"The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveler." (Philip Stanhope, "Letters Written by the Earl of Chesterfield to His Son", 1827)
"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and 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, [Presidential Address to British Association] 1869)
"What are the sciences but maps of universal laws, and universal laws but the channels of universal power; and universal power but the outgoings of a universal mind?" (Edward Thomson, "Evidences of Revealed Religion", 1872)
"Just as, in the map of a half-explored country, we see detached bits of rivers, isolated mountains, and undefined plains, not connected into any complete plan, so a new branch of knowledge consists of groups of facts, each group standing apart, so as not to allow us to reason from one to another." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)
"A fundamental value in the scientific outlook is concern with the best available map of reality. The scientist will always seek a description of events which enables him to predict most by assuming least. He thus already prefers a particular form of behavior. If moralities are systems of preferences, here is at least one point at which science cannot be said to be completely without preferences. Science prefers good maps." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)
"Scientific research was much like prospecting: you went out and you hunted, armed with your maps and your instruments, but in the end your preparations did not matter, or even your intuition. You needed your luck, and whatever benefits accrued to the diligent, through sheer, grinding hard work." (Michael Crichton, "The Andromeda Strain", 1969)
"To do science is to search for repeated patterns, not simply to accumulate facts, and to do the science of geographical ecology is to search for patterns of plants and animal life that can be put on a map." (Robert H. MacArthur, "Geographical Ecology", 1972)
"The orchard of science is a vast globe-encircling monster, without a map, and known to no one man; indeed, to no group of men fewer than the whole international mass of creative scientists. Within it, each observer clings to his own well-known and well-loved clump of trees. If he looks beyond, it is usually with a guilty sigh." (Isaac Asimov, "View from a Height", 1975)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
11 October 2019
Map vs. Territory II
"We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put on paper. What is on the paper map is a representation of what was in the retinal representation of the man who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. […] Always, the process of representation will filter it out so that the mental world is only maps of maps, ad infinitum." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)
"For our purposes, a simple way to understand paradigms is to see them as maps. We all know that ‘the map is not the territory’. A map is simply an explanation of certain aspects of the territory. That’s exactly what a paradigm is. It is a theory, an explanation, or model of something else." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)
"Principles are the territory. Values are maps. When we value correct principles, we have truth - a knowledge of things as they are." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)
"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)
"The ultimate metaphysical secret, if we dare state it so simply, is that there are no boundaries in the universe. Boundaries are illusions, products not of reality but of the way we map and edit reality. And while it is fine to map out the territory, it is fatal to confuse the two." (Ken Wilber, "No Boundary: Eastern and Western Approaches to Personal Growth", 1979)
"Our view of reality is like a map with which to negotiate the terrain of life. If the map is true and accurate, we will generally know how to get there. If the map is false and inaccurate, we generally will be lost." (M Scott Peck, "Wisdom from the Road Less Traveled", 2001)
"A map does not just replicate the shape of a territory; rather, it actively inflects and works over that territory." (Steven Shaviro, "Post Cinematic Affect", 2010)
"An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)
"For our purposes, a simple way to understand paradigms is to see them as maps. We all know that ‘the map is not the territory’. A map is simply an explanation of certain aspects of the territory. That’s exactly what a paradigm is. It is a theory, an explanation, or model of something else." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)
"Principles are the territory. Values are maps. When we value correct principles, we have truth - a knowledge of things as they are." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)
"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)
"The ultimate metaphysical secret, if we dare state it so simply, is that there are no boundaries in the universe. Boundaries are illusions, products not of reality but of the way we map and edit reality. And while it is fine to map out the territory, it is fatal to confuse the two." (Ken Wilber, "No Boundary: Eastern and Western Approaches to Personal Growth", 1979)
"Our view of reality is like a map with which to negotiate the terrain of life. If the map is true and accurate, we will generally know how to get there. If the map is false and inaccurate, we generally will be lost." (M Scott Peck, "Wisdom from the Road Less Traveled", 2001)
"A map does not just replicate the shape of a territory; rather, it actively inflects and works over that territory." (Steven Shaviro, "Post Cinematic Affect", 2010)
"An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)
Map vs. Territory I
"The map appears to us more real than the land." (David W Lawrence, "Study of Thomas Hardy", cca. 1914)
"If words are not things, or maps are not the actual territory, then, obviously, the only possible link between the objective world and the linguistic world is found in structure, and structure alone." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)
"Two important characteristics of maps should be noticed. A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)
"The map is not the territory, and the name is not the thing named." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)
"The most promising words ever written on the maps of human knowledge are terra incognita, unknown territory." (Daniel J Boorstin, "The Discoverers: A History of Man's Search to Know His World and Himself", 1983)
"None of us see the world as it is but as we are, as our frames of reference, or maps, define the territory." (Stephen Covey, "Principle Centered Leadership", 1992)
"But ignorance exists in the map, not in the territory. If I am ignorant about a phenomenon, that is a fact about my own state of mind, not a fact about the phenomenon itself. A phenomenon can seem mysterious to some particular person. There are no phenomena which are mysterious of themselves. To worship a phenomenon because it seems so wonderfully mysterious, is to worship your own ignorance." (Eliezer Yudkowsky, "Mysterious Answers To Mysterious Questions" 2007)
"It is hard to navigate across one’s environment without having some ideas, however coarse, about it. Indeed, to face any situation we must know whether it is real or imaginary, profane or sacred, sensitive or insensitive to our actions, and so on. This is why even lowly organisms develop, if not worldviews, at least rough sensory maps of their immediate environment – as noted by ethologists from the start. But it is generally assumed that only humans can build conceptual models of their environments. And, except for some philosophers, humans distinguish maps from the territories they represent." (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)
"If words are not things, or maps are not the actual territory, then, obviously, the only possible link between the objective world and the linguistic world is found in structure, and structure alone." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)
"Two important characteristics of maps should be noticed. A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)
"The map is not the territory, and the name is not the thing named." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)
"The most promising words ever written on the maps of human knowledge are terra incognita, unknown territory." (Daniel J Boorstin, "The Discoverers: A History of Man's Search to Know His World and Himself", 1983)
"None of us see the world as it is but as we are, as our frames of reference, or maps, define the territory." (Stephen Covey, "Principle Centered Leadership", 1992)
"But ignorance exists in the map, not in the territory. If I am ignorant about a phenomenon, that is a fact about my own state of mind, not a fact about the phenomenon itself. A phenomenon can seem mysterious to some particular person. There are no phenomena which are mysterious of themselves. To worship a phenomenon because it seems so wonderfully mysterious, is to worship your own ignorance." (Eliezer Yudkowsky, "Mysterious Answers To Mysterious Questions" 2007)
"It is hard to navigate across one’s environment without having some ideas, however coarse, about it. Indeed, to face any situation we must know whether it is real or imaginary, profane or sacred, sensitive or insensitive to our actions, and so on. This is why even lowly organisms develop, if not worldviews, at least rough sensory maps of their immediate environment – as noted by ethologists from the start. But it is generally assumed that only humans can build conceptual models of their environments. And, except for some philosophers, humans distinguish maps from the territories they represent." (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)
Alfred Korzybski - Collected Quotes
"If we consider an actual territory (a) say, Paris, Dresden, Warsaw, and build up a map (b) in which the order of these cities would be represented as Dresden, Paris, Warsaw; to travel by such a map would be misguiding, wasteful of effort. In case of emergencies, it might be seriously harmful. We could say that such a map was ‘not true’, or that the map had a structure not similar to the territory, structure to be defined in terms of relations and multidimensional order. We should notice that: A) A map may have a structure similar or dissimilar to the structure of the territory. B) Two similar structures have similar ‘logical’ characteristics. Thus, if in a correct map, Dresden is given as between Paris and Warsaw, a similar relation is found in the actual territory. C) A map is not the territory. D) An ideal map would contain the map of the map, the map of the map of the map, endlessly." (Alfred Korzybski, "Science and Sanity: A Non-Aristotelian System and Its Necessity for Rigour in Mathematics and Physics", 1931)
"A language is like a map; it is not the territory represented, but it may be a good map or a bad map. If the map shows a different structure from the territory represented-for instance, shows the cities in a wrong order, or some places east of others while in the actual territory they are west - then the map is worse than useless, as it misinforms and leads astray. One who made use of it could never be certain of reaching his destination. The use of ellanguage to represent events which operate as-a-whole is, at least, equally misguiding and semantically dangerous." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1933)
"As words are not the things we speak about, and structure is the only link between them, structure becomes the only content of knowledge. If we gamble on verbal structures that have no observable empirical structures, such gambling can never give us any structural information about the world. Therefore such verbal structures are structurally obsolete, and if we believe in them, they induce delusions or other semantic disturbances." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1933)
"Different 'philosophies' represent nothing but methods of evaluation, which may lead to empirical mis-evaluation if science and empirical facts are disregarded." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1933)
"Every language having a structure, by the very nature of language, reflects in its own structure that of the world as assumed by those who evolved the language. In other words, we read unconsciously into the world the structure of the language we use." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1933)
"If words are not things, or maps are not the actual territory, then, obviously, the only possible link between the objective world and the linguistic world is found in structure, and structure alone. The only usefulness of a map or a language depends on the similarity of structure between the empiricalworld and the map-languages. If the structure is not similar, then the traveller or speaker is led astray, which, in serious human life-problems, must become always eminently harmful. If the structures are similar, then the empirical world becomes 'rational' to a potentially rational being, which means no more than that verbal, or map-predicted characteristics, which follow up the linguistic or mapstructure, are applicable to the empirical world." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)
"What we call progress consists in coordinating ideas with realities." (Alfred Korzybski, "Manhood of Humanity", 1974)
"It seems evident that everything which exists in nature, is natural, no matter how simple or complicated a phenomenon it is; and on no occasion can the so-called 'supernatural' be anything else than a completely natural law, though it may, at the moment, be above and beyond the present understanding." (Alfred Korzybski, "Manhood of Humanity", 1974)
"Two important characteristics of maps should be noticed. A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness. If the map could be ideally correct, it would include, in a reduced scale, the map of the map; the map of the map, of the map [...]" (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1933)
"A model which took account of all the variegation of reality would be of no more use than a map at the scale of one to one." (Joan Robinson, "Essays in the Theory of Economic Growth", 1962)
"What we call progress consists in coordinating ideas with realities." (Alfred Korzybski, "Manhood of Humanity", 1974)
"It seems evident that everything which exists in nature, is natural, no matter how simple or complicated a phenomenon it is; and on no occasion can the so-called 'supernatural' be anything else than a completely natural law, though it may, at the moment, be above and beyond the present understanding." (Alfred Korzybski, "Manhood of Humanity", 1974)
04 October 2019
On Truth (1940-1949)
“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)
"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)
"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)
On Truth (1910-1919)
"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)
"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)
On Truth (1930-1939)
“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)
“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)
"When an induction, based on observations, is made, it is not intended that it shall be accepted as a universal truth, but it is advanced as a hypothesis for further study. Additional observations are then made and the results compared with the results expected from the hypothesis. If there is more deviation between the experimental results and the computed results than can be expected from the inaccuracies of observation and measurement, the scientist discards the' hypothesis and tries to formulate another."
"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)
"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)
On Truth (1990-1999)
“[…] 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)
"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)
"When a theory is sufficiently general to cover many fields of application, it acquires some 'truth' from each of them. Thus [...] a positive value for generalization in mathematics."
"Mathematics is good for the soul, getting things right enlivens a sense of truth, efforts to understand automatically purify desires." (Iris Murdoch, "Metaphysics As A Guide To Morals", 1992)
"[...] 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)
"[...] 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)
"Science undercuts ethics because we have made science the measure of all things. Truth means truth of science. Truth means logical truth or factual truth. Truth means math proof or data test. The truth can be a matter of degree. But that does not help ethics." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)
"Somewhere in the process wishful thinking seems to take over. Scientists start to believe they do math when they do science. This holds to greatest degree in an advanced science like physics or at the theoretical frontier of any science where the claims come as math claims. The first victim is truth. What was inaccurate or fuzzy truth all along gets bumped up a letter grade to the all-or-none status of binary logic. Most scientists draw the line at giving up the tentative status of science. They will concede that it can all go otherwise in the next experiment. But most have crossed the bivalent line by this point and believe that in the next experiment a statement or hypothesis or theory may jump from TRUE to FALSE, from 1 to 0." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)
"Truth is also where formal fuzzy logic begins. The mismatch problem - gray world but black-white scientific description - reduces to a truth problem, the problem of gray truth. The bivalence of modem science ignores or denies or whitewashes and blackwashes gray truth. That tactic leads to paradoxes and self-contradictions. The fuzzy view says almost all truth is gray truth, partial truth, fractional truth, fuzzy truth. It lets math truths remain black or white as extreme cases of gray." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 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)
"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)
On Truth (1980-1989)
"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 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)
"[…] 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)
"The mathematical theory of black holes is a subject of immense complexity; but its study has convinced me of the basic truth of the ancient mottoes 'The simple is the seal of the true' and 'Beauty is the splendor of truth.'" (Subrahmanyan Chandrasekhar, [Nobel lecture] 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 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)
"We can assert the truth, attain the truth, often enough. But we can never attain certainty." (Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)
"Deduction is typically distinguished from induction by the fact that only for the former is the truth of an inference guaranteed by the truth of the premises on which it is based. The fact that an inference is a valid deduction, however, is no guarantee that it is of the slightest interest." (John H Holland et al, "Induction: Processes Of Inference, Learning, And Discovery", 1986)
"Most people like to believe something is or is not true. Great scientists tolerate ambiguity very well. They believe the theory enough to go ahead; they doubt it enough to notice the errors and faults so they can step forward and create the new replacement theory. If you believe too much you'll never notice the flaws; if you doubt too much you won't get started. It requires a lovely balance." (Richard W Hamming, "You and Your Research", 1986)
"The task of us all as thinking human beings is to discover the truth. The truth is absolute and objective, but we do not have it in the bag. We are constantly seeking it and often find it only with difficulty; and we keep trying to improve our approximation to the truth. If truth were not absolute and objective, we should not be able to go wrong. Or our mistakes would be as good as our truth." (Karl R Popper, "The Epistemological Position of Evolutionary Epistemology", [in "All Life is Problem Solving", 1999] 1987)
"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)
On Truth (1970-1979)
"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)
"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)
"The idea of approximation to the truth is, in my view, one of the most important ideas in the theory of science. [...] The idea of approximation to the truth - like the idea of truth as are gulative principle - presupposes a realistic view of the world. It does not presuppose that reality is as our scientific theories describe it; but it does presuppose that there is a reality and that we and our theories - which are ideas we have ourselves created and are therefore always idealizations - can draw closer and closer to an adequate description of reality, if we employ the four-stage method of trial and error." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)
"The realistic view ofthe world, together with the idea of approximation to the truth, seem to me indispensable for an understanding of the perpetually idealizing character of science." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)
“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 agreement between a world apart from it is notoriously nebulous." (Nelson Goodman, "Ways of Worldmaking", 1978)
“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 agreement between a world apart from it is notoriously nebulous." (Nelson Goodman, "Ways of Worldmaking", 1978)
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