On Art: Poetry and Mathematics II

"The true spirit of delight, the exaltation...which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry."  (Bertrand Russell, “Mysticism and Logic”, 1917)

"Mathematics is one component of any plan for liberal education. Mother of all the sciences, it is a builder of the imagination, a weaver of patterns of sheer thought, an intuitive dreamer, a poet. The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)

"Mathematics in this sense is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process." (Jacob Bronowski, "Science and Human Values", 1956)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, “Philosophy of Mathematics”, 1999)

"Mathematicians have always appreciated clever notations; but symbolism is usually seen as a tool - it's what the tool does that we really care about. Fair enough. But if we want a richer appreciation of mathematics, we should focus some of our energy on this remarkable tool - notation. Besides mathematics, poetry alone works wonders with it." (James R Brown, “Philosophy of Mathematics”, 1999)

"[...] mathematics bears on poetry not only by analogy, but directly through metrics. Metrics is the science of poetry, and it would be healthy for poetry if that science were more widely and astutely studied." (Kurt Brown, “The Measured Word: On Poetry and Science”, 2001)

"What could mathematics and poetry share, except that the mention of either one is sometimes enough to bring an uneasy chill into a conversation? [...] Both fields use analogies - comparisons of all sorts - to explain things, to express unknown or unknowable concepts, and to teach." (Marcia Birken & Anne C Coon, “Discovering Patterns in Mathematics and Poetry”, 2008)

"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)

"You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things." (Jonathan D Farley)

"Proofs are to mathematics what spelling is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters." (Vladimir Arnold)

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5 Books 10 Quotes I: Mathematics in Its Creation

Walter W Sawyer, "What is Calculus About?”, 1961
	"In mathematics, a certain surprising thing happens again and again. Someone poses a simple question, a question so simple that it seems no useful result can come from answering it. And yet it turns out that  the answer opens the door to all kinds of interesting developments, and  gives great power to the person who understands it."
"Mathematics also is an exploration. As we push out further, we meet new and unexpected situations and we have to revise our ideas. Rules we have used, theorems we have proved turn out to have unforeseen weaknesses."
Ian Stewart, "In Pursuit of the Unknown", 2012
	“Equations have hidden powers. They reveal the innermost secrets of nature. […] The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us.”
“There are two kinds of equations in mathematics, which on the surface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician’s task is to solve it - to make the unknown known.”
Philip J Davis et al, "The Mathematical Experience", 1995
	“Where is the place of mathematics? Where does it exist? On the printed page, of course, and prior to printing, on tablets or on papyri. Here is a mathematical book - take it in your hand; you have a palpable record of mathematics as an intellectual endeavor. But first it must exist in people's minds, for a shelf of books doesn't create mathematics.”
“The definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights.”
David Ruelle, "Chance and Chaos", 1991
	"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner."
"Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole. Mathematics is a great  kingdom, and that kingdom belongs to those who see."
James R Brown,"Philosophy of Mathematics", 1999
	"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations."
“In sum, these are a few of the ingredients in the mathematical image: (1) Mathematical results are certain (2) Mathematics is objective (3) Proofs are essential (4) Diagrams are psychologically useful, but prove nothing (5) Diagrams can even be misleading (6) Mathematics is wedded to classical logic (7) Mathematics is independent of sense experience (8) The history of mathematics is cumulative (9) Computer proofs are merely long and complicated regular proofs (10) Some mathematical problems are unsolvable in principle” ||>> Next Post

Isn’t it Obvious? - Part I

"A thing is obvious mathematically after you see it." (Robert D Carmichael)

"The more original a discovery, the more obvious it seems afterwards." (Arthur Koestler)

"Everything you’ve learned in school as 'obvious' becomes less and less obvious as you begin to study the universe." (Richard B Fuller)

"The scientist is not content to stop at the obvious." (Charles H Mayo)

"There is nothing more deceptive than an obvious fact." (Sir Arthur C Doyle)

"The world is full of obvious things which nobody by any chance ever observes." (Sir Arthur C Doyle)

"The measure of the value of a new hypothesis in physics is not its obviousness but its utility." (Max Plank)

"To spell out the obvious is often to call it in question." (Eric Hoffer)

"No question is so difficult to answer as that to which the answer is obvious." (George B Shaw)

About Teachers

"A teacher affects eternity; he can never tell where his influence stops." (Henry Adams)

"The teacher is like the candle which lights others in consuming itself." (Giovanni Ruffini)

"A teacher is a compass that activates the magnets of curiosity, knowledge, and wisdom in the pupils." (Ever Garrison)

"A teacher is never a giver of truth; he is a guide, a pointer to the truth that each student must find for himself." (Bruce Lee)

"The teacher is of course an artist, but being an artist does not mean that he or she can make the profile, can shape the students. What an educator does in teaching is to make it possible for the students to become themselves." (Paulo Freire, "Pedagogy of the Oppressed")

"For many students, their teachers may be the only adult with whom they have a meaningful conversation all day." (Vickie Gill)

"A teacher is one who makes himself progressively unnecessary." (Thomas Carruthers)

"Teachers are those who help us in resolving problems which, without them, we wouldn’t have." (Anon)

"We need teachers. We need to be teachers. Knowing when for each, is wisdom." (Rick Beneteau)

Teaching vs Learning I

"To teach is to learn." (Japanese saying)

"To teach is to learn twice." (Joseph Joubert)

“One can only learn by teaching.” (John A Wheeler)

"Thoroughly to teach another is the best way to learn for yourself." (Tyron Edwards)

"If you want to learn a subject then teach it" (Paul Sloane)

“Who dares to teach must never cease to learn." (John Cotton Dana)

“No one learns as much about a subject as one who is forced to teach it.” (Peter F Drucker)

"A teacher never stops learning, a learner never stops teaching." (Todd Whitaker)

"Teachers open the door. But you must enter by yourself." (Chinese Proverb)

"We cannot teach people anything; we can only help them discover it within themselves." (Galileo Galilei)

Teaching Children

"Do not train a child to learn by force or harshness; but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each." (Plato)

“Teaching is not about how we see things, it is about how children see things.” (Kavita B Ghosh)

"Children must be taught how to think, not what to think." (Margaret Mead)

“A teacher who cannot explain any abstract subject to a child does not himself thoroughly understand his subject; if he does not attempt to break down his knowledge to fit the child's mind, he does not understand teaching.” (Fulton J. Sheen, “Life Is Worth Living”)

"When you want to teach children to think, you begin by treating them seriously when they are little, giving them responsibilities, talking to them candidly, providing privacy and solitude for them, and making them readers and thinkers of significant thoughts from the beginning. That’s if you want to teach them to think." (Bertrand Russell)

“Everyday classroom teaching is not what children will remember, but how you made a difference in their lives.” (Nita Ambani)

"The object of teaching a child is to enable him to get along without a teacher." (Elbert Hubbard)

"Failure is so much more interesting because you learn from it. That's what we should be teaching children at school, that being successful the first time, there's nothing in it. There's no interest, you learn nothing actually." (Sir James Dyson)

Good Teachers

"One good teacher outweighs a ton of books.” (Chinese saying)

“A good teacher is better than many books.” (German proverb)

"A good teacher can inspire hope, ignite the imagination, and instill a love of learning." (Brad Henry)

"The best teachers are those who show you where to look, but don't tell you what to see.”    (Alexandra K Trenfor)

“A good teacher isn't someone who gives the answers out to their kids but is understanding of needs and challenges and gives tools to help other people succeed.” (Justin Trudeau)

"The teacher, if indeed wise, does not bid you to enter the house of their wisdom, but leads you to the threshold of your own mind." (Kahil Gibran)

“A great teacher is not simply one who imparts knowledge to his students but is one who awakens their interest in the subject and makes them eager to pursue it for themselves. An outstanding teacher is a spark plug, not a fuel line.” (Norman J Berrill)

"Great teachers see challenging students as a reason to try that much harder." (Todd Whitaker)

"Great teachers have high expectations for their students, but higher expectations for themselves."  (Todd Whitaker)

"You do not have to be kind to be a teacher, but you do have to be kind to be a good one." (Todd Whitaker)

"The true teacher defends his pupils against his own personal influence." (Amos B Alcott)

"A good teacher protects his pupils from his own influence."  (Bruce Lee)

"Great teachers do not act important; they make their students feel important."    (Todd Whitaker)

"Great teachers treat their students the way their best teacher treated them."  (Al Burr)

"The best teachers make every decision based on what is best for their students."   (Al Burr)

"The struggle to become a better teacher begins all over again with the advent of each new class."   (Martin Henley)

Education: Ingredients of Education

“Educating the mind without educating the heart is no education at all." (Aristotle)

“Rest assured that literary education is no good without character.” (Gandhi)

“Education without values, as useful as it is, seems rather to make man a more clever devil.” (Clive S Lewis)

“An education which does not cultivate the will is an education that depraves the mind.” (Anatole France, “The Crime of Sylvestre Bonnard”, 1881)

“Education and morals will be found almost the whole that goes to make a good man.” (Aristotle)

“I respect faith, but doubt is what gets you an education.” (Wilson Mizner)

"Thinking is trying to make up the gap in one’s education" (Gilbert Ryle)

“Education comes from within; you get it by struggle and effort and thought. (Napoleon Hill)

"The secret of education is respecting the pupil." (Ralph W Emerson)

“The very spring and root of honesty and virtue lie in good education.” (Plutarch)

“To educate a man in mind and not in morals is to educate a menace to society.” (Theodore Roosevelt)

“Observation more than books and experience more than persons, are the prime educators.” (Amos B Alcott)

“Common sense and education are highly compatible; in fact, neither is worth much without the other.” (Donald G. Smith)

“A sense of curiosity is nature's original school of education.” (Smiley Blanton, “Love or Perish”)

Self-Education

"I have never let my schooling interfere with my education" (Mark Twain)

“There is no greater education than one that is self-driven.” (Neil deGrasse Tyson)

"Self-education is a continuing source of pleasure to me, for the more I know, the fuller my life is and the better I appreciate my own existence" (Isaac Asimov)

"Self-education is, I firmly believe, the only kind of education there is. The only function of a school is to make self-education easier; failing that, it does nothing." (Isaac Asimov)

“Formal education will make you a living; self-education will make you a fortune.” (Jim Rohn)

“A formal education will teach you how to conform with society. Self-education will teach you how to get out of conformity so that you can fill your life with adventure and beauty.” (Debasish Mridha)

“I will simply express my strong belief, that that point of self-education which consists in teaching the mind to resist its desires and inclinations, until they are proved to be right, is the most important of all, not only in things of natural philosophy, but in every department of daily life.” (Michael Faraday)

“The highest culture is not obtained from the teacher when at school or college, so much as by our ever-diligent self-education when we become men.” (Samuel Smiles)

“Only a development of thought achieved through the self-education of the whole man can prevent any body of thought whatsoever from becoming a poison; can prevent enlightenment from becoming an agent of death.” (Karl Jaspers)

“Vitally important for a young man or woman is, first, to realize the value of education and then to cultivate earnestly, aggressively, ceaselessly, the habit of self-education.” (Bertie C Forbes)

"Every man who rises above the common level has received two educations: the first from his teachers; the second, more personal and important, from himself." (Edward Gibbon)

Education at Superlative

“Education is the most powerful weapon which you can use to change the world.” (Nelson Mandela)

“Education is the key to unlock the golden door of freedom.” (George W Carver)

"Education is the best provision for old age." (Aristotle)

"The highest result of education is tolerance." (Helen Keller)

"The most important outcome of education is to help students to become independent of formal education." (Paul E Gray)

"The highest education is that which does not merely give us information but makes our life in harmony with all existence." (Rabindranath Tagore)

“The business of education is not to make the young perfect in any one of the sciences, but so to open their minds as may best make them capable of any, when they shall apply themselves to it.” (John Locke)

"Perhaps the most valuable result of all education is the ability to make yourself do the thing you have to do, when it ought to be done, whether you like it or not; it is the first lesson that ought to be learned; and however early a man's training begins, it is probably the last lesson that he learns thoroughly." (Thomas H Huxley)

“Education is the leading human souls to what is best, and making what is best out of them; and these two objects are always attainable together, and by the same means; the training which makes men happiest in themselves also makes them most serviceable to others.” (John Ruskin, “The Stones of Venice”, 1853)

“A human being is not, in any proper sense, a human being till he is educated.” (Horace Mann)

“All real education is the architecture of the soul.” (William Bennett)

Education at Pejorative

“It is a miracle that curiosity survives formal education.” (Albert Einstein)

“Nothing in education is so astonishing as the amount of ignorance it accumulates in the form of inner facts.” (Henry Adams)

"The present system of education is all wrong. The mind is crammed with facts before it knows how to think. Control of the mind should be taught first. It takes people a long time to learn things because they can't concentrate their minds at will." (Swami Vivekananda)

“Education has produced a vast population able to read but unable to distinguish what is worth reading.” (G M Trevelyan)

“Education appears to be the thing that enables a man to get along without using his intelligence.” (A E Wiggan)

“Education is one of the chief obstacles to intelligence and freedom of thought.” (Bertrand Russel)

"Nothing in education is so astonishing as the amount of ignorance it accumulates in the form of inert facts." (Henry B Adams)

“Education is a method by which one acquires a higher grade of prejudices.”  (Laurence J Peter)

“Education is the inculcation of the incomprehensible into the indifferent by the incompetent.” (John M Keynes)

“Much education today is monumentally ineffective. All too often we are giving young people cut flowers when we should be teaching them to grow their own plants.” (John W Gardner)

"The most dangerous thing about an academic education is that it enables my tendency to over-intellectualize stuff, to get lost in abstract thinking instead of simply paying attention to what’s going on in front of me.” (David F Wallace)

“It is the malady of our age that the young are so busy teaching us that they have no time left to learn.” (Eric Hoffer)

“[…] public education will always be mediocre, for the same reason that in large kitchens the cooking is usually bad.” (Friedrich Nietzsche)

“Education is the process of casting false pearls before real swine.” (Irwin Edman)

“Education does not make us smarter, it merely propels us further and faster in the direction of our native abilities; and if one's ability is to make a fool of himself, education can help him do a magnificent job of that.” (Sydney J Harris)

Out of Context: The Aim of Education

"Education's goal should be to develop a love of learning that stays with students throughout their lives. Education should be a lifetime experience, not limited to youth." (Bob Web)

"Education's purpose is to replace an empty mind with an open one." (Malcolm S Forbes)

"The aim of education is the knowledge not of facts but of values." (William R Inge)

"The aim [of education] must be the training of independently acting and thinking individuals who, however, see in the service to the community their highest life achievement." (Albert Einstein)

"The aim of education should be to convert the mind into a living fountain not a reservoir." (John Mason)

"The chief object of education is not to learn but to unlearn."  (G K Chesterton)

"The function of education is to teach one to think intensively and to think critically. Intelligence plus character - that is the goal of true education." (Martin L King, Jr.)

"The goal of education is not to increase the amount of knowledge but to create the possibilities for a child to invent and discover, to create men who are capable of doing new things." (Jean Piaget)

"The goal of education is the advancement of knowledge and the dissemination of truth." (John F Kennedy)

"The great aim of education is not knowledge but action." (Herbert Spencer)

"The object of education is to give man the unity of truth." (Rabindranath Tagore)

"The object of education is to teach us to love what is beautiful." (Plato, “The Republic”)

"The object of education is to prepare the young to educate themselves throughout their lives." (Robert M Hutchins)

"The principle goal of education is to create men who are capable of doing new things, not simply of repeating what other generations have done – men who are creative, inventive and discoverers.” (Jean Piaget)

"The whole purpose of education is to turn mirrors into windows." (Sydney J Harris)

Education: What is Education Not?

"Education must not simply teach work - it must teach life." (W E B DuBois)

"Education is not preparation for life; education is life itself." (John Dewey)

"Education is not the filling of a pail, but the lightning of a fire." (William B Yeats)

"Education is not learning, but the training of the mind that it may learn." (Sir William W Gull)

"Education is not training but rather the process that equips you to entertain yourself, a friend and an idea." (Wallace Sterling)

"Education is habit-forming rather than information; illumination rather than indoctrination, inspiration rather than compulsion." (Edwin G Conklin, "Science in the World Crisis", 1939)

"Education is not the piling on of learning, information, data, facts, skills, or abilities - that's training or instruction - but is rather making visible what is hidden as a seed." (Thomas W Moore)

"What is education? Is it book-learning? No. Is it diverse knowledge? Not even that. The training by which the current and expression of will are brought under control and become fruitful is called education." (Swami Vivekananda)

"An education isn't how much you have committed to memory, or even how much you know. It's being able to differentiate between what you do know and what you don't." (Anatole France)

"Education does not mean teaching people to know what they do not know; it means teaching them to behave as they do not behave." (John Ruskin)

"The object of education is not only to produce a man who knows, but one who does; who makes his mark in the straggle of life and succeeds well in whatever he undertakes: who can solve the problems of nature and of humanity as they arise, and who, when he knows he is right, can boldly convince the world of the fact." (Henry A Rowland)

"A real education will not teach you to compete; it will teach you to co-operate. It will not teach you to fight and come first. It will teach you to be creative, to be loving, to be blissful without any comparison with the other." (Osho)

"Education can't make us all leaders, but it can teach us which leader to follow." (Bel Kaufman)

What is Education?

“Education is the apprenticeship of life.” (Robert A Willmott)

“Education is teaching our children to desire the right things." (Plato)

“Education is the art of making man ethical.” (Georg W F Hegel)

"Education is the only quality which remains after we have forgotten all we have learned.” (VoItaire)

"Education is simply the soul of a society as it passes from one generation to another." (Gilbert K Chesterton)

“Education is the manifestation of the perfection already in man." (Swami Vivekananda)

“Education is the key to unlock the golden door of freedom." (George W Carver)

“Education is a progressive discovery of our own ignorance.” (Will Durant)

“Education is the movement from darkness to light.” (Allan Bloom)

“Education is the ability to listen to almost anything without losing your temper or your self-confidence.” (Robert Frost)

"Education is the power to think clearly, the power to act well in the world's work, and the power to appreciate life." (Brigham Young)

"Education is the best way to train ourselves that we will secure our own well-being by concerning ourselves with others.” (Dalai Lama)

What is Mathematics not? - Part I

“Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined; it is as limitless as the space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell and is forever ready to burst forth into new forms of vegetable and animal existence. “ (James J Sylvester, "The Educational Times", 1877)

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

"It has been argued that mathematics is not or, at least, not exclusively an end in itself; after all it should also be applied to reality. But how can this be done if mathematics consisted of definitions and analytic theorems deduced from them and we did not know whether these are valid in reality or not. One can argue here that of course one first has to convince oneself whether the axioms of a theory are valid in the area of reality to which the theory should be applied. In any case, such a statement requires a procedure which is outside logic.” (Ernst Zermelo, "Mathematische Logik - Vorlesungen gehalten von Prof. Dr. E. Zermelo zu Göttingen im S. S", 1908)

"Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.” (David Hilbert, “Natur und Mathematisches Erkennen”, 1919–20)

“[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words.” (Charles Nordmann, "Einstein and the Universe", 1922)

“Mathematics is not a world empire where one man can exert a dominating influence over work along all the lines, but it is continually splitting up into self-determining republics.” (Geroge A Miller, “Felix Klein and the History of Modern Mathematics”, 1927)

“Mathematics is not a complete and perfected body of doctrine which one goes to seek in the scriptures whenever one has need of it.” (Karl K Darrow, “The Renaissance of Physics”, 1936)

“[…] mathematics is not an empirical science, or at least that it is practiced in a manner which differs in several decisive respects from the techniques of the empirical sciences.” (John von Neumann, “The Mathematician“, Works of the Mind Vol. I, 1947) [Link]