribution to knowledge - the individual pieces which are fitted together attempt to form a picture of the physical universe in action." (Edwin P Hubble, "The Nature of Science and Other Lectures", 1954)

"Science cannot be based on dogma or authority of any kind, nor on any institution or revelation, unless indeed it be of the Book of Nature that lies open before our eyes. We need not dwell on the processes of acquiring knowledge by observation, experiment, and inductive and deductive reasoning. The study of scientific method both in theory and practice is of great importance. It is inherent in the philosophy that the record may be imperfect and the conceptions erroneous; the potential fallibility of our science is not only acknowledged but also insisted upon." (Sir Robert Robinson, "Science and the Scientist", Nature Vol. 176 (4479), 1955)

"There comes a point where the mind takes a leap - call it intuition or what you will - and comes out upon a higher plane of knowledge, but can never prove how it got there. All great discoveries have involved such a leap." (Albert Einstein, [interview in Life, "Death of a Genius"] 1955)

"There is no correlation between the cause and the effect. The events reveal only an aleatory determination, connected not so much with the imperfection of our knowledge as with the structure of the human world." (Raymond Aron, "The Opium of the Intellectuals", 1955)

"General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure mathematics and the specific theories of the specialized disciplines. Mathematics attempts to organize highly general relationships into a coherent system, a system however which does not have any necessary connections with the 'real' world around us. It studies all thinkable relationships abstracted from any concrete situation or body of empirical knowledge." (Kenneth E Boulding, "General Systems Theory - The Skeleton of Science", Management Science Vol. 2 (3), 1956)

"In no subject is there a rule, compliance with which will lead to new knowledge or better understanding. Skillful observations, ingenious ideas, cunning tricks, daring suggestions, laborious calculations, all these may be required to advance a subject. Occasionally the conventional approach in a subject has to be studiously followed; on other occasions it has to be ruthlessly disregarded. Which of these methods, or in what order they should be employed is generally unpredictable. Analogies drawn from the history of science are frequently claimed to be a guide; but, as with forecasting the next game of roulette, the existence of the best analogy to the present is no guide whatever to the future. The most valuable lesson to be learnt from the history of scientific progress is how misleading and strangling such analogies have been, and how success has come to those who ignored them." (Thomas Gold, "Cosmology", 1956) 

"Knowledge is not something which exists and grows in the abstract. It is a function of human organisms and of social organization. Knowledge, that is to say, is always what somebody knows: the most perfect transcript of knowledge in writing is not knowledge if nobody knows it. Knowledge however grows by the receipt of meaningful information - that is, by the intake of messages by a knower which are capable of reorganising his knowledge." (Kenneth E Boulding, "General Systems Theory - The Skeleton of Science", Management Science Vol. 2 (3), 1956)

"The mathematical formulas indeed no longer portray nature, but rather our knowledge of nature." (Werner K Heisenberg, "The Representation of Nature in Contemporary Physics", Daedalus Vol. 87 (3), 1958)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The New Science", 1959)

On Equilibrium (1900-1919)

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

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

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

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

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

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

On Principles V: Identity

"The topics of ontology, or metaphysic, are cause, effect, action, passion, identity, opposition, subject, adjunct, and sign." (Isaac Watts, "Logic, or The right use of reason, in the inquiry after truth", 1725)

"Science arises from the discovery of Identity amid Diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle." (Henri Poincaré, "Science and Hypothesis", 1901)

"Metaphors deny distinctions between things: problems often arise from taking structural metaphors too literally. Because unexamined metaphors lead us to assume the identity of unidentical things, conflicts can arise which can only be resolved by understanding the metaphor (which requires its recognition as such), which means reconstructing the analogy on which it is based. […] The unexplained extension of concepts can too often result in the destruction rather than the expansion of meaning." (David Pimm,"Metaphor and Analogy in Mathematics", For the Learning of Mathematics Vol. 1 (3), 1981)

"The idea that one can 'introduce' a kind of objects simply by laying down an identity criterion for them really inverts the proper order of explanation. As Locke clearly understood, one must first have a clear conception of what kind of objects one is dealing with in order to extract a criterion of identity for them from that conception. […] So, rather than 'abstract' a kind of object from a criterion of identity, one must in general 'extract' a criterion of identity from a metaphysically defensible conception of a given kind of objects." (Edward J Lowe, The metaphysics of abstract objects, Journal of Philosophy 92 (10), 1995) 

"There is no unique, global, and universal relation of identity for abstract objects. [...] Abstract objects are of different sorts and this should mean, almost by definition, that there is no global, universal identity for sorts. Each sort X is equipped with an internal relation of identity but there is no identity relation that would apply to all sorts." (Jean-Pierre Marquis," Categorical foundations of mathematics, or how to provide foundations for abstract mathematics", The Review of Symbolic Logic Vol. 6 (1), 2012) 

"Reason is indeed all about identity, or, rather, tautology. Mathematics is the eternal, necessary system of rational, analytic tautology. Tautology is not 'empty', as it is so often characterized by philosophers. It is in fact the fullest thing there, the analytic ground of existence, and the basis of everything. Mathematical tautology has infinite masks to wear, hence delivers infinite variety. Mathematical tautology provides Leibniz’s world that is 'simplest in hypothesis and the richest in phenomena'. No hypothesis cold be simpler than the one revolving around tautologies concerning 'nothing'. There is something - existence - because nothing is tautologous, and 'something' is how that tautology is expressed. If we write x = 0, where x is any expression that has zero as its net result, then we have a world of infinite possibilities where something ('x') equals nothing (0)." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Imagination (-1699)

"Sometimes a thing is perceived [via sense-perception] when it is observed; then it is imagined, when it is absent [in reality] through the representation of its form inside, Sense-perception grasps [the concept] insofar as it is buried in these accidents that cling to it because of the matter out of which it is made without abstracting it from [matter], and it grasps it only by means of a connection through position [ that exists] between its perception and its matter. It is for this reason that the form of [the thing] is not represented in the external sense when [sensation] ceases. As to the internal [faculty of] imagination, it imagines [the concept] together with these accidents, without being able to entirely abstract it from them. Still, [imagination] abstracts it from the afore-mentioned connection [through position] on which sense-perception depends, so that [imagination] represents the form [of the thing] despite the absence of the form's [outside] carrier." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"Imagination is accordingly the first activity [movement] of the soul after it is subjected to external stimulation. Imagination  either formulates second judgment, or brings back first judgment by recollection." (John of Salisbury, "Metalogicon", 1159)

"The objection we are dealing with argues from the standpoint of an agent that presupposes time and acts in time, but did not institute time. Hence the question about 'why God's eternal will produces an effect now and and not earlier' presupposes that time exists; for 'now' and 'earlier' are segments of time. With regard to the universal production of things, among which time is also to be counted, we should not ask, 'Why now and not earlier?' Rather we should ask: 'Why did God wish this much time to intervene?' And this depends on the divine will, which is perfectly free to assign this or any other quantity to time. The same may be noted with respect to the dimensional quantity of the world. No one asks why God located the material world in such and such a place rather than higher up or lower down or in some other position; for there is no place outside the world. The fact that God portioned out so much quantity to the world that no part of it would be beyond the place occupied in some other locality, depends on the divine will. However, although there was no time prior to the world and no place outside the world, we speak as if there were. Thus we say that before the world existed there was nothing except God, and that there is no body lying outside the world. But in thus speaking of 'before' and 'outside,' we have in mind nothing but time and place as they exist in our imagination." (Thomas Aquinas, "Compendium Theologiae" ["Compendium of Theology"], cca. 1265 [unfinished])

"[…] the painter cannot produce any form or figure […] if first this form or figure is not imagined and reduced into a mental image (idea) by the inward wits. And to paint, one needs acute senses and a good imagination with which one can get to know the things one sees in such a way that, once these things are not present anymore and transformed into mental images (fantasmi), they can be presented to the intellect. In the second stage, the intellect by means of its judgement puts these things together and, finally, in the third stage the intellect turns these mental images […] into a finished composition which it afterwards represents in painting by means of its ability to cause movement in the body." (Romano Alberti, "Della nobiltà della Pittura", 1585)

"God forbid that we should give out a dream of our own imagination for a pattern of the world." (Francis Bacon, "The Great Instauration", 1620)

"From all this I am beginning to have a rather better understanding of what I am. But it still appears - and I cannot stop thinking this - that the corporeal things of which images are formed in my thought, and which the senses investigate, are known with much more distinctness than this puzzling 'I' which cannot be pictured in the imagination." (René Descartes, "Meditations" II, 1641)

"For after the object is removed, or the eye shut, we still retain an image of the thing seen, though more obscure than when we see it. And this is it the Latins call imagination, from the image made in seeing, and apply the same, though improperly, to all the other senses. But the Greeks call it fancy, which signifies appearance, and is as proper to one sense as to another. IMAGINATION, therefore, is nothing but decaying sense; and is found in men and many other living creatures, as well sleeping as waking." (Thomas Hobbes, "Leviathan: The Matter, Form and Power of a Commonwealth  Ecclesiastical and Civil", 1651)

"Measure, time and number are nothing but modes of thought or rather of imagination." (Baruch Spinoza, [Letter to Ludvicus Meyer] 1663)

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On Conjecture (1975-1999)

"All knowledge, the sociologist could say, is conjectural and theoretical. Nothing is absolute and final. Therefore all knowledge is relative to the local situation of the thinkers who produce it: the ideas and conjectures that they are capable of producing: the problems that bother them; the interplay of assumptions and criticism in their milieu; their purposes and aims; the experiences they have and the standards and meanings they apply." (David Bloor, "Knowledge and Social Imagery", 1976)

"The essential function of a hypothesis consists in the guidance it affords to new observations and experiments, by which our conjecture is either confirmed or refuted." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1976)

"The verb 'to theorize' is now conjugated as follows: 'I built a model; you formulated a hypothesis; he made a conjecture.'" (John M Ziman, "Reliable Knowledge", 1978)

"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true - a preconception that always, and necessarily, goes a little way (sometimes a long way) beyond anything which we have logical or factual authority to believe in. It is the invention of a possible world, or of a tiny fraction of that world. The conjecture is then exposed to criticism to find out whether or not that imagined world is anything like the real one. Scientific reasoning is therefore at all levels an interaction between two episodes of thought - a dialogue between two voices, the one imaginative and the other critical; a dialogue, as I have put it, between the possible and the actual, between proposal and disposal, conjecture and criticism, between what might be true and what is in fact the case." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"So-called scientific knowledge is not knowledge, for it consists only of conjectures or hypotheses - even if some have gone through the crossfire of ingenious tests." (Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)

"Three shifts can be detected over time in the understanding of mathematics itself. One is a shift from completeness to incompleteness, another from certainty to conjecture, and a third from absolutism to relativity." (Leone Burton, "Femmes et Mathematiques: Y a–t–il une?",  Association for Women in Mathematics Newsletter, Intersection 18, 1988)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 1994)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof - which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lan, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

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

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

"Architectural conjectures are mathematically precise assertions, as well milled as minted coins, provisionally usable in the commerce of logical arguments; less than ‘coins’ and more aptly, promissory notes to be paid in full by some future demonstration, or to be contradicted. These conjectures are expected to turn out to be true, as, of course, are all conjectures; their formulation is often away of "formally" packaging, or at least acknowledging, an otherwise shapeless body of mathematical experience that points to their truth." (Barry Mazur, "Conjecture", Synthese 111, 1997)

"The everyday usage of 'theory' is for an idea whose outcome is as yet undetermined, a conjecture, or for an idea contrary to evidence. But scientists use the word in exactly the opposite sense. [In science] 'theory' [...] refers only to a collection of hypotheses and predictions that is amenable to experimental test, preferably one that has been successfully tested. It has everything to do with the facts." (Tony Rothman & George Sudarshan, "Doubt and Certainty: The Celebrated Academy: Debates on Science, Mysticism, Reality, in General on the Knowable and Unknowable", 1998)

"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist, 1999)

Geoff Cumming - Collected Quotes

"A second approach to statistical inference is estimation, which focuses on finding the best point estimate of the population parameter that’s of greatest interest; it also gives an interval estimate of that parameter, to signal how close our point estimate is likely to be to the population value." (Geoff Cumming, "Understanding the New Statistics", 2012)

"An effect is anything we might be interested in, and an effect size is simply the size of anything that may be of interest." (Geoff Cumming, "Understanding the New Statistics", 2012)

"In the long term, randomness is highly predictable. In the short term, true randomness is often lumpy." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Meta-analysis is a set of techniques for the quantitative analysis of results from two or more studies on the same or similar issues. […] Meta-analytic thinking is estimation thinking that considers any result in the context of past and potential future results on the same question. It focuses on the cumulation of evidence over studies." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Meta-analytic thinking is the consideration of any result in relation to previous results on the same or similar questions, and awareness that combination with future results is likely to be valuable. Meta-analytic thinking is the application of estimation thinking to more than a single study. It prompts us to seek meta-analysis of previous related studies at the planning stage of research, then to report our results in a way that makes it easy to include them in future meta-analyses. Meta-analytic thinking is a type of estimation thinking, because it, too, focuses on estimates and uncertainty." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Replication is at the heart of science. If you read a study claiming that toast usually lands buttered side down, you’ll probably want to find at least one replication before you begin to take the result seriously. That’s good scientific practice. A replication experiment inevitably differs a little from the initial experiment - the toast was a little different in shape and dropped in a slightly different way - and so again finding a similar result gives some reassurance that the initial result was not caused by some quirk of the initial experiment." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Statistical cognition is concerned with obtaining cognitive evidence about various statistical techniques and ways to present data. It’s certainly important to choose an appropriate statistical model, use the correct formulas, and carry out accurate calculations. It’s also important, however, to focus on understanding, and to consider statistics as communication between researchers and readers." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Statistical cognition is the empirical study of how people understand, and misunderstand, statistical concepts and presentations." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Statistical inference is the drawing of conclusions about the world (more specifically: about some population) from our sample data." (Geoff Cumming, "Understanding the New Statistics", 2012)

"The law of large numbers is a law of mathematical statistics. It states that when random samples are sufficiently large they match the population extremely closely. […] The 'law' of small numbers is a widespread human misconception that even small samples match the population closely." (Geoff Cumming, "Understanding the New Statistics", 2012)

Mathematics through Students' Eyes III

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline, "Mathematics and the Physical World", 1959)

"Formerly, the beginner was taught to crawl through the underbrush, never lifting his eyes to the trees; today he is often made to focus on the curvature of the universe, missing even the earth." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"I would therefore urge that people be introduced to [the logistic equation] early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems. Not only in research but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties." (Robert M May, "Simple Mathematical Models with Very Complicated Dynamics", Nature Vol. 261 (5560), 1976)

"Students enjoy […] and gain in their understanding of today's mathematics through analyzing older and alternative approaches." (Lucas N H Bunt et al, "The Historical Roots of Elementary Mathematics", 1976)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"Mathematics is often thought to be difficult and dull. Many people avoid it as much as they can and as a result much of the population is mathematically illiterate. This is in part due to the relative lack of importance given to numeracy in our culture, and to the way that the subject has been presented to students." (Julian Havil , "Gamma: Exploring Euler's Constant", 2003)

"As students, we learned mathematics from textbooks. In textbooks, mathematics is presented in a rigorous and logical way: definition, theorem, proof, example. But it is not discovered that way. It took many years for a mathematical subject to be understood well enough that a cohesive textbook could be written. Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson, "Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)

"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)

On Coincidence II

"People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual coincidence is far more unusual than any coincidence could possibly be." (Isaac Asimov, "The Planet That Wasn't", 1976)

"Our form of life depends, in delicate and subtle ways, on several apparent ‘coincidences’ in the fundamental laws of nature which make the Universe tick. Without those coincidences, we would not be here to puzzle over the problem of their existence […] What does this mean? One possibility is that the Universe we know is a highly improbable accident, ‘just one of those things’." (John R Gribbin, "Genesis: The Origins of Man and the Universe", 1981)

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

"Moreover, joint occurrences tend to be better recalled than instances when the effect does not occur. The proneness to remember confirming instances, but to overlook disconfirming ones, further serves to convert, in thought, coincidences into causalities." (Albert Bandura, "Social Foundations of Thought and Action: A social cognitive theory", 1986)

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

"A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence." (John A Paulos, "Innumeracy: Mathematical Illiteracy and its Consequences", 1988)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences", Journal of the American Statistical Association Vol. 84, 1989)

"Most coincidences are simply chance events that turn out to be far more probable than many people imagine." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1997)

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

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

On Generalization (2000-2009)

"The fruitful generalization in mathematics often involves starting from a commonsense concept such as a point on a line. A mathematical framework is then developed within which the particular example of a point in space is seen to be just a very special case of a much broader structure, say a point in three-dimensional space. Further generalizations then show this new structure itself to be only a special case of an even broader framework, the notion of a point in a space of n dimensions. And so it goes, one generalization piled atop another, each element leading to a deeper understanding of how the original object fits into a bigger picture." (John L Casti, "Five More Golden Rules : Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematics", 2000)

"A mental model is a representation of some domain or situation that supports understanding, reasoning, and prediction. Mental models permit reasoning about situations not directly experienced. They allow people to mentally simulate the behavior of a system. Many mental models are based on generalizations and analogies from experience." (D Gentner, "Psychology of Mental Models" [in "International Encyclopedia of the Social & Behavioral Sciences"], 2001)

"Ecology, on the other hand, is messy. We cannot find anything deserving of the term law, not because ecology is less developed than physics, but simply because the underlying phenomena are more chaotic and hence less amenable to description via generalization." (Lev Ginzburg & Mark Colyvan, "Ecological Orbits: How Planets Move and Populations Grow", 2004)

"Limiting factors in population dynamics play the role in ecology that friction does in physics. They stop exponential growth, not unlike the way in which friction stops uniform motion. Whether or not ecology is more like physics in a viscous liquid, when the growth-rate-based traditional view is sufficient, is an open question. We argue that this limit is an oversimplification, that populations do exhibit inertial properties that are noticeable. Note that the inclusion of inertia is a generalization - it does not exclude the regular rate-based, first-order theories. They may still be widely applicable under a strong immediate density dependence, acting like friction in physics." (Lev Ginzburg & Mark Colyvan, "Ecological Orbits: How Planets Move and Populations Grow", 2004)

"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"The concept of symmetry (invariance) with its rigorous mathematical formulation and generalization has guided us to know the most fundamental of physical laws. Symmetry as a concept has helped mankind not only to define ‘beauty’ but also to express the ‘truth’. Physical laws tries to quantify the truth that appears to be ‘transient’ at the level of phenomena but symmetry promotes that truth to the level of ‘eternity’." (Vladimir G Ivancevic & Tijana T Ivancevic, "Quantum Leap", 2008)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

On Technology II

"The 'message' of any medium or technology is the change of scale or pace or pattern that it introduces into human affairs." (Marshall McLuhan, "Understanding Media", 1964)

"Our technology forces us to live mythically, but we continue to think fragmentarily, and on single, separate planes." (Marshall McLuhan, "The Medium is the Massage: An inventory of effects", 1967)

"Modern scientific principle has been drawn from the investigation of natural laws, technology has developed from the experience of doing, and the two have been combined by means of mathematical system to form what we call engineering." (George S Emmerson, "Engineering Education: A Social History", 1973)

"The system of nature, of which man is a part, tends to be self-balancing, self-adjusting, self-cleansing. Not so with technology." (Ernst F Schumacher, "Small is Beautiful", 1973)

"Technology has not advanced because people are starved for instruments to make a better civilization, but because they are starved for entertainment - technology is still mostly a toy factory for grown-ups." (Eugene J Martin, 1977-1978)

"People’s views of the world, of themselves, of their own capabilities, and of the tasks that they are asked to perform, or topics they are asked to learn, depend heavily on the conceptualizations that they bring to the task. In interacting with the environment, with others, and with the artifacts of technology, people form internal, mental models of themselves and of the things with which they are interacting. These models provide predictive and explanatory power for understanding the interaction." (Donald A Norman, "Some observations on Mental Models", 1983)

"With the changes in technological complexity, especially in information technology, the leadership task has changed. Leadership in a networked organization is a fundamentally different thing from leadership in a traditional hierarchy." (Edgar Schein, "Organizational Culture and Leadership", 1985)

"The new information technologies can be seen to drive societies toward increasingly dynamic high-energy regions further and further from thermodynamical equilibrium, characterized by decreasing specific entropy and increasingly dense free-energy flows, accessed and processed by more and more complex social, economic, and political structures." (Ervin László, "Information Technology and Social Change: An Evolutionary Systems Analysis", Behavioral Science 37, 1992) 

"Now that knowledge is taking the place of capital as the driving force in organizations worldwide, it is all too easy to confuse data with knowledge and information technology with information." (Peter Drucker, "Managing in a Time of Great Change", 1995)

"Commonly, the threats to strategy are seen to emanate from outside a company because of changes in technology or the behavior of competitors. Although external changes can be the problem, the greater threat to strategy often comes from within. A sound strategy is undermined by a misguided view of competition, by organizational failures, and, especially, by the desire to grow." (Michael E Porter, "What is Strategy?", Harvard Business Review, 1996)

On Coherence IV

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

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

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

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

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

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

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

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

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

Information Overload III

"Every person seems to have a limited capacity to assimilate information, and if it is presented to him too rapidly and without adequate repetition, this capacity will be exceeded and communication will break down." (R Duncan Luce, "Developments in Mathematical Psychology", 1960)

"People today are in danger of drowning in information; but, because they have been taught that information is useful, they are more willing to drown than they need be. If they could handle information, they would not have to drown at all." (Idries Shah, "Reflections", 1968)

"Everyone spoke of an information overload, but what there was in fact was a non-information overload." (Richard S Wurman, "What-If, Could-Be", 1976)

"The greater the uncertainty, the greater the amount of decision making and information processing. It is hypothesized that organizations have limited capacities to process information and adopt different organizing modes to deal with task uncertainty. Therefore, variations in organizing modes are actually variations in the capacity of organizations to process information and make decisions about events which cannot be anticipated in advance." (John K Galbraith, "Organization Design", 1977)

"We are drowning in information but starved for knowledge." (John Naisbitt, "Megatrends: Ten New Directions Transforming Our Lives", 1982)

"In the Information Age, the first step to sanity is FILTERING. Filter the information: extract for knowledge. Filter first for substance. Filter second for significance. […] Filter third for reliability. […] Filter fourth for completeness." (Marc Stiegler, "David’s Sling", 1988)

"It has become evident time and again that when events become too complex and move too rapidly as appears to be the case today, human beings become demonstrably less able to cope." (Alan Greenspan, "The Structure of the International Financial System", 1998)

"Specialization, once a maneuver methodically to collect information, now is a manifestation of information overloads. The role of information has changed. Once justified as a means of comprehending the world, it now generates a conflicting and contradictory, fleeting and fragmentation field of disconnected and undigested data." (Stelarc, From Psycho-Body to Cyber-Systems: Images as Post-human Entities, 1998)

"Our needs going forward will be best served by how we make use of not just this data but all data. We live in an era of Big Data. The world has seen an explosion of information in the past decades, so much so that people and institutions now struggle to keep pace. In fact, one of the reasons for the attachment to the simplicity of our indicators may be an inverse reaction to the sheer and bewildering volume of information most of us are bombarded by on a daily basis. […] The lesson for a world of Big Data is that in an environment with excessive information, people may gravitate toward answers that simplify reality rather than embrace the sheer complexity of it." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

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

On Economics I (Models I)

"Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this, because, unlike the typical natural science, the material to which it is applied is, in too many respects, not homogeneous through time. The object of a model is to segregate the semi-permanent or relatively constant factors from those which are transitory or fluctuating so as to develop a logical way of thinking about the latter, and of understanding the time sequences to which they give rise in particular cases." (John M Keynes, [letter to Roy Harrod] 1938)

"The striking parallel between the economic models that are currently under discussion and some engineering systems suggests the hope that in some way the rapid progress in the development of the theory and practice of automatic control in the world of engineering may contribute to the solution of the economic problems." (Arnold Tustin "The Mechanism of Economic Systems", 1953) 

"The construction of an economic model, or of any model or theory for that matter (or the writing of a novel, a short story, or a play) consists of snatching from the enormous and complex mass of facts called reality, a few simple, easily-managed key points which, when put together in some cunning way, become for certain purposes a substitute for reality itself." (Evsey Domar, "Essays in the Theory of Economic Growth", 1957)

"One of the most important skills of the economist, therefore, is that of simplification of the model." (Kenneth Boulding, "The Skills of the Economist", Journal of Political Economy 67 (1), 1959)

"In many parts of the economy, stabilizing forces appear not to operate. Instead, positive feedback magnifies the effects of small economic shifts; the economic models that describe such effects differ vastly from the conventional ones. Diminishing returns imply a single equilibrium point for the economy, but positive feedback - increasing returns - makes for many possible equilibrium points. There is no guarantee that the particular economic outcome selected from among the many alternatives will be the 'best' one." (W Brian Arthur, "Increasing Returns and Path Dependence in the Economy", 1994)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

"The long term solution to the financial crisis is to move beyond the ‘growth at all costs’ economic model to a model that recognizes the real costs and benefits of growth." (Robert Costanza, "Toward a New Sustainable Economy", 2008)

"Real economic efficiency implies including all resources that affect sustainable human well-being in the allocation system, not just marketed goods and services. Our current market allocation system excludes most non-marketed natural and social capital assets and services that are critical contributors to human well-being. The current economic model ignores this and therefore does not achieve real economic efficiency. A new, sustainable ecological economic model would measure and include the contributions of natural and social capital and could better approximate real economic efficiency." (Robert Costanza, "Toward a New Sustainable Economy", 2008)

"Economists also use models to learn about the world, but instead of being made of plastic, they are most often composed of diagrams and equations. Like a biology teacher’s plastic model, economic models omit many details to allow us to see what is truly important. Just as the biology teacher’s model does not include all the body’s muscles and capillaries, an economist’s model does not include every feature of the economy." (N Gregory Mankiw, "Principle of Economics" 6th ed., 2012)

"Many of the stories economists tell take the form of models - for whatever else they are, economic models are stories about how the world works." (Paul Krugman & Robin Wells, "Economics" 3rd Ed., 2013)

On Structure: Structure in Mathematics (1990-1999)

"[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost." (Mark Kac,  "Mathematics: Tensions", 1992)

"Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode." (William P Thurston, "On Proof and Progress in Mathematics", 1994)

"The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design." (Freeman J Dyson, [interview] 1994)

"The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions." (Ian Stewart, "Nature's Numbers", 1995)

"Prime numbers are the most basic objects in mathematics. They also are among the most mysterious, for after centuries of study, the structure of the set of prime numbers is still not well understood […]" (Andrew Granville, 1997)

"Mathematics is a product - a discovery - of the human mind. It enables us to see the incredible, simple, elegant, beautiful, ordered structure that lies beneath the universe we live in. It is one of the greatest creations of mankind - if it is not indeed the greatest." (Keith Devlin, "Life By the Numbers", 1998)

"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." (Andreas Knauf, "Number Theory, Dynamical Systems and Statistical Mechanics", 1998)

Systems Thinking V

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

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

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

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

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

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

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

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

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

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Mental Models LXII

"A mental model is a collection of 'connected' autonomous objects. Running  a mental model corresponds to modifying the parameters of the model by propagating information using the internal rules and specified topology. Running a mental model can also occur when autonomous objects change state. For us the definition of state is distinct from the current parameter values of an object. A state change consists of the replacement of one set of behavior rules with another." (Michael D Williams et al, "Human Reasoning About a Simple Physical System", [in "Mental Models", Ed(s). Dedre Gentner & Albert L Stevens], 1983)

"Central to this conception of mental models is the notion of autonomous objects. An autonomous object is a mental object with an explicit representation of state, an explicit representation of its topological connections to other objects, and a set of internal parameters. Associated with each autonomous object is a set of rules which modify its parameters and thus specify its behavior." (Michael D Williams et al, "Human Reasoning About a Simple Physical System", [in "Mental Models", Ed(s). Dedre Gentner & Albert L Stevens], 1983)

"In the consideration of mental models we need really consider four different things: the target system, the conceputal model of that target system, the user’s mental model of the target system, and the scientist's conceptualization of that mental model. The system that the person is learning or using is, by definition, the target system. A conceptual model is invented to provide an appropriate representation of the target system, appropriate in the sense of being accurate, consistent, and complete." (Donald A Norman, "Some Observations on Mental Models" [in "Mental Models", Ed(s). Dedre Gentner & Albert L Stevens], 1983)

"The purpose of a mental model is to allow the person to understand and to anticipate the behavior of a physical system. This means that the model must have predictive power, either by applying rules of inference or by procedural derivation (in whatever manner these properties may be realized in a person); in other words, it should be possible for people to ' run' their models mentally. This means that the conceptual mental model must also include a model of the relevant human information processing and knowledge structures that make it possible for the person to use a mental model to predict and understand the physical system." (Donald A Norman, "Some Observations on Mental Models" [in "Mental Models"], Ed(s). Dedre Gentner & Albert L Stevens], 1983)

"From a functional point of view, mental models can be described as symbolic structures which permit people: to generate descriptions of the purpose of a system, to generate descriptions of the architecture of a system, to provide explanations of the state of a system, to provide explanations of the functioning of a system, to make predictions of future states of a system." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"Under the label 'cognitive maps', mental models have been conceived of as the mental representation of spatial aspects of the environment. A mental model, in this sense, comprises the topology of an area, including relevant districts, landmarks, and paths. [...] Under the label 'naive physics', mental models have been conceived of as the mental representation of natural or technical systems. A mental model, in this sense, comprises the effective determinants, true or not, of the functioning of a physical system. [...] Under the label 'model based reasoning', the mental models notion is featured in yet another area of cognitive science - deductive reasoning. In contrast to the commonly held view that logical competence depends on formal rules of deduction, it has been argued that reasoning is a semantic process based on the manipulation of mental models. [...] Finally, under terms like 'discourse model', 'situation model', or 'scenario', mental models have been conceived of as the mental representation of a verbal description of some real or fictional state of affairs. The role of mental models in the comprehension of discourse is discussed in more detail below." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"A mental model is an internal representation with analogical relations to its referential object, so that local and temporal aspects of the object are preserved. It comes somewhat close to the mental images people report having in their minds whilst processing information. The great advantage of the notion of mental models, however, is its ability to include the notion of a partner model and the notion of a situation model. Thus, mental models can build a bridge to the other two dimensions of communication, namely interaction and situation." (Gert Rickheit et al, "The concept of communicative competence" [in "Handbook of Communication Competence"], 2008)

"Because all mental models or mindsets are incomplete, we can engage in second-order studies, evaluations, judgments, and assessments about our own and other operative mental models. Of course this is highly complex since the act of reflection is itself a further of framing or reframing." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)

"Mental models bind our awareness within a particular scaffold and then selectively can filter the content we subsequently receive. Through recalibration using revised mental models, we argue, we cultivate strategies anew, creating new habits, and galvanizing more intentional and evolved mental models. This recalibration often entails developing a strong sense of self and self-worth, realizing that each of us has a range of moral choices that may deviate from those in authority, and moral imagination." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)

"These framing perspectives or mental models construe the data of our experiences, and it is the construed data that we call 'facts'. What we often call reality, or the world, is constructed or socially construed in certain ways such that one cannot get at the source of the data except through these construals." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)

On Imagination (BC)

"We invoke the imagination and the intervals that it furnishes, since the form itself is without motion or genesis, indivisible and free of all underlying matter, though the elements latent in the form are produced distinctly and individually on the screen of imagination. What projects the images is the understanding; the source of what is projected is the form in the understanding; and what they are projected in is this 'passive nous' that unfolds in revolution about the partlessness of genuine Nous." (Proclus Lycaeus, "A Commentary on the First Book of Euclid’s Elements", cca 5th century)

"But since we have, in our work on the soul, treated of imagination, and the faculty of imagination is identical with that of sense-perception, though the being of a faculty of imagination is different from that of a faculty of sense-perception; and since imagination is the movement set up by a sensory faculty when actually discharging its function, while a dream appears to be an image (for which occurs in sleep - whether simply or in some particular way - is what we call a dream): it manifestly follows that dreaming is an activity of the faculty of sense-perception, but belongs to this faculty qua imaginative." (Aristotle, "On Dreams", 4th century BC)

"It is obvious then, that memory belongs to that part of the soul to which imagination belongs. […] Just as the picture painted on the panel is at once a picture and a portrait, and though one and the same, is both, yet the essence of the two is not the same, and it is possible to think of it both as a picture and as a portrait, so in the same way we must regard the mental picture within us both as an object of contemplation in itself and as a mental picture of something else […]. Insofar as we consider it in relation to something else, e.g. as a likeness, it is also an aid to memory." (Aristotle, "De Memoria et Reminiscentia" [On Memory and Recollection], 4th century BC)

"For imagination is different from either perceiving or discursive thinking, though it is not found without sensation, or judgement without it. That this activity is not the same kind of thinking as judgement is obvious. For imagining lies within our own power whenever we wish (e.g. we can call up a picture, as in the practice of mnemonics by the use of mental images), but in forming opinions we are not free: we cannot escape the alternative of falsehood or truth." (Aristotle, "De Anima", cca. 350 BC)

"[Imagination is] that in virtue of which we say that an image occurs to us and not as we speak of it metaphorically."  (Aristotle, "De Anima" III, cca. 350 BC)

"Since it seems that there is nothing outside and separate in existence from sensible spatial magnitudes, the objects of thought are in the sensible forms, viz. both the abstract objects and all the states and affections of sensible things. Hence no one can learn or understand anything in the absence of sense, and when the mind is actively aware of anything it is necessarily aware of it along with an image; for images are like sensuous contents except in that they contain no matter. Imagination is different from assertion and denial; for what is true or false involves a synthesis of thoughts. In what will the primary thoughts differ from images? Must we not say that neither these nor even our other thoughts are images, though they necessarily involve them?" (Aristotle, "De Anima", cca. 350 BC)

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

"Imagination (VIKALPA) is a thought based on a mental image describable by words but not based on an object directly observable." (Patanjali, "Yoga Sutra" cca. 500 BC - 400 AD)

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