Herbert Stachowiak - Collected Quotes

"All the model-boundedness of human encounter with the world as well as the interhuman communication is equally involved already in the modeling character of the perception process. […] The modeling character of perception forms is also not changed by the circumstance that the access to the original-side, i.e. to the properties of signal constellations from the external world, it is only always possible via the building of internal models of the external world." (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)

"Models are models of something, namely, [they are] reflections, representations of natural and artificial originals, that can themselves be models again. […] Models, in general, do not cover all the attributes of the originals they are representing, but only those [attributes] that seem relevant to the actual model creators and/or model users." (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)

"Models are not assigned per se uniquely to their originals. They perform their replacement function: a) for definite – cognitive and/or handling, model-using – subjects, b) within definite time intervals, c) under restrictions of definite operations of thought or fact. […] Models are not only models of something. They are also models for somebody, a human or an artificial model user. They perform thereby their functions in time, within a time interval. And finally, they are models for a definite purpose." (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)

"The advantage of this way of proceeding is evident: insights and skills obtained on the model-side can be - certain transference criteria satisfied - transferred to the original, [in this way] the model-builder obtains a new knowledge about the modeled original […]" (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)

"The model-based concept of cognition takes the representation/reflection idea of the classical theory of cognition, but relativizes it in the sense of the pragmatic decision. Accordingly, all of cognition is cognition in models or by means of models, and in general, any human encounter with the world needs a ‘model’ as the mediator: focusing on – active or passive – comprehension of something, it [cognition] proceeds relative to certain subjects, further selective – intentionally selecting and centering – and depending on the temporal restriction of its relation to the original." (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)

Lester F Ward - Collected Quotes

"The constructive process inheres in all forms of synergy, and the cooperation of antithetical forces in nature always results in making, that is, in creating something that did not exist before. But in the organic world this character of structure becomes the leading feature, and we have synthetic products consisting of tissues and organs serving definite purposes, which we call functions." (Lester F Ward, "Pure Sociology", 1903)

"[...] there is a universal principle, operating in every department of nature and at every stage of evolution, which is conservative, creative and constructive. [...] I have at last fixed upon the word synergy, as the term best adapted to express its twofold character of ‘energy’ and ‘mutuality’ or the systematic and organic ‘working together’ of the antithetical forces of nature. [...] Synergy is a synthesis of work, or synthetic work, and this is what is everywhere taking place. It may be said to begin with the primary atomic collision in which mass, motion, time, and space are involved, and to find its simplest expression in the formula for force, which implies a plurality of elements, and signifies an interaction of these elements." (Lester F Ward, "Pure Sociology", 1903)

"This compromise among the contending forces of nature was effected through organization and the formation of chemical systems, which are so many reservoirs of power, this power being represented by what we call the properties of matter. These systems store up energy and expend it in work, but the work is always a collaboration or cooperation of all the competing forces involved. It is synergy." (Lester F Ward, "Pure Sociology", 1903)

"It is in the organic world that we can best on begin the study of function. But for the function, organic structures would be worthless. The structures are only means. Function is the end. All natural structures are developed along with their functions, which may be regarded in a sense as the cause of the structures. The effort of nature to accomplish its ends results in material means capable of accomplishing them, and such means are structures." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

"Social structures are the products of social synergy, i.e., of the interaction of different social forces, all of which, in and of themselves, are destructive, but whose combined effect, mutually checking, constraining, and equilibrating one another, is to produce structures. The entire drift is toward economy, conservatism, and the prevention of waste. Social structures are mechanisms for the production of results, and the results cannot be secured without them. They are reservoirs of power." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

"The true nature of the universal principle of synergy pervading all nature and creating all the different kinds of structure that we observe to exist, must now be made clearer. Primarily and essentially it is a process of equilibration, i.e., the several forces are first brought into a state of partial equilibrium. It begins in collision, conflict, antagonism, and opposition, and then we have the milder phases of antithesis, competition, and interaction, passing next into a modus vivendi, or compromise, and ending in collaboration and cooperation." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

"Synergy is the principle that explains all organization and creates all structures. The products of cosmic synergy are found in all fields of phenomena. Celestial structures are worlds and world systems; chemical structures are atoms, molecules, and substances; biotic structures are protoplasm, cells, tissues, organs, and organisms. There are also psychic structures - feelings, emotions, passions, volitions, perceptions, cognitions, memory, imagination, reason, thought, and all the acts of consciousness. And then there are social structures […]. These are the products of the social forces acting under the principle of social synergy." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

On Patterns (1940-1949)

"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1941)

"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"One may generalize upon these processes in terms of group equilibrium. The group may be said to be in equilibrium when the interactions of its members fall into the customary pattern through which group activities are and have been organized. The pattern of interactions may undergo certain modifications without upsetting the group equilibrium, but abrupt and drastic changes destroy the equilibrium." (William F Whyte, "Street Corner Society", 1943)

"Those who are content with a positivist conception of the aims of science will feel that he is in an entirely satisfactory position; he has discovered the pattern of events, and so can predict accurately; what more can he want? A mental picture would be an added luxury, but also a useless luxury. For if the picture did not bear any resemblance at all to the reality it would be valueless, and if it did it would be unintelligible […]" (James H Jeans," Physics and Philosophy" 3rd Ed., 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)

"Science in general […] does not consist in collecting what we already know and arranging it in this or that kind of pattern. It consists in fastening upon something we do not know, and trying to discover it. (Robin G Collingwood, "The Idea of History", 1946)

On Continuity XIII (Trivia)

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." (Galileo Galilei, "The Assayer", 1623)

"Time is the order of existence of those things which are not simultaneous. Thus time is the universal order of changes when we do not take into consideration the particular kinds of change. Duration is magnitude of time. If the magnitude of time is diminished uniformly and continuously, time disappears into moment, whose magnitude is zero. Space is the order of coexisting things, or the order of existence for things which are simultaneous." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"It is also only by virtue of the continual action of God upon us that we have in our soul the ideas of all things; that is to say, since every effect expresses its cause, the essence of our soul is a certain expression, imitation or image of the divine essence, thought, and will and of all the ideas which are comprised in God." (Gottfried W Leibniz, "Discourse on Metaphysics", 1686)

"There are two aspects of statistics that are continually mixed, the method and the science. Statistics are used as a method, whenever we measure something, for example, the size of a district, the number of inhabitants of a country, the quantity or price of certain commodities, etc. […] There is, moreover, a science of statistics. It consists of knowing how to gather numbers, combine them and calculate them, in the best way to lead to certain results. But this is, strictly speaking, a branch of mathematics." (Alphonse P de Candolle, "Considerations on Crime Statistics", 1833)

"The man who is guided by concepts and abstractions only succeeds by such means in warding off misfortune, without ever gaining any happiness for himself from these abstractions. And while he aims for the greatest possible freedom from pain, the intuitive man, standing in the midst of a culture, already reaps from his intuition a harvest of continually inflowing illumination, cheer, and redemption - in addition to obtaining a defense against misfortune. To be sure, he suffers more intensely, when he suffers; he even suffers more frequently, since he does not understand how to learn from experience and keeps falling over and over again into the same ditch." (Friedrich Nietzsche," On Truth and Lie in an Extra-Moral Sense", 1873)

"[…] the statistical prediction of the future from the past cannot be generally valid, because whatever is future to any given past, is in tum past for some future. That is, whoever continually revises his judgment of the probability of a statistical generalization by its successively observed verifications and failures, cannot fail to make more successful predictions than if he should disregard the past in his anticipation of the future. This might be called the ‘Principle of statistical accumulation’." (Clarence I Lewis, "Mind and the World-Order: Outline of a Theory of Knowledge", 1929)

"In time they [physicists] hoped to devise a model which would reproduce all the phenomena of physics, and so make it possible to predict them all. […] To-day we not only have no perfect model, but we know that it is of no use to search for one - it could have no intelligible meaning for us. For we have found out that nature does not function in a way that can be made comprehensible to the human mind through models or pictures. […] Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"Above all, we see from these formulations how difficult it is when we try to push new ideas into an old system of concepts belonging to an earlier philosophy, or, to use an old metaphor, when we attempt to put new wine into old bottles. Such attempts are always distressing, for they mislead us into continually occupying with the inevitable cracks in the old bottles, instead of rejoicing over the new wine." (Werner K Heisenberg, "Physics and Philosophy: The Revolution in Modern Science", 1958)

"Scientific knowledge is not created solely by the piecemeal mining of discrete facts by uniformly accurate and reliable individual scientific investigations. The process of criticism and evaluation, of analysis and synthesis, are essential to the whole system. It is impossible for each one of us to be continually aware of all that is going on around us, so that we can immediately decide the significance of every new paper that is published. The job of making such judgments must therefore be delegated to the best and wisest among us, who speak, not with their own personal voices, but on behalf of the whole community of Science. […] It is impossible for the consensus - public knowledge - to be voiced at all, unless it is channeled through the minds of selected persons, and restated in their words for all to hear." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

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

"Hard though the scientists of mental imagery try, they cannot get around the fact that the representations they deal with are like pictures. […] The methods have to assume, and the experiments continually corroborate, that having imagery is somehow like perceptual seeing, and that it is somehow like seeing pictures. […] The minimal reason for this assumption is that people do naturally talk of seeing pictures before their mind’s eye." (Eva T H Brann,"The World of Imagination", 1991)

"To remedy chaotic situations requires a chaotic approach, one that is non-linear, constantly morphing, and continually sharpening its competitive edge with recurring feedback loops that build upon past experiences and lessons learned. Improvement cannot be sustained without reflection. Chaos arises from myriad sources that stem from two origins: internal chaos rising within you, and external chaos being imposed upon you by the environment. The result of this push/pull effect is the disequilibrium [...]." (Jeff Boss, "Navigating Chaos: How to Find Certainty in Uncertain Situations", 2015)

"[…] we see from these formulations how difficult it is when we try to push new ideas into an old system of concepts belonging to an earlier philosophy, or, to use an old metaphor, when we attempt to put new wine into old bottles. Such attempts are always distressing, for they mislead us into continually occupying with the inevitable cracks in the old bottles, instead of rejoicing over the new wine." (Werner K Heisenberg)

On Nature (-1599)

"They say that the greatest and fairest things are the work of nature and of chance, the lesser of art, which, receiving from nature the greater and primeval creations, molds and fashions all those lesser works which are generally termed artificial." (Plato, "Nomoi" ["Laws"], cca. 360 BC)

"Everything that depends on the action of nature is by nature as good as it can be, and similarly everything that depends on art or any rational cause, and especially if it depends on the best of all causes. To entrust to chance what is greatest and most noble would be a very defective arrangement." (Aristotle, "Nicomachean Ethics", cca. 350 BC)

"The proof that the state is a creation of nature and prior to the individual is that the individual, when isolated, is not self-sufficing; and therefore he is like a part in relation to the whole." (Aristotle, "Politics", 4th century BC)

"Thus, of all the honorable arts, which are carried out either naturally or proceed in imitation of nature, geometry takes the skill of reasoning as its field. It is hard at the beginning and difficult of access, delightful in its order, full of beauty, unsurpassable in its effect. For with its clear processes of reasoning it illuminates the field of rational thinking, so that it may be understood that geometry belongs to the arts or that the arts are from geometry." (Agennius Urbicus, "Controversies about Fields", cca. 4 century BC)

"No species remains constant: that great renovator of matter Nature, endlessly fashions new forms from old: there’s nothing in the whole universe that perishes, believe me; rather it renews and varies its substance. What we describe as birth is no more than incipient change from a prior state, while dying is merely to quit it. Though the parts may be transported hither and thither, the sum of all matter is constant." (Publius Ovidius Naso [Ovid], "Metamorphoses", 8 AD)

"All that is superfluous displeases God and nature. All that displeases God and nature is evil." (Dante Alighieri, "De Monarchia", cca. 1312-1313)

"Given that annihilation of nature in its entirety is impossible, and that death and dissolution are not appropriate to the whole mass of this entire globe or star, from time to time, according to an established order, it is renewed, altered, changed, and transformed in all its parts." (Giordano Bruno, "The Ash Wednesday Supper", 1584)

"Nature that framed us of four elements, Warring within our breasts for regiment, Doth teach us all to have aspiring minds: Our souls, whose faculties can comprehend The wondrous architecture of the world: And measure every wand’ring planet’s course, Still climbing after knowledge infinite, And always moving as the restless spheres, Wills us to wear ourselves and never rest, Until we reach the ripest fruit of all, That perfect bliss and sole felicity, The sweet fruition of an earthly crown.""  (Christopher Marlowe, "Tamburlaine the Great", 1590)

"The diversity of the phenomena of Nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment." (Johannes Kepler, "Mysterium Cosmographicum", 1596)

On Nature: The Nature of Things

"Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed […] Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"How awkward is the human mind in divining the nature of things, when forsaken by the analogy of what we see and touch directly?" (Ludwig Boltzmann, "Certain Questions of the Theory of Gasses", Nature Vol. 51 (1322), 1895)

"The first step, whenever a practical problem is set before a mathematician, is to form the mathematical hypothesis. It is neither needful nor practical that we should take account of the details of the structure as it will exist. We have to reason about a skeleton diagram in which bearings are reduced to points, pieces to lines, etc. and [in] which it is supposed that certain relations between motions are absolutely constrained, irrespective of forces. Some writers call such a hypothesis a fiction, and say that the mathematician does not solve the real problem, but only a fictitious one. That is one way of looking at the matter, to which I have no objection to make: only, I notice, that in precisely the same sense in which the mathematical hypothesis is 'false', so also is this statement 'false', that it is false. Namely, both representations are false in the sense that they omit subsidiary elements of the fact, provided that element of the case can be said to be subsidiary which those writers overlook, namely, that the skeleton diagram is true in the only sense in which from the nature of things any mental representation, or understanding, of the brute existent can be true. For every possible conception, by the very nature of thought, involves generalization; now generalization omits, means to omit, and professes to omit, the differences between the facts generalized." (Charles S Peirce, "Report on Live Loads", cca. 1895)

"A statistical estimate may be good or bad, accurate or the reverse; but in almost all cases it is likely to be more accurate than a casual observer’s impression, and the nature of things can only be disproved by statistical methods." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Great scientific discoveries have been made by men seeking to verify quite erroneous theories about the nature of things." (Aldous L Huxley, "Life and Letters and the London Mercury" Vol. 1, 1928)

"It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper, "The Open Society and Its Enemies", 1945)

"Science does not need mysticism and mysticism does not need science, but man needs both. Mystical experience is necessary to understand the deepest nature of things, and science is essential for modern life. What we need, therefore, is not a synthesis, but a dynamic interplay between mystical intuition and scientific analysis." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"Owing to his lack of knowledge, the ordinary man cannot attempt to resolve conflicting theories of conflicting advice into a single organized structure. He is likely to assume the information available to him is on the order of what we might think of as a few pieces of an enormous jigsaw puzzle. If a given piece fails to fit, it is not because it is fraudulent; more likely the contradictions and inconsistencies within his information are due to his lack of understanding and to the fact that he possesses only a few pieces of the puzzle. Differing statements about the nature of things […] are to be collected eagerly and be made a part of the individual's collection of puzzle pieces. Ultimately, after many lifetimes, the pieces will fit together and the individual will attain clear and certain knowledge." (Alan R Beals, "Strategies of Resort to Curers in South India" [contributed in Charles M. Leslie (ed.), "Asian Medical Systems: A Comparative Study", 1976])

"In strategic thinking, one first seeks a clear understanding of the particular character of each element of a situation and then makes the fullest possible use of human brainpower to restructure the elements in the most advantageous way. Phenomena and events in the real word do not always fit a linear model. Hence the most reliable means of dissecting a situation into its constituent parts and reassembling then in the desired pattern is not a step-by-step methodology such as systems analysis. Rather, it is that ultimate nonlinear thinking tool, the human brain. True strategic thinking thus contrasts sharply with the conventional mechanical systems approach based on linear thinking. But it also contrasts with the approach that stakes everything on intuition, reaching conclusions without any real breakdown or analysis. [...] No matter how difficult or unprecedented the problem, a breakthrough to the best possible solution can come only from a combination of rational analysis, based on the real nature of things, and imaginative reintegration of all the different items into a new pattern, using nonlinear brainpower. This is always the most effective approach to devising strategies for dealing successfully with challenges and opportunities, in the market arena as on the battlefield." (Kenichi Ohmae, "The Mind Of The Strategist", 1982)

"All great theories are expansive, and all notions so rich in scope and implication are underpinned by visions about the nature of things. You may call these visions ‘philosophy’, or ‘metaphor’, or ‘organizing principle’, but one thing they are surely not - they are not simple inductions from observed facts of the natural world." (Stephen J Gould, "Time’s Arrow, Time’s Cycle", 1987)

"If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)

"Metaphor is evidence of the human ability to visualize the universe as a coherent organism. Proof of our capacity, not just to see one thing in another but to change the very nature of things. When a metaphor is accepted as fact, it enters groupthink, taking on an existence in the real world. [...] Metaphor is the default form of thought, providing many angles from which to literally 'see' the world." (Marcel Danesi, "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture", 2004)

Dietrich Dorner - Collected Quotes

"A system of variables is 'interrelated' if an action that affects or meant to affect one part of the system will also affect other parts of it. Interrelatedness guarantees that an action aimed at one variable will have side effects and long-term repercussions. A large number of variables will make it easy to overlook them." (Dietrich Dorner, "The Logic of Failure: Recognizing and Avoiding Error in Complex Situations", 1989)

"Complexity is not an objective factor but a subjective one. Supersignals reduce complexity, collapsing a number of features into one. Consequently, complexity must be understood in terms of a specific individual and his or her supply of supersignals. We learn supersignals from experience, and our supply can differ greatly from another individual's. Therefore there can be no objective measure of complexity." (Dietrich Dorner, "The Logic of Failure: Recognizing and Avoiding Error in Complex Situations", 1989)

"If we want to solve problems effectively […] we must keep in mind not only many features but also the influences among them. Complexity is the label we will give to the existence of many interdependent variables in a given system. The more variables and the greater their interdependence, the greater the system's complexity. Great complexity places high demands on a planner's capacity to gather information, integrate findings, and design effective actions. The links between the variables oblige us to attend to a great many features simultaneously, and that, concomitantly, makes it impossible for us to undertake only one action in a complex system." (Dietrich Dorner, "The Logic of Failure: Recognizing and Avoiding Error in Complex Situations", 1989)

"A system is extremely complex when it consists of a great variety of variables." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"Adapting oneself inadequately to the sequential characteristics of processes may also be attributable to an incredibly simple feature of human data processing, namely, forgetfulness. An important requirement for gaining the correct picture of temporal sequences is having information on the length of time available. If this is not the case, one is also unable to posit hypotheses on temporal patterns. The fact that people forget means that past data are only partially available. This means that there are great difficulties in recognizing the correct form of temporal sequences. A simple means of coping with this difficulty is the 'spatialization' of time. Diagrams of temporal sequences make it possible to treat temporal sequences like 'spatial forms', which are easier to cope with." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"Human beings have a strong tendency to react only to the status quo and to disregard developments and their conditions." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"Humans have a strong tendency to guard their opinion of their own competence in acting. To a certain extent this makes sense, as someone who considers himself to be incapable of acting will hardly act. Guarding one's opinion of one's competence is an important motivation. But it can lead to deformations in the thought process. To maintain a high opinion of one's own competence, people fail to take notice of data that show that their hypotheses are wrong. Or they act 'ballistically' and do not check the effects of their actions so as to maintain the illusion of having solved the corresponding problems by means of their action. The underlying reasons for dispensing with self-reflection may also lie in the tendency to avoid looking at one's own mistakes so as not to endanger one's estimation of one's own competence." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"It is possible to learn strategic flexibility [...] however, that it is difficult to teach it. It is not a matter of learning a few readily grasped general principles, but of learning a lot of small, 'local' rules, each of which is applicable in a limited area. The point is not to learn how to drive a steamroller with which one can flatten all problems in the same way, but to learn the adroitness of a puppeteer, who at one time holds many strings in his hands and who is able to adapt his movements to the given circumstances in the most sophisticated ways." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"Subjects often act 'ballistically'. They take measures without checking the effects of these measures later. As the effects of measures are usually uncertain, this lies in the nature of complex systems, this is a dangerous error. Crisis situations are especially susceptible for ballistic forms of action […]" (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"Subjects' strategies for coping with complex systems are for the most part insufficient, in one respect or another. Self-reflexive examination and critique of one's own way of acting is an essential means of adapting one's own way of acting to the given circumstances. Dispensing with self-reflection is therefore a major error." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"Unlike other living creatures, humans can adapt to uncertainty. They can form hypotheses about situations marked by uncertainty and can anticipate their actions by planning. They can expect the unexpected and take precautions against it." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

On Strategy I

"It is possible to learn strategic flexibility [...] however, that it is difficult to teach it. It is not a matter of learning a few readily grasped general principles, but of learning a lot of small, 'local' rules, each of which is applicable in a limited area. The point is not to learn how to drive a steamroller with which one can flatten all problems in the same way, but to learn the adroitness of a puppeteer, who at one time holds many strings in his hands and who is able to adapt his movements to the given circumstances in the most sophisticated ways." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"[…] a rule for choosing an action is termed a strategy. If the rule says to always take the same action, it's called a pure strategy; otherwise, the strategy is called mixed. A solution to a game is simply a strategy for each player that gives each of them the best possible payoff, in the sense of being a regret-free choice." (John L Casti, "Five Golden Rules", 1995)

"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)

"A strategy is usually expressed by a set of heuristic rules. The heuristic rules ease the process of searching for an optimal solution. The process is usually iterative and at one step either the global optimum for the whole problem (state) space is found and the process stops, or a local optimum for a subspace of the state space of the problem is found and the problem continues, if it is possible to improve." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"Strategy in complex systems must resemble strategy in board games. You develop a small and useful tree of options that is continuously revised based on the arrangement of pieces and the actions of your opponent. It is critical to keep the number of options open. It is important to develop a theory of what kinds of options you want to have open." (John H Holland, [presentation] 2000)

"[...] a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration." Yu-Chi Ho & David L Pepyne, "Simple explanation of the no-free-lunch theorem and its implications", Journal of Optimization Theory and Applications 115, 2002)

"[...] the System may be so thoroughly organized around the familiar response strategy that a new response would require extensive restructuring - something that Systems do with the greatest reluctance and difficulty." (John Gall, "Systemantics: The Systems Bible", 2002)

"We can find the minimax strategy by exploiting the game’s symmetry. Roughly speaking, the minimax strategy must have the same kind of symmetry." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"A heuristic is a strategy we derive from previous experience with a similar problem." (Darius Foroux, "Think Straight", 2017)

On God (-1199)

"God's dice always have a lucky roll." (Sophocles, 5th century BC)

"As infinite kinds of almost identical images arise continually from the innumerable atoms and flow out to us from the gods, so we should take the keenest pleasure in turning and bending our mind and reason to grasp these images, in order to understand the nature of these blessed and eternal beings." (Marcus Tullius Cicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC)

"I say, then, that the universe and all its parts both received their first order from divine providence, and are at all times administered by it." (Marcus T Cicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC)

"Uneven numbers are the god’s delight" (Virgil, "The Eclogues", cca. 40 BC)

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

"Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect." (Saint Augustine, "The City of God", 426 AD)

"In the foregoing you will discover a very remarkable thing. God reserved the truth of things, which is the supreme truth, for Himself, but He conceded to His image the formation of images of things at whatever time." (Richard of St. Victor, "Benjamin Major" [aka "The Mystical Ark"], cca 1162)

"A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth." (Bhaskara II, "Bijaganita", 12th century)

On God (1900-1949)

"There is no such thing as chaos, it tacitly asserts, in the sidereal world or outside of it. For chaos is the negation of law, and law is the expression of the will of God." (Agnes M Clerke, "Problems in Astrophysics",  1903)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature 62 develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor, "A Theory of Motives, Ideals and Values in Education" 1907)

"By the help of God and with His precious assistance I say that algebra is a scientific art. The objects with which it deals are absolute numbers and (geometrical) magnitudes which, though themselves unknown, are related to things which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. […] What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of algebra as stated above." (Omar Khayyam [quoted by Daoud Suleiman Kasir in "The Algebra of Omar Khayyam", 1931)

"Both religion and natural science require a belief in God for their activities, to the former He is the starting point, and to the latter the goal of every thought process. To the former He is the foundation, to the latter, the crown of the edifice of every generalized world view." (Max Planck, "Religion and Natural Science", 1937)

"Science is the organised attempt of mankind to discover how things work as causal systems. The scientific attitude of mind is an interest in such questions. It can be contrasted with other attitudes, which have different interests; for instance the magical, which attempts to make things work not as material systems but as immaterial forces which can be controlled by spells; or the religious, which is interested in the world as revealing the nature of God." (Conrad H Waddington, "The Scientific Attitude", 1941)

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

"If God has made the world a perfect mechanism, He has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success." (Max Born, "Albert Einstein: Philosopher-Scientist", 1949)

On God (Unsourced)

"An equation for me has no meaning unless it expresses a thought of God." (Srinivasa Ramanujan)

"As knowledge advances, science ceases to scoff at religion; and religion ceases to frown on science. The hour of mockery by the one, and of reproof by the other, is passing away. Henceforth, they will dwell together in unity and goodwill. They will mutually illustrate the wisdom, power, and grace of God. Science will adorn and enrich religion; and religion will ennoble and sanctify science." (Oliver W Holmes)

"God created everything by number, weight and measure." (Sir Isaac Newton)

"God is in the details, for mathematicians have plunged deeper and deeper within Pi’s digits with a religious fervor, hoping to find even a hint of understanding." (Ludwig Mies van der Rohe)

"God made the natural numbers. all else is the work of man." (Leopold Kronecker)

"God may not play dice with the universe, but something strange is going on with the prime numbers." (Paul Erdos)

"It is a right, yes a duty, to search in cautious manner for the numbers, sizes, and weights, the norms for everything [God] has created. For He himself has let man take part in the knowledge of these things […] For these secrets are not of the kind whose research should be forbidden; rather they are set before our eyes like a mirror so that by examining them we observe to some extent the goodness and wisdom of the Creator." (Johannes Kepler)

"Man’s first glance at the universe discovers only variety, diversity, multiplicity of phenomena. Let that glance be illuminated by science - by the science which brings man closer to God, - and simplicity and unity shine on all sides." (Louis Pasteur)

"Mathematics is the life supreme. The life of the gods is mathematics. All divine messengers are mathematicians. Pure mathematics is religion. Its attainment requires a theophany." (Friederich von Hardenberg [Novalis])

"Number is the ruler of forms and ideas, and the cause of gods and demons." (Pythagoras)

"That deep emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God." (Albert Einstein)

"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics." (Johannes Kepler)

"The God that reigns in Olympus is Number Eternal." (Carl Gustav Jacobi)

"The laws of nature are but the mathematical thoughts of God." (Euclid)

"The laws of thought, and especially of number, must hold good in heaven, whether it is a place or a state of mind; for they are independent of any particular sphere of existence, essential to Being itself, to God’s being as well as ours, laws of His mind before we learned them. The multiplication table will hold good in heaven […]" (Hilda P Hudson)

"The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods; the gods are there, behind the wall, at play with numbers." (Le Corbusier)

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." (John W N Sullivan)

"[…] there is a God precisely because Nature itself, even in chaos, cannot proceed except in an orderly and regular manner." (Immanuel Kant)

"What else can the human mind hold besides numbers and magnitudes? These alone we apprehend correctly, and if piety permits to say so, our comprehension is in this case of the same kind as God’s, at least insofar as we are able to understand it in this mortal life." (Johannes Kepler)

"What I’m really interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all." (Albert Einstein)

"What one man calls God, another calls the laws of physics." (Nikola Tesla)

"What you can show using physics, forces this universe to continue to exist. As long as you're using general relativity and quantum mechanics you are forced to conclude that God exists." (Frank Tipler)

On God (1975-1999)

"The sciences have started to swell. Their philosophical basis has never been very strong. Starting as modest probing operations to unravel the works of God in the world, to follow its traces in nature, they were driven gradually to ever more gigantic generalizations. Since the pieces of the giant puzzle never seemed to fit together perfectly, subsets of smaller, more homogeneous puzzles had to be constructed, in each of which the fit was better." (Erwin Chargaff, "Voices in the Labyrinth", 1975)

"The vector equilibrium is the true zero reference of the energetic mathematics. Zero pulsation in the vector equilibrium is the nearest approach we will ever know to eternity and god: the zero phase of conceptual integrity inherent in the positive and negative asymmetries that propagate the differentials of consciousness." (Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)

"Whenever the Eastern mystics express their knowledge in words - be it with the help of myths, symbols, poetic images or paradoxical statements-they are well aware of the limitations imposed by language and 'linear' thinking. Modern physics has come to take exactly the same attitude with regard to its verbal models and theories. They, too, are only approximate and necessarily inaccurate. They are the counterparts of the Eastern myths, symbols and poetic images, and it is at this level that I shall draw the parallels. The same idea about matter is conveyed, for example, to the Hindu by the cosmic dance of the god Shiva as to the physicist by certain aspects of quantum field theory. Both the dancing god and the physical theory are creations of the mind: models to describe their authors' intuition of reality." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"If entropy must constantly and continuously increase, then the universe is remorselessly running down, thus setting a limit (a long one, to be sure) on the existence of humanity. To some human beings, this ultimate end poses itself almost as a threat to their personal immortality, or as a denial of the omnipotence of God. There is, therefore, a strong emotional urge to deny that entropy must increase." (Isaac Asimov," Asimov on Physics", 1976)

"Our look is as bound by time-space as our brain. We never look, we never see beyond this limitation; we do not know how to look through and beyond these fragmentary frontiers. But the eyes have to see beyond them, penetrating deeply and widely, without choosing, without shelter; they have to wander beyond man-made frontiers of ideas and values and to feel beyond love. Then there is a benediction which no god can give." (Jiddu Krishnamurti," Krishnamurti's Notebook", 1976)

"Whether mathematical simplicity is God’s affair or our, the fact remains that this feature more than any other remains the mainspring of progress in the physical sciences." (Paul C W Davies, "The Edge of Infinity", 1981)

"The equations of physics have in them incredible simplicity, elegance and beauty. That in itself is sufficient to prove to me that there must be a God who is responsible for these laws and responsible for the universe" (Paul C W Davies, 1984)

"Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God." (Malba Tahan & Patricia R Baquero, "The Man Who Counted", 1993)

"In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world." (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

"Is God a mathematician? Certainly, the world, the universe, and nature can be reliably understood using mathematics. Nature is mathematics." (Clifford A Pickover, "The Loom of God", 1997)

On God (1950-1974)

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

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

"[…] the scientific picture of the real world around me is very deficient. It gives a lot of factual information, puts all our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously." (Erwin Schrödinger, "Nature and the Greeks", 1954)

"In particular, and most importantly, this is the reason why the scientific worldview contains of itself no ethical values, no esthetical values, not a word about our own ultimate scope or destination, and no God, if you please." (Erwin Schrödinger, "Nature and the Greeks", 1954)

"It is of our very nature to see the universe as a place that we can talk about. In particular, you will remember, the brain tends to compute by organizing all of its input into certain general patterns. It is natural for us, therefore, to try to make these grand abstractions, to seek for one formula, one model, one God, around which we can organize all our communication and the whole business of living." (John Z Young, "Doubt and Certainty in Science: A Biologist’s Reflections on the Brain", 1960)

"A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." (Paul Dirac, Scientific American, 1963)

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better." (Paul Dirac, "The Evolution of the Physicist's Picture of Nature", 1963)

"Philosophy is metaphysics. Metaphysics thinks beings as a whole - the world, man, God - with respect to Being, with respect to the belonging together of beings in Being. Metaphysics thinks beings as being in the manner of representational thinking which gives reasons." (Martin Heidegger, "The End of Philosophy and the Task of Thinking", 1964)

"All men seek to be enlightened. Religion is but the most ancient and honorable way in which men have striven to make sense out of God's universe. Scientists seek the lawfulness of events. It is the task of Religion to fit man into this lawfulness." (Frank Herbert, "Dune", 1965)

"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true." (Bertrand Russell, "Autobiography", 1967)

"Nature is a network of happenings that do not unroll like a red carpet into time, but are intertwined between every part of the world; and we are among those parts. In this nexus, we cannot reach certainty because it is not there to be reached; it goes with the wrong model, and the certain answers ironically are the wrong answers. Certainty is a demand that is made by philosophers who contemplate the world from outside; and scientific knowledge is knowledge for action, not contemplation. There is no God’s eye view of nature, in relativity, or in any science: only a man’s eye view." (Jacob Bronowski, "The Identity of Man", 1972)

On God (1700-1899)

"As in mathematics, when there is no maximum nor minimum, in short nothing distinguished, everything is done equally, or when that is not nothing at all is done: so it may be said likewise in respect of perfect wisdom, which is no less orderly than mathematics, that if there were not the best (optimum) among all possible worlds, God would not have produced any." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)

"There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)

"Even as the finite encloses an infinite series And in the unlimited limits appear, So the soul of immensity dwells in minutia And in narrowest limits no limit in here. What joy to discern the minute in infinity! The vast to perceive in the small, what divinity!"  (Jacques Bernoulli, "Ars Conjectandi", 1713)

"God put a secret art into the forces of Nature so as to enable it to fashion itself out of chaos into a perfect world system." (Immanuel Kant, "Universal Natural History and Theory of the Heavens", 1755)

"Nature is the system of laws established by the Creator for the existence of things and for the succession of creatures. Nature is not a thing, because this thing would be everything. Nature is not a creature, because this creature would be God. But one can consider it as an immense vital power, which encompasses all, which animates all, and which, subordinated to the power of the first Being, has begun to act only by his order, and still acts only by his concourse or consent. […] Time, space and matter are its means, the universe its object, motion and life its goal." (Georges-Louis L de Buffon, "Histoire Naturelle, Générale et Particulière, Avec la Description du Cabinet du Roi", 1764)

"Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds." (Johann Wolfgang von Goethe, [Letter to Naumann] 1826)

"God puts his finger in the other scale, And up we bounce, a bubble. Nought is great Nor small, with God; for none but he can make The atom imperceptible, and none But he can make a world; he counts the orbs, He counts the atoms of the universe, And makes both equal; both are infinite."  (Philip James Bailey, "Festus", 1845)

"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)

"Life through many long periods has been manifested in a countless host of varying structures, all circumscribed by one general plan, each appointed to a definite place, and limited to an appointed duration. On the whole the earth has been thus more and more covered by the associated life of plants and animals, filling all habitable space with beings capable of enjoying their own existence or ministering to the enjoyment of others; till finally, after long preparation, a being was created capable of the wonderful power of measuring and weighing all the world of matter and space which surrounds him, of treasuring up the past history of all the forms of life, and considering his own relation to the whole. When he surveys this vast and coordinated system, and inquires into its history and origin, can he be at a loss to decide whether it be a work of Divine thought and wisdom, or the fortunate offspring of a few atoms of matter, warmed by the anima mundi, a spark of electricity, or an accidental ray of sunshine?" (John Phillips, "Life on the Earth: Its Origin and Succession", 1860)

"Everything in nature is a puzzle until it finds its solution in man, who solves it in some way with God, and so completes the circle of creation. " (Theodore T Munger, "The Appeal to Life", 1891)

"Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent." (Arthur M Marshall, 1894)

On God (1000-1699)

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

"All that is superfluous displeases God and nature. All that displeases God and nature is evil." (Dante Alighieri, "De Monarchia", cca. 1312-1313)

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

"There is divinity in odd numbers, either in nativity, chance, or death." (William Shakespeare, "The Merry Wives of Windsor", 1602)

"I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said 'it appears to us', because I am not against thinking that not even for such an effect does Nature make use of straight line motion." (Galileo Galilei, [Letter to Francesco Ingoli] 1624)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"It is also only by virtue of the continual action of God upon us that we have in our soul the ideas of all things; that is to say, since every effect expresses its cause, the essence of our soul is a certain expression, imitation or image of the divine essence, thought, and will and of all the ideas which are comprised in God." (Gottfried W Leibniz, "Discourse on Metaphysics", 1686)

"As in a block of marble all possible figures are potentially contained in it, and can be drawn out of it by the movement or by the action of the chisel, so in the same way all intelligible figures are potentially in intelligible extension and are discovered in it according to the different ways in which this extension is represented to the mind, as a consequence of the general laws which God has established according to which he continuously acts in us. " (Nicolas Malebranche , "Dialogues On Metaphysics And Religion", 1688)

"According to the common system, before the creation of Matter, there was nothing but God, whose essence is immutable, and cannot be the pre-existent subject of Bodies." (Jean C de la Crose, "Memoirs for the ingenious", 1693)

On Computations (-1949)

"Algebra is a general Method of Computation by certain signs and symbols which have been contrived for the Purpose, and found convenient." (Colin Maclaurin, "A Treatise of Algebra", 1748)

"It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind." (Nicolas de Condorcet, "Outlines Of An Historical View Of The Progress Of The Human Mind", 1795)

"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)

"Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey - having been thus constructed - an intuitive perception of the significance of numbers being out of the question." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1908)

"Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a palette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"To humanize the teaching of mathematics means so to present the subject, so to interpret its ideas and doctrines, that they shall appeal, not merely to the computatory faculty or to the logical faculty but to all the great powers and interests of the human mind." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Let it be remarked [...] that an important difference between the way in which we use the brain and the machine is that the machine is intended for many successive runs, either with no reference to each other, or with a minimal, limited reference, and that it can be cleared between such runs; while the brain, in the course of nature, never even approximately clears out its past records. Thus the brain, under normal circumstances, is not the complete analogue of the computing machine but rather the analogue of a single run on such a machine." (Norbert Wiener, "Cybernetics: Or Control and Communication in the Animal and the Machine", 1948)

On Computation (2000-2009)

"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)

"Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall & Carl Pomerance, "Prime Numbers: A Computational Perspective", 2001)

"Indeed a deterministic die behaves very much as if it has six attractors, the steady states corresponding to its six faces, all of whose basins are intertwined. For technical reasons that can't quite be true, but it is true that deterministic systems with intertwined basins are wonderful substitutes for dice; in fact they're super-dice, behaving even more ‘randomly’ - apparently - than ordinary dice. Super-dice are so chaotic that they are uncomputable. Even if you know the equations for the system perfectly, then given an initial state, you cannot calculate which attractor it will end up on. The tiniest error of approximation – and there will always be such an error - will change the answer completely." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Inequalities are useful for bounding quantities that might otherwise be hard to compute." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"Why was progress in computing technology so fast compared with the lack of progress in space travel? The reason is very simple: computing technology is only now approaching scientific limits such as quantum uncertainty and the speed of light, while space technology has already run into its limits that derive from the basic principles of physics and chemistry." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"No investigation of complexity would be complete without a brief summary of what is often considered to be its most extreme form. Beyond the mathematical upper border of complexity lies the deceptively camouflaged notion of chaos. This is not strictly analogous to the classical interpretations of its name involving shear calamity and confusion. Instead, in mathematical or computational terms, chaos relates to much simpler notions of pattern and organization. It may be random to our native observation, certainly, but it is also far more concisely describable than complexity when inspected using modern mathematical techniques." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"[…] statistical thinking, though powerful, is never as easy or automatic as simply plugging numbers into formulas. In order to use statistical methods appropriately, you need to understand their logic, not just the computing rules." (Ann E Watkins et al, "Statistics in Action: Understanding a World of Data", 2007)

"Complexity Theory is concerned with the study of the intrinsic complexity of computational tasks. Its 'final' goals include the determination of the complexity of any well-defined task. Additional goals include obtaining an understanding of the relations between various computational phenomena (e.g., relating one fact regarding computational complexity to another). Indeed, we may say that the former type of goal is concerned with absolute answers regarding specific computational phenomena, whereas the latter type is concerned with questions regarding the relation between computational phenomena." (Oded Goldreich, "Computational Complexity: A Conceptual Perspective", 2008)

"Granular computing is a general computation theory for using granules such as subsets, classes, objects, clusters, and elements of a universe to build an efficient computational model for complex applications with huge amounts of data, information, and knowledge. Granulation of an object a leads to a collection of granules, with a granule being a clump of points (objects) drawn together by indiscernibility, similarity, proximity, or functionality. In human reasoning and concept formulation, the granules and the values of their attributes are fuzzy rather than crisp. In this perspective, fuzzy information granulation may be viewed as a mode of generalization, which can be applied to any concept, method, or theory." (Salvatore Greco et al, "Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach", 2009)

"How are we to explain the contrast between the matter-of-fact way in which v-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

On Computation (2010-2019)

"Could it be that some place out there in the computational universe, we might find our physical universe?" (Stephen Wolfram, "Computing a Theory of Everything", 2010)

"From a historical viewpoint, computationalism is a sophisticated version of behaviorism, for it only interpolates the computer program between stimulus and response, and does not regard novel programs as brain creations. [...] The root of computationalism is of course the actual similarity between brains and computers, and correspondingly between natural and artificial intelligence. The two are indeed similar because the artifacts in question have been designed to perform analogs of certain brain functions. And the computationalist program is an example of the strategy of treating similars as identicals." (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)

"It should also be noted that the novel information generated by interactions in complex systems limits their predictability. Without randomness, complexity implies a particular non-determinism characterized by computational irreducibility. In other words, complex phenomena cannot be known a priori." (Carlos Gershenson, "Complexity", 2011)

"The notion of emergence is used in a variety of disciplines such as evolutionary biology, the philosophy of mind and sociology, as well as in computational and complexity theory. It is associated with non-reductive naturalism, which claims that a hierarchy of levels of reality exist. While the emergent level is constituted by the underlying level, it is nevertheless autonomous from the constituting level. As a naturalistic theory, it excludes non-natural explanations such as vitalistic forces or entelechy. As non-reductive naturalism, emergence theory claims that higher-level entities cannot be explained by lower-level entities." (Martin Neumann, "An Epistemological Gap in Simulation Technologies and the Science of Society", 2011)

"Black Swans (capitalized) are large-scale unpredictable and irregular events of massive consequence - unpredicted by a certain observer, and such un - predictor is generally called the 'turkey' when he is both surprised and harmed by these events. [...] Black Swans hijack our brains, making us feel we 'sort of' or 'almost' predicted them, because they are retrospectively explainable. We don’t realize the role of these Swans in life because of this illusion of predictability. […] An annoying aspect of the Black Swan problem - in fact the central, and largely missed, point - is that the odds of rare events are simply not computable." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"[…] there exists a close relation between design analysis of algorithm and computational complexity theory. The former is related to the analysis of the resources (time and/or space) utilized by a particular algorithm to solve a problem and the later is related to a more general question about all possible algorithms that could be used to solve the same problem. There are different types of time complexity for different algorithms." (Shyamalendu Kandar, "Introduction to Automata Theory, Formal Languages and Computation", 2013)

"These nature-inspired algorithms gradually became more and more attractive and popular among the evolutionary computation research community, and together they were named swarm intelligence, which became the little brother of the major four evolutionary computation algorithms." (Yuhui Shi, "Emerging Research on Swarm Intelligence and Algorithm Optimization", Information Science Reference, 2014)

"The subject of computational complexity theory is focused on classifying problems by how hard they are. […] (1) P problems are those that can be solved by a Turing machine (TM) (deterministic) in polynomial time. (‘P’ stands for polynomial). P problems form a class of problems that can be solved efficiently. (2) NP problems are those that can be solved by non-deterministic TM in polynomial time. A problem is in NP if you can quickly (in polynomial time) test whether a solution is correct (without worrying about how hard it might be to find the solution). NP problems are a class of problems that cannot be solved efficiently. NP does not stand for 'non-polynomial'. There are many complexity classes that are much harder than NP. (3) Undecidable problems: For some problems, we can prove that there is no algorithm that always solves them, no matter how much time or space is allowed." (K V N Sunitha & N Kalyani, "Formal Languages and Automata Theory", 2015)

"When datasets are small, a parametric model may perform well because the strong assumptions made by the model - if correct - can help the model to avoid overfitting. However, as the size of the dataset grows, particularly if the decision boundary between the classes is very complex, it may make more sense to allow the data to inform the predictions more directly. Obviously the computational costs associated with nonparametric models and large datasets cannot be ignored. However, support vector machines are an example of a nonparametric model that, to a large extent, avoids this problem. As such, support vector machines are often a good choice in complex domains with lots of data." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"The higher the dimension, in other words, the higher the number of possible interactions, and the more disproportionally difficult it is to understand the macro from the micro, the general from the simple units. This disproportionate increase of computational demands is called the curse of dimensionality." (Nassim N Taleb, "Skin in the Game: Hidden Asymmetries in Daily Life", 2018)

"Computational complexity theory, or just complexity theory, is the study of the difficulty of computational problems. Rather than focusing on specific algorithms, complexity theory focuses on problems." (Rod Stephens, "Essential Algorithms" 2nd Ed., 2019)

On Computation (1990-1999)

"Illiteracy and innumeracy are social ills created in part by increased demand for words and numbers. As printing brought words to the masses and made literacy a prerequisite for productive life, so now computing has made numeracy an essential feature of today's society. But it is innumeracy, not numeracy, that dominates the headlines: ignorance of basic quantitative tools is endemic […] and is approaching epidemic levels […]." (Lynn A Steen, "Numeracy", Daedalus Vol. 119 No. 2, 1990)

"Mathematics is not just a collection of results, often called theorems; it is a style of thinking. Computing is also basically a style of thinking. Similarly, probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"In spite of the insurmountable computational limits, we continue to pursue the many problems that possess the characteristics of organized complexity. These problems are too important for our well being to give up on them. The main challenge in pursuing these problems narrows down fundamentally to one question: how to deal with systems and associated problems whose complexities are beyond our information processing limits? That is, how can we deal with these problems if no computational power alone is sufficient?"  (George Klir, "Fuzzy sets and fuzzy logic", 1995)

"Small changes in the initial conditions in a chaotic system produce dramatically different evolutionary histories. It is because of this sensitivity to initial conditions that chaotic systems are inherently unpredictable. To predict a future state of a system, one has to be able to rely on numerical calculations and initial measurements of the state variables. Yet slight errors in measurement combined with extremely small computational errors (from roundoff or truncation) make prediction impossible from a practical perspective. Moreover, small initial errors in prediction grow exponentially in chaotic systems as the trajectories evolve. Thus, theoretically, prediction may be possible with some chaotic processes if one is interested only in the movement between two relatively close points on a trajectory. When longer time intervals are involved, the situation becomes hopeless."(Courtney Brown, "Chaos and Catastrophe Theories", 1995)

 "An artificial neural network (or simply a neural network) is a biologically inspired computational model that consists of processing elements (neurons) and connections between them, as well as of training and recall algorithms." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"Beauty is more important in computing than anywhere else in technology because software is so complicated. Beauty is the ultimate defense against complexity." (David Gelernter, "Machine Beauty: Elegance And The Heart Of Technolog", 1998)

"As systems became more varied and more complex, we find that no single methodology suffices to deal with them. This is particularly true of what may be called information intelligent systems - systems which form the core of modern technology. To conceive, design, analyze and use such systems we frequently have to employ the totality of tools that are available. Among such tools are the techniques centered on fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing and related methodologies. It is this conclusion that formed the genesis of the concept of soft computing." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

"In science, it is a long-standing tradition to deal with perceptions by converting them into measurements. But what is becoming increasingly evident is that, to a much greater extent than is generally recognized, conversion of perceptions into measurements is infeasible, unrealistic or counter-productive. With the vast computational power at our command, what is becoming feasible is a counter-traditional move from measurements to perceptions. […] To be able to compute with perceptions it is necessary to have a means of representing their meaning in a way that lends itself to computation." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

On Computation (1950-1969)

"Instead of having a single control unit sequencing the operations of the machine in series (except for certain subsidiary operations as certain input and output functions) as is now done, the idea is to decentralize control with several different control units capable of directing various simultaneous operations and interrelating them when appropriate." (John F Nash, "Parallel Control", 1954)

"It is interesting to consider what a thinking machine will be like. It seems clear that as soon as the machines become able to solve intellectual problems of the highest difficulty which can be solved by humans they will be able to solve most of the problems enormously faster than a human." (John F Nash, "Parallel Control", 1954)

"We could define the intelligence of a machine in terms of the time needed to do a typical problem and the time needed for the programmer to instruct the machine to do it." (John F Nash, "Parallel Control", 1954)

"There are two types of systems engineering - basis and applied. [...] Systems engineering is, obviously, the engineering of a system. It usually, but not always, includes dynamic analysis, mathematical models, simulation, linear programming, data logging, computing, optimating, etc., etc. It connotes an optimum method, realized by modern engineering techniques. Basic systems engineering includes not only the control system but also all equipment within the system, including all host equipment for the control system. Applications engineering is - and always has been - all the engineering required to apply the hardware of a hardware manufacturer to the needs of the customer. Such applications engineering may include, and always has included where needed, dynamic analysis, mathematical models, simulation, linear programming, data logging, computing, and any technique needed to meet the end purpose - the fitting of an existing line of production hardware to a customer's needs. This is applied systems engineering." (Instruments and Control Systems Vol. 31, 1958)

"If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming."  (George Dantzig, "Linear Programming and Extensions", 1959)

"Computers do not decrease the need for mathematical analysis, but rather greatly increase this need. They actually extend the use of analysis into the fields of computers and computation, the former area being almost unknown until recently, the latter never having been as intensively investigated as its importance warrants. Finally, it is up to the user of computational equipment to define his needs in terms of his problems, In any case, computers can never eliminate the need for problem-solving through human ingenuity and intelligence." (Richard E Bellman & Paul Brock, "On the Concepts of a Problem and Problem-Solving", American Mathematical Monthly 67, 1960)

"It is of our very nature to see the universe as a place that we can talk about. In particular, you will remember, the brain tends to compute by organizing all of its input into certain general patterns. It is natural for us, therefore, to try to make these grand abstractions, to seek for one formula, one model, one God, around which we can organize all our communication and the whole business of living." (John Z Young, "Doubt and Certainty in Science: A Biologist’s Reflections on the Brain", 1960)

"The mathematical and computing techniques for making programmed decisions replace man but they do not generally simulate him." (Herbert A Simon, "Management and Corporations 1985", 1960)

"There is the very real danger that a number of problems which could profitably be subjected to analysis, and so treated by simpler and more revealing techniques. will instead be routinely shunted to the computing machines [...] The role of computing machines as a mathematical tool is not that of a panacea for all computational ills." (Richard E Bellman & Paul Brock, "On the Concepts of a Problem and Problem-Solving", American Mathematical Monthly 67, 1960)

"The purpose of computing is insight, not numbers." (Richard W Hamming, "Numerical Methods for Scientists and Engineers", 1962)

"Another thing I must point out is that you cannot prove a vague theory wrong. If the guess that you make is poorly expressed and rather vague, and the method that you use for figuring out the consequences is a little vague - you are not sure, and you say, 'I think everything's right because it's all due to so and so, and such and such do this and that more or less, and I can sort of explain how this works' […] then you see that this theory is good, because it cannot be proved wrong! Also if the process of computing the consequences is indefinite, then with a little skill any experimental results can be made to look like the expected consequences." (Richard P Feynman, "The Character of Physical Law", 1965)

"A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective." (George Spencer-Brown, "Laws of Form", 1969)

On Computation (1980-1989)

"The computational formalism of mathematics is a thought process that is externalised to such a degree that for a time it becomes alien and is turned into a technological process. A mathematical concept is formed when this thought process, temporarily removed from its human vessel, is transplanted back into a human mold. To think […] means to calculate with critical awareness." (Yuri I. Manin, "Mathematics and Physics", 1981)

"At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs." (Philip Johnson-Laird, "Mental Models" 1983)

"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"The formal structure of a decision problem in any area can be put into four parts: (1) the choice of an objective function denning the relative desirability of different outcomes; (2) specification of the policy alternatives which are available to the agent, or decisionmaker, (3) specification of the model, that is, empirical relations that link the objective function, or the variables that enter into it, with the policy alternatives and possibly other variables; and (4) computational methods for choosing among the policy alternatives that one which performs best as measured by the objective function." (Kenneth Arrow, "The Economics of Information", 1984)

"Computational reducibility may well be the exception rather than the rule: Most physical questions may be answerable only through irreducible amounts of computation. Those that concern idealized limits of infinite time, volume, or numerical precision can require arbitrarily long computations, and so be formally undecidable." (Stephen Wolfram, Undecidability and intractability in theoretical physics", Physical Review Letters 54 (8), 1985)

"To experience the joy of mathematics is to realize mathematics is not some isolated subject that has little relationship to the things around us other than to frustrate us with unbalanced check books and complicated computations. Few grasp the true nature of mathematics - so entwined in our environment and in our lives." (Theoni Pappas, "The Joy of Mathematics" Discovering Mathematics All Around You", 1986)

"A mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"We distinguish diagrammatic from sentential paper-and-pencil representations of information by developing alternative models of information-processing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, like the propositions in a text. Diagrammatic representations are indexed by location in a plane. Diagrammatic representations also typically display information that is only implicit in sentential representations and that therefore has to be computed, sometimes at great cost, to make it explicit for use. We then contrast the computational efficiency of these representations for solving several. illustrative problems in mathematics and physics." (Herbert A Simon, "Why a diagram is (sometimes) worth ten thousand words", 1987)