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Mathematics can be hard to define

• What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Stanislaw Ulam
• 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
• What is mathematics? Ask this question of person chosen at random, and you are likely to receive the answer “Mathematics is the study of number.” With a bit of prodding as to what kind of study they mean, you may be able to induce them to come up with the description “the science of numbers.” But that is about as far as you will get. And with that you will have obtained a description of mathematics that ceased to be accurate some two and a half thousand years ago! Keith Devlin

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Some would say mathematics is the science of numbers

• The science of numbers and shapes. Branches of mathematics include arithmetic, algebra, geometry and trigonometry. Oxford Dictionary
• the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them. Cambridge Dictionary
• the science of numbers and their operations. interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.  Merriam Webster

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Mathematics is the economy of counting

• Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting. Ernst Mach

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Mathematics is far more than computation

• Mathematics is no more computation than typing is literature. John Allen Paulos
• Calculators can only calculate – they cannot do mathematics. John A. Van de Walle
• Mathematics is much more than computation with pencil and a paper and getting answers to routine exercises. In fact, it can easily be argued that computation, such as doing long division, is not mathematics at all. Calculators can do the same thing and calculators can only calculate they cannot do mathematics. John A. Van de Walle

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Mathematics is that which establishes the unseen relations between things

• Mathematics is concerned only with the enumeration and comparison of relations. Carl Friedrich Gauss
• 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
• Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them. Joseph Fourier
• Mathematics deals exclusively with the relations of concepts to each other without consideration of their relation to experience. Albert Einstein
• Mathematical science shows what is. It is the language of unseen relations between things… Ada Lovelace

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Mathematics is a language…

• Mathematics is a language J. Willard Gibbs
• Mathematics is the language of size. Lancelot Hogben
• Nature is written in mathematical language. Galileo Galilei
• Mathematics is the universal language of physics. Joseph Selbie
• There is a logic of language and a logic of mathematics. Thomas Merton
• Logic and mathematics are nothing but specialised linguistic structures. Jean Piaget
• Film is one of the three universal languages, the other two: mathematics and music. Frank Capra
• Mathematics to me is like a language I don’t speak though I admire its literature in translation. David Quammen
• The power of mathematics is often to change one thing into another, to change geometry into language. Marcus du Sautoy
• Being a language, mathematics may be used not only to inform but also, among other things, to seduce. Benoit Mandelbrot
• We have to come back to something like ordinary language after all when we want to talk “about” mathematics! Harold Jeffreys
• To understand the Universe, you must understand the language in which it’s written, the language of Mathematics. Galileo Galilei
• Mathematics is a language plus reasoning. It’s like a language plus logic. Mathematics is a tool for reasoning. Richard P. Feynman
• Prayers for the condemned man will be offered on an adding machine. Numbers constitute the only universal language. Nathanael West
• The language of mathematics exhibits every quality of design: balance, harmony, symmetry, and some would say beauty. Deepak Chopra
• The mathematical sciences wield their particular language made of digits and signs, no less subtle than any other. Jorge Luis Borges
• When we talk mathematics, we may be discussing a secondary language built on the primary language of the nervous system. John von Neumann
• Mathematics is a language. We want scientists to be able to read it, speak it, and write it. But we are are not training them to be grammarians. Herbert Simon
• Mathematics is the universal language of physics. Two of the most important mathematical systems in physics—general relativity and quantum theory—are mathematically, deeply incompatible.  Joseph Selbie
• 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
• Our thoughts, our language, are always at a distance from whatever they’re trying to describe. We have other kinds of languages, like mathematics, like music, like art, but there’s always that gap. John Edgar Wideman
• Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say. Bertrand Russell
• The purely formal language of geometry describes adequately the reality of space. We might say, in this sense, that geometry is successful magic. I should like to state a converse: is not all magic, to the extent that it is successful, geometry? Rene Thom
• The idea here, of course, is, you know, mathematics is the language of science, it’s the way that we understand the natural world. And there’s definitely been a push to sort of study advanced math and kind of reawaken the love of advanced math. Anya Kamenetz
• To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. Richard P. Feynman
• It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathematicians have a good ear; some not (and some prefer the slangy expressions such as ‘iff’). That’s life. Jean-Pierre Serre
• I sometimes think we fail to be nearly as surprised and baffled by mathematical knowledge as we ought to be largely because we tend to think of the world as vaguely “arithmetical,” in the sense that its parts are enumerable and measurable, and because we lazily think of pure mathematics as just some sort of immense amplification of arithmetic, rather than the angelic language of nearly limitless intelligibility that it is. David Bentley Hart

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…a language that transcends words

• You can not apply mathematics as long as words still becloud reality. Hermann Weyl
• Mathematics is so much easier than words. Mathematics makes things clear that words merely muddle and confuse and mess up. John Maynard Smith

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Mathematics is a science…

• Mathematics is the queen of the sciences. Carl Friedrich Gauss
• Mathematics is the science which draws necessary conclusions. Benjamin Peirce
• Mathematics is the science which uses easy words for hard ideas. Edward Kasner
• Geometry is the science of correct reasoning on incorrect figures. George Polya
• There is an astonishing imagination, even in the science of mathematics. Voltaire
• Mathematics in general is fundamentally the science of self-evident things. Felix Klein
• Mathematics is an experimental science, and definitions do not come first, but later on. Oliver Heaviside
• … mathematics is the science of skillful operations with concepts and rules invented just for this purpose. Eugene Wigner
• This splendid subject [mathematics], queen of all exact sciences, and the ideal and norm of all careful thinking. G. Stanley Hall
• Mathematics-a wonderful science, but it hasn’t yet come up with a way to divide one tricycle between three small boys. Earl Wilson
• Mathematics is not a deductive science – that’s a I… What you do is trial and error, experimentation, guesswork. Paul Halmos
• Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics. Leonardo da Vinci
• What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics? Benjamin Franklin
• Mathematics may, be briefly defined as the science of quantities, and is one of the most important of disciplining studies which engage the practical student. Rufus Choate
• Mathematics, the non-empirical science par excellence . . . the science of sciences, delivering the key to those laws of nature and the universe which are concealed by appearances. Hannah Arendt
• Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about “any” things or about “some” things, without specifications of definite particular things. Alfred North Whitehead
• What affected me most profoundly was the realization that the sciences of cryptography and mathematics are very elegant, pure sciences. I found that the ends for which these pure sciences are used are less elegant. Jim Sanborn
• Mathematics is the queen of sciences and number theory is the queen of mathematics. 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
• The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results. Carl Gustav Hempel
• Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. George Polya

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…an science of experimentation…

• Mathematics is not a deductive science – that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. Paul Halmos
• The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations. G. H. Hardy

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… the cheapest science

• Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. Vladimir Arnold
• Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper. George Polya

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Mathematics is an art form

• Mathematics is really an art, not a science. Freeman Dyson
• I am interested in mathematics only as a creative art. G. H. Hardy
• Mathematics is the art of giving the same name to different things. Henri Poincare
• Real mathematics must be justified as art if it can be justified at all. G. H. Hardy
• Mathematics is the purest of the arts, as well as the most misunderstood. Paul Lockhart
• Mathematics is only the art of saying the same thing in different words. Bertrand Russell
• For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art. M. C. Escher
• The mathematician’s best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another. Gosta Mittag-Leffler
• Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness. Eric Temple Bell
• One of the most amazing things about mathematics is the people who do math aren’t usually interested in application, because mathematics itself is truly a beautiful art form. It’s structures and patterns, and that’s what we love, and that’s what we get off on. Danica McKellar
• Perhaps the best reason for regarding mathematics as an art is not so much that it affords an outlet for creative activity as that it provides spiritual values. It puts man in touch with the highest aspirations and lofiest goals. It offers intellectual delight and the exultation of resolving the mysteries of the universe. Morris Kline
• 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
• When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination. Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination. W. W. Sawyer
• I have often been surprised that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the cause: viz . that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert. Samuel Taylor Coleridge
• The mathematic, then, is an art. As such it has its styles and style periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes form epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics. Oswald Spengler
• Mathematics is really an art, not a science. You could say science also is an art. So I would say the difference is something you can’t really describe – you can only recognize. You hear somebody playing the violin, and it was Fritz Kreisler or it was somebody else, and you can tell the difference. It is so in almost every art. We just don’t understand why it is that there are just a few people who are just completely off the scale and the rest of them are just mediocre. And we don’t know why. But I say it’s certainly true of mathematics. Freeman Dyson

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Mathematics is the poetry of logical ideas

• Pure mathematics is, in its way, the poetry of logical ideas. Albert Einstein
• Poetry is the mathematics of writing and closely kin to music. John Steinbeck
• It is impossible to be a mathematician without being a poet in soul. Sofia Kovalevskaya
• A mathematician who is not also something of a poet will never be a complete mathematician. Karl Weierstrass
• There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. Paul Lockhart
• What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart? David Eugene Smith
• The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. William James
• 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
• … mathematics is very much like poetry … what makes a good poem – a great poem – is that there is a large amount of thought expressed in very few words. In this sense formulas likeor are poems. Lipman Bers
• … we should not be disappointed that everyday language does not work any longer at the apex of our little theory. It is natural; like poetry, the very reason for the existence of mathematics is that it expresses thoughts and feelings which we cannot express in mundane everyday language. Alexandre Borovik
• 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
• We have heard much about the poetry of mathematics, but very little of it has as yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements of any truth must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both. Henry David Thoreau

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Mathematics is a skill…

• To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess. Paul Halmos

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…and a process

• I once wrote: “In mathematics process and result are equivalent.” Ludwig Wittgenstein

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Mathematics is the queen of disciplines

• Mathematics is the queen of disciplines…. it will drive the nonsense out of your head! Leo Tolstoy
• Mathematics is the queen of science, and arithmetic the queen of mathematics. Carl Friedrich Gauss
• Mathematics is the queen of sciences and number theory is the queen of mathematics. 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
• Now do you not see that the eye embraces the beauty of the whole world? It counsels and corrects all the arts of mankind… it is the prince of mathematics, and the sciences founded on it are absolutely certain. It has measured the distances and sizes of the stars it has discovered the elements and their location… it has given birth to architecture and to perspective and to the divine art of painting. Leonardo da Vinci

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Mathematics is pure intelligence and intellect

• Mathematics is an independent world created out of pure intelligence. William Wordsworth
• Mathematics is persistent intellectual honesty. Moses Aaron Richardson

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Mathematics is timeless

• There is an underlying timelessness in the basic conversation that is mathematics. Barry Mazur
• Mathematics is as old as Man. Stefan Banach

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Mathematics is magnificent and lofty

• Mathematics may humbly help in the market-place, but it also reaches to the stars. Herbert Turnbull
• The study of mathematics, like the Nile, begins in minuteness but ends in magnificence. Charles Caleb Colton
• Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite. Cassius Jackson Keyser
• Modern mathematics, that most astounding of intellectual creations, has projected the mind’s eye through infinite time and the mind’s hand into boundless space. Nicholas Murray Butler
• The world of ideas which mathematics discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance. James Joseph Sylvester

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Mathematics is the highest form of thought

• The highest form of pure thought is in mathematics. Plato
• The knowledge of which geometry aims is the knowledge of the eternal. Plato
• The mathematician has reached the highest rung on the ladder of human thought. Havelock Ellis
• Mathematics is nothing more, nothing less, than the exact part of our thinking. L. E. J. Brouwer
• Either mathematics is too big for the human mind or the human mind is more than a machine. Kurt Gödel
• It is clear that the chief end of mathematical study must be to make the students think. John Wesley Young
• Mathematics is man’s own handiwork, subject only to the limitations imposed by the laws of thought. Edward Kasner
• Mathematics … is indispensable as an intellectual technique. In many subjects, to think at all is to think like a mathematician. Robert M. Hutchins
• Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Richard Courant
• Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there. Meir Shalev
• Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos [mathematics], where pure thought can dwell in its natural home. Bertrand Russell
• What is best in mathematics deserves not merely to be learnt as a task, but to assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Bertrand Russell
• Human thought, flying on the trapezes of the star-filled universe, with mathematics stretched beneath, was like an acrobat working with a net but suddenly noticing that in reality there is no net. Vladimir Nabokov]
• The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks. Benjamin Peirce

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Mathematics is magic and mystical

• The sky is where mathematics and magic become one. Tony Abbott
• I have often admired the mystical way of Pythagoras, and the secret magick of numbers. Thomas Browne
• Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. Vanna Bonta
• If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago… This time, however, it is not only based on intuition, but also on experiments of great precision and sophistication, and on a rigorous and consistent mathematical formalism. Fritjof Capra

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Mathematics may express great beauty

• Beauty in mathematics is seeing the truth without effort. George Polya
• Mathematics rightly viewed possesses not only truth but supreme beauty. Bertrand Russell
• The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. G. H. Hardy
• Beauty is the first test: there is no permanent place in the world for ugly mathematics. G. H. Hardy
• Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty. Archimedes
• As for everything else, so for a mathematical theory: beauty can be perceived but not explained. Arthur Cayley
• I love mathematics not only because it is applicable to technology but also because it is beautiful. Rozsa Peter
• To give emphasis only to beauty makes me think of a mathematics that deals with positive numbers only. Paul Klee
• Mathemagical mathematics combines the beauty of mathematical structure with the entertainment value of a trick. Martin Gardner
• Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty. Deepak Chopra
• Beauty is startling. She wears a gold shawl in the summer and sells seven kinds of honey at the flea market. She is young and old at once, my daughter and my grandmother. In school she excelled in mathematics and poetry. . . Beauty will dance with anyone who is brave enough to ask her. J. Ruth Gendler
• The mathematical sciences particularly exhibit order symmetry and limitations; and these are the greatest forms of the beautiful. Aristotle
• All mathematicians share… a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics. Martin Gardner
• My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful. Tom Stoppard
• One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories. Philip J. Davis
• The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth. George Bernard Shaw
• When I am working on a problem, I never think about beauty but when I have finished, if the solution is not beautiful, I know it is wrong. R. Buckminster Fuller
• Mathematics has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there. Marcus du Sautoy
• There are few humanities that could surpass in discipline, in beauty, in emotional and aesthetic satisfaction, those humanities which are called mathematics, and the natural sciences. Robert Watson-Watt
• The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty. D’Arcy Wentworth Thompson
• Theoretical physicists accept the need for mathematical beauty as an act of faith… For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty. Paul Dirac
• No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful. George Boole
• Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austere … yet sublimely pure and capable of stern perfection such as only the greatest art can show. Bertrand Russell
• 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
• The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. G. H. Hardy
• Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty’s sake and pulls it down to earth. Marston Morse
• Pure mathematics offers no mercenary inducements to its followers, who is attracted to it by the importance and beauty of the truths it contains; and the complete absence of any material advantage to be gained by means of it, adds perhaps another charm to its study. James Whitbread Lee Glaisher
• What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. Paul Dirac
• Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful! James Tanton
• The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events. Morris Kline
• Beauty means a lot of different things to a lot of different people. A lot of different ways in which things can be beautiful. But this really has a very specific meaning and which is more along the lines of elegance which is that we say an idea is beautiful or elegant in mathematics or physics if a very simple principle or a very simple idea, or simple set of ideas, turns out to be very powerful and leads to all sort of unexpected structure and unexpected predictions. Peter Woit
• 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 georgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. 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
• Like music or art, mathematical equations can have a natural progression and logic that can evoke rare passions in a scientist. Although the lay public considers mathematical equations to be rather opaque, to a scientist an equation is very much like a movement in a larger symphony. Simplicity. Elegance. These are the qualities that have inspired some of the greatest artists to create their masterpieces, and they are precisely the same qualities that motivate scientists to search for the laws of nature. Like a work of art or a haunting poem, equations have a beauty and rhythm all their own. Michio Kaku

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Mathematics is the honor of the mind and spirit

• Mathematics exists solely for the honour of the human mind. Carl Gustav Jacob Jacobi
• The object of mathematics is the honor of the human spirit. Carl Gustav Jacob Jacobi
• Mathematics is the most beautiful and most powerful creation of the human spirit. Stefan Banach
• Mathematics may, like poetry or music, “promote and sustain a lofty habit of mind.” G. H. Hardy
• Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself. Claude Fayette Bragdon

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Mathematics exists between constraint and freedom…

• Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature. Hermann Weyl

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…and allows great freedom of expression

• The essence of mathematics lies precisely in its freedom. Georg Cantor
• 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

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Mathematics crates links between theory and practice

• The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms. David Hilbert

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Mathematics inspires love and passion…

• Children display a universal love of mathematics, which is par excellence the science of precision, order, and intelligence. Maria Montessori
• Life is good for only two things, discovering mathematics and teaching mathematics. Simeon Denis Poisson
• It was as though applied mathematics was my spouse, and pure mathematics was my secret lover. Edward Frenkel
• It is always noteworthy that all those who seriously study this science [the theory of numbers] conceive a sort of passion for it. Carl Friedrich Gauss

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…and can be delightful and fun

• The main application of Pure Mathematics is to make you happy. Hendrik Lenstra
• Mathematics is fun if you don’t let mathematicians push you around when you are doing it. Jack Edmonds
• The true spirit of delight…is to be found in mathematics as surely as in poetry. Bertrand Russell
• My special pleasure in mathematics rested particularly on its purely speculative part. Bernard Bolzano
• Somebody came up to me after a talk I had given, and say, “You make mathematics seem like fun.” I was inspired to reply, “If it isn’t fun, why do it?” Ralph P. Boas, Jr.
• Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren’t ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive. Ronald Graham
• If there is anything that can bind the heavenly mind of man to this dreary exile of our earthly home and can reconcile us with our fate so that one can enjoy living,-then it is verily the enjoyment of the mathematical sciences and astronomy. Johannes Kepler

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Mathematics is pure possibility

• Mathematics is a place where you can do things which you can’t do in the real world. Marcus du Sautoy
• Mathematics is concerned with “all possible worlds.” David Malet Armstrong

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Mathematics sharpens the mind and brain

• I take a dose of mathematics every day to prevent my brain from becoming quite soft. George Eliot
• The completion of a rigorous course in mathematics – it is not even necessary that the student does well in such a course – appears to be an excellent means of sharpening the mind and developing mental skills that are of general benefit. Keith Devlin
• Mathematics may be a way of developing physically, that is anatomically, new connections in the brain. Stanislaw Ulam
• Mathematical knowledge adds a manly vigour to the mind, frees it from prejudice, credulity, and superstition. John Arbuthnot
• What I have said may serve to recommend mathematics for acquiring a vigorous constitution of mind; for which purpose they are as useful as exercise is for procuring health and strength to the body. John Arbuthnot

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Mathematics transcends culture and geographic boundaries…

• Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. David Hilbert
• I like to think of mathematicians as forming a nation of our own without distinctions of geographical origin, race, creed, sex, age or even time… all dedicated to the most beautiful of the arts and sciences. Julia Robinson

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…and is truly universal

• Mathematics reflects the universe because the complexities of our minds, our physiologies and the universe all occur within the one Consciousness. Tony Nader
• Mathematics is universal. It’s discovered by human beings, but the rules of mathematics are the same throughout the universe and the laws of the universe. Paul Davies
• Poor teaching leads to the inevitable idea that the subject (mathematics) is only adapted to peculiar minds, when it is the one universal science and the one whose four ground-rules are taught us almost in infancy and reappear in the motions to the universe. Henry John Stephen Smith
• It is the exact experience of mathematics. Not merely the adding up of your grocery bill, or the daily uses that we make of number. But the great concept of a universal exactitude, that numbers are an instrument of magic. And by means of them, men can unlock all the wonders of the world. Manly Hall
• Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility. Charles Sanders Peirce
• It is amusing to discover, in the twentieth century, that the quarrels between two lovers, two mathematicians, two nations, two economic systems, usually assumed insoluble in a finite period should exhibit one mechanism, the semantic mechanism of identification – the discovery of which makes universal agreement possible, in mathematics and in life. Alfred Korzybski
• Besides language and music, it [mathematics] is one of the primary manifestations of the free creative power of the human mind, and it is the universal organ for world understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation. Hermann Weyl
• There cannot be a language more universal and more simple, more free from errors and obscurities…more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes Joseph Fourier

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Mathematics brings progress

• The advance and perfecting of mathematics are closely joined to the prosperity of the nation. Napoleon Bonaparte
• Innovations that drive lasting economic growth emerge from the most advanced science, mathematics and technology. Susan Hockfield
• The progress of mathematics can be viewed as progress from the infinite to the finite. Gian-Carlo Rota
• The study of mathematics is the indispensable basis for all intellectual and spiritual progress. Francis Cornford
• During the past fifty years, more mathematics has been created than in all previous ages put together. Ian Stewart
• When we cannot use the compass of mathematics or the torch of experience…it is certain we cannot take a single step forward. Voltaire
• Mathematicians seem to have no difficulty in creating new concepts faster than the old ones become well understood. Edward Norton Lorenz
• The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future… Eric Temple Bell
• What promotes math progress even more than new ideas are new technical tools and habits of thought that encapsulate existing ideas, so that insights of one generation become the instincts of the next. Jim Propp
• How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments. David Hilbert

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Mathematics provides answers…

• One of the nice things about math and science is it‚ it’s obvious, you get the answer or you don’t get the answer. Lisa Randall

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…though countless questions are not answered yet

• One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics. Leon Henkin
• Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation. Ivars Peterson

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Mathematics brings discoveries…

• Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them. Joseph Fourier
• Every new body of discovery is mathematical in form, because there is no other guidance we can have. Charles Darwin
• One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is. Bertrand Russell
• Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard. Carl Friedrich Gauss
• If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries. Carl Friedrich Gauss
• Mathematics is the surest way to immortality. If you make a big discovery in mathematics, you will be remembered after everyone else will be forgotten Paul Erdos
• 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. Paul Lockhart
• Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences. in mathematics our freedom lies in the questions we ask  and in how we pursue them ‚ but not in the answers awaiting us. Steven Strogatz

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…that accumulate over time

• 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
• In mathematics alone each generations adds a new story to the old structure. Hermann Hankel

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Mathematics is a powerful form of knowledge

• Without mathematics, we are blind. Alain Badiou
• Mathematics is merely the means to a general and ultimate knowledge of man. Friedrich Nietzsche
• For the things of this world cannot be made known without a knowledge of mathematics. Roger Bacon
• The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization. John G. Kemeny
• Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency. Rene Descartes
• If all things have form, and forms can be described by numbers, then number becomes the essence of knowledge, the gateway to superior wisdom. Marcelo Gleiser
• Mathematics is one of the most basic — and most ancient — types of knowledge. Yet the details of its historical development remain obscure to all but a few specialists. Ivor Grattan-Guinness
• 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
• The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another. Henri Poincare
• The word “mathematics” is a Greek word and, by origin, it means “something that has been learned or understood,” or perhaps “acquired knowledge,” or perhaps even, somewhat against grammar, “acquirable knowledge,” that is, “learnable knowledge,” that is, “knowledge acquirable by learning.” Salomon Bochner
• 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

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Mathematics aids understanding…

• Mathematics is an art of human understanding. William Thurston
• The aim of mathematics is to explain as much as possible in simple terms. Michael Atiyah
• Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. William Thurston
• When you can measure what you are speaking about, and express it in numbers, you know something about it. Lord Kelvin
• People who wish to analyze nature without using mathematics must settle for a reduced understanding. Richard P. Feynman
• 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
• The 3-legged stool of understanding is held up by history, languages, and mathematics. Equipped with those three you can learn anything you want to learn. But if you lack any one of them you are just another ignorant peasant with dung on your boots. Robert A. Heinlein
• Art is a form of understanding like philosophy and science and mathematics are an understanding, but the difference is that art has the capacity to hold all these different things. It is the form of understanding that is best suited for the contemporary time. Robert Longo
• In mathematics … we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials … being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. David Hilbert
• The purpose of computing is insight, not numbers. Richard Hamming

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…though some disagree

• In mathematics you don’t understand things, you just get used to them. John von Neumann
• Young man, in mathematics you don’t understand things. You just get used to them. John von Neumann

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Mathematics stimulates invention

• As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention. James Joseph Sylvester

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Mathematics contains the capacity to surprise

• In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy. G. H. Hardy
• Perhaps the most surprising thing about mathematics is that it is so surprising. Edward Charles Titchmarsh
• It seems to me there’s this grand mathematical world out there, and I am wandering through it and discovering fascinating phenomena that often totally suprise me. I do not think of mathemaatics as invented but rather discovered. George Andrews

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Mathematics is all encompassing

• Mathematics is concerned with “all possible worlds.” David Malet Armstrong
• Defined broadly enough, mathematics encompasses everything. John Allen Paulos

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Mathematics involves common sense and transcends it

• Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense. Lord Kelvin
• We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra. William Kingdon Clifford
• Inferences of Science and Common Sense differ from those of deductive logic and mathematics in a very important respect, namely, when the premises are true and the reasoning correct, the conclusion is only probable. Bertrand Russell
• Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense. Edward Kasner

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Mathematics is objective and some might say cold

• Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold. Hermann Weyl
• The writing in mathematics text is not only laconic to a fault; it is cold, monotonous, dry, dull, and even ungrammatical… The books are not only printed by machines; they are written by machines. Morris Kline
• Mathematics is realm altogether, one that is cold, hard, objective, necessary, clear and thus utterly non-human. Nonetheless, it is enormously useful as a means, to “turn the soul around from becoming to being.” David Roochnik
• One may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight – brilliant, sharp but cold … thus we are clearest where knowledge matters least: in mathematics, especially number theory. Hermann Weyl

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Mathematics is perennially young

• It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences. Eric Temple Bell

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Mathematics is creative

• Mathematics is not a contemplative but a creative subject. G. H. Hardy
• All creative people hate mathematics. It’s the most uncreative subject you can study. Alec Guinness
• Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind. Hermann Weyl
• The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Alfred North Whitehead
• Creativity is essential to particle physics, cosmology, and to mathematics, and to other fields of science, just as it is to its more widely acknowledged beneficiaries – the arts and humanities. Lisa Randall
• 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
• Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him. G. H. Hardy
• The question for the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all. “Mathematizing” may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization. Hermann Weyl

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Mathematics is the supreme judge

• Mathematics is the supreme judge; from its decisions there is no appeal. Tobias Dantzig

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Mathematics is a communal pursuit

• The real satisfaction from mathematics is in learning from others and sharing with others. William Thurston
• These days people are much more aware that mathematics is a communal endeavor: even the most brilliant idea gets meaning only from its relation to the whole. Dusa McDuff
• Only dead mathematics can be taught where the attitude of competition prevails: living mathematics must always be a communal possession. Mary Everest Boole

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Mathematics is unending

• Regarding the fundamental investigations of mathematics, there is no final ending … no first beginning. Felix Klein
• Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. … A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. George Polya
• Newton’s theory is not ‘not right’, it just does not cover all distances. Contrary to popular belief, theories in science are not proven wrong, they are just replaced by more complete and convenient theories. To sound provocative, even the geocentric theory was never “proven” wrong, it is just not as convenient as the heliocentric theory, since it requires endless epicycles. Nathan Seiberg
• Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room. Andrew Wiles

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Mathematics can be addictive and consuming

• Number theorists are like lotus-eaters — having once tasted of this food they can never give it up. Leopold Kronecker
• It was not so much that I was doing mathematics, but rather that mathematics had taken possession of me. Stanislaw Ulam

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Mathematics can be complex

• The most complex object in mathematics, the Mandelbrot Set … is so complex as to be uncontrollable by mankind and describable as ‘chaos’. Benoit Mandelbrot
• Because of mathematics precise, formal character, mathematical arguments remain sound even when they are long and complex. In contast, common sense arguments can generally be trusted only if they remain short; even moderately long nonmathematical arguments rapidly becomes farfetched an dubious. Jacob T. Schwartz
• The explanatory power of mathematics rapidly declines as systems become more complex. René Thom

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Mathematics can be immersive

• Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world. Robert Langlands
• I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Louise Bourgeois

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Mathematical discovery is a path to immortality

• Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean. G. H. Hardy
• Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language. Edward Charles Titchmarsh
• Like mathematics and music and cosmology and philosophy, poetry, too, can “infinitize” us, granting us what immortality there is to be had in this mortal life. And all those who vibrate in harmony to language that itself vibrates to the harmonies of the infinite are entitled to inclusion among the “small group of people. Rebecca Goldstein

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Mathematics is an adventure

• Mathematics is not a language, it’s an adventure Paul Lockharture
• Mathematics is a vast adventure; its history reflects some of the noblest thoughts of countless generations. Dirk Jan Struik

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Some describe mathematics as a healthy form of laziness…

• Mathematics is for lazy people. Peter Hilton
• Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself George Polya
• In my experience most mathematicians are intellectually lazy and especially dislike reading experimental papers. Francis Crick

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…that helps replace thought

• The ultimate goal of mathematics is to eliminate all need for intelligent thought. Ronald Graham
• Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations. Ernst Mach

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Yet more reason mathematics is special

• Mathematics seems to endow one with something like a new sense. Charles Darwin
• I am too good for philosophy and not good enough for physics. Mathematics is in between. George Polya
• Mathematics is the only instructional material that can be presented in an entirely undogmatic way. Max Dehn
• Few realize that the world of modern mathematics is rich with vivid images and provocative ideas. Ivars Peterson
• Do not lose your faith. A mighty fortress is our mathematics. Mathematics will rise to the challenge, as it always has. Stanislaw Ulam
• The science of mathematics performs more than it promises, but the science of metaphysics promises more than it performs. Charles Caleb Colton
• Mathematics is an obscure field, an abstruse science, complicated and exact; yet so many have attained perfection in it that we might conclude almost anyone who seriously applied himself would achieve a measure of success. Marcus Tullius Cicero

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Mathematics involves geometry…

• Where there is matter, there is geometry. Johannes Kepler
• Geometry is the archetype of the beauty of the world. Johannes Kepler
• Equations are just the boring part of mathematics. I attempt to see things in terms of geometry. Stephen Hawking
• The first test of potential in mathematics is whether you can get anything out of geometry. John Edensor Littlewood
• Fractal geometry is not just a chapter of mathematics, but one that helps everyman to see the same world differently. Benoit Mandelbrot
• I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect. . . Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics. Carl Friedrich Gauss

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…and algebra…

• Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world. Alfred North Whitehead
• 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

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…and arithmetic…

• Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer. Carl Sandburg
• It is not the job of mathematicians… to do correct arithmetical operations. It is the job of bank accountants. Samuil Shatunovsky
• Cardinal arithmetic will be quite important for us, so we spend some time on it. Since, however, it tends to be trivial, we shall not need to spend much of this time on proofs. Keith Devlin
• A peculiarity of the higher arithmetic is the great difficulty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. Harold Davenport

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…and intimately connects these three disciplines…

• Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul. James Joseph Sylvester
• 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
• The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems — the determination of the diagonal of a square and that of the circumference of a circle — revealed the existence of new mathematical beings for which no place could be found within the rational domain. David van Dantzig
• In fact, Gentlemen, no geometry without arithmetic, no mechanics without geometry… you cannot count upon success, if your mind is not sufficiently exercised on the forms and demonstrations of geometry, on the theories and calculations of arithmetic … In a word, the theory of proportions is for industrial teaching, what algebra is for the most elevated mathematical teaching. Jean-Victor Poncelet

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…into an indivisible whole

• Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. David Hilbert
• Is mathematics doomed to suffer the same fate as other sciences that have split into separate branches?… Mathematics is, in my opinion, an indivisible whole… May the new century bring with it ingenious champions and many zealous and enthusiastic disciples. David Hilbert

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Mathematics involves devising theorems

• Young men should prove theorems, old men should write books. G. H. Hardy
• Theorems are not to mathematics what successful courses are to a meal. Gian-Carlo Rota
• If only I had the Theorems! Then I should find the proofs easily enough. Bernhard Riemann
• We often hear that mathematics consists mainly of “proving theorems.” Is a writer’s job mainly that of “writing sentences?” Gian-Carlo Rota
• A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem. Atle Selberg
• The “seriousness” of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. G. H. Hardy
• The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them. George Polya
• We decided that ‘trivial’ means ‘proved’. So we joked with the mathematicians: We have a new theorem- that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial. Richard P. Feynman
• A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so – and proves a theorem. Steven G. Krantz
• 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, by the logic of proofs and refutations. Imre Lakatos
• To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples. John B. Conway
• Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, “This is from the book!” Ross Honsberger
• The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular. Eric Temple Bell
• For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems. Michael Frayn

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Mathematics involves devising proofs

• Mathematics consists in proving the most obvious thing in the least obvious way. George Polya
• Proof is an idol before which the mathematician tortures himself. Arthur Eddington
• I had made an empirical discovery and it carried all the weight of a mathematical proof. Paul Auster
• A mathematician’s reputation rests on the number of bad proofs he has given. Abram Samoilovitch Besicovitch
• Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning. John Locke
• Many of the proofs in mathematics are very long and intricate. Others, though not long, are very ingeniously constructed. Edward Charles Titchmarsh
• Mathematicians need proofs to keep them honest. All technical areas of human activity need reality checks. It is not enough to believe that something works, that it is a good way to proceed, or even that it is true. We need to know why it’s true. Otherwise, we won’t know anything at all. Ian Stewart

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Mathematics involves asking questions…

• In mathematics, the art of asking questions is more valuable than solving problems. Georg Cantor
• Computation involves going from a question to an answer. Mathematics involves going from an answer to a question. Peter Hilton
• In mathematics, our freedom lies in the questions we ask and in how we pursue them, but not in the answers awaiting us. Steven Strogatz
• The mathematical question is “Why?” It’s always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems. Paul Lockhart

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…and solving problems

• The heart of mathematics is its problems. Paul Halmos
• All problems in mathematics are psychological. Pierre Deligne
• It is obvious that mathematics needs both sorts of mathematicians, theory-builders and problem-solvers. Timothy Gowers
• The definition of a good mathematical problem is the mathematics it generates rather than the problem itself. Andrew Wiles
• Great mathematics is achieved by solving difficult problems not by fabricating elaborate theories in search of a problem. Harold Davenport
• Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems. Carl Pomerance
• It’s fine to work on any problem, so long as it generates interesting mathematics along the way – even if you don’t solve it at the end of the day. Andrew Wiles
• Mathematics, even in its present and most abstract state, is not detached from life. It is just the ideal handling of the problems of life. Cassius Jackson Keyser
• The history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible. Freeman Dyson
• When the difficulty of a problem lies only in finding out what follows from certain fixed premises, mathematical methods furnish invaluable wings for flying over intermediate obstructions. Arthur M. Wellington
• Mathematics is about problems, and problems must be made the focus of a student’s mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process – having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work. Paul Lockhart

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Mathematics involves exploration

• The desire to explore thus marks out the mathematician. This is one of the forces making for the growth of mathematics. The mathematician enjoys what he already knows; he is eager for more knowledge. W. W. Sawyer
• Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley. Yoko Ogawa
• The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution. Kenneth E. Iverson

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Mathematics involves determining and describing patterns

• Mathematics is the science of patterns. Lynn Steen
• A mathematician, like a painter or a poet, is a maker of patterns. G. H. Hardy
• What humans do with the language of mathematics is to describe patterns. Lynn Steen
• The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. G. H. Hardy
• Philosophically, mathematics is not a part of science. Mathematics studies patterns, science studies nature Lynn Steen
• Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations. Paul Lockhart
• There are a lot of very strong connections with music and mathematics. They both can work in patterns and sequences and repetitions. Kate Bush
• Patterns and regularities exist in nature that mathematics can accurately predict. It’s a wonder. It doesn’t need to be that way. Anthony Lambert
• A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. G. H. Hardy
• Mathematics is often defined as the science of space and number . . . it was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns. Lynn Steen
• In mathematics, if a pattern occurs, we can go on to ask, Why does it occur? What does it signify? And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears. W. W. Sawyer
• All mathematics is is a language that is well tuned, finely honed, to describe patterns; be it patterns in a star, which has five points that are regularly arranged, be it patterns in numbers like 2, 4, 6, 8, 10 that follow very regular progression. Brian Greene
• Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guid to this growth is not calculation and formulas, but an open ended search for pattern. Lynn Steen
• What humans do with the language of mathematics is to describe patterns… To grow mathematically children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections. Lynn Steen
• Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty’s sake and pulls it down to earth. Marston Morse
• 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
• But actually a code is a language for translating one thing into another. And mathematics is the language of science. My big thesis is that although the world looks messy and chaotic, if you translate it into the world of numbers and shapes, patterns emerge and you start to understand why things are the way they are. Marcus du Sautoy

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Mathematics involves reason in action

• Mathematics is the music of reason. James Joseph Sylvester
• I studied mathematics which is the madness of reason. Benjamin Moser
• Mathematics is the bold luxury of pure reason, one of the few that remain today. Robert Musil
• Mathematics is the art of accurate reasoning on inaccurately-drawn figures… let that be our motto. Arthur Mattuck
• Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning. John Locke
• Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning. Alfred North Whitehead
• The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning. Alfred North Whitehead
• The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience Immanuel Kant
• The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. George Polya
• The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings. George Washington
• As an exercise of the reasoning faculties, pure mathematics is an admirable exercise, because it consists of reasoning alone and does not encumber the student with any exercise of judgment. Richard Whately
• Mathematics is one of the deepest and most powerful expressions of pure human reason, and, at the same time, the most fundamental resource for description and analysis of the experiential world. Hyman Bass
• Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences. Christopher Zeeman

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Mathematics starts with integers

• God made integers; all else is the work of man. Leopold Kronecker
• Integers are the fountainhead of all mathematics. Hermann Minkowski
• All of mathematics can be deduced from the sole notion of an integer; here we have a fact universally acknowledged today. Emile Borel
• Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc… But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. Abraham Robinson

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Mathematics makes use of rules

• Mathematics is about making up rules and seeing what happens. Vi Hart
• Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives.
• Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert
• Physicists are more like avant-garde composers, willing to bend traditional rules… Mathematicians are more like classical composers. Brian Greene
• Mathematics is an activity governed by the same rules imposed upon the symphonies of Beethoven, the paintings of DaVinci, and the poetry of Homer. Edward Kasner
• One reason why mathematics enjoys special esteem,above all other sciences, is that its laws are absolutely certain and indisputable,while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. Albert Einstein
• What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game. Keith Devlin

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Mathematics makes use of logic

• Mathematicians care no more for logic than logicians for mathematics. Augustus De Morgan
• Logic is the hygiene the mathematician practices to keep his ideas healthy and strong. Hermann Weyl
• The rules of logic are to mathematics what those of structure are to architecture. Bertrand Russell
• Arithmetic is where the content lies, and not logic; but logic prompts certainty, and not arithmetic. David Berlinski
• Mathematicians create by acts of insights and intuition. Logic then sanctions the conquests of intuition. Morris Kline
• Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life. IU?. I. Manin
• Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty. Deepak Chopra
• The subtleties of mathematics defecate the grossness of our apprehension, and supply the elements of a sounder and severer logic. William Godwin
• Inferences of Science and Common Sense differ from those of deductive logic and mathematics in a very important respect, namely, when the premises are true and the reasoning correct, the conclusion is only probable. Bertrand Russell
• Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry. Karl Wilhelm Friedrich Schlegel
• Mathematics and logic have been proved to be one; a fact from which it seems to follow that mathematics may successfully deal with non-quantitative problems in a much broader sense than was suspected to be possible. Alfred Korzybski
• The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists of the analysis of Symbolic Logic itself. Bertrand Russell
• I am convinced that the act of thinking logically cannot possibly be natural to the human mind. If it were, then mathematics would be everybody’s easiest course at school and our species would not have taken several millennia to figure out the scientific method. Neil deGrasse Tyson
• Mathematics is a logical method. . . . Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics. Ludwig Wittgenstein

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Mathematics devises and makes use of equations

• It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations. Felix Klein
• The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us. Ian Stewart
• Mathematicians are proud of the fact that, generally, they do their work with a piece of chalk and a blackboard. They value hand-done proofs above all else. A big question in mathematics today is whether or not computational proofs are legitimate. Some mathematicians won’t accept computational proofs and insist that a real proof must be done by the human hand and mind, using equations. Margaret Wertheim

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Mathematics makes use of symbols

• Some of the greatest advances in mathematics have been due to the invention of symbols, which it afterwards became necessary to explain; from the minus sign proceeded the whole theory of negative quantities. Aldous Huxley

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Mathematics is abstract

• Abstract: Existing as an idea, feeling, or quality, not as a material object. Cambridge Dictionary
• The longer mathematics lives the more abstract – and therefore, possibly also the more practical – it becomes. Eric Temple Bell
• The pendulum of mathematics swings back and forth towards abstraction and away from it with a timing that remains to be estimated. Gian-Carlo Rota
• 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
• There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned. Raymond Louis Wilder
• Thinking – in particular abstract thinking, which most of us are introduced to through the study of mathematics and literature – helps us learn that we can become problem solvers. Kathryn Lasky
• Two contrasting attitudes: non-math person: “Math is so abstract.” i.e. “hard to understand” math person: We abstract in order to understand. Part of our job is to teach people this latter mentality. James Key
• Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics. Eric Temple Bell
• The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work. Keith Devlin
• To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer. Ian Stewart

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Mathematics is hypothetical

• Mathematics is purely hypothetical: it produces nothing but conditional propositions. Charles Sanders Peirce

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Mathematics is exact…

• Mathematics has no symbols for confused ideas. George Stigler
• Mathematics allows for no hypocrisy and no vagueness. Stendhal
• On the ostensible exactitude of certain branches of human knowledge, including mathematics. The exactness is a fake. Alfred North Whitehead
• Mathematics may be the only exception in the sciences that leaves no room for skepicism. But, if mathematical results are exact as no empirical law can ever be, philosophers have discovered that they are not absolutely novel – instead, they are tautological. Luigi Luca Cavalli-Sforza
• In studying mathematics or simply using a mathematical principle, if we get the wrong answer in sort of algebraic equation, we do not suddenly feel that there is an anti-mathematical principle that is luring us into the wrong answers. Eric Butterworth

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…and precise…

• The development of mathematics towards greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. Kurt Godel
• Mathematics is much less formally complete and precise than computer programs. William Thurston

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…and not open to interpretation

• Very few people can communicate with one another. The only language that’s not subject to interpretation is mathematics, chemistry, basic science, engineering principles, and applied agriculture. But other than that, many systems today are subject to interpretation. Jacque Fresco

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Mathematics has no creeds

• Creed: A set of beliefs that influences the way you live. Cambridge Dictionary
• There are no creeds in mathematics. Peter Drucker
• There are no sects in geometry. Voltaire

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Mathematics requires intuition

• Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs. Felix Klein
• Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. Alan Turing

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Mathematics requires imagination

• The moving power of mathematical invention is not reasoning but imagination. Augustus De Morgan
• It is here in mathematics that the artist has the fullest scope of his imagination. Havelock Ellis
• There is an astonishing imagination, even in the science of mathematics. … We repeat, there was far more imagination in the head of Archimedes than in that of Homer. Voltaire
• I’m convinced that imagination is at the heart of everything we do – in art, science, even astrophysics and higher mathematics. Imagination leads us to ask, “What if?” Lloyd Alexander
• Many who have never had an opportunity of knowing any more about mathematics confound it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination. Sofia Kovalevskaya
• Here, where we reach the sphere of mathematics, we are among processes which seem to some the most inhuman of all human activities and the most remote from poetry. Yet it is here that the artist has the fullest scope of his imagination. Havelock Ellis
• Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: ‘I never thought he had enough imagination to be a mathematician.’ George Polya
• What would mathematics have amounted to without the imagination of its devotees-its giants and their followers? There never was a discovery made without the urge of imagination-of imagination which broke the roadway through the forest in order that cold logic might follow. David Eugene Smith
• [Mathematics] unceasingly calls forth the faculties of observation and comparison; one of its principal weapons is induction: it has frequent recourse to trial and verification; and it affords a boundless scope for the exercise of the highest efforts of imagination and invention. James Joseph Sylvester
• Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this . . . . The imagination in a mathematician who creates makes no less difference than in a poet who invents. . . . Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer. Jean le Rond d’Alembert
• In abstract mathematics or abstract art, the purpose is to describe inner states of our mind, and to explore the limits of our own imagination and our capacity for creativity. While this has some applications in the world, I think it leads to a distance from the world. Going to Congo was for me an act of seeking proximity, of breaking that distance. With abstraction, which is brilliant and vain, you divorce yourself from any kind of proximity to other people. Anjan Sundaram

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Mathematics plays an essential role

• Mathematics is not merely an idle art form, it is an essential part of our society. Richard Hamming
• 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 Poincare
• Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologies generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity. Oliver Heaviside

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Mathematics has many practical applications

• Very little mathematics has direct applications – though fortunately most of it has plenty of indirect ones. Gian-Carlo Rota
• The longer mathematics lives the more abstract – and therefore, possibly also the more practical – it becomes. Eric Temple Bell
• The most painful thing about mathematics is how far away you are from being able to use it after you have learned it. James R Newman
• Without mathematics, there’s nothing you can do. Everything around you is mathematics. Everything around you is numbers. Shakuntala Devi
• There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. Nikolai Lobachevsky
• All mathematicians share… a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics. Martin Gardner
• In mathematics I can report no deficiency, except it be that men do not sufficiently understand the excellent use of Pure Mathematics. Francis Bacon
• Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible. Should we say therefore that calculus is bad? Stanislaw Ulam
• The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight. George Dantzig

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Mathematics is used to simplify the complex

• In mathematics the complicated things are reduced to simple things. So it is in painting. Thomas Eakins
• The aim of mathematics is to explain as much as possible in simple terms. Michael Atiyah
• If people do not believe that mathematics is simple it is only because they do not realize how complicated life is. John von Neumann
• All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers. Leopold Kronecker
• In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts. Andre Weil

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Mathematics is used to generalise

• Another characteristic of mathematical thought is that it can have no success where it cannot generalize. Charles Sanders Peirce
• The art of doing mathematics consists in finding that special case which contains all the germs of generality. David Hilbert
• Before one can generalize, formalize, and axiomatize, there must be a mathematical substance. Hermann Weyl
• A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise. George Polya

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Mathematics is used to determine and create order

• Science attempts to find logic and simplicity in nature. Mathematics attempts to establish order and simplicity in human thought. Edward Teller
• I see a certain order in the universe and math is one way of making it visible. May Sarton
• Numbers tend to give the impression that there’s more order in the world than there is. Andy Rooney
• Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty. Deepak Chopra
• The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics. Johannes Kepler

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Mathematics is used in programming…

• Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians. Edsger Dijkstra
• With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable “native speaker” of mathematical notation. Kenneth E. Iverson
• My basic idea is that programming is the most powerful medium of developing the sophisticated and rigorous thinking needed for mathematics, for grammar, for physics, for statistics, for all the “hard” subjects…. In short, I believe more than ever that programming should be a key part of the intellectual development of people growing up. Seymour Papert

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…and especially in devising algorithms

• Mathematics my foot! Algorithms are mathematics too, and often more interesting and definitely more useful. Doron Zeilberger
• Mathematics is as much an aspect of culture as it is a collection of algorithms. Carl Benjamin Boyer

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Mathematics is used in devising models

• I think that mathematics can benefit by acknowledging that the creation of good models is just as important as proving deep theorems. David Mumford
• The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work-that is, correctly to describe phenomena from a reasonably wide area. John von Neumann

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Mathematics is used widely in science…

• All science requires mathematics. Roger Bacon
• Mathematics is the gate and key to science. Roger Bacon
• Mathematics is the key and door to the sciences. Galileo Galilei
• Mathematics is as little a science as grammar is a language. Ernst Mayr
• Numerical precision is the very soul of science. D’Arcy Wentworth Thompson
• The creative principle [of science] resides in mathematics. Albert Einstein
• Proofs exist only in mathematics and logic, not in science. Satoshi Kanazawa
• … mathematics is absolutely necessary and useful to the other sciences. Roger Bacon
• I accept no principles of physics which are not also accepted in mathematics. Rene Descartes
• Mathematics is the queen of science, and arithmetic the queen of mathematics. Carl Friedrich Gauss
• Only mathematics and mathematical logic can say as little as the physicist means to say. Bertrand Russell
• No human investigation can be called real science if it cannot be demonstrated mathematically. Leonardo da Vinci
• If all art aspires to the condition of music, all the sciences aspire to the condition of mathematics. George Santayana
• Mathematics – the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human affairs. Isaac Barrow
• Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics. David Hilbert
• We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry. Maria Montessori
• Over the centuries, monumental upheavals in science have emerged time and again from following the leads set out by mathematics. Brian Greene
• To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls. Pafnuty Chebyshev
• ‘Without the help of mathematics,’ the wise man continued, ‘the art could not advance and all the sciences would perish.’ Julio Cesar de Mello e Souza
• We in science are spoiled by the success of mathematics. Mathematics is the study of problems so simple that they have good solutions Whitfield Diffie
• I assert that, in any particular natural science, one encounters genuine scientific substance only to the extent that mathematics is present. Immanuel Kant
• If you want to be a physicist, you must do three things-first, study mathematics, second, study more mathematics, and third, do the same. Arnold Sommerfeld
• The new art must be based upon science – in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences. Albrecht Durer
• Mathematics as we know it and as it has come to shape modern science could never have come into being without some disregard for the dangers of the infinite. David Bressoud
• One factor that has remained constant through all the twists and turns of the history of physical science is the decisive importance of the mathematical imagination. Freeman Dyson
• It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones. Charles Sanders Peirce
• For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created. Freeman Dyson
• But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain. Albert Einstein
• One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics. Henri Poincare
• It is perplexing to see the flexibility of the so-called ‘exact sciences’ which by cast-iron laws of logic and by the infallible help of mathematics can lead to conclusions which are diametrically opposite to one another. Vasco Ronchi
• In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science, as it grows towards perfection, becomes mathematical in its ideas. Alfred North Whitehead
• The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level. John von Neumann
• When a branch of mathematics ceases to interest any but the specialists, it is very near its death, or at any rate dangerously close to a paralysis, from which it can be rescued only by being plunged back into the vivifying source of the science. Andre Weil
• All science requires mathematics. The knowledge of mathematical things is almost innate in us. This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon. Roger Bacon
• There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured … Of these sciences the gate and key is mathematics … He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world. Roger Bacon
• The more progress physical sciences make, the more they tend to enter the domain of mathematics, which is a kind of centre to which they all converge. We may even judge the degree of perfection to which a science has arrived by the facility with which it may be submitted to calculation. Adolphe Quetelet
• The usefulness of mathematics in furthering the sciences is commonly acknowledged: but outside the ranks of the experts there is little inquiry into its nature and purpose as a deliberate human activity. Doubtless this is due to the inevitable drawback that mathematical study is saturated with technicalities from beginning to end. Herbert Turnbull
• Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music. Art, on the other hand, begins with the world we construct, not with the world we see. It starts with the imagination, and then works toward ordinary experience. Northrop Frye

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…and especially in physics…

• 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
• Mathematics catalogues everything that is not self-contradictory; within that vast inventory, physics is an island of structures rich enough to contain their own beholders. Greg Egan
• But in spite of the obvious effectiveness of mathematics in physics, I have never heard of a good a prioriargument that the world must be organised to mathematical principles. Lee Smolin
• Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover. Bertrand Russell
• 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
• Everyone now agrees that a physics lacking all connection with mathematics … would only be an historical amusement, fitter for entertaining the idle than for occupying the mind of a philosopher. Franz Karl Achard
• Mathematics is not something that you find lying around in your back yard. It’s produced by the human mind. Yet if we ask where mathematics works best, it is in areas like particle physics and astrophysics, areas of fundamental science that are very, very far removed from everyday affairs. Paul Davies
• It seems that every practitioner of physics has had to wonder at some point why mathematics and physics have come to be so closely entwined. Opinions vary on the answer. ..Bertrand Russell acknowledged..’Physics is mathematical not because we know so much about the physical world, but because we know so little.’ ..Mathematics may be indispensable to physics, but it obviously does not constitute physics. Etienne Klein
• So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn’t that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist’s eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality. Henri Poincare

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…though some question its true usefulness

• It would be better for the true physics if there were no mathematicians on earth. Daniel Bernoulli
• If all of mathematics disappeared, physics would be set back by exactly one week. Richard P. Feynman

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Mathematics is used in chemistry

• A chemist who does not know mathematics is seriously handicapped. Irving Langmuir
• Unless the chemist learns the language of mathematics, he will become a provincial and the higher branches of chemical work, that require reason as well as skill, will gradually pass out of his hands. Alexander Crum Brown

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Mathematics is used in biology

• The lack of real contact between mathematics and biology is either a tragedy, a scandal or a challenge, it is hard to decide which. Gian-Carlo Rota
• The basic science is not physics or mathematics but biology — the study of life. We must learn to think both logically and bio-logically. Edward Abbey
• In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry out their researches. Auguste Comte
• I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia. John Forbes Nash
• Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. Israel Gelfand
• The excellent beginning made by quantum mechanics with the hydrogen atom peters out slowly in the sands of approximations in as much as we move towards more complex situations . . . This decline in the efficiency of mathematical algorithms accelerates when we go into chemistry. The interactions between two molecules of any degree of complexity evades precise mathematical description . . . In biology, if we make exceptions of the theory of population and of formal genetics, the use of mathematics is confined to modelling a few local situations (transmission of nerve impulses, blood flow in the arteries, etc.) of slight theoretical interest and limited practical value . . . The relatively rapid degeneration in the possible uses of mathematics when one moves from physics to biology is certainly known among specialists, but there is a reluctance to reveal it to the public at large . . . [T]he feeling of security given by the reductionist approach is in fact illusory. René Thom

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Mathematics is used in economics

• Mathematics has given economics rigor, but alas, also mortis. Robert Heilbroner
• Mathematics brought rigor to Economics. Unfortunately, it also brought mortis. Kenneth E. Boulding
• It is clear that economics, if it is to be a science at all, must be a mathematical science. William Stanley Jevons
• The use of mathematics has brought rigor to economics. Unfortunately, it has also brought mortis. Robert Heilbroner
• Economics has become increasingly an arcane branch of mathematics rather than dealing with real economic problems. Milton Friedman
• The economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth. Paul Krugman
• There can be no question, however, that prolonged commitment to mathematical exercises in economics can be damaging. It leads to the atrophy of judgement and intuition. . . John Kenneth Galbraith
• You will perceive that economy, scientifically speaking, is a very contracted science; it is in fact a sort of vague mathematics which calculates the causes and effects of man’s industry, and shows how it may be best applied. William Stanley Jevons
• The discipline of economics has yet to get over its childish passion for mathematics and for purely theoretical and often highly ideological speculation, at the expense of historical research and collaboration with the other social sciences. Thomas Piketty
• It is clear that Economics, if it is to be a science at all, must be a mathematical science … simply because it deals with quantities… As the complete theory of almost every other science involves the use of calculus, so we cannot have a true theory of Economics without its aid. William Stanley Jevons
• Modern economics is sick. Economics has increasingly become an intellectual game played for its own sake and not for its practical consequences for understanding the economic world. Economists have converted the subject into a sort of social mathematics in which analytical rigour is everything and practical relevance is nothing. Mark Blaug
• Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon common places as French was once used in diplomatic communications. James R Newman

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Mathematics is used in statistics

• Statistics is, or should be, about scientific investigation and how to do it better, but many statisticians believe it is a branch of mathematics. Now I agree that the physicist, the chemist, the engineer, and the statistician can never know too much mathematics, but their objectives should be better physics, better chemistry, better engineering, and in the case of statistics, better scientific investigation. Whether in any given study this implies more or less mathematics is incidental. George E. P. Box

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Mathematics is used in cryptography and online security

• Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers. Vladimir Arnold
• I chose to deal with the science of cryptography. Cryptography began in mathematics. Codes were developed, even from Caesar’s time, based on number theory and mathematical principles. I decided to use those principles and designed a work that is encoded. Jim Sanborn
• Cryptography shifts the balance of power from those with a monopoly on violence to those who comprehend mathematics and security design. Jacob Appelbaum

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Some mathematics has not yet found an application

• Mathematicians boast of their exacting achievements, but in reality they are absorbed in mental acrobatics and contribute nothing to society. Ogyu Sorai
• The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter. Carl Gustav Hempel
• No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. G. H. Hardy

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Mathematics cannot be used to predict human behaviour

• While physics and mathematics may tell us how the universe began, they are not much use in predicting human behavior because there are far too many equations to solve. I’m no better than anyone else at understanding what makes people tick, particularly women. Stephen Hawking
• Whereas, to borrow an illustration from mathematics, life was formerly an equation of, say, 100 unknown quantities, it is now one of 99 only, inasmuch as memory and heredity have been shown to be one and the same thing. Samuel Butler

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Mathematics is intimately related with other disciplines

• Woe be to him who tries to isolate one department of knowledge from the rest. All science is one: language, literature and history, physics, mathematics and philosophy; subjects which seem the most remote from one another are in reality connected, or rather they all form a single system. Jules Michelet
• Mathematics is not a separate entity. It owes a great deal of its power and its beauty to other disciplines. Mark Kac

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Mathematics is related to philosophy

• Mathematics is less related to accounting than it is to philosophy. Leonard Adleman
• Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician. Gottlob Frege
• What has philosophy got to do with measuring anything? It’s the mathematicians you have to trust, and they measure the skies like we measure a field. Galileo Galilei

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Mathematics is related to computer science and is increasingly aided by it

• Nowadays we can do computer experiments using Mathematica, and even solve a system of 42 equations. This offers another route to knowledge, rather than mere ideas. John Forbes Nash
• Without computers we will be stuck only proving theorems that have short proofs. Kenneth Appel
• Computer assisted proofs are getting better and better and computers will play a bigger and bigger role in the future. Enrico Bombieri
• Before a kid learns how to use a computer that can solve mathematical problems, he or she should know how to do arithmetic without a computer. Andy Rooney
• The computer was, to the best of my feelings about the subject, not thinking like a mathematician, and it was much more successful, because it was thinking not like a mathematician. Kenneth Appel

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Mathematics is related to art

• Without mathematics there is no art. Luca Pacioli
• A mathematical point is the most indivisible and unique thing which art can present. John Donne
• But mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius. Marston Morse

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Mathematics is related to music

• At its heart, music is all higher mathematics. Mos Def
• A lot of music is mathematics. It’s balance. Mel Brooks
• Music is what mathematics does on a Saturday night. Aaron Sorkin
• Mathematics: silent harmonies. Music: sounding numbers. Mason Cooley
• Music at times is more like perfume than mathematics. Gabriel Marcel
• Music is mathematics, the mathematics of listening, mathematics for the ears. Karlheinz Stockhausen
• The mathematics of rhythm are universal. They don’t belong to any particular culture. John McLaughlin
• Reading music is something that’s inherently hateful to me. It makes music like mathematics. Robert Quine
• But however measurable, there is much more life in music than mathematics or logic ever dreamed of. Gabriel Marcel
• Music is the pleasure the human mind experiences from counting without being aware that it is counting. Gottfried Leibniz
• There are a lot of very strong connections with music and mathematics. They both can work in patterns and sequences and repetitions. Kate Bush
• I love music. I do play. It’s like mathematics, and it’s also emotional. It’s nice to play, for no other reason than just to play. Jeremy Renner
• The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic. Gottfried Leibniz
• A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful? Martin Gardner
• If you learn music, you’ll learn history. If you learn music, you’ll learn mathematics. If you learn music, you’ll learn most all there is to learn. Edgar Cayce
• There is a beauty in discovery. There is mathematics in music, a kinship of science and poetry in the description of nature, and exquisite form in a molecule. Glenn T. Seaborg
• Through music you can become sad, joyful, loving, you can learn. You can learn mathematics, touch, pacing… Oh my God! Ooh… Wow… You can see colors through music. Anything! Nina Simone
• Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics. Jean-Philippe Rameau
• May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life. James Joseph Sylvester
• Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience. Keith Devlin
• Music, for centuries and centuries, was used to teach everything. It was used to teach language, mathematics, history. The news was music. Everything traveled by song. It was used to teach ethics. It was used to create conscience, probably more than anything. T Bone Burnett
• 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

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Mathematics is related to poetry

• Poetry is a mystic, sensuous mathematics of fire, smoke-stacks, waffles, pansies, people, and purple sunsets. Carl Sandburg
• Poetry is a form of mathematics, a highly rigorous relationship with words. Tahar Ben Jelloun
• I have long admired Ron Whitehead. He is crazy as nine loons, and his poetry is a dazzling mix of folk wisdom and pure mathematics. Hunter S. Thompson
• We have heard much about the poetry of mathematics, but very little of it has yet been sung. The ancients had a juster notion of their poetic value than we. Henry David Thoreau
• 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. Brian Cox
• Just as a poet often has license from the rules of grammar and pronunciation, we should like to ask for ‘physicists’ license from the rules of mathematics in order to express what we wish to say in as simple a manner as possible. Richard P. Feynman
• Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite. Ezra Pound
• Poetry and code – and mathematics – make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations. Stephanie Strickland

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Mathematics is related to prose

• It wasn’t by accident that the Gettysburg address was so short. The laws of prose writing are as immutable as those of flight, of mathematics, of physics. Ernest Hemingway

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Mathematics is related to ethics

• The understanding of mathematics is necessary for a sound grasp of ethics. Socrates
• Mathematics should be mixed not only with physics but with ethics. Henry David Thoreau
• Mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Charles Sanders Peirce
• No code of ethics and no effort are justifiable a priori in the face of the cruel mathematics that command our condition. Albert Camus
• The problem is that no ethical system has ever achieved consensus. Ethical systems are completely unlike mathematics or science. This is a source of concern. Daniel Dennett
• The training one receives when one becomes a technician, like a data scientist – we get trained in mathematics or computer science or statistics – is entirely separated from a discussion of ethics. Cathy O’Neil
• The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. Henry David Thoreau
• …mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be. Charles Sanders Peirce

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…often from what appears random

• Randomness is the true foundation of mathematics. Gregory Chaitin
• Real numbers are good if you add the word ‘random’. Peter Sarnak

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Mathematics is an expression of certainty…

• Numbers are the most certain things we have. Andy Rooney
• Except in pure mathematics, nothing is known for certain (although much is certainly false). Carl Sagan
• He who does not understand the supreme certainty of mathematics is wallowing in confusion. Leonardo da Vincicertain
• And for mathematical science, he that doubts their certainty hath need of a dose of hellebore. Joseph Glanvill
• Those who condemn the supreme certainty of mathematics feed on confusion, and can never silence the contradictions of the sophistical sciences which lead to eternal quackery. Leonardo da Vinci
• The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as “All bachelors are unmarried,” but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter. Carl Gustav Hempel
• In mathematics there are no true controversies. Carl Friedrich Gauss

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…but can be far from obvious…

• Obvious is the most dangerous word in mathematics. Eric Temple Bell
• If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book … it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way. George Polya
• Some persons have contended that mathematics ought to be taught by making the illustrations obvious to the senses. Nothing can be more absurd or injurious: it ought to be our never-ceasing effort to make people think, not feel. Samuel Taylor Coleridge

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…and even vague

• That sometimes clear … and sometimes vague stuff … which is … mathematics. Imre Lakatos
• Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians’ stock of formulae. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless. Lewis Fry Richardson

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Some believe that mathematics is truth…

• Beauty in mathematics is seeing the truth without effort. George Polya
• Mathematics rightly viewed possesses not only truth but supreme beauty. Bertrand Russell
• So mathematical truth prefers simple words since the language of truth is itself simple. Tycho Brahe
• Physicists have come to realize that mathematics, when used with sufficient care, is a proven pathway to truth. Brian Greene
• Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it. Leo Tolstoy
• The calculus is the greatest aid we have to the application of physical truth in the broadest sense of the word. William Fogg Osgood
• There are only two kinds of certain knowledge: Awareness of our own existence and the truths of mathematics. Jean le Rond d’Alembert
• All the truths of mathematics are linked to each other, and all means of discovering them are equally admissible. Adrien-Marie Legendre
• There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered. Theon called this analysis. Francois Viete
• In a mathematical proposition, for example, the objectivity is given, but therefore its truth is also an indifferent truth. Soren Kierkegaard
• 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 is good for the soul, getting things right enlivens a sense of truth, efforts to understand automatically purify desires. Iris Murdoch
• The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth. George Bernard Shaw
• Truth . . . and if mine eyes Can bear its blaze, and trace its symmetries, Measure its distance, and its advent wait, I am no prophet – I but calculate. Charles Mackay
• He that gives a portion of his time and talent to the investigation of mathematical truth, will come to all other questions with a decided advantage over his opponents. Charles Caleb Colton
• Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or “mathematics envy.” Marvin Minsky
• Religion has the same relation to man’s heavenly condition that mathematics has to his earthly one: both the one and the other are merely the rules of the game. Belief in God and belief in numbers: local truth and truth of location. Vladimir Nabokov
• The student of mathematics must get rid of all arbitrary thinking and follow purely the demands of thought. In thinking in this way, the laws of the spiritual world flow into him. This regulated thinking leads to the most spiritual truths. Rudolf Steiner
• The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me – both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension. Paul Halmos
• There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation. Charles Hermite
• You have this world of mathematics, which is very real and which contains all kinds of wonderful stuff. And then we also have the world of nature, which is real, too. Freeman Dyson
• [I]f in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics, in so far as disposed through it we are able to reach certainty in other sciences and truth by the exclusion of error. Roger Bacon
• When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates. Gottfried Leibniz
• Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth. Salomon Bochner
• Mathematical demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick are the only truths that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth according as their Subjects are more or less capable of Mathematical Demonstration. Christopher Wren
• I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato’s world of mathematical concepts… When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of ‘seeing’. Roger Penrose
• Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God. Julio Cesar de Mello e Souza Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God. Julio Cesar de Mello e Souza
• The peculiarity of the evidence of mathematical truths is that all the argument is on one side. John Stuart Mill
• 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 Gardner
• There are several kinds of truths, and it is customary to place in the first order mathematical truths, which are, however, only truths of definition. These definitions rest upon simple, but abstract, suppositions, and all truths in this category are only constructed, but abstract, consequences of these definitions … Physical truths, to the contrary, are in no way arbitrary, and do not depend on us. Georges-Louis Leclerc, Comte de Buffon
• The faith of scientists in the power and truth of mathematics is so implicit that their work has gradually become less and less observation, and more and more calculation…. But the facts which are accepted by virtue of these tests are not actually observed at all. Susanne Katherina Langer
• The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies. A.J. Ayer
• It is from this absolute indifference and tranquillity of the mind, that mathematical speculations derive some of the most considerable advantages; because there is nothing to interest the imagination; because the judgment sits free and unbiased to examine the point. All proportions, every arrangement of quantity, is alike to the understanding, because the same truths result to it from all; from greater from lesser, from equality and inequality. Edmund Burke

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…some would say relative truth…

• Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional. Charles Proteus Steinmetz

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…and others even say absolute truth

• The study of the properties of numbers, Plato tells us, habituates the mind to the contemplation of pure truth, and raises us above the material universe. He would have his disciples apply themselves to this study, not that they may be able to buy or sell, not that they may qualify themselves to be shopkeepers or travelling merchants, but that they may learn to withdraw their minds from the ever-shifting spectacle of this visible and tangible world, and to fix them on the immutable essences of things. Thomas B. Macaulay

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Others argue mathematics is not truth

• Mathematics is a body of knowledge, but it contains no truths. Morris Kline
• Insofar as mathematics is true, it does not describe the real world. Insofar as it describes the real world, it is not true. Albert Einstein
• 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

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Some argue mathematics is reality itself…

• Mathematics is not only real, but it is the only reality. Martin Gardner
• Do mathematics have a relation to reality or are they only a mathematical symbol? Fulton J. Sheen
• Is mathematics, for all its power, really the root of reality? Or is it a product of the human mind?
• 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
• But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed. Albert Einstein
• Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality. Georg Cantor
• Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality. Nikola Tesla
• You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present. Hermann Hesse
• I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply the notes of our observations. G. H. Hardy
• Mathematics is the one language we might conceivably have in common with other forms of intelligent life in the universe. As I understand it, there is no reality more independent of our perception and more true to itself than mathematical reality. Don DeLillo
• …Science and mathematics Run parallel to reality, they symbolize it, they squint at it, They never touch it: consider what an explosion Would rock the bones of men into little white fragments and unsky the world If any mind for a moment touch truth. Robinson Jeffers
• There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation. Charles Hermite
• I believe that numbers and functions of Analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists. Charles Hermite

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…a metaphysical reality…

• Metaphysical: Relating to the transcendent or to a reality beyond what is perceptible to the senses.  Cambridge Dictionary
• Mathematics is the only good metaphysics. Lord Kelvin
• Mathematics has not a foot to stand upon which is not purely metaphysical. Thomas de Quincey

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…a reality that transcends the physical universe

• If math exists beyond all created things, it stabilizes the constants and anchors reality in a place untouched by nature’s chaos, roughness, and irregularity. Deepak Chopra and Menas Kafatos

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Others argue mathematics is not reality…

• In real life, I assure you, there is no such thing as algebra. Fran Lebowitz
• Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality. Nikola Tesla
• Still more astonishing is that world of rigorous fantasy which we call mathematics. Gregory Bateson
• . . . we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed. Edward Kasner
• In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Stanislaw Ulam
• On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say ‘mathematics is just a combination of meaningless symbols,’… Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient. Jean Dieudonne

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…but exists only in the human mind

• The mathematics is not there till we put it there. Arthur Eddington
• Numbers are intellectual witnesses that belong only to mankind. Honore de Balzac
• I firmly believe that mathematics does not exist outside of humans. It is something we, as a species, invent. Keith Devlin
• It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. Percy Williams Bridgman
• 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
• Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man. Maria Goeppert-Mayer
• … the laws of physics and of logic … the number system … the principle of algebraic substitution. These are ghosts. We just believe in them so thoroughly they seem real. Robert M. Pirsig
• Laws of nature are human inventions, like ghosts. Laws of logic, or mathematics are also human inventions, like ghosts. The whole blessed thing is a human invention, including the idea that it isn’t a human invention. Robert M. Pirsig

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The laws of nature are mathematical…

• Nature is written in mathematical language. Galileo Galilei
• Nature’s great book is written in mathematics. Galileo Galilei
• Our external physical reality is a mathematical structure. Max Tegmark
• Mathematics is the abstract key which turns the lock of the physical universe. John Polkinghorne
• Mathematics is the science of patterns, and nature exploits just about every pattern that there is. Ian Stewart
• There are mathematical equations, little squiggles on pieces of paper, that tell us how whole universes behave. John Barrow
• Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry. Vladimir Arnold
• Patterns and regularities exist in nature that mathematics can accurately predict. It’s a wonder. It doesn’t need to be that way. Anthony Lambert
• Physics is essentially an intuitive and concrete science. Mathematics is only a means for expressing the laws that govern phenomena. Albert Einstein
• We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry. Ludwig Wittgenstein
• Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars. Edward Frenkel
• One cannot understand… the universality of laws of nature, the relationship of things, without an understanding of mathematics. There is no other way to do it. Richard P. Feynman
• The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures. James Jeans
• One can understand nature only when one has learned the language and the signs in which it speaks to us; but this language is mathematics and these signs are methematical figures. Galileo Galilei
• When God produces the world, he produces a thoroughly mathematical structure that obeys the laws of number, geometrical figure and quantitative function. Nature is an embodied mathematical system. Galileo
• Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude. Friedrich Nietzsche
• Pure mathematics is, in its way, the poetry of logical ideas. … [By seeking] logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature. Albert Einstein
• All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true. Georg C. Lichtenberg
• The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe. Karl Pearson
• The laws of Nature are written in the language of mathematics…the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word. Galileo Galilei
• Among all of the mathematical disciplines the theory of differential equations is the most important… It furnishes the explanation of all those elementary manifestations of nature which involve time. Sophus Lie
• The tree is made by nature, mathematics by people. And combining the two is creating this beautiful alliance between humanity and nature. That’s why my forests are mathematical expansion systems, all of them. Agnes Denes
• The Universe is a grand book which cannot be read until one first learns to comprehend the language and become familiar with the characters in which it is composed. It is written in the language of mathematics. Galileo Galilei
• … it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case. Richard P. Feynman
• All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. James Clerk Maxwell
• For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems. Michael Frayn
• …nature seems very conversant with the rules of pure mathematics, as our own mathematicians have formulated them in their studies, out of their own inner consciousness and without drawing to any appreciable extent on their experience of the outer world. James Jeans
• Heading off in another direction, a new book by the physicist Max Tegmark suggests that a different ingredient — mathematics — needs to be admitted into science as one of nature’s irreducible parts. In fact, he believes, it may be the most fundamental of all. George Johnson
• It is interesting thus to follow the intellectual truths of analysis in the phenomena of nature. This correspondence, of which the system of the world will offer us numerous examples, makes one of the greatest charms attached to mathematical speculations. Pierre-Simon Laplace
• We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it. Roger Penrose
• Traditionally, scientists have treated the laws of physics as simply ‘given,’ elegant mathematical relationships that were somehow imprinted on the universe at its birth, and fixed thereafter. Inquiry into the origin and nature of the laws was not regarded as a proper part of science. Paul Davies
• All phenomena are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough. Nicolas de Caritat, marquis de Condorcet
• Until now, physical theories have been regarded as merely models which approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord. Paul Davies
• For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the mathematics, of which sort are perspective, music, astronomy, cosmography, architecture, engineery, and divers others. Francis Bacon
• Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework. Melvin Schwartz
• A second argument is that the Universe appears to run on mathematical lines, just as you would expect from a computer program. Ultimately, say some physicists, reality might be nothing but mathematics. Max Tegmark of the Massachusetts Institute of Technology argues that this is just what we would expect if the laws of physics were based on a computational algorithm. Philip Ball
• Three principles – the conformability of nature to herself, the applicability of the criterion of simplicity, and the utility of certain parts of mathematics in describing physical reality – are thus consequences of the underlying law of the elementary particles and their interactions. Those three principles need not be assumed as separate metaphysical postulates. Instead, they are emergent properties of the fundamental laws of physics. Murray Gell-Mann

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…even down to the quantum level

• Modern physics has definitely decided for Plato. For the smallest units of matter are not physical objects in the ordinary sense of the word: they are forms, structures, or – in Plato’s sense – Ideas, which can be unambiguously spoken of only in the language of mathematics. Rupert Sheldrake
• The mathematics of quantum mechanics very accurately describes how our universe works. And it tells us our reality is continually branching into different possibilities, just like a coral. Antony Garrett Lisi
• Quantum physics is a bit of a passion of mine. It’s extraordinary. There’s a branch of mathematics that is based on lunacy, and that’s wonderful. Bob Hoskins
• For me, the important thing about quantum mechanics is the equations, the mathematics. If you want to understand quantum mechanics, just do the math. All the words that are spun around it don’t mean very much. It’s like playing the violin. If violinists were judged on how they spoke, it wouldn’t make much sense. Freeman Dyson

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Why the laws of nature are mathematical is a profound mystery because it does not need to be that way

• Why does mathematics describe nature. That’s a deeper question than most. Terence McKenna
• Why does the natural world always – so far as we know – obey laws of mathematics? Eugene Wigner
• The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation of it. Eugene Wigner
• How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Albert Einstein
• 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. Eugene Wigner
• It remains a mystery to scientists why mathematics has the ability to so accurately describe the structure and behaviors of reality, but it is impossible to be a physicist and not have come to appreciate that it does. Joseph Selbie
• The astonishing fact is that similar mathematics applies so well to planets and to clocks. It needn’t have been this way. We didn’t impose it on the Universe. That’s the way the Universe is. If this is reductionism, so be it. Carl Sagan
• The important point is not merely that there are regularities in nature, but that these regularities are mathematically precise, universal, and “tied together.” Einstein spoke of them as “reason incarnate.” The question we should ask is how nature came packaged in this fashion. Antony Flew
• Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. Israel Gelfand
• You have this world of mathematics, which is very real and which contains all kinds of wonderful stuff. And then we also have the world of nature, which is real, too. And that, by some miracle, the language that nature speaks is the same language that we invented for mathematics. That’s just an amazing piece of luck, which we don’t understand. Freeman Dyson
• As physicists investigate the most fundamental characteristics of nature, they’re tackling issues that have long been the province of philosophers and theologians: Is the universe infinite and eternal? Why does it seem to follow mathematical laws, and are those laws inevitable? And, perhaps most important, why does the universe exist? Why is there something instead of nothing? Mark Alpert
• Even if we should ever succeed at tracing back the entire physical narrative of reality to its origin from “matter” (using that word in the broadest sense, as an alternative to all things “spiritual”) and to a set of immutable physical laws (like the law of gravity) describable by mathematics, we shall still not have dispensed with form or finality. What, after all, are laws, and why can mathematics describe them at all? David Bentley Hart
• I sometimes think we fail to be nearly as surprised and baffled by mathematical knowledge as we ought to be largely because we tend to think of the world as vaguely “arithmetical,” in the sense that its parts are enumerable and measurable, and because we lazily think of pure mathematics as just some sort of immense amplification of arithmetic, rather than the angelic language of nearly limitless intelligibility that it is. David Bentley Hart
• THEY call it the “unreasonable effectiveness of mathematics”. Physicist Eugene Wigner coined the phrase in the 1960s to encapsulate the curious fact that merely by manipulating numbers we can describe and predict all manner of natural phenomena with astonishing clarity, from the movements of planets and the strange behaviour of fundamental particles to the consequences of a collision between two black holes billions of light years away. Michael Brooks
• Our experience teaches us that there are indeed laws of nature, regularities in the way things behave, and that these laws are best expressed using the language of mathematics. This raises the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality, and this has proved to be the case time and again throughout the history of science. We will see this happen during the course of this book, and it is truly one of the wonderful mysteries of our universe that it should be so. Brian Cox

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It is even more mysterious that we can comprehend the mathematical nature of the universe

• Perhaps even more mysterious and wondrous than the mathematical nature of the universe is the fact that we possess the consciousness and intelligence to comprehend this. Anthony Lambert
• It takes the existence of some kind of a god to make the mathematical underpinnings of the universe comprehensible. Michael Peligro
• Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes. Eric Temple Bell
• Nagel finds it astonishing that the human brain — this biological organ that evolved on the third rock from the sun — has developed a science and a mathematics so in tune with the cosmos that it can predict and explain so many things. George Johnson
• Almost everything about the universe is astounding. I think the most amazing thing is how gifted we are – we are only monkeys who came down from the trees just recently. We have these amazing gifts of music and mathematics and painting and Olympic running. Freeman Dyson
• Is it true that simple laws, in these senses, are more probable than complicated ones? I cannot see any reason at all for saying so. We may very much like to have laws that are elegant, comprehensive, parsimonious and mathematically precise. That will suit our aesthetic tastes. If the laws are like that, it will be a great and unexpected gift, as Eugene Wigner famously said. But it is not by any means probable that the laws should be like that. If anything, it is amazingly improbable that the laws of nature should be so finely adjusted to our intellectual capacities. Keith Ward

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Some argue the mathematical nature of the universe is proof of God

• God exists since mathematics is consistent, and the Devil exists since we cannot prove it. Andre Weil
• We know that nature is described by the best of all possible mathematics because God created it. Alexander Polyakov
• God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. Hilda Phoebe Hudson
• For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear. Leonhard Euler
• The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics. Johannes Kepler
• There may be innumerable finite causes of complexity, but a good argument can be made that only a single infinite cause can account for perfect, universal, intelligible, mathematically describable order. David Bentley Hart
• Sir Arthur Eddington deduces religion from the fact that atoms do not obey the laws of mathematics. Sir James Jeans deduces it from the fact that they do. Bertrand Russell
• Thus God himself was too kind to remain idle and began to play the game of signatures signing his likeness unto the world: therefore I chance to think that all nature and the graceful sky are symbolized in the art of Geometria. Johannes Kepler
• Religion has the same relation to man’s heavenly condition that mathematics has to his earthly one: both the one and the other are merely the rules of the game. Belief in God and belief in numbers: local truth and truth of location. Vladimir Nabokov
• There are good reasons to believe in God, including the existence of mathematical principles and order in creation. They are positive reasons, based on knowledge, rather than default assumptions based on a temporary lack of knowledge. Francis Collins
• Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God. Julio Cesar de Mello e Souza
• When Galileo discovered he could use the tools of mathematics and mechanics to understand the motion of celestial bodies, he felt, in the words of one imminent researcher, that he had learned the language in which God recreated the universe. Today we are learning the language in which God created life. We are gaining ever more awe for the complexity, the beauty, the wonder of God’s most devine and sacred gift. William J. Clinton
• Now you may think I have written God entirely out of the picture. Who needs a God when the laws of physics can do such a splendid job? But we are bound to return to that burning question: Where do the laws of physics come from? And why those laws rather than some other set? Most especially: Why a set of laws that drives the searing, featureless gases coughed out of the big bang toward life and consciousness and intelligence and cultural activities such as religion, art, mathematics, and science? Paul Davies

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They point out that the mathematical laws underpinning the universe are extremely precise…

• Precise mathematical laws govern the four fundamental forces, gravity, electromagnetism, and the strong and weak nuclear forces. Deepak Chopra and Menas Kafatos
• If math exists beyond all created things, it stabilizes the constants and anchors reality in a place untouched by nature’s chaos, roughness, and irregularity. Deepak Chopra and Menas Kafatos

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…and seem to have been precisely fine-tuned to allow life to exist

• The fine tuning of the universe provides prima facie evidence of deistic design. Edward Robert Harrison
• The laws of physics accepted by modern science, including the particular values of its fundamental constants, are minutely fine-tuned to permit the evolution of life to go on. Bernard Haisch
• It’s hard to get past the proposition that the constants of nature have their particular values because the universe is built to allow for life to develop. Deepak Chopra and Menas Kafatos
• The Goldilocks Enigma is the idea that everything in the universe is just right for life, like the porridge in the fairy tale. Paul Davies
• A life-giving factor lies at the centre of the whole machinery and design of the world. John Archibald Wheeler
• This is a fine-tuned universe, which has a number of constants in play. If any of these constants had a different value, then life as we know it would not have been possible. Therefore, there must have been an intelligent designer. Alan H. Dawe

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The universe, they argue, thus shows every sign of mathematical and intelligent design

• I hope the foregoing discussion will have convinced the reader that the natural world is not just any old concoction of entities and forces, but a marvelously ingenious and unified mathematical scheme. …these rules look as if they are the product of intelligent design. I do not see how that can be denied. Paul Davies
• The temptation to believe that the Universe is the product of some sort of design, a manifestation of subtle aesthetic and mathematical judgment, is overwhelming. The belief that there is “something behind it all” is one that I personally share with, I suspect, a majority of physicists. Paul Davies
• If the simulation hypothesis is correct, then “there’s a creator, an architect — someone who designed the world.” It’s an ancient idea recast in terms of “mathematics and science rather than just faith. Rich Terrile

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God, they argue, is a consummate mathematician…

• God ever arithmetizes. Carl Gustav Jacob Jacobi
• If there is a God, he’s a great mathematician. Paul Dirac
• God is an awesome mathematician and physicist. Francis Collins
• The laws of nature are but the mathematical thoughts of God. Euclid
• Black holes result from God dividing the universe by zero. Steven Wright
• God has the Big Book, the beautiful proofs of mathematical theorems are listed here. Paul Erdos
• God made integers; all else is the work of man. Leopold Kronecker
• The laws of nature are written by the hand of God in the language of mathematics. Galileo Galilei
• The Good Lord made all the integers; the rest is man’s doing. Leopold Kronecker
• God does not care about our mathematical difficulties. He integrates empirically. Albert Einstein
• A mathematical formula should never be “owned” by anybody! Mathematics belong to God. Donald Knuth
• Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God. Herbert Turnbull
• God is a mathematician of a very high order and He used advanced mathematics in constructing the universe. Paul A. M. Dirac
• Before creation, God did just pure mathematics. Then He thought it would be a pleasant change to do some applied. John Edensor Littlewood
• One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Paul Dirac
• The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers. Le Corbusier
• Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, “This is from the book!” Ross Honsberger
• At the Egyptian city of Naucratis there was a famous old god whose name was Theuth; the bird which is called the Ibis was sacred to him, and he was the inventor of many arts, such as arithmetic and calculation and geometry and astronomy and draughts and dice, but his great discovery was the use of letters. Plato
• God is a pure mathematician!’ declared British astronomer Sir James Jeans. The physical Universe does seem to be organised around elegant mathematical relationships. And one number above all others has exercised an enduring fascination for physicists: 137.0359991…. It is known as the fine-structure constant and is denoted by the Greek letter alpha (α). Paul Davies

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…who used beautiful mathematics in creating the universe

• God built the universe on numbers. Pythagoras
• God used beautiful mathematics in creating the world. Paul Dirac
• Mathematics is the language with which God has written the universe. Galileo Galilei
• In creating the world, God used arithmetic, geometry, and likewise astronomy. Nicholas of Cusa
• All the mathematical sciences are founded on relations between physical laws and laws of numbers. James Clerk Maxwell
• People who wish to analyze nature without using mathematics must settle for a reduced understanding. Richard P. Feynman
• All the effects of Nature are only the mathematical consequences of a small number of immutable laws. Pierre-Simon Laplace
• The creator of the universe works in mysterious ways; but he uses a base ten counting system and likes round numbers. Scott Adams
• The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics. Isaac Newton
• We know that nature is described by the best of all possible mathematics because God created it. So there is a chance that the best of all possible mathematics will be created out of physicists’ attempts to describe nature. Alexander Markovich Polyakov
• If God creates a world of particles and waves, dancing in obedience to mathematical and physical laws, who are we to say that he cannot make use of those laws to cover the surface of a small planet with living creatures? Martin Gardner
• In the beginning (if there was such a thing), God created Newton’s laws of motion together with the necessary masses and forces. This is all; everything beyond this follows from the development of appropriate mathematical methods by means of deduction. Albert Einstein
• Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist. Saint Augustine
• …all of creation was made so that we, men and women, could see and testify to God’s glory in it. It says that all of this, the entire universe, was created for us, for us to see and to observe and to appreciate, for alone in all creation, man his this capacity to see and observe and appreciate. This means that all of the laws of physics, of chemistry, of motion, of mathematics, of thermodynamics, all natural laws, were authored by a God who delights in man’s discovery of them when that discovery is accompanied by an appreciation of the God who created them. Robert W. Faid

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God is an expert in applied geometry

• Geometry is the only science that it hath pleased God hitherto to bestow on mankind. Thomas Hobbes
• Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. They are no exceptions to the rule that God always geometrizes. Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science. D’Arcy Wentworth Thompson
• Thus God himself was too kind to remain idle and began to play the game of signatures signing his likeness unto the world: therefore I chance to think that all nature and the graceful sky are symbolized in the art of Geometria. Johannes Kepler
• One merit of mathematics few will deny: it says more in fewer words than any other science. The formula, e^iπ = -1 expressed a world of thought, of truth, of poetry, and of the religious spirit “God eternally geometrizes.” David Eugene Smith
• Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God. Johannes Kepler
• Why waste words? Geometry existed before the Creation, is co-eternal with the mind of God, is God himself (what exists in God that is not God himself?): geometry provided God with a model for the Creation and was implanted into man, together with God’s own likeness – and not merely conveyed to his mind through the eyes. Johannes Kepler

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Through mathematical laws, God brings order to the world…

• Order is heaven’s first law. Alexander Pope
• I see a certain order in the universe and math is one way of making it visible. May Sarton
• Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty. Deepak Chopra
• The language of mathematics exhibits every quality of design: balance, harmony, symmetry, and some would say beauty. Deepak Chopra and Menas Kafatos
• The wonderful arrangement and harmony of the cosmos would only originate in the plan of an almighty omniscient being. This is and remains my greatest comprehension. Isaac Newton
• The universe contains vastly more order than Earth-life could ever demand. All those distant galaxies, irrelevant for our existence, seem as equally well ordered as our own. Paul Davies
• All nature has come to expect from God a sense of orderliness. Whatever God does carries with it His fingerprint. And in the world around us His fingerprint of orderliness is evident to anybody who is honest with the facts. If you look at nature, you will discover a mathematical exactness. Without this precision, the entire world would be in utter confusion. One plus one always equals two no matter what part of the universe you happen to be in. And the laws of nature operate in beautiful harmony, a harmony that is ordered by God Himself. Aiden Wilson Tozer

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…order and harmony that reveals and expresses God

• God reveals himself in the orderly harmony of what exists. Albert Einstein
• One possibility is: God is nothing but the power of the universe to organize itself. Lee Smolin
• I believe God is a name we give to the mystery of the world that is beyond our capacity to understand. I believe that the energy of the universe is ordered by forces I cannot comprehend. Robert Jensen

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It is God that breathes life into the mathematical equations of the universe

• Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? Stephen Hawkins
• If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed, I would feel that the Creator had been uncharacteristically lacking in imagination. Freeman Dyson
• One cannot escape the feeling that these equations have an existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them. Heinrich Herz
• These are the qualities that have inspired some of the greatest artists to create their masterpieces, and they are precisely the same qualities that motivate scientists to search for the laws of nature. Like a work of art or a haunting poem, equations have a beauty and rhythm all their own. Michio Kaku
• Take away God and what also disappears is a very good reason for the existence of a universe containing intelligent, finite minds. We are left with the question that Stephen Hawking asked in A Brief History of Time: ‘What is it that breathes fire into the equations and makes a universe for them to describe?’ Keith Ward
• In the 1860s, a Scottish physicist named James Clerk Maxwell succeeded in summarizing all that was known about phenomena such as electricity and magnetism in four short equations possessed of as much aesthetic elegance as theoretical impact. But Maxwell did not merely succeed in summarizing in his equations everything people already knew. He also succeeded in predicting phenomena that nobody thought had anything to do with electricity and magnetism— phenomena that were not discovered until after Maxwell’s death in 1879. How could this be possible? Asked Ludwig Boltzmann, a contemporary of Maxwell’s and a colleague in the formulation of important theoretical landmarks. How can a summary of a rich and varied collection of phenomena occur with so few and such powerful symbols as those we find in the four famous lines that constitute Maxwell’s equations? In a sense, this is the very mystery of science: Not only can it pursue the goal of saying as much as possible in as few words or equations as possible, drawing a map of the terrain, a map that simply and clearly summarizes all the important data and thus allows us to find our way; but— and here lies the mystery—it can create a map that enables us to see details of the terrain that were unknown when the map was drawn! Tor Norrethranders

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Mathematics is God’s relation to the realm of nature…

• If memory within nature sounds mysterious, we should bear in mind that mathematical laws transcending nature are more rather than less so; they are metaphysical rather than physical. The way mathematical laws can exist independently of the evolving universe and at the same time act upon it remains a profound mystery. For those who accept God, this mystery is an aspect of God’s relation to the realm of nature; for those who deny God, the mystery is even more obscure: A quasi-mental realm of mathematical laws somehow exists independently of nature, yet not in God, and governs the evolving physical world without itself being physical. Rupert Sheldrake

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…and gives us insight into the mind of God

• 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
• In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven. Edward Everett Hale

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Mathematics is the very language of God

• Mathematics is the language in which the gods talk to people. Plato
• The language that nature speaks is the same language that we invented for mathematics. That’s just an amazing piece of luck, which we don’t understand. Freeman Dyson
• Mathematics is the language with which God has written the universe. Galileo Galilei
• When Galileo discovered he could use the tools of mathematics and mechanics to understand the motion of celestial bodies, he felt, in the words of one imminent researcher, that he had learned the language in which God recreated the universe. Today we are learning the language in which God created life. We are gaining ever more awe for the complexity, the beauty, the wonder of God’s most devine and sacred gift. William J. Clinton
• The laws of nature are written by the hand of God in the language of mathematics. Galileo

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Mathematics is described by some as a branch of theology…

• This is not mathematics; this is theology. Paul Gordan
• God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. Hilda Phoebe Hudson

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…and even a religion…

• The Mathematics are Friends to Religion; inasmuch as they charm the Passions, restrain the Impetuosity of Imagination, and purge the Mind from Error and Prejudice. John Arbuthnot
• One would normally define a “religion” as a system of ideas that contain statements that cannot be logically or observationally demonstrated… Gödel’s theorem not only demonstrates that mathematics is a religion, but shows that mathematics is the only religion that proves itself to be one! John D. Barrow

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…which is ironical because mathematics is also described by others as the religion of atheists

• Atheism is the opium of the mathematicians. Atheism is the religion of Mathematics. Bill Gaede
• Regardless of whether or not God exists, God has no place in mathematics, at least in my book. Doron Zeilberger

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Mathematics belongs to God…

• Mathematics is the Life of the Gods. Novalis
• A mathematical formula should never be “owned” by anybody! Mathematics belong to God. Donald Knuth

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…and is a divine pursuit…

• The pursuit of mathematics is a divine madness of the human spirit. Alfred North Whitehead
• He who can properly define and divide is to be considered a god. Plato
• There still remain three studies suitable for free man. Arithmetic is one of them. Plato
• Now do you not see that the eye embraces the beauty of the whole world? It counsels and corrects all the arts of mankind… it is the prince of mathematics, and the sciences founded on it are absolutely certain. It has measured the distances and sizes of the stars it has discovered the elements and their location… it has given birth to architecture and to perspective and to the divine art of painting. Leonardo da Vinci
• Religion has the same relation to man’s heavenly condition that mathematics has to his earthly one: both the one and the other are merely the rules of the game. Belief in God and belief in numbers: local truth and truth of location. Vladimir Nabokov

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…that can bring us closer to God…

• Science is complex and chilling. The mathematical language of science is understood by very few. The vistas it presents are scary-an enormous universe ruled by chance and impersonal rules, empty and uncaring, ungraspable and vertiginous. How comfortable to turn instead to a small world, only a few thousand years old, and under God’s personal; and immediate care; a world in which you are His peculiar concern. Isaac Asimov

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…and makes us Godlike

• He who can properly define and divide is to be considered a god. Plato

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In ancient times, mathematics was very much associated with God

• In the Pythagorean system, thinking about numbers, or doing mathematics, was an inherently masculine task. Mathematics was associated with the gods, and with transcendence from the material world; women, by their nature, were supposedly rooted in this latter, baser realm. Margaret Wertheim

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Mathematics, like God, is an ineffable a mystery

• What Is Mathematics? This question, if asked in earnest, has no answer. Salomon Bochner
• 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. Bertrand Russell
• The fact that mathematics does such a good job of describing the Universe is a mystery that we don’t understand. And a debt that we will probably never be able to repay. Lord Kelvin
• 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
• There is something so mysterious that it is almost like God. God is in the equations. God is in the fundamental constants. And that’s fine. I mean, that’s just redefinition of that which we find mysterious at the basis of the universe. Richard Dawkins
• I find enough mystery in mathematics to satisfy my spiritual needs. I think, for example, that pi is mysterious enough (don’t get me started!) without having to worry about God. Or if pi isn’t enough, how about fractals? or quantum mechanics? Tom Lehrer
• All our surest statements about the nature of the world are mathematical statements, yet we do not know what mathematics “is”… and so we find that we have adapted a religion strikingly similar to many traditional faiths. Change “mathematics” to “God” and little else might seem to change. The problem of human contact with some spiritual realm, of timelessness, of our inability to capture all with language and symbol-all have their counterparts in the quest for the nature of Platonic mathematics. John D. Barrow

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It is through mathematics that we can come to an understanding of infinity…

• Black holes result from God dividing the universe by zero. Steven Wright
• The infinite in mathematics is always unruly unless it is properly treated. Edward Kasner
• Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite. Cassius Jackson Keyser
• God created infinity, and man, unable to understand infinity, had to invent finite sets. Gian-Carlo Rota
• Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. Blaise Pascal
• Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory. Hermann Weyl
• There is no remainder in the mathematics of infinity. All life is one; therefore, there cannot be God and man, nor a universe and God. A god not in the world is a false god, and a world not in God is unreal. All things return to one, and one operates in all. Nyogen Senzaki
• The study of infinity is much more than a dry academic game. The intellectual pursuit of the absolute infinity is, as Georg Cantor realized, a form of the soul’s quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment. Rudy Rucker
• Zero is powerful because it is infinity’s twin. They are equal and opposite, yin andyang. They are equally paradoxical and troubling. The biggest questions in scienceand religion are about nothingness and eternity, the void and the infinite, zero andinfinity. The clashes over zero were the battles that shook the foundations of philosophy,of science, of mathematics, and of religion. Underneath every revolution lay a zero‚ and an infinity. Charles Seife
• Georg Cantor was the first to prove that there could be a series of infinities; that infinities come in an infinite number of sizes.] Thus Cantor’s Absolute is a perfect image for what we experience of God. When I speak of a Big Enough God I am not merely thinking of an Infinite God, but the God of infinities, the Absolute, which either chooses to reveal itself or remains veiled in mystery. Modern mathematics does begin to feel like the language that God talks. Sara Maitland

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…and zero

• I don’t agree with mathematics; the sum total of zeros is a frightening figure. Stanislaw Jerzy Lec
• Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay zero – and an infinity. Charles Seife
• 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

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God, the mathematician, some would argue, is an impersonal God

• I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe – because, like Spinoza’s God, it won’t love us in return. Bertrand Russell

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God transcends mathematics

• God’s ways are more than Man’s arithmetic. Mahatma Gandhi

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• In Mathematics it is always best to cheat. Herbert Wilf
• Sex is the mathematics urge sublimated. Michael C. Reed
• Lottery: A tax on people who are bad at math. Ambrose Bierce
• My mathematics is simple: one plus one = one. Dejan Stojanovic
• Perfect numbers like perfect men are very rare. Rene Descartes
• Statistics: the mathematical theory of ignorance. Morris Kline
• A mathematician is a device for turning coffee into theorems. Paul Erdos
• Mathematics is a collection of cheap tricks and dirty jokes. Lipman Bers
• Arithmetic is where numbers fly like pigeons in and out of your head. Carl Sandburg
• A math lecture without a proof is like a movie without a love scene. Hendrik Lenstra
• As long as algebra is taught in school, there will be prayer in school. Cokie Roberts
• Are we going to do any thinking today, or is it going to be all math? Philip J. Hanlon
• A Mathematician is someone who can take a cup of coffee and turn it into a theory. Paul Erdos
• Mathematics is like childhood diseases. The younger you get it, the better. Arnold Sommerfeld
• The hardest arithmetic to master is that which enables us to count our blessings. Eric Hoffer
• Mathematics are well and good but Nature keeps dragging us around by the nose. Albert Einstein
• The numbers are a catalyst that can help turn raving madmen into polite humans. Philip J. Davis
• Mathematics is a dangerous profession; an appreciable proportion of us go mad. John Edensor Littlewood
• A mathematician is a blind man in a dark room looking for a black cat which isn’t there. Charles Darwin
• It should seem that it is easier to square the circle than to get round a mathematician. Augustus De Morgan
• God may not play dice with the universe, but something strange is going on with the prime numbers. Paul Erdos
• Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein
• There is a very good saying that if triangles invented a god, they would make him three-sided. Baron de Montesquieu
• The creator of the universe works in mysterious ways; but he uses a base ten counting system and likes round numbers. Scott Adams
• Mathematics-a wonderful science, but it hasn’t yet come up with a way to divide one tricycle between three small boys. Earl Wilson
• If it squirms, it’s biology; if it stinks, it’s chemistry; if it doesn’t work, it’s physics; and if you can’t understand it, it’s mathematics. Magnus Pyke
• Most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity. G. H. Hardy
• It is now quite lawful for a Catholic woman to avoid pregnancy by a resort to mathematics though she is still forbidden to resort to physics or chemistry. L. Mencken
• I was reminded of the Sydney Harris cartoon that said ‘adding two numbers that have not been added before does not constitute a mathematical breakthrough’. Ronald Graham
• Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different. Johann Wolfgang von Goethe
• Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another. Bernard le Bovier de Fontenelle
• I know that two and two make four – and should be glad to prove it too if I could – though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure. Lord Byron

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