Best 579 quotes in «mathematics quotes» category

{Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was 'dull', showing off his spontaneous mathematical genius} No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.

Is everyone with one face called a Milo?" "Oh no," Milo replied; "some are called Henry or George or Robert or John or lots of other things." "How terribly confusing," he cried. "Everything here is called exactly what it is. The triangles are called triangles, the circles are called circles, and even the same numbers have the same name. Why, can you imagine what would happen if we named all the twos Henry or George or Robert or John or lots of other things? You'd have to say Robert plus John equals four, and if the four's name were Albert, things would be hopeless." "I never thought of it that way," Milo admitted. "Then I suggest you begin at once," admonished the Dodecahedron from his admonishing face, "for here in Digitopolis everything is quite precise.

I shall treat the nature and power of the Affects, and the power of the Mind over them, by the same Method by which, in the preceding parts, I treated God and the Mind, and I shall consider human actions and appetites just as if it were a Question of lines, planes, and bodies.

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

[The Old Astronomer to His Pupil] Reach me down my Tycho Brahe, I would know him when we meet, When I share my later science, sitting humbly at his feet; He may know the law of all things, yet be ignorant of how We are working to completion, working on from then to now. Pray remember that I leave you all my theory complete, Lacking only certain data for your adding, as is meet, And remember men will scorn it, 'tis original and true, And the obloquy of newness may fall bitterly on you. But, my pupil, as my pupil you have learned the worth of scorn, You have laughed with me at pity, we have joyed to be forlorn, What for us are all distractions of men's fellowship and smiles; What for us the Goddess Pleasure with her meretricious smiles. You may tell that German College that their honor comes too late, But they must not waste repentance on the grizzly savant's fate. Though my soul may set in darkness, it will rise in perfect light; I have loved the stars too fondly to be fearful of the night. What, my boy, you are not weeping? You should save your eyes for sight; You will need them, mine observer, yet for many another night. I leave none but you, my pupil, unto whom my plans are known. You 'have none but me,' you murmur, and I 'leave you quite alone'? Well then, kiss me, -- since my mother left her blessing on my brow, There has been a something wanting in my nature until now; I can dimly comprehend it, -- that I might have been more kind, Might have cherished you more wisely, as the one I leave behind. I 'have never failed in kindness'? No, we lived too high for strife,-- Calmest coldness was the error which has crept into our life; But your spirit is untainted, I can dedicate you still To the service of our science: you will further it? you will! There are certain calculations I should like to make with you, To be sure that your deductions will be logical and true; And remember, 'Patience, Patience,' is the watchword of a sage, Not to-day nor yet to-morrow can complete a perfect age. I have sown, like Tycho Brahe, that a greater man may reap; But if none should do my reaping, 'twill disturb me in my sleep So be careful and be faithful, though, like me, you leave no name; See, my boy, that nothing turn you to the mere pursuit of fame. I must say Good-bye, my pupil, for I cannot longer speak; Draw the curtain back for Venus, ere my vision grows too weak: It is strange the pearly planet should look red as fiery Mars,-- God will mercifully guide me on my way amongst the stars.

I think a strong claim can be made that the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well constructed theory is in some respects undoubtedly an artistic production. A fine example is the famous Kinetic Theory of Maxwell. ... The theory of relativity by Einstein, quite apart from any question of its validity, cannot but be regarded as a magnificent work of art.

It is a funny paradox of design: utility breeds beauty. There is elegance in efficiency, a visual pleasure in things that just barely work.

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true.

It is mathematics which reveals every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever then has the effrontery to study physics while neglecting mathematics, should know from the start that he will never make his entry into the portals of wisdom.

It is not knowledge, but the act of learning, not the possession of but the act of getting there, which grants the greatest enjoyment.

It’s all accumulation and the aftermath,' she says as I would question her about the failing earth and giants unaware that they are sinking everyday

It is said that Mathematics is the language of nature. If so, Physics is its poetry.

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

It is shown that the golden ratio plays a prominent role in the dimensions of all objects which exhibit five-fold symmetry. It is also showed that among the irrational numbers, the golden ratio is the most irrational and, as a result, has unique applications in number theory, search algorithms, the minimization of functions, network theory, the atomic structure of certain materials and the growth of biological organisms.

Lots of people wrote to the magazine to say that Marilyn vos Savant was wrong, even when she explained very carefully why she was right. Of the letters she got about the problem, 92% said that she was wrong and lots of these were from mathematicians and scientists. Here are some of the things they said: 'I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error.' -Robert Sachs, Ph.D., George Mason University ... 'I am sure you will receive many letters from high school and college students. Perhaps you should keep a few addresses for future columns.' -W. Robert Smith, Ph.D., Georgia State University... 'If all those Ph.D.'s were wrong, the country would be in very serious trouble.' -Everett Harman, Ph.D., U.S. Army Research Institute

Littlewood, on Hardy's own estimate, is the finest mathematician he has ever known. He was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power.

Logic, it is often said, is the study of valid arguments. It is a systematic attempt to distinguish valid arguments from invalid arguments.

Luck is not some esoteric, godlike phenomenon. Luck is countable but undefinable. Luck easily can be explained as number of factors acting in a favour of a person. These factors' behaviour could be statistically proved , and the probability of such result is possible. It is not related to something explainable event. Actually, the miracle would be if these events (luck) are not in presence in our life. The matter as then would be mathematics proved wrong. So, make your luck!"

Magic is like a lot of other disciplines that people have recently begun developing, in historic terms. Working with magic is a way of understanding the universe and how it functions. You can approach it from a lot of different angles, applying a lot of different theories and mental models to it. You can get to the same place through a lot of different lines of theory and reasoning, kind of like really advanced mathematics. There's no truly right or wrong way to get there, either--there are just different ways, some more or less useful than others for a given application. And new vistas of thought, theory, and application open up on a pretty regular basis, as the Art develops and expands through the participation of multiple brilliant minds. But that said, once you have a good grounding in it,you get a pretty solid idea of what's possible and what isn't. No matter how much circumlocution you do with your formulae, two plus two doesn't equal five. (Except maybe very, very rarely, sometimes, in extremely specific and highly unlikely circumstances.)

Mathemata mathematicis scribuntur.

Man had been given a brain that could think in numbers, and it could not be coincidence that the world was unlocked by that very tool. To understand any aspect of the cosmos was to look on the face of God: not directly, but by a species of triangulation, because to think mathematically was to feel the action of God in oneself.

Mathematical truth is immutable; it lies outside physical reality... This is our belief; this is our core motivating force.

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.

Mathematicians call it “the arithmetic of congruences.” You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, … up to 11. If the time is eight o’clock, and you add 9 hours, what do you get? Well, you get five o’clock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 ≡ 5 (mod 12), pronounced “eight plus nine is congruent to five, modulo twelve.

Mathematics is not just a subject of education system, it is the soul of education system.

Mathematics is one of the major modern mysteries. Perhaps it is the leading one, occupying a place in our society similar to the religious mysteries of another age. If we want to know something about what our age is all about, we should have some understanding of what mathematics is, and of how the mathematician operates and thinks.

Mathematics had never had more than a secondary interest for him [her husband, George Boole]; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to give rest had no tendency to prove them.

Mathematics is Open Source.

Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories and anecdotes are important to the young in understanding real life

Mathematics and poetry are the two ways to drink the beauty of truth.

Mathematics is a wonderful common sense, To understand the harmony in nature magnificent; The wondrous language, & popular in terms of @ finance Its’ unique, and abundantly rich in terms of substance,

Mathematics, which most of us see as the most factual of all sciences, constitutes the most colossal metaphor imaginable, and must be judged, aesthetically as well as intellectually in terms of the success of this metaphor.

Math is my Passion. Engineering is my Profession.

Maths is at only one remove from magic.

Mathematics takes us still further from what is human into the region of absolute necessity, to which not only the actual world, but ever possible world, must conform.

More often than not, at the end of the day (or a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It's as if you had spent a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first.

Monty Jones: Dad, is there a word to describe answers that are completely correct but entirely useless under the circumstances? Professor Jones: Yes, yes there is.

Most people would have probably lost count around seven. This was, Harry knew from his extensive reading on logic and arithmetic, the largest number that most people could visually appreciate. Put seven dots on a page, and most people can take a quick glance and declare, “Seven.” Switch to eight, and the majority of humanity was lost.

People who don't like math always accuse mathematicians of trying to make math complicated. (...) But anyone who does love math knows it's really the opposite: math rewards simplicity, and mathematicians value it above all else. So it's no surprise that Walter's favourite axiom was also the most simple in the realm of mathematics: the axiom of the empty set. The axiom of the empty set is the axiom of zero. it states that there must be a concept of nothingness, that there must be the concept of zero: zero value, zero items. Math assumes there's a concept of nothingness, but is it proven? No. But it must exist. And if we're being philosophical—which we today are—we can say that life itself is the axiom of the empty set. It begins in zero and ends in zero. We know that both states exist, but we will not be conscious of either experience: they are states that are necessary parts of life, even as they cannot be experienced as life. We assume the concept of nothingness, but we cannot prove it. But it must exist. So I prefer to think that Walter has not died but has instead proven for himself the axiom of the empty set, that he has proven the concept of zero. I know nothing else would have made him happier. An elegant mind wants elegant endings, and Walter had the most elegant mind. So I wish him goodbye; I wish him the answer to the axiom he so loved.

Mr Baley", said Quemot, "you can't treat human emotions as though they were built about a positronic brain". "I'm not saying you can. Robotics is a deductive science and sociology an inductive one. But mathematics can be made to apply in either case.

My sub doesn't pay for me,” he says, pulling me to my feet. “That just doesn't happen.” “But we ordered so much,” I say helplessly. “It made you happy,” he says simply. “Now I get to play with you. And that makes me happy.” “I don't think it's that simple an equation.” “Maybe not,” he concedes. “But then, if if sex were the same thing as math, a lot more people would be lining up to take calculus.

Newton had considered the calculus as a scientific description of the generation of magnitudes, and Leibniz had viewed it as a metaphysical explanation of such generation. The formalism of the nineteenth century took from the calculus any such preconceptions, leaving only the bare symbolic relationships between abstract mathematical entities.

No one shall expel us from the paradise which Cantor has created for us. {Expressing the importance of Georg Cantor's set theory in the development of mathematics.}

...no thought, if it be non-mathematical in spirit, can be trusted, and, although mathematicians sometimes make mistakes, the spirit of mathematics is always right and always sound.

One must divide one's time between politics and equations. But our equations are much more important to me, because politics is for the present, while our equations are for eternity.

Only three constants are significant for star formation: the gravitational constant, the fine structure constant, and a constant that governs nuclear reaction rates.

Origin of the Logical. Where has logic originated in men’s heads? Undoubtedly out of the illogical, the domain of which must originally have been immense. But numberless beings who reasoned otherwise than we do at present, perished; albeit that they may have come nearer to truth than we! Whoever, for example, could not discern the "like" often enough with regard to food, and with regard to animals dangerous to him, whoever, therefore, deduced too slowly, or was too circumspect in his deductions, had smaller probability of survival than he who in all similar cases immediately divined the equality. The preponderating inclination, however, to deal with the similar as the equal - an illogical inclination, for there is no thing equal in itself - first created the whole basis of logic. It was just so (in order that the conception of substance should originate, this being indispensable to logic, although in the strictest sense nothing actual corresponds to it) that for a long period the changing process in things had to be overlooked, and remain unperceived; the beings not seeing correctly had an advantage over those who saw everything "in flux." In itself every high degree of circumspection in conclusions, every sceptical inclination, is a great danger to life. No living being might have been preserved unless the contrary inclination - to affirm rather than suspend judgment, to mistake and fabricate rather than wait, to assent rather than deny, to decide rather than be in the right - had been cultivated with extraordinary assiduity. - The course of logical thought and reasoning in our modern brain corresponds to a process and struggle of impulses, which singly and in themselves are all very illogical and unjust; we experience usually only the result of the struggle, so rapidly and secretly does this primitive mechanism now operate in us.

Overstimulation has been the real drawback. I need to find ways to stop thinking about analysis of algorithms, in order to do various other things that human beings ought to do.

Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth.

Please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life. [Having himself spent a lifetime unsuccessfully trying to prove Euclid's postulate that parallel lines do not meet, Farkas discouraged his son János from any further attempt.]