On Energy (1960-1969)

"Science is usually understood to depict a universe of strict order and lawfulness, of rigorous economy - one whose currency is energy, convertible against a service charge into a growing common pool called entropy." (Paul A Weiss,"Organic Form: Scientific and Aesthetic Aspects", 1960)

"Entropy is a measure of the heat energy in a substance that has been lost and is no longer available for work. It is a measure of the deterioration of a system." (William B Sill & Norman Hoss" (Eds.), "Popular Science Encyclopedia of the Sciences", 1963)

"'Conservation' (the conservation law) means this […] that there is a number, which you can calculate, at one moment - and as nature undergoes its multitude of changes, this number doesn't change. That is, if you calculate again, this quantity, it'll be the same as it was before. An example is the conservation of energy: there's a quantity that you can calculate according to a certain rule, and it comes out the same answer after, no matter what happens, happens." (Richard P. Feynman, "The Great Conservation Principles", The Messenger Series of Lectures No. 3, Cornell University, 1964) 

"Social organizations are flagrantly open systems in that the input of energies and the conversion of output into further energy input consists of transactions between the organization and its environment." (Daniel Katz, "The motivational basis of organizational behavior", 1964)

"They [archetypes] are, at the same time, both images and emotions. One can speak of an archetype only when these two aspects are simultaneous. When there is merely the image, then there is simply a word picture of little consequence. But by being charged with emotion, the image gains numinosity" (or psychic energy); it becomes dynamic, and consequences of some kind must flow from it." (Carl G Jung,"Man and His Symbols", 1964)

"By squandering nuclear energy, polluting asteroids and planets, ravaging the Preserve, and leaving litter everywhere we go, we shall ruin outer space and turn it into one big dump. It is high time we came to our senses and enforced the laws. Convinced that every minute of delay is dangerous, I sound the alarm: Let us save the Universe." (Stanislaw Lem, "Let Us Save the Universe (An Open Letter from Ijon Tichy, Space Traveller", 1966)

"Higher, directed forms of energy" (e.g., mechanical, electric, chemical) are dissipated, that is, progressively converted into the lowest form of energy, i.e., undirected heat movement of molecules; chemical systems tend toward equilibria with maximum entropy; machines wear out owing to friction; in communication channels, information can only be lost by conversion of messages into noise but not vice versa, and so forth." (Ludwig von Bertalanffy, "Robots, Men and Minds", 1967)

"The general notion in communication theory is that of information. In many cases, the flow of information corresponds to a flow of energy, e. g. if light waves emitted by some objects reach the eye or a photoelectric cell, elicit some reaction of the organism or some machinery, and thus convey information." (Ludwig von Bertalanffy, "General System Theory", 1968)

On Energy (1950-1959)

"It is my thesis that the physical functioning of the living individual and the operation of some of the newer communication machines are precisely parallel in their analogous attempts to control entropy through feedback. Both of them have sensory receptors as one stage of their cycle of operation: that is, in both of them there exists a special apparatus for collecting information from the outer world at low energy levels, and for making it available in the operation of the individual or of the machine. In both cases these external messages are not taken neat, but through the internal transforming powers of the apparatus, whether it be alive or dead. The information is then turned into a new form available for the further stages of performance. In both the animal and the machine this performance is made to be effective on the outer world. In both of them, their performed action on the outer world, and not merely their intended action, is reported back to the central regulatory apparatus." (Norbert Wiener, "The Human Use of Human Beings", 1950)

"To seek not for end but for antecedents is the way of the physicist, who finds causes" in what he has learned to recognise as fundamental properties, or inseparable concomitants, or unchanging laws, of matter and of energy." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

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

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

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"Cybernetics might, in fact, be defined as the study of systems that are open to energy but closed to information and control-systems that are 'information-tight'." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"In conventional statistical mechanics the energy plays a preferred role among all dynamical quantities because it is conserved both in the time development of isolated systems and in the interaction of different systems. Since, however, the principles of maximum-entropy inference are independent of any physical properties, it appears that in subjective statistical mechanics all measurable quantities may be treated on the same basis, subject to certain precautions." (Edwin T Jaynes, "Information Theory and Statistical Mechanics" I, 1956)

"But in addition to what we decide to do by way of transformation, there are certain tendencies in the way systems behave of their own volition when left to their own devices. The convenient analogy for one of these processes is found in the second law of thermodynamics: an 'ordering' process goes on, for which the name is entropy. This can be explained without technicalities as the tendency of a system to settle down to a uniform distribution of its energy. The dissipation of local pockets of high energy is measured by an increase in entropy, until at maximum entropy all is uniform. According to this model, order is more 'natural' than chaos. This is the reason why it is convenient to discuss cybernetic systems, with their self-regulating tendency to attain stability or orderliness, in terms of entropy - a term which has been taken over to name a key tool of cybernetics." (Stafford Beer, "Cybernetics and Management", 1959)

On Energy (1920-1929)

"For a modern ruler the laws of conservation and transformation of energy, when the vivifing stream takes its source, the ways it wends its course in nature, and how, under wisdom and knowledge, it may be intertwined with human destiny, instead of careering headlong to the ocean, are a study at least as pregnant with consequences to life as any lesson taught by the long unscientific history of man." (Frederick Soddy, "Science and Life", 1920)

"The history of man is dominated by, and reflects, the amount of available energy." (Frederick Soddy, "Science and Life", 1920)

"Revolution is everywhere, in everything. It is infinite. There is no final revolution, no final number. The social revolution is only one of an infinite number of numbers; the law of revolution is not a social law, but an immeasurably greater one. It is a cosmic, universal law - like the laws of the conservation of energy and of the dissipation of energy (entropy)." (Yevgeny Zamiatin, "On Literature, Revolution, Entropy, and Other Matters", 1923)

"Revolution is everywhere, in everything. It is infinite. There is no final revolution, no final number. The social revolution is only one of an infinite number of numbers; the law of revolution is not a social law, but an immeasurably greater one. It is a cosmic, universal law - like the laws of the conservation of energy and of the dissipation of energy" (entropy)." (Yevgeny Zamiatin, "On Literature, Revolution, Entropy, and Other Matters", 1923)

"[Ignorance] of the principle of conservation of energy […] does not prevent inventors without background from continually putting forward perpetual motion machines… Also, such persons undoubtedly have their exact counterparts in the fields of art, finance, education, and all other departments of human activity […] persons who are unwilling to take the time and to make the effort required to find what the known facts are before they become the champions of unsupported opinions - people who take sides first and look up facts afterward when the tendency to distort the facts to conform to the opinions has become well-nigh irresistible." (Robert A Millikan, "Evolution in Science and Religion", 1927) 

"In classical science, it was strange to find that action [...] should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time." (Aram D'Abro, "The Evolution of Scientific Thought from Newton to Einstein", 1927)

"An 'empty world', i. e., a homogeneous manifold at all points at which equations" (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy." (Howard P. Robertson, "On Relativistic Cosmology", 1928)

"Social organizations are flagrantly open systems in that the input of energies and the conversion of output into further energy input consists of transactions between the organization and its environment." (Daniel Katz, "The Social Psychology of Organizations", 1929)

On Energy (1900-1909)

"After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible; and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended." (David Hilbert, "Mathematische Probleme", [lecture opening] 1900)

"[...] 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)

"By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy" (and statements concerning the form of the energy) alone." (Hermann Minkowski, "The Fundamental Equations for Electromagnetic Processes in Moving Bodies", 1907)

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

"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 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)

On Energy (1875-1899)

"In no way can we get such an overwhelming idea of the grandeur of Nature than when we consider, that in accordance with the law of the conservation of energy, throughout the Infinite, the forces are in a perfect balance, and hence the energy of a single thought may determine the motion of a universe." (Nikola Tesla, "On Light and Other High Frequency Phenomena", [Lecture delivered before the Franklin Institute, Philadelphia] 1893) 

"The man of science, who, forgetting the limits of philosophical inquiry, slides from these formulæ and symbols into what is commonly understood by materialism, seems to me to place himself on a level with the mathematician, who should mistake the x's and y's with which he works his problems for real entities - and with this further disadvantage, as compared with the mathematician, that the blunders of the latter are of no practical consequence, while the errors of systematic materialism may paralyse the energies and destroy the beauty of a life." (Thomas H Huxley, "Method and Results", 1893)

"Throughout the infinite, the forces are in a perfect balance, and hence the energy of a single thought may determine the motion of a universe." (Nikola Tesla, "The Inventions, Researches and Writings of Nikola Tesla", 1894)

"We have to evolve means for obtaining energy from stores which are forever inexhaustible, to perfect methods which do not imply consumption and waste of any material whatever. I now feel sure that the realization of that idea is not far off....the possibilities of the development I refer to, namely, that of the operation of engines on any point of the earth by the energy of the medium." (Nikola Tesla, [speech] 1897)

"[…] only a part of the whole intrinsic energy of the system is capable of being converted into mechanical work by actions going on within the vessel, and without any communication with external space by the passage either of matter or of heat. This part is sometimes called the Available Energy of the system. Clausius has called the remainder of the energy, which cannot be converted into work, the Entropy of the system. We shall find it more convenient to adopt the suggestion of Professor Tait, and give the name of Entropy to the part which can be converted into mechanical work." (James C Maxwell, "Theory of Heat", 1899)

"[…] the result of the conduction and radiation of heat from one part of a system to another is to diminish the entropy of the system, or the energy, available as work, which can be obtained from the system. The energy of the system, however, is indestructible, and as it has not been removed from the system, it must remain' in it. Hence the intrinsic energy of the system, when the entropy is exhausted by thermal communication, conduction, and radiation, is equal to its original energy, and is of course greater than in the case in which the entropy is exhausted by means of the reversible engine." (James C Maxwell, "Theory of Heat", 1899)

"The Entropy of a system is the mechanical work it can perform without communication of heat, or alteration of its total volume, all transference of heat being performed by reversible engines. When the pressure and temperature of the system have become uniform the entropy is exhausted. The original energy of the system is equal to the sum of the entropy and the energy remaining in the state of uniform pressure and temperature. The entropy of a system consisting of several component systems is the same in whatever order the entropy of the parts is exhausted. It is therefore equal to the sum of the entropy of each component system, together with the entropy of the system consisting of the component systems, each with its own entropy exhausted." (James C Maxwell, "Theory of Heat", 1899)

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

"Every fact is a logarithm; one added term ramifies it until it is thoroughly transformed. In the general aspect of things, the great lines of creation take shape and arrange themselves into groups; beneath lies the unfathomable." (Victor Hugo, "The Toilers of the Sea", 1866)

'It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts." (Sir Arthur C Doyle, "The Adventures of Sherlock Holmes", 1892)

"If the fresh facts which come to our knowledge all fit themselves into the scheme, then our hypothesis may gradually become a solution." (Arthur C Doyle, "The Adventure of Wisteria Lodge", 1908)

"The facts of life do not penetrate to the sphere in which our beliefs are cherished; as it was not they that engendered those beliefs, so they are powerless to destroy them." (Marcel Proust, "In Search of Lost Time", 1913)

"The superstitions of today are the scientific facts of tomorrow." (Hamilton Deane & John L. Balderston, "Dracula", 1927)

"The point is that we are all capable of believing things which we know to be untrue, and then, when we are finally proved wrong, impudently twisting the facts so as to show that we were right. Intellectually, it is possible to carry on this process for an indefinite time: the only check on it is that sooner or later a false belief bumps up against solid reality, usually on a battlefield." (George Orwell, "In Front of Your Nose", Tribune, 1946)

"One thinks one’s something unique and wonderful at the center of the universe. But in fact one’s merely a slight delay in the ongoing march of entropy." (Aldous Huxley, "Island", 1962)

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

"These dwarfs amass knowledge as others do treasure; for this reason they are called Hoarders of the Absolute. Their wisdom lies in the fact that they collect knowledge but never use it." (Stanislaw Lem, "How Erg the Self-Inducing Slew a Paleface", 1965)

"[Human] communication is rendered more complex by the use of differing sets of sound-symbols, called languages and by the fact that a given set of symbols tends to change with the passage of years to become an entirely new language." (Howard L Myers, "The Creatures of Man", 1968)

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

"Many scientists deeply involved in the exploration of the solar system (myself among them) were first turned in that direction by science fiction. And the fact that some of that science fiction was not of the highest quality is irrelevant. Ten-year-olds do not read the scientific literature." (Carl Sagan, "Broca's Brain", 1979)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)

"The fact is that thresholds exist throughout reality, and that things on their far sides are altogether different from things on their hither sides." (Poul Anderson, "The Saturn Game", 1981)

"This is an exercise in fictional science, or science fiction, if you like that better. Not for amusement: science fiction in the service of science. Or just science, if you agree that fiction is a part of it, always was, and always will be as long as our brains are only miniscule fragments of the universe, much too small to hold all the facts of the world but not too idle to speculate about them." (Valentino Braitenberg," Vehicles: Experiments in Synthetic Psychology", 1984)

"A science fiction writer is - or should be - constrained by what is, or logically might be. That can mean simple fidelity to facts (which, in science, are always more important than theories - though Lord knows the two help shape each other, undermining the convenient, complacent separation of observer and observed). To me it also means heeding the authentic, the actual and concrete. Bad fiction uses the glossy generality; good writing needs the smattering of detail, the unrelenting busy mystery of the real." (Gregory Benford, "Afterword to Exposures", [in Alien Flesh] 1986)

"In the end, each life is no more than the sum of contingent facts, a chronicle of chance intersections, of flukes, of random events that divulge nothing but their own lack of purpose." (Paul Auster, "The Locked Room", 1988)

"The rationality of our universe is best suggested by the fact that we can discover more about it from any starting point, as if it were a fabric that will unravel from any thread." (George Zebrowski, "Is Science Rational?", OMNI Magazine, 1994)

"The old knowledge had been difficult but not distressing. It had been all paradox and myth, and it had made sense. The new knowledge was all fact and reason, and it made no sense." (Ursula K Le Guin," "A Man of the People", 1995)

"Because the question for me was always whether that shape we see in our lives was there from the beginning or whether these random events are only called a pattern after the fact. Because otherwise we are nothing." (Cormac McCarthy, "All the Pretty Horses", 2010)

"Science fiction these days is only half a step ahead of science. Astrophysicists and scientists are working in the same way as science fiction writers. They’re working things out in their imagination based on the slim scientific facts that they know. Hawking imagines a black hole and then discovers the mathematics that support his theory, and new possibilities come to light. That’s the imaginative flair that scientists have to have. For me as a sci-fi writer, spinning those ideas in your mind brings you to the point where you dream in science fiction. Suddenly you think of something in the middle of the night, and it’s so vivid you don’t need to write it down because you know you’ll remember it in the morning. That’s what these books, Zero G, reflect: a vivid imagination." (William Shatner, "William Shtner on Sci-Fi, Aging and the Environment", Saturday Evening Post, [interview] 2017)

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

"My desire for knowledge is intermittent; but my desire to bathe my head in atmospheres unknown to my feet is perennial and constant. The highest that we can attain to is not Knowledge, but Sympathy with Intelligence. I do not know that this higher knowledge amounts to anything more definite than a novel and grand surprise on a sudden revelation of the insufficiency of all that we called Knowledge before - a discovery that there are more things in heaven and earth than are dreamed of in our philosophy. It is the lighting up of the mist by the sun." (Henry D Thoreau, "Walking", 1851)

"How do you attack the unknown, or defend yourself from it?" (Jules Verne, "Twenty Thousand Leagues under the Sea", 1870)

"Sitting there on the heather, on our planetary grain, I shrank from the abysses that opened up on every side, and in the future. The silent darkness, the featureless unknown, were more dread than all the terrors that imagination had mustered." (Olaf Stapledon, "Star Maker", 1937)

"All unknowns are organically inimical to man, and homo sapiens is human in the full sense of the word only when his grammar is entirely free of question marks." (Yevgeny Zamiatin, "We", 1924)

"The unknown is a terrible place. There are monsters out there." (Philip K Dick, "Solar Lottery", 1955)

"There are people who run at the sight of the unknown, others who advance to meet it." (Fredric Brown, "Puppet Show", 1962)

"The exploration of the unknown is always a fraud." (James Gunn, "Station in Space", 1958)

"He knew the unknown. It was beyond all comprehension and Guy thought that it was no more than a set of words: a black, empty World preceding the appearance of the World Light; a dead, icy World when the World Light was extinguished; an endless Wasteland with many World Lights. No one could explain what this meant." (Arkady Strugatsky &Boris Strugatsky, "Prisoners of Power", 1969)

"The unknown is always worse than the known." (Eando Binder, "Menace of the Saucers", 1969)

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

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"So together they left the office and walked into the uncertainty of the rest of their lives. That, in the final analysis, is the great adventure in which each of us takes part; what more courageous thing is there, after all, than facing the unknown we all share, the danger and joy that awaits us in the unread pages of the Book of the Future [...]" (George Alec Effinger," The World of Pez Pavilion: Preliminary to the Groundbreaking Ceremony", 1983)

"The Sioux had outlived their way of life, had turned it over to the white technicians, who would map everything. That was what he most disliked about them: that they sought to know all things, and did not realize that a forest without dark places has value only to the woodcutter." (Jack McDevitt, "Ancient Shores", 1996)

"'Do you really believe in physics?'  'I dont know what that means. Physics tries to draw a numerical picture of the world. I dont know that it actually explains anything. You cant illustrate the unknown. Whatever that might mean.'" (Cormac McCarthy, "Stella Maris", 2022)

On Models: On Mathematical Models (1950-1959)

"Mathematical models for empirical phenomena aid the development of a science when a sufficient body of quantitative information has been accumulated. This accumulation can be used to point the direction in which models should be constructed and to test the adequacy of such models in their interim states. Models, in turn, frequently are useful in organizing and interpreting experimental data and in suggesting new directions for experimental research." (Robert R. Bush & Frederick Mosteller, "A Mathematical Model for Simple Learning", Psychological Review 58, 1951)

"There are at least four fundamental purposes that the study of mathematics should attain. First, it should serve as a functional tool in solving our individual everyday problems. [...] In the second place, mathematics serves as a handmaiden for the explanation of the quantitative situations in other subjects, such as economics, physics, navigation, finance, biology, and even the arts. [...] In the third place, mathematics, when properly conceived, becomes a model for thinking, for developing scientific structure, for drawing conclusions, and for solving problems. [...] In the fourth place, mathematics is the best describer of the universe about us." (Howard F Fehr,  "Reorientation in Mathematics Education", Teachers Record 54, 1953)

"The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work" (John Von Neumann, "Method in the Physical Sciences", 1955)

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

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958)

"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)

Robert M Axelrod - Collected Quotes

"A cognitive map has only two basic types of elements: concepts and causal beliefs. The concepts are treated as variables, and the causal beliefs are treated as relationships between the variables." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"A cognitive map is a particular kind of mathematical model of a person's belief system; in actual practice, cognitive maps are derived from assertions of beliefs. [...] Like all mathematical models, a cognitive map can be useful in two quite distinct ways: as a normative model and as an empirical model. Interpreted as a normative model, a cognitive map makes no claims to reflect accurately how a person deduces new beliefs from old ones, how he makes decisions, and so on, but instead claims to show how he should do these things. Interpreted as an empirical model, a cognitive map claims to indicate how a person actually does perform certain cognitive operations, in the sense that the results of the various operations that are possible with the model do, in fact, correspond to the behavior of the per son who is being modeled." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"A cognitive map is designed to capture the structure of the causal assertions of a person with respect to a particular policy domain, and generate the consequences that follow from this structure." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"A decision maker must simplify the manifest complexities of the external world. He must be able to construct a manageable representation of the external world so that he can describe and cope with his environment. The use of this representation for the purposes of making reasoned decisions requires some beliefs that link possible choices with potential outcomes." (Robert M Axelrod, "Decision for Neoimperialism: The Deliberations of the British Eastern Committee in 1918", 1976)

"A mathematical model is a tremendous simplification of what it represents. But it does not simplify everything about its object, or there would be nothing left to model. Instead, it simplifies everything that is not to be examined, and leaves in the model what is to be examined." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"Cognitive mapping is one specific approach to belief systems. It focuses on causal beliefs and values and their structural relationships. Cognitive mapping is, therefore especially suitable to the study of the means-ends arguments people use when they try to evaluate the policy alternatives that they perceive are available to them." (Robert Axelrod, "Structure of Decision: The Cognitive Maps of Political Elites", 1976)

"The concepts a person uses are represented as points, and the causal links between these concepts are represented as arrows between these points. This gives a pictorial representation of the causal assertions of a person as a graph of points and arrows. This kind of representation of assertions as a graph will be called a cognitive map. The policy alternatives, all of the various causes and effects, the goals, and the ultimate utility of the decision maker can all be thought of as concept variables, and represented as points in the cognitive map. The real power of this approach ap pears when a cognitive map is pictured in graph form; it is then relatively easy to see how each of the concepts and causal relation ships relate to each other, and to see the overall structure of the whole set of portrayed assertions." (Robert Axelrod, "The Cognitive Mapping Approach to Decision Making" [in "Structure of Decision: The Cognitive Maps of Political Elites"], 1976)

"What cognitive mapping offers is a systematic way to proceed in our search for understanding how others will act. Its real strength (especially as compared to other formal approaches to decision making) is that it is able to employ the concepts of the decision maker who is being predicted, rather than the concepts of the person who is doing the predicting." (Robert Axelrod, "Structure of Decision: The Cognitive Maps of Political Elites", 1976)

"What difference does it make which cognitive map a person has on a policy issue? Why would we want to know about a person's cognitive map? There are two broad answers to these questions. The first is that we want to know about cognitive maps so that we can better understand the decision-making process. The second is that we want to know about cognitive maps so that we can improve the decision-making process." (Robert Axelrod, "Structure of Decision: The Cognitive Maps of Political Elites", 1976)

On Models: On Mathematical Models (1980-1989)

"Let me start by posing what I tike to call 'the fundamental problem of equilibrium theory': how is economic equilibrium attained? A dual question more commonly raised is: why is economic equilibrium stable? Behind these questions lie the problem of modeling economic processes and introducing dynamics into equilibrium theory. A successful attack here would give greater validity to equilibrium theory. It may be however that a resolution of this fundamental problem will require a recasting of the foundations of equilibrium theory." (Steven Smale, "Some Dynamics Questions in Mathematical Economics", 1980)

"In physics it is usual to give alternative theoretical treatments of the same phenomenon. We construct different models for different purposes, with different equations to describe them. Which is the right model, which the 'true' set of equations? The question is a mistake. One model brings out some aspects of the phenomenon; a different model brings out others. Some equations give a rougher estimate for a quantity of interest, but are easier to solve. No single model serves all purposes best." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"Physics is like that. It is important that the models we construct allow us to draw the right conclusions about the behaviour of the phenomena and their causes. But it is not essential that the models accurately describe everything that actually happens; and in general it will not be possible for them to do so, and for much the same reasons. The requirements of the theory constrain what can be literally represented. This does not mean that the right lessons cannot be drawn. Adjustments are made where literal correctness does not matter very much in order to get the correct effects where we want them; and very often, as in the staging example, one distortion is put right by another. That is why it often seems misleading to say that a particular aspect of a model is false to reality: given the other constraints that is just the way to restore the representation." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

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

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

"The large-scale computer models of systems ecology do not fit under the heading of holism at all. Rather they are forms of large-scale reductionism: the objects of study are the naively given 'parts' -abundances or biomasses of populations. No new objects of study arise at the community level. The research is usually conducted on a single system - a lake, forest, or prairie - and the results are measurements of and projections for that lake, forest, or prairie, with no attempts to find the properties of lakes, forests, or prairies in general. Such modeling requires vast amounts of data for its simulations, and much of the scientific effort goes into problems of estimation." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"One of the features that distinguishes applied mathematics is its interest in framing important questions about the observed world in a mathematical way. This process of translation into a mathematical form can give a better handle for certain problems than would be otherwise possible. We call this the modeling process. It combines formal reasoning with intuitive insights. Understanding the models devised by others is a first step in learning some of the skills involved, and that is how we proceed in this text, which is an informal introduction to the mathematics of dynamical systems." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"The essence of modeling, as we see it, is that one begins with a nontrivial word problem about the world around us. We then grapple with the not always obvious problem of how it can be posed as a mathematical question. Emphasis is on the evolution of a roughly conceived idea into a more abstract but manageable form in which inessentials have been eliminated. One of the lessons learned is that there is no best model, only better ones." (Edward Beltrami,"Mathematics for Dynamic Modeling", 1987)

"Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?" (Stephen W Hawking, "A Brief History of Time: From the Big Bang to Black Holes", 1988)

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

“The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?” (Stephen Hawking, "A Brief History of Time", 1988)

"In a real experiment the noise present in a signal is usually considered to be the result of the interplay of a large number of degrees of freedom over which one has no control. This type of noise can be reduced by improving the experimental apparatus. But we have seen that another type of noise, which is not removable by any refinement of technique, can be present. This is what we have called the deterministic noise. Despite its intractability it provides us with a way to describe noisy signals by simple mathematical models, making possible a dynamical system approach to the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)