Math Quotations

Here are a few quotations which show how mathematicians viewed their subject.

Mathematics

"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field" - Paul Dirac. It stems from his observation of mathematics's foundational role in formulating and understanding the laws of physics and beyond.

“Mathematics is the music of reason; unsolved problems are the untapped melodies, waiting to be discovered.” — George Cantor

“In mathematics, the art of proposing a question must be held of higher value than solving it.” - Georg Cantor.

"In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generations adds a new story to the old structure." - Hermann Hankel

“Mathematics is the language in which gods talk to people” — Plato

"All is Number" -the Pythagorean's motto, carved above the entrance of his school.

“Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules.” ― Roger Penrose

“All the truths of mathematics are linked to each other, and all means of discovering them are equally admissible.” — Adrien-Marie Legendre (1752–1833)

"Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." - Martin Gardener

“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.” - Neil deGrasse Tyson, astrophysicist

"Mathematics is about rigor and elegance" – Unknown

"The highest form of pure thought is in mathematics." - Plato

"Concepts are Caught. They cannot be Taught" - P K Srinivasan

"Materials (for learning math) are Immaterial" - P K Srinivasan

"Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty" - Archimedes

"Don’t disturb my circles" - Archimedes to a Roman soldier before he was killed.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. - Aristotle

"Proof is an idol before which the mathematician tortures himself" - Arthur Eddington

"Obvious" is the most dangerous word in mathematics. - E.T Bell

"Mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism" - Imre Lakatos

"I wanted certainty in the kind of way in which people want religious faith. I thought certainty is more likely to be found in mathematics than elsewhere. But after twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable" - Bertrand Russel in 1959

"All Mathematics is Symbolic Logic" - Bertrand Russel in Principles of Mathematics

“People think that mathematics is complicated. Mathematics is the simple bit, it’s the stuff we CAN understand. It’s cats that are complicated.” — John Horton Conway

"One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts." - Albert Einstein

How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? - Albert Einstein

"Pure mathematics is, in its way, the poetry of logical ideas"-Albert Einstein, in his obituary for Emmy Noether (1935)

"Either mathematics is too big for the human mind, or the human mind is more than a machine." - Kurt Gödel. This thought-provoking statement posits an intriguing question- is the human mind truly capable of comprehending the vastness of mathematics?

“Mathematics is like oxygen. If it is there, you do not notice it. If it would not be there, you realize that you cannot do without.” - Lex Schrijver

Mathematics is the glory of the human mind - Liebnitz

“Mathematics is the music of reason.” - Paul Lockhart

"The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such."- Paul Lockhart

"Math is not about following directions, it's about making new directions." - Paul Lockhart

"Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping things simple." - Paul Lockhart

Mathematics is the most marvelous instrument created by the genius of man for the discovery of truth. - Laisant.

From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician -Sir James Jeans

Universe is written in the language of mathematics and its characters are triangles, circles and other geometrical figures – Galileo

The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience -Emmanuel Kant

[Mathematics] is an independent world created out of pure intelligence -William Wordsworth

Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning -John Locke

The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit – Alfred North Whitehead

Mathematics is not about numbers, equations, computations or algorithms, it is about understanding - William Paul Thurston

So Mathematical Truth prefers simple words since the language of Truth is itself simple - Tycho Brahe, Danish Astronomer (translated from Latin)

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show - Bertrand Russel

“Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as a sensible intelligible world.” - Bertrand Russell

“Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.” -Bertrand Russell

“The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.” -Bertrand Russell

“Mathematics is the poetry of the universe.”- Jonathan David Farley

“Mathematics is the discipline dealing with nontrivial and interesting things.”-Jody Azzouni

“Mathematics is the queen of the sciences.”- Carl Friedrich Gauss

“Mathematics is a game played according to certain simple rules with meaningless marks on paper.” - David Hilbert

"Physics is becoming too difficult for the physicists." - David Hilbert. This statement reflected Hilbert's observation of the increasingly complex mathematical tools and methods required to advance in the field of physics. As physics delved deeper into the realms of quantum mechanics, relativity, and other frontier theories, the mathematical challenges became more sophisticated. Hilbert suggested that physicists, primarily trained in empirical and experimental sciences, might struggle without advanced mathematical knowledge, thus hinting at a growing need for interdisciplinary collaboration.

“All that is correct thinking is mathematics.”- Grigore Moisil

“How does the mathematician — closer to the artist than to the explorer — by turning away from nature, arrive at its most appropriate descriptions?” – Mark Steiner

“Mathematics is the door and key of the sciences.” - Roger Bacon

“Mathematics is the art of giving the same name to different things.” - Henri Poincaré

"Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose." - Henri Poincaré

“Mathematics is the tool especially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.” - Paul Dirac

“Mathematics is pure language — the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms- Alfred Adler

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

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

“Mathematics is not only real, but it is the only reality.” -Martin Gardner

“How does the mathematician — closer to the artist than to the explorer — by turning away from nature, arrive at its most appropriate descriptions?” – Mark Steiner

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning" - Eugene Wigner

“Mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomenon in these connections.” - Eugene Wigner

“Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.” ― Jordan Ellenberg, How Not to Be Wrong

“Mathematics is the extension of common sense by other means.” - ― Jordan Ellenberg, How Not to Be Wrong

"The heart of mathematics is its problems" - Paul Holmos

“The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization “ - John Kemeny

"Go down deep enough into anything and you will find mathematics" - Dean Schlicter

"Mathematics makes the invisible visible" – Keith Devlin

“Mathematics is the art of giving the same name to different things.” - – Henry Poincaré

"The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” – Stan Gudder

"A mathematician is a maker of patterns, like a painter or poet." - G H Hardy

“The mathematician’s patterns, like the painter’s or poet’s, must be beautiful … Beauty is the first test: there is no permanent place in the world for ugly mathematics.” - G H Hardy

"Only by giving up math’s connection to reality, could it guide us to a deeper truth about the way the universe works" -Derek Muller

"Mathematics is about relationships, relationships & relationships' - Christina Tondewold

"I love abstract mathematics in somewhat the same way that I love boat trips. It’s not just about getting to a destination.… It’s about the fun, the mental exertion, communing with mathematical nature and seeing the mathematical sights" - Eugenia Cheng

"Stories play crucial roles in our discovering, creating, explaining, and organizing knowledge, and thus mathematics also has a great need for narrative, even though its taste for general ideas might make one forget this" - Apostolos Doxiadis and Barry Mazur in "Circles Disturbed: The Interplay of Mathematics and Narrative"

“It is impossible to be a mathematician without being a poet in soul … the poet must see what others do not see, must see more deeply …. And the mathematician must do the same.” — Sofia Kovalevskaya (1850-1891)

“What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of the mathematics?” — Benjamin Franklin (1706-1790)

"The moving power of mathematical invention is not reasoning but imagination" - Augustus De Morgan.

"Mathematics is the study of anything that obeys the rules of logic, using the rules of logic"- Eugenia Cheng

"Mathematics is not playing with numbers and doing accounting. Mathematics is dealing with ideas in a creative and yet very precise way." - Przemysław Prusinkiewicz

"Go down deep enough into anything and you will find mathematics" - Dean Schlicter

Infinity

"A fractal is a way of seeing infinity" - Benois Mandelbrot

Mathematics, Logic & Proof

“What has been affirmed without proof can also be denied without proof.” ― Euclid of Alexandria

Study logic and math, because once you’ve mastered them, you won’t fear any book. To think clearly, understand the basics. If you’re memorizing advanced concepts without being able to re-derive them as needed, you’re lost. — Naval Ravikant

Number Sense

“Number sense is not a term that all researchers define in exactly the same way, but includes counting and comparing skills and the flexible ability to compute and represent numbers. It can be improved through teaching.” Number sense – making sense of numbers – is the foundation for higher conceptual understanding and mathematical knowledge and relies on the links between number relationships, operations, and processes- Cambridge Mathematics Express (Issue 4, February 2017)

Numbers & Arithmetic

“Numbers constitute the only universal language.” - Nathanael West

"Most people who never had a chance to learn what mathematics is confuse it with Arithmetic and consider it a dry and arid science" - Sofia Kovalevskaya

“Reducing math in elementary school to arithmetic steals the joy of it. Students need arithmetic, but embedded in fascinating problems and beautiful patterns” – Laurie Walsh, Teacher

"Numbers are intellectual witnesses that belong only to mankind" - Unknown

"No society, however small or remote, has ever lacked the basic curiosity and “number sense” that is part of the global mathematical experience" – George Ghevarghese Joseph

"By number we understand not so much a multitude of unities, as the abstracted ratio of any quantity to another quantity of the same kind, which we take for unity" - Newton, 1728 in his Universal Arithmetic.

God created the natural numbers. All the rest is the work of man - Leopold Kronecker; German mathematician and logician; 1823-1891.

God does arithmetic -Karl Friedrich Gauss

“Mathematics is the queen of sciences, and arithmetic is the queen of mathematics.” Karl Friedrich Gauss

Wherever there is number there is beauty - Proclus

“Why are numbers beautiful? It’s like asking why is Beethoven’s ninth symphony beautiful. If you don’t see why someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” - Paul Erdos

"As far as I’m concerned, prime numbers constitute a more stable reality than the material reality that surrounds us" A.Connes

"If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe.” - Nikola Tesla. (While we may not fully grasp what Tesla meant by this, it underscores the enduring fascination with these numbers, their connections to the natural world, and the intriguing figure who brought them to our attention.)

The Number 1

"I love how simple it is — everyone understands what it says — yet how provocative it is. Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity." - Steven Strogatz on the equation 1 = 0.999999999999....

Prime Numbers

"God may not play dice with the universe, but something strange is going on with the prime numbers." - Paul Erdös

"It will be another million years at least, before we understand Prime Numbers" - Paul Erdos

“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate “- Leonhard Euler

"The succession of primes is unpredictable. We don’t know if they will obey any rule or order that we have not been able to discover still. For centuries, the most illustrious minds tried to put an end to this situation, but without success. Leonhard Euler commented on one occasion. Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate" - Julian Havil

“There are two facts about the distribution of prime numbers that I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that [they are] the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.” - Julian Havil

Logarithms

“You have no idea, how much poetry there is in the calculation of a table of logarithms!” — Carl Friedrich Gauss

Number π

“Pi is God’s telephone number." - Alexander Graham Bell

Number Theory

"Number theory is the art of seeing the relationships of whole numbers" - Manjul Bhargava

Imaginary Numbers

“For the rest, neither the false nor the true roots are always real, sometimes they are only imaginary, that is to say one may imagine as many as I said in each equation, but sometimes there exists no quantity corresponding to those one imagines.” – Rene Descartes in his book La Geometrie. (Descartes stressed that this is an alternate system, a way of solving a “what-if” scenario. These imaginary roots, although useful, aren’t real in the sense that they are not true solutions on a graph. They’re imagined solutions.)

“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.” — Friedrich Gauss

“Like a Shakespearean sonnet that captures the very essence of life or a painting that brings out the beauty of the human form that is far more than skin deep, Euler’s Equation reaches down into the very depths of existence”- Keith Devlin

Geometry

“Architecture is geometry made visible in the same sense that music is number made audible.” — Claude F. Bragdon

"In all of Nature's infinite variety of forms there is a mathematical logic expressed geometrically" - Unknown

“I think the universe is pure geometry — basically, a beautiful shape twisting around and dancing over space-time.” — Antony Garrett Lisi

"Inspiration is needed in geometry, just as much as in poetry." - Alexander Pushkin

"God geometrises continually" - Plato

“The knowledge at which geometry aims is knowledge of the eternal, and not of anything transient which will decay. ... it must not be studied for its practical uses but for training the mind." – Plato in the Republic

"Geometry will draw the soul towards truth and create the spirit of philosophy." - Plato in the Republic

There is no royal road to Geometry - Euclid

All my physics is nothing else than geometry - Descartes

If the Greeks had not cultivated conic sections, Kepler would not have superseded Ptolemy - Whewell.

Geometry is nothing if it is not rigorous – H J S Smith

“Let no one unversed in geometry enter here” - Inscription at the gate of Plato's Academy

"There is geometry in the humming of the strings, there is music in the spacing of the spheres." - Pythagoras

"Geometry is knowledge of the eternally existent" - Pythagoras

"I attach special importance to the view of geometry, which I have just set forth, because without it I should have been unable to formulate the theory of relativity." - Albert Einstein

"Geometry is the archetype of the beauty of the world." – Johannes Kepler

"Where there is matter, there is geometry." - Johannes Kepler (This statement may initially seem cryptic, but it carries a profound significance in physics and mathematics. Kepler believed everything in the physical universe was based on mathematics and geometry. From the trajectories of celestial bodies to the shape of crystals, everything is connected by universal principles of proportion, symmetry, and harmony. Kepler's statement reminds us of the deep intellectual and philosophical connections between seemingly disparate fields of knowledge. It encourages us to seek out the hidden patterns and laws that govern our world.)

Statistics

“Statistics is the grammar of science.” - Karl Pearson

Algebra

"Algebra is like sheet of music. The important thing isn’t can you read music, it’s can you hear it. Can you hear the music, Robert?" — Niels Bohr to J. Robert Oppenheimer, Oppenheimer (2023)

Recreational/ Popular Mathematics

"I go up to calculus, and beyond that I don’t understand any of the papers that are being written. I consider that that was an advantage for the type of column I was doing because I had to understand what I was writing about, and that enabled me to write in such a way that an average reader could understand what I was saying. If you are writing popularly about math, I think it’s good not to know too much math." - Martin Gardener

Mathematics & Sciences

“Math is a powerful tool that allows us to understand physiology in a very precise way, and it will continue to accelerate progress in the life sciences — the applications are limitless.” -Fadil Santosa - Director, Institute of Mathematics

"The universe is under no obligation to make sense to you" - Neil deGrasse Tyson

"The two eyes of exact science of mathematics and logic" - Augustus De Morgan

"All scientific education which does not commence with mathematics is, of necessity, defective at its foundation" - Compte.

"A natural science is a science only in so far as it is mathematical" - Kant.

"Mathematics is the gate and key to science" – Roger Bacon

"Mathematics is the queen of sciences She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to the first rank" - Carl Friedrich Gauss; German mathematician, physicist, and prodigy; 1777-1855.

"The great book of nature can be read-only by those who know the language in which it was written. And this language is mathematics"- Galileo

"There is no branch of mathematics, however abstract, which may not someday be applied to phenomena of the real world" - Nikolai Lobatchevsky

"Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore" - Albert Einstein

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

“Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics" - Leonardo Da Vinci

"No investigation can claim to be scientific if it does not pass the test of mathematical proof" – Leonardo da Vinci

Mathematics & Physics

"A physical law must possess mathematical beauty” - ― Paul Dirac

“It is more important to have beauty in one’s equations than to have them fit experiment.” - Paul Dirac

"The Fine Structure Constant determines how stars burn, how chemistry happens and even whether atoms exist at all. It is one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding” - Richard Feynman

"Mathematics is the indispensable instrument of all physical research" - Berthelot

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

When Freeman Dyson remarked that the marriage between mathematics and physics had ended in divorce, he highlighted how the two fields, which were once intertwined, have drifted apart. Historically, physics advances have been propelled by the powerful tools of mathematics and vice versa. However, in recent times, we have witnessed a gradual shift towards specialization in each field, with physics becoming more focused on experiments and observations and mathematics delving further into abstract concepts. Despite this, some physicists and mathematicians are still working to bring the two fields back together to unlock new mysteries of the universe

Mathematics & Ultimate Reality

“A quark is a purely mathematical construct. It has no meaning apart from its mathematical definition. The properties of quarks — charm, colour, strangeness — are mathematical properties that have no analogue in the macroscopic world we inhabit.” — John Horgan

“[If] this electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure — indeed, that same mathematical structure.”— Max Tegmark in "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality"

Mathematics & Society

Entities should not be multiplied unnecessarily. — William of Ockham (about 1288-1347), barber of science.

I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate -David Mumford

The progress & the improvement of mathematics are linked to the prosperity of the state - Napoleon.

No subject loses more than mathematics by any attempt to dissociate it from its history - Glaisher.

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction - Leo Tolstoy

The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head - G K Chesterton

The study of mathematics is the indispensable basis for all intellectual and spiritual progress — F.M. Cornford (1874–1943)

“Music is the pleasure the human mind experiences from counting without being aware that it is counting” - Gottfried Leibniz

“May not music be described as the mathematics of the senses, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life.” - James Joseph Sylvester

The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization. -John Kemeny

The essence of mathematics is not to make simple things complicated, but to make complicated things simple. -S. Gudder

Go down deep enough into anything and you will find mathematics. -Dean Schlicter

Realizing that mathematics was developed with contributions from every race and culture helps students see mathematics as being more accessible – L Shirley (‘Using Ethnomathematics to Find Multicultural Mathematics Connections’)

”An equation for me has no meaning unless it expresses the mind of God.” - Ramanujan Srinivasa

Mathematics - Other Branches

A topologist is one who doesn't know the difference between a doughnut and a coffee cup -John Kelley

Statistics: the mathematical theory of ignorance -Morris Kline

Math & Humanities

"Poetry says more with fewer words. So does mathematics.” - Unknown

Music is the pleasure the human mind experiences from counting without being aware that it is counting. -Gottfried Leibniz

“I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history" - James Whitbread Lee Glaisher

Mathematicians - Leonhard Euler

"The pure mathematician, like the musician, is a free creator of his world of ordered beauty" - Bertrand Russel

"It is impossible to be a mathematician without being a poet in soul" - Sofya Kovalevskaya

"It is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician"- Karl Weierstrass

“Read Euler: he is our master in everything" - Laplace

“Our jewel ... one of the most remarkable, almost astounding, formulas in all of mathematics” - Richard Feynman, 20th century American physicist on Euler’s identity formula.

“Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence”– Keith J. Devlin, 20th century British mathematician and popular science writer on Euler’s identity formula.

“He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air" – François Arago, French mathematician, physicist, and astronomer.

Doing Mathematics

"While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a numb er of elliptic integrals. They stuck to my mind. As soon as I woke up. I committed them to writing" - Srinivasa Ramanujan (on his spiritual experiences)

"Mathematics is not a spectator sport. Learning must be active" - Anonymous

"I don't have any magical ability. I look at a problem, play with it, work out a strategy." - Terence Tao

"Mathematics has immense practical value. But many people do not realize that mathematicians actually “enjoy” mathematics. Take my word for it, there is as much satisfaction in knocking over an interesting problem with well-aimed thought as there is in knocking over ten wooden pins with a well-aimed bowling ball." - Martin Gardener

"The line between entertaining math and serious math is a blurry one. Many professional mathematicians regard their work as a form of play, in the same way professional golfers or basketball stars might." - Martin Gardener

“There are two ways to do great mathematics. The first way is to be smarter than everybody else. The second way is to be stupider than everybody else -- but persistent.” — Raoul Bott (1923-2005),

"What we need are notions, not notations." - Karl Frederic Gauss

“In mathematics, the art of proposing a question must be held of higher value than solving it.” - Georg Cantor.

“The yearning for and the satisfaction gained from mathematical insight brings the subject near to art.” - Olga Taussky-Todd

It (Math) can help us see the big picture and realize that we’re “just scratching the surface of something really profound.” Dr. Francis Su, the author of “Mathematics for Human Flourishing"

"Math does not mean that you have to do it all 'in your head'. Mental math means that you do it 'with your head'"– Cathy Fosnot

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street - David Hilbert

“Every good mathematician is at least half a philosopher, and every good philosopher at least half a mathematician" - Gottlieb Frege

"In mathematics you don't understand things. You just get used to them." – John von Neumann

"I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then … I realized that playing games is math" - John Conway

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive - Jo Boaler

Maryam Mirzakhani (1977–2017), the first woman to have won the Fields Medal, admitted that when she had a result she felt as if she were on the top of a hill from which the entire plain could be contemplated and enjoyed; however, when she was in the midst of research the feeling was rather like one of “being on a long hike with no trail and no end in sight” - Mirzakhani's Field Medal Acceptance Speech 2014

Some times I feel like I’m in a big forest and I don’t know where I’m going. But then somehow I come to the top of a hill and can see everything more clearly. When it happens, it’s really exciting – Maryam Mirzakhani

School Mathematics Curriculum

"It is important to make problem-solving the heart of math instruction so that students do not ask "Why do I need to know this?" - Yveline Germain Macarthy in the Education Week Teacher

"Focus on processes and connections between processes rather than just finding the answer. The answer is important, but the processes that lead to the answer are far more important" - David Wees in the Education Week Teacher

"For 40 years I have done my best to convince educators that recreational math should be incorporated into the standard curriculum. It should be regularly introduced as a way to interest young students in the wonders of mathematics. So far, though, movement in this direction has been glacial" - Martin Gardener

"All this fussing and primping about which 'topics' should be taught in what order, or the use of this notation instead of that notation, or which model of calculator to use, for god's sake -- it's like rearranging the deck chairs on the Titanic. Mathematics is the music of reason." - Paul Lockhart

"The mathematics curriculum doesn't need to be reformed, it needs to be scrapped." - Paul Lockhart

"There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it. School boards do not understand what math is, neither do educators, textbook authors, publishing companies, and sadly, neither do most of our math teachers. The scope of the problem is so enormous, I hardly know where to begin." - Paul Lockhart, mathematician and school teacher)

"SIMPLICIO: So you would remove mathematics from the school curriculum? SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas. SIMPLICIO: But how many math teachers know enough about their subject to teach it that way? SALVIATI: Very few. And that's just the tip of the iceberg ..." - Paul Lockhart, mathematician and school teacher in "A Mathematician's Lament"

"The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. " - Paul Lockhart

Teaching Math in Schools

“Although kinesthetic experiences can enhance perception and thinking, understanding does not travel through the fingertips and up the arm” – Deborah Ball in Magical hopes: Manipulatives and the reform of math education. American Educator: The Professional Journal of the American Federation of Teachers. She argued strongly against a constructivist view of manipulatives and the idea that children can independently develop an understanding of mathematics concepts by interacting with concrete materials:

"In order to come up with the most effective approach for teaching math, we need to agree on the goals of math instruction. Do we want kids to be able to compute accurately? Yes, but not everyone agrees that this should be the main goal of mathematics education. “The public needs to understand that the goals of math education are contested” - Deborah Lowenberg Ball

"Most of my work as a Math teacher isn't even math. It is helping students to believe they can do math. We do not talk about it enough" - Educator Jose Wilson

“Mathematics facts are important but the memorization of math facts through times table repetition practice and timed testing is unnecessary and damaging…. Math facts, themselves, are a small part of mathematics and they are best learned through the use of numbers in different ways and situations.”- Jo Boler

Mathematical Mindset/ Thinking

"A mathematical mindset makes “the complex world a bit more manageable.” - Terrance Tao

Mathematicians

“A blind man in a dark room looking for a black cat which isn’t there” - Charles Darwin of a mathematician

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