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)

On Imagination (1800-1849)

"The philosopher who is really useful to the cause of science, is he who, uniting to a fertile imagination, a rigid severity in investigation and observation, is at once tormented by the desire of ascertaining the cause of the phenomena, and by the fear of deceiving himself in that which he assigns." (Pierre-Simon Laplace, "System of the World" Vol. 2, 1809)

"When the eye or the imagination is struck with an uncommon work, the next transition of an active mind is to the means by which it was performed." (Samuel Johnson, 1810)

"The imagination […] that reconciling and mediatory power, which incorporating the reason in images of the sense and organizing (as it were) the flux of the senses by the permanence and self-circling energies of the reason, gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors." (Samuel T Coleridge, "The Statesman's Manual", 1816)

"It seems to be like taking the pieces of a dissected map out of its box. We first look at one part, and then at another, then join and dove-tail them; and when the successive acts of attention have been completed, there is a retrogressive effort of mind to behold it as a whole. The poet should paint to the imagination, not to the fancy; and I know no happier case to exemplify the distinction between these two faculties." (Samuel T Coleridge," Biographia Literaria", 1817)

"Whilst chemical pursuits exalt the understanding, they do not depress the imagination or weaken genuine feeling; whilst they give the mind habits of accuracy, by obliging it to attend to facts, they likewise extend its analogies; and, though conversant with the minute forms of things, they have for their ultimate end the great and magnificent objects of nature." (Sir Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher", 1830)

"No occupation is more worthy of an intelligent and enlightened mind, than the study of Nature and natural objects; and whether we labour to investigate the structure and function of the human system, whether we direct our attention to the classification and habits of the animal kingdom, or prosecute our researches in the more pleasing and varied field of vegetable life, we shall constantly find some new object to attract our attention, some fresh beauties to excite our imagination, and some previously undiscovered source of gratification and delight." (Sir Joseph Paxton, "A Practical Treatise on the Cultivation of the Dahlia", 1838)

"But a thousand unconnected observations have no more value, as a demonstrative proof, than a single one. If we do not succeed in discovering causes by our researches, we have no right to create them by the imagination; we must not allow mere fancy to proceed beyond the bounds of our knowledge."(Justus von Liebig, "The Lancet", 1844)

"The nose of a mob is its imagination. By this, at any time, it can be quietly led." (Edgar A Poe, "The Works of Edgar Allan Poe", 1849)

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On Imagination (1900-1924)

"This is the greatest degree of impoverishment; the [mental] image, deprived little by little of its own characteristics, is nothing more than a shadow. […] Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others." (Théodule-Armand Ribot, "Essay on the Creative Imagination", 1900)

"This means that it is not a dead thing; it is not at all like a photographic plate with which one may reproduce copies indefinitely. Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others." (Théodule-Armand Ribot, "Essay on the Creative Imagination" , 1900)

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"Imagination is as vital to any advance in science as learning and precision are essential for starting points." (Percival Lowell, "The Solar System", 1903)

"Nature talks in symbols; he who lacks imagination cannot understand her." (Abraham Miller, "Unmoral Maxims", 1906)

"Mathematics makes constant demands upon the imagination, calls for picturing in space (of one, two, three dimensions), and no considerable success can be attained without a growing ability to imagine all the various possibilities of a given case, and to make them defile before the mind's eye." (Jacob W A Young, "The Teaching of Mathematics", 1907)

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

"The beautiful has its place in mathematics as elsewhere. The prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put, but we should be poor indeed without the literature of imagination. Mathematics too has its triumphs of the Creative imagination, its beautiful theorems, its proofs and processes whose perfection of form has made them classic. He must be a 'practical' man who can see no poetry in mathematics." (Wiliam F White, "A Scrap-book of Elementary Mathematics: Notes, Recreations, Essays", 1908)

"No system would have ever been framed if people had been simply interested in knowing what is true, whatever it may be. What produces systems is the interest in maintaining against all comers that some favourite or inherited idea of ours is sufficient and right. A system may contain an account of many things which, in detail, are true enough; but as a system, covering infinite possibilities that neither our experience nor our logic can prejudge, it must be a work of imagination and a piece of human soliloquy: It may be expressive of human experience, it may be poetical; but how should anyone who really coveted truth suppose that it was true?" (George Santayana, "The Genteel Tradition in American Philosophy", 1911)

"Only in men’s imagination does every truth find an effective and undeniable existence." (Joseph Conrad, "Some Reminiscences", 1912)

"What is the imagination? Only an arm or weapon of the interior energy; only the precursor of the reason." (Ralph W Emerson, "Miscellanies, Natural history of intellect", 1912)

"The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s ‘blank form of a universe’ without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition." (Lawrence J Henderson, "The Order of Nature: An Essay", 1917)

"[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination." (James B Shaw, "Lectures on the Philosophy of Mathematics", 1918)

"Nature uses human imagination to lift her work of creation to even higher levels." (Luigi Pirandello, "Six Characters in Search of an Author", 1921)

"The story of scientific discovery has its own epic unity - a unity of purpose and endeavour - the single torch passing from hand to hand through the centuries; and the great moments of science when, after long labour, the pioneers saw their accumulated facts falling into a significant order - sometimes in the form of a law that revolutionised the whole world of thought - have an intense human interest, and belong essentially to the creative imagination of poetry." (Alfred Noyes, "Watchers of the Sky", 1922)

George Santayana - Collected Quotes

"Symmetry is evidently a kind of unity in variety, where a whole is determined by the rhythmic repetition of similar." (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)

"The scientific value of truth is not, however, ultimate or absolute. It rests partly on practical, partly on aesthetic interests. As our ideas are gradually brought into conformity with the facts by the painful process of selection, - for intuition runs equally into truth and into error, and can settle nothing if not controlled by experience, - we gain vastly in our command over our environment. This is the fundamental value of natural science." (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)

"No system would have ever been framed if people had been simply interested in knowing what is true, whatever it may be. What produces systems is the interest in maintaining against all comers that some favourite or inherited idea of ours is sufficient and right. A system may contain an account of many things which, in detail, are true enough; but as a system, covering infinite possibilities that neither our experience nor our logic can prejudge, it must be a work of imagination and a piece of human soliloquy: It may be expressive of human experience, it may be poetical; but how should anyone who really coveted truth suppose that it was true?" (George Santayana, "The Genteel Tradition in American Philosophy", 1911)

"If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics." (George Santayana, Some Turns of Thought in Modern Philosophy: Five Essays, 1933)

"[…] mathematics is like music, freely exploring the possibilities of form. And yet, notoriously, mathematics holds true of things; hugs and permeates them far more closely than does confused and inconstant human perception; so that the dream of many exasperated critics of human error has been to assimilate all science to mathematics, so as to make knowledge safe by making it, as Locke wished, direct perception of the relations between ideas […]" (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

 "Theory helps us to bear our ignorance of facts." (George Santayana)

Complexity vs Mathematics II

"[Mathematics] guides our minds in an orderly way, and furnishes us simple and rational principles by means of which ambiguities are clarified, disorder is converted into order, and complexities are analyzed into their component parts." (Johann B Mencken, "The Charlatanry of the Learned", 1715)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

"The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines but of its methods. Mathematics will ever remain the past perfect type of the deductive method in general; and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex." (John S Mill, "System of Logic", 1843)

"It is certainly true that all physical phenomena are subject to strictly mathematical conditions, and mathematical processes are unassailable in themselves. The trouble arises from the data employed. Most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it. So that experiment is rather the criterion of mathematical conclusions and must lead the way." (Amos E Dolbear, "Matter, Ether, Motion", 1894)

"Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex [...]" (William F White, "A Scrap-book of Elementary Mathematics", 1908)

"A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to  applications to other departments of science." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of  mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought." (Jules Henri Poincaré, "The Future of Mathematics", Monist Vol. 20, 1910)

"The mathematical laws presuppose a very complex elaboration. They are not known exclusively either a priori or a posteriori, but are a creation of the mind; and this creation is not an arbitrary one, but, owing to the mind’s resources, takes place with reference to experience and in view of it. Sometimes the mind starts with intuitions which it freely creates; sometimes, by a process of elimination, it gathers up the axioms it regards as most suitable for producing a harmonious development, one that is both simple and fertile. The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)

"No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect." (Arthur C Clarke, "Profiles of the Future: An Inquiry into the Limits of the Possible", 1973)

"Economists are all too often preoccupied with petty mathematical problems of interest only to themselves. This obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in." (Thomas Piketty, Capital in the Twenty-First Century, 2013)

On Entropy (From Fiction to Science-Fiction)

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

"No structure, even an artificial one, enjoys the process of entropy. It is the ultimate fate of everything, and everything resists it." (Philip K Dick, "Galactic Pot-Healer", 1969)

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

"In the wastes of nonbeing it is born, flickers out, is born again and holds together, swells and spreads. In lifelessness it lives, against the gray tide of entropy it strives, improbably persists, gathering itself into ever richer complexities until it grows as a swelling wave. (James Tiptree Jr., "SheWaits for All Men Born", 1976)

"Her dance spoke of nothing more and nothing less than the tragedy of being alive, and being human. It spoke, most eloquently, of pain. It spoke, most knowingly, of despair. It spoke of the cruel humor of limitless ambition yoked to limited ability, of eternal hope invested in an ephemeral lifetime, of the driving need to try and create an inexorably predetermined future. It spoke of fear, and of hunger, and, most clearly, of the basic loneliness and alienation of the human animal. It described the universe through the eyes of man: a hostile environment, the embodiment of entropy, into which we are all thrown alone, forbidden by our nature to touch another mind save secondhand, by proxy. It spoke of the blind perversity which forces man to strive hugely for a peace which, once attained, becomes boredom. And it spoke of folly, of the terrible paradox by which man is simultaneously capable of reason and unreason, forever unable to cooperate even with himself." Spider Robinson and Jeanne Robinson, "Stardance", 1977)

"We see the universe as it is, Father Damien, and these naked truths are cruel ones. We who believe in life, and treasure it, will die. Afterward there will be nothing, eternal emptiness, blackness, nonexistence. In our living there has been no purpose, no poetry, no meaning. Nor do our deaths possess these qualities. When we are gone, the universe will not long remember us, and shortly it will be as if we had never lived at all. Our worlds and our universe will not long outlive us. Ultimately entropy will consume all, and our puny efforts cannot stay that awful end." (George R R Martin, "The Way of Cross and Dragon", 1979)

"But no longer were they always obedient to the mandates of their creators; like all material things, they were not immune to the corruptions of Time and its patient, unsleeping servant, Entropy." (Arthur C Clark, "3001: The Final Odyssey", 1997)

"Out of twinkling stardust all came, into dark matter all will fall. Death mocks us as we laugh defiance at entropy, yet ignorance birthed mortals sail forth upon time’s cruel sea." (Peter F Hamilton, "The Temporal Void", 2008)

"Yet, in the end, entropy will always emerge victorious, snuffing out the very last glimmer of heat and light. After that there is only darkness. When that state is reached even eternity will cease to exist, for one moment will be like every other and nothingness will claim the universe." (Peter F Hamilton, "The Temporal Void", 2008)

"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)

"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)

"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)

"Entropy is just a fancy way of saying: things fall apart." (Dan Brown, "Origin", 2017)

Mental Models L

"[…] the mind orders nothing by its own motions, but lies merely receptive under the impressions of bodies, reflecting empty images in a mirror in place of reality." (Anicius Manlius Severinus Boethius, "The Consolation of Philosophy", cca. 524)

"The likeness of a visible thing is that in virtue of which sight sees. And the likeness of an intellectively cognized thing, an intelligible species, is the form in virtue of which intellect cognizes. […] That which is intellectively cognized first is the thing of which the intelligible species is a likeness." (Thomas Aquinas, "Quaestiones disputatae de veritate", cca. 1256-1259)

"One who doesn't perceive the essence and quiddity of a thing, but only its image, can't know the thing. For one who has seen only a picture of Hercules doesn't know Hercules. A human being, however, perceives nothing of a thing, except only its image, that is, a species received through the senses, which is an image of the thing and not the thing itself. For not the stone but a species of the stone is in the soul." (Thomas Aquinas, "Summa Theologiae", cca. 1266-1273)

"[…] one ought to say that one may perceive the image of a thing in two ways. In one way, as the object of cognition. In this way it is true that one perceiving only the thing's image does not cognize the thing; for example, someone seeing the image of Hercules painted on a wall does not thereby either see or cognize Hercules. In another way, as the basis [ratio] of cognizing, and in this way the claim is not true. For through only a species perceived of a thing the thing is truly cognized - as a stone is truly seen through its sensible species alone, received in the eye, and is truly intellectively cognized through its intelligible species alone, received in intellect." (Thomas Aquinas, "Summa Theologiae", cca. 1266-1273)

"Phantasms don't have the same manner of existing that the human intellect has [...] and so they cannot through their own power make an impression on the possible intellect. But through the power of the agent intellect, a kind of likeness results in the possible intellect as a result of agent intellect's turning toward the phantasms […] And this is how intelligible species are said to be abstracted from phantasms. It's not that some form that is numerically the same is first in phantasms and then produced in the possible intellect."(Thomas Aquinas, "Summa Theologiae", cca. 1266-1273)

"The attention will tend toward the species either in such a way that it would not pass beyond so as to attend to the object, or in such a way that it would pass beyond. If in the first way, then the thing will not be seen in itself but only its image will be seen as if it were the thing itself." (Peter J Olivi, "Quaestiones in secundum librum Sententiarum", cca. 1280- 1282)

"No prior assimilation through a species is required before an act of intellectively cognizing. Rather, the assimilation suffices that comes about through the act of intellectively cognizing, which is [itself] a likeness of the thing cognized." (William Ockham," Expositio in librum Perihermenias", cca. 1321-1324)

"Such an image or fictum was postulated for no other reason than to supposit for a thing in such a way that both a proposition might be composed out of it and it might be common to things. For these are denied of things." (William Ockham, "Expositio in librum Perihermenias", cca, 1321-1324)

"The thing represented needs to be cognized in advance - otherwise the representative would never lead to a cognition of the thing represented as to something similar." (William Ockham, "Expositio in librum Perihermenias", cca. 1321-1324)

"Within image theory, it is suggested that important components of decision-making processes are the different 'images' that a person may use to evaluate choice options. Images may represent a person's principles, goals, or plans. Decision options may then match or not match these images and be adopted, rejected, considered further, depending on circumstances." (Deborah J Terry & Michael A Hogg, "Attitudes, Behavior, and Social Context: The Role of Norms and Group Membership", 1999) 

Mental Models XLVII (Limitations VI)

"Every presentation of philosophy, whether oral or written, is to be taken and can only be taken in the sense of a means. Every system is only an expression or image of reason, and hence only an object of reason, an object which reason - a living power that procreates itself in new thinking beings - distinguishes from itself and posits as an object of criticism. Every system that is not recognized and appropriated as just a means, limits and warps the mind for it sets up the indirect and formal thought in the place of the direct, original and material thought." (Ludwig A Feuerbach, "Towards a Critique of Hegel's Philosophy", 1839) 

"[…] we can only study Nature through our senses - that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (William C Dampier, "The Recent Development of Physical Science", 1904)

"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931) 

"The model of the natural world we build in our minds by such a process will forever be inadequate, just a little cathedral in the mountains. Still it is better than no model at all." (Timothy Ferris, "The Red Limit: The Search for the Edge of the Universe", 1977)

"A person who thinks by images becomes less and less capable of thinking by reasoning, and vice versa. The intellectual process based on images is contradictory to the intellectual process of reasoning that is related to the word. There are two different ways of dealing with an object. They involve not only different approaches, but even more important, opposing mental attitudes. This is not a matter of complementary processes, such as analysis and synthesis or logic and dialectic. These processes lack any qualitative common denominator." (Jacques Ellul, "The Humiliation of the Word", 1981) 

"Whenever I have talked about mental models, audiences have readily grasped that a layout of concrete objects can be represented by an internal spatial array, that a syllogism can be represented by a model of individuals and identities between them, and that a physical process can be represented by a three-dimensional dynamic model. Many people, however, have been puzzled by the representation of abstract discourse; they cannot understand how terms denoting abstract entities, properties or relations can be similarly encoded, and therefore they argue that these terms can have only 'verbal' or propositional representations." (Philip Johnson-Laird,"Mental Models: Towards a Cognitive Science of Language, Inference and Consciousness", 1983)

"Perhaps we all lose our sense of reality to the precise degree to which we are engrossed in our own work, and perhaps that is why we see in the increasing complexity of our mental constructs a means for greater understanding, even while intuitively we know that we shall never be able to fathom the imponderables that govern our course through life." (Winfried G Sebald, "The Rings of Saturn", 1995) 

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

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

Marcus Aurelius - Collected Quotes

"Always take the short cut; and that is the rational one. Therefore say and do everything according to soundest reason." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"And in the case of superior things like stars, we discover a kind of unity in separation. The higher we rise on the scale of being, the easier it is to discern a connection even among things separated by vast distances." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"Constantly regard the universe as one living being, having one substance and one soul; and observe how all things have reference to one perception, the perception of this one living being; and how all things act with one movement; and how all things are the cooperating causes of all things which exist; observe too the continuous spinning of the thread and the contexture of the web." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"Either all things proceed from one intelligent source and come together as in one body, and the part ought not to find fault with what is done for the benefit of the whole; or there are only atoms, and nothing else than a mixture and dispersion. Why, then, art thou disturbed? Say to this ruling faculty, Art thou dead, art thou corrupted, art thou playing the hypocrite, art thou become a beast, dost thou herd and feed with the rest?" (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"Look to the essence of a thing, whether it be a point of doctrine, of practice, or of interpretation." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

"In the series of things those which follow are always aptly fitted to those which have gone before; for this series is not like a mere enumeration of disjointed things, which has only a necessary sequence, but it is a rational connection: and as all existing things are arranged together harmoniously, so the things which come into existence exhibit no mere succession, but a certain wonderful relationship." (Marcus Aurelius, "Meditations", cca. 180 AD)

 

"Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life." (Marcus Aurelius, Meditations, cca. 121–180)

 

"Observe the movements of the stars as if you were running their courses with them, and let your mind constantly dwell on the changes of the elements into each other." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"Time is a sort of river of passing events, and strong is its current; no sooner is a thing brought to sight than it is swept by and another takes its place, and this too will be swept away.“ (Marcus Aurelius, "Meditations", cca. 121–180 AD)

"The other reason is that what happens to the individual is a cause of well-being in what directs the world - of its well-being, its fulfillment, or its very existence, even. Because the whole is damaged if you cut away anything - anything at all - from its continuity and its coherence. Not only its parts, but its purposes. And that's what you're doing when you complain: hacking and destroying." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"The universal intelligence puts itself in motion for every separate effect [...] or it puts itself in motion once, and everything else comes by way of a sequence in a manner; or individual elements are the origin of all things. In a word, if there is a god, all is well; and if chance rules, do not thou be governed by it." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"The universe is change; our life is what our thoughts make it." (Marcus Aurelius)

On Chance (BC)

"No human being will ever know the Truth, for even if they happen to say it by chance, they would not even known they had done so." (Xenophanes, 5th century BC)

"The art of war teaches us to rely not on the likelihood of the enemy's not coming, but on our own readiness to receive him; not on the chance of his not attacking, but rather on the fact that we have made our position unassailable." (Sun Tzu, "The Art of War", cca. 5th century BC)

"[...] the freaks of chance are not determinable by calculation." (Thucydides, History of the Peloponnesian War, cca. 5th century)

"Everything existing in the universe is the fruit of chance." (Democritus, 4th century BC)

"All human actions have one or more of these seven causes: chance, nature, compulsions, habit, reason, passion, desire." (Aristotle, 4th century BC)

"Chance seldom interferes with the wise man; his greatest and highest interests have been, are, and will be, directed by reason throughout his whole life." (Epicurus, cca. 4th century)

"Thus all the action of men must necessarily be referred to seven causes: chance, nature, compulsion, habit, reason, anger, and desire." (Aristotle, "The Art of Rhetoric", 4th century BC)

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

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


"How often things occur by mere chance which we dared not even hope for." (Publius Terentius Afer, "Phormio", cca. 161 BC)

"Suam habet fortuna rationem.’
"Chance has its reasons." (Gaius Petronius, "Satryicon liber" ["The Book of Satyrlike Adventures"], 1st century BC)

"Valor is of no service, chance rules all, and the bravest often fall by the hands of cowards." (Cornelius Tacitus, cca. 69-100 AD)

"But things that happen by chance cannot be certain." (Marcus T Cicero, "De Divinatione", cca 44 BC)

"Chance joins with force to guide the steel." (Virgil, "Aeneid", cca. 29–19 BC)

Mental Models XLI

"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", [discourse] 1769) 

"Taste is the intermediate faculty which connects the active with the passive powers of our nature, the intellect with the senses; and its appointed function is to elevate the images of the latter, while it realizes the ideas of the former."(Samuel T Coleridge, "On the Principles of Genial Criticism", 1814)

"The imagination […] that reconciling and mediatory power, which incorporating the reason in images of the sense and organizing (as it were) the flux of the senses by the permanence and self-circling energies of the reason, gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors." (Samuel T Coleridge, "The Statesman's Manual", 1816) 

"Theories usually result from the precipitate reasoning of an impatient mind which would like to be rid of phenomena and replaces them with images, concepts, indeed often with mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

"Word and picture are correlatives which are continually in quest of each other, as is sufficiently evident in the case of metaphors and similes. So from all time what was said or sung inwardly to the ear had to be presented equally to the eye. And so in childish days we see word and picture in continual balance; in the book of the law and in the way of salvation, in the Bible and in the spelling-book. When something was spoken which could not be pictured, and something pictured which could not be spoken, all went well; but mistakes were often made, and a word was used instead of a picture; and thence arose those monsters of symbolical mysticism, which are doubly an evil." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

"This language controls by reducing the linguistic forms and symbols of reflection, abstraction, development, contradiction; by substituting images for concepts. It denies or absorbs the transcendent vocabulary; it does not search for but establishes and imposes truth and falsehood." (Herbert Marcuse, "One-Dimensional Man", 1964) 

"Imagination is the outreaching of mind […] the bombardment of the conscious mind with ideas, impulses, images and every sort of psychic phenomena welling up from the preconscious. It is the capacity to ‘dream dreams and see visions’" (Rollo May, "The Courage to Create", 1975) 

"Myth is the system of basic metaphors, images, and stories that in-forms the perceptions, memories, and aspirations of a people; provides the rationale for its institutions, rituals and power structure; and gives a map of the purpose and stages of life." (Sam Keen, "The Passionate Life", 1983) 

"We must begin by distinguishing between visual mental imagery and visual perception: Visual perception occurs while a stimulus is being viewed, and includes functions such as visual recognition (i. e., registering that a stimulus is familiar) and identification (i. e., recalling the name, context, or other information associated with the object). Two types of mechanisms are used in visual perception: ‘bottom-up’ mechanisms are driven by the input from the eyes; in contrast, ‘top-down’ mechanisms make use of stored information (such as knowledge, belief, expectations, and goals). Visual mental imagery is a set of representations that gives rise to the experience of viewing a stimulus in the absence of appropriate sensory input. In this case, information in memory underlies the internal events that produce the experience. Unlike afterimages, mental images are relatively prolonged." (Stephen M Kosslyn, "Mental images and the brain", Cognitive Neuropsychology 22, 2005) 

Mental Models XXXV

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

"It may sound quite strange, but for me, as for other scientists on whom these kinds of imaginative images have a greater effect than other poems do, no science is at its very heart more closely related to poetry, perhaps, than is chemistry." (Just Liebig, 1854)

"But I thoroughly believe myself, and hope to prove to you, that science is full of beautiful pictures, of real poetry, and of wonder-working fairies; and what is more […] though they themselves will always remain invisible, yet you will see their wonderful power at work everywhere around you. […] There is only one gift we must have before we can learn to know them - we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind, of that which is, though it is invisible to us." (Arabella Buckley, Fairyland, 1879)

"Ask your imagination if it will accept a vibrating multiple proportion - a numerical ratio in a state of oscillation? I do not think it will. You cannot crown the edifice with this abstraction. The scientific imagination, which is here authoritative, demands, as the origin and cause of a series of ether-waves, a particle of vibrating matter quite as definite, though it may be excessively minute, as that which gives origin to a musical sound. Such a particle we name an atom or a molecule. I think the intellect, when focused so as to give definition without penumbral haze, is sure to realize this image at the last." (John Tyndall, "Fragments of Science", 1892)

"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)

"Many people believe that reasoning, and therefore science, is a different activity from imagining. But this is a fallacy […] Reasoning is constructed with movable images just as certainly as poetry is." (Jacob Bronowski, "Visionary Eye", 1978)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"Mathematicians have always needed to ‘see’ the complex concepts they work with in order to reason with them effectively. In the past, they conjured up mental images as best they could, but the wonders of computer graphics provide them with far more detailed pictures to think with." (Richard Palais and Luc Bernard, "2006 Visualization Project", 2006)

"Time is to eternity as an image is to its exemplar, and those things which are temporal bear a resemblance to those things which are eternal." (Nicholas of Cusa)

"You know how the divine Simplicity enfolds all things. Mind is the image of this enfolding Simplicity. If, then, you called this divine Simplicity infinite Mind, it will be the exemplar of our mind. If you called the divine mind the totality of the truth of things, you will call our mind the totality of the assimilation of things, so that it may be a totality of ideas. In the divine Mind conception is the production of things; in our mind conception is the knowledge of things. If the divine Mind is absolute Being, then its conception is the creation of beings; and conception in the human mind is the assimilation of beings." (Nicholas of Cusa)

On Abstraction (1960-1969)

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

"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)

"Scientists, it should already be clear, never learn concepts, laws, and theories in the abstract and by themselves. Instead, these intellectual tools are from the start encountered in a historically and pedagogically prior unit that displays them with and through their applications." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractions, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range for its applications." (Aleksandr D Aleksandrov, 1963)

"A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense..." (Hermann Bondi. "Relativity and Common Sense", 1964)

"If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details." (John G Kemeny, "Random Essays on Mathematics, Education, and Computers", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Learning is any change in a system that produces a more or less permanent change in its capacity for adapting to its environment. Understanding systems, especially systems capable of understanding problems in new task domains, are learning systems." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The more we are willing to abstract from the detail of a set of phenomena, the easier it becomes to simulate the phenomena. Moreover we do not have to know, or guess at, all the internal structure of the system but only that part of it that is crucial to the abstraction." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

On Simplicity IV (Complexity vs Simplicity II)

"The first obligation of Simplicity is that of using the simplest means to secure the fullest effect. But although the mind instinctively rejects all needless complexity, we shall greatly err if we fail to recognise the fact, that what the mind recoils from is not the complexity, but the needlessness." (George H Lewes, "The Principles of Success in Literature", 1865)

"If we study the history of science we see happen two inverse phenomena […] Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities. […] No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations." (Henri Poincaré, "Science and Hypothesis", 1901)

"Nor does complexity deny the valid simplification which is part of the process of analysis, and even a method of achieving complex architecture itself." (Robert Venturi, "Complexity and Contradiction in Architecture", 1966)

"The systems view is the emerging contemporary view of organized complexity, one step beyond the Newtonian view of organized simplicity, and two steps beyond the classical world views of divinely ordered or imaginatively envisaged complexity."  (Ervin László, "Introduction to Systems Philosophy", 1972)

"Do not be alarmed by simplification, complexity is often a device for claiming sophistication, or for evading simple truths." John K Galbraith, "The Age of Uncertainty", 1977)

"Simplicity is a great virtue but it requires hard work to achieve it and education to appreciate it. And to make matters worse: complexity sells better." (Edsger W Dijkstra, "On the nature of Computing Science", 1984)

"Somehow the breathless world that we witness seems far removed from the timeless laws of Nature which govern the elementary particles and forces of Nature. The reason is clear. We do not observe the laws of Nature: we observe their outcomes. Since these laws find their most efficient representation as mathematical equations, we might say that we see only the solutions of those equations not the equations themselves. This is the secret which reconciles the complexity observed in Nature with the advertised simplicity of her laws." (John D Barrow, "New Theories of Everything", 1991)

"People who pride themselves on their ‘complexity’ and deride others for being "simplistic" should realize that the truth is often not very complicated. What gets complex is evading the truth." (Thomas Sowell, "Barbarians inside the Gates and Other Controversial Essays", 1999)

"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît Mandelbrot, "The Fractalist", 2012)

"Decentralized systems are the quintessential patrons of simplicity. They allow complexity to rise to a level at which it is sustainable, and no higher." (Lawrence K. Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

On Metaphors VII

"The greatest thing by far is to be a master of metaphor. It is the one thing that cannot be learnt from others; it is also a sign of genius, since a good metaphor implies an intuitive perception of the similarity in dissimilars." (Aristotle, "Poetics", cca. 335 BC)

"[Metaphor] is a loose word, at best, and we must beware of attributing to it stricter rules of usage than are usually found in practice." (Max Black, “Models and Metaphors”, 1962)

"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)

"The very nature of science is such that scientists need the metaphor as a bridge between old and new theories." (Earl R MacCormac, "Metaphor and Myth in Science and Religion", 1976)

“[…] the use of analogies, particularly with metaphor, adds richness and dimension to arguments and descriptions not possible with ordinary discourse or with propositional reasoning.” (Jeanette M Gallagher, “The Future of Formal Thought Research: The Study of Analogy and Metaphor”, 1978)

"The use of metaphor is one of many devices available to the scientific community to accomplish the task of accommodation of language to the causal structure of the world." (Richard Boyd, "Metaphor and theory change: what is ‘metaphor’ a metaphor for?", 1979)

"Metaphor is for most people a device of the poetic imagination and the rhetorical flourish - a matter of extraordinary rather than ordinary language. Moreover, metaphor is typieully viewed as characteristic of language alone, a matter of words rather than thought or action. For this reason, most people think they can get along perfectly well without metaphor. We have found, on the contrary, that metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature." (George Lakoff & Mark Johnson, "Metaphors we Live by", 1980)

"Metaphor plays an essential role in establishing a link between scientific language and the world. Those links are not, however, given once and for all. Theory change, in particular, is accompanied by a change in some of the relevant metaphors and in the corresponding parts of the network of similarities through which terms attach to nature." (Thomas S Kuhn, "Metaphor in science", 1993)

“Metaphorizing is a  manner of thinking, not a  property of thinking. It is a  capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects.” (Ivan Mladenov, “Conceptualizing Metaphors: On Charles Peirce’s marginalia”, 2006)

On Metaphors V

"Metaphor consists in giving the thing a name that belongs to something else; the transference being either from genus to species, or from species to genus, or from species to species, or on grounds of analogy." (Aristotle, "Poetics", cca. 335 BC)
 
"Mathematical research can lend its organisational characteristics to poetry, whereby disjointed metaphors take on a universal sense. Similarly, the axiomatic foundations of group theory can be assimilated into a larger moral concept of a unified universe. Without this, mathematics would be a laborious Barbary." (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts. " (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"[…] theoretical science is essentially disciplined exploitation of metaphor." (Anatol Rapoport, "Operational Philosophy", 1953)

"Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities." (Erwin Schrödinger, "Mind and Matter: the Tarner Lectures", 1956)

"The symbol and the metaphor are as necessary to science as to poetry." (Jacob Bronowski, "Science and Human Values", 1956) 

"The model is only a suggestive metaphor, a fiction about the messy and unwieldy observations of the real world. In order for it to be persuasive, to convey a sense of credibility, it is important that it not be too complicated and that the assumptions that are made be clearly in evidence. In short, the model must be simple, transparent, and verifiable." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)
 
"People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like something else (association); and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. Personally, I put a lot of effort into ‘listening’ to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations." (William P Thurston, "On proof and progress in mathematics", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"If we are to have meaningful, connected experiences; ones that we can comprehend and reason about; we must be able to discern patterns to our actions, perceptions, and conceptions. Underlying our vast network of interrelated literal meanings (all of those words about objects and actions) are those imaginative structures of understanding such as schema and metaphor, such as the mental imagery that allows us to extrapolate a path, or zoom in on one part of the whole, or zoom out until the trees merge into a forest." (William H Calvin, "The Cerebral Code", 1996)

"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)

On Concepts VI

"The analysis of concepts is for the understanding nothing more than what the magnifying glass is for sight." (Moses Mendelssohn, 1763)

"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)

"The principle of contradiction establishes merely the agreement of concepts, but does not itself produce concepts." (Arthur Schopenhauer, "The World as Will and Representation", 1819)

"To repeat abstractly, universally, and distinctly in concepts the whole inner nature of the world, and thus to deposit it as a reflected image in permanent concepts always ready for the faculty of reason, this and nothing else is philosophy." (Arthur Schopenhauer, "The World as Will and Representation", 1819)

"The relation of word to thought, and the creation of new concepts is a complex, delicate and enigmatic process unfolding in our soul." (Lev N Tolstoy, "Pedagogical Writings", 1903)

"The reason why new concepts in any branch of science are hard to grasp is always the same; contemporary scientists try to picture the new concept in terms of ideas which existed before." (Freeman J Dyson, "Innovation in Physics", Scientific American, 1958)

"This language controls by reducing the linguistic forms and symbols of reflection, abstraction, development, contradiction; by substituting images for concepts. It denies or absorbs the transcendent vocabulary; it does not search for but establishes and imposes truth and falsehood." (Herbert Marcuse, "One-Dimensional Man", 1964)

"A concept is a brick. It can be used to build a courthouse of reason. Or it can be thrown through the window." (Gilles Deleuze, "EPZ Thousand Plateaus", 2004)

“Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics.” (Johann F Herbart)

"Scientific understanding is an essential step to our finding a home for ourselves in the universe. Through understanding the universe, we become at home in it. In a certain sense we have made this universe out of human concepts and human discoveries. It ceases to be a lonely place, because we can to some extent actually navigate in it." (Isidor Isaac Rabi)

Mental Models XXII

"The impossibility of separating the nomenclature of a science from the science itself is owing to this, that every branch of physical science must consist of three things: the series of facts which are the objects of the science, the ideas which represent these facts, and the words by which these ideas are expressed. Like three impressions of the same seal, the word ought to produce the idea, and the idea to be a picture of the fact." (Antoine L Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

"Therefore, the great business of the scientific teacher is, to imprint the fundamental, irrefragable facts of his science, not only by words upon the mind, but by sensible impressions upon the eye, and ear, and touch of the student, in so complete a manner, that every term used, or law enunciated, should afterwards call up vivid images of the particular structural, or other, facts which furnished the demonstration of the law, or the illustration of the term." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870) 

"The sciences bring into play the imagination, the building of images in which the reality, of the past is blended with the ideals for the future, and from the picture there springs the prescience of genius." (William J Mayo, "Contributions of Pure Science to Progressive Medicine", The Journal of the American Medical Association Vol. 84 (20), 1925)

"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." (Werner K Heisenberg, "The Physical Principles of the Quantum Theory", 1930)

"A picture of reality drawn in a few sharp lines cannot be expected to be adequate to the variety of all its shades. Yet even so the draftsman must have the courage to draw the lines firm." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1949)

"Man tries to make for himself in the fashion that suits him best a simplified and intelligible picture of the world; he then tries to some extent to substitute this cosmos of his for the world of experience, and thus to overcome it. This is what the painter, the poet, the speculative philosopher, and the natural scientist do, each in his own way." (Albert Einstein, "The World as I See It", 1949)

"Thus a word or an image is symbolic when it implies something more than its obvious and immediate meaning. It has a wider ‘unconscious’ aspect that is never precisely defined or fully explained. […] As the mind explores the symbols it is led to ideas that lie beyond the grasp of reason." (Carl G Jung, "Man and His Symbols", 1964)

"It is hard for us today to assimilate all the new ideas that are being suggested in response to the new information we have. We must remember that our picture of the universe is based not only on our scientific knowledge but also on our culture and our philosophy. What new discoveries lie ahead no one can say. There may well be civilizations in other parts of our galaxy or in other galaxies that have already accomplished much of what lies ahead for mankind. Others may just be beginning. The universe clearly presents an unending challenge." (Necia H Apfel & J Allen Hynek, "Architecture of the Universe", 1979)

“Simple observation generally gets us nowhere. It is the creative imagination that increases our understanding by finding connections between apparently unrelated phenomena, and forming logical, consistent theories to explain them. And if a theory turns out to be wrong, as many do, all is not lost. The struggle to create an imaginative, correct picture of reality frequently tells us where to go next, even when science has temporarily followed the wrong path.” (Richard Morris, “The Universe, the Eleventh Dimension, and Everything: What We Know and How We Know It”, 1999)

Mental Models XXI

"You may read any quantity of books, and you may be almost as ignorant as you were at starting, if you don’t have, at the back of your minds, the change for words in definite images which can only be acquired through the operation of your observing faculties on the phenomena of nature." (Thomas H Huxley, "Science and Education", 1877)

"[…] we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind of that which is, though it is invisible to us."  (Arabella B Buckley, "The Fairy-Land of Science", 1899)

"We wish to obtain a representation of phenomena and form an image of them in our minds. Till now, we have always attempted to form these images by means of the ordinary notions of time and space. These notions are perhaps innate; in any case they have been developed by our daily observations. For me, these notions are clear, and I confess that I am unable to gain any idea of physics without them. […] I would like to retain this ideal of other days and describe everything that occurs in this world in terms of clear pictures." (Hendrik A Lorentz, [Fifth Solvay Conference] 1927)

"Just as the eye sees details that are not there if they fit in with the sense of the picture, or overlooks them if they make no sense, so also very little inherent certainty will suffice to secure the highest scientific value to an alleged fact, if only it fits in with a great scientific generalization, while the most stubborn facts will be set aside if there is no place for them in the established framework of science." (Michael Polanyi, "Personal Knowledge", 1962)

"If we are to have meaningful, connected experiences; ones that we can comprehend and reason about; we must be able to discern patterns to our actions, perceptions, and conceptions. Underlying our vast network of interrelated literal meanings (all of those words about objects and actions) are those imaginative structures of understanding such as schema and metaphor, such as the mental imagery that allows us to extrapolate a path, or zoom in on one part of the whole, or zoom out until the trees merge into a forest." (William H Calvin, "The Cerebral Code", 1996)

"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)

“We start from vague pictures or ideas […] which we encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialog between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs” (E Brian Davies, “Science in the Looking Glass”, 2003)

“What is consciousness? Our brain simulates reality. So, our everyday experiences are a form of dreaming, which is to say, they are mental models, simulations, not the things they appear to be.” (Stephen LaBerge, “Losi in Lucidity”, 2014)

“A mathematician possesses a mental model of the mathematical entity she works on. This internal mental model is accessible to her direct observation and manipulation. At the same time, it is socially and culturally controlled, to conform to the mathematics community's collective model of the entity in question. The mathematician observes a property of her own internal model of that mathematical entity. Then she must find a recipe, a set of instructions, that enables other competent, qualified mathematicians to observe the corresponding property of their corresponding mental model. That recipe is the proof. It establishes that property of the mathematical entity.” (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)

“Men of broader intellect know that there is no clear distinction between the real and the unreal, that things appear as they seem only by virtue of the delicate physical and mental instruments through which we perceive them.” (Howard P Lovecraft)