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

"Experience is the child of Thought, and Thought is the child of Action. We cannot learn men from books." (Benjamin Disraeli, "Vivian Grey", 1826)

"Phenomena may well be suspected of anything, are capable of anything. Hypothesis proclaims the infinite; that is what gives hypothesis its greatness. Beneath the surface fact it seeks the real fact. It asks creation for her thoughts, and then for her second thoughts. The great scientific discoverers are those who hold nature suspect." (Victor Hugo, "The Toilers of the Sea", 1866)

"This history of culture will explain to us the motives, the conditions of life, and the thought of the writer or reformer." (Lev N Tolstoy, "War and Peace", 1867)

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

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language." (Herbert G Wells, "Mankind in the Making", 1903)

"Doublethink means the power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them." (George Orwell, "Nineteen Eighty-Four", 1949

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

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

"To my way of thinking it is precisely because time travel involves such fascinating paradoxes that we can conclude, even in the absence of other evidence, that time travel is impossible." (Isaac Asimov, 1954)

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

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

"Science offers a sounder basis on which to formulate systems of thought and ethics." (Michael Moorcock, "Behold the Man", 1967)

"Thought itself is a disease of the brain, a degenerative condition of matter. [...] The mind defends itself against the disintegrative process of creativity. It begins to jell, notions solidify into inalterable systems, which simply refuse to be broken down and reformed." (Thomas M Disch, "Camp Concentration", 1968)

"Whether a thought is spoken or not it is a real thing and has powers of reality." (Frank Herbert, "Dune Messiah", 1969)

"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)

"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)

"In a way, science might be described as paranoid thinking applied to Nature: we are looking for natural conspiracies, for connections among apparently disparate data." (Carl Sagan, "The Dragons of Eden", 1977)

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

"There're two kinds of people - those who think there are two kinds of people and those who have more sense." (James Tiptree Jr, "Up the Walls of the World", 1978)

"We take foul medicines to improve our health; so we must entertain foul thoughts on occasion, to strengthen wisdom." (Brian W Aldiss, "The Small Stones of Tu Fu", 1978)

"Thought is a matrix which engenders its own reality. The ideas, concepts, belief-systems that your ancestors trapped have become your trap." (Alfred A Attanasio, "Radix", 1981)

"A stray thought, wandering through the dimensions in search of a mind to harbour it, slid into his brain." (Terry Pratchett, "The Colour of Magic", 1983)

"Universities are meant to pass the torch of civilization, not just download data into student skulls, and the values of the academic community are strongly at odds with those of all would-be information empires. Teachers at all levels, from kindergarten up, have proven to be shameless and persistent software and data pirates. Universities do not merely ‘'leak information' but vigorously broadcast free thought." (Bruce Sterling, "The Hacker Crackdown", 1992)

"We think we are the victims of time. In reality, the way of the world isn't fixed anywhere. How could that be possible? We are our own journey. And therefore we are time as well. We are the same. Fugitive. Inscrutable. Ruthless." (Cormac McCarthy, "The Crossing", 1994)

 "Nothing up there tonight but entropy, and the same imaginary shapes that people had been imposing on nature since they’d first thought to wonder at the heavens." (Peter Watts, "Echopraxia", 2014)

"And don’t ever make the mistake of thinking that things you didn’t intend or plan don’t matter. It’s a big, disorganised multiverse out there - an accident of stars. Almost nothing ever works out like we want it to, and when it does, there’s guaranteed to be unexpected consequences. Randomness is what separates life from entropy, but it’s also what makes it fun." (Foz Meadows, "An Accident of Stars", 2016)

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

"If mathematical objects exist independently of human thought what else are they independent of? The universe, I suppose." (Cormac McCarthy, "Stella Maris", 2022)

🧩Out of Context: On Thought (Definitions)

"It is probable that what we call thought is not an actual being, but no more than the relation between certain parts of that infinitely varied mass, of which the rest of the universe is composed, and which ceases to exist as soon as those parts change their position with regard to each other." (Percy B Shelley, "On a Future State", 1815)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises." (George H Lewes "Problems of Life and Mind", 1873)

"[...] thought is the representative or cognitive apprehension of relations among notions; imagination is the affective or felt apprehension of relations among images." (James M Baldwin,"Handbook of Psychology: Senses and Intellect", 1890)

"Thought is existence. More than that, so far as we are concerned, existence is thought, all our conceptions of existence being some kind or other of thought." (Thomas H Huxley, "Method and Results", 1893)

"Consequently, all truly strict and exact thought is sustained by the symbolic and semiotics on which it is based." (Ernst Cassirer, "The Philosophy of Symbolic Forms", 1923)

"Thought is prior to language and consists in the simultaneous presentation to the mind of two different images." (Thomas E Hulme, "Notes on Language and Style", 1929)

"Analytic thought is based on detailed defined relations between two elements at a time. Intuitive thought is based on an emotional state associated with all the elements in the field of knowledge (overall impression). " (Tony Bastick, "Intuition: How we think and act", 1982) 

On Continuity XV (Thought)

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

"But mathematics, certainly, does not play the smallest part in the charm and movement of the mind produced by music. Rather is it only the indispensable condition (conditio sine qua non) of that proportion of the combining as well as changing impressions which makes it possible to grasp them all in one and prevent them from destroying one another, and to let them, rather, conspire towards the production of a continuous movement and quickening of the mind by affections that are in unison with it, and thus towards a serene self-enjoyment." (Immanuel Kant, "The Critique of Judgment", 1790)

"Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1814)

"But how can we avoid the use of human language? The [....] symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason - only thus may we hope to build mathematics on the solid foundation of logic." (Tobias Dantzig, "Number: The Language of Science", 1930)

"In this way things, external objects, are assimilated to more or less ordered motor schemas, and in this continuous assimilation of objects the child's own activity is the starting point of play. Not only this, but when to pure movement are added language and imagination, the assimilation is strengthened, and wherever the mind feels no actual need for accommodating itself to reality, its natural tendency will be to distort the objects that surround it in accordance with its desires or its fantasy, in short to use them for its satisfaction. Such is the intellectual egocentrism that characterizes the earliest form of child thought." (Jean Piaget, "The Moral Judgment of the Child", 1932)

"Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

📁 On DIKW: On Knowledge (1940-1949)

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

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

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

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

"Causality, which requires description of states and a law of necessary connection relating them, has appeared in several forms in several sciences, but never, except in statistics, has it been as irreciprocal as in this theory. Specification for any one time of afferent stimulation and of the activity of all constituent neurons, each an 'all-or-none' affair, determines the state. Specification of the nervous net provides the law of necessary connection whereby one can compute from the description of any state that of the succeeding state, but the inclusion of disjunctive relations prevents complete determination of the one before. Moreover, the regenerative activity of Constituent circles renders reference indefinite as to time past. Thus our knowledge of the world, including ourselves, is incomplete as to space and indefinite as to time." (Warren S McCulloch & Walter Pitts, "A logical calculus of the ideas immanent in nervous activity" Vol. 5, 1943)

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

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

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

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

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

"Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders." (William L Schaaf, "Mathematics: Our Great Heritage: Essays on the Nature and Cultural Significance of Mathematics", 1948)

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

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

📁 On DIKW: On Knowledge (1775-1799)

"Cultivate that kind of knowledge which enables us to discover for ourselves in case of need that which others have to read or be told of." (Georg C Lichtenberg, Notebook D, 1773-1775)

"Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it." (Samuel Johnson, 1775)

"Our knowledge springs from two fundamental sources of the mind; the first is the capacity of receiving representations (receptivity for impressions), the second is the power of knowing an object through these representations (spontaneity [in the production] of concepts)." (Immanuel Kant, "Critique of Pure Reason", 1781)

"Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts." (Immanuel Kant, "Critique of Pure Reason", 1781)

"Thoughts without content are empty, intuitions without concepts are blind. The understanding can intuit nothing, the senses can think nothing. Only through their unison can knowledge arise." (Immanuel Kant, "Critique of Pure Reason", 1781)

"The word ‘chance’ then expresses only our ignorance of the causes of the phenomena that we observe to occur and to succeed one another in no apparent order. Probability is relative in part to this ignorance, and in part to our knowledge." (Pierre-Simon Laplace, "Mémoire sur les Approximations des Formules qui sont Fonctions de Très Grands Nombres", 1783)

"The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"On completing one discovery we never fail to get an imperfect knowledge of others of which you could have no idea before […]" (Joseph Priestley, 1786)

"As there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves, by their own exertions, to distinction and eminence. […] Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"The power of Reason […] is unquestionably the most important by far of those which are comprehended under the general title of Intellectual. It is on the right use of this power that our success in the pursuit of both knowledge and of  happiness depends; and it is by the exclusive possession of it that man is distinguished, in the most essential respects, from the lower animals. It is, indeed, from their subserviency to its operations, that the other faculties […] derive their chief value." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"Conjecture may lead you to form opinions, but it cannot produce knowledge. Natural philosophy must be built upon the phenomena of nature discovered by observation and experiment." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

On Continuity XI (Thought II)

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

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

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

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

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

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

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

Peter B Medawar - Collected Quotes

"The formulation of a hypothesis carries with it an obligation to test it as rigorously as we can command skills to do so." (Peter Medawar, "Hypothesis and Imagination", 1963)

"Scientific discovery, or the formulation of scientific theory, starts in with the unvarnished and unembroidered evidence of the senses. It starts with simple observation - simple, unbiased, unprejudiced, naive, or innocent observation - and out of this sensory evidence, embodied in the form of simple propositions or declarations of fact, generalizations will grow up and take shape, almost as if some process of crystallization or condensation were taking place. Out of a disorderly array of facts, an orderly theory, an orderly general statement, will somehow emerge." (Sir Peter B Medawar, "Is the Scientific Paper Fraudulent?", The Saturday Review, 1964)

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

"Every discovery, every enlargement of the understanding, begins as an imaginative preconception of what the truth might be. The imaginative preconception - a ‘hypothesis’ - arises by a process as easy or as difficult to understand as any other creative act of mind; it is a brainwave, an inspired guess, a product of a blaze of insight. It comes anyway from within and cannot be achieved by the exercise of any known calculus of discovery." (Sir Peter B Medawar, "Advice to a Young Scientist", 1979)

"I cannot give any scientist of any age better advice than this: the intensity of a conviction that a hypothesis is true has no bearing over whether it is true or not. The importance of the strength of our conviction is only to provide a proportionately strong incentive to find out if the hypothesis will stand up to critical evaluation." (Sir Peter B Medawar, "Advice to a Young Scientist", 1979)

"The intensity of a conviction that a hypothesis is true has no bearing on whether it is true or false." (Peter Medawar, "Advice to a Young Scientist", 1979)

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

"If the purpose of scientific methodology is to prescribe or expound a system of enquiry or even a code of practice for scientific behavior, then scientists seem able to get on very well without it." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"In a modern professional vocabulary a hypothesis is an imaginative preconception of what might be true in the form of a declaration with verifiable deductive consequences. It no longer tows ‘gratuitous’, ‘mere’, or ‘wild’ behind it, and the pejorative usage (‘Evolution is a mere hypothesis’, ‘It is only a hypothesis that smoking causes lung cancer’) is one of the outward signs of little learning." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"In all sensation we pick and choose, interpret, seek and impose order, and devise and test hypotheses about what we witness. Sense data are taken, not merely given: we learn to perceive. […] The teacher has forgotten, and the student himself will soon forget, that what he sees conveys no information until he knows beforehand the kind of thing he is expected to see."  (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Intuition takes many different forms in science and mathematics, though all forms of it have certain properties in common: the suddenness of their origin, the wholeness of the conception they embody, and the absence of conscious premeditation." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Observation is the generative act in scientific discovery. For all its aberrations, the evidence of the senses is essentially to be relied upon - provided we observe nature as a child does, without prejudices and preconceptions, but with that clear and candid vision which adults lose and scientists must strive to regain." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Scientific theories (I have said) begin as imaginative constructions. The begin, if you like, as stories, and the purpose of the critical or rectifying episode in scientific reasoning is precisely to find out whether or not these stories are stories about real life. Literal or empiric truthfulness is not therefore the starting-point of scientific enquiry, but rather the direction in which scientific reasoning moves. If this is a fair statement, it follows that scientific and poetic or imaginative accounts of the world are not distinguishable in their origins. They start in parallel, but diverge from one another at some later stge. We all tell stories, but the stories differ in the purposes we expect them to fulfil and in the kinds of evaluations to which they are exposed." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Scientific discovery is a private event, and the delight that accompanies it, or the despair of finding it illusory, does not travel. One scientist may get great satisfaction from another’s work and admire it deeply; it may give him great intellectual pleasure; but it gives him no sense of participation in the discovery, it does not carry him away, and his appreciation of it does not depend on his being carried away. If it were otherwise the inspirational origin of scientific discovery would never have been in doubt." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Simultaneous discovery is utterly commonplace, and it was only the rarity of scientists, not the inherent improbability of the phenomenon, that made it remarkable in in the past." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The ballast of factual information, so far from being just about to sink us, is growing daily less. The factual burden of a science varies inversely with its degree of maturity. As a science advances, particular facts are comprehended within, and therefore in a sense annihilated by, general statements of steadily increasing explanatory power and compass - whereupon the facts need no longer be known explicitly, that is, spelled out and kept in mind. In all sciences we are being progressively relieved of the burden of singular instances, the tyranny of the particular. We need no longer record the fall of every apple." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 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)

"The formulation of a natural ‘law’ always begins as an imaginative exploit, and without imagination scientific thought is barren." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The scientific method is a potentiation of common sense, exercised with a specially firm determination not to persist in error if any exertion of hand or mind can deliver us from it. Like other exploratory processes, it can be resolved into a dialogue between fact and fancy, the actual and the possible; between what could be true and what is in fact the case. The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticise and modify as we go along, so that it ends by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"There is no such thing as a Scientific Mind. Scientists are people of very dissimilar temperaments doing different things in very different ways. Among scientists are collectors, classifiers and compulsive tidiers-up; many are detectives by temperament and many are explorers; some are artists and others artisans. There are poet-scientists and philosopher-scientists and even a few mystics. What sort of mind or temperament can all these people be supposed to have in common? Obligative scientists must be very rare, and most people who are in fact scientists could easily have been something else instead." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"What shows a theory to be inadequate or mistaken is not, as a rule, the discovery of a mistake in the information that led us to propound it; more often it is the contradictory evidence of a new observation which we were led to make because we held that theory." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"A scientist is no more a collector and classifier of facts than a historian is a man who complies and classifies a chronology of the dates of great battles and major discoveries." (Sir Peter B Medawar, "Aristotle to Zoos: A Philosophical Dictionary of Biology", 1983)

"The attempt to discover and promulgate the truth is nevertheless an obligation upon all scientists, one that must be persevered in no matter what the rebuffs - for otherwise what is the point in being a scientist?" (Sir Peter B Medawar, "Aristotle to Zoos: A Philosophical Dictionary of Biology", 1983)

⚛On Physics (1900-1909)

"The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results." (Josiah W Gibbs, "Elementary Principles in Statistical Mechanics", 1902)

"Chemistry and physics are experimental sciences; and those who are engaged in attempting to enlarge the boundaries of science by experiment are generally unwilling to publish speculations; for they have learned, by long experience, that it is unsafe to anticipate events. It is true, they must make certain theories and hypotheses. They must form some kind of mental picture of the relations between the phenomena which they are trying to investigate, else their experiments would be made at random, and without connection." (William Ramsay, "Radium and Its Products", Harper’s Magazine, 1904)

"The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect, - the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae." (John T Merz, "History of European Thought in the 19th Century" Vol. 1, 1904)

"The science of physics does not only give us [mathematicians] an opportunity to solve problems, but helps us also to discover the means of solving them, and it does this in two ways: it leads us to anticipate the solution and suggests suitable lines of argument." (Henri Poincaré, "La valeur de la science" ["The Value of Science"], 1905)

"Geometry, accordingly, consists of the application of mathematics to experiences concerning space. Like mathematical physics, it can become an exact deductive science only on the condition of its representing the objects of experience by means of schematizing and idealizing concepts." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"[...] as for physics, it has developed remarkably as a precision science, in such a way that we can justifiably claim that the majority of all the greatest discoveries in physics are very largely based on the high degree of accuracy which can now be obtained in measurements made during the study of physical phenomena. [... Accuracy of measurement] is the very root, the essential condition, of our penetration deeper into the laws of physics - our only way to new discoveries." (K Bernhard Hasselberg, [Nobel Lecture] 1907)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Energy really is only an integral; now, what we want to have is a substantial definition, like that of Leibniz, and this demand is justifiable to a certain degree, since our very conviction of the conservation of energy rests in great part on this foundation. [..] And so the manuals of physics contain really two discordant definitions of energy, the first which is verbal, intelligible, capable of establishing our conviction, and false; and the second which is mathematical, exact, but lacking verbal expression." (Emile Meyerson, "Identity & Reality", 1908)

"If the aim of physical theories is to explain experimental laws, theoretical physics is not an autonomous science; it is subordinate to metaphysics." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

"It is impossible to follow the march of one of the greatest theories of physics, to see it unroll majestically its regular deductions starting from initial hypotheses, to see its consequences represent a multitude of experimental laws down to the smallest detail, without being charmed by the beauty of such a construction, without feeling keenly that such a creation of the human mind is truly a work of art." (Pierre-Maurice-Marie DuhemDuhem, "The Aim and Structure of Physical Theory", 1908)

"[...] physics makes progress because experiment constantly causes new disagreements to break out between laws and facts, and because physicists constantly touch up and modify laws in order that they may more faithfully represent the facts." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

"The laws of physics are therefore provisional in that the symbols they relate too simple to represent reality completely." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

On Entropy (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)

"The basic objection to attempts to deduce the unidirectional nature of time from concepts such as entropy is that they are attempts to reduce a more fundamental concept to a less fundamental one." (Gerald J Whitrow, "The Natural Philosophy of Time", 1961)

"Both the uncertainty principle and the negentropy principle of information make Laplace's scheme [of exact determinism] completely unrealistic. The problem is an artificial one; it belongs to imaginative poetry, not to experimental science." (Léon Brillouin, "Science and Information Theory" 2nd Ed., 1962)

"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)

"The homeostatic principle does not apply literally to the functioning of all complex living systems, in that in counteracting entropy they move toward growth and expansion." (Daniel Katz, "The Social Psychology of Organizations", 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)

On Symbols (-1799)

"Natural justice is a symbol or expression of usefulness, to prevent one person from harming or being harmed by another."  (Epicurus, "Sovereign Maxims"

"All teems with symbol; the wise man is the man who in any one thing can read another." Plotinus, The Enneads c. 250)

"Letters are signs of things, symbols of words, whose power is so great that without a voice they speak to us the words of the absent; for they introduce words by the eye, not by the ear." (Isidore of Seville, "Etymologiae", cca. 800-625)

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, The New Organon, 1620)

"Nature's great book is written in mathematical symbols." (Galileo Galilei, "The Assayer", 1623)

"It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters, insofar as they are subject to reasoning, all that we can do in arithmetic and geometry." (Gottfried W Leibniz, 1677)

"In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished." (Gottfried W Leibniz, [letter to Ehrenfried W von Tschirnhaus] cca. 1680)

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

As a general rule - never substitute the symbol for the thing signified, unless it is impossible to show the thing itself; for the child's attention is so taken up with the symbol that he will forget what it signifies." (Jean Jacques Rousseau, "Emile, or On Education", 1762) 

On Engineering VI

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

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

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

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

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

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

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

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

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

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

On Symbols (2000-2009)

"Precision is greatly enhanced by the human capacity to symbolize. Symbols can be devised to stand for mathematical ideas, entities, operations, and relations. Symbols also permit precise and repeatable calculation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"A symbol is a mental representation regarding the internal reality referring to its object by a convention and produced by the conscious interpretation of a sign. In contrast to signals, symbols may be used every time if the receiver has the corresponding representation. Symbols also relate to feelings and thus give access not only to information but also to the communicator’s motivational and emotional state. The use of symbols makes it possible for the organism using it to evoke in the receiver the same response it evokes in himself. To communicate with symbols is to use a language." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"In the definition of meaning, it is assumed that both the source and receiver have previously coded (and stored) signals of the same or similar referents, such that the messages may have meaning and relate to behaviour. That is, the used symbols must have the same signification for both sender and receiver. If not, the receiver will create a different mental picture than intended by the transmitter. Meaning is generated by individuals in a process of social interaction with a more or less common environment. It is a relation subsisting within a field of experience and appears as an emergent property of a symbolic representation when used in culturally accepted interaction. The relation between the symbolic representation and its meaning is random. Of this, however, the mathematical theory has nothing to say. If human links in the chain of communication are missing, of course no questions of meaning will arise." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"A person thinking in the nonverbal mode is actually thinking with the meaning of the language in the form of mental pictures of the concepts and ideas it contains. Nonverbal thought doesn't require literacy. An illiterate person can communicate without knowing what the symbols look like. [...] Literacy, then, is established as the person learns how the symbols look and becomes able to recognize them as representing certain things or concepts." (Ronald D Davis & Eldon M Braun, "The Gift of Learning", 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. […] [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"I often told the fanatics of realism that there is no such thing as realism in art: it only exists in the mind of the observer. Art is a symbol, a thing conjuring up reality in our mental image. That is why I don't see any contradiction between abstract and figurative art either." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)

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

"But because of the way in which depictions represent, there is a correspondence between parts and spatial relations of the representation and those of the object; this structural mapping, which confers a type of resemblance, underlies the way images convey specific content. In this respect images are like pictures. Unlike words and symbols, depictions are not arbitrarily paired with what they represent." (Stephen Kosslyn et al," The Case for Mental Imagery", 2006)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"[...] the scientific models of concrete things are symbolic rather than iconic: they are systems of propositions, not pictures. Besides, such models are seldom if ever completely accurate, if only because they involve more or less brutal simplifications, such as pretending that a metallic surface is smooth, a crystal has no impurities, a biopopulation has a single predator, or a market is in equilibrium.  These are all fictions. However, they are stylizations rather than wild fantasies. Hence, introducing and using them to account for real existents does not commit us to fictionism, just as defending the role of experience need not make us empiricists, nor is admitting the role of intuition enough to qualify as intuitionist." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"But notice, a subatomic particle is itself a holon [hole/parts]. And so is a cell. And so is a symbol, and an image, and a concept. What all of those entities are, before they are anything else, is a holon. So the world is not composed of atoms or symbols or cells or concepts. It is composed of holons." (Ken Wilber, "A Brief History of Everything", 2007)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

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

"Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre O Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

🗺️On Knowledge Representation: On Diagrams (1925-1949)

"The preliminary examination of most data is facilitated by the use of diagrams. Diagrams prove nothing, but bring outstanding features readily to the eye; they are therefore no substitutes for such critical tests as may be applied to the data, but are valuable in suggesting such tests, and in explaining the conclusions founded upon them." (Sir Ronald A Fisher, "Statistical Methods for Research Workers", 1925)

"What the diagram has in common with the symbolic schema is the fact that the diagram spatially represents an abstract and unextended object. But there is here nothing other than a determinate location in space. This location serves as a mooring, an attachment, an orientation for our memory, but does not play any role in our thought." (Jean-Paul Sartre, "The Imaginary: A phenomenological psychology of the imagination", 1940)

"[…] statistical literacy. That is, the ability to read diagrams and maps; a 'consumer' understanding of common statistical terms, as average, per cent, dispersion, correlation, and index number." (Douglas Scates, “Statistics: The Mathematics for Social Problems”, 1943)

"When I undertake some geometrical research, I have generally a mental view of the diagram itself, though generally an inadequate or incomplete one, in spite of which it affords the necessary synthesis - a tendency which, it would appear, results from a training which goes back to my very earliest childhood." (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field”, 1949)

"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […] The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Polya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])

🪷On Mind: Mental Models LIV (Limitations VII)

"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 abide by the principle of parsimony: They represent only possibilities compatible with the premises, and they represent clauses in the premises only when they hold in a possibility. Fully explicit models represent clauses when they do not hold too. The advantage of mental models over fully explicit models is that they contain less information, and so they are easier to work with. But they can lead reasoners astray. The occurrence of these systematic and compelling fallacies is shocking. The model theory predicts them, and they are a 'litmus' test for mental models, because no other current theory predicts them. They have so far resisted explanation by theories of reasoning based on formal rules of inference, because these theories rely on valid rules." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"People don’t need to know all the details of how a complex mechanism actually works in order to use it, so they create a cognitive shorthand for explaining it, one that is powerful enough to cover their interactions with it, but that doesn’t necessarily reflect its actual inner mechanics. […] In the digital world, however, the differences between a user’s mental model and the implementation model are often quite distinct. The discrepancy between implementation and mental models is particularly stark in the case of software applications, where the complexity of implementation can make it nearly impossible for the user to see the mechanistic connections between his actions and the program’s reactions." (Alan Cooper et al,  "About Face 3: The Essentials of Interaction Design", 2007)

"The closer the represented model comes to the user’s mental model, the easier he will find the program to use and to understand. Generally, offering a represented model that follows the implementation model too closely significantly reduces the user’s ability to learn and use the program, assuming (as is almost always the case) that the user’s mental model of his tasks differs from the implementation model of the software." (Alan Cooper et al,  "About Face 3: The Essentials of Interaction Design", 2007)

"Gaining awareness about how the system is built up and how it works can also help us to avoid solutions that only treat the symptoms of an underlying problem without curing the problem itself. System thinking is powerful because it helps us to see our own mental models and how these models color our perception of the world. In many cases, it is difficult for us to alter our mental models. There are always some beliefs or viewpoints that we are not willing to change, no matter what evidence is presented against it. This causes a certain resistance to new concepts. Problems can occur, however, when a rigid mental model stands in the way of a solution that might solve a problem. In such situations, adherence to mental models can be dangerous to the health of the organization." (Akhilesh Bajaj & Stanisław Wrycza, "Systems Analysis and Design for Advanced Modeling Methods: Best Practices", 2009)

"We all use mental models every day. Our minds do not contain real economic or social systems. Instead, they contain representations - models - of reality. We use these models in all aspects of decision-making. Being explicitly aware of our mental models can help us in understanding why we make the decisions we do and how we can improve our decision-making processes. If everyone’s mental models are brought to light in the context of an organization, we can begin to see where, how, and why the models diverge. This is the first step in building a shared understanding within an organization. As long as mental models remain hidden, they constitute an obstacle to building shared understanding." (Akhilesh Bajaj & Stanisław Wrycza, "Systems Analysis and Design for Advanced Modeling Methods: Best Practices", 2009)

"Each person has a different mental model and, therefore, potentially a different interpretation of the Facts. The danger comes when we start to assume that our interpretation of the Facts is the only interpretation and we believe that what we see and think is the Truth, and that there is only one Truth." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

Our mental model governs what we see, how we think about things and how we act, but is this just an individual phenomenon, or can whole communities and professions share a common mental model, a shared view of reality? […] What is beyond doubt is that all professions have their own vocabularies, lexicons, tools and techniques. These models, techniques and ways of analysing things are not value-free; they represent a way of seeing the world. When we select and use them we are subconsciously buying into that way of thinking and, inevitably, we are also reinforcing our own mental model of cause and effect. We tend to expend significant effort in understanding the inner workings of a new tool or practice, but we give scant attention to the thought paradigm that underpins it. (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

"Our past (mental model) governs what we perceive (See) and how we make sense of it (Assess). This, in turn, prescribes our range of possible actions (Do), which leads to results (Get). We are therefore trapped into repeating well-understood, but possibly ineffective, patterns of behaviour and can Get sub-optimal results. The cycle is both self-fulfilling and self-reinforcing – we tend to apply the same assessment techniques to review what we get, thus further reinforcing our mental model and further limiting our ability to perceive the unusual or the remarkable." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

"[…] our strong mental models tend to make us blind to certain possibilities, and therefore we unknowingly engage in biased listening. Whenever we interpret information, we subconsciously access three filters based upon how we feel about the content, the information source and situation (or context) in which we receive the information." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

🪷On Mind: Mental Models LVII

"Thoughts are the images of things, words are of thoughts; and we all know that images and pictures are only so true as they are true representations men and things. […] For poets as well painters think it their business to take likeness of things from their appearance."(Joseph Trapp, "Lectures on Poetry", 1711)

"By reducing imagination to the power of forming images, and by  insisting that no image can be formed except out of the elements furnished by experience, I do not mean to confound imagination with memory; indeed, the frequent occurrence of great strength of memory with comparative feebleness of imagination, would suffice to warn us against such a conclusion." (George H Lewes, "The Principles of Success in Literature", 1865)

"However rapid and remote their flight of thought, it is a succession of images, not of abstractions. The details which give significance, and which by us are seen vaguely as through a vanishing mist, are by them seen in sharp outlines. The image which to us is a mere suggestion, is to them almost as vivid as the object. And it is because they see vividly that they can paint effectively." (George H Lewes, "The Principles of Success in Literature", 1865)

"The strangest and most wonderful constructions in the whole animal world are the amazing, intricate constructions made by the primate Homo sapiens. Each normal individual of this species makes a self. Out of its brain it spins a web of words and deeds, and, like the other creatures, it doesn't have to know what it's doing; it just does it. This web protects it, just like the snail's shell. […] As such, it plays a singularly important role in the ongoing cognitive economy of that living body, because, of all the things in the environment an active body must make mental models of, none is more crucial than the model the agent has of itself." (Daniel Dennett, "Consciousness Explained", 1991)

"[Language comprehension] involves many components of intelligence: recognition of words, decoding them into meanings, segmenting word sequences into grammatical constituents, combining meanings into statements, inferring connections among statements, holding in short-term memory earlier concepts while processing later discourse, inferring the writer’s or speaker’s intentions, schematization of the gist of a passage, and memory retrieval in answering questions about the passage. [… The reader] constructs a mental representation of the situation and actions being described. […] Readers tend to remember the mental model they constructed from a text, rather than the text itself." (Gordon H Bower & Daniel G Morrow, 1990)

"We build mental models that represent significant aspects of our physical and social world, and we manipulate elements of those models when we think, plan, and try to explain events of that world. The ability to construct and manipulate valid models of reality provides humans with our distinctive adaptive advantage; it must be considered one of the crowning achievements of the human intellect." (Gordon H Bower & Daniel G Morrow, 1990)

"[For] us to be able to speak and understand novel sentences, we have to store in our heads not just the words of our language but also the patterns of sentences possible in our language. These patterns, in turn, describe not just patterns of words but also patterns of patterns. Linguists refer to these patterns as the rules of language stored in memory; they refer to the complete collection of rules as the mental grammar of the language, or grammar for short." (Ray Jackendoff, "Patterns in the Mind", 1994)

"When we visually perceive the world, we do not just process information; we have a subjective experience of color, shape, and depth. We have experiences associated with other senses (think of auditory experiences of music, or the ineffable nature of smell experiences), with bodily sensations (e.g., pains, tickles, and orgasms), with mental imagery (e.g., the colored shapes that appear when one tubs one's eyes), with emotion (the sparkle of happiness, the intensity of anger, the weight of despair), and with the stream of conscious thought." (David Chalmers, "The Puzzle of Conscious Experience", Scientific American, 1995)

"Actually, around 80% of the data we use to make decisions is already in our heads before we engage with a situation. Our power to perceive is governed and limited by cognitive filters, sometimes termed our ‘mental model’. Mental models are formed as a result of past experience, knowledge and attitudes. They are deeply ingrained, often subconscious, structures that limit what we perceive and also colour our interpretation of supposed facts." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

"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)

🪷On Mind: Mental Models LIII (Conceptual Models II)

"But metaphor is an indispensable tool of thought and expression - a characteristic of all human communication, even of that of the scientist. The conceptual models of cybernetics and the energy theories of psycho-analysis are, after all, only labeled metaphors." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"[…] conceptual models, even when incorrect, are useful to the extent that criticism of the model may point to new theoretical developments."  (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"As the least conscious layer of the user experience, the conceptual model has the paradoxical quality of also having the most impact on usability. If an appropriate conceptual model is faithfully represented throughout the interface, after users recognize and internalize the model, they will have a fundamental understanding of what the application does and how to operate it." (Bob Baxley, "Making the Web Work: Designing Effective Web Applications", 2002) 

"A conceptual model of an interactive application is, in summary: the structure of the application - the objects and their operations, attributes, and relation-ships; an idealized view of the how the application works – the model designers hope users will internalize; the mechanism by which users accomplish the tasks the application is intended to support." (Jeff Johnson & Austin Henderson, "Conceptual Models", 2011)

"Conceptual models are best thought of as design-tools - a way for designers to straighten out and simplify the design and match it to the users’ task-domain, thereby making it clearer to users how they should think about the application. The designers’ responsibility is to devise a conceptual model that seems natural to users based on the users’ familiarity with the task domain. If designers do their job well, the conceptual model will be the basis for users’ mental models of the application." (Jeff Johnson & Austin Henderson, "Conceptual Models", 2011)

"The conceptual model is not the users’ mental model of the application. […] users of an application form mental models of it to allow them to predict its behavior. A mental model is the user’s high-level understanding of how the application works; it allows the user to predict what the application will do in response to various user-actions. Ideally, a user’s mental model of an application should be similar to the designers’ conceptual model, but in practice the two models may differ significantly. Even if a user’s mental model is the same as the designer’s conceptual model, they are distinct models." (Jeff Johnson & Austin Henderson, "Conceptual Models", 2011)

"Briefly, a conceptual model is the configuration of conceptual elements and the navigation between them. As such, a conceptual model is the foundation of the user interface of any interactive system." (Avi Parush, "Conceptual Design for Interactive Systems", 2015)

"A conceptual model is a framework that is initially used in research to outline the possible courses of action or to present an idea or thought. When a conceptual model is developed in a logical manner, it will provide a rigor to the research process." (N Elangovan & R Rajendran, "Conceptual Model: A Framework for Institutionalizing the Vigor in Business Research", 2015) 

"Once we understand our user's mental model, we can capture it in a conceptual model. The conceptual model is a representation of the mental model using elements, relationships, and conditions. Our design and final system will be the tangible result of this conceptual model." (Pau Giner & Pablo Perea, "UX Design for Mobile, 2017) 

"A model or conceptual model is a schematic or representation that describes how something works. We create and adapt models all the time without realizing it. Over time, as you gain more information about a problem domain, your model will improve to better match reality." (James Padolsey, "Clean Code in JavaScript", 2020)

On Definitions V

"The use of a mathematical definition is, to deduce from it the properties of the thing defined […]" (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"The language of mathematics, permitting great sharpness and accuracy of definition, conduces largely to their power of drawing necessary conclusions. Language is not only a means of recording the results of our thinking; it is an instrument of thought, and that of the highest value." (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32, 1875)

"The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the concepts with which it deals. What is that distinctive characteristic? I answer: precision, sharpness, completeness of definition. But how comes your mathematician by such completeness? There is no mysterious trick involved; some ideas admit of such precision, others do not; and the mathematician is one who deals with those that do." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal Vol. 3, 1904–1905)

"[…] mathematics is a science whose concepts are too breakable, too dry, too precisely limited. The disciplines of life and society, of human thinking, are fluid disciplines, with some flexibility, with concepts that are not clearly defined, but which are able to include things less strictly delimited than a mathematical definition does it." (Grigore C Moisil, 1968)

"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)

"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)

"The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument." (Yuri I Manin, "Mathematics as Metaphor: Selected Essays of Yuri I. Manin", 2007)

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

🪷On Mind: Mental Models XLIX

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

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

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

"As regards the question, therefore, what memory or remembering is, it has now been shown that it is the having of an image, related as a likeness to that of which it is an image; and as to the question of which of the faculties within us memory is a function, it has been shown that it is a function of the primary faculty of sense-perception, i.e. of that faculty whereby we perceive time."  (Aristotle," De Memoria et Reminiscentia" ["On Memory and Recollection"], 4th century BC)

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

"For according to the arguments from the existence of the sciences there will be Forms of all things of which there are sciences, and according to the argument that there is one attribute common to many things there will be Forms even of negations, and according to the argument that there is an object for thought even when the thing has perished, there will be Forms of perishable things; for we can have an image of these." (Aristotle, "Metaphysics", 4th century BC)

"All that is required between cognizer and cognized is a likeness in terms of representation, not a likeness in terms of an agreement in nature. For it's plain that the form of a stone in the soul is of a far higher nature than the form of a stone in matter. But that form, insofar as it represents the stone, is to that extent the principle leading to its cognition." (Thomas Aquinas, "Quaestiones disputatae de veritate", cca. 1256-1259) 

"Imagery is not past but present. It rests with what we call our mental processes to place these images in a temporal order." (George H Mead, 1929)

"It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience. Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme - the quantum theory - which seems entirely adequate for the treatment of atomic processes; for visualisation, however, we must content ourselves with two incomplete analogies - the wave picture and the corpuscular picture." (Werner Heisenberg, "On Quantum Physics", 1930)

"The solution of the difficulty is that the two mental pictures which experiment lead us to form - the one of the particles, the other of the waves - are both incomplete and have only the validity of analogies which are accurate only in limiting cases." (Werner Heisenberg,"On Quantum Mechanics", 1930)

🚀On Invention (-1799)

"[...] because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. Nor ought It to seem wonderful, that the invention of this as well as of other sciences, should receive its commencement from convenience and opportunity. Since whatever is carried in the circle of generation proceeds from the imperfect to the perfect." (Proclus Lycaeus, cca 5th century)

"Reflexion is careful and laborious thought, and watchful attention directed to the agreeable effect of one's plan. Invention, on the other hand, is the solving of intricate problems and the discovery of new principles by means of brilliancy and versatility." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture], cca. 15BC) 

"To invent is to discover that we know not, and not to recover or resummon that which we already know." (Francis Bacon, "Advancement of Learning", 1605)

"Moreover, the works already known are due to chance and experiment rather than to sciences; for the sciences we now possess are merely systems for the nice ordering and setting forth of things already invented; not methods of invention or directions for new works." (Sir Francis Bacon, "Novum Organum", 1620)

"To apply oneself to great inventions, starting from the smallest beginnings, is no task for ordinary minds; to divine that wonderful arts lie hid behind trivial and childish things is a conception for superhuman talents." (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632) 

"Invention, strictly speaking, is little more than a new combination of those images which have been previously gathered and deposited in the memory: nothing can come of nothing." (Joshua Reynolds, "Discourses on Art", 1769) 

"Nothing, in truth, has such a tendency to weaken not only the powers of invention, but the intellectual powers in general, as a habit of extensive and various reading without reflection." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792) 

🚀On Invention (1950-1974)

"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem." (Milton Friedman, "Essays in Positive Economics", 1953) 

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya,"Induction and Analogy in Mathematics", 1954)

"Invention is not the product of logical thought, even though the final product is tied to a logical structure." (Albert Einstein, "Autobiographische Skizze", 1955) 

"No part of the aim of normal science is to call forth new sorts of phenomena; indeed those that will not fit the box are often not seen at all. Nor do scientists normally aim to invent new theories, and they are often intolerant of those invented by others." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962) 

"We must now ask how changes of this sort can come about, considering first discoveries, or novelties of fact, and then inventions, or novelties of theory. That distinction between discovery and invention or between fact and theory will, however, immediately prove to be exceedingly artificial." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962) 

"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) 

"When the problems in physics become difficult we may often look to the mathematician who may already have studied such things and have prepared a line of reasoning for us to follow. On the other hand they may not have, in which case we have to invent our own line of reasoning, which we then pass back to the mathematician." (Richard Feynman,"The Character of Physical Law" , 1965) 

"The essential functions of the mind consist in understanding and in inventing, in other words, in building up structures by structuring reality." (Jean Piaget, 1971)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)