"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)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
04 June 2019
On Truth (1900-1909)
"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 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)
"Science is the only truth and it is the great lie. It knows nothing, and people think it knows everything. It is misrepresented. People think that science is electricity, automobilism, and dirigible balloons. It is something very different. It is life devouring itself. It is the sensibility transformed into intelligence. It is the need to know stifling the need to live. It is the genius of knowledge vivisecting the vital genius." (Rémy de Gourmont, "Art and Science", cca. 1905-1909)
On Truth (1800-1849)
“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)
"[...] 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)
"Facts, however numerous, do not constitute a science. Like innumerable grains of sand on the sea shore, single facts appear isolated, useless, shapeless; it is only when compared, when arranged in their natural relations, when crystallised by the intellect, that they constitute the eternal truths of science." (William Farr, "Observation", Br. Ann. Med. 1, 1837)
“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 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)
On Truth (until 1599)
“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)
“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)
"In creation, on the other hand, truth is one thing, reason another. For in creation, truth is an image of the divinity, which is sought and found by reason in created things. Reason is a virtue or activity of the mind, whose object is to discern truth. Truth, like reason, does not have any contrary, and this for the same cause that was given and explained above in regard to reason." (John of Salisbury, "Metalogicon", 1159)
"[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)
"Truth is both the light of the mind and the subject matter of reason." (John of Salisbury, "Metalogicon", 1159)
“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)
“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)
03 June 2019
Mathematics as Language II
"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language." (Herbert G Wells, “Mankind in the Making”, 1904)
”Mathematics is the language of definiteness, the necessary vocabulary of those who know. Hence the intimate connection between mathematics and science.” (William F White, “A Scrap-book of Elementary Mathematics”, 1908)
“[…] one does not know a foreign language unless on is able to think in that language; one does not know mathematics unless one is able to think mathematically.” (John A L Waddell, 1908)
”One of the difficulties which a mathematician has in describing his work to non-mathematicians is that the present day language of mathematics has become so esoteric that a well educated layman, or even a group of scientists, can comprehend essentially nothing of the discourse which mathematicians hold with each other, or of the accounts of their latest researches which are published in their professional journals.” (Angus E Taylor, “Some Aspects of Mathematical Research”, American Scientist , Vol. 35, No. 2, 1947)
“The study of mathematics cultivates the reason; that of the languages, at the same time, the reason and the taste. The former gives the grasp and power to the mind; the latter both power and flexibility. The former by itself, would prepare us for a state of certainties, which nowhere exists; the later, for a state of probabilities which is that of common life. Each, by itself, does but an imperfect work: in the union of both, is the best discipline for the mind, and the best mental training for the world as it is.” (Tyron Edwards, “The New Dictionary of Thoughts”, 1948)
”[…] 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 infinitely wide, while the language that describes it is finite. It follows from the pigeonhole principle that there exist distinct concepts that are referred to by the same name. Mathematics is also infinitely deep and sometimes entirely different concepts turn out to be intimately and profoundly related.” (Doron Zeilberger, 1988)
”Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding.” (Robert Kanigel, “The Man Who Knew Infinity”, 1991)
”Mathematics is not a way of hanging numbers on things so that quantitative answers to ordinary questions can be obtained. It is a language that allows one to think about extraordinary questions.” (James O Bullock, “Literacy in the Language of Mathematics”, The American Mathematical Monthly, Vol. 101, No. 8, October, 1994)
"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain "recipes" for calculating things." (Ian Stewart, “Why Beauty Is Truth”, 2007)
"Mathematics is not a language, it's an adventure." (Paul Lockhart, “A Mathematician's Lament", 2009)
”Mathematics is the language of definiteness, the necessary vocabulary of those who know. Hence the intimate connection between mathematics and science.” (William F White, “A Scrap-book of Elementary Mathematics”, 1908)
“[…] one does not know a foreign language unless on is able to think in that language; one does not know mathematics unless one is able to think mathematically.” (John A L Waddell, 1908)
”One of the difficulties which a mathematician has in describing his work to non-mathematicians is that the present day language of mathematics has become so esoteric that a well educated layman, or even a group of scientists, can comprehend essentially nothing of the discourse which mathematicians hold with each other, or of the accounts of their latest researches which are published in their professional journals.” (Angus E Taylor, “Some Aspects of Mathematical Research”, American Scientist , Vol. 35, No. 2, 1947)
“The study of mathematics cultivates the reason; that of the languages, at the same time, the reason and the taste. The former gives the grasp and power to the mind; the latter both power and flexibility. The former by itself, would prepare us for a state of certainties, which nowhere exists; the later, for a state of probabilities which is that of common life. Each, by itself, does but an imperfect work: in the union of both, is the best discipline for the mind, and the best mental training for the world as it is.” (Tyron Edwards, “The New Dictionary of Thoughts”, 1948)
”[…] 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 infinitely wide, while the language that describes it is finite. It follows from the pigeonhole principle that there exist distinct concepts that are referred to by the same name. Mathematics is also infinitely deep and sometimes entirely different concepts turn out to be intimately and profoundly related.” (Doron Zeilberger, 1988)
”Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding.” (Robert Kanigel, “The Man Who Knew Infinity”, 1991)
”Mathematics is not a way of hanging numbers on things so that quantitative answers to ordinary questions can be obtained. It is a language that allows one to think about extraordinary questions.” (James O Bullock, “Literacy in the Language of Mathematics”, The American Mathematical Monthly, Vol. 101, No. 8, October, 1994)
"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain "recipes" for calculating things." (Ian Stewart, “Why Beauty Is Truth”, 2007)
"Mathematics is not a language, it's an adventure." (Paul Lockhart, “A Mathematician's Lament", 2009)
02 June 2019
On Metaphors (1900-1949)
"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)
"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", 1930)
"Thought is metaphoric, and proceeds by comparison, and the metaphors of language derive therefrom." (Ivor A Richards, "The Philosophy of Rhetoric", 1936)
"Mathematical research can lend its organisational characteristics to poetry, whereby disjointed metaphors take on a universal sense. Similarly, the axiomatic foundations of group theory can be assimilated into a larger moral concept of a unified universe. Without this, mathematics would be a laborious Barbary." (Dan Barbilian, "The Autobiography of the Scientist", 1940)
"[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts. " (Dan Barbilian, "The Autobiography of the Scientist", 1940)
"At the final stage you teach me that this wondrous and multicolored universe can be reduced to the atom and that the atom itself can be reduced to the electron. All this is good and I wait for you to continue. But you tell me of an invisible planetary system in which electrons gravitate around a nucleus. You explain this world to me with an image. I realize then that you have been reduced to poetry: I shall never know. Have I the time to become indignant? You have already changed theories. So that science that was to teach me everything ends up in a hypothesis, that lucidity founders in metaphor, that uncertainty is resolved in a work of art." (Albert Camus, "The Myth of Sisyphus", 1942)
"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)
"The price of metaphor is eternal vigilance." (Arturo Rosenblueth & Norbert Wiener, "The Role of Models in Science", Philosophy of Science Vol. 12, 1945)
"A man desiring to understand the world looks about for a clue to its comprehension. He pitches upon some area of commonsense fact and tries to understand other areas in terms of this one. The original area becomes his basic analogy or root metaphor." (Stephen Pepper, "World Hypotheses: A Study in Evidence", 1948)
Isn't it Obvious? - Part II
“It [science] involves an intelligent and persistent endeavor to revise current beliefs so as to weed out what is erroneous, to add to their accuracy, and, above all, to give them such shape that the dependencies of the various facts upon one another may be as obvious as possible.” (John Dewey, “Democracy and Education”, 1916)
“The reason the obvious in Nature goes unrecognized so long is because artful man misdoubts her plain message and in a spirit of subtlety reads in a farrago of irrelevancies between the lines.” (George H Lepper, “From Nebula to Nebula”, 1917)
”It requires a very unusual mind to undertake the analysis of the obvious.” (Alfred N Whitehead, “Science in the Modern World”, 1925)
“’Obvious’ is the most dangerous word in mathematics.” (Eric T Bell, “Mathematics: Queen and Servant of Science”, 1951)
”The obvious is that which is never seen until someone expresses it simply.” (Kahlil Gibran, “Sand and Foam: A Book of Aphorisms”, 1959)
“Significant advances in science often have a peculiar quality: they contradict obvious, commonsense opinions.” (Salvador E Luria, “A Slot Machine, a Broken Test Tube”, 1984)
“[…] creativity is the ability to see the obvious over the long term, and not to be restrained by short-term conventional wisdom.” (Arthur J Birch, “To See the Obvious”, 1995)
“The reason the obvious in Nature goes unrecognized so long is because artful man misdoubts her plain message and in a spirit of subtlety reads in a farrago of irrelevancies between the lines.” (George H Lepper, “From Nebula to Nebula”, 1917)
”It requires a very unusual mind to undertake the analysis of the obvious.” (Alfred N Whitehead, “Science in the Modern World”, 1925)
“’Obvious’ is the most dangerous word in mathematics.” (Eric T Bell, “Mathematics: Queen and Servant of Science”, 1951)
”The obvious is that which is never seen until someone expresses it simply.” (Kahlil Gibran, “Sand and Foam: A Book of Aphorisms”, 1959)
“Significant advances in science often have a peculiar quality: they contradict obvious, commonsense opinions.” (Salvador E Luria, “A Slot Machine, a Broken Test Tube”, 1984)
“[…] creativity is the ability to see the obvious over the long term, and not to be restrained by short-term conventional wisdom.” (Arthur J Birch, “To See the Obvious”, 1995)
What is Mathematics not? - Part IV
"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)
"It is sometimes said that mathematics is not an experimental subject. This is not true! Mathematicians often use the evidence of lots of examples to help form a conjecture, and this is an experimental approach. Having formed a conjecture about what might be true, the next task is to try to prove it." (George M Phillips, "Mathematics Is Not a Spectator Sport", 2000)
"Mathematics is not monolithic in its general subject matter. There is no such thing as the geometry or the set theory or the formal logic. Rather, there are mutually inconsistent versions of geometry, set theory, logic, and so on. Each version forms a distinct and internally consistent subject matter." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is not just about numbers. As well as numbers, modern mathematics also looks at the relations between them. The passage from pure numerology to this new vision has derived from the realization that the most profound meaning is not in the numbers but in the relations between them. Mathematical investigation is precisely the exploration and the study of the different possible relations; some of them find a concrete and immediate application in the environment in which they are immersed, others just ‘live’ in the minds of those that conceive them." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos and Complexity: The Dynamics of Natural and Social Systems", 2003)
"Contrary to popular belief, mathematics is not a universal language. Rather, mathematics is based on a strict set of definitions and rules that have been instated and to which meaning has been given." (Christopher Tremblay, "Mathematics for Game Developers", 2004)
"Mathematics is not about abstract entities alone but is about relation of abstract entities with real entities. […] Adequacy relations between abstract and real entities provide space or opportunity where mathematical and logical thought operates parsimoniously." (Navjyoti Singh, "Classical Indian Mathematical Thought", 2005)
"Mathematics is not a matter of ‘anything goes,’ and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate – whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time, mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)
"Logic moves in one direction, the direction of clarity, coherence, and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. Mathematics is not a fixed, static entity that can be structured definitively. It is dynamic, alive: its dynamism a function of the relationship between the two poles that have been described above. It is the interactions between these different aspects that give mathematics its power." (William Byers, "How Mathematicians Think", 2007)
"[…] mathematics is not only to teach the algorithms and skills of mathematics - which we will agree are very important - but also to teach for understanding, with an emphasis on reasoning." (Alfred S Posamentier et al, "Exemplary Practices for Secondary Math Teachers", 2007)
"[…] mathematics is not only an essential tool for science and technology, but also for humanities, in particular for art. And out of art we may say that mathematics gains one of its main reasons for developing and changing in time. Mathematics contributes to our way of conceiving and shaping the world we live in, while art develops the means to harmonize, describe, represent aesthetically - or even to transcend and transfigure - the world of our sensations and perception." (Mauro Francaviglia, "Art and Mathematics", 2008)
"Mathematics is not an inevitable body of knowledge. Understanding it and doing it requires a consciousness of the ‘rules’ and the awareness that they are rules or conventions. Such awareness is particularly needed at the early stages where we often act as if there is nothing to be surprised about." (Bill Barton, "The Language of Mathematics: Telling Mathematical Tales", 2008)
What is Mathematics not? - Part III
“[…] mathematics is not a science – it is not capable of proving or disproving the existence of real things. In fact, a mathematician’s ultimate concern is that his or her inventions be logical, not realistic.” (Michael Guillen, “Bridges to Infinity: The Human Side of Mathematics”, 1983)
“[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics.” (Mary Tiles, “Bachelard: Science and Objectivity”, 1984)
“Mathematics is not a deductive science – that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.” (Paul R Halmos, “I Want to Be A Mathematician”, 1985)
“Mathematics is not a topic that one can easily approach with a virgin mind.” (Wilfrid Hodges, “Building Models by Games”, 1985)
“Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it.” (Morris Kline, “Mathematics and the Search for Knowledge”, 1985)
“[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost.” (Mark Kac, “Mathematics: Tensions”, 1992)
“Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations.” (Anna Sierpinska, “Understanding in Mathematics”, 1994)
“Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle.” (Ian Stewart, “Nature’s Numbers”, 1995)
“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis et al, “The Mathematical Experience”, 1995)
“There is no end to discoveries in mathematics just as there is no end to the mystery of the universe. Both are boundless. Hence mathematics is not so much a body of knowledge as a way of thought with inexhaustible possibilities.” (Karma V Mital, “Understanding Mathematics And Computers” , 1997)
“[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics.” (Mary Tiles, “Bachelard: Science and Objectivity”, 1984)
“Mathematics is not a deductive science – that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.” (Paul R Halmos, “I Want to Be A Mathematician”, 1985)
“Mathematics is not a topic that one can easily approach with a virgin mind.” (Wilfrid Hodges, “Building Models by Games”, 1985)
“Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it.” (Morris Kline, “Mathematics and the Search for Knowledge”, 1985)
“[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost.” (Mark Kac, “Mathematics: Tensions”, 1992)
“Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations.” (Anna Sierpinska, “Understanding in Mathematics”, 1994)
“Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle.” (Ian Stewart, “Nature’s Numbers”, 1995)
“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis et al, “The Mathematical Experience”, 1995)
“There is no end to discoveries in mathematics just as there is no end to the mystery of the universe. Both are boundless. Hence mathematics is not so much a body of knowledge as a way of thought with inexhaustible possibilities.” (Karma V Mital, “Understanding Mathematics And Computers” , 1997)
What is Mathematics not? - Part II
“Mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.” (David Hilbert, “Geometry and the Imagination”, 1952)
“Mathematics is not only the model along the lines of which the exact sciences are striving to design their structure; mathematics is the cement which holds the structure together.” (Tobias Dantzig, “Number: The Language of Science”, 1954)
“Mathematics is not, then, merely a bunch of calculations that could be done much faster and better by an electronic machine; nor is it a science developed in ancient times, with nothing added to it since then except a bit of polish here and there .” (C Stanley Ogilvy, “Excursions in Mathematics”, 1956)
“The eternal lesson is that Mathematics is not something static, closed, but living and developing. Try as we may to constrain it into a closed form, it finds an outlet somewhere and escapes alive.” (Péter Rózsa, “Playing with Infinity”, 1961)
“Mathematics is not a question of calculation perforce but rather the presence of royalty: a law of infinite resonance, consonance and order.” (Le Corbusier, "Architecture and the Mathematical Spirit", 1962)
“[…] although mathematical concepts and operations are formulated to represent aspects of the physical world, mathematics is not to be identified with the physical world. However, it tells us a good deal about that world if we are careful to apply it and interpret it properly.” (Morris Kline, “Mathematics for the Nonmathematician”, 1967)
“[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization.” (Morris Kline, “Mathematics for the Nonmathematician”, 1967)
“The introduction and gradual acceptance of concepts that have no immediate counterparts in the real world certainly forced the recognition that mathematics is a human, somewhat arbitrary creation, rather than an idealization of the realities in nature, derived solely from nature. But accompanying this recognition and indeed propelling its acceptance was a more profound discovery - mathematics is not a body of truths about nature.” (Morris Kline, “Mathematical Thought from Ancient to Modern Times” Vol. III, 1972)
“Mathematics is not a compendium or memorizable formula and magically manipulated figures.” (Scott Buchanan, “Poetry and Mathematics”, 1975)
"The essence of mathematics is not to make simple things complicated, but to make complicated things simple." (Stan Gudder, “A Mathematical Journey”, 1976)
“Mathematics is not a branch of aesthetics. The mistake, which is common enough, probably stems from the requirement of aesthetic unity but is not identical with that unity.” (J K Feibleman, “Assumptions of Grand Logics”, 1979)
“Mathematics is not only the model along the lines of which the exact sciences are striving to design their structure; mathematics is the cement which holds the structure together.” (Tobias Dantzig, “Number: The Language of Science”, 1954)
“Mathematics is not, then, merely a bunch of calculations that could be done much faster and better by an electronic machine; nor is it a science developed in ancient times, with nothing added to it since then except a bit of polish here and there .” (C Stanley Ogilvy, “Excursions in Mathematics”, 1956)
“The eternal lesson is that Mathematics is not something static, closed, but living and developing. Try as we may to constrain it into a closed form, it finds an outlet somewhere and escapes alive.” (Péter Rózsa, “Playing with Infinity”, 1961)
“Mathematics is not a question of calculation perforce but rather the presence of royalty: a law of infinite resonance, consonance and order.” (Le Corbusier, "Architecture and the Mathematical Spirit", 1962)
“[…] although mathematical concepts and operations are formulated to represent aspects of the physical world, mathematics is not to be identified with the physical world. However, it tells us a good deal about that world if we are careful to apply it and interpret it properly.” (Morris Kline, “Mathematics for the Nonmathematician”, 1967)
“[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization.” (Morris Kline, “Mathematics for the Nonmathematician”, 1967)
“The introduction and gradual acceptance of concepts that have no immediate counterparts in the real world certainly forced the recognition that mathematics is a human, somewhat arbitrary creation, rather than an idealization of the realities in nature, derived solely from nature. But accompanying this recognition and indeed propelling its acceptance was a more profound discovery - mathematics is not a body of truths about nature.” (Morris Kline, “Mathematical Thought from Ancient to Modern Times” Vol. III, 1972)
“Mathematics is not a compendium or memorizable formula and magically manipulated figures.” (Scott Buchanan, “Poetry and Mathematics”, 1975)
"The essence of mathematics is not to make simple things complicated, but to make complicated things simple." (Stan Gudder, “A Mathematical Journey”, 1976)
“Mathematics is not a branch of aesthetics. The mistake, which is common enough, probably stems from the requirement of aesthetic unity but is not identical with that unity.” (J K Feibleman, “Assumptions of Grand Logics”, 1979)
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🅷James S Hodges - Collected Quotes
"A bad model is a combination of assertions, some fac tual, others conjectural, and others plainly false but convenient. [..] By defi...