Quotes & Sayings About Mathematics By Mathematicians

Although I am even now still a layman in the area of mathematics, and although I lack theoretical knowledge, the mathematicians, and in particular the crystallographers, have had considerable influence on my work of the last twenty years. The laws of the phenomena around us
order, regularity, cyclical repetition, and renewals
have assumed greater and greater importance for me. The awareness of their presence gives me peace and provides me with support. I try in my prints to testify that we live in a beautiful and orderly world, and not in a formless chaos, as it sometimes seems. — M.C. Escher

Mathematical thinking is not the same as doing mathematics - at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability in today's world. — Keith Devlin

By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists. — M.C. Escher

It is obvious that mathematics needs both sorts of mathematicians, theory-builders and problem-solvers. — Timothy Gowers

We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of science are mathematics and logic; the mathematical set puts out the logical eye, the logical set puts out the mathematical eye; each believing that it sees better with one eye than with two. Note that De Morgan, himself, only had sight with only one eye. — Augustus De Morgan

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. — Bernhard Riemann

These estimates may well be enhanced by one from F. Klein (1849-1925), the leading German mathematician of the last quarter of the nineteenth century. 'Mathematics in general is fundamentally the science of self-evident things.' ... If mathematics is indeed the science of self-evident things, mathematicians are a phenomenally stupid lot to waste the tons of good paper they do in proving the fact. Mathematics is abstract and it is hard, and any assertion that it is simple is true only in a severely technical sense - that of the modern postulational method which, as a matter of fact, was exploited by Euclid. The assumptions from which mathematics starts are simple; the rest is not. — Eric Temple Bell

It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. — Carl Friedrich Gauss

A typical mathematician does not actively try to be useful. Individual mathematicians are motivated primarily by a subtle mixture of ambition and intellectual curiosity, and not by a wish to benefit society, nevertheless, mathematics as a whole does benefit society. — Timothy Gowers

It may be true that people who are merely mathematicians have certain specific shortcomings; however that is not the fault of mathematics, but is true of every exclusive occupation. Likewise a mere linguist, a mere jurist, a mere soldier, a mere merchant, and so forth. One could add such idle chatter that when a certain exclusive occupation is often connected with certain specific shortcomings, it is on the other hand always free of certain other shortcomings. — Carl Friedrich Gauss

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics. — Henri Poincare

Two centuries ago Carl Friedrich Gauss, one of the greatest mathematicians and a founder of number theory, described his brainchild as "the queen of mathematics." Queens are regal, but they are also largely decorative, and this nuance was not lost on Gauss. — Ian Stewart

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

[Adams] supposed that, except musicians, everyone thought Beethoven a bore, as every one except mathematicians thought mathematics a bore. — Henry Adams

Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand ... So Greek mathematics is 'permanent', more permanent even than Greek literature. — G.H. Hardy

If the NSF had never existed, if the government had never funded American mathematics, we would have half as many mathematicians as we now have, and I don't see anything wrong with that. — Paul Halmos

I count Maxwell and Einstein, Eddington and Dirac, among "real" mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are at present at any rate, almost as "useless" as the theory of numbers. — G.H. Hardy

There is only one universal language, which is the language of numbers and proportions that are so striking and stunningly built into the Great Pyramid and to which our current science has no appropriate response. We can no longer ignore that this ancient civilization was aware of our units used in modern mathematics and physics and were even aware of our metric system. Our metric system originating in the eighteenth century, designed and implemented by a committee of mathematicians and physicists commissioned by the French revolutionary government. — Willem Witteveen

Abel has left mathematicians enough to keep them busy for five hundred years. — Charles Hermite

There are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as "pure mathematicians" (though "purist" would be more adequate) while the second is, with equal inadequacy, referred to as "applied". — Mark Kac

How do you quantify love? Can you weigh it, measure it, pin it down with equations? If the sum of all experiences is really just the interaction of a finite soup of chemicals copulating in nerve endings, how did this even dare articulate the infinite?
Mathematicians will tell you there are different types of infinities. Some are countable, some are not. We can love someone more and more; we can stop loving. But we can never guess how much all this is. Love has no units. — Sean J Halford

It appears that the solution of the problem of time and space is reserved to philosophers who, like Leibniz, are mathematicians, or to mathematicians who, like Einstein, are philosophers. — Hans Reichenbach

Mathematics is concerned with "all possible worlds." — David Malet Armstrong

Mathematics is written for mathematicians. — Nicolaus Copernicus

How happy is the lot of the mathematician! He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. No cashier writes a letter to the press complaining about the incomprehensibility of Modern Mathematics and comparing it unfavorably with the good old days when mathematicians were content to paper irregularly shaped rooms and fill bathtubs without closing the waste pipe. — W. H. Auden

Outside observers often assume that the more complicted a piece of mathematics is, the more mathematicians admire it. Nothing could be further from the truth. Mathematicians admire elegance and simplicity above all else, and the ultimate goal in solving a problem is to find the method that does the job in the most efficient manner. Though the major accolades are given to the individual who solves a particular problem first, credit (and gratitude) always goes to those who subsequently find a simpler solution. — Keith Devlin

So-called professional mathematicians have, in their reliance on the relative incapacity of the rest of mankind, acquired for themselves a reputation for profundity very similar to the reputation for sanctity possessed by theologians. — Georg C. Lichtenberg

The product of mathematics is clarity and understanding. Not theorems, by themselves ... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. — William Thurston

Mathematics is a world created by the mind of men, and mathematicians are people who devote their lives to what seems to me a wonderful kind of play! — Constance Reid

Mathematicians seem to have no difficulty in creating new concepts faster than the old ones become well understood. — Edward Norton Lorenz

One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it's not worth doing mathematics unless you're the best at mathematics, because those special few are the only ones whose contributions matter. We don't treat any other subject that way! I've never heard a student say, "I like Hamlet, but I don't really belong in AP English - that kid who sits in the front row knows all the plays, and he started reading Shakespeare when he was nine!" Athletes don't quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was "ahead" of them. — Jordan Ellenberg

Today, it is not only that our kings do not know mathematics, but our philosophers do not know mathematics and - to go a step further - our mathematicians do not know mathematics. — J. Robert Oppenheimer

Most of the arts, as painting, sculpture, and music, have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians, and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose. — Cornelius Lanczos

The final story of randomness - utter chaos - has not yet been told to us by the mathematicians. It seems remarkable that something so fundamental for probability theory has not been defined and even more remarkable that we can go so far in mathematics lacking a definition. By simply assuming randomness exists, mathematicians assign elementary probabilities to events, and that is their starting point. But they have not captured chaos and looked it in the eye. — Heinz R. Pagels

In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life. — Michael Atiyah

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. — G.H. Hardy

A result once generally accepted by mathematicians is seldom retracted, and then only with great pangs.
The Nature of Mathematics — Max Black

On sighting mathematicians poetry should unhook the algebra from their minds and replace it with poetry; on sighting poets it should unhook poetry from their minds and replace it with algebra. — Brian Patten

There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein. — Carl Friedrich Gauss

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. — E. T. Bell

I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics. — Jo Boaler

The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design. — Freeman Dyson

It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together. — Richard Courant

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 — Mark Haddon

It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics ... required ... had already been worked out by pure mathematicians for their own amusement ... The moral for statesmen would seem to be that, for proper scientific "planning", pure mathematics should be endowed fifty years ahead of scientists. — R. B. Braithwaite

Douady and Hubbard used a brilliant chain of new mathematics to prove that every floating molecule does indeed hang on a filigree that binds it to all the rest, a delicate web springing from tiny outcroppings on the main set, a "devil's polymer," in Mandelbrot's phrase. The mathematicians proved that any segment-no matter where, and no matter how small-would, when blown up by the computer microscope, reveal new molecules, each resembling the main set and yet not quite the same. Every new molecule would be surrounded by its own spirals and flame-like projections, and those, inevitably, would reveal molecules tinier still, always similar, never identical, fulfilling some mandate of infinite variety, a miracle of miniaturization in which every new detail was sure to be a universe of its own, diverse and entire. — James Gleick

A mathematician says that an electromagnetic wave travels from Andromeda to your eye and that it also extends from Andromeda to your eye. — Bill Gaede

Everyone knows that physicists are concerned with the laws of the universe and have the audacity sometimes to think they have discovered the choices God made when He created the universe in thus and such a pattern. Mathematicians are even more audacious. What they feel they discover are the laws that God Himself could not avoid having to follow. — David Mumford

Of course, the set of logically consistent mathematical structures is many times larger than the set of physical principles. Therefore, some mathematical structures, such as number theory (which some mathematicians claim to be the purest branch of mathematics), have never been incorporated into any physical theory. Some argue that this situation may always exist: Perhaps the human mind will always be able to conceive of logically consistent structures that cannot be expressed through any physical principle. However, there are indications that string theory may soon incorporate number theory into its structure as well. — Michio Kaku

Like musicians who can read and write complicated scores in a world without sounds, for us mathematics is a source of delight, excitement, and even controversy which are hard to share with non mathematicians. In our small micro-cosmos we should ever seek the right balance between competition and solidarity, criticism and empathy, exclusion and inclusion. — Gil Kalai

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. — Mario Livio

Mathematics is fun if you don't let mathematicians push you around when you are doing it. — Jack Edmonds

College is the grinding machine of the Mathematical Establishment, a conveyor belt that takes individuals from one cookie cutter to another so that the product comes within tight control limits out of the assembly line. — Bill Gaede

The things that fascinate me the most about mathematics are logical thought and the great importance attached to the correctness of propositions. Every step made during calculations is conclusive and mathematicians don't like to make false statements. This is the reason why people from this particular domain contemplate longer before they respond to questions. Recently I read a sentence in a book which summarizes all this fascinating stuff to me succinctly: 'Mathematics is the purest form of thought. — Barbara Meier

All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But they also live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to another. They're adults in the crystalline world, infants in the real one. — Sylvain Cappell

A mathematician tells you that the wall of warped space prevents the Moon from flying out of its orbit yet can't tell you why an astronaut can go back and forth across that same space. — Bill Gaede

The only reason psychology students don't have to do more and harder mathematics than physics students is because the mathematicians haven't yet discovered ways of dealing with problems as hard as those in psychology. — John G. Kemeny

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated ... The importance of this invention is more readily appreciated when one considers that it was beyod the two greatest men of antiquity, Archimedes and Apollonius. — Pierre-Simon Laplace

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

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds. — Paul Cohen

Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought. For me, mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos. — Clifford A. Pickover

This special was followed one month later by "Bart the Genius." This was the first genuine episode of The Simpsons , inasmuch as it premiered the famous trademark opening sequence and included the debut of Bart's notorious catchphrase "Eat my shorts." Most noteworthy of all, "Bart the Genius" contains a serious dose of mathematics. In many ways, this episode set the tone for what was to follow over the next two decades, namely a relentless series of numerical references and nods to geometry that would earn The Simpsons a special place in the hearts of mathematicians. — Simon Singh

Mathematicians are a bit like the laconic Vermonter who, when asked if he's lived in the state his whole life, replies, "Not yet." — John Allen Paulos

All mathematicians share ... a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics. — Martin Gardner

I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future. — Andrew Wiles

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

If there should chance to be any mathematicians who, ignorant in mathematics yet pretending to skill in that science, should dare, upon the authority of some passage of Scripture wrested to their purpose, to condemn and censure my hypothesis, I value them not, and scorn their inconsiderate judgement.
De Revolutionibus Coelestibus — Nicolaus Copernicus

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

People tend to think that mathematicians always work in sterile conditions, sitting around and staring at the screen of a computer, or at a ceiling, in a pristine office. But in fact, some of the best ideas come when you least expect them, possibly through annoying industrial noise. — Edward Frenkel

Some things that satisfy the rules of algebra can be interesting to mathematicians even though they don't always represent a real situation. — Richard P. Feynman

The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never decided whether they should feel excited or upset by such events. — Gian-Carlo Rota

The mathematicians are the priests of the modern world. — Bill Gaede

Just as all things speak about God to those that know Him, and reveal Him to those that love Him, they also hide Him from all those that neither seek nor know Him. — Blaise Pascal

Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract. — Benoit Mandelbrot

I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn. — Archimedes

The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities. — George Sarton