Mathematics What Is It Quotes & Sayings

To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. — Leonhard Euler

It seems perfectly clear that Economy, if it is to be a science at all, must be a mathematical science. There exists much prejudice against attempts to introduce the methods and language of mathematics into any branch of the moral sciences. Most persons appear to hold that the physical sciences form the proper sphere of mathematical method, and that the moral sciences demand some other method-I know not what. — William Stanley Jevons

There are two generic and invariable features that characterize utopias. One is the content: the authors of utopias paint what they consider to be ideal societies; translating this into the language of mathematics, we might say that utopias bear a + sign. The other feature, organically growing out of the content, is to be found in the form: a utopia is always static; it is always descriptive and has no, of almost no, plot dynamics. — Yevgeny Zamyatin

What I know is that if I was asked to teach mathematics in French for a week to young kids, I would do my homework and I think I could do a decent job. I don't think a degree in education would make me a better teacher. I sometimes teach in college. I don't teach for long periods of time, but I give workshops and I think I can communicate stuff. So, it's about communicating. — Philippe Falardeau

Mowaljarlai rarely answered questions with an abstract explanation; he always told a story. His was not a fragmented world, divided into the convenient disciplinary languages and jargon that seem to be required for the understanding of concepts and principles in, for example, mathematics, physics, art and literature. Not only did he not have these languages; he thought this was a strange way to arrive at understanding the way in which the world lives in itself. It baffled him that whitefellas developed their knowledge by busting things up, reducing things to little pieces separate from everything else that contributes to their nature. For him, everything in creation is not only living and interconnected, but exists in a story and story cycle. Yet his knowledge of what whitefellas call 'science' was extraordinary."
p80-1. — Hannah Rachel Bell

And this is what I mainly learned up there, that the Parthenon was not a thing to study but to feel. It wasn't aloof, rational, timeless, pure. I couldn't locate the serenity of the place, the logic and steady sense. It wasn't a relic species of dead Greece but part of the living city below it. This was a surprise. I'd thought it was a separate thing, the sacred height, intact in its Doric order. I hadn't expected a human feeling to emerge from the stones but this is what I found, deeper than the art and mathematics embodied in the structure, the optical exactitudes. I found a cry for pity. — Don DeLillo

At schools, the children who are too stupid or lazy to learn languages, mathematics and elementary science can be set to doing the things that children used to do in their spare time. Let them, for example, make mud pies and call it modelling. But all the time there must be no faintest hint that they are inferior to the children who are at work. Whatever nonsense they are engaged in must have - I believe the English already use the phrase - "parity of esteem." An even more drastic scheme is not impossible. Children who are fit to proceed to a higher class may be artificially kept back, because the others would get a trauma - Beelzebub, what a useful word! - by being left behind. The bright pupil thus remains democratically fettered to his own age group throughout his school career, and a boy who would be capable of tackling Aeschylus or Dante sits listening to his coeval's attempts to spell out 'A Cat Sat On A Mat'. — C.S. Lewis

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing-one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me. — Paul Halmos

Well, of
course one must have concentration. Courage. Self-control. That goes
without saying. But more important than these, one must have ... I
don't know how to say it. One must be both a mathematician and a
poet. As though poetry were a science; or mathematics an art. One
must have an affection for proportion to play Go at all well.Ah ... what Go is to philosophers
and warriors, chess is to accountants and merchants. — Trevanian

That is, for a mathematical Platonist, what the C.H. proofs really show is that set theory needs to find a better set of core axioms than classical ZFS, or at least it will need to add some further postulates that are-like the Axiom of Choice-both "self-evident" and Consistent with classical axioms. If you're interested, Godel's own personal view was that the Continuum Hypothesis is false, that there are actually a whole (Infinity Symbol) of Zeno-type (Infinity Symbol)s nested between (Aleph0) and c, and that sooner or later a principle would be found that proved this. As of now no such principle's ever been found. Godel and Cantor both died in confinement, bequeathing a world with no finite circumference. One that spins, now, in a new kind of all-formal Void. Mathematics continues to get out of bed. — David Foster Wallace

Look you wanna know the truth? I don't really care about the
stats or the cup or the trophy or anything like that. In fact even the games aren't that important to me. What matters to me is the perfect throw, making the perfect catch, the perfect step and block. Perfection. That's what it's about. It's those moments. When you can feel the perfection of creation. The beauty the physics you know the wonder of mathematics. The elations of action and reaction and that is the kind of perfection that I want to be connected to. — Samuel Anders

Everything useful in mathematics has been devised for a purpose. Even if you don't know it, the guy who did it first, he knew what he was doing. Banach didn't just develop Banach spaces for the sake of it. He wanted to put many spaces under one heading. Without knowing the examples, the whole thing is pointless. — Michael Atiyah

The science of mathematics applies to the clouds; the radiance of starlight nourishes the rose; no thinker will dare say that the scent of hawthorn is valueless to the constellations ... The cheese-mite has its worth; the smallest is large and the largest is small ... Light does not carry the scents of earth into the upper air without knowing what it is doing with them; darkness confers the essence of the stars upon the sleeping flowers ... Where the telescope ends the microscope begins, and which has the wider vision? You may choose. A patch of mould is a galaxy of blossom; a nebula is an antheap of stars. There is the same affinity, if still more inconceivable, between the things of the mind and material things. — Victor Hugo

Take this neat little equation here. It tells me all the ways an electron can make itself comfortable in or around an atom. That's the logic of it. The poetry of it is that the equation tells me how shiny gold is, how come rocks are hard, what makes grass green, and why you can't see the wind. And a million other things besides, about the way nature works. — Richard Feynman

What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. — Stanislaw Ulam

What is the use of aesthetics if they can neither teach how to produce beauty nor how to appreciate it in good taste? It exists because it behooves rational human beings to provide reasons for their actions and assessments. Even if aesthetics are not the mathematics of beauty, they are the proof of the calculation. — Franz Grillparzer

The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction ... Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about. — Alfred North Whitehead

Deep Blue didn't win by being smarter than a human; it won by being millions of times faster than a human. Deep Blue had no intuition. An expert human player looks at a board position and immediately sees what areas of play are most likely to be fruitful or dangerous, whereas a computer has no innate sense of what is important and must explore many more options. Deep Blue also had no sense of the history of the game, and didn't know anything about its opponent. It played chess yet didn't understand chess, in the same way a calculator performs arithmetic bud doesn't understand mathematics. — Jeff Hawkins

It is certain that the real function of art is to increase our self-consciousness; to make us more aware of what we are, and therefore of what the universe in which we live really is. And since mathematics, in its own way, also performs this function, it is not only aesthetically charming but profoundly significant. It is an art, and a great art. — J. W. N. Sullivan

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [ ... ] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate. — Bertrand Russell

I think of myth and magic as the hieroglyphics of the human psyche. They are a special language that circumvents conscious thought and goes straight to the subconscious.
Non-fiction uses the medium of information. It tells us what we need to know.
Science fiction primarily uses the medium of physics and mathematics. It tells us how things work, or could work.
Horror taps into the darker imagery of the psychology, telling us what we should fear.
Fantasy, magic and myth, however, tap into the spiritual potential of the human life. Their medium is symbolism, truth made manifest in word pictures, and they tell us what things mean on a deep, internal level. I have always been a meaning-maker. I have always been someone who strives to make sense of everything and perhaps that is where my life as a storyteller first began. Life doesn't always make sense, but story must. And so I write stories, and the world comes right again. — Ripley Patton

I guess a sock is also a geometric shape - technically - but I don't know what you'd call it. A socktagon? — Stephen King

For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the what and the why of the big picture. — Keith Devlin

Worth as I use it here is immeasurable, not as in mathematics towards infinity. But that it can not be measured. There are no measurable parameters for it! Certainly not a material-communal measurable parameter for it! Such, it is what the being holds that cannot and should never be traded. When it is there, every essence of your being knows it, and takes commands from it that will be able to override any personal or imposed sense of value. — Dew Platt

Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems. — George Polya

This attitude [the abstract method in mathematics] can be encapsulated in the following slogan: a mathematical object is what it does. — Timothy Gowers

What's the best part of being a mathematician? I'm not a religious man, but it's almost like being in touch with God when you're thinking about mathematics. God is keeping secrets from us, and it's fun to try to learn some of the secrets. — Paul Halmos

"I think you're begging the question," said Haydock, "and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that, you would go round the bend. Let me assure you of that!" — Agatha Christie

The sciences are not sectarian. People do not persecute each other on account of disagreements in mathematics. Families are not divided about botany, and astronomy does not even tend to make a man hate his father and mother. It is what people do not know, that they persecute each other about. Science will bring, not a sword, but peace. — Robert G. Ingersoll

The exciting thing about mathematics and science and music and literature is what they can tell us about the workings of the human mind. For these disciplines are literally models (extensions) of at least certain parts of the mind. Just as the knife cuts but does not chew, while the lens does only a portion of what the eye can do, extensions are reductionist in their capability. No matter how hard it tries, the human race can never fully replace what was left out of extensions in the first place. Also, it is just as important to know what is left out of a given extension system as it is to know what the system will do. Yet the extension-omissions side is frequently overlooked. — Edward T. Hall

All those formal systems, in mathematics and physics and the philosophy of science, which claim to give foundations for certain truth are surely mistaken. I am tempted to say that we do not look for truth, but for knowledge. But I dislike this form of words, for two reasons. First of all, we do look for truth, however we define it, it is what we find that is knowledge. And second, what we fail to find is not truth, but certainty; the nature of truth is exactly the knowledge that we do find. — Jacob Bronowski

The research reported on in our book "A=B", has moved a whole active field of mathematics from the province of human thought to the realm of computer-fodder. It is quite exciting to think about what other fields of pure mathematics, hitherto thought to be reserved to human intelligence, might be moved to that realm next. The goal is to put ourselves out of business completely, and the work is well underway. — Herbert Wilf

Language is remarkable, except under the extreme constraints of mathematics and logic, it never can talk only about what it's supposed to talk about but is always spreading around. — Howard Nemerov

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction - only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. — Carl B. Boyer

There is no great religion without a great schism. All of them have it. And that's because you're dealing with something called faith. And faith is not something you can prove; faith is personal opinion. Uh, when you're dealing with something with certainty, like, y'know, science or logic, you don't have the
there's no wiggle room; that's why history is not filled with warring math cults, y'know, because you can settle the issue; you can prove something to be right or wrong, and that's the end of the argument: next case. Whereas, when you're dealing with faith, you can forever argue your point, or another point, because you're dealing with intangibles. Personally, I think, faith is what you ask of somebody when you don't have the goods to prove your point. — Tom Quinn

We should reject the view that high culture, as the possession of an elite, is of no use to those who don't possess it. This is as false as the view that science or higher mathematics are useless to those who don't understand them. Scientific knowledge exists because a few talented people are prepared to devote their energy to pursuing it. That is what a university is for: and since you cannot pass on difficult knowledge without discriminating between the students who can absorb it and those who cannot, discrimination is a social good. The same is true of high culture. Those able to acquire it will be a minority and the process of cultural transmission will be critically impeded if that teacher must teach Mozart and Lady Gaga side by side to satisfy some egalitarian agenda. — Roger Scruton

To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about. — James R Newman

Why do we care about singers? Wherein lies the power of songs? Maybe it derives from the sheer strangeness of there being singing in the world. The note, the scale, the chord; melodies, harmonies, arrangements; symphonies, ragas, chinese operas,jazz, the blues: that such things should exist, that we should have discovered the magical intervals and distances that yield the poor cluster of notes, all within the span of a human hand from which we can build our cathedrals of sound, is alchemical a mystery as mathematics, or wine, or love. Maybe the birds taught us. Maybe not. Maybe we are just creatures in search of exaltation. We don't have much of it. Our lives are not what we deserve; they are, let us agree, in many painful ways deficient. Song turns them into something else. Song shows us a world that is worthy of our yearning, it shows us our selves as they might be, if we were worthy of the world. — Salman Rushdie

At the beginning of the new millennium, we still do not know why mathematics is true and whether it is certain. But we know what we do not know in an immeasurably richer way than we did. And learning this has been a remarkable achievement-among the greatest and least-known of the modern era. — David Berlinski

Our teaching of mathematics revolves around a fundamental conflict. Rightly or wrongly, students are required to master a series of mathematical concepts and techniques, and anything that might divert them from doing so is deemed unnecessary. Putting mathematics into its cultural context, explaining what is has done for humanity, telling the story of its historical development, or pointing out the wealth of unsolved problems or even the existence of topics that do not make it into school textbooks leaves less time to prepare for the exam. So most of these things aren't discussed. — Ian Stewart

What would it be like, a world without snow? I cannot imagine such a place. It would be like a world devoid of numbers. Every snowflake, unique as every number, tells us something about complexity. Perhaps that is why we will never tire of its wonder. — Daniel Tammet

Mere subtlety may qualify you as a sceptic but not as a philosopher. On the other hand, scepticism is in philosophy what the Opposition is in Parliament; it is just as beneficial, and indeed necessary. It rests everywhere on the fact that philosophy is not capable of producing the kind of evidence mathematics produces. — Arthur Schopenhauer

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. — Herbert S. Gaskill

Distrust even Mathematics; albeit so sublime and highly perfected, we have here a machine of such delicacy it can only work in vacuo, and one grain of sand in the wheels is enough to put everything out of gear. One shudders to think to what disaster such a grain of sand may bring a Mathematical brain. Remember Pascal. — Anatole France

I think math is a hugely creative field, because there are some very well-defined operations that you have to work within. You are, in a sense, straightjacketed by the rules of the mathematics. But within that constrained environment, it's up to you what you do with the symbols. — Brian Greene

They were very upset when I said that the thing of greatest importance to mathematics in Europe was the discovery by Tartaglia that you can solve a cubic equation-which, altho it is very little used, must have been psychologically wonderful because it showed a modern man could do something no ancient Greek could do, and therefore helped in the renaissance which was the freeing of man from the intimidation of the ancients-what they are learning in school is to be intimidated into thinking they have fallen so far below their super ancestors. — Richard Feynman

How do we change the way science is taught?
Ask anybody how many teachers truly made a difference in their life, and you never come up with more than the fingers on one hand. You remember their names, you remember what they did, you remember how they moved in front of the classroom. You know why you remember them? Because they were passionate about the subject. You remember them because they lit a flame within you. They got you excited about a subject you didn't previously care about, because they were excited about it themselves. That's what turns people on to careers in science and engineering and mathematics. That's what we need to promote. Put that in every classroom, and it will change the world. — Neil DeGrasse Tyson

Neglect of mathematics work injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy. — Roger Bacon

Though mathematics may teach a man how to build a bridge, it is what the Scotch Universities call the humanities, that teach him to be civil and sweet-tempered. — Amelia Barr

The philosophers make still another objection: "What you gain in rigour," they say, "you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application. — Henri Poincare

There is no thing as a man who does not create mathematics and yet is a fine mathematics teacher. Textbooks, course material-these do not approach in importance the communication of what mathematics is really about, of where it is going, and of where it currently stands with respect to the specific branch of it being taught. What really matters is the communication of the spirit of mathematics. It is a spirit that is active rather than contemplative-a spirit of disciplined search for adventures of the intellect. Only as adventurer can really tell of adventures. — Alfred Adler

It has been said at various points in history that the current sum of human
knowledge is but a fraction of that which was once known, yet now is lost.
Likewise, it is argued with simple mathematics that any sum of knowledge we
may yet accrue must always equate to virtually nothing when compared to the
infinity of what is. Apparently our's is a fate of perpetual ignorance. What then is truly lost in the course of human events?
(attrib: 'R.I.B. Ushguriud', Note On The Text) — Robert Robert

Mr S. got angry.
'Yes, I do have a son. He's a good-for-nothing. A dead loss.'
I couldn't ask which prison he was in, so I put it more tactfully: 'What is he doing?'
He sighed deeply: 'He's a professor of mathematics at London University. — George Mikes

Axiomatization is what one does last, it's rubbish. It's the hygiene of mathematics, axiomatization. — Serge Lang

More important than any one new application is the new 'materials' concept itself. It marks a shift from concern with substances to concern with structures, a shift from artisan to scientist as man's artificer, a shift from chemistry to physics as the basic discipline, and a shift, above all, from the concrete experience of the workshop to abstract mathematics, a shift from starting with what nature provides to what man wants to accomplish.
-The Age of Discontinuity, 1969 — Peter F. Drucker

Mathematics as an innate ability. You either have "it" or you don't. But to Schoenfeld, it's not so much ability as attitude. You master mathematics if you are willing to try. That's what Schoenfeld attempts to teach his students. Success is a function of persistence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds. Put a bunch of Renees in a classroom, and give them the space and time to explore mathematics for themselves, — Malcolm Gladwell

The reasons why anthropologists haven't been able to come up with a simple, compelling story for the origins of money is because there's no reason to believe there could be one. Money was no more ever "invented" than music or mathematics or jewelry. What we call "money" isn't a "thing" at all; it's a way of comparing things mathematically, as proportions: of saying one of X is equivalent to six of Y. As such it is probably as old as human thought. — David Graeber

Game theory is a branch of, originally, applied mathematics, used mostly in economics and political science, a little bit in biology, that gives us a mathematical taxonomy of social life, and it predicts what people are likely to do and believe others will do in cases where everyone's actions affect everyone else. — Colin Camerer

It is almost as hard to define mathematics as it is to define economics, and one is tempted to fall back on the famous old definition attributed to Jacob Viner, "Economics is what economists do," and say that mathematics is what mathematicians do. A large part of mathematics deals with the formal relations of quantities or numbers. — Kenneth E. Boulding

Every action of giving creates an opposite action of receiving and what you receive is always equal to what you've given. Whatever you give out in life, must return to you. It is the physics and the mathematics of the universe. — Rhonda Byrne

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. — G.H. Hardy

What this committee needs, what this media center needs, is a good dose of Jeeves."
"I'm sorry," said Mr. Peabody, a mathematics lecturer who sat hunched at the far end of the table taking the minutes. "How do you spell that?"
"Is it possible," said Arthur, raising both his shoulders and his voice, "that we are working in a university where lecturers are not aware of the identity of one Reginald Jeeves, the gentleman's personal gentleman and the personal gentleman's gentleman? What has happened to cultural literacy, my fellow members of the Advisory Committee for the Media Center? This sort of ignorance is exactly what needs addressing. What I mean, Mr. Peabody, when I say that we need a dose of Jeeves, is that we need quiet and reasoned wisdom that leads to prompt and directed action. — Charlie Lovett

Freeman denied the claim that he was a "man of God", saying that "the question of faith is whatever you actually believe is. We take a lot of what we're talking about in science on faith; we posit a theory, and until it's dis-proven we have faith that it's true. If the mathematics work out, then it's true, until it's proven to be untrue. — Morgan Freeman

And I'll go even further and say that mathematics, this art of abstract pattern-making - even more than storytelling, painting, or music - is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are. — Paul Lockhart

This book is about physics and its about physics and its relationship with mathematics and how they seem to be intimately related and to what extent can you explore this relationship and trust it. — Roger Penrose

How to describe the excitement I felt when I saw this beautiful work and realized its potential? I guess it's like when, after a long journey, suddenly a mountain peak comes in full view. You catch your breath, take in its majestic beauty, and all you can say is "Wow!" It's the moment of revelation. You have not yet reached the summit, you don't even know yet what obstacles lie ahead, but its allure is irresistible, and you already imagine yourself at the top. It's yours to conquer now. But do you have the strength and stamina to do it? — Edward Frenkel

A famous Japanese Zen master, Hakuun Yasutani Roshi, said that unless you can explain Zen in words that a fisherman will comprehend, you don't know what you're talking about. Some fifty years ago a UCLA professor told me the same thing about applied mathematics. We like to hide from the truth behind foreign-sounding words or mathematical lingo. There's a saying: The truth is always encountered but rarely perceived. If we don't perceive it, we can't help ourselves and we can't much help anyone else. — Jeff Bridges

There is no place for [the Bantu] in the European community above the level of certain forms of labour ... What is the use of teaching the Bantu child mathematics when it cannot use it in practice? That is quite absurd. Education must train people in accordance with their opportunities in life, according to the sphere in which they live. — Hendrik Verwoerd

Thousands of years ago the ancients had an advanced mathematical understanding of universe that is revealed in many sources. There is a consistent link to knowledge of the golden mean, but the way in which the ancients were able to formulate and use this information speaks of a technical grasp of the subject that exceeds what we know about it in the present day. — Alison Charlotte Primrose

Did you know that in mathematics they determined what was real by what was not imaginary?" Finn's voice was just a soft rumble beneath my fingertips that had found his lips
"What?"
"When mathematicians came up with imaginary numbers, accepted them, defined them, they had to come up with a name for everything that wasn't imaginary. Everything that wasn't an imaginary number from that point on became a 'real' number."
What's an imaginary number?"
"The square root of negative one is an imaginary number."
"Is that all?"
"Any number that was once the square root of a negative number becomes an imaginary number. Square root of -4 becomes 2i, square root of -100 becomes 10i."
"Is infinity an imaginary number?"
"No."
"Is it a real number?"
"No. It isn't a number at all. It's a concept of endlessness, unreachableness.
"I knew it. See? You are just a figment of my imagination. — Amy Harmon

As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry. — Blaise Pascal

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight. — Johann Wolfgang Von Goethe

If you can't test it, it's not theorics
it's metatheorics. A branch of philosophy. So, if you want to think of it this way, our test equipment is what defines the boundary separating theorics from philosophy. — Neal Stephenson

However great a man's fear of life, suicide remains the courageous act, the clear-headed act of a mathematician. The suicide has judged by the laws of chance - so many odds against one that to live will be more miserable than to die. His sense of mathematics is greater than his sense of survival. But think how a sense of survival must clamor to be heard at the last moment, what excuses it must present of a totally unscientific nature. — Graham Greene

For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics? — Richard Courant

Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other. — Abraham De Moivre

He had a particular facination with the I Ching, that art of tossing three coins three times and divining a pattern out of heads and tails. Pete would begin with questions:
What determined the pattern? Was it random? Was it a higher power? Was is mathematical? Wasn't poker based on mathematical probability and not just luck? Did that mean randomness was actually mathematical? And if the I Ching was governed by mathematics, hey, wouldn't that mean the I Ching was actually predictable, a prescribed answer? And if it was prescribed, did that mean that your life followed the I Ching, like some sort of equation? Or did the I Ching simply capture correctly what had already been determined as the next series of events in your life? — Amy Tan

Emotion is not a defect in an otherwise perfect reasoning machine. Reason, unfettered from human feeling, has led to as many horrors as any crusader's zeal. What use is pity in a world devoted to maximizing efficiency and productivity? Scientific husbandry tells us to weed out the sick, the infirm, the weak. The ruthless efficiency of euthanasia initiatives and ethnic cleansing are but the programmatic application of Nietzsche's point: from any quantifiable cost-benefit analysis, the principles of animal husbandry should apply to the human race. Charles Darwin himself acknowledged that strict obedience to "hard reason" rather than sympathy for fellow humans would represent a sacrifice of "the noblest part of our nature."6 It is the human heart resonating with empathy, not the logical brain attuned to the mathematics of efficiency, that revolts at cruelty and inhumanity. In — Terryl L. Givens

I suspect gentlemen, that you're regarding me with pity; you keep repeating to me that an enlightened and cultured man
such as, in short, as the man of the future will be
cannot knowingly desire anything unprofitable for himself
that that's mathematics. I agree totally that it really is mathematics. But I repeat to you for the hundredth time: there is only one case, only one, when a man can intentionally and consciously desire for himself even what is harmful and stupid, even what is extremely stupid: namely, in order to have the right to desire for himself even what is extremely stupid and not be constrained by the obligation to desire for himself only what is intelligent. — Fyodor Dostoyevsky

The question as to which of these two theories applies to the actual world is, like all questions concerning the actual world, in itself irrelevant to pure mathematics.* But the argument against absolute position usually takes the form of maintaining that a space composed of points is logically inadmissible, and hence issues are raised which a philosophy of mathematics must discuss. In what follows, I am concerned only with the question: Is a space composed of points self-contradictory? It is true that, if this question be answered in the negative, the sole ground for denying that such a space exists in the actual world is removed; but this is a further point, which, being irrelevant to our subject, will be left entirely to the sagacity of the reader. — Bertrand Russell

It is hard to know what you are talking about in mathematics, yet no one questions the validity of what you say. There is no other realm of discourse half so queer. — James Newman

... This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.

On the other hand, once you have made your choices then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back! — Paul Lockhart

Mathematics, natural science, laws, arts, even morality, etc. do not completely fill the soul; there is always a space left over reserved for pure and speculative reason, the emptiness of which prompts us to seek in vagaries, buffooneries, and mysticism for what seems to be employment and entertainment, but what actually is mere pastime undertaken in order to deaden the troublesome voice of reason, which, in accordance with its nature, requires something that can satisfy it and does not merely subserve other ends or the interests of our inclinations. — Immanuel Kant

The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was "out there" all along I just couldn't see it. And now I can! This is really what keeps me in the math game
the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods. — Paul Lockhart

Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don't happen to have all the data. In mathematics we have all the data and yet we don't understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence. — Simone Weil

Roger Bacon, a disciple of the Arabs, also insisted on the primary necessity of Mathematics, without which no other science can be known; yet by Mathematics it is clear that he meant something very different from what we mean, including under that head even dancing, singing, gesticulation, and performance on musical instruments. — George Henry Lewes

Add Snow White and her seven dwarfs,
2 droids for Luke Skywalker, of course.
1 true ring to rule them all.
A decimal is a place to stall.
Snow White's gone, the dwarfs alone.
This system your next clue has shown.

Now you might ask, this little key, Just what does it mean for me?
Hold on tight and you will see,
Someday it will set clues free. — Megan Frazer Blakemore

How does the industry seek to master the risks? Through mathematics. The common-sense version of what sophisticated investors do is diversification, a technique so old it's mentioned in the Talmud, where the strategy advocated is to have a third of your assets in trade, a third in cash and a third in land. — John Lanchester

I said Jeannine why are you unhappy?
I'm not unhappy.
You have everything (I said). What is there that you want and haven't got?
I want to die.
Do you want to be an airplane pilot? is that it? And they won't let you? Did you have a talent for mathematics, which they squelched? Did they refuse to let you be a truck driver? What is it?
I want to live. — Joanna Russ

Because the observer's only freedom is the choice of which question to pose (Shall I look up at the sky?), it is here that the mind of the observer has a chance to affect the dynamics of the brain. An examination of the mathematics, Stapp argued, shows that "the conscious intentions of a human being [reflected in the choices he makes about what question to put to nature] can influence the activities of his brain....Each conscious event picks out from the multitude of...possibilities that comprise the quantum brain the subensemble that is compatible with the conscious experience." The physical event reduces the state of the brain to that branch of it that is compatible with the particular experience or observation. — Jeffrey M. Schwartz

First of all, when we are all finished, and we have a mathematical theory by which we can compute consequences, what can we do? It really is an amazing thing. In order to figure out what an atom is going to do in a given situation we make up rules with marks on paper, carry them into a machine which has switches that open and close in some complicated way, and the result will tell us what the atom is going to do! If the way that these switches open and close were some kind of model of the atom, if we thought that the atom had switches in it, then I would say that I understood more or less what is going on. I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with what is going on in the original thing. The closing and opening of switches in a computer is quite different from what is happening in nature. — Anonymous

Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. — Paul Lockhart

A man craves ultimate truths. Every mortal mind, I think, is that way. But what is ultimate truth? It's the end of the road, where there is no more mystery, no more hope. And no more questions to ask, since all the answers have been given. But there is no such place.
The Universe is a labyrinth made of labyrinths. Each leads to another. And wherever we cannot go ourselves, we reach with mathematics. Out of mathematics we build wagons to carry us into the nonhuman realms of the world. — Stanislaw Lem

But is it not already an insult to call chess anything so narrow as a game? Is it not also a science, an art, hovering between these categories like Muhammad's coffin between heaven and earth, a unique yoking of opposites, ancient and yet eternally new, mechanically constituted and yet an activity of the imagination alone, limited to a fixed geometric area but unlimited in its permutations, constantly evolving and yet sterile, a cogitation producing nothing, a mathematics calculating nothing, an art without an artwork, an architecture without substance and yet demonstrably more durable in its essence and actual form than all books and works, the only game that belongs to all peoples and all eras, while no one knows what god put it on earth to deaden boredom, sharpen the mind, and fortify the spirit? — Stefan Zweig

The world is colors and motion, feelings and thoughts and what does math have to do with it? Not much, if 'math' means being bored in high school, but in truth mathematics is the one universal science. Mathematics is the study of pure pattern and everything in the cosmos is a kind of pattern. — Rudy Rucker

I was reasonably interested in mathematics in school. Typically what happens is ... when you start playing chess, it takes up a lot of your attention. But about 10 years ago, I found that the Internet is very good to start learning about a lot of subjects. — Viswanathan Anand

Mathematical objects states "only the relationships between mathematically 'undefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics - what — Richard Courant

To calculate 'the' fine structure constant, 1/137, we would need a realistic model of just about everything, and this we do not have. In this talk I want to return to the old question of what it is that determines gauge couplings in general, and try to prepare the ground for a future realistic calculation. — Steven Weinberg

That Marxism is not a science is entirely clear to intelligent people in the Soviet Union. One would even feel awkward to refer to it as a science. Leaving aside the exact sciences, such as physics, mathematics, and the natural sciences, even the social sciences can predict an event - when, in what way and how an event might occur. Communism has never made any such forecasts. It has never said where, when, and precisely what is going to happen. Nothing but declamations. Rhetoric to the effect that the world proletariat will overthrow the world bourgeoisie and the most happy and radiant society will then arise. — Aleksandr Solzhenitsyn

One of the biggest problems of mathematics is to explain to everyone else what it is all about. The technical trappings of the subject, its symbolism and formality, its baffling terminology, its apparent delight in lengthy calculations: these tend to obscure its real nature. A musician would be horrified if his art were to be summed up as "a lot of tadpoles drawn on a row of lines"; but that"s all that the untrained eye can see in a page of sheet music ... In the same way, the symbolism of mathematics is merely its coded form, not its substance. — Ian Stewart