On Structure: Structure in Knowledge (1960-1969)

"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one’s craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful." (Jerome S Bruner, "On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)

"Teaching specific topics or skills without making clear their context in the broader fundamental structure of a field of knowledge is uneconomical." (Jerome Bruner, "The Process of Education", 1960)

"For Science in its totality, the ultimate goal is the creation of a monistic system in which - on the symbolic level and in terms of the inferred components of invisibility and intangibly fine structure - the world’s enormous multiplicity is reduced to something like unity, and the endless successions of unique events of a great many different kinds get tidied and simplified into a single rational order. Whether this goal will ever be reached remains to be seen. Meanwhile we have the various sciences, each with its own system coordinating concepts, its own criterion of explanation." (Aldous Huxley, "Literature and Science", 1963)

"This other world is the so-called physical world image; it is merely an intellectual structure. To a certain extent it is arbitrary. It is a kind of model or idealization created in order to avoid the inaccuracy inherent in every measurement and to facilitate exact definition." (Max Planck, "The Philosophy of Physics", 1963)

"[...] 'information' is not a substance or concrete entity but rather a relationship between sets or ensembles of structured variety." (Walter F Buckley, "Sociology and modern systems theory", 1967)

"It [knowledge] is clearly related to information, which we can now measure; and an economist especially is tempted to regard knowledge as a kind of capital structure, corresponding to information as an income flow. Knowledge, that is to say, is some kind of improbable structure or stock made up essentially of patterns - that is, improbable arrangements, and the more improbable the arrangements, we might suppose, the more knowledge there is." (Kenneth E Boulding, "Beyond Economics: Essays on Society", 1968)

"Knowing reality means constructing systems of transformations that correspond, more or less adequately, to reality. They are more or less isomorphic to transformations of reality. The transformational structures of which knowledge consists are not copies of the transformations in reality; they are simply possible isomorphic models among which experience can enable us to choose. Knowledge, then, is a system of transformations that become progressively adequate." (Jean Piaget, "Genetic Epistemology", 1968)

"The idea of knowledge as an improbable structure is still a good place to start. Knowledge, however, has a dimension which goes beyond that of mere information or improbability. This is a dimension of significance which is very hard to reduce to quantitative form. Two knowledge structures might be equally improbable but one might be much more significant than the other." (Kenneth E Boulding, "Beyond Economics: Essays on Society", 1968)

"Modern science is characterized by its ever-increasing specialization, necessitated by the enormous amount of data, the complexity of techniques and of theoretical structures within every field. Thus science is split into innumerable disciplines continually generating new subdisciplines. In consequence, the physicist, the biologist, the psychologist and the social scientist are, so to speak, encapusulated in their private universes, and it is difficult to get word from one cocoon to the other." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Visual thinking calls, more broadly, for the ability to see visual shapes as images of the patterns of forces that underlie our existence - the functioning of minds, of bodies or machines, the structure of societies or ideas." (Rudolf Arnheim, "Visual Thinking", 1969)

Science vs. Myths III

"Today the function of the artist is to bring imagination to science and science to imagination, where they meet, in the myth." (Cyril Connolly, The Unquiet Grave, 1945)

"For what are myths if not the imposing of order on phenomena that do not possess order in themselves? And all myths, however they differ from philosophical systems and scientific theories, share this with them, that they negate the principle of randomness in the world." (Stanisław Lem, "Highcastle: A Remembrance", 1965)

"Both the myths of religion and the laws of science, it is now becoming apparent, are not so much descriptions of facts as symbolic expressions of cosmic truths." (René J Dubos, A God Within, 1972)

"A scientist can not be measured quantitatively by the number of degrees or the accumulation of information. A true scientist should have a measure of courage to correct error and seek truth - no matter how painful. The alternative is more painful. To build error upon error is to drift into dogmas, metaphysics, science fiction, and mythology." (Alexander Wilf, "Origin and Destiny of the Moral Species", 1969)

"Science never starts from scratch; it can never be described as free from assumptions; for at every instant it presupposes a horizon of expectations - yesterday’s horizon of expectations, as it were. Today’s science is built upon yesterday’s science (and so it is the result of yesterday’s searchlight); and yesterday’s science, in turn, is based on the science of the day before. And the oldest scientific theories are built on pre- scientific myths, and these, in their turn, on still older expectations." (Karl R Popper, "Objective Knowledge: An Evolutionary Approach", 1972)

"The mythology of science asserts that with many different scientists all asking their own questions and evaluating the answers independently, whatever personal bias creeps into their individual answers is canceled out when the large picture is put together." (Ruth Hubbard, "Women Look at Biology Looking at Women", 1979)

"However, for most of us, science functions like myth in that we have no personal experience in the matter. We put our trust in the scientific view given us by our culture and enshrined in its myths. If asked why leaves are green, most of us would probably mutter something about “chlorophyll.” But unless we were specialists, we would simply be repeating the story of someone else’s experience." (Wallace B Clift, "Jung and Christianity", 1982)

"It is important that we, as working scientists, combat these myths of our profession as something superior and apart. The myths may serve us well in the short and narrow as rationale for a lobbying strategy - give us the funding and leave us alone, for we know what we’re doing and you don’t understand anyway. But science can only be harmed in the long run by its self-proclaimed separation as a priesthood guarding a sacred rite called the scientific method." (Stephen J Gould, "Time’s Arrow, Time’s Cycle: Myth and Metaphor in the Discovery of Geological Time", 1987)

"Myths are errors that result [both from] scientists bringing societal preconceptions into science and […] scientists feeding society ideas that masquerade as science." (MN Mahadeva, "From Misinterpretations to Myths", Science Teacher Vol. 56 (4), 1989)

"There is no good reason to discard the scientific method as an ideal; rather, there is good reason to keep it so. Myths, after all, even if not literally true, are stories that embody moral truths." (Henry H Bauer, "Scientific Literacy and the Myth of the Scientific Method", 1992)

Science vs. Myths II

"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)

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

"The critical task of science is not complete and never will be, for it is the merest truism that we do not abandon mythologies and superstitions, but merely substitute new variants for old." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Today’s quarks and leptons can be viewed as metaphors of the underlying reality of nature, though metaphors that are objectively and rationally defied and are components of theories that have great predictive power. And that’s the difference between the metaphors of science and those of myth: scientific metaphors work." (Victor J Stenger, "Physics and Psychics: The Search for a World Beyond the Senses", 1990)

"The ability of a scientific theory to be refuted is the key criterion that distinguishes science from metaphysics. If a theory cannot be refuted, if there is no observation that will disprove it, then nothing can prove it - it cannot predict anything, it is a worthless myth." (Eric Lerner, "The Big Bang Never Happened", 1991)

"But our ways of learning about the world are strongly influenced by the social preconceptions and biased modes of thinking that each scientist must apply to any problem. The stereotype of a fully rational and objective ‘scientific method’, with individual scientists as logical (and interchangeable) robots, is self-serving mythology." (Stephen J Gould, "This View of Life: In the Mind of the Beholder", "Natural History", Vol. 103, No. 2, 1994)

"Mythology and science both extend the scope of human beings. Like science and technology, mythology, as we shall see, is not about opting out of this world, but about enabling us to live more intensely within it." (Karen Armstrong, "A Short History Of Myth", 2004)

"The truth is more magical - in the best and most exciting sense of the word - than any myth or made-up mystery or miracle. Science has its own magic: the magic of reality." (Richard Dawkins, "The Magic of Reality: How We Know What's Really True", 2011)

"Intellectual inquiry begins with myth, religion and philosophy. Originally, philosophy (or perhaps theology or metaphysics) is the queen of the sciences, other intellectual disciplines having only a highly subservient, specialized role to play within philosophy. [...] Instead of being the queen of the sciences, overarching all other sciences, philosophy has been transformed into a highly specialized, technical, somewhat meagre enterprise, concerned not with improving our knowledge and understanding of the world - for that is the business of the empirical sciences - but rather with clarifying concepts and solving conceptual problems." (Nicholas Maxwell, "Karl Popper, Science and Enlightenment", 2017)

Science vs. Myths I

"The philosophical study of nature rises above the requirements of mere delineation, and does not consist in the sterile accumulation of isolated facts. The active and inquiring spirit of man may therefore be occasionally permitted to escape from the present into the domain of the past, to conjecture that which cannot yet be clearly determined, and thus to revel amid the ancient and ever-recurring myths of geology." (Alexander von Humboldt, "Views of Nature: Or Contemplation of the Sublime Phenomena of Creation", 1850)

"In order to depict nature in its exalted sublimity, we must not dwell exclusively on its external manifestations, but we must trace its image, reflected in the mind of man, at one time filling the dreamy land of physical myths with forms of grace and beauty, and at another developing the noble germ of artistic creations." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe" Vol. 2, 1869)

"Science, in one aspect, is ordered technique; in another, it is rationalized mythology." (John D Bernal, "Science in History", 1957)

"Myth is more individual and expresses life more precisely than does science. Science works with concepts of averages which are far too general to do justice to the subjective variety of an individual life." (Carl G Jung, "Memories, Dreams, Reflections", 1963)

"The mathematicians and physics men have their mythology; they work alongside the truth, never touching it; their equations are false But the things work. Or, when gross error appears, they invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers, "The Beginning and the End and Other Poems, The Great Wound", 1963) 

"Thus science must begin with myths, and with the criticism of myths; neither with the collection of observations, nor with the invention of experiments, but with the critical discussion of myths, and of magical techniques and practices." (Karl Popper, "Conjectures and Refutations: The Growth of Scientific Knowledge", 1963)

"Innocent, unbiased observation is a myth." (Sir Peter B Medawar, "Induction and Intuition in Scientific Thought", 1969)

"[…] science is not sacrosanct. The restrictions it imposes (and there are many such restrictions though it is not easy to spell them out) are not necessary in order to have general coherent and successful views about the world. There are myths, there are the dogmas of theology, there is metaphysics, and there are many other ways of constructing a worldview. It is clear that a fruitful exchange between science and such ‘nonscientific’ world-views will be in even greater need of anarchism than is science itself. Thus, anarchism is not only possible, it is necessary both for the internal progress of science and for the development of our culture as a whole." (Paul Feyerabend, "Against Method", 1975)

"In the ultimate analysis science is born of myth and religion, all three being expressions of the ordering spirit of the human mind." (Lancelot L Whyte, "The Unconscious Before Freud", 1978)

"To try to write a grand cosmical drama leads necessarily to myth. To try to let knowledge substitute ignorance in increasingly larger regions of space and time is science." (Hannes Alfven, 1978)

📒🅶Ursula K Le Guin - Collected Quotes

"But you must not change one thing, one pebble, one grain of sand, until you know what good and evil will follow on that act. The world is in balance, in Equilibrium. […] It is dangerous, that power. [...] It must follow knowledge, and serve need." (Ursula K Le Guin, "A Wizard of Earthsea", 1968)

"He thought no more of performing the lesser arts of magic than a bird thinks of flying. Yet a greater, unlearned skill he possessed, which was the art of kindness." (Ursula K. Le Guin, "A Wizard of Earthsea", 1968)

"It is yin and yang. Light is the left hand of darkness... how did it go? Light, dark. Fear, courage. Cold, warmth. Female, male. It is yourself... both and one. A shadow on snow." (Ursula K. Le Guin, "The Left Hand of Darkness", 1969)

"When action grows unprofitable, gather information; when information grows unprofitable, sleep". (Ursula K. Le Guin, "The Left Hand of Darkness", 1969)

"A man can endure the entire weight of the universe for eighty years. It is unreality that he cannot bear." (Ursula K. Le Guin, "The Lathe of Heaven", 1971)

"Isn’t the measure of complexity the measure of the eternal joy?" (Ursula K Le Guin, "Vaster Than Empires and More Slow", 1971)

"When things don't change any longer, that's the end result of entropy, the heat-death of the universe. The more things go on moving, interrelating, conflicting, changing, the less balance there is - and the more life." (Ursula K Le Guin, "The Lathe of Heaven", 1971)

"Nothing is immortal. But only to us is it given to know that we must die. And that is a great gift: the gift of selfhood. For we have only what we know we must lose, what we are willing to lose . . . That selfhood, which is our torment, and our treasure, and our humanity, does not endure. It changes; it is gone, a wave on the sea. Would you have the sea grow still and the tides cease, to save one wave, to save yourself ?" (Ursula K Le Guin, "The Farthest Shore", 1972)

"What man would not live forever, if he could?" (Ursula K. Le Guin, "The Farthest Shore", 1972)

"It is only when science asks why, instead of simply describing how, that it becomes more than technology. When it asks why, it discovers Relativity. When it only shows how, it invents the atomic bomb, and then puts its hands over its eyes and says, 'My God what have I done?'" (Ursula K Le Guin, "The Stalin in the Soul", 1973)

"It is of the nature of idea to be communicated: written, spoken, done. The idea is like grass. It craves light, likes crowds, thrives on crossbreeding, grows better for being stepped on." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"Music is a cooperative art, organic by definition, social. It may be the noblest form of social behavior we’re capable of. It’s certainly one of the noblest jobs an individual can undertake. And by its nature, by the nature of any art, it’s a sharing. The artist shares, it’s the essence of his act." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"Our model of the cosmos must be as inexhaustible as the cosmos. A complexity that includes not only duration but creation, not only being but becoming, not only geometry but ethics. It is not the answer we are after, but only how to ask the question." (Ursula K Le Guin, "The Dispossessed", 1974)

"Scientific truth will out, you can’t hide the sun under a stone." (Ursula K. Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"The duty of the individual is to accept no rule, to be the initiator of his own acts, to be responsible. Only if he does so will the society live, and change, and adapt, and survive." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"You can’t crush ideas by suppressing them. You can only crush them by ignoring them. By refusing to think, refusing to change." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"The best maze is the mind." (Ursula K Le Guin, "Mazes", 1975)

"All fiction is metaphor. Science fiction is metaphor. What sets it apart from older forms of fiction seems to be its use of new metaphors, drawn from certain great dominants of our contemporary life - science, all the sciences, and technology, and the relativistic and the historical outlook, among them. Space travel is one of these metaphors; so is an alternative society, an alternative biology; the future is another." (Ursula K Le Guin, "The Left Hand of Darkness", [introduction] 1976)

"Intellectualism tends to foster negative thinking and may lead to psychosis, and those suffering from it should ideally be treated, as Professor Arca was treated, and released if still competent." (Ursula K Le Guin, "The Diary of the Rose", 1976)

"It is amazing how banal most people’s minds are. [...] I used to think how wonderful other people’s minds would be, how wonderful it was going to be to share in all the different worlds, the different colors of their passions and ideas. How naïve I was!" (Ursula K. Le Guin, ‘"The Diary of the Rose", 1976)

"Time is no longer a line along which history, past or future, lies neatly arranged, but a field of great mystery and complexity, in the contemplation of which the mind perceives an immense terror, and an indestructible hope." (Ursula K Le Guin, 1977) 

"It is only when science asks why, instead of simply describing how, that it becomes more than technology. When it asks why, it discovers Relativity. When it only shows how, it invents the atomic bomb." (Ursula K Le Guin, "Language of the Night", 1979)

"If science fiction is the mythology of modern technology, then its myth is tragic." (Ursula K Le Guin, "The Carrier Bag Theory of Fiction", 1986)

"Science fiction properly conceived, like all serious fiction, however funny, is a way of trying to describe what is in fact going on, what people actually do and feel, how people relate to everything else in this vast sack, this belly of the universe, this womb of things to be and tomb of things that were, this unending story." (Ursula K Le Guin, "Dancing at the Edge of the World", 1986)

"A wrong that cannot be repaired must be transcended." (Ursula K Le Guin, "Tehanu: The Last Book of Earthsea", 1990)

"My imagination makes me human and makes me a fool; it gives me all the world and exiles me from it." (Ursula . Le Guin, "The Creatures on My Mind", 1990)

"All knowledge is local, all truth is partial. [...] No truth can make another truth untrue. All knowledge is part of the whole knowledge. A true line, a true color. Once you have seen the larger pattern, you cannot get back to seeing the part as the whole." (Ursula K Le Guin, "A Man of the People", 1995)

On Synergy II

"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)

"Social equilibration under the principle of social synergy, while it involves a perpetual and vigorous struggle among the antagonistic social forces, still works out social structures and conserves them, and these structures perform their prescribed functions. Upon the perfection of these structures and the consequent success with which they perform their functions depends the degree of social efficiency. In the organic world the struggle has the appearance of a struggle for existence. The weaker species go to the wall and the stronger persist. There is a constant elimination of the defective and survival of the fittest. On the social plane it is the same, and weak races succumb in the struggle while strong races persist. But in both cases it is the best structures that survive." (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)

"[...] synergy is the consequence of the energy expended in creating order. It is locked up in the viable system created, be it an organism or a social system. It is at the level of the system. It is not discernible at the level of the system. It is not discernible at the level of the system’s components. Whenever the system is dismembered to examine its components, this binding energy dissipates." (J-C Spender, "Organizational Knowledge, Collective Practice and Penrose Rents", 1999)

[synergy:] "Measure describing how one agent or system increases the satisfaction of other agents or systems." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"To develop a Control, the designer should find aspect systems, subsystems, or constraints that will prevent the negative interferences between elements (friction) and promote positive interferences (synergy). In other words, the designer should search for ways of minimizing frictions that will result in maximization of the global satisfaction" (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

💠On Problem Solving 15: On Representation II

"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. These questions: How much? How many? What form or shape? and Can you prove it? arise every day in the lives of every citizen.
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. The mathematics used in these areas of practice is exactly the same mathematics and involves the same mathematical concepts and skills. It is only the things to which the mathematics is applied that are different, and this is immaterial if one really understands the mathematics.
In the third place, mathematics, when properly conceived, becomes a model for thinking, for developing scientific structure, for drawing conclusions, and for solving problems. Its postulational nature, that is, accepted relations axioms or postulates), undefined terms, definitions, theorems, and logic. aids ail other areas of knowledge to approach scientific This same structure aids us in problem solving methods in which we collect, organize, and analyze data, and deduce conclusions for future action. For example, one who understands the mathematical method can easily frame the problem, [...]
In the fourth place, mathematics is the best describer of the universe about us. In an age that has become statistical and scientific in much of its human endeavor, the need for people to understand these phenomena is not only a cultural necessity but to some extent a necessity for intelligent action." (Howard F Fehr,  "Reorientation in Mathematics Education", Teachers Record 54, 1953)

"Every problem-solving effort must begin with creating a representation for the problem - a problem space in which the search for the solution can take place. Of course, for most of the problems we encounter in our daily personal or professional lives, we simply retrieve from memory a representation that we have already stored and used on previous occasions. Sometimes, we have to adapt the representation a bit to the new situation, but that is usually a rather simple matter." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Solving a problem simply means representing it so as to make the solution transparent." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"For the mathematician, the physical way of thinking is merely the starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)

"The difficulty facing us when we have to make inferences is two-fold. First, we may build entirely the wrong mental model from the information we read or hear. […] The second difficulty facing us is that we may well build a reasonably correct initial representation of a problem, but this representation may be impoverished in some way because we have no idea what inferences are relevant […]" (S Ian Robertson, "Problem Solving", 2001)

"Thinking involves reasoning about a situation, and to do that we must have some kind of dynamic "model" of the situation in our heads. Any changes we make to this mental model of the world should ideally mirror changes in the real world." (S Ian Robertson, "Problem Solving", 2001)

"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)

"Good numeric representation is a key to effective thinking that is not limited to understanding risks. Natural languages show the traces of various attempts at finding a proper representation of numbers. [...] The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)

“A conceptual model is a mental image of a system, its components, its interactions. It lays the foundation for more elaborate models, such as physical or numerical models. A conceptual model provides a framework in which to think about the workings of a system or about problem solving in general. An ensuing operational model can be no better than its underlying conceptualization.” (Henry N Pollack, “Uncertain Science … Uncertain World”, 2005)

"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)

"Mental models are formed over time through a deep enculturation process, so it follows that any attempt to align mental models must focus heavily on collective sense making. Alignment only happens through a process of socialisation; people working together, solving problems together, making sense of the world together." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

"Mathematics does not merely describe the problem in an abstract way, it allows us to find a previously unknown 'solution' from the abstract description. It is surprising that the unknown can be transformed into the well known when we succeed in describing the problem mathematically." (Waro Iwane, "Mathematics in Our Company: What Does It Describe?", [in "What Mathematics Can Do for You"] 2013)

"Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it." (Sandip Banerjee, "Mathematical Modeling: Models, Analysis and Applications", 2014)

"Mental imagery is often useful in problem solving. Verbal descriptions of problems can become confusing, and a mental image can clear away excessive detail to bring out important aspects of the problem. Imagery is most useful with problems that hinge on some spatial relationship. However, if the problem requires an unusual solution, mental imagery alone can be misleading, since it is difficult to change one’s understanding of a mental image. In many cases, it helps to draw a concrete picture since a picture can be turned around, played with, and reinterpreted, yielding new solutions in a way that a mental image cannot." (James Schindler, "Followership", 2014)

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On Numbers (1600-1674)

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

"Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers. [...] I began therefore to consider in my mind by what certain  and ready art I might remove those hindrances." (John Napier, "Mirifici logarithmorum canonis descriptio", 1614)

"Thus we can give three names to the other solutions, seeing that there are some which are greater than nothing, other less than nothing, and other enveloped, as those which have like √- or √-3 or other similar numbers." (Albert Girard, "L'Invention nouvelle de l'Algébre", 1629)

"The length of strings is not the direct and immediate reason behind the forms [ratios] of musical intervals, nor is their tension, nor their thickness, but rather, the ratios of the numbers of vibrations and impacts of air waves that go to strike our eardrum." (Galileo Galilei, "Two New Sciences", 1638)

"[…] I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers […]" (René Descartes, [a letter to Marin Mersenne] 1638)

"What is big is easy to perceive: what is small is difficult to perceive. In short, it is difficult for large numbers of men to change position, so their movements can be easily predicted. An individual can easily change his mind, so his movements are difficult to predict."(Miyamoto Musashi, "The Book of Five Rings", 1645)

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." (Pierre de Fermat, [Letter to Frénicle] 1657)

"[…] the knowledge we have of the Mathematicks, hath no reason to elate us; since by them we know but numbers, and figures, creatures of our own, and are yet ignorant of our Maker’s." (Joseph Glanvill, "The Vanity of Dogmatizing", 1661)

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

"For any number there exists a corresponding even number which is its double. Hence the number of all numbers is not greater than the number of even numbers, that is, the whole is not greater than the part." (Gottfried W Leibniz, "De Arte Combinatoria", 1666)

"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)

On Statisticians (1975 - 1999)

"Competent statisticians will be front line troops in our war for survival-but how do we get them? I think there is now a wide readiness to agree that what we want are neither mere theorem provers nor mere users of a cookbook. A proper balance of theory and practice is needed and, most important, statisticians must learn how to be good scientists; a talent which has to be acquired by experience and example." (George E P Box, "Science and Statistics", Journal of the American Statistical Association 71, 1976)

"When the statistician looks at the outside world, he cannot, for example, rely on finding errors that are independently and identically distributed in approximately normal distributions. In particular, most economic and business data are collected serially and can be expected, therefore, to be heavily serially dependent. So is much of the data collected from the automatic instruments which are becoming so common in laboratories these days. Analysis of such data, using procedures such as standard regression analysis which assume independence, can lead to gross error. Furthermore, the possibility of contamination of the error distribution by outliers is always present and has recently received much attention. More generally, real data sets, especially if they are long, usually show inhomogeneity in the mean, the variance, or both, and it is not always possible to randomize." (George E P Box, "Some Problems of Statistics and Everyday Life", Journal of the American Statistical Association, Vol. 74 (365), 1979)

"Science usually amounts to a lot more than blind trial and error. Good statistics consists of much more than just significance tests; there are more sophisticated tools available for the analysis of results, such as confidence statements, multiple comparisons, and Bayesian analysis, to drop a few names. However, not all scientists are good statisticians, or want to be, and not all people who are called scientists by the media deserve to be so described." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Another reason for the applied statistician to care about Bayesian inference is that consumers of statistical answers, at least interval estimates, commonly interpret them as probability statements about the possible values of parameters. Consequently, the answers statisticians provide to consumers should be capable of being interpreted as approximate Bayesian statements." (Donald B Rubin, "Bayesianly justifiable and relevant frequency calculations for the applied statistician", Annals of Statistics 12(4), 1984)

"Stepwise regression is probably the most abused computerized statistical technique ever devised. If you think you need stepwise regression to solve a particular problem you have, it is almost certain that you do not. Professional statisticians rarely use automated stepwise regression." (Leland Wilkinson, "SYSTAT", 1984)

"The result is that non-statisticians tend to place undue reliance on single ‘cookbook’ techniques, and it has for example become impossible to get results published in some medical, psychological and biological journals without reporting significance values even if of doubtful validity. It is sad that students may actually be more confused and less numerate at the end of a ‘service course’ than they were at the beginning, and more likely to overlook a descriptive approach in favor of some inferential method which may be inappropriate or incorrectly executed." (Christopher Chatfield, "The initial examination of data", Journal of the Royal Statistical Society, Series A 14, 1985)

"Too much of what all statisticians do [...] is blatantly subjective for any of us to kid ourselves or the users of our technology into believing that we have operated ‘impartially’ in any true sense. [...] We can do what seems to us most appropriate, but we can not be objective and would do well to avoid language that hints to the contrary." (Steve V Vardeman, Comment, Journal of the American Statistical Association 82, 1987)

"[In statistics] you have the fact that the concepts are not very clean. The idea of probability, of randomness, is not a clean mathematical idea. You cannot produce random numbers mathematically. They can only be produced by things like tossing dice or spinning a roulette wheel. With a formula, any formula, the number you get would be predictable and therefore not random. So as a statistician you have to rely on some conception of a world where things happen in some way at random, a conception which mathematicians don’t have." (Lucien LeCam, [interview] 1988)

"It is clear that a statistician who is involved at the start of an investigation, advises on data collection, and who knows the background and objectives, will generally make a better job of the analysis than a statistician who was called in later on." (Christopher Chatfield, "Problem solving: a statistician’s guide", 1988)

"The statistician should not always remain in his or her own office: not only is relevant information more likely to be on hand in the experimenter’s department, but in the longer term the statistician stands to gain immeasurably in understanding of agricultural problems by often visiting other departments and their laboratories and fields." (David J Finney, "Was this in your statistics textbook?", Experimental Agriculture 24, 1988)

"A little thought reveals a fact widely understood among statisticians: The null hypothesis, taken literally (and that’s the only way you can take it in formal hypothesis testing), is always false in the real world. [...] If it is false, even to a tiny degree, it must be the case that a large enough sample will produce a significant result and lead to its rejection. So if the null hypothesis is always false, what’s the big deal about rejecting it?" (Jacob Cohen, "Things I Have Learned (So Far)", American Psychologist, 1990)

"Statisticians classically asked the wrong question–and were willing to answer with a lie, one that was often a downright lie. They asked 'Are the effects of A and B different?' and they were willing to answer “no”. All we know about the world teaches us that the effects of A and B are always different–in some decimal place–for every A and B. Thus asking 'Are the effects different?' is foolish. What we should be answering first is 'Can we tell the direction in which the effects of A differ from the effects of B?' In other words, can we be confident about the direction from A to B? Is it 'up', 'down' or 'uncertain'?" (John W Tukey, "The Philosophy of Multiple Comparisons", Statistical Science 6, 1991)

"Statistics is a very powerful and persuasive mathematical tool. People put a lot of faith in printed numbers. It seems when a situation is described by assigning it a numerical value, the validity of the report increases in the mind of the viewer. It is the statistician's obligation to be aware that data in the eyes of the uninformed or poor data in the eyes of the naive viewer can be as deceptive as any falsehoods." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"A careful and sophisticated analysis of the data is often quite useless if the statistician cannot communicate the essential features of the data to a client for whom statistics is an entirely foreign language." (Christopher J Wild, "Embracing the ‘Wider view’ of Statistics", The American Statistician 48, 1994)

"We have to teach non-statisticians to recognize where statistical expertise is required. No one else will. We teach students how to solve simple statistical problems, but how often do we make any serious effort to teach them to recognize situations that call for statistical expertise that is beyond the technical content of the course." (Christopher J Wild, "Embracing the ‘Wider view’ of Statistics", The American Statistician 48, 1994)

"Because no one becomes statistically self-sufficient after one semester of study, I try to prepare students to become intelligent consumers of the assistance that they will inevitably seek. Service courses train future clients, not future statisticians." (Michael W Tosset, "Statistical Science", 1998)

"There are aspects of statistics other than it being intellectually difficult that are barriers to learning. For one thing, statistics does not benefit from a glamorous image that motivates students to persist through tedious and frustrating lessons[...]there are no TV dramas with a good-looking statistician playing the lead, and few mothers’ chests swell with pride as they introduce their son or daughter as 'the statistician'." (Chap T Le & James R Boen, "Health and Numbers: Basic Statistical Methods", 1995)

"When an analyst selects the wrong tool, this is a misuse which usually leads to invalid conclusions. Incorrect use of even a tool as simple as the mean can lead to serious misuses. […] But all statisticians know that more complex tools do not guarantee an analysis free of misuses. Vigilance is required on every statistical level."  (Herbert F Spirer et al, "Misused Statistics" 2nd Ed, 1998)

On Axioms (1975-1999)

"Mathematics has been trivialized, derived from indubitable, trivial axioms in which only absolutely clear trivial terms figure, and from which truth pours down in clear channels." (Imre Lakatos, "Mathematics, Science and Epistemology", 1980)

"The axiom of choice has many important consequences in set theory. It is used in the proof that every infinite set has a denumerable subset, and in the proof that every set has at least one well-ordering. From the latter, it follows that the power of every set is an aleph. Since any two alephs are comparable, so are any two transfinite powers of sets. The axiom of choice is also essential in the arithmetic of transfinite numbers." (R Bunn, "Developments in the Foundations of Mathematics, 1870-1910", 1980)

"If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book […] it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way." (George Pólya, "Mathematical Discovery: on Understanding, Learning, and Teaching Problem Solving", 1981)

"A slight variation in the axioms at the foundation of a theory can result in huge changes at the frontier." (Stanley P Gudder, "Quantum Probability", 1988)

"The mathematical theories generally called 'mathematical theories of chance' actually ignore chance, uncertainty and probability. The models they consider are purely deterministic, and the quantities they study are, in the final analysis, no more than the mathematical frequencies of particular configurations, among all equally possible configurations, the calculation of which is based on combinatorial analysis. In reality, no axiomatic definition of chance is conceivable." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Nobel lecture] 1988)

"Whenever we axiomitize a real-world system, we always, of necessity, oversimplify. Frequently, the oversimplification will adequately describe the system for the purposes at hand. In many other cases, the oversimplification may seem deceptively close to reality, when in fact it is far wide of the mark. The best hope, of course, is the use of a model adequate to explain observation. However, when we are unable to develop an adequate model, we would generally be well advised to stick with empiricism and axiomatic imprecision." (James R Thompson, "Empirical Model Building", 1989)

"It is not surprising to find many mathematical ideas interconnected or linked. The expansion of mathematics depends on previously developed ideas. The formation of any mathematical system begins with some undefined terms and axioms (assumptions) and proceeds from there to definitions, theorems, more axioms and so on. But history points out this is not necessarily the route that creativity" (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro-assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

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

"In view of the developments of abstract mathematics, the first thing mathematicians studied was how to extract the property of 'nearness' from the set of numbers. If the property of nearness could be extracted using a few axioms, and if it was possible to associate the extracted property with a set, then the resulting set would provide an abstract scene to study 'nearness'." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"In everyday language, an axiom is often held to be a 'self-evident truth'. That phrase betrays some awfully sloppy thinking. A truth can be evident, but self-evident? Evidence is what convinces people that something is true. So what on earth is 'self-evident' supposed to mean? A truth that convinces itself? [...] To those who were seeking foundations for mathematics, axioms were much more prosaic. They weren't truths at all, let alone evident ones, and certainly not self-evident ones - assuming that such a slogan means anything at all. Axioms were assumptions. A place to start. A collection of statements that mathematicians agreed to accept. You are free to challenge them if you wish, but even if you do, you won't change mathematics (though you might create some more, branching off in a new direction). From the axiomatic viewpoint, mathematics consists of the deductions that are made, once the axioms are accepted as a starting point. It's like a game. If you want to play football, then you follow the rules of football. Of course you are free to change the rules - but then you're playing a different game." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Let us regard a proof of an assertion as a purely mechanical procedure using precise rules of inference starting with a few unassailable axioms. This means that an algorithm can be devised for testing the validity of an alleged proof simply by checking the successive steps of the argument; the rules of inference constitute an algorithm for generating all the statements that can be deduced in a finite number of steps from the axioms." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)