📓On Literature: On Thinking (From Fiction to Science-Fiction)

"The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." (Herbert G Wells, "Mankind in the Making", 1903)

"Old men tend to forget what thought was like in their youth; they forget the quickness of the mental jump, the daring of the youthful intuition, the agility of the fresh insight. They become accustomed to the more plodding varieties of reason, and because this is more than made up by the accumulation of experience, old men think themselves wiser than the young." (Isaac Asimov, "Pebble in the Sky", 1950)

"At least half of mankind [...] still makes an unconscious equation in its thinking, and assumes that change - any sort of change - is identical with progress. It is not so; and any student of the course of evolutionary history on Terra could tell you of change which has been regressive, change which has led to an ultimately fatal specialization, change which has been overadaptation to an ecological niche which no longer existed, or did not yet exist." (Margaret St. Clair, "Agent of the Unknown", 1952)

"A lot of human thinking occurred beneath the level of consciousness, downin the darker regions of the mind where it was not allowed to become conscious lest it bring shame to the thinker. And perhaps he had reasoned it all out in that mental half-world where thoughts are inner ghosts, haunting the possessed man with vague stirrings of uneasiness, leading him into inexplicable behavior." (Walter M Miller Jr, "Way of a Rebel", 1954)

"We’re free out here, really free for the first time. We’re floating, literally. Gravity can’t bow our backs or break our arches or tame our ideas. You know, it’s only out here that stupid people like us can really think. The weightlessness gets our thoughts and we can sort them. Ideas grow out here like nowhere else - it’s the right environment for them. Anyone can get into space, if he wants to hard enough. The ticket is a dream." (Fritz Leiber," The Beat Cluster", 1961)

"Science fiction is, very strictly and literally, analogous to science facts. It is a convenient analog system for thinking about new scientific, social, and economic ideas - and for re-examining old ideas." (John W Campbell Jr., "Prologue to Analog", 1962)

"[…] it took men about five thousand years, counting from the beginning of number symbols, to think of a symbol for nothing." (Isaac Asimov, "Of Time and Space and Other Things", 1965)

"The mind can go either direction under stress - toward positive or toward negative: on or off. Think of it as a spectrum whose extremes are unconsciousness at the negative end and hyperconsciousness at the positive end. The way the mind will lean under stress is strongly influenced by training." (Frank Herbert, "Dune", 1965)

"What are the facts? Again and again and again - what are the facts? Shun wishful thinking, ignore divine revelation, forget what ‘the stars foretell,’ avoid opinion, care not what the neighbors think, never mind the unguessable ‘verdict of history', - what are the facts, and to how many decimal places? You pilot always into an unknown future; facts are your only clue. Get the facts!" (Robert A Heinlein, "Time Enough for Love", 1973)

"The idea of making machines that think has an unfailing fascination, not only for science fiction readers, but for all who can see it is a possible way of gaining some understanding of the working of our own minds. Thinking, however, is not an easily defined phenomenon, although it is often considered to be the process of solving problems." (Edward Ihnatowicz, "The Relevance of Manipulation to the Process of Perception", 1977)

"It is change, continuing change, inevitable change, that is the dominant factor in society today. No sensible decision can be made any longer without taking into account not only the world as it is, but the world as it will be - and naturally this means that there must be an accurate perception of the world as it will be. This, in turn, means that our statesmen, our businessmen, our everyman must take on a science fictional way of thinking, whether he likes it or not, or even whether he knows it or not. Only so can the deadly problems of today be solved." (Isaac Asimov, [foreword to Robert Holdstock (Ed.), "Encyclopedia of Science Fiction] 1978)

"In the face of all this, many of the standard ideas of science fiction seem to me to pale by comparison. I see the relative absence of these things and the distortions of scientific thinking often encountered in science fiction as terrible wasted opportunities. Real science is as amenable to exciting and engrossing fiction as fake science, and I think it is important to exploit every opportunity to convey scientific ideas in a civilization which is both based upon science and does almost nothing to ensure that science is understood." (Carl Sagan, "Broca's Brain", 1979)

"How often people speak of art and science as though they were two entirely different things, with no interconnection. An artist is emotional, they think, and uses only his intuition; he sees all at once and has no need of reason. A scientist is cold, they think, and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong. The true artist is quite rational as well as imaginative and knows what he is doing; if he does not, his art suffers. The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly; if he does not, his science suffers." (Isaac Asimov, "The Roving Mind", 1983)

"Science is a process. It is a way of thinking, a manner of approaching and of possibly resolving problems, a route by which one can produce order and sense out of disorganized and chaotic observations. Through it we achieve useful conclusions and results that are compelling and upon which there is a tendency to agree." (Isaac Asimov, "'X' Stands for Unknown", 1984)

"Most people don’t listen. They use the time when someone else is speaking to think of what they’re going to say next. True Listeners have always been revered among oral cultures, and prized for their rarity value." (Terry Pratchett, "Pyramids", 1989

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"The process of thinking itself requires us to view the universe in the direction of entropy, since an abstraction always involves information loss, since symbols 'abstract' complexity from observed objects." (John C Wright, "Awake in the Night Land", 2014)

On Mathematical Thinking

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

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

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

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

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

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

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

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

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

Edmund C Berkeley - Collected Quotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

On Knowledge (-1699)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

On Continuity XI (Thought II)

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

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

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

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

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

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

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

On Continuity XIV (Progress)

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

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

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

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

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

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

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

On Creativity (Mathematics I)

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

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

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

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

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

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

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

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

Daniel Kahneman - Collected Quotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

On Generalization (1800-1849)

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

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

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

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

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

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

On Technology IV

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

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

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

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

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

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

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

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

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

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

Richard E Nisbett - Collected Quotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

On Coherence IV

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

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

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

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

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

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

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

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

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

Ralph D Stacey - Collected Quotes

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

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

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

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

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

"Systems are wholes consisting of parts interacting with each other in a self-generating, self-organising way and it is in this interaction that both parts and whole emerge without prior design." (Ralph D Stacey, "Strategic Management and Organisational Dynamics", 2007)

"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate."  (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity" 5th Ed. , 2007)

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

Richard L Daft - Collected Quotes

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

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

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

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

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

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

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

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

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

Jamshid Gharajedaghi - Collected Quotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fuzzy Logic I

"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply denned criteria of class membership rather than the presence of random variables." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)

"Let me say quite categorically that there is no such thing as a fuzzy concept. [...] We do talk about fuzzy things but they are not scientific concepts. Some people in the past have discovered certain interesting things, formulated their findings in a non-fuzzy way, and therefore we have progressed in science." (Rudolf E Kálmán, 1972)

"[Fuzzy logic is] a logic whose distinguishing features are (1) fuzzy truth-values expressed in linguistic terms, e. g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact." (Lotfi A. Zadeh, "Fuzzy logic and approximate reasoning", 1975)

"[...] much of the information on which human decisions are based is possibilistic rather than probabilistic in nature, and the intrinsic fuzziness of natural languages - which is a logical consequence of the necessity to express information in a summarized form - is, in the main, possibilistic in origin." (Lotfi A Zadeh, "Fuzzy Sets as the Basis for a Theory of Possibility", Fuzzy Sets and Systems, 1978) 

"Philosophical objections may be raised by the logical implications of building a mathematical structure on the premise of fuzziness, since it seems (at least superficially) necessary to require that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extent that mathematical structures are used to model physical actualities, it is often an unrealistic requirement. [...] Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox." (James Bezdek, 1981)

"Fuzziness, then, is a concomitant of complexity. This implies that as the complexity of a task, or of a system for performing that task, exceeds a certain threshold, the system must necessarily become fuzzy in nature. Thus, with the rapid increase in the complexity of the information processing tasks which the computers are called upon to perform, we are reaching a point where computers will have to be designed for processing of information in fuzzy form. In fact, it is the capability to manipulate fuzzy concepts that distinguishes human intelligence from the machine intelligence of current generation computers. Without such capability we cannot build machines that can summarize written text, translate well from one natural language to another, or perform many other tasks that humans can do with ease because of their ability to manipulate fuzzy concepts." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"It is important to observe that there is an intimate connection between fuzziness and complexity. Thus, a basic characteristic of the human brain, a characteristic shared in varying degrees with all information processing systems, is its limited capacity to handle classes of high cardinality, that is, classes having a large number of members. Consequently, when we are presented with a class of very high cardinality, we tend to group its elements together into subclasses in such a way as to reduce the complexity of the information processing task involved. When a point is reached where the cardinality of the class of subclasses exceeds the information handling capacity of the human brain, the boundaries of the subclasses are forced to become imprecise and fuzziness becomes a manifestation of this imprecision." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy act to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are very often represented by real-number values ranging in the closed interval between 0 and 1." (George J Klir & Bo Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Applications", 1995)

On Symbols (1880-1889)

"Symbolical reasoning may be said to have pretty much the same relation to ordinary reasoning that machine-labour has to manual labour. In the case of machine labour we see some ingeniously contrived arrangement of wheels, levers, &c., producing with speed and facility results which the hands of man without such aid could only accomplish slowly and with difficulty, or which they would be utterly powerless to accomplish at all. In the case of symbolical reasoning we find in an analogous manner some regular system of rules and formulae, easy to retain in the memory from their general symmetry and interdependence, economizing or superseding the labour of the brain, and enabling any ordinary mind to obtain by simple mechanical processes results which would be beyond the reach of the strongest intellect if left entirely to its own resources." (Hugh MacColl, Symbolical reasoning. Mind 5 (17), 1880)

"We need a system of symbols from which every ambiguity is banned, which has a strict logical form from which the content cannot escape." (Gottlob Frege, "Über die wissenschaftliche berechtigung einer begriffsschrift", Zeitschrift für Philosophie und philosophische Kritik 81, 1882)

"[...] without symbols we could scarcely lift ourselves to conceptual thinking. Thus, in applying the same symbol to different but similar things, we actually no longer symbolize the individual thing, but rather what [the similars] have in common: the concept. This concept is first gained by symbolizing it; for since it is, in itself, imperceptible, it requires a perceptible representative in order to appear to us." (Gottlob Frege, "Über die wissenschaftliche berechtigung einer begriffsschrift", Zeitschrift für Philosophie und philosophische Kritik 81, 1882)

"While all that we have is a relation of phenomena, a mental image, as such, in juxtaposition with or soldered to a sensation, we can not as yet have assertion or denial, a truth or a falsehood. We have mere reality, which is, but does not stand for anything, and which exists, but by no possibility could be true. […] the image is not a symbol or idea. It is itself a fact, or else the facts eject it. The real, as it appears to us in perception, connects the ideal suggestion with itself, or simply expels it from the world of reality. […] you possess explicit symbols all of which are universal and on the other side you have a mind which consists of mere individual impressions and images, grouped by the laws of a mechanical attraction." (Francis H Bradley, "Principles of Logic", 1883)

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

"Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols." (Eduard Dillmann, "Die Mathematik die Fackelträgerin einer neuen Zeit", 1889)

On Engineering VI

"Can one think that because we are engineers, beauty does not preoccupy us or that we do not try to build beautiful, as well as solid and long lasting structures? Aren’t the genuine functions of strength always in keeping with unwritten conditions of harmony? [...] Besides, there is an attraction, a special charm in the colossal to which ordinary theories of art do not apply." (Gustave Eiffel, [interview in 'Le Temps'] 1887)

"The characteristic feature of our age results from the wedding of science and engineering. It is the working together of disciplined curiosity and purposeful ingenuity to create new materials, new forces, and new opportunities which powerfully affect our manner of living and ways of thinking." (Karl T Compton, "A Scientist Speaks: Excerpts from Addresses by Karl Taylor Compton - During the Years 1930-1949", 1955)

"[An engineer's] invention causes things to come into existence from ideas, makes world conform to thought; whereas science, by deriving ideas from observation, makes thought conform to existence." (Carl Mitcham, "Types of Technology", Research in Philosophy & Technology Vol. 1, 1978)

"Engineers use knowledge primarily to design, produce, and operate artifacts. [...] Scientists, by contrast, use knowledge primarily to generate more knowledge." (Walter Vincenti, What Engineers Know and How They Know It, 1990)

"Engineering is quite different from science. Scientists try to understand nature. Engineers try to make things that do not exist in nature. Engineers stress invention. To embody an invention the engineer must put his idea in concrete terms, and design something that people can use. That something can be a device, a gadget, a material, a method, a computing program, an innovative experiment, a new solution to a problem, or an improvement on what is existing. Since a design has to be concrete, it must have its geometry, dimensions, and characteristic numbers. Almost all engineers working on new designs find that they do not have all the needed information. Most often, they are limited by insufficient scientific knowledge. Thus they study mathematics, physics, chemistry, biology and mechanics. Often they have to add to the sciences relevant to their profession. Thus engineering sciences are born." (Yuan-Cheng Fung & Pin Tong, "Classical and Computational Solid Mechanics", 2001)

"Engineering isn't about perfect solutions; it's about doing the best you can with limited resources." (Randy Pausch, "The Last Lecture", 2008)

"The central activity of engineering, as distinguished from science, is the design of new devices, processes and systems." (Myron Tribus, "Rational Descriptions, Decisions and Designs", 2016)

"Engineering is a living branch of human activity and its frontiers are by no means exhausted." (Igor I Sikorsky)

"Engineering is not merely knowing and being knowledgeable, like a walking encyclopedia; engineering is not merely analysis; engineering is not merely the possession of the capacity to get elegant solutions to non-existent engineering problems; engineering is practicing the art of the organizing forces of technological change. [...] Engineers operate at the interface between science and society." (Gordon S Brown)

"The scientist describes what is; the engineer creates what never was." (Theodore von Kármán)

On Engineering III

"[...] without imagination, heightened awareness, moral sense, and some reference to the general culture, the engineering experience becomes less meaningful, less fulfilling than it should be." (Samuel C Florman, "The Civilized Engineer", 1985)

"Science can amuse and fascinate us all, but it is engineering that changes the world." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Engineering knowledge reflects the fact that design does not take place for its own sake and in isolation." (Walter G Vincenti, "What Engineers Know and How They Know It", 1990)

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"No matter how vigorously a 'science' of design may be pushed, the successful design of real things in a contingent world will always be based more on art than on science. Unquantifiable judgments and choices are the elements that determine the way a design comes together. Engineering design is simply that kind of process. It always has been; it always will be. (Eugene S Ferguson , "Engineering and the Mind’s Eye", 1992)

"Good engineering is not a matter of creativity or centering or grounding or inspiration or lateral thinking, as useful as those might be, but of decoding the clever, even witty, messages the solution space carves on the corpses of the ideas in which you believed with all your heart, and then building the road to the next message." (Fred Hapgood, "Up the infinite Corridor: MIT and the Technical Imagination", 1993)

"Engineering isn't about perfect solutions; it's about doing the best you can with limited resources." (Randy Pausch, "The Last Lecture", 2008)

"Engineering is the art or science of utilizing, directing or instructing others in the utilization of the principles, forces, properties and substances of nature in the production, manufacture, construction, operation and use of things [...] or of means, methods, machines, devices and structures [...]"  (Alfred W Kiddle)

"The essence of engineering consists not so much in the mere construction of the spectacular layouts or developments, but in the invention required - the analysis of the problem, the design, the solution by the mind which directs it all." (William Hood)

On Symbols (1940-1949)

"Symbols have a trick of stealing the show away from the thing they stand for." (Henry S Haskins, "Meditations in Wall Street", 1940) 

"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for […] The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)

"A formula is simply a mathematical statement of a principle or a rule describing the relation between two or more quantities. This mathematical statement shows that there is an equality between certain quantities; in other words, the formula translates a verbal rule into algebraic symbols. Thus a formula is very similar to an equation." (William L Schaaf, "Mathematics for Mechanics", 1942)

"Nothing is harder to understand than a symbolic work. A symbol always transcends the one who makes use of it and makes him say in reality more than he is aware of expressing." (Albert Camus, "The Myth of Sisyphus", 1942)

"The methods of algebra are essentially an extension of arithmetic. In other words, the numbers, symbols and operations used in algebra are the same as those used in arithmetic, only they are more general in character. This means (1) that letters as well as numbers are are used te to represent quantities, and (2) that numbers are regarded as having quality as well as quantity." (William L Schaaf, "Mathematics for Mechanics", 1942)

"[The power of understanding symbols] issues in an unconscious, spontaneous process of abstraction, which goes on all the time in the human mind: a process of recognizing the concept in any configuration given to experience, and forming a conception accordingly." (Suzanne K Langer, "Philosophy in a New Key: A Study in the Symbolism of Reason, Rite, and Art", 1942)

"It is generally agreed that thought employs symbols such as written or spoken words or tokens; but it is not generally considered whether the whole of thought may not consist of a process of symbolism, nor is the nature of symbolism and its presence or absence in the inorganic world discussed." (Kenneth Craik, "The Nature of Explanation", 1943)

"My hypothesis then is that thought models, or parallels, reality - that its essential feature is not ‘the mind’, ‘the self’, ‘sense-data’, nor propositions but symbolism, and that this symbolism is largely of the same kind as that which is familiar to us in mechanical devices which aid thought and calculation." (Kenneth Craik, "The Nature of Explanation", 1943)

"Thus there are instances of symbolisation in nature; we use such instances as an aid to thinking; there is evidence of similar mechanisms at work in our own sensory and central nervous systems; and the function of such symbolisation is plain. If the organism carries a ’small-scale model’ of external reality and of its own possible actions within its head, it is able to try out various alternatives, conclude which is the best of them, react to future situations before they arise […]" (Kenneth Craik, "The Nature of Explanation", 1943)

"Thus we do not try to prove the existence of the external world – we discover it, because the fundamental power of words or other symbols to represent events [...] permits us to put forward hypotheses and test their truth by reference to experience. [..] A particular type of symbolism may always fail in a particular case, as Euclidean geometry apparently fails to represent stellar space; but if all types of symbolism always failed, we should be unable to recognise any objects or exist at all." (Kenneth Craik, "The Nature of Explanation", 1943)

"Without falling into the trap of attempting a precise definition, we may suggest a theory as to the general nature of symbolism, viz. that it is the ability of processes to parallel or imitate each other, or the fact that they can do so since there are recurrent patterns in reality." (Kenneth Craik, "The Nature of Explanation", 1943)

"Man has [...] discovered a new method of adapting himself to his environment. Between the receptor system and the effector system, which are to be found in all animal species, we find in man a third link which we may describe as the symbolic system." (Ernst Cassirer, "An Essay on Man", 1944)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

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

"For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations -even a calculating machine can often do so - but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols - subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes." (Ralph W Gerard, "The Biological Basis of Imagination", American Thought, 1947)

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

"Belief has its structures, and its symbols change. Its tradition changes. All the relationships within these forms are inter-dependent. We look at the symbols, we hope to read them, we hope for sharing and communication." (Muriel Rukeyser, "The Life of Poetry", 1949)

"However obvious these facts may appear at first glance, they are actually not so obvious as they seem except when we take special pains to think about the subject. Symbols and things symbolized are independent of each other; nevertheless, we all have a way of feeling as if […] there were necessary connections." (Samuel I Hayakawa, "Language in Thought and Action", 1949)

"Men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations." (Arthur Koestler, "Crossman", 1949)

"The first of the principles governing symbols is this: The symbol is NOT the thing symbolized; the word is NOT the thing; the map is NOT the territory it stands for." (Samuel I Hayakawa, "Language in Thought and Action", 1949)