Great Throughts Treasury

"Whenever, for any reason, we wish to think of the world, not as it appears to common sense, but as a continuum, we find that our traditional syntax and vocabulary are quite inadequate. Mathematicians have therefore been compelled to invent radically new symbol-systems for this express purpose. But the divine Ground of all existence is not merely a continuum, it is also out of time, and different, not merely in degree, but in kind from the worlds to which traditional language and the languages of mathematics are adequate." - Aldous Leonard Huxley

"There exists a passion for comprehension, just as there exists a passion for music. That passion is rather common in children but gets lost in most people later on. Without this passion, there would be neither mathematics nor natural science." - Albert Einstein

"All the commandments follow three ways: faith, word, and deed. As one is basic in mathematics, so the essence of every commandment, whether it depends on speech or action, is faith of heart. If it has not that , all else is meaningless." - Abraham ben Meir ibn Ezra, also known as Ben Ezra or Abenezra

"Music has always had its own syntax, its own vocabulary and symbolic means. Indeed, it is with mathematics the principal language of the mind when the mind is in a condition of non-verbal feeling." - George Steiner, fully Francis George Steiner

"That the truths of the Bible have the power of awakening an intense moral feeling in every human being; that they make bad men good, and send a pulse of healthful feeling through all the domestic, civil, and social relations; that they teach men to love right, and hate wrong, and seek each other's welfare as children of a common parent; that they control the baleful passions of the heart, and thus make men proficient in self-government; and finally that they teach man to aspire after conformity to a being of infinite holiness, and fill him with hopes more purifying, exalted, and suited to his nature than any other book the world has ever known - these are facts as incontrovertible as the laws of philosophy, or the demonstrations of mathematics." - Francis Wayland

"Evolution is not necessarily a reductive theory: it does not explain away or reduce meaningfulness and value, any more than it explains away or reduces mathematics, economics, or even sociobiology itself. It aims to provide a naturalistic explanation of biological characteristics, including the capacities that enable us to recognize value and meaning. Giving a causal explanation of the origin of capacities is not the same as giving an account of the relevant meaning or content." - Garrett Thomson

"Computational irreducibility tends to make infinite questions undecidable. The presence of universality implies that there must at some level be computational irreducibility… This means that today’s mathematics will be viewed as small and surprisingly uncharacteristic sample of what is possible. If a system is computationally irreducible this means that there is in effect a tangible separation between the underlying rules for the system and its overall behavior associated with the irreducible amount of computational work needed to go from one to the other. And it is this separation that the basic origin of the apparent freedom we see in all sorts of system lie – whether those systems are abstract cellular automata or actual living brains." - Stephen Wolfram

"The overall similarity between mathematics and nature must have a deeper origin, both involving processes that can be thought of as computations." - Stephen Wolfram

"When it comes to more complex behavior mathematics has never in fact done well at explaining most of what we see every day in nature." - Stephen Wolfram

"If there should be any [persons], who though ignorant in Mathematics, yet pretending a skill in those Learnings, should dare, upon the authority of some place of Scripture wrested to their purpose, to condemn and censure my Hypothesis, I value them not, but shall slight their inconsiderate judgment." - Nicholas Copernicus

"Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."" - Albert Einstein

"To see what students learn in school, look at how they leave school. If they leave thinking that reading and writing are difficult and pointless, that mathematics is confusing, that history is irrelevant, and that art is a bore, then that is what they have been taught. People learn what is demonstrated to them, and this reality will not change to suit the convenience of politicians and educations administrators." - Frank Smith

"Mathematics... possesses not only truth, but supreme beauty - a beauty cold and austere, like that of a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." - Bertrand Russell, fully Bertrand Arthur William Russell, 3rd Earl Russell

"Reading maketh a full man; conference a ready man; histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral philosophy, grace; logic and rhetoric, able to contend." - Francis Bacon

"Mathematics is the language with which God has written the universe." - Galileo Galilei, known simply as Galileo

"If you help others, you will be helped, perhaps tomorrow, perhaps in one hundred years, but you will be helped. Nature must pay off the debt... It is a mathematical law and all life is mathematics." - George Gurdjieff, fully George Ivanovich Gurdjieff

"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness. " - Eric Temple Bell

"I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce." - Freeman John Dyson

"I must study politics and war, that my sons may have liberty to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain." - John Quincy Adams

"Science and mathematics run parallel to reality, they symbolize it, they squint at it, they never touch it: consider what an explosion would rock the bones of men into little white fragments and unsky the world if any mind for a moment touch truth." - Robinson Jeffers, fully John Robinson Jeffers

"A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." - John Von Newmann

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." - John Von Newmann

"If it were true [that there are mathematical problems undecidable by the human mind] it would mean that human reason is utterly irrational in asking questions it cannot answer, while asserting emphatically that only reason can answer them. Human reason would then be very imperfect and, in some sense, even inconsistent, in glaring contradiction to the fact that those parts of mathematics which have been systematically and completely developed show an amazing degree of beauty and perfection. In these fields, by entirely unexpected laws and procedures, means are provided not only for solving all relevant problems, but also solving them in a most beautiful and perfectly feasible manner." - Kurt Gödel, also Goedel

"If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies" - Nicholas Copernicus

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." - Nikola Tesla

"God used beautiful mathematics in creating the world. " - Paul Dirac, fully Paul Adrien Maurice Dirac

"I must say that I am very dissatisfied with the situation, because this so called good theory does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small - not neglecting it just because it is infinitely great and you do not want it! " - Paul Dirac, fully Paul Adrien Maurice Dirac

"If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics lead me by the hand. It has lead me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress." - Paul Dirac, fully Paul Adrien Maurice Dirac

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." - Paul Dirac, fully Paul Adrien Maurice Dirac

"Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future. A good many people are working on the mathematical basis of quantum theory, trying to understand the theory better and to make it more powerful and more beautiful. If someone can hit on the right lines along which to make this development, it may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply them." - Paul Dirac, fully Paul Adrien Maurice Dirac

"One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better." - Paul Dirac, fully Paul Adrien Maurice Dirac

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. ... It was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and non-commutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalization of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation. " - Paul Dirac, fully Paul Adrien Maurice Dirac

"There are, at present, fundamental problems in theoretical physics… the solution of which… will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely, these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will, therefore, have to proceed in a more direct way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities." - Paul Dirac, fully Paul Adrien Maurice Dirac

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." - Paul Klee

"No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language was mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. And relationships, he had told her, contained the essential meaning of life." - Pearl S. Buck, fully Pearl Sydenstricker Buck, also known by her Chinese name Sai Zhenzhu

"It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention." - Percy W. Bridgman, fully Percy Williams Bridgman

"Mathematics, which as a student by nature he will take very easily, will be prescribed to train him to abstract thought and to faith in the unembodied; a moral being by native disposition, he must be led to make his virtue perfect; after the Mathematics he must be put through a course in Dialectic and made an adept in the science." - Plotinus NULL

"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality." - Albert Einstein

"Looking back at the wake of my ship one day in 1917, I became interested in its beautiful white path. I said to myself, That path is white because of the different refractions of light by the bubbles of water-H20 (not H0). The bubbles are beautiful little spheres. I wonder how many bubbles I am looking at stretching miles astern. I began to make calculations of how many bubbles there were per cubic foot of water. I began to find that in calculating the ship's white wake I was dealing in quintillions to the fourth power times quintillions to the fourth power or some such fantastically absurd number of bubbles. And nature was making those bubbles in sublimely swift ease! Any time one looks carefully at a bubble, one is impressed with the beauty of its structure, its’ beautiful sphericity glinting with the colors of the spectrum. It is ephemeral-elegantly conceived, beautifully manufactured and readily broken. Inasmuch as the kind of mathematics I had learned of in school required the use of the XYZ coordinate system and the necessity of employing in calculating the spheres, I wondered, to how many decimal places does nature carry out before she decides the computation can't be concluded? Next I wondered, to how many arbitrary decimal places does nature carry out the transcendental irrational before she decides to say it's a bad job and call it off? If nature uses she has to do what we call fudging of her design which means improvising, compromisingly. I thought sympathetically of nature's having to make all those myriad frustrating decisions each time she made a bubble. I didn't see how she managed to formulate the wake of every ship while managing the rest of the universe if she had to make all those decisions. So I said to myself, I don't think nature uses. I think she has some other mathematical way of coordinating her undertakings." - Buckminster Fuller, fully Richard Buckminster "Bucky" Fuller

"Pure mathematics is, in its way, the poetry of logical ideas." - Albert Einstein

"In life there are so many factors involved that mathematical enumeration is the smallest and often the least important element involved. No illustration is more apt than the time-worn example of the logics, wherein it is presumed that if one man could dig a well in ten days, ten men could dig it in one. The mathematics is, of course, perfect, but worthless as overlooking the fact that ten men would, in that kind of a task, be in each other’s way." - Ralph Tyler Flewelling

"As I considered the matter carefully it gradually came to light that all those matters only were referred to mathematics in which order and measurements are investigated, and that it makes no difference whether it be in numbers, figures, stars, sounds or any other object that the question of measurement arises. I saw consequently that there must be some general science to explain that element as a whole which gives rise to problems about order and measurement, restricted as these are to no special subject matter. This, I perceived was called 'universal mathematics'." - René Descartes

"I knew that the languages which one learns there are necessary to understand the works of the ancients; and that the delicacy of fiction enlivens the mind; that famous deeds of history ennoble it and, if read with understanding, aid in maturing one's judgment; that the reading of all the great books is like conversing with the best people of earlier times; it is even studied conversation in which the authors show us only the best of their thoughts; that eloquence has incomparable powers and beauties; that poetry has enchanting delicacy and sweetness; that mathematics has very subtle processes which can serve as much to satisfy the inquiring mind as to aid all the arts and diminish man's labor; that treatises on morals contain very useful teachings and exhortations to virtue; that theology teaches us how to go to heaven; that philosophy teaches us to talk with appearance of truth about things, and to make ourselves admired by the less learned; that law, medicine, and the other sciences bring honors and wealth to those who pursue them; and finally, that it is desirable to have examined all of them, even to the most superstitious and false in order to recognize their real worth and avoid being deceived thereby" - René Descartes

"Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning." - Richard Feynman, fully Richard Phillips Feynman

"The real problem in speech is not precise language. The problem is clear language. The desire is to have the idea clearly communicated to the other person. It is only necessary to be precise when there is some doubt as to the meaning of a phrase, and then the precision should be put in the place where the doubt exists. It is really quite impossible to say anything with absolute precision, unless that thing is so abstracted from the real world as to not represent any real thing.?Pure mathematics is just an abstraction from the real world, and pure mathematics does have a special precise language for dealing with its own special and technical subjects. But this precise language is not precise in any sense if you deal with real objects of the world, and it is only pedantic and quite confusing to use it unless there are some special subtleties which have to be carefully distinguished." - Richard Feynman, fully Richard Phillips Feynman

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in." - Richard Feynman, fully Richard Phillips Feynman

"As an exercise of the reasoning faculties, pure mathematics is an admirable exercise, because it consists of reasoning alone and does not encumber the student with any exercise of judgment." - Richard Whately

"We do not learn by inference and deduction and the application of mathematics to philosophy, but by direct intercourse and sympathy." - Richard Nixon, fully Richard Milhous Nixon

"There is a largeness about mathematics that transcends race and time; mathematics may humbly help in the market-place, but it also reaches to the stars." - Robert James Turnbull

"The greatest mathematics has the simplicity and inevitableness of supreme poetry and music, standing on the borderland of all that is wonderful in Science, and all that is beautiful in Art. " - Robert James Turnbull