19 June 2012

Knowledge Representation: On Truth (Quotes)

“As being is to become, so is truth to belief” (Plato, “Timaeus”, cca. 360 BC)

“The first duty of man is the seeking after and the investigation of truth.” (Marcus Tullius Cicero, “De Officiis”, [“On Duties”], cca. 44 BC)

“The exact kind of language we employ in philosophical analyses of abstract truth is one thing, and the language used in attempts to popularize the subject is another.” (Marcus Tullius Cicero, “De officiis” [“On Duties”], cca.44 BC)

“Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.” (Marcus Aurelius, "Meditations", cca. 2nd century)

“We are meant to take them [the words ‘increase and multiply’] in a figurative sense. […] It is only in the case of signs outwardly given that we find increase and multiplication in the sense that a single truth can be expressed by several different means […] that a single expression can be interpreted in several different ways.” (St. Augustine, “Confessions”, 397- 400)

"Truth is sought for itself, but the truths are immersed in uncertainties." (Abu Ali al-Hasan ibn al-Haytham [Alhazen], "Aporias against Ptolemy", 1025-1028)

“If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” (Roger Bacon, “Opus Majus” Book 1, 1267)

"Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience."(Roger Bacon, "Opus Majus", cca. 1267) 

“The truth of voice perishes with the sound; truth latent in the mind is hidden wisdom and invisible treasure; but the truth which illuminates books desires to manifest itself to every disciplinable sense. Let us consider how great a commodity of doctrine exists in books, - how easily, how secretly, how safely, they expose the nakedness of human ignorance without putting it to shame. These are the masters that instruct us without rods and ferules, without hard words and anger, without clothes or money. If you approach them, they are not asleep; if, investigating, you interrogate them, they conceal nothing; if you mistake them, they never grumble; if you are ignorant, they cannot laugh at you.” (Richard de Burry, “Philobiblon”, 1344)

"Man's mind is so formed that it is far more susceptible to falsehood than to truth." (Desiderius Erasmus, "Praise of Folly", 1509)

“[…] no pleasure is comparable to the standing upon the vantage ground of truth […]” (Sir Francis Bacon, “Essays”, 1597)

“The first and most ancient inquirers into truth were wont to throw their knowledge into aphorisms, or short, scattered, unmethodical sentences.” (Lord Bacon, “Novum Organum”, 1620)

"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)

“[…] thus each truth discovered was a rule available in the discovery of subsequent ones.” (René Descartes, “Discourse on Method”, 1637)

 “Every man is not a proper champion for truth, nor fit to take up the gauntlet in the cause of verity: many from the ignorance of these maxims, and an inconsiderate zeal for truth, have too rashly charged the troops of error, and remain as trophies unto the enemies of truth. A man may be in as just possession of truth as of a city, and yet be forced to surrender: ’tis therefore far better to enjoy her with peace than to hazard her on a battle: if therefore there rise any doubts in my way, I do forget them, or at least defer them, till my better settled judgment and more manly reason be able to resolve them.” (Sir Thomas Browne, ”Religio Medici”, 1643)

“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.” (René Descartes, “Principles of Philosophy”, 1644)

“Knowledge is made by oblivion, and to purchase a clear and warrantable body of truth, we must forget and part with much we know.” (Sir Thomas Browne, “Pseudodoxia Epidemica”, 1646)

“Men are apt to prefer a prosperous error before an afflicted truth.” (Jeremy Taylor, “The Rule and Exercises of Holy Living”, 1650)

 “All things being double-handed, and having the appearances both of truth and falsehood, where our affections have engaged us we attend only to the former.” (Joseph Glanvill, “Scepsis”, 1665)

“Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” (Blaise Pascal, “Pensées”, 1670)

“We see neither justice nor injustice which does not change its nature with change in climate. Three degrees of latitude reverse all jurisprudence: a meridian decides the truth.” (Blaise Pascal, “Pensées”, 1670)

“In logic, they teach that contraries laid together more evidently appear: it follows, then, that all controversy being permitted, falsehood will appear more false, and truth the more true; which must needs conduce much to the general confirmation of an implicit truth.” (John Milton, “True Religion, Heresy, Schism, Toleration, and what best means may be used against the Growth of Popery”, 1673)

“In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow.” (Baruch Spinoza, [letter to Hugo Boxel], 1674)

“Truth is always consistent with itself, and needs nothing to help it out; it is always near at hand, and sits upon our lips, and is ready to drop out before we are aware; whereas a lie is troublesome, and sets a man’s invention upon the rack, and one trick needs a great many more to make it good.” (John Tillotson, “Sermons”, 1682)

"And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Two things are identical if one can be substituted for the other without affecting the truth." (Gottfried W Leibniz, "Table de definitions", 1704)

“Not only the investigation of truth, but the communication of it also, is often practised in such a method as neither agrees precisely to synthetic or analytic.”  (Isaac Watts, “Logic, or The right use of reason, in the inquiry after truth”, 1725)

“He that would make a real progress in knowledge must dedicate his age as well as first fruits - the latter growth as well as the first-fruits - at the altar of truth.” (Bishop George Berkeley, “Siris”, 1744)

“If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty.” (Edmund Burke, “Essay on the Sublime and Beautiful”, 1756)

"There are truths which are not for all men, nor for all times." (Voltaire, [Letter to François-Joachim de Pierre] 1764)

"Ignorance is preferable to error; and he is less remote from the truth who believes nothing, than he who believes what is wrong." (Thomas Jefferson, "Notes on the State of Virginia", 1781)

"General abstract truth is the most precious of all blessings; without it, man is blind; it is the eye of reason." (Jean-Jacques Rousseau, "The Confessions of J. J. Rousseau", 1783)

“The discovery of truth by slow, progressive meditation is talent. Intuition of the truth, not preceded by perceptible meditation, is genius.” (Johann K Lavater, 1787) 

“Everything possible to be believed is an image of truth.” (William Blake, “The Marriage of Heaven and Hell”, 1790)

“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world.” (Immanuel Kant, “Critique of Judgment”, 1790)

"We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoine Lavoisier, "Elements of Chemistry", 1790)

"It is an acknowledged truth in philosophy that a just theory will always be confirmed by experiment." (Thomas R Malthus, "An Essay on The Principle of Population", 1798)
“Forgetting that the only eternal part for man to act is man, and that the only immutable greatness is truth.” (Alphonse Lamartine, “The History of the Restoration of Monarchy in France”, 1851)

"Accuracy of language is one of the bulwarks of truth." (Anna B Jameson, "A Commonplace Book of Thoughts, Memories, and Fancies", 1854)

"We must therefore discover some method of investigation which allows the mind at every step to lay hold of a clear physical conception, without being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis." (James C Maxwell, "On Faraday’s lines of force", 1855)

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

“The excellence of every art is its intensity, capable of making all disagreeables evaporate from their being in close relationship with beauty and truth.” (John Keats. [letter to George and Thomas Keats] 1817)

"[...] all knowledge, and especially the weightiest knowledge of the truth, to which only a brief triumph is allotted between the two long periods in which it is condemned as paradoxical or disparaged as trivial." (Arthur Schopenhauer, "The World as Will and Representation", 1819)

"We are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it." (Thomas Jefferson, [Letter to William Roscoe] 1820)

"Mathematics, like dialectics, is an instrument of the inner higher sense, while in practice it is an art like rhetoric. For both of these, nothing has value but form; content is immaterial. Whether mathematics is adding up pennies or guineas, whether rhetoric is defending truth or falsehood, makes no difference to either.” (Johann Wolfgang von Goethe, "Wilhelm Meisters Wanderjahre" ["Reflections in the Spirit of the Wanderers"], 1821)

“Facts are the mere dross of history. It is from the abstract truth which interpenetrates them, and lies latent among them, like gold in the ore, that the mass derives its whole value: and the precious particles are generally combined with the baser in such a manner that the separation is a task of the utmost difficulty.” (Thomas B Macaulay, “History”, 1828)

"Truth in itself is rarely sufficient to make men act. Hence the step is always long from cognition to volition, from knowledge to ability. The most powerful springs of action in men lie in his emotions." (Carl von Clausewitz, "On War", 1832)

“It is difficult to discriminate the voice of truth from amid the clamour raised by heated partisans.” (Friedrich Schiller, “Schillers Sammtliche Werke”, 1834)

“The most important and lasting truths are the most obvious ones. Nature cheats us with her mysteries, one after another, like a juggler with his tricks; but shews us her plain honest face, without our paying for it.” (William Hazlitt, “Characteristics: In the Manner of Rochefoucault's Maxims”, 1837)

“In truth, ideas and principles are independent of men; the application of them and their illustration is man's duty and merit.” (Edward Forbes, 1847)

“The peculiarity of the evidence of mathematical truths is that all the argument is on one side.” (John Stuart Mill, “On Liberty”, 1859)

[…] the besetting danger is not so much of embracing falsehood for truth, as of mistaking a part of the truth for the whole.” (John Stuart Mill, “Dissertations and Discussions: Political, Philosophical, and Historical”, 1864) 

"No departure from the truth of nature shall be discovered by the closest scrutiny." (Henry P Robinson, "Pictorial Effect in Photography", 1869)

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

“Pure truth cannot be assimilated by the crowd; it must be communicated by contagion.” (Henri-Frédéric Amiel, [journal entry] 1875)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)

“Convictions are more dangerous enemies of truth than lies.” (Friedrich Nietzsche, “Human, All Too Human: A book for Free Spirits”, 1878) 

“It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors.” (Thomas H Huxley, “The Progress of Science”, 1887)

“How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?” (Sir Arthur Conan Doyle, “The Sign of Four”, 1890)

“Accuracy of statement is one of the first elements of truth; inaccuracy is a near kin to falsehood.” (Tyron Edwards, “A Dictionary of Thoughts”, 1891)

"There is no short cut to truth, no way to gain a knowledge of the universe except through the gateway of scientific method." (Karl Pearson, “The Grammar of Science”, 1892)

"It is they who hold the secret of the mysterious property of the mind by which error ministers to truth, and truth slowly but irrevocably prevails. Theirs is the logic of discovery, the demonstration of the advance of knowledge and the development of ideas, which as the earthly wants and passions of men remain almost unchanged, are the charter of progress, and the vital spark in history." (Lord John Acton, "The Study of History", [lecture delivered at Cambridge] 1895)

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valery, "Introduction to the Method of Leonardo da Vinci", 1895)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notations, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. (William S Jevons, "The Principles of Science" 2nd Ed., 1900)

"The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe." (Karl Pearson, “The Grammar of Science”, 1900)

“Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty.” (Henri Poincaré, “Science and Hypothesis”, 1902) 

“Logic, then, is not necessarily an instrument for finding truth; on the contrary, truth is necessarily an instrument for using logic - for using it, that is, for the discovery of further truth and for the profit of humanity. Briefly, you can only find truth with logic if you have already found truth without it.” (Gilbert K Chesterton, Daily News, 1905)

"The forceps of our minds are clumsy forceps, and crush the truth a little in taking hold of it." (Herbert G Wells, "Scepticism of the Instrument: A Modern Utopia", 1905)

“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 truth of an idea is not a stagnant property inherent in it. Truth happens to an idea. It becomes true, is made true by events. Its verity is in fact an event, a process: the process namely of its verifying itself, its verification. Its validity is the process of its validation.” (William James, “Pragmatism: A New Name for Some Old Ways of Thinking”, 1907)

"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré," Science and Method", 1908)

"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned, these, and such like, are the surest groimds of its title of human regard, and would remain unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.” (James J Sylvester, "A Plea for the Mathematician", Nature, 1908)

“Truth is on a curve whose asymptote our spirit follows eternally.” (Léo Errera, “Recueil d'Œuvres de Léo Errera: Botanique Générale”, 1908) 

“Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts.” (Paul Carus, “Philosophy as a Science”, 1909)

“The pursuit of truth is chimerical. […] There is no permanent absolute unchangeable truth;  what we should pursue is the most convenient arrangement of our ideas.” (Samuel Butler, “Notebooks”, 1912)

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

“The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics.” (J Craig, “A Course of Study for the Preparation of Rural School Teachers”, 1912)

"The conception of logical laws must be the decisive factor in the treatment of logic, and that conception depends upon what we understand by the word ‘true’. It is generally admitted at the very beginning that logical laws must be rules of conduct to guide thought to truth […]" (Gottlob Frege," Grundgesetze", The Monist, 1915) 

“As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies. “ (Albert Einstein: “Relativity: The Special and General Theory”, 1916)

“It may be impossible for human intelligence to comprehend absolute truth, but it is possible to observe Nature with an unbiased mind and to bear truthful testimony of things seen.” (Sir Richard A Gregory, “Discovery, Or, The Spirit and Service of Science”, 1916)

“[…] 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)

“Ignorance may find a truth on its doorstep that erudition vainly seeks in the stars.” (George Iles, “Canadian Stories”, 1918)

"It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability." (John M Keynes, "A Treatise on Probability", 1921)

"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)

“We all know that Art is not truth. Art is a lie that makes us realize truth.” (Pablo Picasso, “The Arts”, 1923)

“Science does not aim at establishing immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved.” (Bertrand Russell, “The ABC of Relativity”, 1925)

"Progress in truth - truth of science and truth of religion - is mainly a progress in the framing of concepts, in discarding artificial abstractions or partial metaphors, and in evolving notions which strike more deeply into the root of reality." (Alfred N Whitehead, "Religion in the Making", 1926)

"The scientist is a lover of truth for the very love of truth itself, wherever it may lead." (Luther Burbank, "Why I Am An Infidel", 1926)

“If our so-called facts are changing shadows, they are shadows cast by the light of constant truth.” (Sir Arthur S Eddington, “Science and the Unseen World”, 1929) 

“Try to be conspicuously accurate in everything, pictures as well as text. Truth is not only stranger than fiction, it is more interesting.” (William R Hearst, “Letter of Instruction to Hearst Publishers”, 1929)

“Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man.” (Bertrand Russell, “The Scientific Outlook”, 1931)

“It is not the possession of truth, but the success which attends the seeking after it, that enriches the seeker and brings happiness to him.” (Max Planck, “Where is Science Going?”, 1932) 

“Apart from blunt truth, our lives sink decadently amid the perfume of hints and suggestions.” (Alfred N Whitehead, “Adventures of Ideas”, 1933)

”[…] the merit of mathematics, in all its forms, consists in its truth; truth conveyed to the understanding, not directly by words but by symbols which serve as the world’s only universal written language.” (David Eugene Smith, “The Poetry of Mathematics and Other Essays”,  1934)

“Mathematics is the science of number and space. It starts from a group of self-evident truths and by infallible deduction arrives at incontestable conclusions […] the facts of mathematics are absolute, unalterable, and eternal truths.” (E Russell Stabler, “An Interpretation and Comparison of Three Schools of Thought in the Foundations of Mathematics”, The Mathematics Teacher Vol 26, 1935)

"Science makes no pretension to eternal truth or absolute truth; some of its rivals do. That science is in some respects inhuman may be the secret of its success in alleviating human misery and mitigating human stupidity." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1938)

"Even if all parts of a problem seem to fit together like the pieces of a jigsaw puzzle, one has to remember that the probable need not necessarily be the truth and the truth not always probable." (Sigmund Freud, "Moses and Monotheism", 1939)

“When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic.” (Jean Rostand, “Pensées d'un Biologiste”, 1939)

“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)

"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) 

"Thus we do not try to prove the existence of the external world – we discover it, because the fundamental power of words or other symbols to represent events [...] permits us to put forward hypotheses and test their truth by reference to experience." (Kenneth Craik, "The Nature of Explanation", 1943)

“When two hypotheses are possible, we provisionally choose that which our minds adjudge to the simpler on the supposition that this Is the more likely to lead in the direction of the truth.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

"After all, the ultimate goal of all research is not objectivity, but truth." (Helene Deutsch, "The Psychology of Women", 1944)

"The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists." (Erwin Schrödinger, „What is Life?", 1944)

“I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.” (John von Neumann,  "The Mathematician", The Works of the Mind Vol. I (1), 1947)

"Science condemns itself to failure when, yielding to the infatuation of the serious, it aspires to attain being, to contain it, and to possess it; but it finds its truth if it considers itself as a free engagement of thought in the given, aiming, at each discovery, not at fusion with the thing, but at the possibility of new discoveries; what the mind then projects is the concrete accomplishment of its freedom." (Simone de Beauvoir, "The Ethics of Ambiguity", 1947)

"Any useful logic must concern itself with Ideas with a fringe of vagueness and a Truth that is a matter of degree.” (Norbert Wiener, “Cybernetics”, 1948)

"A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." (Max Planck, "A Scientific Autobiography", 1949)

“Representation of the world, like the world itself, is the work of men; they describe it from their own point of view, which they confuse with absolute truth.” (Simone de Beauvoir, “The Second Sex”, 1949)
"Science makes no pretension to eternal truth or absolute truth; some of its rivals do." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"It is true that the grasping of truth is not possible without empirical basis. However, the deeper we penetrate and the more extensive and embracing our theories become the less empirical knowledge is needed to determine those theories." (Albert Einstein, 1952)

“Logic and truth are two very different things, but they often look the same to the mind that’s performing the logic. " (Theodore Sturgeon, “More Than Human”, 1953)

"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)

"There are no whole truths; all truths are half-truths. It is trying to treat them as whole truths that plays the devil.” (Alfred North Whitehead, “Dialogues”, 1954) 

"The history of science is rich in the example of the fruitfulness of bringing two sets of techniques, two sets of ideas, developed in separate contexts for the pursuit of new truth, into touch with one another." (J. Robert Oppenheimer, "Science and the common understanding", 1954)

"Science is the creation of concepts and their exploration in the facts. It has no other test of the concept than its empirical truth to fact." (Jacob Bronowski, "Science and Human Values", 1956)

“Starting from statistical observations and applying to them a clear and precise concept of probability it is possible to arrive at conclusions which are just as reliable and ‘truth-full’ and quite as practically useful as those obtained in any other exact science.” (Richard von Mises, “Probability, Statistics, and Truth”2nd Ed., 1957)

“Uncertainty is introduced, however, by the impossibility of making generalizations, most of the time, which happens to all members of a class. Even scientific truth is a matter of probability and the degree of probability stops somewhere short of certainty.” (Wayne C Minnick, “The Art of Persuasion”, 1957)

"We speak in terms of ‘acceptance’, ‘confidence’, and ‘probability’, not ‘proof’. If by proof it is meant the establishment of eternal and absolute truth, open to no possible exception or modification, then proof has no place in the natural sciences." (George G Simpson, “Life: An Introduction to Biology”, 1957)

“We can never achieve absolute truth but we can live hopefully by a system of calculated probabilities. The law of probability gives to natural and human sciences - to human experience as a whole - the unity of life we seek.” (Agnes E Meyer, “Education for a New Morality”, 1957)

"It will never be possible by pure reason to arrive at some absolute truth." (Werner K Heisenberg, "Physics and Philosophy: The revolution in modern science", 1958)

"Scientific method is the way to truth, but it affords, even in principle, no unique definition of truth. Any so-called pragmatic definition of truth is doomed to failure equally." (Willard v O Quine, "Word and Object", 1960) 

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)

"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)

"When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic." (Jean Rostand, "The substance of man", 1962)

“Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected.” (Richard Feynman, “The Feynman Lectures on Physics” Vol. 1,1964)

“[…] in the statistical world you can multiply ignorance by a constant and get truth.” (Raymond F Jones, “The Non-Statistical Man”, 1964)

"The belief that there is only one truth and that oneself is in possession of it, seems to me the deepest root of all that is evil in the world." (Max Born, "Natural Philosophy of Cause and Chance", 1964)

“The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links.” (Arthur Koestler, “The Act of Creation”, 1964)

“All views are only probable, and a doctrine of probability which is not bound to a truth dissolves into thin air. In order to describe the probable, you must have a firm hold on the true. Therefore, before there can be any truth whatsoever, there must be absolute truth.” (Jean-Paul Sartre, “The Philosophy of Existentialism”, 1965)

“Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth.” (Salomon Bochner, “Why Mathematics Grows”, Journal of the History of Ideas, 1965)

“[…] truth is the intersection of independent lies.” (Richard Levins, “The Strategy of Model Building in Population Biology”, 1966)

"Primary scientific papers are not meant to be final statement of indisputable truths; each is merely a tiny tentative step forward, through the jungle of ignorance." (Erwin Schrödinger, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)

"At root what is needed for scientific inquiry is just receptivity to data, skill in reasoning, and yearning for truth. Admittedly, ingenuity can help too." (Willard v O Quine, "The Web of Belief", 1970)

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn," The Structure of Scientific Revolutions", 1970)

“Probability is truth in some degree […]” (Errol E Harris, “Hypothesis and Perception: The Roots of Scientific Method”, 1970)

“In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.” (Martin Gardner, “Mathematical Games”, Scientific American, 1973)

“No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect.” (Arthur C Clarke, “Profiles of the Future”, 1973)

“No matter how much reverence is paid to anything purporting to be ‘statistics’, the term has no meaning unless the source, relevance, and truth are all checked.” (Tom Burnam, “The Dictionary of Misinformation”, 1975)

“It seems that truth
Is progressive approximation
In which the relative fraction
Of our spontaneously tolerated residual error
Constantly diminishes.” (R Buckminster Fuller, “And It Came to Pass - Not to Stay”, 1976)

„[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)

"Truth cannot be defined or tested by agreement with 'the world'; for not only do truths differ for different worlds but the nature of „[…] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)

“Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous.” (Nelson Goodman, “Ways of Worldmaking”, 1978)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural.” (Stephen J Gould, “The Mismeasure of Man”, 1980)

"[…] the truth or likeness to truth that much of science pursues is of a rather special kind – we might call it 'physically necessary truth'" (L Jonathan Cohen, "What has science to do with truth?", Synthese 45, 1980)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, “The Role of Mathematics in the Rise of Science”, 1981)

"True, the initial ideas are in general those of an individual, but the establishment of the reality and truth is in general the work of more than one person." (Willard Libby, "Talking to people", 1981)

“In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it.” (Rudy Rucker, “Infinity and the Mind: The science and philosophy of the infinite”, 1982)

"Scientific theories must tell us both what is true in nature, and how we are to explain it. […] Scientific theories are thought to explain by dint of the descriptions they give of reality." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite, The Mathematical Intelligencer, Vol. 5, No. 4, 1983)

"The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me - both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension." (Paul R Halmos, “I Want to Be a Mathematician”, 1985)

"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend, “Farewell to Reason”, 1987)

“Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it.” (Isaac Asimov, [Interview by Bill Moyers] 1988)

“[…] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.” (Ivars Peterson, “Islands of Truth”, 1990)

"It is often the scientist’s experience that he senses the nearness of truth when such connections are envisioned. A connection is a step toward simplification, unification. Simplicity is indeed often the sign of truth and a criterion of beauty.” (Mahlon B Hoagland, “Toward the Habit of Truth”, 1990)

"It is not merely the truth of science that makes it beautiful, but its simplicity.” (Walker Percy, “Signposts in a Strange Land”, 1991)

„[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less ‚beautiful‘. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula.“ (Jean Dieudonné, „Mathematics - The Music of Reason“, 1992)

“Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth.” (Clifford Truesdell and Walter Noll, “The Non-Linear Field Theories of Mechanics” 2nd Ed., 1992)

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

“The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth’.” (Richard Feynman, “Six Easy Pieces”, 1994)

“[…] equations are like poetry: They speak truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend.” (Michael Guillen, “Five Equations That Changed the World”, 1995)

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

“Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith.” (K C Cole, “The Universe and the Teacup: The Mathematics of Truth and Beauty”, 1997)

"Mathematical logic deals not with the truth but only with the game of truth.” (Gian-Carlo Rota, “Indiscrete Thoughts”, 1997)

“Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics.” (Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty”, 1997)

“Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net.” (John Polkinghorne, “Belief in God in an Age of Science”, 1998)

“Mathematics has no privileged road to the truth.”(Donald C Benson, “The Moment of Proof: Mathematical Epiphanies”, 1999)
“Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!” (Gregory J Chaitin, “A century of controversy over the foundations of mathematics”, 2000)

"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)

“Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley.” (Yōko Ogawa, "The Housekeeper and the Professor", 2003)

“A model is a simplification or approximation of reality and hence will not reflect all of reality. […] Box noted that ‘all models are wrong, but some are useful’. While a model can never be ‘truth’, a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless.” (Kenneth P Burnham & David R Anderson, “Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach” 2nd Ed., 2005)

"It is art which invents the lies that raise falsehood to its highest affirmative power, that turns the will to deceive into something which is affirmed in the power of falsehood. For the artist, appearance no longer means the negation of the real in this world but this kind of selection, correction, redoubling and affirmation. Then truth perhaps takes on a new sense. Truth is appearance." (Gilles Deleuze, "Nietzsche as Philosopher", 2005)

“Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception.“ (David Ruelle, “The Mathematician's Brain”, 2007)

"It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject.” (Steven G Krantz, "The History and Concept of Mathematical", 2007)

“Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty.” (William Byers, “How Mathematicians Think”, 2007)

“Geometrical truth is (as we now speak) synthetic: it states facts about the world. Such truths are not ordinary truths but essential truths, giving the reality of the empirical world in which they are imperfect embodied.” (Fred Wilson, “The External World and Our Knowledge of It”, 2008)

"The concept of symmetry (invariance) with its rigorous mathematical formulation and generalization has guided us to know the most fundamental of physical laws. Symmetry as a concept has helped mankind not only to define ‘beauty’ but also to express the ‘truth’. Physical laws tries to quantify the truth that appears to be ‘transient’ at the level of phenomena but symmetry promotes that truth to the level of ‘eternity’.” (Vladimir G Ivancevic & Tijana T Ivancevic, “Quantum Leap”, 2008)

"Mathematicians, like priests, seek ‘ideal’, immutable truths and then often try to apply these truths to the real world." (Clifford A Pickover, "The Loom of God: Tapestries of Mathematics and Mysticism", 2009)

"Philosophers have sometimes made a distinction between analytic and synthetic truths. Analytic truths are not verified by observation; true analytic statements are tautologies and are true by virtue of the definitions of their terms and their logical structure. Synthetic truths relate to the material world; the truth of synthetic statements depends on their correspondence to how physical reality works. Mathematics, according to this distinction, deals exclusively with analytic truths. Its statements are all tautologies and are (analytically) true by virtue of their adherence to formal rules of construction." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"There is an absolute nature to truth in mathematics, which is unmatched in any other branch of knowledge. A theorem, once proven, requires independent checking but not repetition or independent derivation to be accepted as correct. […] Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity” (James Glimm, "Reflections and Prospectives", 2009)

“A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms.” (Raymond S Nickerson, “Mathematical Reasoning”, 2010)

“What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?” (Raymond S Nickerson, “Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs”, 2010)

“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri  & Jan von Plato, “Proof Analysis”, 2011)

“[…] statistics is a method of pursuing truth. At a minimum, statistics can tell you the likelihood that your hunch is true in this time and place and with these sorts of people. This type of pursuit of truth, especially in the form of an event’s future likelihood, is the essence of psychology, of science, and of human evolution.” (Arthur Aron et al, "Statistics for Psychology" 6th Ed., 2012)

“Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.” (Edward Frenkel, “Love and Math”, 2014)

"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)

“Scientists generally agree that no theory is 100 percent correct. Thus, the real test of knowledge is not truth, but utility.” (Yuval N Harari, “Sapiens: A brief history of humankind”, 2017)

Knowledge Representation: On Wisdom (Quotes)

"Things […] are some of them continuous […] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes', a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end […] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude […] or with multitude […] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus of Gerasa, "Introductio Arithmetica", cca. 100 AD)

"[Intuitive] Understanding is consequent upon deliberation, and firmly embraces the better part. For [intuitive] understanding concerns itself with divine truths, and the relish, love, and observance of the latter constitutes true wisdom. Rather than being the [mere] product of nature, these successive steps are the result of grace. The latter, according to its own free determination, derives the various rivulets of the sciences and wisdom from the fountainhead of sense perception. Grace reveals hidden divine truths by means of those things which have been made, and by that unity which belongs to love, communicates what it has made manifest, thus uniting man to God." (John of Salisbury, "Metalogicon", 1159

"Of all things the most desirable is wisdom, whose fruit consists in the love of what is good and the practice of virtue. Consequently the human mind must apply itself to the quest of wisdom, and thoroughly study and investigate questions in order to formulate clear and sound judgments concerning each." (John of Salisbury, "Metalogicon", 1159)

"There can only be one wisdom. For if it were possible that there be several wisdoms, then these would have to be from one. Namely, unity is prior to all plurality." (Nicholas of Cusa, "De Pace Fidei" ["The Peace of Faith"], 1453)

"Our wisdom and deliberation for the most part follow the lead of chance." (Michel de Montaigne, "Essays", 1580)

"Look round the world: contemplate the whole and every part of it: You will find it to be nothing but one great machine, subdivided into an infinite number of lesser machines, which again admit of subdivisions, to a degree beyond what human senses and faculties can trace and explain. All these various machines, and even their most minute parts, are adjusted to each other with an accuracy, which ravishes into admiration all men, who have ever contemplated them. The curious adapting of means to ends, throughout all nature, resembles exactly, though it much exceeds, the productions of human contrivance; of human design, thought, wisdom, and intelligence." (David Hume, "Dialogues Concerning Natural Religion Dialogues Concerning Natural Religion", 1779)

"It should be noted that the seeds of wisdom that are to bear fruit in the intellect are sown less by critical studies and learned monographs than by insights, broad impressions, and flashes of intuition." (Carl von Clausewitz, "On War", 1832)

"Religion is the metaphysics of the masses […] Just as they have popular poetry, and the popular wisdom of proverbs, so they must have popular metaphysics too: for mankind absolutely needs an interpretation of life; and this, again, must be suited to popular comprehension." (Arthur Schopenhauer, "The Horrors and Absurdities of Religion", 1851)

“We learn wisdom from failure much more than from success. We often discover what will do, by finding out what will not do; and probably he who never made a mistake never made a discovery.” (Samuel Smiles, “Facilities and Difficulties”, 1859)

"Collective wisdom, alas, is no adequate substitute for the intelligence of individuals. Individuals who opposed received opinions have been the source of all progress, both moral and intellectual. They have been unpopular, as was natural." (Bertrand Russell, "Why I Am Not a Christian", 1927)

"Accidental discoveries of which popular histories of science make mention never happen except to those who have previously devoted a great deal of thought to the matter. Observation unilluminated by theoretic reason is sterile. […] Wisdom does not come to those who gape at nature with an empty head. Fruitful observation depends not as Bacon thought upon the absence of bias or anticipatory ideas, but rather on a logical multiplication of them so that having many possibilities in mind we are better prepared to direct our attention to what others have never thought of as within the field of possibility." (Morris R Cohen, "Reason and Nature", 1931)

"[…] that all science is merely a game can be easily discarded as a piece of wisdom too easily come by. But it is legitimate to enquire whether science is not liable to indulge in play within the closed precincts of its own method. Thus, for instance, the scientist’s continuous penchant for systems tends in the direction of play." (Johan Huizinga, "Homo Ludens", 1938)

"Wisdom is your perspective on life, your sense of balance, your understanding of how the various parts and principles apply and relate to each other." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

“[…] the human brain must work in models. The trick is to have your brain work better than the other person’s brain because it understands the most fundamental models: ones that will do most work per unit. If you get into the mental habit of relating what you’re reading to the basic structure of the underlying ideas being demonstrated, you gradually accumulate some wisdom."  (Charles T Munger, “Poor Charlie’s Almanack”, 2005)

"The fact that cognitive diversity matters does not mean that if you assemble a group of diverse but thoroughly uninformed people, their collective wisdom will be smarter than an expert's. But if you can assemble a diverse group of people who possess varying degrees of knowledge and insight, you're better off entrusting it with major decisions rather than leaving them in the hands of one or two people, no matter how smart those people are." (James Surowiecki, "The Wisdom of Crowds", 2005)

"[…] many-model thinking produces wisdom through a diverse ensemble of logical frames. The various models accentuate different causal forces. Their insights and implications overlap and interweave. By engaging many models as frames, we develop nuanced, deep understandings." (Scott E Page, "The Model Thinker", 2018)

"A ray of imagination or of wisdom may enlighten the universe, and glow into remotest centuries." (George Berkeley)

"Collective intelligence is where the whole is smarter than any one individual in it. You can think of it that in a predictive context, this could be the wisdom of crowds, sort of thing where people guessing the weight of a steer, the crowd’s guess is going to be better than the average guess of the person in it." (Scott E Page [interview])

"Common sense in an uncommon degree is what the world calls wisdom." (Samuel T Coleridge)

"It is not in the nature of things for any one man to make a sudden violent discovery; science goes step by step, and every man depends on the work of his predecessors. When you hear of a sudden unexpected discovery - a bolt from the blue, as it were - you can always be sure that it has grown up by the influence of one man on another, and it is this mutual influence which makes the enormous possibility of scientific advance. Scientists are not dependent on the ideas of a single man, but on the combined wisdom of thousands of men, all thinking of the same problem, and each doing his little bit to add to the great structure of knowledge which is gradually being erected." (Ernest Rutherford)

"Real wisdom is not the knowledge of everything, but the knowledge of which things in life are necessary, which are less necessary, and which are completely unnecessary to know." (Lev N Tolstoy)

"The highest wisdom has but one science-the science of the whole-the science explaining the whole creation and man's place in it." (Lev N Tolstoy)

10 June 2012

Knowledge Representation: On Theories (Quotes)

“Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results […]” (Saint Thomas Aquinas, “Summa Theologica”, cca. 1266-1273)

“No one has yet been found so firm of mind and purpose as resolutely to compel himself to sweep away all theories and common notions, and to apply the understanding, thus made fair and even, to a fresh examination of particulars. Thus it happens that human knowledge, as we have it, is a mere medley and ill-digested mass, made up of much credulity and much accident, and also of the childish notions which we at first imbibed.” (Sir Francis Bacon, “Novum Organum” Book 2, 1620)

“We are under obligation to the ancients for having exhausted all the false theories that could be formed.” (Bernard le Bovier de Fontenelle, “Digression sur les Anciens et les Modernes", 1688)

“The moment a person forms a theory, his imagination sees, in every object, only the traits which favor that theory.” (Thomas Jefferson, [letter to Charles Thompson] 1787)

“It is not possible to feel satisfied at having said the last word about some theory as long as it cannot be explained in a few words to any passerby encountered in the street.” (Joseph D Gergonne, [letter] 1825)

“[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us.” (Auguste Comte, “Cours de Philosophie Positive”, 1830-1842)

“[Precision] is the very soul of science; and its attainment afford the only criterion, or at least the best, of the truth of theories, and the correctness of experiments.” (John F W Herschel, “A Preliminary Discourse on the Study of Natural Philosophy”, 1830)

“Theories are always very thin and unsubstantial; experience only is tangible.” (Hosea Ballou, ”Universalist Expositor”, 1831)

"The function of theory is to put all this in systematic order, clearly and comprehensively, and to trace each action to an adequate, compelling cause. […] Theory should cast a steady light on all phenomena so that we can more easily recognize and eliminate the weeds that always spring from ignorance; it should show how one thing is related to another, and keep the important and the unimportant separate. If concepts combine of their own accord to form that nucleus of truth we call a principle, if they spontaneously compose a pattern that becomes a rule, it is the task of the theorist to make this clear." (Carl von Clausewitz, "On War", 1832)

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action."(Carl von Clausewitz, "On War", 1832)

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

“How wonderful it is to me the simplicity of nature when we rightly interpret her laws and how different the convictions which they produce on the mind in comparison with the uncertain conclusions which hypothesis or even theory present.” (Michael Faraday, [letter to Svanberg] 1850)

“Every detection of what is false directs us towards what is true: every trial exhausts some tempting form of error. Not only so; but scarcely any attempt is entirely a failure; scarcely any theory, the result of steady thought, is altogether false; no tempting form of error is without some latent charm derived from truth.” (William Whewell, “Lectures on the History of Moral Philosophy in England”, 1852)

“The dimmed outlines of phenomenal things all merge into one another unless we put on the focusing-glass of theory, and screw it up sometimes to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world.” (James C Maxwell, “Are There Real Analogies in Nature?”, 1856) 

“[…] ideas may be both novel and important, and yet, if they are incorrect – if they lack the very essential support of incontrovertible fact, they are unworthy of credence. Without this, a theory may be both beautiful and grand, but must be as evanescent as it is beautiful, and as unsubstantial as it is grand.” (George Brewster, “A New Philosophy of Matter”, 1858)
“The world little knows how many of the thoughts and theories which have passed through the mind of a scientific investigator have been crushed in silence and secrecy; that in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary conclusions have been realized.” (Michael Faraday, “The Forces of Matter”, 1860)

“Observation is so wide awake, and facts are being so rapidly added to the sum of human experience, that it appears as if the theorizer would always be in arrears, and were doomed forever to arrive at imperfect conclusion; but the power to perceive a law is equally rare in all ages of the world, and depends but little on the number of facts observed.” (Henry Thoreau, “A Week on the Concord and Merrimack Rivers”, 1862) 

“If an idea presents itself to us, we must not reject it simply because it does not agree with the logical deductions of a reigning theory.” (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained - a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas." (Chauncey Wright, "The Philosophy of Herbert Spencer", North American Review, 1865)

 “Step by step we cross great eras in the development of thought: there is no sudden gigantic stride; a theory proceeds by slow evolution until it dominates or is destroyed.” (George F Rodwell, “Theory of Phlogiston”, ‘The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science’ 35, 1868)

"Isolated facts and experiments have in themselves no value, however great their number may be. They only become valuable in a theoretical or practical point of view when they make us acquainted with the law of a series of uniformly recurring phenomena, or, it may be, only give a negative result showing an incompleteness in our knowledge of such a law, till then held to be perfect." (Hermann von Helmholtz, "The Aim and Progress of Physical Science", 1869)

 “The triumph of a theory is to embrace the greatest number and the greatest variety of facts.” (Charles A Wurtz, “A History of Chemical Theory from the Age of Lavoisier to the Present Time”, 1869)

“The aim of natural science is to obtain connections among phenomena. Theories, however, are like withered leaves, which drop off after having enabled the organism of science to breathe for a time." (Ernst Mach, “Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit”, 1871)
“The possession of an original theory which has not yet been assailed must certainly sweeten the temper of a man who is not beforehand ill-natured.” (George Eliot, “Theophrastus Such”, 1879)

"Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics.” (Benjamin Peirce, “Linear Associative Algebra”, American Journal of Mathematics, Vol. 4, 1881)

"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [president's address] 1883)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)

“Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent.” (William S Jevons, “The Principles of Science”, 1887)

“The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet been formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe.” (Sir James G Frazer, “The Golden Bough: A Study in Magic and Religion”, 1890) 

“One is almost tempted to assert that quite apart from its intellectual mission, theory is the most practical thing conceivable, the quintessence of practice as it were, since the precision of its conclusions cannot be reached by any routine of estimating or trial and error; although given the hidden ways of theory, this will hold only for those who walk them with complete confidence.” (Ludwig E Boltzmann, “On the Significance of Theories”, 1890) 

“Facts are not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter. Theory is of the essence of facts. Without theory scientific knowledge would be only worthy of the mad house.” (Oliver Heaviside, “Electromagnetic Theory”, 1893)

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

"A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." (David Hilbert [paraphrasing Joseph D Gergonne], "Mathematical Problems", 1900)

“One does not ask whether a scientific theory is true, but only whether it is convenient.” (Henri Poincaré, “La Science et l'Hypothèse”, 1902) 

“With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.” (David Hilbert. “Mathematical Problems”, Bulletin of the American Mathematical Society Vol. 8, 1902)

"But surely it is self-evident that every theory is merely a framework or scheme of concepts together with their necessary relations to one another, and that the basic elements can be constructed as one pleases." (Gottlob Frege, "On the Foundations of Geometry and Formal Theories of Arithmetic" , cca. 1903-1909)

“It [a theory] ought to furnish a compass which, if followed, will lead the observer further and further into previously unexplored regions. Whether these regions will be barren or fertile experience alone will decide; but, at any rate, one who is guided in this way will travel onward in a definite direction, and will not wander aimlessly to and fro.” (Sir Joseph J Thomson, “The Corpuscular Theory of Matter”, 1907)

"Things and events explain themselves, and the business of thought is to brush aside the verbal and conceptual impediments which prevent them from doing so. Start with the notion that it is you who explain the Object, and not the Object that explains itself, and you are bound to end in explaining it away. It ceases to exist, its place being taken by a parcel of concepts, a string of symbols, a form of words, and you find yourself contemplating, not the thing, but your theory of the thing." (Lawrence P Jacks, "The Usurpation Of Language", 1910)

“The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features." (Eliakim H Moore, “Introduction to a Form of General Analysis”, 1910)

“The discovery which has been pointed to by theory is always one of profound interest and importance, but it is usually the close and crown of a long and fruitful period, whereas the discovery which comes as a puzzle and surprise usually marks a fresh epoch and opens a new chapter in science.” (Sir Oliver J Lodge, [Becquerel Memorial Lecture] Journal of the Chemical Society, Transactions 101 (2), 1912) 

"There is no great harm in the theorist who makes up a new theory to fit a new event. But the theorist who starts with a false theory and then sees everything as making it come true is the most dangerous enemy of human reason." (Gilbert K Chesterton, "The Flying Inn", 1914)

"Theory is the best guide for experiment - that were it not for theory and the problems and hypotheses that come out of it, we would not know the points we wanted to verify, and hence would experiment aimlessly" (Henry Hazlitt,  “Thinking as a Science”, 1916)

“As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies. “ (Albert Einstein: “Relativity: The Special and General Theory”, 1916)

 “No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.” (Albert Einstein: “Relativity, The Special and General Theory”, 1916)

“To come very near to a true theory, and to grasp its precise application, are two very different things, as the history of a science teaches us. Everything of importance has been said before by somebody who did not discover it.” (Alfred N Whitehead, “The Organization of Thought”, 1917)

”Facts are carpet-tacks under the pneumatic tires of theory.” (Austin O’Malley, “Keystones of Thought”, 1918)

"Philosophy, like science, consists of theories or insights arrived at as a result of systemic reflection or reasoning in regard to the data of experience. It involves, therefore, the analysis of experience and the synthesis of the results of analysis into a comprehensive or unitary conception. Philosophy seeks a totality and harmony of reasoned insight into the nature and meaning of all the principal aspects of reality." (Joseph A Leighton, "The Field of Philosophy: An outline of lectures on introduction to philosophy", 1919)

“[…] analogies are not ‘aids’ to the establishment of theories; they are an utterly essential part of theories, without which theories would be completely valueless and unworthy of the name. It is often suggested that the analogy leads to the formulation of the theory, but that once the theory is formulated the analogy has served its purpose and may be removed or forgotten. Such a suggestion is absolutely false and perniciously misleading.” (Norman R Campbell, “Physics, the Elements”, 1920) 

“Nothing is more interesting to the true theorist than a fact which directly contradicts a theory generally accepted up to that time, for this is his particular work.” (Max Planck, “A Survey of Physics”, 1925)

“[…] the mere collection of facts, without some basis of theory for guidance and elucidation, is foolish and profitless.” (Gamaliel Bradford, “Darwin”, 1926)

"It is characteristic of a good scientific theory that it makes no more assumptions than are needed to explain the facts under consideration and predict a few more." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

“Often a liberal antidote of experience supplies a sovereign cure for a paralyzing abstraction built upon a theory.” (Benjamin N Cardozo, “Paradoxes of Legal Science”, 1928)

“[…] facts are too bulky to be lugged about conveniently except on the wheels of theory.” (Julian Huxley, “Essays of a Biologist”, 1929)

 “We can invent as many theories we like, and any one of them can be made to fit the facts. But that theory is always preferred which makes the fewest number of assumptions.” (Albert Einstein [interview] 1929)

"Every theory of the course of events in nature is necessarily based on some process of simplification and is to some extent, therefore, a fairy tale." (Sir Napier Shaw, “Manual of Meteorology”, 1932)

“[…] the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well-constructed theory is in some respects undoubtedly an artistic production.” (Ernest Rutherford, 1932)

"It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience." (Albert Einstein, [lecture] 1933)

“All the theories and hypotheses of empirical science share this provisional character of being established and accepted ‘until further notice’ [...]” (Carl G Hempel, "Geometry and Empirical Science”, 1935)

”[while] the traditional way is to regard the facts of science as something like the parts of a jig-saw puzzle, which can be fitted together in one and only one way, I regard them rather as the tiny pieces of a mosaic, which can be fitted together in many ways. A new theory in an old subject is, for me, a new mosaic pattern made with the pieces taken from an older pattern. [...] Theories come into fashion and theories go out of fashion, but the facts connected with them stay.” (William H George, “The Scientist in Action”, 1936)

“Every new theory as it arises believes in the flush of youth that it has the long sought goal; it sees no limits to its applicability, and believes that at last it is the fortunate theory to achieve the 'right' answer.” (Percy W Bridgman, “The Nature of Physical Theory”, 1936)

"When an active individual of sound common sense perceives the sordid state of the world, desire to change it becomes the guiding principle by which he organizes given facts and shapes them into a theory. The methods and categories as well as the transformation of the theory can be understood only in connection with his taking of sides. This, in turn, discloses both his sound common sense and the character of the world. Right thinking depends as much on right willing as right willing on right thinking." (Max Horkheimer, "The Latest Attack on Metaphysics", 1937)

“Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up.” (Albert Einstein & Leopold Infeld, ”The Evolution of Physics”, 1938)

“With the help of physical theories we try to find our way through the maze of observed facts, to order and understand the world of our sense impressions.” (Albert Einstein & Leopold Infeld, ”The Evolution of Physics”, 1938)
"There is nothing as practical as a good theory” (Kurt Z Lewin, "Psychology and the process of group living", Journal of Social Psychology 17, 1943)

“To a scientist a theory is something to be tested. He seeks not to defend his beliefs, but to improve them. He is, above everything else, an expert at ‘changing his mind’.” (Wendell Johnson, 1946)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

“One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects ‘elegance’ in its ‘architectural’ structural makeup.” (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I (1), 1947]) 

”We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts.” (James B Conant, “On Understanding Science”, 1947)

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability." (Albert Einstein, "Autobiographical Notes", 1949)

“When a scientific theory is firmly established and confirmed, it changes its character and becomes a part of the metaphysical background of the age: a doctrine is transformed into a dogma.” (Max Born, “Natural Philosophy of Cause and Chance”, 1949)
"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure." (André Weil, "De la Métaphysique aux Mathématiques", 1960)

”In imagination there exists the perfect mystery story. Such a story presents all the essential clews, and compels us to form our own theory of the case. If we follow the plot carefully we arrive at the complete solution for ourselves just before the author’s disclosure at the end of the book. The solution itself, contrary to those of inferior mysteries, does not disappoint us; moreover, it appears at the very moment we expect it.” (Leopold Infeld, “The Evolution of Physics”, 1961)

"It seems to be one of the fundamental features of nature the fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. " (Paul A M Dirac , “The Evolution of the Physicist’s Picture of Nature”, Scientific American, 1963)

“A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. ” (Paul A M Dirac, Scientific American, 1963)

"The final test of a theory is its capacity to solve the problems which originated it." (George Dantzig, "Linear Programming and Extensions", 1963)

“It is easy to obtain confirmations, or verifications, for nearly every theory - if we look for confirmations. Confirmations should count only if they are the result of risky predictions. […] A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice. Every genuine test of a theory is an attempt to falsify it, or refute it.” (Karl R Popper, “Conjectures and Refutations: The Growth of Scientific Knowledge”, 1963)

"One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.” (Philip J Davis, "Number", Scientific American, No 211 (3), 1964)

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

“This is the key of modern science and it was the beginning of the true understanding of Nature - this idea to look at the thing, to record the details, and to hope that in the information thus obtained might lie a clue to one or another theoretical interpretation.” (Richard P Feynman, “The Character of Physical Law”, 1965)

"Theories are usually introduced when previous study of a class of phenomena has revealed a system of uniformities. […] Theories then seek to explain those regularities and, generally, to afford a deeper and more accurate understanding of the phenomena in question. To this end, a theory construes those phenomena as manifestations of entities and processes that lie behind or beneath them, as it were." (Carl G Hempel, "Philosophy of Natural Science", 1966)

“A theory is scientific only if it can be disproved. But the moment you try to cover absolutely everything the chances are that you cover nothing. “ (Sir Hermann Bondi, “Assumption and Myth in Physical Theory”, 1967) 

 ”As soon as we inquire into the reasons for the phenomena, we enter the domain of theory, which connects the observed phenomena and traces them back to a single ‘pure’ phenomena, thus bringing about a logical arrangement of an enormous amount of observational material.” (Georg Joos, “Theoretical Physics”, 1968)

"It makes no sense to say what the objects of a theory are, beyond saying how to interpret or reinterpret that theory in another." (Willard v O Quine, "Ontological Relativity and Other Essays", 1969)

“One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’.” (Thomas S Kuhn, “The Structure of Scientific Revolutions”, 1970)

“Blind commitment to a theory is not an intellectual virtue: it is an intellectual crime.” (Imre Lakatos, [radio Lecture] 1973) 

“For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.” (Michael Frayn, “Constructions”, 1974)

“No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions.” (Paul K Feyerabend, “Against Method: Outline of an Anarchistic Theory of Knowledge”, 1975) 

“A physical theory remains an empty shell until we have found a reasonable physical interpretation.” (Peter Bergmann, [conference] 1976)

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

 “A good scientific law or theory is falsifiable just because it makes definite claims about the world. For the falsificationist, If follows fairly readily from this that the more falsifiable a theory is the better, in some loose sense of more. The more a theory claims, the more potential opportunities there will be for showing that the world does not in fact behave in the way laid down by the theory. A very good theory will be one that makes very wide-ranging claims about the world, and which is consequently highly falsifiable, and is one that resists falsification whenever it is put to the test.” (Alan F Chalmers,  “What Is This Thing Called Science?”, 1976)

”Facts do not ‘speak for themselves’; they are read in the light of theory. Creative thought, in science as much as in the arts, is the motor of changing opinion. Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation.” (Stephen J Gould, “Ever Since Darwin”, 1977)

“Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world.” (Steven Weinberg, “The First Three Minutes”, 1977)

"The theory of our modern technic shows that nothing is as practical as the theory." (J Robert Oppenheimer, "Reflex", 1977)

“Science has so accustomed us to devising and accepting theories to account for the facts we observe, however fantastic, that our minds must begin their manufacture before we are aware of it.” (Gene Wolfe, “Seven American Nights”, 1978) 

"For mathematicians, only one test was necessary: once the elements of any mathematical theory were seen to be consistent, then they were mathematically acceptable. Nothing more was required." (Joseph W  Dauben, "Georg Cantor: His Mathematics and Philosophy of the Infinite", 1979)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural.” (Stephen J Gould, “The Mismeasure of Man”, 1980)

"Facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world's data. Theories are structures of ideas that explain and interpret facts. Facts do not go away while scientists debate rival theories for explaining them." (Stephen J Gould "Evolution as Fact and Theory", 1981)

"A real change of theory is not a change of equations - it is a change of mathematical structure, and only fragments of competing theories, often not very important ones conceptually, admit comparison with each other within a limited range of phenomena." (Yuri I Manin, "Mathematics and Physics", 1981)

"The principal aim of physical theories is understanding. A theory's ability to find a number is merely a useful criterion for a correct understanding." (Yuri I Manin, "Mathematics and Physics", 1981)

“Data in isolation are meaningless, a collection of numbers. Only in context of a theory do they assume significance […]” (George Greenstein, “Frozen Star”, 1983)

“In all scientific fields, theory is frequently more important than experimental data. Scientists are generally reluctant to accept the existence of a phenomenon when they do not know how to explain it. On the other hand, they will often accept a theory that is especially plausible before there exists any data to support it.” (Richard Morris, 1983) 

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

"Scientific theories must tell us both what is true in nature, and how we are to explain it. […] Scientific theories are thought to explain by dint of the descriptions they give of reality." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

“The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.” (Paul Halmos, “Selecta: Expository writing”, 1983)

“Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord.” (Paul C W Davies, “Superforce”, 1984)

“Nature is disordered, powerful and chaotic, and through fear of the chaos we impose system on it. We abhor complexity, and seek to simplify things whenever we can by whatever means we have at hand. We need to have an overall explanation of what the universe is and how it functions. In order to achieve this overall view we develop explanatory theories which will give structure to natural phenomena: we classify nature into a coherent system which appears to do what we say it does.” (James Burke, “The Day the Universe Changed”, 1985) 

”Experience without theory teaches nothing.” (William E Deming, “Out of the Crisis”, 1986)

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

"Facts do not 'speak for themselves'. They speak for or against competing theories. Facts divorced from theory or visions are mere isolated curiosities." (Thomas Sowell, "A Conflict of Visions: Ideological Origins of Political Struggles", 1987)

“[…] no good model ever accounted for all the facts, since some data was bound to be misleading if not plain wrong. A theory that did fit all the data would have been ‘carpentered’ to do this and would thus be open to suspicion.” (Francis H C Crick, “What Mad Pursuit: A Personal View of Scientific Discovery”, 1988)

“Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory.” (Stephen Hawking,  “A Brief History of Time”, 1988)

“Theories are not so much wrong as incomplete." (Isaac Asimov, "The Relativity of Wrong", 1988)

”A discovery in science, or a new theory, even where it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalyzed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow: it takes a vast world unchallenged and for granted.” (James R Oppenheimer, “Atom and Void”, 1989)

 “Model is used as a theory. It becomes theory when the purpose of building a model is to understand the mechanisms involved in the developmental process. Hence as theory, model does not carve up or change the world, but it explains how change takes place and in what way or manner. This leads to build change in the structures.” (Laxmi K Patnaik, “Model Building in Political Science”, The Indian Journal of Political Science Vol. 50 (2), 1989)
“A law explains a set of observations; a theory explains a set of laws. […] Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism.” (John L Casti, “Searching for Certainty”, 1990)

“It is in the nature of theoretical science that there can be no such thing as certainty. A theory is only ‘true’ for as long as the majority of the scientific community maintain the view that the theory is the one best able to explain the observations.” (Jim Baggott, “The Meaning of Quantum Theory”, 1992)

“Scientists use mathematics to build mental universes. They write down mathematical descriptions - models - that capture essential fragments of how they think the world behaves. Then they analyse their consequences. This is called 'theory'. They test their theories against observations: this is called 'experiment'. Depending on the result, they may modify the mathematical model and repeat the cycle until theory and experiment agree. Not that it's really that simple; but that's the general gist of it, the essence of the scientific method.” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

"Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about  reverence, not mastery." (Richard Power, “Gold Bug Variations”, 1993) 

“Clearly, science is not simply a matter of observing facts. Every scientific theory also expresses a worldview. Philosophical preconceptions determine where facts are sought, how experiments are designed, and which conclusions are drawn from them.” (Nancy R Pearcey & Charles B. Thaxton, “The Soul of Science: Christian Faith and Natural Philosophy”, 1994)

“The amount of understanding produced by a theory is determined by how well it meets the criteria of adequacy - testability, fruitfulness, scope, simplicity, conservatism - because these criteria indicate the extent to which a theory systematizes and unifies our knowledge.” (Theodore Schick Jr.,  “How to Think about Weird Things: Critical Thinking for a New Age”, 1995)

“Scientists, being as a rule more or less human beings, passionately stick up for their ideas, their pet theories. It's up to someone else to show you are wrong.” (Niles Eldredge, “Reinventing Darwin”, 1995)

"There is no sharp dividing line between scientific theories and models, and mathematics is used similarly in both. The important thing is to possess a delicate judgement of the accuracy of your model or theory. An apparently crude model can often be surprisingly effective, in which case its plain dress should not mislead. In contrast, some apparently very good models can be hiding dangerous weaknesses." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory." (Gian-Carlo Rota, [lecture] 1996)

"Paradigms are the most general-rather like a philosophical or ideological framework. Theories are more specific, based on the paradigm and designed to describe what happens in one of the many realms of events encompassed by the paradigm. Models are even more specific providing the mechanisms by which events occur in a particular part of the theory's realm. Of all three, models are most affected by empirical data - models come and go, theories only give way when evidence is overwhelmingly against them and paradigms stay put until a radically better idea comes along." (Lee R Beach, "The Psychology of Decision Making: People in Organizations", 1997)

"Ideas about organization are always based on implicit images or metaphors that persuade us to see, understand, and manage situations in a particular way. Metaphors create insight. But they also distort. They have strengths. But they also have limitations. In creating ways of seeing, they create ways of not seeing. There can be no single theory or metaphor that gives an all-purpose point of view, and there can be no simple 'correct theory' for structuring everything we do." (Gareth Morgan, ”Imaginization”, 1997)

“An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)

“[…] philosophical theories are structured by conceptual metaphors that constrain which inferences can be drawn within that philosophical theory. The (typically unconscious) conceptual metaphors that are constitutive of a philosophical theory have the causal effect of constraining how you can reason within that philosophical framework.” (George Lakoff, “Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought”, 1999)
"All scientific theories, even those in the physical sciences, are developed in a particular cultural context. Although the context may help to explain the persistence of a theory in the face of apparently falsifying evidence, the fact that a theory arises from a particular context is not sufficient to condemn it. Theories and paradigms must be accepted, modified or rejected on the basis of evidence." (Richard P Bentall,  "Madness Explained: Psychosis and Human Nature", 2003)

“Do not be afraid of the word 'theory'. Yes, it can sound dauntingly abstract at times, and in the hands of some writers can appear to have precious little to do with the actual, visual world around us. Good theory however, is an awesome thing. [...] But unless we actually use it, it borders on the metaphysical and might as well not be used at all.” (Richard Howells,  Visual Culture, 2003)

"A scientific theory is a concise and coherent set of concepts, claims, and laws (frequently expressed mathematically) that can be used to precisely and accurately explain and predict natural phenomena." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

“It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers - not by logical deduction, but by something like a sense of smell.” (Steven Weinberg, “Physics Today”, 2005)

 "In science, for a theory to be believed, it must make a prediction - different from those made by previous theories - for an experiment not yet done. For the experiment to be meaningful, we must be able to get an answer that disagrees with that prediction. When this is the case, we say that a theory is falsifiable - vulnerable to being shown false. The theory also has to be confirmable, it must be possible to verify a new prediction that only this theory makes. Only when a theory has been tested and the results agree with the theory do we advance the statement to the rank of a true scientific theory." (Lee Smolin, “The Trouble with Physics”, 2006)

"A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have." (Murray Gell-Mann, "Beauty and Truth in Physics", 2007)

"In science we try to explain reality by using models (theories). This is necessary because reality itself is too complex. So we need to come up with a model for that aspect of reality we want to understand – usually with the help of mathematics. Of course, these models or theories can only be simplifications of that part of reality we are looking at. A model can never be a perfect description of reality, and there can never be a part of reality perfectly mirroring a model." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"It is also inevitable for any model or theory to have an uncertainty (a difference between model and reality). Such uncertainties apply both to the numerical parameters of the model and to the inadequacy of the model as well. Because it is much harder to get a grip on these types of uncertainties, they are disregarded, usually." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty.” (William Byers, “How Mathematicians Think”, 2007)

“A theory is a speculative explanation of a particular phenomenon which derives it legitimacy from conforming to the primary assumptions of the worldview of the culture in which it appears. There can be more than one theory for a particular phenomenon that conforms to a given worldview. […]  A new theory may seem to trigger a change in worldview, as in this case, but logically a change in worldview must precede a change in theory, otherwise the theory will not be viable. A change in worldview will necessitate a change in all theories in all branches of study.” (M G Jackson, “Transformative Learning for a New Worldview: Learning to Think Differently”, 2008)

“All scientific theories, even those in the physical sciences, are developed in a particular cultural context. Although the context may help to explain the persistence of a theory in the face of apparently falsifying evidence, the fact that a theory arises from a particular context is not sufficient to condemn it. Theories and paradigms must be accepted, modified or rejected on the basis of evidence.”  (Richard P Bentall,  “Madness Explained: Psychosis and Human Nature”, 2003)

“A theory is like medicine (or government): often useless, sometimes necessary, always self-serving, and on occasion lethal. So, it needs to be used with care, moderation and close adult supervision.” (Nassim N Taleb, “The Black Swan: The Impact of the Highly Improbable”, 2007)

“With each theory or model, our concepts of reality and of the fundamental constituents of the universe have changed.” (Stephen Hawking & Leonard Mlodinow, “The Grand Design”, 2010)

 “A theory is a set of deductively closed propositions that explain and predict empirical phenomena, and a model is a theory that is idealized.” (Jay Odenbaugh, “True Lies: Realism, Robustness, and Models”, Philosophy of Science, Vol. 78, No. 5, 2011)

"Science would be better understood if we called theories ‘misconceptions’ from the outset, instead of only after we have discovered their successors." (David Deutsch, "Beginning of Infinity", 2011)

"Complexity has the propensity to overload systems, making the relevance of a particular piece of information not statistically significant. And when an array of mind-numbing factors is added into the equation, theory and models rarely conform to reality." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

“[…] if one has a theory, one needs to be willing to try to prove it wrong as much as one tries to provide that it is right […]” (Lawrence M Krauss et al, A Universe from Nothing, 2013)

"Another way to secure statistical significance is to use the data to discover a theory. Statistical tests assume that the researcher starts with a theory, collects data to test the theory, and reports the results - whether statistically significant or not. Many people work in the other direction, scrutinizing the data until they find a pattern and then making up a theory that fits the pattern." (Gary Smith, "Standard Deviations", 2014)

"Data clusters are everywhere, even in random data. Someone who looks for an explanation will inevitably find one, but a theory that fits a data cluster is not persuasive evidence. The found explanation needs to make sense and it needs to be tested with uncontaminated data." (Gary Smith, "Standard Deviations", 2014)

"Data without theory can fuel a speculative stock market bubble or create the illusion of a bubble where there is none. How do we tell the difference between a real bubble and a false alarm? You know the answer: we need a theory. Data are not enough. […] Data without theory is alluring, but misleading." (Gary Smith, "Standard Deviations", 2014)

"We are hardwired to make sense of the world around us - to notice patterns and invent theories to explain these patterns. We underestimate how easily pat - terns can be created by inexplicable random events - by good luck and bad luck." (Gary Smith, "Standard Deviations", 2014)

"Mathematical modeling is the modern version of both applied mathematics and theoretical physics. In earlier times, one proposed not a model but a theory. By talking today of a model rather than a theory, one acknowledges that the way one studies the phenomenon is not unique; it could also be studied other ways. One's model need not claim to be unique or final. It merits consideration if it provides an insight that isn't better provided by some other model." (Reuben Hersh,"Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

“Scientists generally agree that no theory is 100 percent correct. Thus, the real test of knowledge is not truth, but utility.” (Yuval N Harari, “Sapiens: A brief history of humankind”, 2017) 

"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)

“A theory is nothing but a tool to know the reality. If a theory contradicts reality, it must be discarded at the earliest.” (Awdhesh Singh, “Myths are Real, Reality is a Myth”, 2018)

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